THE NECESSARY AND SUFFICIENT CONDITIONS IN WEIGHTED
INEQUALITIES FOR SINGULAR INTEGRALS AND A LOCAL T B

THEOREM WITH AN ENERGY SIDE CONDITION.

By

Christos Grigoriadis

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics — Doctor of Philosophy

2021

ABSTRACT

THE NECESSARY AND SUFFICIENT CONDITIONS IN WEIGHTED
INEQUALITIES FOR SINGULAR INTEGRALS AND A LOCAL T B

THEOREM WITH AN ENERGY SIDE CONDITION.

By

Christos Grigoriadis

We provide an essentially complete dictionary of all implications among the basic and

fundamental conditions in weighted theory such as the doubling, one weight Ap(w), A∞
and Cp conditions as well as the two weight Ap(ω, σ) and the “buffer" Energy and Pivotal
conditions. The most notable implication is that in the case of A∞ weights the two weight
Ap condition implies the p−Pivotal condition hence giving an elegant and short proof of the
known NTV-conjecture with p = 2 for A∞ weights in terms of existing T1 theory. We also
provide a quite technical construction inspired by [15] proving that we can have doubling

weights satisfying the Cp condition which are not in A∞.

We obtain a local two weight T b theorem with an energy side condition for higher di-

mensional fractional Calderón-Zygmund operators. The proof follows the general outline

of the proof for the corresponding one-dimensional T b theorem in [69], but encountering a

number of new challenges, including several arising from the failure in higher dimensions of

T. Hytönen’s one-dimensional two weight A2 inequality.

Hytönen used this inequality to deal with estimates for measures living in adjacent in-

tervals. Hytönen’s theorem states that the off-testing condition for the Hilbert transform is
controlled by the Muckenhoupt’s A2 and A∗
weight Tb theorem to higher dimensions, it is natural to ask if a higher dimensional analogue
of Hytönen’s theorem holds that permits analogous control of terms involving measures that

conditions. So in attempting to extend the two

2

live on adjacent cubes.We show that it is not the case even in the presence of the energy

conditions used in one dimension [69]. Thus, in order to obtain a local Tb theorem in higher
dimensions, it was necessary to find some substantially new arguments to control the noto-

riously difficult nearby form. More precisely, we show that Hytönen’s off-testing condition

for the two weight fractional integral and the Riesz transform inequalities is not controlled

by Muckenhoupt’s Aα
2

and Aα,∗

2

conditions and energy conditions.

ACKNOWLEDGMENTS

I have been studying in Michigan State University for five years and looking back at these

years, is very clear that it is something I could have never accomplished alone. I want to

thank everyone who has supported me in this journey of excitement, self discovery, surprise,

hardship, work, kindness, joy, love.

First and foremost, I want to thank my advisors Ignacio Uriarte-Tuero and Eric Sawyer.

From my very first year in Michigan, I met with Ignacio and in the second semester of my

studies he said: "I do pure math because I like it and I like solving hard problems. That’s

it." Yeah that’s it. That is when I knew who I wanted my advisor to be. Thank you for

accepting me as your student! A couple of years later the collaboration with professor Sawyer

also started and all of us were meeting together. All of the meetings were full of hope and

positivity in contrast to my casual "oh this probably does not work" mentality. Thank you

both professors for cheering me up every time I could not see the light at the end of the

tunnel and for being next to me in this whole trip.

I am very grateful to professor Alexander Volberg for his teaching and really helpful

discussions throughout these years. I also want to thank the rest of my committee members,

professors Russell Schwab, Jeffrey Schenker and Dapeng Zhan.

A very important part of my work at Michigan State has been my teaching. I was in

complete ignorance of what are the duties and responsibilities of a teacher. I want to express

my gratitude to Andy Krause, Tsveta Sendova, Alec Drachman and Jane Zimmerman for

their advice, trust, support and mostly for communicating the joy of teaching to me and

everyone in the math department.

Graduate school is not easy and I think it is impossible if you go through it alone. I

iv

want to thank all my friends for accompanying me in this trip. First, I want to express my

gratitude to the whole Greek gang. My roommate and office mate Ioannis for these years of

friendship and sharing. My academic brother Mihalis without whom I would need a couple

more years to graduate. Georgios for the thousand hours of studying together. Dimitrios

for sharing his board-game excitement. Andriana for her infinite patience. Ana-Maria for

being a great listener.

Ilias and Eleni for bringing Maria to the group and for all these

movie&popcorn nights that we shared.

Escaping to Ann-Arbor was quite common for our group these years mainly due to

Alexandros and Christina who were always ready for our surprise visits. Thank you for all

the good time we had together.

I also want to thank Abhishek for sharing his passion for Barcelona, Wenchuan for all the

movie nights, Rodrigo for playing football together, Reshma for sharing her love for teaching

and Tyler for initiating me to the world of climbing.

I will always remember Beggar’s

Thursday nights with Reshma, Tyler, Hitesh, Charlotte and Rami. There are so many more

reasons to thank all these people that I would have to double the length of this dissertation.

Thank you all for being in my life all these years!

Last, I want to express my gratitude to my family, without whom nothing could have

been made possible. Thank you Mom and Dad. Thank you for teaching me values, hard

work, the importance of a hug, love. Thank you for trusting in me, for pushing me, for

letting me go, for being always there in joy and sadness. Sister, thank you for your trust,

support and understanding at all times. Grandma, I know you cannot read this, but thank

you for all these beautiful childhood memories that I have with you.

I thank God for everyone who came in my life and in some direct or indirect way has

helped me complete this dissertation. Thank you!

v

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Harmonic functions in the upper half plane . . . . . . . . . . . . . . . . . . .
1.2 The Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The Fourier transform and tempered distributions . . . . . . . . . . . . . . .
1.4 The Hilbert transform on Lp spaces . . . . . . . . . . . . . . . . . . . . . . .
1.5 Singular integrals and the maximal operator. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
1.6 Extending the Lebesgue measure

Chapter 2 On weighted theory.

. . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 One weight theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Two weight theory.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The testing conditions. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The “buffer" Pivotal and Energy conditions.
. . . . . . . . . . . . . .
2.2.3 The relationship between the two weight Ap and “buffer" conditions.
2.2.4 Known cases of the NTV conjecture.
. . . . . . . . . . . . . . . . . .
2.3 T b theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Challenges in higher dimensional two weight T b theory and a coun-
terexample to Hytönen’s off testing condition . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Organization of the paper
2.5 Open problems, lattices and a historic diagram . . . . . . . . . . . . . . . . .

3.1 One weight conditions

3.2 Two weight conditions

Chapter 3 Necessary and sufficient conditions in weighted theory . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 The A1 and A∞ conditions.
. . . . . . . . . . . . . . . . . . . . . . .
3.1.2 A Cp and doubling weight that is not in A∞ . . . . . . . . . . . . . .
3.1.3 Doubling Cp weights are in A∞ for small doubling constants . . . . .
3.1.4 The Aα∞ condition.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Non doubling Ap examples . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Two weight Ap equivalence for doubling measures . . . . . . . . . . .
3.2.3 The T1 theorem for A∞ weights.
. . . . . . . . . . . . . . . . . . . .
3.2.3.1 The Sawyer testing condition. . . . . . . . . . . . . . . . . .
3.2.4 The “buffer" conditions do not imply the tailed Ap conditions.
. . . .
. . . . . . . .

3.2.4.1 Doubling measures and the Pivotal condition.

Chapter 4 Counterexample to Hytönen’s off-testing condition in two di-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 The One-Dimensional Construction . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 The Testing Constant is Unbounded . . . . . . . . . . . . . . . . . .

mensions

vi

ix

1
1
3
5
7
8
12

15
15
17
20
21
23
23
24

26
30
31

35
35
35
36
43
44
46
46
52
53
54
57
59

61
61
62

4.1.2 The ¨A2 Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 The Energy Constants ¨E and ¨E∗ . . . . . . . . . . . . . . . . . . . . .
4.2 The Two Dimensional Construction . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The A2 conditions.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Off-Testing Constant . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 The Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 The Riesz transform lemma . . . . . . . . . . . . . . . . . . . . . . . . . . .

63
67
70
71
72
72
74

5.1.1

5.1.4.1 Punctured Aα
2

conditions

5.1.9 Three corona decompositions

5.1 The local T b theorem and proof preliminaries

Chapter 5 A two weight local T b theorem for n-dimensional Fractional
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
. . . . . . . . . . . . . . . . .
79
Standard fractional singular integrals . . . . . . . . . . . . . . . . . .
79
5.1.1.1 Defining the norm inequality . . . . . . . . . . . . . . . . .
80
5.1.2 Weakly accretive functions . . . . . . . . . . . . . . . . . . . . . . . .
81
5.1.3
b-testing conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
. . . . . . . . . .
5.1.4 Poisson integrals and the Muckenhoupt conditions
81
. . . . . . . . . . . . . . . . . . .
83
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.5 Energy Conditions
85
5.1.6 The two weight local T b Theorem . . . . . . . . . . . . . . . . . . . .
86
5.1.7 Reduction to real bounded accretive families . . . . . . . . . . . . . .
89
5.1.8 Reverse Hölder control of children . . . . . . . . . . . . . . . . . . . .
99
5.1.8.1 Control of averages over children . . . . . . . . . . . . . . .
99
5.1.8.2 Control of averages in coronas . . . . . . . . . . . . . . . . . 105
. . . . . . . . . . . . . . . . . . . . . . 108
5.1.9.1 The Calderón-Zygmund corona decomposition . . . . . . . . 109
5.1.9.2 The accretive/testing corona decomposition . . . . . . . . . 110
5.1.9.3 The energy corona decompositions
. . . . . . . . . . . . . . 115
5.1.10 Iterated coronas and general stopping data . . . . . . . . . . . . . . . 117
5.1.11 Reduction to good functions . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.11.1 Parameterizations of dyadic grids . . . . . . . . . . . . . . . 123
5.1.12 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.1.13 Monotonicity Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.1.13.1 The smaller Poisson integral . . . . . . . . . . . . . . . . . . 137
5.1.13.2 The Energy Lemma . . . . . . . . . . . . . . . . . . . . . . 139
. . . . . . . . . . . . . . . . . . . . . . . . 144
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.2.1 The Hytönen-Martikainen decomposition and weak goodness . . . . . 151
5.2.1.1 Good cubes with ‘body’
. . . . . . . . . . . . . . . . . . . . 152
5.2.1.2 Grid probability . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2.1.3 Weak goodness . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.3 Disjoint form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3.1 Long range form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3.2
Short range form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.4 Nearby form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.4.1 The case of δ-separated cubes. . . . . . . . . . . . . . . . . . . . . . . 182

5.1.14 Organization of the proof

5.2 Form splittings

vii

5.4.2 The case of δ-close cubes.

5.5.1 The canonical splitting and local below forms
5.5.2 Diagonal and far below forms
5.5.3
5.5.4 Paraproduct, neighbour and broken forms

. . . . . . . . . . . . . . . . . . . . . . . . 184
5.4.2.1 Return to the original testing functions . . . . . . . . . . . . 201
5.4.2.2 A finite iteration and a final random surgery.
. . . . . . . . 205
5.5 Main below form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
. . . . . . . . . . . . . 220
. . . . . . . . . . . . . . . . . . . . . . 224
Intertwining Proposition . . . . . . . . . . . . . . . . . . . . . . . . . 225
. . . . . . . . . . . . . . . 239
5.5.4.1 The paraproduct form . . . . . . . . . . . . . . . . . . . . . 243
5.5.4.2 The neighbour form . . . . . . . . . . . . . . . . . . . . . . 245
5.6 The stopping form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
5.6.1 The bound for the second sublinear inequality . . . . . . . . . . . . . 257
5.6.2 The bound for the first sublinear inequality . . . . . . . . . . . . . . 261
5.6.3
(cid:91)Straddling, Substraddling, Corona-Straddling Lemmas . . . . . . . . 268
5.6.4 The bottom/up stopping time argument of M. Lacey . . . . . . . . . 284
5.6.5 The indented corona construction . . . . . . . . . . . . . . . . . . . . 291
5.6.5.1 Flat shifted coronas
. . . . . . . . . . . . . . . . . . . . . . 293
Size estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

5.6.6

5.7 Finishing the proof

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

viii

LIST OF FIGURES

Figure 2.5.1:Theory development diagram . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.1.1:Building construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.2.1:Splitting for n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.2.1:Positioning of measures

. . . . . . . . . . . . . . . . . . . . . . . . . . .

34

37

50

70

Figure 5.4.1:Nearby form diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

ix

Chapter 1

Introduction

The goal of this section is to first provide the reader with an exposition to the basic tools

needed to work in weighted theory. We start with a few basic facts about harmonic functions

and the Poisson kernel which leads to the definition of the Hilbert transform for a special class

of functions, the Schwartz functions. Then, in the attempt to extend the Hilbert transform

to Lp spaces we talk about the Fourier transform and tempered distributions. Last, we

generalize from the Hilbert transform to singular integrals and the Maximal operator.

1.1 Harmonic functions in the upper half plane

Let f : R → R be a real valued smooth function, i.e. it has derivatives of all orders. We can
extend f to a harmonic function u in the upper half plane H = {(x, y) : y > 0, x ∈ R} by
convolution with the Poisson kernel:

Pt(x) =

1
π

t

t2 + x2

(1.1.1)

and we get

(cid:90)

R

u(x, t) =

Pt(x − y)f (y)dy, t > 0

1

It is simple to prove that u is harmonic since Pt(x) is harmonic (with respect to the variables
x, t). Indeed

∂Pt(x)
∂x2 =

−2t3 + 6x2t
(x2 + t2)3 ,

∂Pt(x)
∂t2 =

2t3 − 6x2t
(x2 + t2)3

and since f is smooth we can move the derivative inside the integral and hence we have

(cid:18) ∂Pt(x)
Using the fact(cid:82)R Pt(x)dx = 1 we can also check that

uxx(x, t) + utt(x, t) =

∂x2 +

(cid:90)

R

(cid:19)

∂Pt(x)

∂t2

f (y)dy = 0.

u(x, t) = f (x)

lim
t→0

Singular integrals arise when we try to find the harmonic conjugate of u, i.e. the function

v such that g(z) = u(z) + iv(z) is analytic.

First we notice that the Poisson kernel Pt(x) is the real part of the analytic function

h(z) = i
πz

. Indeed, writing z = x + it we have

h(z) =

i
πz

=

i¯z
π|z|2 =

t + ix
t2 + x2 = Pt(x) + iQt(x)

where

and hence v(x, t) =(cid:82)R Qt(x− y)f (y)dy is the harmonic conjugate of u. But now it is not

t2 + x2

Qt(x) =

(1.1.2)

x

1
π

easy to define limt→0 v(x, t). We can see there is a problem by taking limt→0 Qt(x) = 1
πx
since this is not even locally integrable since the integral around 0 does not exist. The

singularity at x = 0 is the motivation for the name singular integrals. So we cannot just talk

2

about (cid:82)R 1

πxf (x)dx since this integral is not properly defined so let’s see how to interpret

that integral.

1.2 The Hilbert transform

It is clear that for(cid:82)R 1

πxf (x)dx to make sense f cannot be any function (just try any constant

function on R). For this reason we define the Schwartz space.

Definition 1.2.1. Let f be a real valued function. We say f ∈ S if and only if f is infinitely

differentiable and

|xnf (k)(x)| ≤ cn,k = cn,k(f ) < ∞, n, k ∈ N

sup
x∈R

(1.2.1)

where f (k) denotes the k − th derivative of f.

We can now define the principal value of 1
x

.

Definition 1.2.2. We define the principal value of 1

x and we write p.v. 1

x, by

(cid:90)

p.v.

1
x

(f ) = lim
ε→0

|x|>ε

x

f (x)

dx,

f ∈ S

(1.2.2)

So p.v. 1

x maps Schwartz functions to the real numbers.

To see that this definition makes sense, we have to verify that the limit in the definition

actually exists. So let f ∈ S, we have

p.v.

1
x

(f ) = lim
ε→0

= lim
ε→0

|x|>ε

f (x)

dx = lim
ε→0
x
f (x) − f (0)

ε<|x|<1

(cid:90)

(cid:90)

(cid:90)

f (x)

|x|≥1

x

dx

f (x)

x

dx +

dx +

|x|≥1

x

f (x)

dx

(cid:90)
(cid:90)

ε<|x|<1

x

3

and the last equality holds since(cid:82)

ε<|x|<1

(1.2.1) we get

|p.v.

1
x

(f )| ≤

sup

|f(cid:48)(x)| + sup
|x|≥1
≤ c0,1 + c0,0c1,0 < ∞

ε<|x|<1

1

xdx = 0. Now since f is a Schwartz function, by

(cid:90)

|x|≥1

|f (x)|

c1,0

x2 dx ≤ ||f(cid:48)||∞ + c1,0||f||∞

where ||f||∞ = sup
x∈R

|f (x)|. So the limit makes sense.

This brings us to the definition of the Hilbert transform which is the archetype of singular

integrals.

Definition 1.2.3. Let f ∈ S. We define the Hilbert transform by

(cid:90)

Hf (x) = lim
t→0

Qt ∗ f (x) = lim
ε→0

1
π

f (y)
x − y

dy

(1.2.3)

|x−y|>ε

The Hilbert transform is extremely important mainly for the reason of mapping the

boundary values (f) of a harmonic function (u) to it’s harmonic conjugate (v) and it’s been

extensively studied since then.

It is quite restrictive though that right now we can talk about the Hilbert transform only

for the limited class of Schwartz functions. To extend the definition of the Hilbert transform

to bigger classes of functions we need to talk about the Fourier transform of a function and

tempered distributions.

4

1.3 The Fourier transform and tempered distributions

In the class of Schwartz functions S we can define a topology by (1.2.2). We say a sequence
{fm}m∈N, where f ∈ S converges to 0 if and only if for all n, k ∈ N

lim
n,k→∞ cn,k(f ) = 0.

(1.3.1)

Definition 1.3.1. We say a function f ∈ Lp(Rn) if and only if

(cid:90)

Rn

|f (x)|pdx < ∞

It is easy to see using (1.2.2) that S ⊂ Lp(Rn) for all p > 1. The Schwartz class is also
dense in Lp(Rn) meaning that for any f ∈ Lp(Rn) there exist a sequence {fm} such that

m→∞||fm − f||p = 0.

lim

With this metric now in the Schwartz class we can define the space of bounded linear func-
tionals in S, we call it S(cid:48) the space of tempered distributions.
Definition 1.3.2. We say a map T : S → R (or S → C) is in S(cid:48), the space of tempered

distributions, if

m→∞ fm = 0 ⇒ lim

lim

m→∞ T (fm) = 0

Remember the limit on the left hand side is in the sense of (1.3.1). One can easily check

that the principal value defined in (1.2.2) is a tempered distribution.

Let’s now define the Fourier transform which we will see is a tempered distribution itself.

We first define the Fourier transform for L1 functions.

5

Definition 1.3.3. Let f ∈ L1(Rn). The Fourier transform F of f is given by

(cid:90)

Rn

F(f )(ξ) =

f (x)e−2πix·ξdx

(1.3.2)

where x · ξ = x1ξ1 + x2ξ2 + ... + xnξn. We will also write F(f )(ξ) = ˆf (ξ).

Here are some properties of the Fourier transform that we need:

1. Linearity: (cid:92)(af + bg) = a ˆf + bˆg.

2. || ˆf||∞ ≤ ||f||1 and hence ˆf is continuous.

3. (cid:91)f ∗ g = ˆf ˆg.

We can also define the Fourier transform of tempered distributions.

Definition 1.3.4. The Fourier transform of T ∈ S(cid:48) is the tempered distribution ˆT given by

ˆT (f ) = T ( ˆf ), f ∈ S.

The following theorems can be found in [12], pg 13-16.

Theorem 1.3.5. The Fourier transform is a continuous map from S to S such that

(cid:90)

(cid:90)

ˆf g =

Rn

f ˆg

Rn

and

(cid:90)

Rn

f (x) =

ˆf (ξ)e2πix·ξdξ

Theorem 1.3.6. The Fourier transform is a bounded linear bijection from S(cid:48) to S(cid:48) whose

inverse is also bounded.

6

Theorem 1.3.7. The Fourier transform is an isometry on L2 meaning for f ∈ S, ˆf ∈ L2
and || ˆf||2 = ||f||2.

Now we can extend the Fourier transform to functions in L2. Let f ∈ L2(Rn). By theorem
1.3.7 and using the density of S in L2 there exist {fm}m∈N such that ||fm||2 → ||f||2 as
m → ∞ hence there is a function g such that || ˆfm||2 → ||g||2 as m → ∞. We call then
g = ˆf the Fourier transform of f and we have equality of norms, i.e. ||f||2 = || ˆf||2.

1.4 The Hilbert transform on Lp spaces

Now we have all the tools we need to extend the Hilbert transform from the space of Schwartz

functions to the space of Lp functions.

First, by (1.1.2) and calculating it’s Fourier transform we get

ˆQt(ξ) = −isgn(ξ)e−2πt|ξ|

(1.4.1)

Now by (1.2.3) we get for f ∈ S:

ˆHf (x) = lim
t→0

(cid:92)Qt ∗ f (x) = lim
t→0

ˆQt(ξ) ˆf (ξ) = −isgn(ξ) ˆf (ξ).

(1.4.2)

The first equality holds by the continuity of the Fourier transform on tempered distri-

butions (theorem 1.3.6), the second equality from property (2) after (1.3.2) and the last

equality by (1.4.1).

Now by the density of S in L2 and using theorem 1.3.7 we can extend the definition of

the Hilbert transform to L2 functions and we also have

||Hf||2 = || ˆHf||2 = ||f||2,

f ∈ L2(Rn)

7

Next, the extension to Lp, p > 1 came from Riesz in 1928 with the following theorem.

Theorem 1.4.1 (Riesz 1928). Let f ∈ S. The Hilbert transform is strong (p, p), 1 < p < ∞

i.e.

||Hf||p ≤ Cp||f||p

Again, like in the case for L2 functions, using the density of S in Lp we extend the

definition of the Hilbert transform to Lp functions.

1.5 Singular integrals and the maximal operator.

The Hilbert transform is the archetype of a more general class of operators that we want to

study the so called singular integral operators.

Definition 1.5.1. A singular integral of convolution type is an operator T defined by con-
volution with a kernel K that is locally integrable on Rn\{0}, in the sense that

(cid:90)

T (f )(x) = lim
ε→0

|x−y|>ε

K(x − y)f (y)dy

Typically the kernel satisfies the following conditions:

1. || ˆK||∞ ≤ C < ∞

(cid:82)|x|>2|y| |K(x − y) − K(x)|dx ≤ C

2. sup
y(cid:54)=0

For K(x) = 1

πx we get the Hilbert transform.

The first property, usually referred as a size condition, on ˆK is used to ensure that the

8

tempered distribution p.v.K given by the principal value integral

p.v.K[φ] = lim
ε→0

is well defined on L2.

φ(x)K(x)dx

|x|>ε

The second property, usually referred as the Hörmander condition or smoothness condi-

tion, is used to ensure the boundedness of the operators in Lp(Rn) for p > 1.

Example 1.5.2. An important example of singular integrals is the Riesz transforms:

(cid:90)

(cid:90)

Rjf (x) = cn lim
ε→0

|y|>ε

where

(cid:18) n + 1

2

cn = Γ

yj

|y|n+1 f (x − y)dy
(cid:19)

− n+1
2 .

π

The Riesz transforms are the extension of the Hilbert transform in higher dimensions. The
constant cn is so that ˆRjf (ξ) = −i

ξj|ξ| ˆf (ξ).

We have the following theorem for singular integrals of convolution type.

Theorem 1.5.3 (Calderon-Zygmund 1952, Hörmander 1960). Let f ∈ S. Assume K is

a kernel as in definition 1.5.1. Then the operator T associated with K is strong (p, p),
1 < p < ∞ i.e.

||Hf||p ≤ Cp||f||p

These singular integrals are called of convolution type because T f = K ∗ f (with an
abuse of notation since we do not write the limit here). We can extend to operators T that

have a kernel K(x, y) and the resulting operator is not of convolution type. First we need

9

to define what a standard kernel is.

Definition 1.5.4. We say K(x, y) : Rn × Rn → R is a standard kernel when:

1. |K(x, y)| ≤ C|x−y|n
2. |K(x, y) − K(x(cid:48), y)| ≤

3. |K(x, y) − K(x, y(cid:48))| ≤

C|x−x(cid:48)|δ

(|x−y|+|x(cid:48)−y|)n+δ , for |x − x(cid:48)| ≤ 1
(|x−y|+|x−y(cid:48)|)n+δ , for |y − y(cid:48)| ≤ 1

C|y−y(cid:48)|δ

2 max(cid:0)|x − y|,|x(cid:48) − y|(cid:1)
2 max(cid:0)|x − y|,|x − y(cid:48)|(cid:1)

property (1) here is the analog of || ˆK||∞ ≤ C < ∞ and is again a size condition on the
kernel. Properties (2) and (3) are the analog of condition (2) and are again called Hörmander

conditions.

Definition 1.5.5. A general singular integral is an operator T associated with a standard

kernel K, in the sense that

T (f )(x) =

(cid:90)

Rn

K(x, y)f (y)dy, x /∈ suppf, f ∈ S

(1.5.1)

An important note here is that the operator uniquely defines the Kernel but the ker-

nel does not uniquely define the operator. The classical example on that is the operators

Tb(f )(x) = b(x)f (x) where b is a bounded function. All these are general singular integrals
with kernel K(x, y) = 0. This comes from the fact that the integral formula holds only for
x /∈ suppf so we have

(cid:90)

0 = Tb(f )(x) =

Rn

K(x, y)f (y)dy ⇒ K(x, y) = 0

and the implication comes from setting f1(x) = 1A(x), f2(x) = 1B(x) where A = {K(x, y) >
0} and B = {K(x, y) < 0}.

10

Example 1.5.6. A general singular integral studied a lot is the Cauchy integral. Let’s
remember the Cauchy integral theorem. Let U be an open subset of C and Dz0 = {z :
|z − z0| ≤ r} be contained in U. Given a holomorphic function f : U → C and any w in the
interior of Dz0 we have

(cid:90) 2π

0

f (w) =

1
2πi

f (z0 + reiθ)ireiθ
z0 + reiθ − w

dθ

Inspired by the classic Cauchy integral, we get the Cauchy integral on a Lipschitz curve.
Let L : R → R be Lipschitz or equivalently ||L(cid:48)||∞ ≤ C < ∞ and let Γ = (t, L(t)) be a curve
on the plane. Let f ∈ S(R). The Cauchy integral along Γ is given by

CΓf (z) =

1
2πi

f (t)(1 + iL(cid:48)(t))
t + iL(t) − z

dt

(cid:90) ∞

−∞

CΓf (z) is a holomorphic function on C \ Γ.

Another operator that often appears together with the Hilbert transform and singular

integrals, as we will later see, is the Maximal operator of Hardy and Littlewood.

Definition 1.5.7. The Maximal operator is given by

(cid:90)

I

M f (x) = sup
I(cid:51)x

1
|I|

|f (y)|dy

(1.5.2)

where the supremum is taken over all cubes containing x ∈ Rn.

Hardy and Littlewood proved that the maximal operator is bounded in Lp(R), p > 1 and

Wiener later extended the result to higher dimensions. We have:

Theorem 1.5.8 (Hardy-Littlewood-Wiener 1930,1939). The Maximal operator is a bounded

11

operator on Lp(Rn) → Lp(Rn), p > 1.

(cid:18)(cid:90)

(cid:19) 1

p ≤ C

(cid:18)(cid:90)

Rn

(cid:19) 1

p

.

|f (x)|pdx

|M f (x)|pdx

Rn

(1.5.2) is not the only way to define the maximal operator. We could also use

M(cid:48)f (x) = sup

r>0

1

(r)n

|f (y)|dy

(1.5.3)

(cid:90)

Ir

where Ir is the cube centered at x with side length r. We call M(cid:48) the centered maximal
operator and M is the non-centered maximal operator. One can use either of them since

they are pointwise equivalent, i.e. there exist constants cn, Cn such that

cnM f (x) ≤ M(cid:48)f (x) ≤ CnM f (x).

1.6 Extending the Lebesgue measure

All the theorems presented until now were using the Lebesgue measure. The work in this

monograph is concerned on extending all these results to more general measures. The fol-

lowing definition contains all the different measures that will concern us.

Definition 1.6.1.

1. The measure µ is called inner regular if for any open set U, µ(U ) = supK⊂U µ(K)

where the supremum is taken over all compact subsets of U.

2. The measure µ is called outer regular if for any Borel set B, µ(B) = infB⊂U µ(U )

where the infimum is taken over all open sets containing B.

12

3. The measure µ is called locally finite if for every point x ∈ Rn there is an open set U
containing x such that µ(U ) < ∞. Equivalently µ is called locally finite if for every K
compact µ(K) < ∞.

4. The measure µ is called Radon if it satisfies all the above, i.e. it is inner regular, outer

regular and it is locally finite.

5. The measure µ is called absolutely continuous if there exists w : Rn → R+ such that

for any Borel set B, µ(B) = (cid:82)

B w(x)dx. The function w(x) in this case is called a

weight.

6. The measure µ is called doubling if there exists a constant C > 0 such that for any

cube I we have

µ(2I) ≤ Cµ(I)

(1.6.1)

7. The measure µ on Rn is called upper doubling if there exists a constant C > 0 such

that for any ball Br of radius r we have

µ(Br) ≤ M rn.

(1.6.2)

8. We say a measure σ is reverse doubling if there exists ε > 0 depending only on the

measure σ such that for all cubes I:

σ(2I) ≥ (1 + ε)σ(I).

(1.6.3)

Doubling measures satisfy the reverse doubling property as the following lemma from

[52] proves.

13

Lemma 1.6.2. Let σ be a doubling measure with doubling constant Kσ. Then there

exist a constant δσ > 0 depending only on the doubling constant of σ such that for all
cubes I we have σ(2I) ≥ (1 + δσ)σ(I).

14

Chapter 2

On weighted theory.

The goal of this section is to provide the reader with the development of the theory from

Muckenhoupt’s Ap condition for one weight results up to 2020 where we have general singular
integrals and two Radon measures σ, ω . Weighted theory tries to answer questions as the

following.

Question 1. Given two locally finite positive Borel measures ω, σ in Rn and a general

singular integral operator T , what are the necessary and sufficient conditions so that the

following inequality holds:

||T (f dσ)||Lp(ω)

(cid:46) ||f||Lp(σ),

∀f ∈ Lp(σ), p > 1.

(2.0.1)

2.1 One weight theory

(2.0.1) is a generalization of the one weight inequality for the Hilbert transform where T = H,
dω(x) = w(x)dx, dσ(x) = w(x)1−p(cid:48)dx and f ∈ Lp(w)

||Hf||Lp(ω)

(cid:46) ||f||Lp(ω)

(2.1.1)

which was shown by Hunt, Muckenhoupt and Wheeden [20] to be equivalent to the finiteness

of the Muckenhoupt one weight Ap condition, namely ω has to be absolutely continuous to

15

Lebesgue measure dω = w(x)dx and

(cid:90)

I

(cid:18) 1

|I|

(cid:90)

I

(cid:19)p−1 ≤ C < ∞

1

1−p dx

w(x)

(2.1.2)

Ap(w) = sup

I

1
|I|

w(x)dx

where the supremum is taken uniformly over all cubes in Rn. There has been a huge amount

of work in harmonic analysis and boundary value problems around the Ap condition, check
Stein [71], Duoandikoetxea [12], Garnett [14] and references there.

Coifman and Fefferman in [7] proved (2.1.1) using the following inequality, which holds

for any w ∈ A∞ =(cid:83)

p≥1 Ap,

(cid:90)

Rn

|T f (x)|pw(x)dx ≤ C

(cid:90)

Rn

|M f (x)|pw(x)dx

(2.1.3)

where T is any singular integral operator and f is bounded and compactly supported. We

can extend to any locally integrable f for which the right hand side is finite (since otherwise

there is nothing to prove) using the dominated convergence theorem.

Muckenhoupt in [39] proved, for n = 1, that a more general class of weights than the

Ap weights, namely the Cp weights (see (2.1.4)), are necessary for (2.1.3) to hold. This was
generalized in higher dimensions by Sawyer in [54]. Sawyer in [54] also shows that the Cq
condition for q > p is sufficient for (2.1.3) to hold. It is still unknown if the Cp condition is
sufficient for (2.1.3) to hold.

We say the measure ω satisfies the Cp condition, 1 < p < ∞ if it is absolutely continuous

to the Lebesgue measure, i.e. dω = w(x)dx, and there exist C, 
 > 0 such that

(cid:82)Rn |M 1I (x)|pw(x)dx

|E|w

≤ C

(cid:18)|E|

(cid:19)


|I|

, for E compact subset of I cube

(2.1.4)

16

with(cid:82)Rn (M 1I (x))p w(x)dx < ∞, where |E|w =(cid:82)

E w(x)dx. Here M denotes the classical

Hardy-Littlewood maximal operator. We will call w(x) a Cp weight.

We prove that Cp weights is a strictly larger class than the A∞ weights. We actually
show that there exist even doubling weights (see (1.6.1)) that are also Cp weights that are
not in A∞. Check the diagram at the end of the introduction.

Theorem 2.1.1. (Cp ∩ D (cid:59) A∞) There exist a weight w that is doubling and satisfies the
Cp condition but w is not an A∞ weight.

The weight w used in theorem 2.1.1. has a doubling constant Cw (cid:38) 3np. We show that
if the doubling constant Cw of the weight w does not satisfy Cw ≥ 3np

this is sharp, i.e.

then the Cp condition is equivalent to A∞.

Theorem 2.1.2. (Cp+small doubling ⇒ A∞) Let w be a doubling Cp weight with doubling
constant Cw < 3np in Rn. Then w ∈ A∞.

2.2 Two weight theory.

The generalization of the one weight Ap condition was naturally modified to the two weight
problem by:

(cid:18) ω(I)

p(cid:18) σ(I)
(cid:19) 1

(cid:19) 1

p(cid:48)

Ap(ω, σ) = sup

I

|I|

|I|

< ∞

(2.2.1)

where the supremum is taken over all cubes in Rn and the weight w gives its place to two
1−p dx, dσ = w(x)dx

positive locally finite Borel measures. Notice that by setting dω = w(x)

1

we retrieve the one weight Ap condition (2.1.2).

The two weight problem could have applications in a number of problems connected to

higher dimensional analogs of the Hilbert transform. For example, questions regarding sub-

17

spaces of the Hardy space invariant under the inverse shift operator (see [75], [46]),questions

concerning orthogonal polynomials (see [76], [49], [50]) and some questions in quasiconfor-

mal theory for example the conjecture of Iwaniec and Martin (see [25]) or higher dimensional

analogues of the Astala conjecture (see [30]).

The classical Ap condition (2.2.1) is necessary for (2.0.1) to hold but is no longer sufficient,
which is an indication that makes two weight theory much more complicated. F. Nazarov
in [40] has shown that even the strengthened Ap(ω, σ) conditions with one or two tails of
Nazarov, Treil and Volberg

(cid:18) ω(I)

(cid:19) 1

p

|I|

At1
p (ω, σ) = sup

I

(P (I, σ))

1

p(cid:48) < ∞

At2
p (ω, σ) = sup

(P (I, ω))

1
p (P (I, σ))

1

p(cid:48) < ∞

where

(cid:90)

Rn

P (I, ω) ≡

I



n

|I| 1

n

(|I| 1

n + dist(x, I))2

ω(dx)

(2.2.4)

(ω, σ),At2,∗
along with their duals At1,∗
are no longer sufficient for (2.0.1) to hold.

p

p

(ω, σ), where the roles of σ and ω are interchanged,

When the operator T in (2.0.1) is a fractional operator such as the Cauchy transform

or the fractional Riesz transforms then the fractional analogs of (2.2.1), (2.2.2), (2.2.3) are

(2.2.2)

(2.2.3)

(2.2.5)

(2.2.6)

used

Aα
p (ω, σ) = sup
I

(cid:32)
ω(I)
|I|1− α
(cid:32)

n

(cid:33) 1
p(cid:32)
(cid:33) 1

p

ω(I)
|I|1− α

n

(cid:33) 1

p(cid:48)

σ(I)
|I|1− α

n

< ∞

(Pα(I, σ))

1

p(cid:48) < ∞

At1,α

p

(ω, σ) = sup

I

18

At2,α

p

(ω, σ) = sup

(Pα(I, ω))

1

p (Pα(I, σ))

1

p(cid:48) < ∞

(2.2.7)

I

where Pα is the reproducing Poisson integral and is given by

Pα(I, ω) ≡

(|I| 1
The standard Poisson integral, is given by

Rn

|I| 1

n

n + dist(x, I))2

n−α

ω(dx)

(cid:90)



(cid:90)

P α(I, ω) ≡

|I| 1

n

Rn

(|I| 1

n + dist(x, I))n+1−α

ω(dx)

and is used for the definition of the fractional “buffer" conditions. The two Poisson integrals

agree for n = 1, α = 0. We refer the reader to [64] for more details. All the results that we are
proving here for the Ap conditions hold for their fractional analogs without any modification
in the proofs.

We show that the classical Ap condition is weaker than the tailed conditions, but the two

tailed Ap condition holding is equivalent to both one tailed Ap conditions holding.

Theorem 2.2.1. We have the following implications:

1. (Ap (cid:59) At1

p ∩At1,∗

p

Ap conditions.

) The two weight classical Ap condition does not imply the one tailed

p ) The one tailed At1

p condition does not imply the two tailed At2

p condition.

2. (At1

3. (At1

p (cid:59) At2
p ∩ At1,∗

p ⇔ At2
p ,At1,∗

p ) The two tailed At2
conditions holding.

p

tailed At1

p condition holding is equivalent to both one

The measures that we use for the proof of theorem 2.2.1. are non doubling and we show

that this is the only case. All doubling measures are reverse doubling (see lemma 1.6.2.). So

19

the previous sentence is justified by the following theorem:

Theorem 2.2.2. (ω, σ ∈ D , Ap ⇒ At1
the classical two weight classical Ap implies the tailed Ap conditions.

p ⇒ At2

p ) If ω, σ are reverse doubling measures, then

A proof of theorem 2.2.2 for the one tailed conditions can be found in [70].

2.2.1 The testing conditions.

Since the two weight Ap conditions are not sufficient for (2.0.1) to hold, some other necessary
conditions are required, namely the 1-testing conditions

||T (1I dσ)||Lp(ω) ≤ Tp|I|σ
||T∗(1I dω)||Lp(σ) ≤ (T∗)p|I|ω

(2.2.8)

where I runs over all cubes and T, T∗ are the best constants so that (2.2.8) holds.

The testing conditions were first introduced by Sawyer in [56] in 1982 for the boundedness

of the maximal operator and they involved two weights, u, v namely Sawyer proved that the
Maximal operator is bounded on Lp(u) → Lq(w) if and only if the Sawyer testing condition
is satisfied, i.e. if and only if

(cid:18)(cid:90)

p (cid:18)(cid:90)
(cid:19)−1

(cid:104)
M (1I u1−p(cid:48)

(cid:105)q

(cid:19) 1

q

Sp,q(w, u1−p(cid:48)

) = sup

u(x)1−p(cid:48)

dx

I

I

I

)(x)

w(x)dx

< ∞ (2.2.9)

where the supremum is taken over all cubes I ⊂ Rn.

Two years later in 1984, David and Journé in [10] using (2.2.8) for dσ = dω = dx proved

that a general singular integral is bounded in Lp as long as (2.2.8) holds.

The testing conditions alone are trivially not sufficient for (2.0.1) to hold for general

20

measures since as pointed out in [48] for example, the second Riesz transform R2 of any
measure supported on the real line is the zero element in Lp(ω) for any measure ω carried

by the upper half plane. On the other hand, such a pair of measures need not satisfy the

Muckenhoupt conditions, which are necessary for (2.0.1) to hold.

The famous Nazarov-Treil-Volberg conjecture (NTV conjecture), states that Ap(ω, σ)

and testing conditions are necessary and sufficient for (2.0.1) to hold.

2.2.2 The “buffer" Pivotal and Energy conditions.

Nazarov, Treil and Volberg in a series of very clever papers assumed the pivotal condition,

for p = 2, and proved (2.0.1) (see [42],[45],[75]).

The Pivotal condition V is given by

V(ω, σ)p = sup
I0=∪Ir

1

σ(I0)

(cid:88)

r≥1

ω(Ir)P (Ir, 1I0

σ)p < ∞

(2.2.10)

where the supremum is taken over all possible decompositions of I0 in disjoint cubes {Ir}r∈N
and all cubes I0 such that σ(I0) (cid:54)= 0, and its dual V∗ where σ and ω are interchanged.

Lacey, Sawyer and Uriarte-Tuero in [32] proved, again for p = 2, that (2.0.1) for the

Hilbert transform implies the weaker Energy condition E

E(ω, σ)p = sup
I0=∪Ir

1

ω(Ir)E(Ir, ω)2P (Ir, 1I0

σ)p < ∞

(2.2.11)

(cid:88)

σ(I0)

r≥1

where the supremum is taken over all possible decompositions of I0 in disjoint cubes {Ir}r∈N

21

and all cubes I0 such that σ(I0) (cid:54)= 0, where

E(I, ω)2 ≡ 1
2

Eω(dx)
I

Eω(dx(cid:48))

I

(x − x(cid:48))2

|I|2

(2.2.12)

and its dual E∗ where σ and ω are interchanged.

In the same paper, Lacey, Sawyer and Uriarte-Tuero proved that a hybrid of the Pivotal

and Energy conditions was sufficient but not necessary in the two weight inequality for the

Hilbert transform.

Both the energy and the pivotal conditions, sometimes referred to as “buffer conditions",

are used to approximate certain forms that appear in the proofs of almost all two weight

inequalities. The NTV conjecture states that we can prove (2.0.1) without assuming them.
It is true though that if both ω, σ are individually A∞ weights, the classical Ap(ω, σ)
condition implies the Pivotal condition providing a short and elegant proof of the NTV-
conjecture for A∞ weights assuming existing T 1 theory. Earlier, Sawyer in [58] gave a proof
using different methods for the case of smooth kernels.

Theorem 2.2.3. (T 1 theorem for A∞ weights) Assume ω, σ are in A∞, T is an α-fractional
singular integral and we have the T 1 testing and the fractional Aα
2 (ω, σ) conditions to hold,
along with their duals. Then, T is bounded on L2(Rn).

We can get another proof of this theorem for singular integrals by using theorem 3 in

[47] by inserting an Ap weight w between ω, σ and then using the one weight results in [7]
or [20].

22

2.2.3 The relationship between the two weight Ap and “buffer" con-

ditions.

It is shown in [32] that we can have a pair of measures satisfying the tailed A2 conditions
(2.2.2), (2.2.3) but failing to satisfy the Pivotal condition (2.2.10), hence proving the impli-
cation At2

(cid:59) V2.

2

We show here that the Pivotal condition (2.2.10) does not imply the tailed A2 conditions

(2.2.2), (2.2.3).

Theorem 2.2.4. (Vp (cid:59) At1
p ) Let 1 < p ≤ 2. The Pivotal condition Vp does not imply the
one tailed Ap condition At1
p .

Remark 2.2.5. It is immediate from (2.2.12) that the Energy condition (2.2.11) is dominated

by the Pivotal condition (2.2.10) hence we immediately get the following important corollary.

Corollary 2.2.6. (E (cid:59) At1
2 ) Let 1 < p ≤ 2. The Energy condition E does not imply the
one tailed Ap condition At1
p .

2.2.4 Known cases of the NTV conjecture.

While the general case of the NTV conjecture in Rn is still not completely understood,

several important special cases have been completely solved.

First, in the two part paper by Lacey, Sawyer, Shen and Uriarte-Tuero [34] and Lacey
[27] proved the NTV conjecture, namely that Ap(ω, σ) and testing conditions are necessary
and sufficient for (2.0.1) to hold, assuming also that the measures σ and ω had no common

23

point masses, for the Hilbert Transform. Hytönen [22] with his new offset version of A2

n

σ(dx) < ∞

(2.2.13)

(cid:90)



Aoffset

2

(ω, σ) = sup

I

ω(I)
|I|

Rn\I

|I| 1

n

(|I| 1

n + dist(x, I))2

removed the restriction of common point masses on σ, ω. An alternate approach using
“punctured" versions of A2 appears in [67].

Other important cases include first Sawyer, Shen, Uriarte-Tuero [64] for α-fractional

singular integrals, Lacey-Wick [37] for the Riesz transforms, Lacey, Sawyer, Shen, Uriarte-

Tuero and Wick [35] for the Cauchy transform and Sawyer, Shen, Uriarte-Tuero [65] for the

Riesz tranform when a measure is supported on a curve in Rn and recently [58] for general
Calderon-Zygmund operators and doubling measures that also satisfy the fractional Aα∞
condition, check (3.1.16). The NTV conjecture is yet to be proven for a general operator T .

2.3 T b theory

The original T 1 theorem of David and Journé [10], which characterized boundedness of a

singular integral operator by testing over indicators 1Q of cubes Q, was quickly extended to
a T b theorem by David, Journé and Semmes [11], in which the indicators 1Q were replaced
by testing functions b1Q for an accretive function b, i.e. 0 < c ≤ Reb ≤ |b| ≤ C < ∞. Here
the accretive function b could be chosen to adapt well to the operator at hand, resulting in

almost immediate verification of the b-testing conditions, despite difficulty in verifying the

1-testing conditions. One motivating example of this phenomenon is the boundedness of the

Cauchy integral on Lipschitz curves, easily obtained from the above T b theorem1. See e.g.
1The problem reduces to boundedness on L2 (R) of the singular integral operator CA with kernel
, where the curve has graph {x + iA (x) : x ∈ R}. Now b (x) ≡ 1 + iA(cid:48) (x) is

1

KA (x, y) ≡

x−y+i(A(x)−A(y))

24

[72, pages 310-316].

Subsequently, M. Christ [5] obtained a far more robust local T b theorem in the setting of

homogeneous spaces, in which the testing functions could be further specialized to bQ1Q ,
where now the accretive functions bQ can be chosen by the reader to differ for each cube
Q. Applications of the local T b theorem included boundedness of layer potentials, see e.g.

[2] and references there; and the Kato problem, see [19], [18] and [3]: and many authors,

including G. David [8]; Nazarov, Treil and Volberg [43], [42]; Auscher, Hofmann, Muscalu,

Tao and Thiele [4], Hytönen and Martikainen [24], and more recently Lacey and Martikainen

[29], set about proving extensions of the local T b theorem, for example to include a single

upper doubling weight together with weaker upper bounds on the function b. But these

extensions were modelled on the ‘nondoubling’ methods that arose in connection with upper

doubling measures in the analytic capacity problem, see Mattila, Melnikov and Verdera [38],

G. David [8], [9], X. Tolsa [74], and alsoVolberg [75], and were thus constrained to a single

weight - a setting in which both the Muckenhoupt and energy conditions follow from the

upper doubling condition.

More recently, in a precursor to the present result, [69] obtained a general two weight

T b theorem for the Hilbert transform on the real line. Here, we extend this precursor

to higher dimensions. As in [69], we adapt methods from the theory of two weight T 1

theorems, which arose from [44], [75], [34], [27], [64] and [66], and were used in [24] as well,

to prove a two weight local T b theorem. These methods involve the ‘testing’ perspective

toward characterizing two weight norm inequalities for an operator T . As suggested by

work originating in [10] and [55], it is plausible to conjecture that a given operator T is

accretive and the b-testing condition(cid:82)
,
β−x
for x ∈ I = [α, β]. In the case of a C1,δ curve, the kernel KA is C1,δ and a T b theorem applies with T = CA
and σ = ω = dx, to show that CA is bounded on L2 (R).

(cid:0)1I b(cid:1) (x)(cid:12)(cid:12)2 ≈ ln x−α

I

(cid:12)(cid:12)CA

(cid:0)1I b(cid:1) (x)(cid:12)(cid:12)2 dx ≤ Tb

H |I| follows from(cid:12)(cid:12)CA

25

bounded from one weighted space to another if and only if both it and its dual are bounded

when tested over a suitable family of functions related geometrically to T , e.g. testing over

indicators of intervals for fractional integrals T as in [55].

The main two weight local T b theorem: Here is a brief statement of our main

theorem.

Theorem 2.3.1 (local T b in higher dimensions). Let T α denote a Calderón-Zygmund op-
erator on Rn, and let σ and ω be locally finite positive Borel measures on Rn that satisfy
σ f ≡ T α (f σ), is bounded

the energy and Muckenhoupt buffer conditions. Then T α
from L2 (σ) to L2 (ω) if and only if the b-testing and b∗-testing conditions

σ , where T α

(cid:90)

σ bI|2 dω ≤(cid:16)

dσ ≤(cid:16)
(cid:12)(cid:12)2
taken over two families of test functions {bI}I∈P and (cid:8)b∗
(cid:9)

(cid:17)2 |I|σ and

(cid:12)(cid:12)T α,∗

ω b∗

Tb
T α

J

|T α

I

J are only
required to be nondegenerate in an average sense, and to be just slightly better than L2

J

J∈P, where bI and b∗

(cid:90)

J

(cid:17)2 |J|ω ,

Tb∗,∗

T α

(2.3.1)

functions themselves, namely Lp for some p > 2.

2.3.1 Challenges in higher dimensional two weight T b theory and a

counterexample to Hytönen’s off testing condition

A number of difficulties arise in generalizing to higher dimensions the work that was done

in [69] for dimension n = 1. The main difficulty lies in the strictly-one dimensional nature

of a fundamental inequality of Hytönen, namely that local testing, i.e. testing the integral

(cid:12)(cid:12)2 over the cube Q, together with the A2 condition, imply full testing, meaning
(cid:12)(cid:12)2 is integrated over the entire space Rn. For the proof of full testing, Hytönen

of (cid:12)(cid:12)Tσ1Q
that(cid:12)(cid:12)Tσ1Q

uses an inequality for the Hardy operator that is true only in dimension n = 1 - in fact we

26

prove that this property of the Hardy operator is not available in higher dimensions. Before

stating the theorem we need to define the fractional energy and the off testing conditions.

Definition 2.3.2. We say that the pair (σ, ω) satisfies the energy (resp. dual energy) con-

dition if

(E α
2 )2 ≡ sup
Q= ˙∪Qr

1

σ(Q)

(cid:0)E α,∗

2

(cid:1)2 ≡ sup

Q= ˙∪Qr

1

ω(Q)

∞(cid:88)

Pα(cid:0)Qr, 1Qσ(cid:1)
Pα(cid:0)Qr, 1Qω(cid:1)
∞(cid:88)

|Qr| 1

2(cid:13)(cid:13)(cid:13)x − mω
2(cid:13)(cid:13)(cid:13)x − mσ

r=1

n

Qr

Qr

(cid:13)(cid:13)(cid:13)2
L2(cid:16)
(cid:13)(cid:13)(cid:13)2
L2(cid:16)

r=1

|Qr| 1

n

(cid:17) < ∞

1Qr ω

(cid:17) < ∞

1Qr σ

where the supremum is taken over arbitrary decompositions of a cube Q using a pairwise

disjoint union of subcubes Qr, where Pα(Q, µ) is the standard Poisson integral and

(cid:90)

(cid:90)

(cid:28) 1

|I|µ

I ≡ 1
mµ

µ(I)

xdµ(x) =

x1dµ(x), ...,

1
|I|µ

xndµ(x)

.

(cid:29)

Definition 2.3.3. The off-testing constants Toff,α and Rj,off,α in R2 by

(cid:90)

(cid:19)2

(cid:19)2

T 2
off,α = sup
Q

1

|x − y|2−α dω(y)

Q

dσ(x)

(cid:90)

(cid:18)(cid:90)

1

ω(Q)

(cid:90)

R2\Q

(cid:18)(cid:90)

R2
m,off,α = sup
Q

1

ω(Q)

R2\Q

Q

tm − xm
|x − t|3−α dω(t)

dσ(x),

1 ≤ m ≤ 2

for all cubes Q ⊂ R2 whose sides are parallel to the axes.

Theorem 2.3.4. For 0 ≤ α < 2, there exists a pair of locally finite Borel measures σ, ω in
R2 such that the fractional Muckenhoupt Aα
finite but the off-testing constant Toff,α is not.

and the energy E α

2 , E α,∗

constants are

2 ,Aα,∗

2

2

27

Theorem 2.3.5. For 0 ≤ α < 2, there exists a pair of locally finite Borel measures σ, ω in
2 ,Aα,∗
R2 such that the fractional Muckenhoupt Aα
finite but the off-testing constants Rm,off,α are not.

and the energy E α

2 , E α,∗

constants are

2

2

With full testing in hand, we obtain a number of properties that greatly simplify matters

but we no longer have this tool. Here are the main challenges encountered in passing from

the one-dimensional setting to the higher dimensional analog.

1. The nearby form. The main difficulty in proving the T b theorem in dimensions

n > 1 arises in treating the nearby form in Chapter 5. Full testing is used repeatedly

everywhere in this chapter, and a demanding technical approach involving random

surgery and averaging, is needed throughout this chapter.

In particular, to obtain

estimates over adjacent cubes, we decomposed one of the cubes into a smaller rectangle

that is separated from the other cube by a halo. The separated part is estimated by

the Muckenhoupt’s A2 condition, while the halo is estimated by applying probability
over grids. A typical example is the following: Let I be a cube in the grid associated
to the function f and J a cube in the grid associated to the function g. Let also bI , b∗
be the testing functions used in the theorem for these cubes.

J

We would like to estimate(cid:82) T α

(cid:16)

(cid:17)

b∗
J 1J dω. The domains of integration inside
the operator and inside the integral are adjacent. In dimension n = 1 we could use

bI 1I\J

σ

Hytönen’s result. Now we instead argue by splitting the integral as follows:

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

(cid:16)

(cid:17)

T α
σ

bI 1I\J

b∗
J 1J dω

(cid:17)

(cid:12)(cid:12)(cid:12)(cid:12) +
(cid:12)(cid:12)(cid:12)(cid:12) .

b∗
(cid:17)
J 1J dω
b∗
J 1J dω

bI 1I\(1+δ)J

bI 1(I\J)∩(1+δ)J

(cid:16)

T α
σ

(cid:16)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12) ≤
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

T α
σ

28

The first term on the right hand side, where the domains inside the operator and

the integral are disjoint with positive distance, is bounded by a constant multiple,

depending on δ and n, times the A2 constant. Using averaging over grids, the second
term on the right hand side is bounded by δNT α where the small δ gain comes from
the fact that |(I\J)∩ (1 + δ)J| 1
n ≈ δ|I| where |·| denotes the Lebesque measure of the
cube.

2. Splitting forms. Here we begin with a pair of smooth compactly supported func-

tions (f, g) and we would like to decompose the functions into their Haar expansions.
However, when we select a grid G for f, the support of f may not be contained in any
of the dyadic cubes in the grid G, with a similar problem when selecting a grid H for
g. To deal with this, we follow NTV by adding and subtracting certain averages for

these terms, resulting in four integrals to be controlled by our hypotheses. In the one

dimensional setting, full testing was used to eliminate three out of the four such inte-

grals that appear after decomposing the functions in sums of martingale differences.

Here in this paper, the argument was adjusted to avoid using full testing by averaging
over the two grids G and H associated with f and g.

3. Pointwise Lower Bound Property (PLBP). In [69] for n = 1, the P LBP was

used to control terms involving certain ‘modified dual martingale differences’ in which

a factor bQ had been removed. Moreover, it was proved there that, without loss of
generality, the p-weakly accretive families of testing functions bQ and b∗
for Q ∈ P
could be assumed to satisfy the pointwise lower bound property, written P LBP :

Q

(cid:12)(cid:12)bQ (x)(cid:12)(cid:12) ≥ c1 > 0

for Q ∈ P and σ-a.e. x ∈ R,

29

for some positive constant c1. However, this reduction to assuming P LBP depended
heavily on Hytönen’s A2 characterization for supports on disjoint intervals, something
that is unavailable in higher dimensions as the following theorem shows:

To circumvent this difficulty we used an observation (that goes back to Hytönen and

Martikainen) that under the additional assumption that the breaking cubes Q, those
for which there is a dyadic child Q(cid:48) of Q with bQ(cid:48)
(cid:54)= 1Q(cid:48)bQ, satisfy an appropriate
Carleson measure condition.

4. Indented corona.

In chapter 8 (dealing with the stopping form) we construct an

‘indented corona’. In dimension n = 1 this construction simply reduces to consideration

of the ‘left and right ends’ of the intervals. In the absence of ‘right and left ends’ in

higher dimensions, this simple construction is replaced by a more intricate tower of

Carleson cubes.

2.4 Organization of the paper

In chapter 3, section 3.1 we prove theorems 2.1.1 and 2.1.2. In section 3.2 we prove theorem
2.2.1 and theorem 2.2.2. We prove the T 1 theorem for A∞ weights, theorem 2.2.3, in
In
subsection 3.2.3, using the Sawyer testing condition (see (2.2.9) and theorem 3.2.6).

subsection 3.2.4 we prove theorem 2.2.4 and give a partial answer to question 2 in theorem

3.2.11. In chapter 4 we prove theorems 2.3.4 and 2.3.5. In chapter 5 we prove theorem 2.3.1.

Check the lattices and the graph in section 2.5 for a summary of the theorems presented in

this paper and the history of weighted theory.

30

2.5 Open problems, lattices and a historic diagram

Here is a list of open questions.

1. The most difficult and important problem in the theory of T 1 and T b arises from the

fact that, while the Muckenhoupt buffer conditions are necessary for boundedness of

a wide range of singular integrals, the energy buffer conditions are only necessary for

boundedness of the Hilbert transform and some perturbations in dimension n = 1, see

[57], [68]. What is a reasonable substitute for the energy buffer conditions in a T 1 or

T b theorem?

2. Does Theorem 2.3.1 remain true in the case p = 2, i.e. when b = (cid:8)bQ

(cid:9)

2-weakly σ-accretive family of functions, and b∗ =
family of functions?

(cid:111)

(cid:110)

b∗

Q

Q∈P is a
is a 2-weakly ω-accretive

Q∈P

3. In the special case of the Hilbert transform in dimension n = 1, are the energy con-

ditions in Theorem 2.3.1 already implied by the Muckenhoupt, b-testing and dual
b∗-testing conditions for a pair of p-weakly accretive families, p > 2?

The following lattices provide a summary of the implications among the necessary and

sufficient conditions in weighted theory.

One weight conditions

Combining (3.1.3), theorem 2.1.1, remark 3.1.3, remark 3.1.4, remark 3.1.5 and theorem

3.1.6 we get, for p < q, the following lattice of inclusions for the conditions used in one

31

weighted theory

(cid:40)
A1(ω) (cid:40) Ap(ω) (cid:40) Aq(ω) (cid:40) A∞(ω)

Aα∞(ω) ∩ D(ω) (cid:40)

(cid:40)

Cp(ω) ∩ D(ω) (cid:40)

Two weight conditions

D(ω)
D(ω)

Cp(ω)

Aα∞(ω)

Combining remark 3.2.1, theorem 2.2.1, theorem 2.2.2, remark 3.2.9, theorem 2.2.4, remark

2.2.5, corollary 2.2.6, theorem 3.1.6, theorem 3.2.6, corollary 3.2.7 and the example in [32]

we get the following lattice of inclusions for the conditions used in two weighted theory.

For general Radon measures:

Theorem 2.2.1,

remark 3.2.1, remark 3.2.9, [32]:

p (ω, σ) ∪ At1

p (σ, ω)

V(ω, σ)p (cid:40) Ap(ω, σ) (cid:40) At1
p (ω, σ) ∩ A
t1
t1
p (σ, ω) = A
p (ω, σ) (cid:40) A
t1

t2
p (ω, σ)

A

A

t2
p (ω, σ)

(cid:54)=⇒ At1

p (ω, σ) (cid:40) At2

p (ω, σ)

Remark 2.2.5,

theorem 2.2.4, corollary 2.2.6:

E(ω, σ)p (cid:40) V(ω, σ)p

32

For doubling measures:

Theorem 2.2.2:

Theorem 3.2.6, corollary 3.2.7:

Theorem 3.2.11:

p (ω, σ)

p (ω, σ) = At2
t1

Ap(ω, σ) = A
Ap(ω, σ) ∩ A∞(ω) (cid:40) Sd(ω, σ) ⊆ V(ω, σ)p
Ap(ω, σ) ∩ D(σ) ∩ D(ω) (cid:40) V(ω, σ)p

(small doubling constant)

We end the introduction with a diagram detailing the relevant history of two weight

theory for this paper. Many important contributions are omitted, such as those dealing with

Lp, Lq assumptions in the case of Lebesgue measure, see for example [17] and references

there, and results for dyadic operators, see for example [4] and references there.

33

Figure 2.5.1: Theory development diagram

As is evident from the diagram, theorem 2.3.1 (and its precursor for n = 1) is the first

local T b theorem for two weights.

34

Chapter 3

Necessary and sufficient conditions in

weighted theory

3.1 One weight conditions

3.1.1 The A1 and A∞ conditions.

We say the weight w(x) is an A1 weight if and only if

M w(x) ≤ [w]A1

w(x)

(3.1.1)

and we call [w]A1
for p > 1.

the A1 constant of w. A1 is a stronger condition than the Ap condition

(cid:91)

If we take the union of all the Ap weights for the different p we get the larger class of
Ap (check [12], chapter 7). Another equivalent and commonly
A∞ weights, i.e. A∞ =
used characterization for A∞ weights is the following: We say w ∈ A∞, if for all I ⊂ Rn
and E ⊂ I, there exist uniform constants C, ε > 0 such that

p>1

.

(3.1.2)

(cid:18)|E|

(cid:19)ε

w(E)
w(I)

≤ C

|I|

35

Remark 3.1.1. We have the following linear lattice for 1 < p < q < ∞:

A1 (cid:40) Ap (cid:40) Aq (cid:40) A∞

(3.1.3)

The power weights w(x) = |x|α show that all the inclusions are proper.

In particular we have the following known lemma.

Lemma 3.1.2. Let w(x) = |x|α, x ∈ Rn. Then

(α + n)−1(−α p(cid:48)

otherwise

∞,

[w]Ap ≈

− p
p(cid:48) , −n < α < n(p − 1)

p + n)

3.1.2 A Cp and doubling weight that is not in A∞

For the rest of this section, we are going to say that a measure ω is doubling if

ω(3I) ≤ Cω(I).

(3.1.4)

This definition is equivalent to (1.6.1). In this subsection we give the proof for theorem 2.1.1.

The construction is a very involved variation of the construction in [15].

Proof of theorem 2.1.1: Let I0 = [− 1
2] and In = 3In−1 = 3nI0, the intervals centered
at 0 with length 3n. We call G the triadic grid created by the intervals In. Define the
measure w as follows: w (x) = 1, x ∈ I0 and w (In) = 1
3 > δ1 > 0 to be determined later.
, 1
δn
1

2 , 1

36

Il
kn

I0

Figure 3.1.1: Building construction

n the left, middle and right third of In respectively. Let w(x) =

3−n+1(1−δ1)

2δn
1

,

n , Ir

Call Il
n, Im
x ∈ Ir
n.
Fix k ∈ N and nk ∈ N to be determined later. Let Il,m
nk
(cid:17)
nk−m

and |Il,0
nk
determined later. For m ≥ 2 let w (x) =

| = 3m, 0 ≤ m ≤ nk − 1. Let w(Il,m
nk
3(1−δ2)
| w
2|I
l,m
nk

Il,m
nk

) = δ

(cid:16)

2

defines w completely outside 3Il,0
nk

. Check the figure below.

to have the same center as Il
nk
3 > δ2 > 0 to be
. This

w(Il
nk
, for all x ∈ Il,m
nk

\Il,m−1
nk

), where 1

Now let I ⊂ Il,0
nk
determined later. Let

be any triadic interval such that |I| ≥ 3−ik, and ik ∈ N will be

 δ2w (πI)

1−δ2
2 w (πI)

w (I) =

if ∂I ∩ ∂πI = ∅
if ∂I ∩ ∂πI (cid:54)= ∅

where πI is the triadic parent of I in the grid G. Let w (x) be constant for any triadic interval
I ⊂ Il,0
nk

with |I| ≤ 3−ik.

We are left with defining w on 3Il,0
nk

\Il,0
nk

. Call J l
nk

l,ik
nk

be the right most triadic ik child of J l
nk

third. Let J
x ∈ J
middle and right thirds of I and define w(x) =

. Now for all triadic I such that J

l,ik
nk

l,ik
nk

the left third of 3Il,0
nk
and let w(x) =

and J r
nk

(cid:16) 1−δ2

(cid:17)ik w(J l

2

its right

),

nk

⊂ I ⊂ J l
nk
3(1−δ2)
2|I| w(I), x ∈ Il, w(x) =

, let Il, Im, Ir denote the left,
3δ2|I| w(I),

37

x ∈ Im and w(Ir) =
construction on J l
nk

1−δ2
2 w(I). Similarly (but on the left end) we define w on J r
nk

. This

and J r
nk

is done so that w is doubling.

Indeed, to see that w is doubling, let J1, J2 be two triadic intervals of the same length
. If not,

that touch. If they have the same triadic parent then w(J1)/w(J2) (cid:46)
we apply the first case to their common ancestor and get again w(J1)/w(J2) (cid:46)
.
min(δ1,δ2)
For an arbitrary interval I, let 3m ≤ |I| ≤ 3m+1. Then I ⊂ J1 ∪ J2 triadic intervals with
|J1| = |J2| = 3m+1. Then w(3I) (cid:46)

min(δ1,δ2)

1

1

1

min(δ1,δ2) w(I).

Allowing ik → ∞ makes w singular to Lebesgue. Check ([15], Lemma 2.2). Choose ik so

that there exists an interval Jnk

(cid:17)
(cid:17) ≈ 1

2

and Enk ⊂ Jnk ⊂ 3Il,0
nk

,such that

,

|Enk|
|Jnk| ≈ 1

2k

and

w (E)
w (I)

(cid:46) 2k|E|
|I|

(3.1.5)

(cid:16)
(cid:16)

w

w

Enk
Jnk

and E ⊂ I. This can be done by following in [15] definition 2.1.
for all intervals I ⊂ 3Il,0
nk
and lemma 2.2. Note that because we stop at height ik, (3.1.5) tells us that there is a “worst
interval" Jnk

.

By letting k → ∞ it is clear that A∞ fails to hold for w. So we now need to prove that

the Cp condition holds.

By the end of the next calculation we will determine δ1, δ2. We want to prove w is Cp

and for that we need to show that (2.1.4) holds with (cid:82)R (M 1I (x))p w (x) dx < ∞ for any

interval I. Let first, I = Il,0
nk

.

38

We have

(cid:32)

M 1

(cid:33)p

(x)

l,0
I
nk

(cid:32)

w (x) dx =

(cid:33)p

(cid:90)

l,0
I
nk

(cid:90)
(cid:90)

R

(cid:32)
(cid:90)

(cid:33)p

M 1

(x)

l,0
I
nk

(cid:32)

w (x) dx +

(cid:33)p

(3.1.6)

\I

+
≡ A + B + C

Il
nk

l,0
nk

(x)

M 1
I

l,0
nk

w (x) dx +

R\Il
nk

(x)

w (x) dx

M 1

l,0
I
nk

We have immediately A = w

(cid:90)

Il
nk

\I

l,0
nk

B =



≈ 2−p (1 − δ2)

(cid:16)

Il,0
nk

(cid:17), for B we get
p
(cid:17)

(cid:16)
(cid:16)

|Il,0
|
nk
| + 2dist
3−mp
δm
2

w

2|Il,0
nk

nk−1(cid:88)

m=1

w (x) dx

(cid:17)

x, Il,0
nk
= 2−p (1 − δ2) w

Il,0
nk

(cid:16)

Il,0
nk

(cid:17) nk−1(cid:88)

(cid:18)3−p

(cid:19)m

m=1

δ2

Now choose δ2 = 3−p
want nk so that

2

so that the series above diverges (any δ2 ≤ 3−p works here). We also

2−p (1 − δ2) w

(cid:16)

Il,0
nk

(cid:17) nk−1(cid:88)

(cid:18)3−p

m=1

δ2

(cid:19)m (cid:38) 2kw

(cid:16)

(cid:17)

.

Il,0
nk

(3.1.7)

(3.1.8)

We are only left with calculating term C. We have,

(cid:90)



C =

R\Il
nk

2|Il,0
nk

≈ 2−p3−nkp (1 − δ1)

(cid:16)
|Il,0
|
nk
| + 2dist
1 − δ2
nk−1
δ

2

x, Il,0
nk

(cid:16)

w

Il,0
nk

p
(cid:17)
(cid:17) ∞(cid:88)

w (x) dx

(cid:18)3−p

(cid:19)m

δ1

m=1

Choose δ1 > 3−p so that the infinite series converges. Combining the estimates for A, B and

39

C we get:

and

(cid:33)p

(cid:32)

(cid:90)

R

(x)

M 1
I

l,0
nk

w (x) dx < ∞

(cid:82)R

(cid:32)

w (E)

(cid:33)p

(x)

w (x) dx

M 1

l,0
I
nk

≤ w (E)
2kw(Il,0
nk

)

(cid:46) 2k
2k

|E|
|Il,0
nk

| =

|E|
|Il,0
nk

| .

(3.1.9)

(3.1.10)

for E ⊂ Il,0
. We want to extend (3.1.9) and (3.1.10) to all triadic intervals. Note that (3.1.9)
nk
holds for any interval I. To see that, choose n big enough so that I ⊂ In. Then, following
the calculations for estimating C in (3.1.8) we get that

(cid:90)

R\In

(M 1I (x))p w (x) dx < ∞

which of course gives us

(cid:90)

R

(M 1I (x))p w (x) dx < ∞

(3.1.11)

, note that we can follow the same calculations

To get (3.1.10) for any triadic I ⊂ 3Il,0
nk
that led to (3.1.7) and just choose nk big enough so that we get the gain 2kw(I). This
is possible since the construction is finite and it stops at some height ik. For that finite
number of intervals, we choose nk big enough so that all the intervals get the gain 2k, i.e.

(cid:82)R (M 1I (x))p w (x) dx ≥ 2kw(I). So we have for any E ⊂ I, using (3.1.5),

(cid:82)R (M 1I (x))p w (x) dx

w (E)

≤ w(E)
2kw(I)

(cid:46) |E|
|I| .

(3.1.12)

We will use the following calculation for triadic intervals I ⊂ Il
nk

. Let I = 3Il,0
nk

, following

40

(3.1.6) and using δ2 = 3−p

2

,

(cid:90)

R

(M 1I)pdw ≡ A(cid:48) + B(cid:48) + C(cid:48).

and A(cid:48) + B(cid:48) ≈ 3p(A + B) hence

(cid:90)

Il
nk

(M 1I (x))p w (x) dx ≈ 3p

(cid:90)

Il
nk

(cid:32)

(cid:33)p

(x)

w (x) dx

M 1

l,0
I
nk

and

(cid:82)R (M 1I (x))p w (x) dx

w (E)

(cid:46)

w(E)

3p2kw(Il,0
nk

)

(cid:46) 31−p|E|

|I| ≤ |E|
|I|

(3.1.13)

, so we don’t lose any of the “gain" necessary for (3.1.12) to hold. We can
B(cid:48) = 0. To

for any E ⊂ 3Il,0
nk
⊂ I ⊂ Il
repeat for all triadic intervals I such that Il,0
nk
nk
extend (3.1.13) to triadic intervals I ⊃ Il
notice that
nk

. Note that for I = Il
nk

w(E)

w(π(I))

(cid:46) δ1

w(E)
w(I)

(cid:46) δ1

|E|
|I| ≤ 3δ1

for any E ⊂ 3Il,0
nk

, where we used δ1 < 1
3

.

|π(I)| ≤ |E|
|E|

|π(I)| =⇒ w(E)

w(π(I))

(cid:46) |E|
|π(I)|

To get (3.1.12) for an arbitrary triadic interval, let I be a triadic interval not contained

in any Il,0
nk

and E any subset of I. We write

 (cid:91)

⊂I

I

l,0
nk

E =

(cid:16)

E ∩ 3Il,0
nk

(cid:17)(cid:91)E(cid:15) (cid:91)

 = E1 ∪ E2

3Il,0
nk

⊂I

l,0
nk

I

41

Using (3.1.13) we see that

w(E1) =

(cid:88)

⊂I

l,0
I
nk

w(E ∩ 3Il,0
nk

To deal with E2, note that for x ∈ I(cid:15) (cid:91)

) (cid:46) (cid:88)

(cid:90)

|

R

|E ∩ 3Il,0
nk

|I|

(M 1I)pdw

(3.1.14)

(cid:90)

⊂I
l,0
I
nk
|E1|
|I|

=

(M 1I)pdw

R

, w(x) (cid:46) 3(1−δ2)

2|I| w(I) so we get

3Il,0
nk

⊂I

l,0
I
nk

w(E2)
w(I)

≈ |E2|
|I|

(3.1.15)

combining (3.1.14), (3.1.15) we get (3.1.12) for a triadic interval I.

We are left with extending (3.1.12) to an arbitrary interval I. Let 3m ≤ |I| ≤ 3m+1 and

E ⊂ I. Then I ⊂ J1 ∪ J2, J1, J2 triadic intervals such that |J1| = |J2| = 3m+1. Since

M 1I (x) ≈ M 1J1

(x) ≈ M 1J2

(x), for all x ∈ R.

we get

(cid:82)R (M 1I (x))p w (x) dx

w (E)

(cid:16)
(cid:82)R
≈
M 1J1
(cid:46) |E ∩ J1|
|J1| +

(cid:17)p
w (E ∩ J1)
(x)
|E ∩ J2|
|J2| ≈ |E|
|I|

w (x) dx

+

(cid:16)

(cid:82)R

(cid:17)p
w (E ∩ J2)

M 1J2

(x)

w (x) dx

This shows that w satisfies (2.1.4) and hence w is a Cp weight and the proof is complete.

42

3.1.3 Doubling Cp weights are in A∞ for small doubling constants

Note that the construction in the proof of theorem 2.1.1 we depended heavily on the big

doubling constant of the weight w. Here we show that this is the only case by proving

theorem 2.1.2.

Proof of theorem 2.1.2: It will be enough to show that

(cid:90)

Rn

|M 1I|pw(x)dx ≈ w(I)

the result then follows immediately from (2.1.4). Let In = 3mI the cubes with same center
as I and side length (cid:96)(In) = 3m(cid:96)(I). We write

(cid:90)

Rn

|M 1I|pwdx =

≈

(cid:90)

∞(cid:88)
∞(cid:88)

m=0

m=0

(cid:90)

∞(cid:88)

m=0

|M 1I|pwdx ≈

Im\Im−1
|I|p w(Im)
|Im|p

∞(cid:88)

(cid:46)

Im\Im−1
(cid:46) w(I)

(Cw)mw(I)

(3np)m

m=0

|I|pw(x)dx
n + dist(x, I))np

(|I| 1

since Cσ < 3np by hypothesis and the series converges.

Remark 3.1.3. Not all doubling weights are Cp weights. For an example just choose δ1 <
3−p in the construction of theorem 2.1.1.

Remark 3.1.4. There exist non-doubling Cp weights. For an example choose δ2,k = 1

5k in
each Ink in the construction of theorem 2.1.1. A much simpler example is given by getting
the Lebesgue measure in Rn and setting the measure of the unit ball equal to 0, i.e. define
w(E) = m(E\B(0, 1)) where B(0, 1) is the unit ball in Rn.

43

3.1.4 The Aα∞ condition.
To complete the picture for the one weight conditions we are introducing the fractional Aα∞
condition. We are following very closely [58] where Aα∞ was introduced.

First we define the α−relative capacity of a measure Capα(E; I) of a compact subset E

of a cube I by

(cid:110)(cid:90)

Capα(E; I) = inf

h(x)dx : h ≥ 0, Supph ⊂ 2I and Iαh ≥ (diam2I)α−n on E

(cid:111)

Check [1] for more properties on capacity.

We say that a locally finite positive Borel measure ω is an Aα∞ measure if

ω(E)
ω(2I)

≤ η(Capα(E, I))

(3.1.16)

when ω(2I) > 0, for all compact subsets E of a cube I, for some function η : [0, 1] → [0, 1]
with lim
t→0

η(t) = 0.

Note that omitting the factor 2 in ω(2I) above makes the condition more restrictive in
general, but remains equivalent for doubling measures. It is shown in [58] that ω ∈ Aα∞
implies the Wheeden-Muckenhoupt inequality

(cid:90)

(cid:90)

|Iαf|p dω ≤

|Mαf|p dω

(3.1.17)

for all f positive Borel measures.

Remark 3.1.5. Aα∞ measures are not necessarily doubling. Take for example the Lebesgue

44

measure in Rn and set the measure of the unit ball equal to 0, i.e. define ω(E) = m(E\B(0, 1))
where B(0, 1) is the unit ball in Rn. This measure is clearly non-doubling and hence not in
A∞ but it is an Aα∞ measure.

There exist also doubling fractional A∞ measures that are not in A∞. The example
we use for that is exactly the one used in [15] but here we have to calculate the relative

capacities of the sets used.

Theorem 3.1.6. (Aα∞∩D (cid:59) A∞) There exist a measure µ singular to the Lebesgue measure
that is doubling and satisfies the Aα∞ condition with η(t) = t but µ is not an A∞ weight.

Proof. Let µ([0, 1]) = 1, 0 < δ < 3−1 to be determined later, and for any triadic I ⊂ [0, 1]
let

 δµ (πI)

1−δ
2 µ (πI)

µ (I) =

if ∂I ∩ ∂πI = ∅
if ∂I ∩ ∂πI (cid:54)= ∅

It was shown in [15] that µ is a doubling measure. It is also shown that it is singular to the

Lebesgue measure hence it does not satisfy the A∞ condition.

To show that it satisfies the Aα∞ condition, let I ⊂ [0, 1] be a triadic interval and E ⊂ I

be compact.

We claim that ||IαµI||L∞(I) = Cα,δµ(I)|I|α−1, where µI is the restriction of µ on the

set I and the constant Cα,δ is independent of I. For any x ∈ I we have
(cid:19)k

(cid:18)1 − δ

|x − y|α−1dµ(y) (cid:46) µ(I)|I|α−1

∞(cid:88)

(cid:90)

IαµI (x) =

I

k=0

3k(1−α)

2

= Cα,δµ(I)|I|α−1

as long as 31−α 1−δ

2 < 1 ⇒ α > 1 − ln( 2
1−δ
ln 3

)

.

45

Now for any f ≥ 0, Suppf ⊂ 2I and Iαf ≥ |2I|α−1 on E, using Fubini’s theorem we

have

(cid:90)
≤||f||1||IαµE||∞|2I|1−α ≤ ||f||1||IαµI||∞|2I|1−α (cid:46) ||f||1µ(I)

|2I|1−αIαf (x)1E(x)dµ(x) =

1Edµ ≤

(cid:90)

I

(cid:90)

I

|2I|1−αIαµE(x)f (x)dx

µ(E) =

So Capα(E, I) (cid:38) µ(E)

µ(I)

hence Aα∞ holds with η(t) = t and the proof is complete.

3.2 Two weight conditions

We start this section with the proofs of theorems 2.2.1 and 2.2.2.

3.2.1 Non doubling Ap examples
Remark 3.2.1. Note first that we have the following simple implications At2

p ⇒ At1

p ⇒ Ap.

Indeed it is easy to see:

P (I, σ) =

=

(cid:90)

|I|

(cid:90)

I
σ(I)
|I| +

(cid:90)

R/I

|I|

(|I| + dist(x, I))2 σ(dx)

(|I| + dist(x, I))2 σ(dx) +

|I|

(|I| + dist(x, I))2 σ(dx) ≥ σ(I)
|I|

R/I

and so immediately from the definitions (2.2.1), (2.2.2) and (2.2.3) we get

Ap(ω, σ) ⊆ At1

p (ω, σ) ⊆ At2

p (ω, σ).

Remark 3.2.2. We work with p = 2 for simplicity. The examples we use work with trivial

modifications for any p > 1.

46

Proof of theorem 2.2.1:
(1) We want to construct two measures ω, σ such that the two weight classical A2 con-
dition holds but both one tailed A2 conditions fail. First, we construct measures uk and vn
that satisfy

k

uk(I)vn
|I|2

k (I)

≤ M,

uk(I)
|I| P (I, vn

k ) (cid:38) n

where the constant M does not depend on k, n. Then we will combine the measures uk and
to create ω, σ such that the two weight classical A2 condition holds and both one tailed

vn
k
A2 conditions fail for the pair ω, σ. Let

uk(E) = m(E ∩ [k, k + 1]),

vn
k (E) =

(cid:16)

E ∩(cid:104)

k + 2i, k + 2i+1(cid:105)(cid:17)

2im

n(cid:88)

i=0

where m is the classic Lebesgue measure on R. Let I = (a, b), a < k + 1 and k + 2i−1 ≤ b <
k + 2i, for some i ≥ 0 (of course if the interval does not intersect [k, k + 1] then uk(I) = 0).
Then

uk(I)vn
|I|2

k (I)

≤

4i+1 − 1

(4 − 1)(2i−1 − 2)2 = M

(3.2.1)

which is bounded for i > 2 (the cases i = 0, 1, 2 can be seen directly). Now let I = [k, k + 1].

We have then:

(cid:90)

R

uk(I)
|I| P (I, vn

k ) =

vn
k (dx)

(1 + dist(x, [k, k + 1]))2 =

n(cid:88)

i=0

(cid:90)

Ii

vn
k (dx)

(1 + dist(x, [k, k + 1]))2

47

where Ii = [k + 2i, k + 2i+1]. We get:

k(cid:88)

i=0

(cid:90)

Ii

vn
k (dx)

(1 + dist(x, [1, 2]))2 ≥ 1 +

= 1 +

n(cid:88)
n(cid:88)

i=1

i=1

vn
k (dx)
1 + 2i − 2)

(cid:90)
(cid:16)
(cid:17)2
n(cid:88)
(cid:0)2i − 1(cid:1)2 = 1 +

vn
k (Ii)

Ii

i=1

(3.2.2)

(cid:0)2i − 1(cid:1)2 ≈ n.

22i

Now we define ω, σ as follows:

ω(E) =

σ(E) =

∞(cid:88)
∞(cid:88)

k=1

k=1

∞(cid:88)
∞(cid:88)

k=1

k=1

u

100k (E) +

vk−100k (E)

u−100k (E) +

vk
100k (E)

It is easy to see that with I = Ik = [100k, 100k + 1] both one tailed A2 conditions fail

using (3.2.2).

To see that the classical A2 condition holds, let I = (a, b) be any interval. It is simple to

check that if I is big enough such that |a| ≈ 100k,|b| ≈ 100n for k (cid:54)= n then

ω(I)σ(I)

|I|2 ≤ 1.

While if |a| ≈ |b| ≈ 100k for some k then using (3.2.1) we get

ω(I)σ(I)

|I|2 ≤ 2M

hence the classical two weight A2 condition holds but both one tailed A2 conditions fail.

48

(2) Now we turn to proving At1

2

(cid:59) At2

2

. Let the new measures be:

∞(cid:88)

(cid:16)

E ∩(cid:104)

2n, 2n+1(cid:105)(cid:17)

2nm

ω(E) =
σ(E) = m(E ∩ [0, 1])

n=1

From the construction above we can see that with I = [0, 1] we get:

(cid:90)

ω(dx)

(cid:90) 1

(cid:90)

(cid:90)

=

P (I, ω)P (I, σ) =

R

(1 + dist(x, [0, 1]))2

R

(1 + dist(x, [0, 1]))2

ω(dx)

R

(1 + dist(x, [0, 1]))2

0

dx

(1 + dist(x, [0, 1]))2 =

σ(dx)

(cid:90)

ω(dx)

R

(1 + dist(x, [0, 1]))2

22n
22n = ∞

∞(cid:88)

n=1

Now from the definition of ω the last expression is equal to:

(cid:90) 2n+1

∞(cid:88)

2n

n=1

2n

(1 + dist(x, [0, 1]))2 dx ≥

To prove that At1
dist(I, [0, 1]) < 2k+1 − 1 with k ≥ 0. We have two cases:

2

hold let I be an interval such that 2n ≤ |I| < 2n+1 and 2k − 1 ≤

(i) n ≥ k.

ω(I)

|I| P (I, σ) ≤

n+1(cid:88)

l=1

(cid:16)

I ∩(cid:104)
2l, 2l+1(cid:105)(cid:17)

2lm

n+1(cid:88)

22l

|I|2

≤

l=1

22n =

22(n+2) − 1
(4 − 1)22n < M < ∞

where the first inequality uses the fact that the interval cannot intersect any point in
[2n+2,∞) otherwise n ≥ k would not be satisfied.

49

(ii) n < k. If k = 0 then I ∩ [2,∞) = ∅ and there is nothing to prove. So assume k > 0.

k+1(cid:88)

(cid:16)

I ∩(cid:104)
2l, 2l+1(cid:105)(cid:17)

2lm

ω(I)

|I| P (I, σ) ≤

l=k

22k

≤ 22k + 22(k+1)

22k

= 5 < ∞

where the first inequality now holds because I cannot contain neither any point in (0, 2k)
for otherwise dist(I, (0, 1)) < 2k − 1 nor any point in [2k+1,∞) because n < k would not be
satisfied and the proof is complete.
(3) Last, for the equivalence of the two tailed A2 condition to both one tailed A2 conditions
let I ∈ Rn be a cube. We have:

P (I, σ) ≈ σ(I)

|I| +

∞(cid:88)

3n−1(cid:88)

k=1

m=1

σ(Ik
m)
3kn|Ik

m| , P (I, ω) ≈ ω(I)
|I| +

∞(cid:88)

3n−1(cid:88)

k=1

m=1

ω(Ik
m)
3kn|Ik
m|

I2
m

I1
m

I

Figure 3.2.1: Splitting for n = 2

where |Ik

m| 1

n = 3k|I| 1

n and d(Ik

m, I) ≈ 3k, and all the implied constants depend only on

the dimension, check Figure 3.2.1. There exist k1, k2 ≥ 0 such that
3n−1(cid:88)
∞(cid:88)

3n−1(cid:88)

k1(cid:88)

 ≈ 2

σ(I)

|I| +

P (I, σ) ≈ 2

σ(Ik
m)
3kn|Ik
m|

k=1

m=1

k=k1

m=1

σ(Ik
m)
3kn|Ik
m|

50

P (I, ω) ≈ 2

|I| +

ω(I)
3n−1(cid:88)
k2(cid:88)
, hence |J| 1
3n−1(cid:91)

Ik
m

m=1

k=1

 k1(cid:91)

We can assume without loss of generality that k1 ≤ k2.
Let J = I ∪

n ≈ 3k1|I| 1

 ≈ 2

∞(cid:88)

3n−1(cid:88)

k=k2

m=1

ω(Ik
m)
3kn|Ik
m|

ω(Ik
m)
3kn|Ik
m|

n where again the implied constant

depends only on dimension. We calculate

k=1

m=1

σ(J)

|J| P (J, ω) ≈ 1
|J|

σ(Ik

m)

1

k=1

m=1

3k1n|I|

σ(I) +
3n−1(cid:88)
k1(cid:88)
σ(I) +
3n−1(cid:88)
k1(cid:88)
σ(I) +
3n−1(cid:88)
k1(cid:88)
σ(I) +
3n−1(cid:88)
k1(cid:88)
σ(I)
3n−1(cid:88)
k1(cid:88)

3k1n|I|

m=1

m=1

m=1

k=1

k=1

k=1

1

|I| +

σ(Ik
m)
3kn|Ik
m|

k=1

m=1

σ(Ik

k=1

m)

|J| +

|J| +

ω(J)
∞(cid:88)
 ω(J)
 ω(J)
 ω(J)
∞(cid:88)
 ω(J)
∞(cid:88)

|J| +

|J| +

m)

|J| +

k=k2

σ(Ik

σ(Ik

m)

k=1

m=1

m=1

3n−1(cid:88)
3n−1(cid:88)
∞(cid:88)
3n−1(cid:88)
∞(cid:88)
3n−1(cid:88)
3n−1(cid:88)

k=k1

m=1

m=1

k=k2

m=1

ω(Ik
m)
3kn|Ik
m|

ω(Ik
m)
3kn|Ik
m|






ω(J k
m)
3kn|J k
m|




ω(I
3kn|I

k+k1
m )
m |
k+k1
3k1nω(Ik
m)
3kn|Ik
m|

≈

≈

(cid:38)

(cid:38) 1
|I|

≈ P (I, σ)P (I, ω)

hence showing that the one tailed Ap conditions bound the two tailed Ap condition and the
proof is complete.

Remark 3.2.3. From the above construction we see that the same measures could work to
prove the same implications for Aoffset

(2.2.13), and it’s two tailed analogue since it’s exactly

p

the nature of the tail that we take advantage of in the construction.

51

3.2.2 Two weight Ap equivalence for doubling measures

Remark 3.2.4. We are going to use p = 2 in the proof for simplicity. The general case

follows immediately since 1

p , 1

p(cid:48) < 1 and hence

 ∞(cid:88)

3n−1(cid:88)

k=1

j=1

 1

p

(cid:32)

∞(cid:88)

3n−1(cid:88)

k=1

j=1

≤

ω(Ik
i )
32kn|I|

(cid:33) 1

p

ω(Ik
i )
32kn|I|

P (I, ω)

1

p ≈

and from here the proof follows the same way as for p = 2.

Proof of theorem 2.2.2: Let ω, σ be reverse doubling measures with reverse doubling

constants 1 + δω and 1 + δσ respectively. It is enough to prove that we can bound the two
tailed At2

p (ω, σ) from the classical Ap(ω, σ). Let I be a cube. We then have,
3n−1(cid:88)

P (I, ω)P (I, σ) (cid:46) ω(I)σ(I)

ω(Ik
i )
32kn|I|

|I|2 +

∞(cid:88)

3n−1(cid:88)

+

m=1

i=1

ω(I)
|I|

σ(Im
i )
32mn|I|

i=1

m=1

σ(I)
|I|

∞(cid:88)

σ(Im
i )
32mn|I| +

3n−1(cid:88)
∞(cid:88)
3n−1(cid:88)
∞(cid:88)
32kn|I| ≡ A + B + C + D
3n−1(cid:91)
n , (cid:91)

ω(Ik
j )

k=1

k=1

j=1

j=1

j | = 3mn|I|, dist(Im

where |Im
depends only on dimension. A is bounded immediately by A2(ω, σ). For B we have:

j = Rn\I and the implied constant
Im

j , I) ≈ 3m|I| 1

m∈N

j=1

∞(cid:88)

m=1

B =

3n−1(cid:88)
 m(cid:91)

i=1

(cid:46)

ω(Im)σ(Im)

3n−1(cid:88)

(1 + δω)−m

∞(cid:88)
 and the implied constant again depends only on dimension

(cid:46) A2(ω, σ) < ∞

|Im|2

m=1

i=1

ω(I)σ(Im
i )
32mn|I|2

3n−1(cid:91)

where Im = I ∪
and the reverse doubling constant of ω. The bound for C is similar to B.

I(cid:96)
j

j=1

(cid:96)=1

52

For D we have:

∞(cid:88)

m(cid:88)

3n−1(cid:88)

3n−1(cid:88)

i )ω(Ik
j )

σ(Im
32mn|I|32kn|I| +

∞(cid:88)

k−1(cid:88)

3n−1(cid:88)

3n−1(cid:88)

k=1

m=1

j=1

i=1

i )ω(Ik
σ(Im
j )
32mn|I|32kn|I|

D =

k=1

m=1

≡ I + II

i=1

j=1

We will get the bound for I, the calculations for II are identical.

∞(cid:88)

m(cid:88)

I (cid:46)

3n−1(cid:88)

3n−1(cid:88)
∞(cid:88)

i=1

m=1

k=1

j=1

(cid:46) A2(ω, σ)

(1 + δω)−m

m=1

k=1

m(cid:88)

(1 + δω)k−m σ(Im)ω(Im)
32kn|Im|2

(cid:46)

(1 + δω)k

32kn

≤ Cn,σA2(ω, σ) < ∞

Combining all the above bounds and getting supremum over the cubes I we get

At2
2 (ω, σ) ≤ Cn,ω,σA2(ω, σ)

which completes the proof of the theorem.

Remark 3.2.5. The same proof works for the fractional Ap(ω, σ) conditions as defined in
[64].

3.2.3 The T1 theorem for A∞ weights.

The goal of this subsection is to prove theorem 2.2.3. For that we are going to use the Sawyer

testing condition.

53

3.2.3.1 The Sawyer testing condition.

Sawyer in [56] proved that the Maximal operator is bounded on Lp(u) → Lq(w) if and only
if 2.2.9 holds, i.e. if and only if

(cid:18)(cid:90)

p (cid:18)(cid:90)
(cid:19)−1

(cid:104)
M (1I u1−p(cid:48)

(cid:105)q

(cid:19) 1

q

)(x)

w(x)dx

< ∞ (3.2.3)

Sp,q(w, u1−p(cid:48)

) = sup

u(x)1−p(cid:48)

dx

I

I

I

where the supremum is taken over all cubes I ⊂ Rn. Replacing the weights w, u with the
measures ω, σ we call Sp,q
d (ω, σ) the dyadic Sawyer testing condition where the Maximal
operator in (3.2.3) is replaced by the dyadic Maximal operator Md where the supremum in
the operator is taken over only dyadic cubes.

Theorem 3.2.6. (σ ∈ A∞, Ap(ω, σ) ⇒ Sp,p
d (ω, σ)) Let ω, σ be Radon measures in Rn such
that σ ∈ A∞. If ω, σ satisfy the Ap(ω, σ) condition then the dyadic Sawyer testing condition
Sp,p
d (ω, σ) holds.

Proof. Let I be a cube in Rn. Let Ωm = {x ∈ I : (Md1I σ) (x) > Km} = ˙(cid:83)Im

, where K is

j

are the maximal, disjoint dyadic cubes such that

a constant to be determined later and Im
j
σ(Im
j )
j | > Km. We have
|Im
(cid:90)

(Md1I σ)p (x)dω(x) (cid:46)(cid:88)
(cid:88)

m,j

I

j )
|Im
j |

(cid:32)σ(Im
σ(Im

j )

=

m,j

(cid:33)p

ω(Im
j )

1
p

1

p(cid:48)
ω(Im
j )
|Im
j |

p

σ(Im

j ) ≤ Ap(ω, σ)

(cid:88)

m,j

σ(Im
j )

54

t =(cid:83)

Call Am

Im+1
j

⊂Im
t

Im+1
j

. Since σ ∈ A∞ we get

(cid:19)ε

(cid:18)|Am
t |
|Im
t |

σ (Am

t ) ≤ C

σ(Im
t )

for some C positive and ε like in (3.1.2). From the maximality of Im
j

we obtain

|Am

t | =

Im+1
j

Km+1 σ (Am

t ) ≤ 2n

t |
|Im

K

Choose K big enough that C

. Fix m ∈ N, k ≥ −m, then

(cid:12)(cid:12)(cid:12) ≤ 1

j

⊂Im
t

t |
|Im
t |

(cid:12)(cid:12)(cid:12)Im+1
(cid:88)
(cid:18)|Am
(cid:19)ε ≤ 1
(cid:18)1
(cid:19)m+k(cid:88)
∞(cid:88)

j ) ≤
(cid:88)

σ(Ik

j ) ≤

2

2

j

k=−m

(cid:18)1

(cid:19)m+k

σ(I)

2

σ(I−m

j

) ≤

2−m−kσ(I) ≤ 2σ(I)

∞(cid:88)

(cid:88)

k=−m

j

σ(Ik

j ) ≤ 2σ(I)

σ(Ik

j ) = lim
m→∞

σ(Ik

(cid:88)
∞(cid:88)

j

(cid:88)

k,j

k=−m
and by taking m → ∞ we get

j

and this completes the proof of the theorem.

With theorem 3.2.6. at hand we get the following corollary.

Corollary 3.2.7. (ω ∈ A∞, Ap(ω, σ) ⇒ V(ω, σ)p) Let ω, σ be Radon measures in Rn such
that σ ∈ A∞. Then the Ap(ω, σ) condition implies the pivotal condition V(ω, σ)p.

55

Proof. Let I be a cube in Rn.

(cid:90)
(cid:18)
∞(cid:88)

m=0

P(I, σ) =

(cid:46)

|I|

(cid:19)2n dσ(x)(cid:46)

|I| 1

n + |x − xI|
Mdσ(x)2−m (cid:46) inf
x∈I

inf
x∈I

Mdσ(x)

σ(cid:0)(2m + 1)I(cid:1)

2m|2mI|

∞(cid:88)

m=0

(3.2.4)

where Md denotes the dyadic maximal function.

Using (3.2.4) we get

Let I0 be a cube in Rn. Let I0 = (cid:83)
(cid:88)

σ) ≤(cid:88)

ω(Ir)Pp(Ir, 1I0

ω(Ir) inf
x∈Ir

r≥1

r≥1

r≥1 Ir be a decomposition of I0 in disjoint cubes.

(cid:16)

Md1I0

σ

(cid:17)p

(x) ≤

(cid:90)

I0

(cid:16)

Md1I0

(cid:17)p

σ

(x)dω(x)

and using theorem 3.2.6. the last expression is bounded by a constant multiple of σ(I0). So
we have

(cid:88)

r≥1

ω(Ir)Pp(Ir, 1I0

σ) ≤ Kσ(I0)

and that completes the proof of the corollary.

Question 2. A∞ is a special class of doubling measures.
measure, Ap(ω, σ) ⇒ V(ω, σ)p?

Is it true that for ω doubling

Question 3. In corollary 3.2.7 we prove that for ω ∈ A∞ dyadic Sawyer testing implies
pivotal. Is it true that Sp,p

d (ω, σ) = V(ω, σ)p?

Remark 3.2.8. The proof of corollary 3.2.7, holds also for the fractional Ap(ω, σ) and
pivotal conditions as stated in [64] (stated for p = 2 but extends immediately to any p > 1).

Proof of theorem 2.2.3: If both the measures ω, σ are in the one weight A∞, then by

56

corollary 3.2.7, the the two weight A2(ω, σ) condition implies both the pivotal conditions
V(ω, σ)2 ((2.2.10) and it’s dual) and we can apply the main theorem from [64] (or the one
in [37]) to get the result.

3.2.4 The “buffer" conditions do not imply the tailed Ap conditions.

The goal of this subsection is to give a proof of theorem 2.2.4. First we make the following

simple remark.

Remark 3.2.9. It is immediate to see that the Pivotal condition implies the classical Ap
condition. Just let the decomposition in (2.2.10) be just a single cube.

Remark 3.2.10. We are going to use p = 2 in the proof for simplicity. The proof works for
1 < p ≤ 2, without any modifications.

Proof of theorem 2.2.4: We construct measures ω and σ so that the pivotal condition V2

(2.2.10) holds, but At1

2

(2.2.2) does not. Let

ω(E) = δ0,

σ(E) =

∞(cid:88)

n=2

nδn(E)

where δn denotes the point mass at x = n. First we check At1
Then

2

does not hold. Let I = [0, 1].

(cid:90)

R

ω(I)
|I| P (I, σ) =

1

(1 + dist(x, [0, 1]))2 σ(dx) =

∞(cid:88)

n=2

n

(n)2 = ∞

To show the pivotal condition holds, let I0 = (a, b) where a < 0 and n ≤ b < n + 1, n ≥ 2
(we need I0 to contain some masses from σ and 0 ∈ I0 for otherwise there is nothing to

57

prove). Decomposing I0 = ˙∪Ir, only the Ir such that 0 ∈ Ir contributes to the pivotal
condition. Call that cube I1. We consider the cases:

(i) |I1| ≤ 1. We calculate:

(cid:32)(cid:90)

ω(I1)P (I1, I0σ)2

σ(I0)

I0

=

(cid:33)2

|I1|σ(dx)

(|I1| + dist(x, I1)))2

σ(I0)

where the constant M does not depend on n.

(ii) |I1| ≥ n. We get:

(cid:32)(cid:90)

ω(I1)P (I1, I0σ)2

σ(I0)

I0

=

(cid:33)2

|I1|σ(dx)

(|I1| + dist(x, I1)))2

σ(I0)

(iii) 1 ≤ |I1| ≤ n. We have:

ω(I1)P (I1, I0σ)2

σ(I0)

(cid:46) |I1|2
n2

(cid:32) |I1|(cid:88)

k=2

k
|I1|2 +

(cid:33)2

n(cid:88)

k=|I1|

1
k

58

(cid:33)2(cid:44) n(cid:88)

k

k=2

k
k2

(cid:33)2(cid:44) n(cid:88)

k

k=2

k
|I1|2

(cid:32) n(cid:88)

≤ |I1|2
≤ M < ∞

k=2

≤ |I1|2

(cid:32) n(cid:88)
≤ n2
|I1|2 ≤ 1

k=2

(cid:18)

(cid:46) |I1|2
n2
(cid:46) |I1|2

n2 +

1 + log

(cid:17)(cid:19)2
(cid:16) n
|I1|
(cid:17)
n2 log2(cid:16) n

|I1|

|I1|2

, x ≥ 1 which
Now, on the last expression setting x = n|I1| we get the function f (x) = log2 x
is bounded independent of n. Combining all three cases we see that the pivotal condition is

x2

bounded.

Question 4. In the above example, one can check that the dual pivotal condition does not
hold. Is it true that V(σ, ω)p ∩ V(ω, σ)p ⇒ At1

p (ω, σ)?

3.2.4.1 Doubling measures and the Pivotal condition.

The result in this subsection is essentially in [58], equation (4.4), but we include it for

completeness. We partially answer positively question 2.

If the measures ω, σ are doubling but not in A∞ then we do not in general know if the
Pivotal condition can be controlled by the Ap(ω, σ) condition. For measures with small
doubling constant though the Ap(ω, σ) condition implies V(ω, σ)p.

Theorem 3.2.11. (Small doubling+Ap(ω, σ) ⇒ V(ω, σ)p) Let ω, σ be doubling measures in
Rn with doubling constants Kω, Kσ and reverse doubling constants 1 + δω, 1 + δσ respectively.
If Kσ < 2p(1 + δω) then the Ap(ω, σ) condition implies the pivotal condition V(ω, σ)p.

Proof. Let I0 be a cube in Rn and I0 = ∪r≥1Ir be a decomposition of I0 in disjoint cubes.

(cid:33)p
p

(cid:32) mr(cid:88)

m=1

(Im
r )

σ(Im
r )
2m|Im
r |

1
p (Im
r )

(cid:33)p

σ

(cid:19) m

p

(cid:88)
≤ (cid:88)

r≥1

r≥1

ω(Ir)

ω(Ir)P (Ir, I0σ)p ≈(cid:88)
 mr(cid:88)
(cid:88)

(cid:32) mr(cid:88)

(1 + δω)

− m

p ω

m=1

r≥1
1
p (Im

1

p(cid:48)
r )σ
2m|Im
r |

(cid:18) Kσ

≤ Ap(ω, σ)

σ(Ir)

r≥1

2p(1 + δω)

m=1

59

(cid:46) Ap(ω, σ)σ(I0)

(cid:17) 1

(cid:16)|I0|

|Ir|

n =

n , Im
r

is the cube with same center as that of Ir and |Im
r | 1

where mr = log2
2m|Ir| 1
reverse doubling constant of ω. This completes the proof of the theorem.

n . where the implied constant depends only on the doubling constant of σ and the

Remark 3.2.12. For a doubling measure ω and a cube I ⊂ Rn we have that in (2.2.12)
E(I, ω)2 ≥ cω > 0 since ω(I1) ≈ ω(I2) where |I1| = |I2| = 2−n|I| and I1 is in the top left
corner of I, I2 in the bottom right corner of I. Hence for ω doubling the Pivotal condition
V(ω, σ)p is equivalent to the Energy condition E(ω, σ)p.

60

Chapter 4

Counterexample to Hytönen’s off-testing

condition in two dimensions

We begin with the proof of Theorem 2.3.4. The proof of Theorem 2.3.5 will be very similar

and we will only have to deal with the cancellation occurring in the kernel with Lemma 4.3.1

being useful.

Proof of Theorem 2.3.4. First we build two measures in R, generalizing the work done in

[32], and then they will be used for our two dimensional construction.

4.1 The One-Dimensional Construction

9 ≤(cid:16) 1−b

2

(cid:17)2−α ≤ 1

(cid:16) 1−b

(cid:17)2−α.

2

. Let s−1

0 =

3

3 ≤ b < 1 such that 1

Given 0 ≤ α < 2, choose 1
Recall the middle-b Cantor set Eb and the Cantor measure ¨ω on the closed interval I0
At the kth generation in the construction, there is a collection {Ik
closed intervals of length |Ik
j=1 Ik
j
and the Cantor measure ¨ω is the unique probability measure supported in E with the property
that is equidistributed among the intervals {Ik

(cid:17)k. The Cantor set is defined by Eb =(cid:84)∞

1 = [0, 1].
of 2k pairwise disjoint

at each scale k, i.e

(cid:16) 1−b

(cid:83)2k

j }2k

j=1

j | =

j }2k

j=1

2

k=1

¨ω(Ik

j ) = 2−k,

k ≥ 0, 1 ≤ j ≤ 2k.

61

We denote the removed open middle bth of Ik
j

by Gk
j

and by ¨zk
j

its center. Following closely

[32], we define

(cid:88)

k,j

sk
j δ

¨zk
j

¨σ =

where the sequence of positive numbers sk
j

is chosen to satisfy sk
j ¨ω(Ik
j )
j |4−2α
|Ik

= 1, i.e.

(cid:32)

(cid:33)k

sk
j =

2
s2
0

, k ≥ 0, 1 ≤ j ≤ 2k.

4.1.1 The Testing Constant is Unbounded

. Consider the following operator

¨T f (x) =

(cid:90)

R

f (y)

|x − y|2−α dy

Note that

¨T ¨ω(¨zk

1 ) =

(cid:90)

I0
1

d¨ω(y)

1 − y|2−α
|¨zk

≥

(cid:90)

Ik
1

d¨ω(y)

1 − y|2−α
|¨zk

≥

(cid:18)

1
2

(cid:17)k(cid:19)2−α ≈(cid:16)s0

2

¨ω(Ik
1 )

(cid:16) 1−b

2

(cid:17)k

since |¨zk

1 − y| ≤ |¨zk

1| for y ∈ Ik

1

and ¨zk

1 = 1

2( 1−b

2 )k. Similar inequalities hold for the rest of

¨zk
j

. This implies that the following testing condition fails:

j ·(cid:16)s0

2

sk

(cid:17)2k

∞(cid:88)

2k(cid:88)

k=1

j=1

=

= ∞

1
2k

(4.1.1)

(cid:18)

¨T (1

(cid:90)

I0
1

(cid:19)2

¨ω)(x)

I0
1

∞(cid:88)

2k(cid:88)

k=1

j=1

d¨σ(y) (cid:38)

62

4.1.2 The ¨A2 Condition

. Let us now define

(cid:32)

(cid:90)

R

¨P(I, µ) =

|I|

(|I| + |x − xI|)2

(cid:33)2−α

dµ(x)

and the following variant of the Aα
2

condition:

¨Aα
2 (¨σ, ¨ω) = sup
I

¨P(I, ¨σ) · ¨P(I, ¨ω)

where the supremum is taken over all intervals in R. We verify that ¨Aα
(¨σ, ¨ω). The starting point is the estimate

2

is finite for the pair

(cid:88)

¨σ(I(cid:96)

r) =

sk
j =
j ∈I(cid:96)

r

(k,j):¨zk

∞(cid:88)

k=l

2k−(cid:96)sk

j = 2−(cid:96)

∞(cid:88)

k=l

(cid:33)(cid:96)

(cid:32)

(cid:33)k

(cid:32)

4
s2
0

≈

2
s2
0

= s(cid:96)
r

and from this, it immediately follows,

j )¨ω(I(cid:96)
¨σ(I(cid:96)
j )
j|4−2α
|I(cid:96)

j ¨ω(I(cid:96)
≈ s(cid:96)
j )
j|4−2α
|I(cid:96)

= 1, for (cid:96) ≥ 0, 1 ≤ j ≤ 2(cid:96).

(4.1.2)

63

Now from the definition of ¨σ we get,

¨P(I(cid:96)

r, ¨σ) ≤ ¨σ(I(cid:96)
r)
r|2−α
|I(cid:96)

≤ ¨σ(I(cid:96)
r)
r|2−α
|I(cid:96)

+

+

(cid:46) ¨σ(I(cid:96)
r)
r|2−α
|I(cid:96)

+

=

¨σ(I(cid:96)
r)
r|2−α
|I(cid:96)
(cid:46) ¨σ(I(cid:96)
r)
r|2−α
|I(cid:96)

+

+

and using the uniformity of ¨ω,

¨P(I(cid:96)

r, ¨ω) ≤ ¨ω(I(cid:96)
r)
r|2−α
|I(cid:96)

+

≤ ¨ω(I(cid:96)
r)
r|2−α
|I(cid:96)

+

≤ ¨ω(I(cid:96)
r)
r|2−α
|I(cid:96)

+

d¨σ(x)

1\I(cid:96)
I0

r

(cid:90)
(cid:96)(cid:88)
(cid:96)(cid:88)


∞(cid:88)

|I(cid:96)
r|

2−α
|(cid:17)2
(cid:16)|I(cid:96)
r| + |x − x
I(cid:96)
(cid:17)m(cid:17)4−2α
(cid:16) 1−b
(cid:16)|I(cid:96)
r
2k−msk
r|2−α
j |I(cid:96)
(cid:19)m
(cid:18)
r| + b
r|2−α
(cid:19)4−2α
(cid:16) 1−b
(cid:17)m−(cid:96) |I(cid:96)
4
s2
0
(cid:33)(cid:96)
(cid:32)
r|
(cid:96)(cid:88)

2

2

2−m|I(cid:96)

(cid:18)

m=0

k=m

m=0

b
b2α−4
r|2−α
|I(cid:96)
s(cid:96)
r
r|2−α
|I(cid:96)

2m

1
s2
0
m=0
≈ ¨σ(I(cid:96)
r)
r|2−α
|I(cid:96)

d¨ω(x)

1\I(cid:96)
I0

r

(cid:90)
(cid:96)(cid:88)
(cid:96)(cid:88)

k=1

k=1

|I(cid:96)
r|



(cid:16)|I(cid:96)

2−α
|(cid:17)2
r| + |x − x
(cid:17)k−1(cid:19)4−2α
(cid:16) 1−b
r|2−α ¨ω(Ik
|I(cid:96)
jk
|I(cid:96)
r| + b
(cid:19)4−2α
(cid:17)k−1−(cid:96) |I(cid:96)
(cid:16) 1−b
r|2−α ¨ω(Ik
|I(cid:96)
)
jk
r|

I(cid:96)
r
)

b

2

2

(cid:18)
(cid:18)

(4.1.3)

(4.1.4)

(cid:46) ¨ω(I(cid:96)
r)
r|2−α
|I(cid:96)

+

2−(cid:96)
r|2−α
|I(cid:96)

= 2

¨ω(I(cid:96)
r)
r|2−α
|I(cid:96)

,

where Ik
jk

⊂ Ik−1

t

, I(cid:96)

r ⊂ Ik−1

t

and Ik
jk

∩ I(cid:96)

r = ∅, and where all the implied constants in the

64

above calculations depend only on α. From (4.1.3), (4.1.4) and (4.1.2), we see that

¨P(I(cid:96)

r, ¨σ) ¨P(I(cid:96)

r, ¨ω) (cid:46) 1.

and let A > 1 be fixed. Then, let k be the smallest

j ∈ AI; if there is no such k, then AI (cid:36) G(cid:96)

j

, for some (cid:96). We have the

Let us now consider an interval I ⊂ I0
integer such that ¨zk

1

following cases:
Case 1. Assume that I ⊂ AI (cid:36) Gk

. If |xI − ¨zk

j | ≤ dist(xI , ∂Gk

j ) then,

(cid:90)

j

j ⊂ Ik
(cid:90)

I0
1

¨P(I, ¨σ) ¨P(I, ¨ω) = |I|4−2α

 sk
(cid:32) sk

j

j

|I|4−2α +

|I|4−2α +

d¨σ(x)

(|I| + |x − xI|)4−2α
(cid:90)
(cid:33) ¨ω(Ik
j )
j |2−α
|Ik

I0
1
j |2−αd¨σ(x)
|Ik
j | + |x − x
Ik
j
j )¨ω(Ik
(cid:46) ¨σ(Ik
j )
j |4−2α
|Ik

1\Gk
I0

(|Ik

j

|)4−2α

≈ 1

1

j |2−α
|Ik
¨σ(Ik
j )
j |4−2α
|Ik

(cid:46) |I|4−2α

(cid:46) |I|4−2α
j |2−α
|Ik

(|I| + |x − xI|)4−2α

(4.1.5)

d¨ω(x)

 ¨P(Ik

j , ¨ω)
j |2−α
|Ik

where in the first inequality we used the fact that |x − xI| ≈ |x − ¨zk
since xI is “close" to the center of Gk
j
(4.1.4).

j | when x /∈ Gk
j
, and for the second inequality we used (4.1.3) and

j | (cid:38) |Ik

,

(cid:16) 1−b
j |, |x− xI| (cid:38) |x− x

2

(cid:17)m−1 ≤ |I| ≤ b

| for x /∈ Gk

j

Ik
j

(cid:16) 1−b

2

(cid:17)m for some

and we can repeat

If |xI − ¨zk

j | > dist(xI , ∂Gk

j ), we can assume b

m > k, since for m = k we have |I| ≈ |Ik
the proof of (4.1.5). Now let Im
t

be the m-th generation interval that is closer to I that
j|, for all (cid:96) ≥ 1, 1 ≤ j ≤ 2(cid:96),

j| (cid:46) |xI − ¨z(cid:96)

. We have, using |xIm

− ¨z(cid:96)

t

touches the boundary of Gk
j
¨P(I, ¨σ) (cid:46) ¨P(Im

t , ¨σ) and ¨P(I, ¨ω) (cid:46) ¨P(Im

t , ¨ω), which imply

¨P(I, ¨σ) ¨P(I, ¨ω) (cid:46) 1.

65

Case 2. Now assume Gk

j ⊂ AI. If Ik

j ∩ I = ∅, then, using the minimality of k, I ⊂ Gm
j | since
, contradicting the minimality of k if we fix A big enough

j ∩ I (cid:54)= ∅ then |I| (cid:46) |Ik

t

for some m < k and we can repeat the proof of (4.1.5). If Ik
otherwise AI would contain ¨zk−1
depending only on α. Hence we have:

t

|Gk

j| + |x − ¨zk

j | ≤ |Gk

j| + |xI − ¨zk

j | + |x − xI| ≤

(cid:18)

A +

A
2

(cid:19)

|I| + |x − xI|

which implies that

(cid:90)

¨P(I, ¨σ)(cid:46)

and similarly

(cid:16)|Gk

|I|2−α
j| + |x − ¨zk

j |(cid:17)4−2α d¨σ(x)(cid:46) |I|2−α

j |2−α
|Ik

I0
1

(cid:90)

I0
1

(cid:16)|Ik

j |2−α
|Ik
j | + |x − ¨zk

j |(cid:17)4−2α d¨σ(x)

¨P(I, ¨ω) (cid:46) |I|2−α
t |2−α
|Ik

¨P(Ik

j , ¨ω) ≤ ¨P(Ik

j , ¨ω).

which implies

¨P(I, ¨σ) ¨P(I, ¨ω) (cid:46) 1

Case 3. If neither Gk

j ∩ AI (cid:54)= Gk

j

nor Gk

j ∩ AI (cid:54)= AI, note that Gk

j ⊂ 3AI and we repeat

again the proof of Case 2.

Thus, for any interval I ⊂ I0

1

, we have shown that ¨P(I, ¨σ) ¨P(I, ¨ω) (cid:46) 1, which implies

¨Aα
2 (¨σ, ¨ω) < ∞

(4.1.6)

66

4.1.3 The Energy Constants ¨E and ¨E∗

. Now define the following variant of the energy constants

¨E = sup

I= ˙(cid:83)Ir
I= ˙(cid:83)Ir
¨E∗ = sup

1

¨σ(I)

1

¨ω(I)

(cid:88)
(cid:88)

r≥1

r≥1

¨ω(Ir)E(Ir, ¨ω)2 ¨P(Ir, 1I ¨σ)2

¨σ(Ir)E(Ir, ¨σ)2 ¨P(Ir, 1I ¨ω)2

where the supremum is taken over the different intervals I and all the different decompositions

of I = ˙(cid:83)

r≥1Ir, and

(cid:90)

|I|

¨P(I, µ) =

(cid:90)

(cid:90)

R
(x − x(cid:48))2

|I|2

(|I| + |x − xI|)3−α dµ(x),
dµ(x(cid:48))dµ(x) =

·(cid:13)(cid:13)x − mµ

1

I

µ(I)

(cid:13)(cid:13)2
L2(1I µ) ≤ 1.

E(I, µ)2 =

1
2

1

µ(I)2

I

I

We first show that ¨E is bounded. We have

(cid:90)

¨P(I, ¨σ) =

|I|

(|I| + |x − xI|)3−α d¨σ(x) (cid:46)
≤

¨σ(cid:0)(2n + 1)I(cid:1)

(2n)|2nI|2−α

M α¨σ(x)2−n (cid:46) inf
x∈I

inf
x∈I

M α¨σ(x)

∞(cid:88)
∞(cid:88)

n=0

n=0

(cid:90)
where M αµ(x) = sup
I(cid:51)x
an interval I = ˙∪r≥1Ir, we have:

|I|2−α

1

I

(cid:88)

r≥1

¨ω(Ir)¨P2(Ir, 1I ¨σ) ≤(cid:88)

r≥1

dµ and the implied constants depend only on α. Thus, given

¨ω(Ir) inf
x∈I

(M α1I ¨σ)2 (x) ≤

(cid:90)

I

(M α1I ¨σ)2 (x)d¨ω(x)

67

and so we are left with estimating the right hand term of the above inequality. We will prove

the inequality

(cid:90)

(cid:16)

I(cid:96)
r

(cid:17)2

M α1

¨σ

Il
r

(x)d¨ω(x) ≤ C ¨σ(I(cid:96)
r).

(4.1.7)

where the constant C depends only on α. This will be enough, since for an interval I

containing a point mass ¨z(cid:96)

r but no masses ¨zk

j

for k < (cid:96), we have

(cid:90)

I

(cid:90)

(cid:16)

(M α¨σ)2 (x)d¨ω(x) =

M α1

I∩I(cid:96)
r

¨σ

I∩I(cid:96)
r

(cid:17)2

(x)d¨ω(x) ≤

(cid:17)2

(x)d¨ω(x)

(cid:90)

(cid:16)
M α1
¨σ
I(cid:96)
r
r) ≈ ¨σ(I)

I(cid:96)
r
≤ ¨σ(I(cid:96)

Since the measure ¨ω is supported in the Cantor set Eb, we can use the fact that for x ∈ I(cid:96)
r∩Eb,
(cid:19)(cid:96)

(cid:90)

(cid:12)(cid:12)(cid:12)Ik

j

1

(cid:12)(cid:12)(cid:12)2−α

d¨σ ≈ sup

s
(k,j):x∈Ik
j

j ∩I(cid:96)
Ik

r

2k∨(cid:96)

−2(k∨(cid:96))
0
s−k

0

≈ ¨σ(I(cid:96)
r)
r|2−α
|I(cid:96)

≈

M α(1

I(cid:96)
r

¨σ)(x) (cid:46) sup

(k,j):x∈Ik
j

(cid:18) 2

s0

Fix m and let the approximations ¨ω(m) and ¨σ(m) to the measures ω and ¨σ given by

2m(cid:88)

d¨ω(m) (x) =

2−m 1(cid:12)(cid:12)Im

(cid:12)(cid:12)1Im

i

i

(x) dx and ¨σ(m) =

(cid:88)

2k(cid:88)

sk
j δ

.

zk
j

i=1

For these approximations we have in the same way the estimate for x ∈(cid:83)2m
(cid:17)k∨(cid:96)
M α(cid:16)

¨σ(m)(cid:17)

d¨σ ≈ sup

(x) (cid:46) sup

(cid:16) 1

(cid:90)

s0

s0

1

k<m

j=1

I(cid:96)
r

(k,j):x∈Ik
j

j ∩I(cid:96)
Ik

r

(k,j):x∈Ik
j

(cid:17)k∨(cid:96)(cid:16) 2
(cid:17)k
(cid:16) 1

s0

(cid:12)(cid:12)(cid:12)Ik

j

1

(cid:12)(cid:12)(cid:12)2−α

i=1 Im
i

,

(cid:18) 2

(cid:19)(cid:96)

s0

≤ C

68

Thus for each m ≥ n ≥ (cid:96) we have

(cid:90)

M α(cid:16)

I(cid:96)
r

1

I(cid:96)
r

¨σ(n)(cid:17)2

(cid:88)

(cid:18) 2

s0

(cid:19)2(cid:96)
2−m = C2m−(cid:96)

(cid:18) 2

s0

(cid:19)2(cid:96)
2−m = Cs(cid:96)

r ≈ C

(cid:90)

I(cid:96)
r

d¨σ

d¨ω(m)≤ C

i ⊂I(cid:96)
i:Im

r

Now since ¨ωm converges weakly to ¨ω and using the fact that M α is lower semi-continuous

we get:

(cid:90)

I(cid:96)
r

M α(cid:16)

¨σ(n)(cid:17)2

d¨ω ≤ lim inf
m→∞

1

I(cid:96)
r

(cid:90)

I(cid:96)
r

M α(cid:16)

¨σ(n)(cid:17)2

d¨ω(m) ≤ C ¨σ(I(cid:96)
r)

1

I(cid:96)
r

Now, taking n → ∞, by monotone convergence we get (4.1.7). This proves

(cid:88)

r≥1

¨ω(Ir)¨P2(Ir, 1I ¨σ) ≤ C ¨σ(I)

(4.1.8)

which in turn implies ¨E < ∞ as E(Ir, ¨ω) ≤ 1.

Finally, we show that the dual energy constant ¨E∗ is finite. Let us show that for I ⊂ I0

1

¨σ(I)E(I, ¨σ)2 ¨P(I, ¨ω)2 (cid:46) ¨ω(I).

(4.1.9)

as if we let {Ir : r ≥ 1} be any partition of I, (4.1.9) gives

¨σ(Ir)E(Ir, ¨σ)2 ¨P(Ir, ¨ω)2 (cid:46)(cid:88)

(cid:88)

r≥1

¨ω(Ir) = ¨ω(I) .

r≥1

Now let us establish (4.1.9). We can assume that E(I, ¨σ) (cid:54)= 0. Let k be the smallest
r ∈ I. And let n be the smallest integer so that for some

integer for which there is a r with ¨zk

69

s we have ¨zk+n

s

∈ I and ¨zk+n

(cid:54)= ¨zk

s

r . We have that

(cid:90)

(cid:90)

|x − x(cid:48)|2

E(I, ¨σ)2 =

1
2

1

=

¨σ(I)2
(cid:46) ¨σ(¨zk

(cid:35)

(cid:34)

I

1

I

I

(cid:90)

¨σ(I)2

|x − x(cid:48)|2

d¨σ(x)d¨σ(x(cid:48))
(cid:90)
(cid:90)
|I|2
r|2
|x − ¨zk
(cid:33)n
(cid:32)
|I|2
d¨σ(x) +
¨σ(I\{¨zk
r})
¨σ(I)
(cid:17)k, which proves (4.1.9).
, ¨ω(I) ≈ 2−k−n, and ¨P(I, ¨ω) ≈(cid:16) s0

¨σ(¨zk
r )
r )¨σ(I\{¨zk
r})
¨σ(I)2
(cid:19)k

I\{¨zk
r }

|I|2

I

2
s2
0

(cid:46)

+

d¨σ(x)d¨σ(x(cid:48))

2

Finally, ¨σ(I) ≈

(cid:18)

2
s2
0

4.2 The Two Dimensional Construction

It is time now to define the two dimensional measures that prove the statement of Theorem
2.3.4. For any set E ⊂ R2 let

ω(E) =

¨ωn(E)

where ¨ω0(E) = ¨ω(Ex ∩ I0
1 ), Ex the projection of E on the x-axis, and ¨ωn are copies of ¨ω0
at the intervals [an, an + 1] × {0} with kn = an+1 − (an + 1) to be determined later. In the
same way, let

σ(E) =

¨σn(E)

where ¨σ0(E) = ¨σ([E∩(I0
where the height γn will be determined later. Check Figure 4.2.1.

1×{γ0})]x), and ¨σn are copies of ¨σ0 at the intervals [an, an+1]×{γn},

n=0

γn

σ

ω

σ

ω

σ
ω

σ
ω

kn
Figure 4.2.1: Positioning of measures

∞(cid:88)

n=0

∞(cid:88)

70

4.2.1 The A2 conditions.
We will now prove that both Aα
0 = [an, an + 1] × {0} and J n
J n
only one of the intervals J n
0

2

and Aα,∗

2

constants are bounded. Let Q be a cube in R2,
γn = [an, an + 1] × {γn}. We take cases for Q. If Q intersects
, say J 0
0

for convenience, and (Q ∩ J 0

0 )x =: J0 we have:

Pα(Q, 1Qcσ)

ω(Q)
|Q|1− α

2

¨ω(J0)

|J0|2−α + Pα(Q, 1

(J1
γ1

)cσ)

¨ω(I0
1 )
|Q|1− α
2

(cid:46) ¨P(J0, ¨σ)
≤ ¨Aα

2 (¨σ, ¨ω) + C < ∞

using (4.1.6) and taking kn large enough so that the second summand is bounded indepen-
dently of the interval (kn = 42n·max{(2−α)−1,1} would do here). If Q intersects more than
, it is easy to see, using that Q is very big (since it intersects more
one of the intervals J n
0

than one of the intervals) and that kn is also large, that:

Pα(Q, 1Qcσ)

ω(Q)
|Q|1− α

2

(cid:46) 1

which of course shows that Aα
that Aα,∗
is bounded as well.

2

2

is bounded. Essentially using the same calculations we see

71

4.2.2 Off-Testing Constant

. Let us now check that the off-testing constant is not bounded. Choose the cube Qn =
[an, an + 1] × [0,−1]. Then,

(cid:90)

(cid:20)(cid:90)

1

(cid:21)2

ω(Qn)

Qc
n

Qn

dω(y)

|x − y|2−α

dσ(x)≥ 1
¨ω(I0
1 )

(cid:90)

(cid:20)(cid:90)

I0
1

I0
1

(cid:112)(x1 − y1)2 + γ2

d¨ω(y1)

n

2−α

(cid:21)2

d¨σ(x1)

for x = (x1, x2) and y = (y1, y2). Taking γn such that the last expression on the display
above equals n (note that this is feasible, since for γn = 0, (4.1.1) gives infinity in the latter
expression above) we have

(cid:90)

(cid:20)(cid:90)

Qc
n

Qn

(cid:21)2

off,α ≥ 1
T 2

ω(Qn)

dω(y)

|x − y|2−α

dσ(x) ≥ n

and by letting n → ∞ we obtain that the off-testing constant is not bounded.

4.2.3 The Energy Conditions

. For the energy condition E α
decomposition of Q. Then we have

2

∞(cid:88)

r=1

1

σ(Q)

Pα(cid:0)Qr, 1Qσ(cid:1)

2(cid:13)(cid:13)(cid:13)x − mω

Qr

|Qr| 1

2

first, let Q be a cube and Q = ˙∪Qr, where {Qr}∞

is a

r=1

(cid:13)(cid:13)(cid:13)2
L2(cid:16)

(cid:17)≤ 2

σ(Q)

1Qr ω

∞(cid:88)

r=1

ω(Qr)(cid:0)Pα(cid:0)Qr, 1Qσ(cid:1)(cid:1)2

. Then we have m − 2 (cid:46) σ(Q) (cid:46) m.
Assume that Q intersects m intervals of the form J n
0
The case m = 1 is exactly the same as the one dimensional analog for ¨E. Assume m = 2.

72

Now we need to take cases for Qr:

(i) Let Q1 be the set of cubes Qr that intersect only one of the intervals J n
0

. Then we

have, following the proof of (4.1.8), that

ω(Qr)(cid:0)Pα(cid:0)Qr, 1Qσ(cid:1)(cid:1)2 ≤ Cσ(Q)

(cid:88)

Qr∈Q1

(ii) If Qr intersects both of the intervals J n
0

then this Qr is unique since the family {Qr}r∈N

forms a decomposition of Q. Therefore we have:

ω(Qr)(cid:0)Pα(cid:0)Qr, 1Qσ(cid:1)(cid:1)2 (cid:46) ω(Qr)σ(Q)

|Qr|2−α σ(Q) (cid:46) σ(Q)

using the fact that |Qr| (cid:38) 42 since it intersect two of the intervals J n
2, σ(Q) (cid:46) 2.

0

and ω(Qr) (cid:46)

For m ≥ 3, again we take cases for Qr:

(i) If Qr intersects only one J n
0

(cid:88)

Qr∈Q1

we again have, following the proof of (4.1.8), that

ω(Qr)(cid:0)Pα(cid:0)Qr, 1Qσ(cid:1)(cid:1)2 ≤ Cσ(Q)

(ii) If Qr intersects more than one of the intervals J n
0

n0
, the last one being J
0

we have

ω(Qr)(cid:0)Pα(cid:0)Qr, 1Qσ(cid:1)(cid:1)2 (cid:46) ω(Qr)σ(Q−

r )2

|Qr|2−α

m(cid:88)

k=1

+ ω(Qr)

1

42k|Qr|2−α

(cid:46) 2

r contains all the intervals J n

where Q−
we use the fact that Qr is very big since it intersects at least two intervals J n
0

such that n ≤ n0. Again in the last inequality
. Now

0

73

since Qr form a decomposition of Q we can have at most m − 1 of these.

Combining the above cases, we obtain

ω(Qr)(cid:0)Pα(cid:0)Qr, 1Qσ(cid:1)(cid:1)2 ≤ Cσ(Q) + 2m − 2 ≤ 2Cσ(Q)

∞(cid:88)

r=1

and that proves the energy condition is bounded.

The dual energy E α,∗

2

can also be proved bounded with the same calculations as in the

energy condition following the proof of (4.1.9) instead of (4.1.8) as in the first case above.

This completes the proof of the Theorem 2.3.4.

4.3 The Riesz transform lemma

To obtain the same result for the Riesz transforms, we need to deal with the fact that the

kernel is not positive. This prevents us from placing the masses for ¨σ at the center of the

intervals Gk
j

, as we did in the proof of Theorem 2.3.4. Since otherwise, if the point-mass ¨σ

is located at the center of Gk
j

, it would result in the cancellation of much of the mass not

letting us deduce that the off testing condition for the Riesz transform is unbounded. The

following lemma, whose proof follows closely the work in [32] but with a two dimensional

twist, helps us overcome this problem, showing that, while not being able to place the point

masses in the middle of Gk
, we can place them far from the boundary. This enables us to
j
show that the ¨A2 condition is bounded, like in the proof of Theorem 2.3.4. First we need to
define the operator

(cid:90)

¨Rf (x) =

(x − y)f (y)
|x − y|3−α dy

R

Lemma 4.3.1. For k ≥ 1, 1 ≤ j ≤ 2k, write Gk

j = (ak

j , bk

j ). Then there exists 0 < c < 1

74

that depends only on α such that

(cid:32)

¨R¨ω

ak
j +c

(cid:18)1 − b

(cid:19)k

2

(cid:33)

b

(cid:17)k

≈(cid:16)s0

2

where ¨ω is the measure defined above.

Proof. Fix k. We have

(cid:32)

¨R¨ω

ak
1 +c

(cid:18)1 − b

2

(cid:33)

(cid:19)k

b

≤ ¨R¨ω

(cid:32)

ak
j +c

(cid:18)1 − b

(cid:19)k

2

(cid:33)

b

≤ ¨R¨ω

(cid:32)

ak
2k +c

(cid:18)1 − b

(cid:19)k

2

(cid:33)

b

from monotonicity. So it is enough to prove the following:

(cid:32)

(cid:17)k (cid:46) ¨R¨ω

(cid:16) s0

2

(cid:18)1 − b

(cid:19)k

(cid:33)

(cid:32)

ak
1 +c

2

≤ ¨R¨ω

b

ak
2k +c

(cid:18)1 − b

(cid:19)k

2

(cid:33)

b

(cid:17)k

(cid:46)(cid:16) s0

2

We start with right hand inequality. Following the definitions of ¨R, ¨ω we get

(cid:32)

¨R¨ω

ak
2k +c

(cid:18)1 − b

(cid:19)k

(cid:33)

b

2

(cid:18)

≤

(cid:90)
≤ k(cid:88)

[0,ak
2k

]

(cid:96)=1

(cid:18)
≈ 2−k
c2−αs−k

0

ak
2k +c

d¨ω(y)

(cid:17)k
(cid:20)

b−

ak
2k +c

2

(cid:16) 1−b
(cid:17)k
(cid:16) 1−b
k−1(cid:88)
(cid:20)
k−1(cid:88)
(cid:20)

s−(cid:96)

(cid:96)=1

2

0

+

b−y
2−(cid:96)

(cid:19)2−α
1−(cid:16) 1−b
(cid:17)(cid:96)−1(cid:16) 1+b
(cid:17)k−(cid:96)+1(cid:104)
(cid:16) 1−b
(cid:17)k−(cid:96)+1(cid:21)2−α
(cid:16) 1−b

cb− 1+b

2−(cid:96)

2

2

2

2

(cid:17)(cid:21)(cid:19)2−α
(cid:105)(cid:21)2−α

1+b
2 +

0

+

2−(cid:96)

≤ 2−k
c2−αs−k
(cid:17)k. The square bracket inside the last fraction is minimized for

2 − 1+b

s−(cid:96)

1+b

(cid:96)=1

0

2

2

75

2k = 1−(cid:16) 1+b

2

since ak

(cid:17)(cid:16) 1−b

2

(cid:96) = k − 1 and we get the inequality

(cid:32)

¨R¨ω

ak
2k +c

(cid:18)1 − b

(cid:19)k

(cid:33)

b

2

(cid:46) 2−k
c2−αs−k

0

+

k−1(cid:88)

(cid:16) s0

2

(cid:96)=1

(cid:17)(cid:96) (cid:46) 1

c2−α

(cid:16)s0

(cid:17)k

2

where the implied constants depend again only on α. We should note here that the summand

with (cid:96) = k is the dominant one in the above inequality.

Now we consider the left hand inequality. We have that ¨R¨ω

(cid:33)

(cid:19)k
(cid:18)1 − b

2

b

+

(cid:32)

k+1(cid:88)

(cid:96)=1

¨R¨ω1

I(cid:96)
2

ak
1 +c

¨R¨ω1

Ik+1
1

ak
1 +c

(cid:19)

b

ak
1 +c

(cid:18)
(cid:16)1−b
(cid:17)k
(cid:33)
(cid:18)1 − b
(cid:19)k

2

b

2

equals

(4.3.1)

(cid:32)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)k+1(cid:88)

(cid:96)=1

and following the argument for the previous inequality we see that

(cid:32)

(cid:19)k
(cid:18)1 − b

(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A

b

(cid:16)s0

(cid:17)k

2

¨R¨ω1

I(cid:96)
2

ak
1 +c

2

where A depends only on α but not on c. The first summand of (4.3.1) gives

(cid:90)

(cid:18)

ak
1 +c

d¨ω(y)

(cid:16) 1−b

(cid:17)k

2

b−y

(cid:19)2−α ≥

Ik+1
1

∞(cid:88)

(cid:96)=k+1

(cid:18)(cid:16) 1−b

2

(cid:17)(cid:96)

2−(cid:96)−1

(cid:16) 1−b

2

+ c

(cid:19)2−α

(cid:17)k

b

∞(cid:88)
(cid:18)(cid:16) 1−b
∞(cid:88)
(cid:18)(cid:16) 1−b
(cid:17)(cid:96)

(cid:96)=k+1

(cid:96)=1

2

2−(cid:96)+k−1

(cid:19)2−α
(cid:17)(cid:96)−k
(cid:19)2−α .

+cb

2−(cid:96)−1

+cb

2

≈ sk
0
2k

=

sk
0
2k

Choosing c small enough not depending on k (since the last sum does not depend on k), we

76

obtain

(cid:90)

Ik+1
1

(cid:18)

ak
1 +c

with C1 > 2A and we conclude our lemma.

Proof of Theorem 2.3.5. Set ˙zk

j = ak

j+cb

2

(cid:18)

(cid:19)k

(cid:16) s0

(cid:17)k

2

2

d¨ω(y)

b−y

(cid:16) 1−b
(cid:16)1−b

(cid:19)2−α ≥ C1

(cid:17)k
(cid:17)k and define the measure ˙σ =

(cid:88)

k,j

where

sk
j δ

˙zk
j

2
s2
0

as before. Following verbatim the calculations of Theorem 2.3.4, one can show
sk
j =
that ¨A2( ˙σ, ¨ω) < ∞. Now define the measures ω and σ, as before, for any measurable set
E ⊂ R2 by

∞(cid:88)

n=0

ω(E) =

¨ωn(E) and σ(E) =

∞(cid:88)

n=0

˙σn(E)

where ˙σ0(E) = ˙σ([E∩(I0
1×{γ0})]x), and ˙σn are copies of ˙σ0 at the intervals [an, an+1]×{γn},
and where the height γn will be determined later. Again, as before, it is easy to see that both
Aα
are bounded. Let us now finish the proof by showing
that the off-testing constant for the Riesz transforms are unbounded. From Lemma 4.3.1 we

and both E α

and Aα,∗

and E α,∗

2

2

2

2

(cid:17)k which implies

have ¨R¨ω( ˙zk

(cid:19)2

¨ω)(x)

I0
1

d ˙σ(y) (cid:38)

∞(cid:88)

2k(cid:88)

k=1

j=1

j ·(cid:16)s0

2

sk

(cid:17)2k

∞(cid:88)

2k(cid:88)

k=1

j=1

=

= ∞.

1
2k

(4.3.2)

j ) (cid:38)(cid:16) s0
(cid:18)
(cid:90)

2

¨R(1

I0
1

Now choose the cube Qn = [an, an + 1] × [0,−1]. Then,

R2
1,off,α ≥

≥

1

ω(Qn)

1

(cid:90)
(cid:90)

(cid:20)(cid:90)
(cid:20)(cid:90)

Qc
n

Qn

(cid:21)2
(x1 − y1)dω(y)
|x − y|3−α
(cid:112)(x1 − y1)2 + γ2
(x1 − y1)d¨ω(y1)

n

ω(Qn)

I0
1

I0
1

dσ(x)

(cid:21)2

3−α

d ˙σ(x1) =

n

ω(Qn)

77

by choosing the height γn so that(cid:82)

(cid:20)(cid:82)

3−α
n → ∞, we see that the off-testing constant is unbounded.

I0
1

I0
1

(cid:113)
(x1−y1)d¨ω(y1)
(x1−y1)2+γ2

n

(cid:21)2

d ˙σ(x1) = n by (4.3.2). Letting

78

Chapter 5

A two weight local T b theorem for

n-dimensional Fractional Integrals

5.1 The local T b theorem and proof preliminaries

5.1.1 Standard fractional singular integrals

Let 0 ≤ α < n. We define a standard α-fractional CZ kernel Kα(x, y) to be a real-valued
function defined on Rn×Rn satisfying the following fractional size and smoothness conditions
of order 1 + δ for some δ > 0: For x (cid:54)= y,

|Kα (x, y)| ≤ CCZ |x − y|α−n
(cid:33)δ
|∇Kα (x, y)| ≤ CCZ |x − y|α−n−1

(cid:32)(cid:12)(cid:12)x − x(cid:48)(cid:12)(cid:12)

(cid:12)(cid:12)∇Kα (x, y) − ∇Kα(cid:0)x(cid:48), y(cid:1)(cid:12)(cid:12) ≤ CCZ

|x − y|

(5.1.1)

(cid:12)(cid:12)x − x(cid:48)(cid:12)(cid:12)
|x − y| ≤ 1

2

,

|x − y|α−n−1 ,

and the last inequality also holds for the adjoint kernel in which x and y are interchanged.

We note that a more general definition of kernel has only order of smoothness δ > 0, rather

than 1 + δ, but the use of the Monotonicity and Energy Lemmas in arguments below involves
first order Taylor approximations to the kernel functions Kα (·, y).

79

5.1.1.1 Defining the norm inequality

We now turn to a precise definition of the weighted norm inequality

For this we introduce a family(cid:110)

the truncated kernels Kα

(cid:107)T α

σ

f ∈ L2 (σ) .

(cid:107)f(cid:107)L2(σ) ,

σ f(cid:107)L2(ω) ≤ NT α
(cid:111)
of nonnegative functions on [0,∞) so that
0<δ<R<∞
δ,R (|x − y|) Kα (x, y) are bounded with compact sup-

δ,R (x, y) = ηα

(5.1.2)

ηα
δ,R

port for fixed x or y. Then the truncated operators

(cid:90)

σ,δ,Rf (x) ≡
T α

Kα

δ,R (x, y) f (y) dσ (y) ,

are pointwise well-defined, and we will refer to the pair (cid:16)

Rn

x ∈ Rn,

(cid:110)

(cid:111)

(5.1.3)

(cid:17) as an α-

Kα,

ηα
δ,R

0<δ<R<∞

fractional singular integral operator, which we typically denote by T α, suppressing the de-

pendence on the truncations.

Definition 5.1.1. We say that an α-fractional singular integral operator T α satisfies the

norm inequality (5.1.2) provided

(cid:13)(cid:13)(cid:13)T α

σ,δ,Rf

(cid:13)(cid:13)(cid:13)L2(ω)

≤ NT α

σ

(cid:107)f(cid:107)L2(σ) ,

f ∈ L2 (σ) , 0 < δ < R < ∞.

It turns out that, in the presence of the Muckenhoupt conditions (5.1.7) below, the

norm inequality (5.1.2) is essentially independent of the choice of truncations used, and this

is explained in some detail in [67]. Thus, as in [67], we are free to use the tangent line

truncations described there throughout the proofs of our results.

80

5.1.2 Weakly accretive functions

Denote by P the collection of cubes in Rn. Note that we include an Lp upper bound in our
definition of ‘p-weakly accretive family’ of functions.

say that a family b =(cid:8)bQ

Definition 5.1.2. Let p ≥ 2 and let µ be a locally finite positive Borel measure on Rn. We
Q∈P of functions indexed by P is a p-weakly µ-accretive family

(cid:9)

of functions on Rn if for Q ∈ P,

supp bQ ⊂ Q

(cid:90)

0 < cb ≤

1
|Q|µ

Q

bQdµ ≤

(cid:32)

(cid:90)

Q

1
|Q|µ

(cid:12)(cid:12)bQ

(cid:12)(cid:12)p dµ

(cid:33) 1
p ≤ Cb < ∞.

(5.1.4)

5.1.3 b-testing conditions

Suppose σ and ω are locally finite positive Borel measures on Rn. The b-testing conditions
for T α and b∗-testing conditions for the dual T α,∗ are given by

(cid:90)

Q

(cid:90)

(cid:12)(cid:12)T α
(cid:12)(cid:12)(cid:12)T α,∗

σ bQ
ω b∗

Q

(cid:12)(cid:12)2 dω ≤ (cid:16)
(cid:12)(cid:12)(cid:12)2
dσ ≤ (cid:16)

Q

(cid:17)2 |Q|σ ,
(cid:17)2 |Q|ω ,

Tb
T α
Tb∗
T α,∗

for all cubes Q,

(5.1.5)

for all cubes Q.

5.1.4 Poisson integrals and the Muckenhoupt conditions

Let µ be a locally finite positive Borel measure on Rn, and suppose Q is a cube in Rn. Recall
that |Q| 1

n = (cid:96) (Q) for a cube Q. The two α-fractional Poisson integrals of µ on a cube Q are

81

given by the following expressions:

Pα (Q, µ) ≡

Pα (Q, µ) ≡

(cid:90)

Rn

(cid:90)

Rn

(cid:18)


|Q| 1

n

|Q| 1

n +(cid:12)(cid:12)x − xQ
n +(cid:12)(cid:12)x − xQ

(cid:12)(cid:12)(cid:19)n+1−α dµ (x) ,

(cid:12)(cid:12)(cid:19)2

|Q| 1

n−α

n

|Q| 1

(cid:18)

dµ (x) ,

where (cid:12)(cid:12)x − xQ

(cid:12)(cid:12) denotes distance between x and the center xQ of Q and |Q| denotes the

Lebesgue measure of the cube Q. We refer to Pα as the standard Poisson integral and to
Pα as the reproducing Poisson integral. Note that these two kernels satisfy for all cubes Q
and positive measures µ,

0 ≤ Pα (Q, µ) ≤ CPα (Q, µ) ,
0 ≤ Pα (Q, µ) ≤ CPα (Q, µ) ,

n − 1 ≤ α < n,
0 ≤ α < n − 1.

We now define the one-tailed constant with holes Aα

using the reproducing Poisson kernel
Pα. On the other hand, the standard Poisson integral Pα arises naturally throughout the
proof of the T b theorem in estimating oscillation of the fractional singular integral T α, and

2

in the definition of the energy conditions below.

Definition 5.1.3. Suppose σ and ω are locally finite positive Borel measures on Rn. The
one-tailed constants Aα

2 with holes for the weight pair (σ, ω) are given by

2 and Aα,∗

2 ≡ sup
Aα
Q∈P
≡ sup
Q∈P

2

Aα,∗

(cid:17) |Q|ω
(cid:17) |Q|σ

|Q|1− α

n

|Q|1− α

n

< ∞,

< ∞.

Pα(cid:16)
Pα(cid:16)

Q, 1Qcσ

Q, 1Qcω

82

Note that these definitions are the conditions with ‘holes’ introduced by Hytönen [22] -

the supports of the measures 1Qcσ and 1Qcω in the definition of Aα
any common point masses of σ and ω do not appear simultaneously in the factors of any

are disjoint, and so

2

. Recall, the definition of the classical Muckenhoupt

of the products Pα(cid:16)

Q, 1Qcσ

(cid:17) |Q|ω

|Q|1− α

n

condition

Aα

2 = sup
Q∈P

|Q|ω
|Q|1− α

n

|Q|σ
|Q|1− α

n

but it will find no use in the two weight setting with common point masses permitted.

(cid:16)
λw, (λw)−1(cid:17)

(cid:0)w, w−1(cid:1) is

= Aα

2

Initially, these definitions of Muckenhoupt type were given in the following ‘one weight’

case, dω (x) = w (x) dx and dσ (x) = 1

w(x) dx, where Aα

2

homogeneous of degree 0. Of course the two weight version is homogeneous of degree 2 in
the weight pair, Aα
2 (σ, ω), while all of the other conditions we consider
in connection with two weight norm inequalities, including the operator norm NT α (σ, ω)
itself, are homogeneous of degree 1 in the weight pair. This awkwardness regarding the

2 (λσ, λω) = λ2Aα

homogeneity of Muckenhoupt conditions could be rectified by simply taking the square root
of Aα
particular in connection with the A2 conjecture, that we will leave it as is.

and renaming it, but the current definition is so entrenched in the literature, in

2

5.1.4.1 Punctured Aα

2 conditions

The classical Aα
2

characteristic fails to be finite when the measures σ and ω have a common

point mass - simply let Q in the sup above shrink to a common mass point. But there is a

substitute that is quite similar in character that is motivated by the fact that for large cubes

Q, the sup above is problematic only if just one of the measures is mostly a point mass when

restricted to Q.

83

Given an at most countable set P = {pk}∞

k=1

in Rn, a cube Q ∈ P, and a positive locally

finite Borel measure µ, define

µ (Q, P) ≡ |Q|µ − sup{µ (pk) : pk ∈ Q ∩ P} ,

(5.1.6)

where the supremum is actually achieved since(cid:80)
pk∈Q∩P µ (pk) < ∞ as µ is locally finite.
The quantity µ (Q, P) is simply the (cid:101)µ measure of Q where (cid:101)µ is the measure µ with its

largest point mass from P in Q removed. Given a locally finite positive measure pair (σ, ω),
let P(σ,ω) = {pk}∞
the weighted norm inequality (5.1.2) typically implies finiteness of the following punctured

be the at most countable set of common point masses of σ and ω. Then

k=1

Muckenhoupt conditions:

Aα,punct

2

(σ, ω) ≡ sup
Q∈P

Aα,∗,punct

2

(σ, ω) ≡ sup
Q∈P

(cid:17)

ω

(cid:16)
Q, P(σ,ω)
(cid:16)
|Q|1− α
|Q|ω
σ
|Q|1− α

n

n

n

|Q|σ
(cid:17)
|Q|1− α
Q, P(σ,ω)
|Q|1− α

n

,

.

In particular, all of the above Muckenhoupt conditions Aα
are necessary for boundedness of an elliptic α-fractional singular integral T α

, Aα,punct

2

2

2

, Aα,∗

and Aα,∗,punct
σ from L2 (σ) to

2

L2 (ω). It is convenient to define

2 ≡ Aα
Aα

2 + Aα,∗

2 + Aα,punct

2

+ Aα,∗,punct

2

.

(5.1.7)

84

5.1.5 Energy Conditions

Here is the definition of the strong energy conditions, which we sometimes refer to simply as

the energy conditions. Let

(cid:90)

(cid:90)

(cid:28) 1

|I|µ

xdµ(x) =

x1dµ(x), ...,

1
|I|µ

(cid:90)

(cid:29)

xndµ(x)

I ≡ 1
mµ
|I|µ

be the average of x with respect to the measure µ, which we often abbreviate to mI when
the measure µ is understood.

Definition 5.1.4. Let 0 ≤ α < n. Suppose σ and ω are locally finite positive Borel measures
on Rn. Then the strong energy constant E α

2 is defined by

(E α
2 )2 ≡ sup
I= ˙∪Ir

1
|I|σ

∞(cid:88)

r=1

Pα (Ir, 1I σ)

2(cid:13)(cid:13)(cid:13)x − mω

Ir

|Ir| 1

n

(cid:13)(cid:13)(cid:13)2
L2(cid:16)

(cid:17) ,

1Ir ω

(5.1.8)

where the supremum is taken over arbitrary decompositions of a cube I using a pairwise
disjoint union of subcubes Ir. Similarly, we define the dual strong energy constant E α,∗
switching the roles of σ and ω:

by

2

(cid:0)E α,∗

2

(cid:1)2 ≡ sup

I= ˙∪Ir

1
|I|ω

∞(cid:88)

r=1

Pα (Ir, 1I ω)

2(cid:13)(cid:13)(cid:13)x − mσ

Ir

(cid:13)(cid:13)(cid:13)2
L2(cid:16)

|Ir| 1

n

(cid:17) .

1Ir σ

(5.1.9)

These energy conditions are necessary for boundedness of elliptic and gradient elliptic

operators, including the Hilbert transform (but not for for certain elliptic singular operators

that fail to be gradient elliptic) - see [68] and [69]. It is convenient to define

2 ≡ E α
Eα

2 + E α,∗

2

85

as well as

NT Vα ≡ Tb

T α + Tb∗

T α,∗ +

(cid:113)

2 + Eα
Aα
2 .

(5.1.10)

5.1.6 The two weight local T b Theorem

Here we derive a local T b theorem based in part on the proof of the T 1 theorem in [63], and

in part on the proof of a one weight T b theorem in Hytönen and Martikainen [24]. Recall

from [68] that an α-fractional singular integral T α with kernel Kα is said to be elliptic if
|Kα (x, y)| ≥ c|x − y|α−1 and gradient elliptic if the kernel Kα (x, y) satisfies

|∇Kα (x, y)| ≥ c|x − y|α−n−1 .

(5.1.11)

The Hilbert transform kernel K (x, y) = 1
y−x

satisfies (5.1.11) with α = 0, n = 1.

In

dimension n = 1 the Muckenhoupt conditions are necessary for norm boundedness of elliptic

operators by results in [32], [22] and [66], and the energy conditions are necessary for norm

boundedness of gradient elliptic operators by results in [68]. Moreover, in dimension n =

1, Hytönen [22, Corollary 3.10] proves that full testing is controlled by testing and the
Muckenhoupt conditions for the Hilbert transform, and this is easily extended to 0 ≤ α < 1:

(cid:113)Aα

2 +

(cid:113)Aα,∗

2

FTb

T α (cid:46) Tb

T α +

and FTb∗

T α,∗ (cid:46) Tb∗

T α,∗ +

(cid:113)Aα

2 +

(cid:113)Aα,∗

2 .

Theorem 5.1.5. Suppose that σ and ω are locally finite positive Borel measures on Eu-
clidean space Rn. Suppose that T α is a standard α-fractional singular integral operator
on Rn, and set T α

σ is apriori
bounded from L2 (σ) to L2 (ω). Assume the Muckenhoupt and energy conditions hold, i.e.

σ f = T α (f σ) for any smooth truncation of T α

σ , so that T α

86

2 < ∞. Finally, let p > 2 and let b = (cid:8)bQ
2 ,E α,∗
,E α

2 , Aα,punct

, Aα,∗,punct

2 ,Aα,∗
Aα
a p-weakly σ-accretive family of functions on Rn, and let b∗ =
ω-accretive family of functions on Rn. Then for 0 ≤ α < n, the operator T α

(cid:111)
Q∈P be
Q∈P be a p-weakly
σ is bounded from

(cid:110)

2

2

b∗

Q

(cid:9)

L2 (σ) to L2 (ω) with operator norm NT α
σ

, i.e.

(cid:107)T α

σ f(cid:107)L2(ω) ≤ NT α

σ

(cid:107)f(cid:107)L2(σ) ,

f ∈ L2 (σ) ,

uniformly in smooth truncations of T α if and only if the b-testing conditions for T α and
the b∗-testing conditions for the dual T α,∗ both hold. Moreover, we have

(cid:113)

NT α (cid:46) Tb

T α + Tb∗

T α +

2 + Eα
Aα
2 .

Remark 5.1.6. In the special case that σ = ω = µ, the classical Muckenhoupt Aα

2 condition

is

|Q|µ
|Q|1− α

n

|Q|µ
|Q|1− α

n

sup
Q∈P

< ∞,

which is the upper doubling measure condition with exponent n − α, i.e.

|Q|µ ≤ C(cid:96) (Q)n−α ,

for all cubes Q,

87

which of course prohibits point masses in µ. Both Poisson integrals are then bounded,

Pα (Q, µ)(cid:46)

Pα (Q, µ)(cid:46)

(cid:16)

∞(cid:88)

k=0

∞(cid:88)

k=0

n

|Q| 1
2k |Q| 1

(cid:12)(cid:12)(cid:12)µ
(cid:12)(cid:12)(cid:12)2kQ
(cid:19)n+1−α
(cid:18)

 |Q| 1
n−α(cid:12)(cid:12)(cid:12)2kQ
(cid:12)(cid:12)(cid:12)µ
(cid:19)2
(cid:18)

n

n

2k |Q| 1

n

n

k=0

|Q| 1
2k |Q| 1

∞(cid:88)
(cid:18)
 |Q| 1
(cid:18)

(cid:19)n+1−α

n−α(cid:16)
(cid:19)2

∞(cid:88)

2k |Q| 1

k=0

n

n

n

(cid:46)

(cid:46)

(cid:17)n−α

2k(cid:96)(Q)

= 2

(cid:17)n−α

2k(cid:96)(Q)

= Cα

and it follows easily that the equal weight pair (µ, µ) satisfies not only the Muckenhoupt Aα
2

condition, but also the strong energy condition Eα
2 :

(cid:18)Pα (Ir, 1I σ)

|Ir|

(cid:19)2(cid:13)(cid:13)(cid:13)x − mω

Ir

(cid:13)(cid:13)(cid:13)2

∞(cid:88)

r=1

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)x − mω

|Ir|

Ir

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2

L2(ω)

|Ir|ω ≤ C |I|ω = C |I|σ ,

∞(cid:88)
∞(cid:88)

r=1

r=1

≤ C

≤ C

since ω = σ. Thus Theorem 5.1.5, when restricted to a single weight σ = ω, recovers a

slightly weaker, due to our assumption that p > 2, version of the one weight theorem of

Lacey and Martikainen [29, Theorem 1.1] for dimension n = 1. On the other hand, the

possibility of a two weight theorem for a 2-weakly µ-accretive family is highly problematic, as

one of the key proof strategies used in [29] in the one weight case is a reduction to testing
over f and g with controlled L∞ norm, a strategy that appears to be unavailable in the two

weight setting.

In order to prove Theorem 5.1.5, it is convenient to establish some improved properties

for our p-weakly µ-accretive family, and also necessary to establish some improved energy
conditions related to the families of testing functions b and b∗. We turn to these matters in
the next two subsections.

88

5.1.7 Reduction to real bounded accretive families

We begin by noting that if bQ satisfies (5.1.4) with µ = σ, and satisfies a given b-testing
condition for a weight pair (σ, ω), then RebQ satisfies

(cid:32)

(cid:90)

(cid:12)(cid:12)RebQ

(cid:12)(cid:12)p dµ

(cid:33) 1
p ≤ Cb (p)

1
|Q|µ

Q

and the given b-testing condition for (σ, ω) with RebQ in place of bQ.

Thus we may assume throughout the proof of Theorem 5.1.5 that our p-weakly µ-accretive

(cid:9)

(cid:111)

Q∈D and b∗ ≡(cid:110)

families b ≡(cid:8)bQ
p-weakly µ-accretive testing functions b =(cid:8)bQ

Q∈G

b∗

Q

consist of real-valued functions.

(cid:9)
Q∈P and b∗ =

(cid:111)

(cid:110)

b∗

Next we show that the assumption of testing conditions for a fractional integral T α and

with p > 2 can
always be replaced with real-valued ∞-weakly µ-accretive testing functions, thus reducing
the T b theorem for the case p > 2 to the case when p = ∞. We now proceed to develop a
precise statement. We extend (5.1.4) to 2 < p ≤ ∞ by

Q∈P

Q

supp bQ ⊂ Q ,

(cid:90)

1 ≤

1
|Q|µ

Q

bQdµ ≤



Q ∈ P,

(cid:18)

1|Q|µ

(cid:82)
(cid:13)(cid:13)bQ

(cid:19) 1
(cid:12)(cid:12)p dµ
(cid:12)(cid:12)bQ
p ≤ Cb (p) < ∞ for 2 < p < ∞
(cid:13)(cid:13)L∞(µ) ≤ Cb (∞) < ∞
for p = ∞

Q

(5.1.12)

Proposition 5.1.7. Let 0 ≤ α < 1, and let σ and ω be locally finite positive Borel measures
on Rn, and let T α be a standard α-fractional elliptic and gradient elliptic singular integral
operator on Rn. Set T α

σ f = T α (f σ) for any smooth truncation of T α

σ , so that T α

σ is apriori

89

bounded from L2 (σ) to L2 (ω). Finally, define the sequence of positive extended real numbers



(cid:17)m

1 −(cid:16) 2

2

3

(cid:26)

∞

m=0

{pm}∞

m=0 =

=

∞, 6,

18
5

,

162
65

, ...

.

(cid:27)

Suppose that the following statement is true:

(S∞) If b =(cid:8)bQ

(cid:9)

(cid:110)

(cid:111)
Q∈P is an ∞-weakly σ-accretive family of functions on Rn and if b∗ =
Q∈P is an ∞-weakly ω-accretive family of functions on Rn, then the operator

b∗

Q

norm NT α
σ

of T α

σ from L2 (σ) to L2 (ω), i.e. the best constant in

(cid:107)T α

σ f(cid:107)L2(ω) ≤ NT α

σ

(cid:107)f(cid:107)L2(σ) ,

f ∈ L2 (σ) ,

uniformly in smooth truncations of T α, satisfies

NT α (cid:46) (Cb (∞) + Cb∗ (∞))

T α + Tb∗
Tb

T α +

(cid:16)

(cid:113)

(cid:17)

,

2 + Eα
Aα
2

where Cb (∞) , Cb∗ (∞) are the accretivity constants in (5.1.12), and the constants
implied by (cid:46) depend on α and the constant CCZ in (5.1.1).
Then for each m ≥ 0, the following statements hold:

(Sm) Let p ∈ (pm+1, pm]. If b =(cid:8)bQ

(cid:9)

(cid:110)

(cid:111)

b∗

Rn, and if b∗ =

Q
the operator norm NT α
σ

Q∈P is a p-weakly σ-accretive family of functions on
Q∈P is a p-weakly ω-accretive family of functions on Rn, then
of T α
σ from L2 (σ) to L2 (ω), uniformly in smooth truncations

of T α, satisfies

NT α (cid:46) (Cb (p) + Cb∗ (p))3m+1(cid:16)

(cid:113)

2 + Eα
Aα
2

(cid:17)

,

T α + Tb∗
Tb

T α +

90

where Cb (p) , Cb∗ (p) are the accretivity constants in (5.1.4), and the constants implied
by (cid:46) depend on p, α, and the constant CCZ in (5.1.1).

(p1, p0) = (6,∞), and let b =(cid:8)bQ

Proof of Proposition 5.1.7. We will prove it by induction. We first prove (S0). So fix p ∈
Q∈P be a p-weakly σ-accretive family of functions on Rn,
be a p-weakly ω-accretive family of functions on Rn. Let 0 < ε < 1

(cid:111)

and let b∗ =
(to be chosen differently at various points in the argument below) and define

Q∈P

(cid:110)

(cid:9)

b∗

Q

(cid:18) p

(cid:19) 1

p−2

λ = λ (ε) =

p − 2

Cb (p)p 1
ε

(5.1.13)

and a new collection of test functions,

(cid:111)(cid:33)
(cid:12)(cid:12)(cid:12)>λ

,

Q ∈ P,

(5.1.14)

We compute

(cid:90)(cid:110)(cid:12)(cid:12)(cid:12)bQ

(cid:111)(cid:12)(cid:12)bQ

(cid:12)(cid:12)(cid:12)>λ

(cid:12)(cid:12)(cid:12)≤λ
1(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:111) +
(cid:90) (cid:12)(cid:12)(cid:12)bQ

(cid:111)

(cid:12)(cid:12)(cid:12)

0

(cid:12)(cid:12)1(cid:110)(cid:12)(cid:12)(cid:12)bQ
λ(cid:12)(cid:12)bQ
 dσ

2tdt

(cid:32)

(cid:98)bQ ≡ 2bQ

(cid:90)(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:12)(cid:12)2 dσ=
(cid:12)(cid:12)(cid:12)>λ
(cid:90) (cid:90)(cid:110)
(cid:90) λ
(cid:90)(cid:110)
=λ2(cid:12)(cid:12)(cid:8)(cid:12)(cid:12)bQ

=

=

0

(x,t)∈Rn×(0,∞):max{t,λ}<

(cid:12)(cid:12)(cid:12)bQ(x)
(cid:12)(cid:12)(cid:12)(cid:111) 2tdtdσ (x)
(cid:90) ∞
(cid:90)(cid:110)
(cid:12)(cid:12)(cid:12)bQ(x)
(cid:12)(cid:12)(cid:12)(cid:111)dσ (x) 2tdt +
(cid:90) ∞
(cid:12)(cid:12) > t(cid:9)(cid:12)(cid:12)σ 2tdt,
(cid:12)(cid:12)(cid:8)(cid:12)(cid:12)bQ
(cid:12)(cid:12) > λ(cid:9)(cid:12)(cid:12)σ +

x∈Rn:λ<

λ

x∈Rn:t<

λ

(cid:12)(cid:12)(cid:12)bQ(x)

(cid:12)(cid:12)(cid:12)(cid:111)dσ (x) 2tdt

91

and hence

(cid:90)(cid:110)(cid:12)(cid:12)(cid:12)bQ

(cid:111)(cid:12)(cid:12)bQ
(cid:12)(cid:12)(cid:12)>λ

(cid:18)(cid:90) (cid:12)(cid:12)bQ

(cid:19)

(cid:12)(cid:12)p dσ

2tdt (5.1.15)

(cid:19)

(cid:12)(cid:12)2 dσ ≤ λ2 1
(cid:26)
λp
λ2−p +
p
p − 2

(cid:90) ∞
(cid:27)
2t1−pdt

(cid:18)(cid:90) (cid:12)(cid:12)bQ
(cid:12)(cid:12)p dσ
(cid:90) ∞
λ2−pCb (p)p |Q|σ = ε|Q|σ ,

1
tp

≤

=

+

λ

λ

Cb (p)p |Q|σ

by (5.1.13). Thus we have the lower bound,

(cid:12)(cid:12)(cid:12)(cid:12) 1

|Q|σ

(cid:90)

Q

(cid:98)bQdσ

(cid:12)(cid:12)(cid:12)(cid:12) = 2

(cid:90)

Q

bQ

(cid:90)

(cid:32)
(cid:33)
1 − λ(cid:12)(cid:12)bQ
(cid:12)(cid:12)
(cid:12)(cid:12)2 1(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:12)(cid:12)bQ

(5.1.16)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)>λ
1(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:111)dσ
(cid:33) 1
(cid:12)(cid:12)(cid:12)>λ
(cid:111)dσ

2

ε ≥ 1 > 0,

Q ∈ P.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:12)(cid:12)(cid:12)(cid:12) 1

|Q|σ

Q

(cid:90)
(cid:90)
(cid:18) 1

Q

≥ 2

|Q|σ
≥ 2 − 2

|Q|σ

bQdσ − 1
(cid:32)
|Q|σ
(cid:12)(cid:12)(cid:12)(cid:12) − 2
(cid:19) 1

bQdσ

2

ε|Q|σ

1
|Q|σ
√
= 2 − 2

Q

For an upper bound we have

(cid:13)(cid:13)(cid:13)(cid:98)bQ

(cid:13)(cid:13)(cid:13)L∞(σ)

≤ 2λ = 2λ (ε) = 2

(cid:18) p

p − 2

Cb (p)p 1
ε

(cid:19) 1

p−2

,

which altogether shows that

C(cid:98)b

(∞) ≤ 2

(cid:18) p

p − 2

(cid:19) 1

p−2

(cid:19) 1

p−2

(cid:18) p

p − 2

= 2

Cb (p)p 1
ε

p

p−2 ε

− 1
p−2

Cb (p)

(5.1.17)

92

if we choose 0 < ε ≤ 1

. Similarly we have

4

(cid:18) p

C(cid:98)b∗ (∞) ≤ 2

p − 2

Cb∗ (p)p 1
ε∗

(cid:19) 1

p−2

(cid:19) 1

p−2

(cid:18) p

p − 2

= 2

Cb∗ (p)

p

p−2 (ε∗)

− 1
p−2

for 0 < ε∗ ≤ 1

4

(cid:115)(cid:90)

(cid:12)(cid:12)(cid:12)T α
σ(cid:98)bQ

(cid:12)(cid:12)(cid:12)2

Q

(cid:115)(cid:90)

dω ≤ 2

. Moreover, we also have, using (5.1.15),

(cid:12)(cid:12)T α
(cid:12)(cid:12)2 dω + 2
(cid:113)|Q|σ + 2NT α

σ bQ

(cid:118)(cid:117)(cid:117)(cid:116)(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)T α
(cid:111)(cid:32)
(cid:12)(cid:12)(cid:12)>λ
σ 1(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:115)(cid:90)(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:111)(cid:12)(cid:12)bQ
(cid:12)(cid:12)2 dσ
(cid:12)(cid:12)(cid:12)>λ
(cid:111)(cid:113)|Q|σ ,

Q

Q

≤ 2Tb
T α

(cid:110)

≤ 2

Tb

T α +

√

εNT α

for all cubes Q,

(cid:33)
(cid:12)(cid:12) − 1

λ(cid:12)(cid:12)bQ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

bQ

dω

which shows that

T(cid:98)b
T α ≤ 2Tb

T α + 2

√

εNT α .

Now we apply the fact that (S∞) holds to obtain
(cid:17)(cid:110)
T(cid:98)b
T α + T(cid:98)b∗

NT α (cid:46)(cid:16)

(∞) + C(cid:98)b∗ (∞)

C(cid:98)b

T α,∗ +

(cid:113)

2 + Eα
Aα
2

(cid:111)

and take ε = ε∗ to conclude, using (5.1.17) and (5.1.18), that

NT α (cid:46) Cimplied (Cb (p) + Cb∗ (p))

p

p−2 ε

p−2(cid:110)

− 1

T α + Tb∗
T α,∗ +
Tb
2− 1
p−2 ε

p

1

(cid:113)
p−2 NT α

2 + Eα
Aα
2

+Cimplied (Cb (p) + Cb∗ (p))

(5.1.18)

(cid:111)(5.1.19)

Now we choose

−
(Cb (p) + Cb∗ (p))

p
p−2
2− 1
p−2

1

ε =

1
Γ

93

with Γ =(cid:0)2Cimplied

(cid:1)4, which satisfies Γ ≥ 1, so that the final term on the right satisfies

Cimplied (Cb (p) + Cb∗ (p))

p

p−2 ε

1

2− 1

p−2 NT α ≤ Cimplied

(cid:18) 1

(cid:19) 1

4

Γ

NT α =

1
2

NT α

where we have used 1

2 − 1

p−2 ≥ 1

4

hand side of (5.1.19) to obtain

for p > 6. This term can then be absorbed into the left

NT α (cid:46) (Cb (p) + Cb∗ (p))
(cid:41)

(cid:40)

p
p − 2

1 +

1
p−2
2 − 1
p−2

1

(cid:18)

Since

we get

(cid:113)

2 + Eα
Aα
2

(cid:111)

T α,∗ +

1+

p
p−2

1
p−2
2− 1
p−2

1

(cid:110)
(cid:19)(cid:18)

T α + Tb∗
Tb
(cid:19)

2
p − 4

=

1 +

1 +

2
p − 2

≤ 3 for p > 6,

NT α (cid:46) (Cb (p) + Cb∗ (p))3(cid:110)

T α + Tb∗
Tb

T α,∗ +

(cid:113)

2 + Eα
Aα
2

(cid:111)

,

which completes the proof of (S0).

We now show that(cid:0)Sp
(cid:1) holds for all p∈ (pm+1, pm]. So fix m ≥ 1, p∈ (pm+1, pm], and
suppose that b = (cid:8)bQ
(cid:9)
(cid:110)
Q∈P is a p-weakly σ-accretive family of functions on Rn and that
(cid:41)∞
b∗ =
b∗
is a p-weakly ω-accretive family of functions on Rn. Note that the sequence
1−(cid:16) 2
(cid:17)m
{pm}∞

satisfies the recursion relation

m=0 =

(cid:40)

Q∈P

(cid:111)

Q

2

3

m=0

pm+1 =

6

1 + 4
pm

, equivalently, pm =

, m ≥ 0.

4

6

pm+1

− 1

94

Choose q ∈ (pm, pm−1] so that

p >

6

1 + 4
q

=

6q

q + 4

, i.e. q <

4

6

p − 1

=

4p
6 − p

,

(5.1.20)

which can be done since p > pm+1 =

(cid:17)m+1
which leaves room to choose q satisfying pm < q < 4
p−1
6

(cid:17)m < 4
p−1
Now let 0 < ε < 1 (to be fixed later), define λ = λ (ε) as in (5.1.13), and define(cid:98)bQ as in

is equivalent to pm =

1−(cid:16) 2

6

1−(cid:16) 2

2

3

3

2

.

,

(5.1.14). Recall from (5.1.15) and (5.1.16) that we then have

(cid:90)(cid:110)(cid:12)(cid:12)(cid:12)bQ

(cid:111)(cid:12)(cid:12)bQ

(cid:12)(cid:12)2 dσ ≤ ε|Q|σ

and

(cid:12)(cid:12)(cid:12)>λ

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12) 1

|Q|σ

Q

(cid:98)bQdσ

(cid:12)(cid:12)(cid:12)(cid:12) ≥ 1,

Q ∈ P ,

if we choose 0 < ε ≤ 1

. We of course have the previous upper bound

4

(cid:13)(cid:13)(cid:13)(cid:98)bQ

(cid:13)(cid:13)(cid:13)L∞(σ)

≤ 2λ = 2λ (ε) = 2

(cid:18) p

p − 2

Cb (p)p 1
ε

(cid:19) 1

p−2

(cid:16) 1

(cid:17) 1
and while this turned out to be sufficient in the case m = 0, we must do better than
p−2 in the case m ≥ 1. In fact we compute the Lq norm instead, recalling that q > p

O

ε

95

and using Chebysev’s inequality,

(cid:32)

(cid:90)

Q

(cid:12)(cid:12)(cid:12)(cid:98)bQ

(cid:12)(cid:12)(cid:12)q

1
|Q|µ

(cid:33)1

q

1
|Q|µ

|Q|µ

(cid:32)
 1
 1
(cid:32)

|Q|µ

1
|Q|µ

dµ

= 2

= 2

≤ 2

≤ 2

≤ 2Cb (p)

= 2Cb (p)

p
q

p
q

1

q

qtq−1dt

q

dµ

(cid:111)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)q
(cid:33) 1
λ(cid:12)(cid:12)bQ
(cid:12)(cid:12)1(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:12)(cid:12)(cid:12)>λ
 dσ +
λq(cid:12)(cid:12)(cid:8)(cid:12)(cid:12)bQ
(cid:12)(cid:12) > λ(cid:9)(cid:12)(cid:12)µ
 1
 qtq−1dt + Cb (p)p λq−p
(cid:33) 1

|Q|µ

q

q

qtq−1dt + Cb (p)p λq−p
(cid:33) 1

q

0

Q

(cid:32)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)bQ
(cid:90)
(cid:12)(cid:12)(cid:12)≤λ
1(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:111) +
(cid:90) (cid:12)(cid:12)(cid:12)bQ
(cid:12)(cid:12)(cid:12)
(cid:90)(cid:110)(cid:12)(cid:12)(cid:12)bQ
(cid:12)(cid:12)(cid:12)≤λ
(cid:111)
(cid:90)(cid:110)
(cid:90) λ
(cid:12)(cid:12)(cid:12)≤λ
(cid:12)(cid:12)(cid:12)bQ
(cid:111) dσ
(cid:20) 1
(cid:21)
(cid:90) (cid:12)(cid:12)bQ
(cid:90) λ
(cid:12)(cid:12)p dσ
(cid:32)(cid:90) λ
(cid:18)2q − p

(cid:19) 1

tp

t<

0

0

0

q

λq−p

q − p

qtq−p−1dt + λq−p

which shows that C(cid:98)b

(q) satisfies the estimate

C(cid:98)b

(q) ≤ 2Cb (p)

(cid:46) Cb (p)

p
q

p
q

q

(cid:19) 1

(cid:18) p

(cid:18)2q − p
q − p
(cid:17)
(cid:16) q−2
− 1− p
p−2 (cid:46) Cb (p)
(cid:18)
(cid:19)

p − 2

p−2

ε

q

− 1
p−2

ε

(cid:19) 1

p−2

1− p

q

Cb (p)p 1
ε
− 1− p
p−2 ,

3
2 ε

q

a significant improvement over the bound O

. Here we have used that if p > 6q
q+4

,

then

p
q

(cid:19)

(cid:18) q − 2

p − 2

<

6q
q−4
q−4 − 2
6q

q − 2
q

<

3
2

96

as the function x (cid:55)→ x
x−2

is decreasing when x > 2. Moreover, from (5.1.18) we also have

T(cid:98)b
T α ≤ 2Tb

T α + 2

√

εNT α .

We can do the same for the dual testing functions b∗ =

provided 0 < ε ≤ 1

4

, we have both

1 ≤

(cid:12)(cid:12)(cid:12)(cid:12) 1

|Q|σ

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:13)(cid:13)(cid:13)(cid:98)bQ

(cid:98)bQdσ
T(cid:98)b
T α ≤ 2Tb

Q

T α + 2

(cid:13)(cid:13)(cid:13)Lq(σ)

√
εNT α ,

(cid:110)

b∗

Q

(cid:111)

Q∈P

and then altogether,

− 1− p
p−2 ,

q

3
2 ε

Q ∈ P ,

≤ Cb (p)

as well as

(cid:12)(cid:12)(cid:12)(cid:12) 1

|Q|ω

1 ≤

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:13)(cid:13)(cid:13)(cid:98)b∗

(cid:98)b∗
T(cid:99)b∗
T α ≤ 2Tb∗

Qdω

Q

(cid:13)(cid:13)(cid:13)Lq(ω)

Q

√

T α + 2

εNT α

≤ Cb∗ (p)

3
2 ε

− 1− p
p−2 ,

q

Q ∈ P ,

We now use these estimates, together with the fact that (Sm−1) holds, to obtain

NT α(cid:46)(cid:16)

C(cid:98)b
(q)+C(cid:98)b∗ (q)
(cid:46)(Cb (p)+Cb∗ (p))

(cid:46)(Cb (p) + Cb∗ (p))

(cid:113)

(cid:17)3n(cid:110)
T(cid:98)b
T α + T(cid:98)b∗
p−2(cid:110)(cid:104)
− 1− p
ε
p−2(cid:110)
− 1− p

3
2 3n
ε

3
2 3n

Tb

q

q

(cid:111)
(cid:104)
2 + Eα
Aα
2
Tb∗
T α,∗ +
(cid:113)

(cid:105)

+

T α,∗ +

2 +Eα
Aα
2

T α,∗ +
√
εNT α

T α +
T α +Tb∗
Tb

(cid:113)
(cid:105)
2 +Eα
Aα
2
− 1− p
p−2 NT α

(cid:111)


+

q

εNT α
√
εε

+

√

(cid:111)

We can absorb the term (Cb (p) + Cb∗ (p))

2 3n √

3

εε

− 1− p
p−2 NT α into the left hand side as

q

97

before, by choosing

ε =

1
Γ

 3

2 3n
1− p
p−2 − 1
q

2

(Cb (p) + Cb∗ (p))



with Γ sufficiently large, depending only on the implied constant, since (5.1.20) gives

6

p−1
2 <

(cid:16)

(cid:17) − 4

1 + 2
q
2p − 4

− 1 − p
p − 2

q

1
2

p

=

(cid:32)

p

1 +

(cid:33)

− 4

6

p−1

2

>

2p − 4

=

1
4

.

(5.1.21)

, and hence

2
q

Thus,

NT α (cid:46) (Cb (p) + Cb∗ (p))

2 3n(1+1)(cid:110)

3

T α + Tb∗
Tb

T α,∗ +

(cid:113)

2 + Eα
Aα
2

(cid:111)

.

Here we have used that (5.1.21) implies

1− p
q
p−2
2 − 1− p
q
p−2
NT α (cid:46) (Cb (p) + Cb∗ (p))3n+1(cid:110)

1

1 − p
p − 2

q

< 4

≤ 1. So we finally have

(cid:113)

2 + Eα
Aα
2

(cid:111)

,

T α + Tb∗
Tb

T α,∗ +

which completes the proof of Proposition 5.1.7.

Thus we may assume for the proof of Theorem 5.1.5 given below that p = ∞ and that

the testing functions are real-valued and satisfy

(cid:90)

suppbQ ⊂ Q ,
1
|Q|µ

bQdµ ≤(cid:13)(cid:13)bQ

Q

Q ∈ P,

(cid:13)(cid:13)L∞(µ) ≤ Cb (∞) < ∞,

1 ≤

(5.1.22)

Q ∈ P .

98

5.1.8 Reverse Hölder control of children

Here we begin to further reduce the proof of Theorem 5.1.5 to the case of bounded real

Q∈P having reverse Hölder control

(cid:9)
testing functions b =(cid:8)bQ
(cid:12)(cid:12)(cid:12)(cid:12) 1

(cid:13)(cid:13)(cid:13)1Q(cid:48)bQ
for all children Q(cid:48) ∈ C (Q) with(cid:12)(cid:12)Q(cid:48)(cid:12)(cid:12)σ > 0 and Q ∈ P.

(cid:90)
Q(cid:48) bQdσ

(cid:12)(cid:12)(cid:12)(cid:12) ≥ c

|Q(cid:48)|σ

(cid:13)(cid:13)(cid:13)L∞(σ)

> 0,

(5.1.23)

5.1.8.1 Control of averages over children

Lemma 5.1.8. Suppose that σ and ω are locally finite positive Borel measures on Rn. As-

sume that T α is a standard α-fractional elliptic and gradient elliptic singular integral oper-
ator on Rn, and set T α
σ is apriori
bounded from L2 (σ) to L2 (ω). Let Q ∈ P and let NT α (Q) be the best constant in the local

σ f = T α (f σ) for any smooth truncation of T α

σ , so that T α

inequality (cid:115)(cid:90)

Q(cid:48)

(cid:12)(cid:12)T α

σ

(cid:0)1Qf(cid:1)(cid:12)(cid:12)2 dω ≤ NT α (Q)

(cid:115)(cid:90)

Q

|f|2 dσ ,

f ∈ L2(cid:0)1Qσ(cid:1) .

Suppose that bQ is a real-valued function supported in Q such that

1 ≤ 1
|Q|σ

(cid:115)(cid:90)

Q

(cid:90)

bQdσ ≤(cid:13)(cid:13)1QbQ
(cid:12)(cid:12)2 dω ≤ T

(cid:13)(cid:13)L∞(σ) ≤ Cb ,
(cid:113)|Q|σ .

Q

σ bQ

(cid:12)(cid:12)T α
, there exists a real-valued function(cid:101)bQ supported in Q such

bQ
T α (Q)

Then for every 0 < δ <

that

1

2n+1C3
b

99

Q

(cid:115)(cid:90)

(cid:90)
(cid:101)bQdσ ≤(cid:13)(cid:13)(cid:13)1Q(cid:101)bQ
(1). 1 ≤ 1
|Q|σ
(cid:12)(cid:12)(cid:12)T α
(cid:12)(cid:12)(cid:12)2
σ(cid:101)bQ
(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQ
(cid:13)(cid:13)(cid:13)L∞(σ)

≤ 16Cb

(3). 0 <

dω ≤

(cid:34)

(2).

Q

δ

bQ
T
T α (Q) + 2C

Cb ,

(cid:17)
(cid:35)(cid:113)|Q|σ ,

(cid:16)

3
4
b δ

≤ 2

1
4 NT α (Q)

(cid:13)(cid:13)(cid:13)L∞(σ)
1 +(cid:112)Cb
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:90)
|Qi|σ
(cid:90)
bQdσ ≤(cid:13)(cid:13)1QbQ

(cid:101)bQdσ

Qi

Proof. Let 0 < δ < 1 and fix Q ∈ P. By assumption we have

1 ≤ 1
|Q|σ

Q

(cid:90)

|Qi|σ

Qi

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:12)(cid:12)(cid:12)(cid:12) ≤ 2n(cid:88)

i=1

≤ δ
Cb

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

Qi
|Q|σ

bQdσ

δ
Cb

(cid:13)(cid:13)(cid:13)1Qi

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) <
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) <
2n(cid:88)
(cid:13)(cid:13)L∞(σ) ≤ δ
(cid:13)(cid:13)1QbQ

δ
Cb

bQdσ

i=1

|Qi|σ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

bQdσ

Q

cannot hold for all Qi, since otherwise we obtain the contradiction

Qi ∈ C (Q) .

(cid:13)(cid:13)L∞(σ) ≤ Cb.

(cid:13)(cid:13)(cid:13)L∞(σ)

bQ

(5.1.24)

(cid:13)(cid:13)(cid:13)1Qi

bQ

(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12) <

Q

bQdσ

Q

bQdσ

(cid:12)(cid:12)(cid:12)(cid:12) .

Let Qi be the children of Q. We now define(cid:101)bQ. First we note that the inequality

If (5.1.24) holds for none of the Qi, then we simply define(cid:101)bQ = bQ, and trivially all the
(cid:12)(cid:12) dσ
(cid:12)(cid:12)bQ
then we define(cid:101)bQ differently according to how large the L1 (σ)-average

conclusions of the Lemma 5.1.8 hold. If (5.1.24) holds for at least one of the children, say Qi0

(cid:82)

,

1(cid:12)(cid:12)(cid:12)Qi0

(cid:12)(cid:12)(cid:12)σ

Qi0

is. In this case, define ˜G to be the set of indices for which (5.1.24) holds and G the set of

100

indices for which (5.1.24) fails. We define

(cid:101)bQ ≡ (cid:88)
(cid:88)

i∈G

+

(cid:88)

(cid:32)
(cid:88)
(cid:17)(cid:17)
1 +(cid:112)Cbδ

i∈G+

δ1Qi

+

(cid:16)

bQ1Q +

(cid:16)

i∈G0
pi − ni

1
|Qi|σ

1Qi

+

i∈B−

(cid:90)
(cid:88)

Qi

(cid:33)
(cid:12)(cid:12)bQ
(cid:12)(cid:12) dσ
(cid:16)(cid:16)
1 +(cid:112)Cbδ

1Qi

i∈B+

(cid:17)

(cid:17)

1Qi

pi − ni

where

G0 ≡

G+ ≡

B− ≡

B+ ≡

(cid:40)
(cid:40)
(cid:40)
(cid:40)

(cid:90)

Qi
1
(cid:90)
|Qi|σ
(cid:90)

Qi

Qi

(cid:41)
(cid:12)(cid:12) dσ = 0
(cid:41)
(cid:12)(cid:12) dσ ≤(cid:112)Cbδ
(cid:12)(cid:12)bQ
(cid:90)
(cid:12)(cid:12) dσ >(cid:112)Cbδ and
(cid:90)
(cid:12)(cid:12) dσ >(cid:112)Cbδ and

(cid:12)(cid:12)bQ
(cid:90)
(cid:12)(cid:12)bQ
(cid:12)(cid:12)bQ

Qi

,

Qi

Qi

(cid:41)
(cid:41)

,

.

pidσ

nidσ

(cid:90)
(cid:90)

Qi

Qi

nidσ >

pidσ ≥

i ∈ ˜G :

1
|Qi|σ
i ∈ ˜G : 0 <

i ∈ ˜G :

i ∈ ˜G :

1
|Qi|σ
1
|Qi|σ

and pi, ni are the positive and negative parts of bQ respectively on Qi, i.e.

1Qi

(x) bQ (x) = pi (x) − ni (x) ,

(x)(cid:12)(cid:12)bQ (x)(cid:12)(cid:12) = pi (x) + ni (x) ,

1Qi

Now let us check the conclusions of the Lemma 5.1.8. For (1) we have

(cid:90)
(cid:90)
(cid:90)

Q

Q

Q

1 ≤

≤

≤

1
|Q|σ
1
|Q|σ
1
|Q|σ

bQdσ

(cid:88)
(cid:101)bQdσ +
(cid:101)bQdσ +(cid:112)CbδCb

1
|Q|σ

i∈B−

(cid:90)

Qi

ni

(cid:112)Cbδdσ − 1
(cid:88)

|Qi|σ ≤ 1
|Q|σ

|Q|σ

(cid:90)
(cid:88)
(cid:101)bQdσ + C

Qi

pi

i∈B+

(cid:90)

Q

1
|Q|σ

(cid:112)Cbδdσ

√

δ

3
2
b

i∈B−

101

and choosing δ small enough we get

1
2

≤ 1
|Q|σ

(cid:90)

Q

(cid:101)bQdσ ≤(cid:13)(cid:13)(cid:13)1Q(cid:101)bQ

(cid:13)(cid:13)(cid:13)L∞(σ)

,

1 +(cid:112)Cb

(cid:17)

Cb

which in turn is bounded by

sup

(cid:16)

≤ 2

Qi∈C(Q)

(cid:13)(cid:13)(cid:13)L∞(σ)

by taking the different cases on Qi:

(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQi
(a) For i ∈ G0,(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQi
(cid:13)(cid:13)(cid:13)L∞ ≤ δ,
(cid:13)(cid:13)(cid:13)L∞ ≤ Cb,
(b) For i ∈ G+,(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQi
(cid:13)(cid:13)(cid:13)L∞ ≤ 2(1 +(cid:112)Cb)Cb.
(c) For i ∈ B− ∪ B+,(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQi

This completes the proof for (1).

For (2), we have from Minkowski’s inequality

(cid:115)

(cid:90)

(cid:12)(cid:12)(cid:12)T α
σ(cid:101)bQ

(cid:12)(cid:12)(cid:12)2

1
|Q|σ

Q

(cid:115)

(cid:90)

(cid:12)(cid:12)T α

dω ≤

Q

σ bQ

1
|Q|σ
bQ
T α (Q) + NT α (Q)

≤ T

= T

bQ
T α (Q) + NT α (Q)

(cid:115)

(cid:12)(cid:12)2dω +
(cid:115)
(cid:118)(cid:117)(cid:117)(cid:116) 1

1
(cid:90)
|Q|σ
1
|Q|σ

Q

|Q|σ

σ

Q

(cid:90)
(cid:12)(cid:12)(cid:12)T α
(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ
(cid:90)
(cid:88)

(cid:17)(cid:12)(cid:12)(cid:12)2
(cid:16)(cid:101)bQ − bQ
(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ

dσ

Qi∈C(Q)

Qi

dω

dσ

and this last term is bounded by:

(cid:88)

i∈G

(cid:88)

i∈G0

(cid:88)

i∈G+

+

(cid:88)

i∈B−

+

+

+

102

(cid:115)

(cid:88)

i∈B+

(cid:90)

Qi

(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ

(cid:12)(cid:12)(cid:12)2

dσ

1
|Q|σ

and since we have:

(a) for i ∈ G,

(cid:90)

Qi

1
|Q|σ

(b) for i ∈ G0,

1
|Q|σ

(cid:90)

Qi

(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ

(cid:12)(cid:12)(cid:12)2

dσ ≤

≤

(cid:12)(cid:12)(cid:12)2

dσ = 0

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ
(cid:32)(cid:90)
(cid:32)

(cid:33)
(cid:33)

1
|Q|σ
1
|Q|σ

δ2dσ +

Qi
Qi
δ2|Qi|σ + Cb

(cid:90)

|bQ|2dσ

|bQ|dσ

Qi

= δ2|Qi|σ
|Q|σ

by the accretivity of bQ and the definition of G0.

(c) for i ∈ G+,

(cid:90)

1
|Q|σ

Qi

(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ

(cid:12)(cid:12)(cid:12)2

dω =

≤

1
|Q|σ

1
|Q|σ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
− bQ
(cid:90)
(cid:33)

(cid:12)(cid:12) dσ

dσ +

Qi

(cid:90)
(cid:90)
(cid:32)(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

1
|Qi|σ

(cid:32)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:112)Cbδ

|Qi|σ

Qi

(cid:90)
(cid:90)

(cid:33)
(cid:12)(cid:12)bQ
(cid:12)(cid:12) dσ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12) dσ
(cid:12)(cid:12)bQ
(cid:90)
(cid:12)(cid:12)bQ
(cid:17) |Qi|σ

Qi
≤ 2C

Qi

|Q|σ

Cbδdσ + Cb

Qi

Qi

3
2
b δ

1
2

|Qi|σ
|Q|σ

.

dσ


(cid:12)(cid:12)2 dσ

(cid:12)(cid:12)bQ

Qi

1
|Q|σ

≤

≤ (cid:16)

Cbδ + Cb

(d) for i ∈ B−,

1
|Q|σ

(cid:90)

Qi

(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ

(cid:12)(cid:12)(cid:12)2

dσ =

1
|Q|σ
≤ C3
bδ

(cid:90)
Qi
|Qi|σ
|Q|σ

.

|Cbδni|2 dσ = Cbδ

(cid:90)

Qi

1
|Q|σ

|ni|2 dσ

103

(e) and for i ∈ B+, the same estimate as in the previous case,

we obtain

(cid:115)

(cid:90)

Q

(cid:12)(cid:12)(cid:12)T α
σ(cid:101)bQ

(cid:12)(cid:12)(cid:12)2

1
|Q|σ

dω ≤ T

T α (Q) + 2 · 2nC
bQ

3
4
b δ

1
4 NT α (Q) .

where the dimensional constant comes from

1(cid:112)|Q|σ

2n(cid:88)

i=1

(cid:112)|Qi|σ ≤ 2n.

Now we are left with verifying (3). Note that

(a) for i ∈ G, the inequality (5.1.24) does not hold and as(cid:101)bQ = bQ there, immediately we

obtain

(b) for i ∈ G0 ∪ G+,

(c) for i ∈ B−,

Qi

≤

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

δ

Qi

(cid:90)

Cb
δ

1|Qi|σ

(cid:101)bQdσ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cb
(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQ
(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQ
(cid:12)(cid:12)(cid:12)(cid:12) = 1 <
(cid:12)(cid:12)(cid:12)(cid:12)
(cid:82)
(cid:101)bQdσ
(cid:0)1 +(cid:112)Cbδ(cid:1) Cb
(cid:12)(cid:12)(cid:12)(cid:12)
(cid:0)1 +(cid:112)Cbδ(cid:1)(cid:3) dσ
(cid:2)pi − ni
(cid:82)
(cid:0)1 +(cid:112)Cbδ(cid:1) Cb
(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:112)Cbδ
(cid:82)
2(1 +(cid:112)Cbδ)Cb
(cid:12)(cid:12)bQ
(cid:12)(cid:12) dσ
(cid:112)Cbδ
(cid:82)

1|Qi|σ

1|Qi|σ

nidσ

Qi

≤

≤

(cid:12)(cid:12)(cid:12)(cid:12)

1|Qi|σ

Qi

≤ 4Cb
Cbδ

=

4
δ

,

104

(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQ
(cid:12)(cid:12)(cid:12)(cid:12)
(cid:82)

1|Qi|σ

(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:12)(cid:12)(cid:12)(cid:12) ≤
(cid:101)bQdσ

Qi

Qi

as, by taking 0 < δ < 1
4C3
b

, we have 1 +(cid:112)Cbδ < 2.

(d) and for i ∈ B+ similarly as in the previous case.

In order to obtain the inequalities for(cid:101)bQ in the conclusion of Lemma 5.1.8, we simply
multiply the above function(cid:101)bQ by a factor of 2.
(cid:12)(cid:12) ≥ c1 > 0, we easily see that(cid:12)(cid:12)(cid:12)(cid:101)bQ
Finally, if(cid:12)(cid:12)bQ
(cid:12)(cid:12) ≥ c1 > 0 as well. This completes

(cid:12)(cid:12)(cid:12) ≥(cid:12)(cid:12)bQ

the proof of Lemma 5.1.8.

5.1.8.2 Control of averages in coronas

Let DQ be the grid of dyadic subcubes of Q. In the construction of the triple corona below,
we will need to repeat the construction in the previous subsubsection for a subdecomposition
{Qi}∞
of dyadic subcubes Qi ∈ DQ of a cube Q. Define the corona corresponding to the
subdecomposition {Qi}∞

by

i=1

i=1

∞(cid:91)

i=1

CQ ≡ DQ\

DQi

.

Lemma 5.1.9. Suppose that σ and ω are locally finite positive Borel measures on Rn. As-

sume that T α is a standard α-fractional elliptic and gradient elliptic singular integral oper-
ator on Rn, and set T α
σ is apriori
bounded from L2 (σ) to L2 (ω). Let Q ∈ P and let NT α (Q) be the best constant in the local

σ f = T α (f σ) for any smooth truncation of T α

σ , so that T α

inequality (cid:115)(cid:90)

Q

(cid:12)(cid:12)T α

σ

(cid:0)1Qf(cid:1)(cid:12)(cid:12)2 dω ≤ NT α (Q)

(cid:115)(cid:90)

Q

|f|2 dσ ,

f ∈ L2(cid:0)1Qσ(cid:1) .

Let {Qi}∞

i=1 ⊂ DQ be a collection of pairwise disjoint dyadic subcubes of Q. Suppose that

105

(cid:34)

(cid:16)

σ bQ

Q

≤ 2

bQ is a real-valued function supported in Q such that

≤ Cb ,

Then for every 0 < δ < 1
4C3
b

bQ
T α (Q)

(cid:115)(cid:90)

(cid:12)(cid:12)T α

Q(cid:48) ∈ CQ ,

1 ≤ 1
|Q(cid:48)|σ

(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:113)|Q|σ .

(cid:90)
Q(cid:48) bQdσ ≤(cid:13)(cid:13)(cid:13)1Q(cid:48)bQ
(cid:12)(cid:12)2 dω ≤ T
, there exists a real-valued function(cid:101)bQ supported in Q such that
(cid:90)
Q(cid:48)(cid:101)bQdσ ≤(cid:13)(cid:13)(cid:13)1Q(cid:48)(cid:101)bQ
(cid:13)(cid:13)(cid:13)L∞(σ)
1 ≤ 1
(cid:12)(cid:12)(cid:12)T α
(cid:12)(cid:12)(cid:12)2
|Q(cid:48)|σ
σ(cid:101)bQ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:90)
(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQ
(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:12)(cid:12)(cid:12)(cid:101)bQ
(cid:12)(cid:12) ≥ c1 > 0, then we may take

1 +(cid:112)Cb
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,
(cid:101)bQdσ
(cid:12)(cid:12)(cid:12) ≥ c1 as well.

(cid:17)
(cid:35)(cid:113)|Q|σ ,

bQ
T α (Q) + 4C

Q(cid:48) ∈ CQ ,

1 ≤ i < ∞.

1
4 NT α (Q)

(cid:115)(cid:90)

≤ 16Cb

Moreover, if(cid:12)(cid:12)bQ
The additional gain in the lemma is in the final line that controls the degeneracy of(cid:101)bQ
at the ‘bottom’ of the corona CQ by establishing a reverse Hölder control. Note that if we
combine this control with the accretivity control in the corona CQ, namely
(cid:90)
Q(cid:48)(cid:101)bQdσ,

1 +(cid:112)Cb

1 +(cid:112)Cb

(cid:13)(cid:13)(cid:13)L∞(σ)

(cid:13)(cid:13)(cid:13)1Q(cid:48)(cid:101)bQ

Cb ≤ 2

≤ 2

(cid:16)

(cid:17)

(cid:17)

(cid:16)

Cb

|Q(cid:48)|σ

1

|Qi|σ

dω ≤

2T

Cb ,

3
2
b δ

Qi

Q

0 <

δ

we obtain reverse Hölder control throughout the entire collection CQ ∪ {Qi}∞

:

i=1

(cid:13)(cid:13)(cid:13)1I(cid:101)bQ(cid:48)

(cid:13)(cid:13)(cid:13)L∞(σ)

≤ Cδ,b

(cid:12)(cid:12)(cid:12)(cid:12) 1

|I|σ

(cid:90)

I

(cid:101)bQ(cid:48)dσ

(cid:12)(cid:12)(cid:12)(cid:12) ,

I ∈ C(cid:0)Q(cid:48)(cid:1) , Q(cid:48) ∈ CQ .

This has the crucial consequence that the martingale and dual martingale differences (cid:52)σ,b
Q(cid:48)

106

and (cid:3)σ,b

Q(cid:48) associated with these functions as defined in (5.1.38), satisfy

(cid:12)(cid:12)(cid:12)(cid:52)σ,b

Q(cid:48) h

(cid:12)(cid:12)(cid:12) ,

(cid:12)(cid:12)(cid:12)(cid:3)σ,b

Q(cid:48) h

(cid:12)(cid:12)(cid:12) ≤ Cδ,b

(cid:88)

I∈C(Q(cid:48))

(cid:90)

(cid:18) 1

|I|σ

(cid:90)
Q(cid:48) |h| dσ

(cid:19)

1I .

(5.1.25)

|h| dσ +

1

|Q(cid:48)|σ

I

However, the defect in this lemma is that we lose the weak testing condition for(cid:101)bQ in the

corona even if we had assumed it at the outset for bQ.

Proof. The proof of Lemma 5.1.9 is similar to that of the Lemma 5.1.8. Indeed, we define

δ1Qi

+

i∈G0

(cid:101)bQ ≡ (cid:88)
(cid:88)
(cid:88)

+

i∈B−

+

(cid:32)
(cid:32)

(cid:88)
(cid:90)
(cid:90)

i∈G+
1
|Qi|σ

Qi

(cid:90)

(cid:32)
1
|Qi|σ
(cid:104)
Qi
pi − ni
(cid:104)(cid:16)

(cid:16)
1 +(cid:112)Cbδ

(cid:33)
(cid:12)(cid:12) dσ
(cid:12)(cid:12)bQ
1 +(cid:112)Cbδ
(cid:17)

pi − ni

1Qi

(cid:17)(cid:105)
(cid:105)

(cid:33)
(cid:33)

1Qi

dσ

dσ

1Qi

1
|Qi|σ

i∈B+

+bQ1Q\∪∞

i=1Qi

Qi

,

where

G0 ≡

G+ ≡

B− ≡

B+ ≡

(cid:40)
(cid:40)
(cid:40)
(cid:40)

(cid:90)

i :

1
|Qi|σ

i : 0 <

i :

i :

1
|Qi|σ
1
|Qi|σ

Qi
1
(cid:90)
|Qi|σ
(cid:90)

Qi

Qi

,

(cid:41)
(cid:12)(cid:12) dσ = 0
(cid:41)
(cid:12)(cid:12) dσ ≤(cid:112)Cbδ
(cid:12)(cid:12)bQ
(cid:90)
(cid:12)(cid:12) dσ >(cid:112)Cbδ and
(cid:90)
(cid:12)(cid:12) dσ >(cid:112)Cbδ and

(cid:12)(cid:12)bQ
(cid:90)
(cid:12)(cid:12)bQ
(cid:12)(cid:12)bQ

Qi

,

Qi

Qi

(cid:41)
(cid:41)

,

.

pidσ

nidσ

(cid:90)
(cid:90)

Qi

Qi

nidσ >

pidσ ≥

and pi, ni the positive and negative parts of bQ on each Qi. The proof of Lemma 5.1.8 can be

107

applied verbatim. We emphasise only that when estimating the testing condition, we need

(cid:90)

Q

the bound

1

4 |Q|σ .

∞(cid:88)

i=1

dσ ≤ C (Cb) δ

1
4

|Qi|σ ≤ C (Cb) δ

(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ
Remark 5.1.10. The estimate(cid:82)
4(cid:80)∞
the above proof is of course too large in general to be dominated by a fixed multiple of(cid:12)(cid:12)Q(cid:48)(cid:12)(cid:12)σ
i=1 |Qi|σ in the last line of
for Q(cid:48) ∈ CQ, and this is the reason we have no control of weak testing for(cid:101)bQ in the rest of
the corona even if we assume weak testing for bQ in the corona CQ. This defect is addressed
in the next subsection below.

(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ

dσ ≤ C (Cb) δ

1

(cid:12)(cid:12)(cid:12)2

Q

5.1.9 Three corona decompositions

We will use multiple corona constructions, namely a Calderón-Zygmund decomposition, an

accretive/testing decomposition, and an energy decomposition, in order to reduce matters

to the stopping form, which is treated in Section 5.6 by adapting the bottom/up stopping

time and recursion of M. Lacey in [27]. We will then iterate these corona decompositions

into a single corona decomposition, which we refer to as the triple corona. More precisely, we

iterate the first generation of common stopping times with an infusion of the reverse Hölder

condition on children, followed by another iteration of the first generation of weak testing

stopping times. Recall that we must show the bilinear inequality

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12) ≤ NT α (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) ,

(T α

σ f ) gdω

f ∈ L2 (σ) and g ∈ L2 (ω) .

108

5.1.9.1 The Calderón-Zygmund corona decomposition

In this section, we introduce the Calderón-Zygmund stopping times F for a function φ ∈
L2 (µ) relative to a cube S0 and a positive constant C0 ≥ 4. Let F = {F}F∈F be the
collection of Calderón-Zygmund stopping cubes for φ defined so that F ⊂ S0, S0 ∈ F, and
for all F ∈ F with F (cid:36) S0 we have

(cid:90)
(cid:90)
|φ| dµ > C0
F(cid:48) |φ| dµ ≤ C0

F

1
|F|µ
1
|F(cid:48)|µ

1

|πF F|µ
|πF F|µ

1

(cid:90)
(cid:90)

F

F

|φ| dµ;

|φ| dµ

for F (cid:36) F(cid:48) ⊂ πF F.

We denote by πF F be the smallest member of F that strictly contains F . For a cube I ∈ D
let πDI be the D-parent of I in the grid D. For F, F(cid:48) ∈ F, we say that F(cid:48) is an F-child of F

if πF(cid:0)F(cid:48)(cid:1) = F (it could be that F = πDF(cid:48)), and we denote by CF (F ) the set of F-children
of F . We call πF(cid:0)F(cid:48)(cid:1) the F-parent of F(cid:48) ∈ F.

To achieve the construction above we use the following definition.

Definition 5.1.11. Let C0 ≥ 4. Given a dyadic grid D and a cube S0 ∈ D, define S (S0)
to be the maximal D-subcubes I ⊂ S0 such that

(cid:90)

1
|I|µ

|φ| dµ > C0

1
|S0|µ

I

|φ| dµ ,

(cid:90)

S0

∞(cid:91)

and then define the Calderón-Zygmund stopping cubes of S0 to be the collection

where S0 = S (S0) and Sm+1 = (cid:83)

F = {S0} ∪

Sm

m=0

S (S) for m ≥ 0.

109

S∈Sm

Define the corona of F by

CF ≡(cid:8)F(cid:48) ∈ D : F ⊃ F(cid:48) (cid:37) H for some H ∈ CF (F )(cid:9) .

The stopping cubes F above satisfy a Carleson condition:

(cid:88)

F∈F: F⊂Ω

|F|µ ≤ C |Ω|µ ,

for all open sets Ω.

Indeed,

(cid:88)
F(cid:48)∈CF (F )

(cid:12)(cid:12)F(cid:48)(cid:12)(cid:12)µ ≤ (cid:88)

F(cid:48)∈CF (F )

(cid:82)
(cid:82)
F(cid:48) |φ| dµ
F |φ| dµ
1|F|µ

C0

|F|,

≤ 1
C0

and standard arguments now complete the proof of the Carleson condition.

We emphasize that accretive functions b play no role in the Calderón-Zygmund corona

decomposition.

5.1.9.2 The accretive/testing corona decomposition

We use a corona construction modelled after that of Hytönen and Martikainen [24], that

delivers a weak corona testing condition that coincides with the testing condition itself only

at the tops of the coronas. This corona decomposition is developed to optimize the choice of

a new family of real valued testing functions(cid:110)(cid:98)bQ

(cid:111)

taken from the vector b ≡(cid:8)bQ

(cid:9)

Q∈D

Q∈D

so that we have

1. the telescoping property at our disposal in each accretive corona,

2. a weak corona testing condition remains in force for the new testing functions(cid:98)bQ that

coincides with the testing condition at the tops of the coronas,

110

3. the tops of the coronas, i.e. the stopping cubes, enjoy a Carleson condition.

We will henceforth refer to the old family as the original family, and denote it by(cid:110)

borig
Q

(cid:111)

Q∈D

.

The original family will reappear later in helping to estimate the nearby form.

Let σ and ω be locally finite Borel measures on Rn . We assume that the vector of

‘testing functions’ b ≡(cid:8)bQ

(cid:9)
Q∈D is a ∞-weakly σ-accretive family, i.e. for Q ∈ D

(cid:90)
supp bQ ⊂ Q,

bQdσ ≤(cid:12)(cid:12)(cid:12)(cid:12)bQ

(cid:12)(cid:12)(cid:12)(cid:12)L∞(σ) ≤ Cb < ∞

Q∈D is a ∞-weakly ω-accretive family, and we assume in addition

0 < cb≤ 1
|Q|µ
(cid:9)

Q

the testing conditions

and also that b∗ ≡(cid:8)bQ
(cid:90)
(cid:1)(cid:12)(cid:12)2 dω ≤ (cid:16)
(cid:12)(cid:12)T α
(cid:0)1QbQ
(cid:12)(cid:12)(cid:12)T α,∗
(cid:17)(cid:12)(cid:12)(cid:12)2
(cid:16)
dσ ≤ (cid:16)

1Qb∗

(cid:90)

Q

Q

ω

σ

Q

(cid:17)2 |Q|σ ,
(cid:17)2 |Q|ω ,

Tb
T α
Tb∗
T α,∗

for all cubes Q,

for all cubes Q.

Definition 5.1.12. Given a cube S0, define S (S0) to be the maximal subcubes I ⊂ S0 such
that satisfy one of the following

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:90)
(cid:90)
|I|µ
(cid:12)(cid:12)(cid:12)T α

σ

I

(a).

(b).

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < γ, or
(cid:17)(cid:12)(cid:12)(cid:12)2

dω > Γ

dσ

bS0

I

(cid:16)

bS0

(cid:16)

Tb
T α

(cid:17)2 |I|σ

where the positive constants γ, Γ satisfy 0 < γ < 1 < Γ < ∞. Then define the b-accretive

stopping cubes of S0 to be the collection

F = {S0} ∪

111

∞(cid:91)

m=0

Sm

S (S) for m ≥ 0.

where S0 = S (S0) and Sm+1 = (cid:83)
a σ-Carleson condition relative to the measure σ, and the new testing functions(cid:110)(cid:101)bQ
defined by(cid:102)bS = 1SbS0

Q∈D
, satisfy weak testing inequalities. The following lemma

For ε > 0 chosen small enough depending on p > 2, the b-accretive stopping cubes satisfy

for S ∈ CS0

S∈Sm

(cid:111)

,

is essentially in [24], but we include a proof for completeness.

Lemma 5.1.13. For γ small enough and Γ large enough, we have the following:

(1). For every open set Ω we have we have the inequality,

(cid:88)

S∈F: S⊂Ω

|S|σ ≤ C |Ω|σ .

(5.1.26)

(2). For every cube S ∈ CS0

we have the weak corona testing inequality,

(cid:90)

S

(cid:12)(cid:12)(cid:12)T α

(cid:12)(cid:12)(cid:12)2

(cid:16)

(cid:17)2 |S|σ .

σ bS0

dω ≤ C

Tb
T α

(5.1.27)

Proof. Inequality (5.1.27) is immediate from the definition of F in the definition 5.1.12. We
now address the Carleson condition (5.1.26). A standard argument reduces matters to the
case where Ω is a cube Q ∈ F with |Q|σ > 0. It suffices to consider each of the two stopping

criteria separately. We first address the stopping condition(cid:12)(cid:12)(cid:12) 1|I|σ
this proof we will denote the union of these children S (Q) of Q by E (Q) ≡ (cid:83)
(cid:88)

(cid:12)(cid:12)(cid:12) < γ. Throughout

S. Then

S∈S(Q)

we have

I bS0

(cid:90)

(cid:82)

dσ

bQdσ

S

S∈S(Q)

|S|σ ≤ γ |Q|σ ,

(cid:12)(cid:12)(cid:12)(cid:12) < γ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88)

S∈S(Q)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

bQdσ

E(Q)

112

which together with our hypotheses on bQ gives

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

|Q|σ ≤

bQdσ

Q

E(Q)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

bQdσ

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12) ≤
(cid:115)(cid:90)
(cid:113)|Q|σ

bQdσ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) +
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:90)
(cid:113)|Q\E (Q)|σ
(cid:12)(cid:12)bQ
(cid:12)(cid:12)2 dσ
(cid:113)|Q\E (Q)|σ.

Q\E(Q)

Q\E(Q)

≤ γ |Q|σ +
≤ γ |Q|σ + Cb

Rearranging the inequality yields

(1 − γ)|Q|σ ≤ Cb

(cid:113)|Q|σ

(cid:113)|Q\E (Q)|σ

or

which in turn gives

(1 − γ)2

C2
b

|Q|σ ≤ |Q\E (Q)|σ ,

(cid:88)

S∈S(Q)

|S|σ = |E(Q)| = |Q|σ − |Q\E (Q)|σ
(cid:18)
1 − (1 − γ)2
C2
b

≤ |Q|σ − (1 − γ)2

|Q|σ =

C2
b

(cid:19)

|Q|σ ≡ β |Q|σ ,

113

where 0 < β < 1 since 1 ≤ Cb. If we now iterate this inequality, we obtain for each k ≥ 1,

(cid:88)

(cid:88)

(cid:88)

(cid:12)(cid:12)S(cid:48)(cid:12)(cid:12)σ ≤ (cid:88)

|S|σ =

π

S∈F: S⊂Q
(k)F (S)=Q
π

S(cid:48)∈S(S)

S∈F: S⊂Q
(k−1)
F
(S)=Q

β |S|σ

S∈F: S⊂Q
(k−1)
F
(S)=Q

π

...

≤ (cid:88)

S∈F: S⊂Q
(1)F (S)=Q
π

βk−1 |S|σ ≤ βk |Q|σ .

Finally then

(cid:88)

S∈F: S⊂Q

∞(cid:88)

k=0

|S|σ ≤

(cid:88)

∞(cid:88)

k=0

|S|σ ≤

S∈F: S⊂Q
(k)F (S)=Q
π

Now we turn to the second stopping criterion(cid:82)
(cid:17)2
(cid:17)2

S∈CF (S0)

|S|σ ≤

(cid:88)

(cid:16)
(cid:16)

T α
1

(cid:90)

≤

Γ

Tb

1

I

Γ

Tb

T α

S0

1
1 − β

|Q|σ =
(cid:16)
(cid:17)(cid:12)(cid:12)(cid:12)2

bS0

(cid:16)

dω

dω > Γ

Tb
T α

βk |Q|σ =
(cid:16)

σ

(cid:12)(cid:12)(cid:12)T α
(cid:88)
(cid:12)(cid:12)(cid:12)T α

σ

(cid:17)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)T α
(cid:17)(cid:12)(cid:12)(cid:12)2

σ

bS0

(cid:90)

(cid:16)

S∈CF (S0)

S

bS0

dω ≤ 1
Γ

|S0|σ .

C2
b

(1 − γ)2 |Q|σ .
(cid:17)2 |I|σ . We have

Iterating this inequality gives

∞(cid:88)

k=0

|S|σ ≤

(cid:88)

S∈F
S⊂S0

1
Γk

|S0|σ =

Γ
Γ − 1

|S0|σ ,

114

and then

(cid:88)

S∈F
S⊂Ω

|S|σ =

(cid:88)

(cid:88)

maximal S0∈F

S0⊂Ω

S∈F
S⊂S0

|S|σ ≤ Γ
Γ − 1

(cid:88)

maximal S0∈F

S0⊂Ω

|S0|σ =

Γ
Γ − 1

|Ω|σ .

This completes the proof of Lemma 5.1.13.

5.1.9.3 The energy corona decompositions

Given a weight pair (σ, ω), we construct an energy corona decomposition for σ and an energy

corona decomposition for ω, that uniformize estimates (c.f. [43], [32], [63] and [64]). In order
to define these constructions, we recall that the energy condition constant E α
(cid:17) ,

Pα(cid:0)Jr, 1Qσ(cid:1)

2(cid:13)(cid:13)x − mJr

is given by

∞(cid:88)

(cid:13)(cid:13)2
L2(cid:16)

2

1Jr ω

(E α
2 )2 ≡ sup
Q∈P
Q= ˙∪Jr

1
|Q|σ

r=1

|Jr| 1

n

where ˙∪Jr is an arbitrary subdecomposition of Q into cubes Jr ∈ P and interchanging the
roles of σ and ω we have the constant E α,∗
. In the next
definition we restrict the cubes Q to a dyadic grid D, but keep the subcubes Jr unrestricted.

. Also recall that Eα

2 + E α,∗

2 = E α

2

2

Definition 5.1.14. Given a dyadic grid D and a cube S0 ∈ D, define S (S0) to be the
maximal D-subcubes I ⊂ S0 such that

∞(cid:88)

r=1

sup
I⊃ ˙∪Jr

Pα (Jr, 1I σ)

2(cid:13)(cid:13)x − mJr

|Jr| 1

n

(cid:13)(cid:13)2
L2(cid:16)

1Jr ω

(cid:17) ≥ Cen

(cid:104)

(cid:105) |I|σ ,

(Eα

2 )2 + Aα
2

(5.1.28)

where the cubes Jr ∈ P are pairwise disjoint in I, Eα
2 is the energy condition constant, and
Cen is a sufficiently large positive constant depending only on α. Then define the σ-energy

115

stopping cubes of S0 to be the collection

F = {S0} ∪

∞(cid:91)

m=0

Sm

where S0 = S (S0) and Sm+1 = (cid:83)

S (S) for m ≥ 0.

S∈Sm

(cid:88)

We now claim that from the energy condition Eα

2 < ∞, we obtain the σ-Carleson estimate,

|S|σ ≤ 2|I|σ ,

I ∈ D.

(5.1.29)

S∈S: S⊂I

Indeed, for any S1 ∈ F we have

(cid:88)

|S|σ≤

S∈CF (S1)

1
Aα

(cid:1)2(cid:17) (cid:88)
(cid:16)
2 +(cid:0)E α
(cid:0)E α
(cid:1)2 (E α

S∈CF (S1)
2 )2 |S1|σ =

1

2

2

Cen
≤
Cen

sup
S⊃ ˙∪Jr

1
Cen

|S1|σ ,

Pα(Jr, 1Sσ)

2(cid:13)(cid:13)x−mJr

|Jr| 1

n

∞(cid:88)

r=1

(cid:13)(cid:13)2

L2(1Jr ω)

upon noting that the union of the subdecompositions ˙∪Jr ⊂ S over S ∈ CF (S1) is a
subdecomposition of S1, and the proof of the Carleson estimate is now finished by iteration
in the standard way.

Finally, we record the reason for introducing energy stopping times. If

Xα (CS)2 ≡ sup
I∈CS

1
|I|σ

sup
I⊃ ˙∪Jr

∞(cid:88)

r=1

Pα (Jr, 1I σ)

2(cid:13)(cid:13)x − mJr

|Jr| 1

n

(cid:13)(cid:13)2
L2(cid:16)

(cid:17)

1Jr ω

(5.1.30)

is (the square of) the α-stopping energy of the weight pair (σ, ω) with respect to the corona

116

CS , then we have the stopping energy bounds

Xα (CS) ≤(cid:112)

Cen

(cid:113)(cid:0)Eα

2

(cid:1)2 + Aα

2 ,

S ∈ F,

(5.1.31)

where Aα
2

and the energy constant Eα
2

are controlled by the assumptions in Theorem 5.1.5.

5.1.10

Iterated coronas and general stopping data

We will use a construction that permits iteration of the above three corona decompositions

by combining Definitions 5.1.11, 5.1.12 and 5.1.14 into a single stopping condition. However,

(cid:82)

there is one remaining difficulty with the triple corona constructed in this way, namely if a
stopping cube I ∈ A is a child of a cube Q in the corona CA, then the modulus of the average

(cid:12)(cid:12)(cid:12) of bQ on I may be far smaller than the sup norm of(cid:12)(cid:12)bQ

(cid:12)(cid:12) on the child I, indeed

I bQdσ

(cid:82)

(cid:12)(cid:12)(cid:12) 1|I|σ

1|I|σ

Q f and (cid:3)σ,b

it may be that
martingale and dual martingale differences (cid:52)σ,b

I bQdσ = 0. This of course destroys any reasonable estimation of the
Q f used in the proof of Theorem

5.1.5, and so we will use Lemma 5.1.9 on the function bA to obtain a new function(cid:101)bA for
which this problem is circumvented at the ‘bottom’ of the corona, i.e. for those A(cid:48) ∈ CA (A).
We then refer to the stopping times A(cid:48) ∈ CA (A) as ‘shadow’ stopping times since we have
lost control of the weak testing condition relative to the new function(cid:101)bA. Thus we must
redo the weak testing stopping times for the new function(cid:101)bA, but also stopping if we hit
Definition 5.1.15. Let C0 ≥ 4, 0 < γ < 1 and 1 < Γ < ∞. Suppose that b =(cid:8)bQ

Q∈P
is an ∞-weakly σ-accretive family on Rn. Given a dyadic grid D and a cube Q ∈ D, define
the collection of ‘shadow’ stopping times Sshadow (Q) to be the maximal D-subcubes I ⊂ Q
such that one of the following holds:

one of the shadow stopping times. Here are the details.

(cid:9)

117

(cid:90)

bQdσ

I

σ

I

(a).

(b).

(c).

(cid:16)

Tb
T α

1
|I|σ

I

1
|Q|σ

Q

|f| dσ ,

sup
I⊃ ˙∪Jr

r=1

(cid:17)2 |I|σ ,

(cid:1)(cid:12)(cid:12)2 dω > Γ

(cid:90)
|f| dσ > C0
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < γ or
(cid:90)
(cid:90)
(cid:12)(cid:12)T α
(cid:0)bQ
|I|µ
2(cid:13)(cid:13)x − mJr
Pα (Jr, σ)
∞(cid:88)
(cid:13)(cid:13)2
L2(cid:16)
Now we apply Lemma 5.1.9 to the function bQ with Sshadow (Q) ≡ {Qi}∞
new function(cid:101)bQ satisfying the properties
supp(cid:101)bQ ⊂ Q ,
(cid:90)
Q(cid:48)(cid:101)bQdσ ≤(cid:13)(cid:13)(cid:13)1Q(cid:48)(cid:101)bQ
(cid:115)(cid:90)
(cid:12)(cid:12)2 dω ≤
(cid:12)(cid:12)T α
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQ

(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:90)
(cid:101)bQdσ

(cid:17)
(cid:35)(cid:113)|Q|σ ,

Cb ,

1 +(cid:112)Cb

(cid:16)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,

(cid:17) ≥ Cen

3
2
b δ

1
4 NT α (Q)

1 ≤ 1
|Q(cid:48)|σ

2Tb

T α (Q) + 4C

1 ≤ i < ∞.

≤ 16Cb

δ

(Eα

2 )2 + Aα
2

σ bQ

Q

|Qi|σ

Qi

|Jr| 1

n

1Jr ω

(cid:34)

≤ 2

Q(cid:48) ∈ CQ ,

(cid:104)

(cid:105) |I|σ .

i=1

to obtain a

(5.1.32)

Note that each of the functions(cid:101)bQ(cid:48) ≡ 1Q(cid:48)(cid:101)bQ, for Q(cid:48) ∈ CQ, now satisfies the crucial reverse

Hölder property

(cid:13)(cid:13)(cid:13)1I(cid:101)bQ(cid:48)

(cid:13)(cid:13)(cid:13)L∞(σ)

≤ Cδ,b

(cid:12)(cid:12)(cid:12)(cid:12) 1

|I|σ

(cid:90)

I

(cid:101)bQ(cid:48)dσ

(cid:12)(cid:12)(cid:12)(cid:12) ,

for all I ∈ C(cid:0)Q(cid:48)(cid:1) , Q(cid:48) ∈ CQ.

Indeed, if I equals one of the Qi then the reverse Hölder condition in the last line of (5.1.32)
applies, while if I ∈ CQ then the accretivity in the second line of (5.1.32) applies.

118

Since we have lost the weak testing condition in the corona for this new function(cid:101)bQ,
starting with the new function(cid:101)bQ, and also stopping if we hit one of the ‘shadow’ stopping

the next step is to run again the weak testing construction of stopping times, but this time

times Qi. Here is the new stopping criterion.
Definition 5.1.16. Let C0 ≥ 4 and 1 < Γ < ∞. Let Sshadow (Q) ≡ {Qi}∞
i=1 be as in
Definition 5.1.15. Define Siterated (Q) to be the maximal D-subcubes I ⊂ Q such that either

(cid:90)

(cid:12)(cid:12)(cid:12)T α

σ

(cid:16)(cid:101)bQ

(cid:17)(cid:12)(cid:12)(cid:12)2

(cid:16)

T(cid:101)b

T α

(cid:17)2 |I|σ ,

dω > Γ

I

or

I = Qi for some 1 ≤ i < ∞.

Thus for each cube Q we have now constructed iterated stopping children Siterated (Q)
by first constructing shadow stopping times Sshadow (Q) using one step of the triple corona
construction, then modifying the testing function to have reverse Hölder controlled children,
and finally running again the weak testing stopping time construction to get Siterated (Q).
These iterated stopping times Siterated (Q) have control of CZ averages of f and energy
control of σ and ω, simply because these controls were achieved in the shadow construction,

and were unaffected by either the application of Lemma 5.1.9 or the rerunning of the weak

testing stopping criterion for(cid:101)bQ. And of course we now have weak testing within the corona
determined by Q and Siterated (Q), and we also have the crucial reverse Hölder condition on
all the children of cubes in the corona. With all of this in hand, here then is the definition

of the construction of iterated coronas.

Definition 5.1.17. Let C0 ≥ 4, 0 < γ < 1 and 1 < Γ < ∞. Suppose that b =(cid:8)bQ

(cid:9)

Q∈P is

119

an ∞-weakly σ-accretive family on Rn. Given a dyadic grid D and a cube S0 in D, define
the iterated stopping cubes of S0 to be the collection

∞(cid:91)

F = {S0} ∪

Sm

where S0 = Siterated (S0) and Sm+1 = (cid:83)

m=0

Siterated (S) for m ≥ 0, and where Siterated (Q)

S∈Sm

is defined in Definition 5.1.16.

It is useful to append to the notion of stopping times S in the above σ-iterated corona
decomposition a positive constant A0 and an additional structure αS called stopping bounds
for a function f. We will refer to the resulting triple (A0,F, αF ) as constituting stopping
data for f. If F is a grid, we define F(cid:48) ≺ F if F(cid:48) (cid:36) F and F(cid:48), F ∈ F. Recall that πF F(cid:48) is
the smallest F ∈ F such that F(cid:48) ≺ F .

Suppose we are given a positive constant A0 ≥ 4, a subset F of the dyadic grid D (called
the stopping times), and a corresponding sequence αF ≡ {αF (F )}F∈F of nonnegative
numbers αF (F ) ≥ 0 (called the stopping bounds). Let (F,≺, πF ) be the tree structure on
F inherited from D, and for each F ∈ F denote by CF = {I ∈ D : πF I = F} the corona
associated with F :

CF =(cid:8)I ∈ D : I ⊂ F and I (cid:54)⊂ F(cid:48) for any F(cid:48) ≺ F(cid:9) .

Definition 5.1.18. We say the triple (A0,F, αF ) constitutes stopping data for a function
f ∈ L1

loc (σ) if
I |f| ≤ αF (F ) for all I ∈ CF and F ∈ F,

(cid:12)(cid:12)F(cid:48)(cid:12)(cid:12)σ ≤ A0 |F|σ for all F ∈ F,

F(cid:48)(cid:22)F

(1). Eσ

(2). (cid:80)

120

(3). (cid:80)
(4). αF (F ) ≤ αF(cid:0)F(cid:48)(cid:1) whenever F(cid:48), F ∈ F with F(cid:48) ⊂ F .

F∈F αF (F )2 |F|σ ≤A2

0 (cid:107)f(cid:107)2

,

L2(σ)

Property (1) says that αF (F ) bounds the averages of f in the corona CF , and property
(2) says that the cubes at the tops of the coronas satisfy a Carleson condition relative to the

weight σ. Note that a standard ‘maximal cube’ argument extends the Carleson condition in

property (2) to the inequality

(cid:88)

F(cid:48)∈F: F(cid:48)⊂A

(cid:12)(cid:12)F(cid:48)(cid:12)(cid:12)σ ≤ A0 |A|σ

for all open sets A ⊂ Rn.

(5.1.33)

Property (3) is the quasi-orthogonality condition that says the sequence of functions

{αF (F ) 1F}F∈F is in the vector-valued space L2(cid:0)(cid:96)2; σ(cid:1) with control and is often referred

to as a Carleson embedding theorem, and property (4) says that the control on stopping
data is nondecreasing on the stopping tree F. We emphasize that we are not assuming in

this definition the stronger property that there is C > 1 such that αF(cid:0)F(cid:48)(cid:1) > CαF (F )

whenever F(cid:48), F ∈ F with F(cid:48) (cid:36) F . Instead, the properties (2) and (3) substitute for this
lack. Of course the stronger property does hold for the familiar Calderón-Zygmund stopping

data determined by the following requirements for C > 1,

F(cid:48) |f| > CEσ
Eσ

F |f| whenever F(cid:48), F ∈ F with F(cid:48) (cid:36) F,

I |f| ≤ CEσ
Eσ

F |f| for I ∈ CF ,

which are themselves sufficiently strong to automatically force properties (2) and (3) with
αF (F ) = Eσ

F |f|.

121

We have the following useful consequence of (2) and (3) that says the sequence

{αF (F ) 1F}F∈F has a quasi-orthogonal property relative to f with a constant C(cid:48)
only on C0:

0

depending

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

(cid:88)

F∈F

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2

L2(σ)

αF (F ) 1F

≤ C(cid:48)

0 (cid:107)f(cid:107)2

L2(σ)

.

(5.1.34)

(cid:82)

Proposition 5.1.19. Let f ∈ L2 (σ), let F be as in Definition 5.1.17, and define stopping
F |f| dσ. Then there is A0 ≥ 4, depending only on the constant
data αF by αF = 1|F|σ
C0 in Definition 5.1.11 , such that the triple (A0,F, αF ) constitutes stopping data for the
function f.

Proof. This is an easy exercise using (5.1.26) and (5.1.29), and is left for the reader.

5.1.11 Reduction to good functions

We begin with a specification of the various parameters that will arise during the proof, as

well as the extension of goodness introduced in [24].

Definition 5.1.20. The parameters r, τ and ρ will be fixed below to satisfy

τ > r and ρ > r + τ,

where r is the goodness parameter fixed in (5.2.16).

Let 0 < ε < 1 to be chosen later. Define J to be ε − good in a cube K if

d (J, skelK) > 2|J|ε |K|1−ε ,

122

where the skeleton skelK ≡ (cid:83)

where k = log2

(cid:96)(K)
(cid:96)(J)

K(cid:48)∈C(K)
(k,ε)−good

∂K(cid:48) of a cube K consists of the boundaries of all the
children K(cid:48) of K. Define GD
to consist of those J ∈ G such that J is good in every
supercube K ∈ D that lies at least k levels above J. We also define J to be ε−good in a cube
K and beyond if J ∈ GD
(k,ε)−good
if and only if J is ε − good in πkJ and beyond. As the goodness parameter ε will eventually
be fixed throughout the proof, we sometimes suppress it, and simply say "J is good in a cube
K and beyond" instead of "J is ε − good in a cube K and beyond".

. We can now say that J ∈ GD

(k,ε)−good

As pointed out on page 14 of [24] by Hytönen and Martikainen, there are subtle difficulties

associated in using dual martingale decompositions of functions which depend on the entire

dyadic grid, rather than on just the local cube in the grid. We will proceed at first in the

spirit of [24]. The goodness that we will infuse below into the main ‘below’ form B(cid:98)ρ (f, g)
will be the Hytönen-Martikainen ‘weak’ goodness: every pair (I, J) ∈ D × G that arises in
the form B(cid:98)ρ (f, g) will satisfy J ∈ GD

where (cid:96) (I) = 2k(cid:96) (J).

(k,ε)−good

It is important to use two independent random grids, one for each function f and g

simultaneously, as this is necessary in order to apply probabilistic methods to the dual

martingale averages (cid:3)µ,b

I

that depend, not only on I, but also on the underlying grid in

which I lives. The proof methods for functional energy from [64] and [63] relied heavily on

the use of a single grid, and this must now be modified to accomodate two independent grids.

5.1.11.1 Parameterizations of dyadic grids

It is important to use two independent grids, one for each function f and g simultaneously,

as it is necessary in order to apply probabilistic methods to the dual martingale averages

(cid:3)µ,b

I

that depend not only on I but also on the underlying grid in which I lives.

Now we recall the construction from the paper [67]. We momentarily fix a large positive

123

integer M ∈ N, and consider the tiling of Rn by the family of cubes DM ≡(cid:110)

(cid:111)

0 + α · 2−M where IM
side length 2−M and given by IM
D built on DM is defined to be a family of cubes D satisfying:

α ≡ IM

0 =

(cid:104)

IM
α

0, 2−M(cid:17). A dyadic grid

having

α∈Z

1. Each I ∈ D has side length 2−(cid:96) for some (cid:96) ∈ Z with (cid:96) ≤ M, and I is a union of

2n(M−(cid:96)) cubes from the tiling DM ,

2. For (cid:96) ≤ M, the collection D(cid:96) of cubes in D having side length 2−(cid:96) forms a pairwise

disjoint decomposition of the space Rn,

3. Given I ∈ Di and J ∈ Dj with j ≤ i ≤ M, it is the case that either I ∩ J = ∅ or I ⊂ J.

We now momentarily fix a negative integer N ∈ −N, and restrict the above grids to cubes

of side length at most 2−N :

DN ≡(cid:110)

I ∈ D : side length of I is at most 2−N(cid:111) .

We refer to such grids DN as a (truncated) dyadic grid D built on DM of size 2−N . There are
now two traditional means of constructing probability measures on collections of such dyadic

grids, namely parameterization by choice of parent, and parameterization by translation.

Construction #1: Consider first the special case of dimension n = 1. For any

β = {βi}

i∈ZN
M

∈ ωN

m ≡ {0, 1}ZN

M ,

where ZN

M ≡ {(cid:96) ∈ Z : N ≤ (cid:96) ≤ M}, define the dyadic grid Dβ built on Dm of size 2−N by

2−(cid:96)

[0, 1) + k +

Dβ =

(cid:88)

i: (cid:96)<i≤M

124



2−i+(cid:96)βi

(5.1.35)

N≤(cid:96)≤M, k∈Z

Place the uniform probability measure ρN
M
that which charges each β ∈ ωN
space Rn by taking products in the usual way and using the product index space ΩN

equally. This construction is then extended to Euclidean
M ≡

on the finite index space ωN

M , namely

M

M = {0, 1}ZN

(ωN

M )n and the uniform product probability measure µN
Construction #2: Momentarily fix a (truncated) dyadic grid D built on DM of size

M = ρN

M

M × ... × ρN

.

2−N . For any

M ≡(cid:110)

γ ∈ ΓN

2−MZn

+ : |γi| < 2−N(cid:111)

,

where Zn

+ = (N ∪ {0})n, define the dyadic grid Dγ built on Dm of size 2−N by

Dγ ≡ D + γ.

Place the uniform probability measure νN
M

on the finite index set ΓN
M

, namely that which

(cid:19)

(cid:18)

and

{Dγ}

(cid:19)

γ∈ΓN
M

are isomorphic

, νN
M
of all (truncated)

charges each multiindex γ in ΓN
M

The two probability spaces

since both collections(cid:8)Dβ

(cid:9)

equally.

(cid:18)(cid:8)Dβ

(cid:9)

, µN
M

β∈ΩN
M
and {Dγ}

M

γ∈ΓN
M

β∈ΩN
M

describe the set AN
M
dyadic grids Dγ built on Dm of size 2−N , and since both measures µN
uniform measure on this space. The first construction may be thought of as being parame-
terized by scales - each component βi in β = {βi}
two possible tilings at level i that respect the choice of tiling at the level below - and since

amounting to a choice of the

and νN
M

i∈ZN
M

∈ ωN

are the

is determined by a choice of scales , we see that(cid:8)Dβ

. The
any grid in AN
M
second construction may be thought of as being parameterized by translation - each γ ∈ ΓN
amounting to a choice of translation of the grid D fixed in construction #2 - and since any
is determined by any of the cubes at the top level, i.e. with side length 2−N , we
grid in AN
M

β∈ΩN
M

= AN
M

M

(cid:9)

M

125

see that {Dγ}
Q + γ for some γ ∈ ΓN

γ∈ΓN
M

= AN
M

as well, since every cube at the top level in AN
M

has the form

M

and Q ∈ D at the top level in AN

(i.e. every cube at the top level in
is a union of small cubes in Dm, and so must be a translate of some Q ∈ D by an amount
M = 2n(M−N ).
to denote expectation with respect to this common probability measure

AN
M
2−M times an element of Z+). Note also that in all dimensions, #ΩN
We will use E

M = #ΓN

M

ΩN
M

on AN
M

.

Notation 5.1.21. For purposes of notation and clarity, we now suppress all reference to M

and N in our families of grids, and in the notations Ω and Γ for the parameter sets, and

we use P Ω and EΩ to denote probability and expectation with respect to families of grids,

and instead proceed as if all grids considered are unrestricted. The careful reader can supply
the modifications necessary to handle the assumptions made above on the grids D and the

functions f and g regarding M and N.

5.1.12 Formulas

We need the following formulas defined on Appendix A of [69].

(cid:82)

Q bQdµ

(cid:82)

1

Q f (x) ≡ 1Q (x)
Eµ,b
Q f (x) ≡ 1Q (x) bQ (x)
Fµ,b
(cid:98)Fµ,b
(cid:82)
Q f (x) ≡ 1Q (x)

1

(cid:90)

Q
1

(cid:90)

f bQdµ,

(cid:90)

Q bQdµ

Q

Q ∈ P ,

(5.1.36)

f dµ,

Q ∈ P ,

f dµ,

Q ∈ P .

(5.1.37)

Q bQdµ

Q

126

and

(cid:52)µ,b
Q f (x)≡

Q f (x)≡
(cid:3)µ,b

 (cid:88)
 (cid:88)

Q(cid:48)∈C(Q)

Q(cid:48)∈C(Q)

We also need

− Eµ,b
− Fµ,b

Eµ,b
Q(cid:48) f (x)

Fµ,b
Q(cid:48) f (x)

(cid:88)
(cid:88)

Q f (x) =

1Q(cid:48)(x)

Q(cid:48)∈C(Q)

Q f (x) =

1Q(cid:48) (x)

Q(cid:48)∈C(Q)

(cid:16)Eµ,b
Q(cid:48) f (x)−Eµ,b
(cid:16)Fµ,b
Q(cid:48) f (x) − Fµ,b

Q f (x)

Q f (x)

(cid:17) (5.1.38)
(cid:17)

and

(cid:33)

(cid:90)
Q(cid:48) |f| dµ
(cid:90)
Q(cid:48) |f| dµ +

1

|Q(cid:48)|µ

1

|Q(cid:48)|µ

L2(µ)

(cid:13)(cid:13)(cid:13)2
 − Fµ,b

f

(cid:32)
(cid:32)
(cid:13)(cid:13)(cid:13)(cid:98)(cid:53)µ

(cid:53)µ

Q∈D

Q(cid:48)∈Cbrok(Q)

Qf ≡ (cid:88)
Qf ≡ (cid:88)
(cid:98)(cid:53)µ
Q(cid:48)∈Cbrok(Q)
(cid:88)
 (cid:88)
(cid:82)
Fµ,π,b
f = 1Q
Q
(cid:88)
Q = (cid:3)µ,π,b
(cid:3)µ,b
(cid:12)(cid:12)(cid:12) ,
Q(cid:48)∈Cbrok(Q)

(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:98)(cid:53)µ

Q(cid:48)∈C(Q)
bπQ

Q,brokf =

f =

Qf

Q

Q

(cid:3)µ,π,b

Q bπQdµ
(cid:3)µ,π,b

Q

Fµ,π,b
Q(cid:48)

(cid:90)

(cid:3)µ,b

(cid:12)(cid:12)(cid:12)(cid:3)µ,π,b

Q,brokf

1Q(cid:48),

1
|Q|µ

(5.1.39)

(cid:33)

|f| dµ

(cid:90)

Q

1Q(cid:48),

Qf

(cid:46) (cid:107)f(cid:107)2

L2(µ)

.

(5.1.40)

(cid:88)

Q(cid:48)∈C(Q)

Q(cid:48) f − Fµ,bQ
Fµ,bQ
Q f,

(5.1.41)

Q f =

f dµ,

Q
+ (cid:3)µ,b
Fµ,bQ(cid:48)
Q(cid:48)

f − Fµ,bQ

Q(cid:48) f,

and (cid:3)µ,b

Q = (cid:3)µ,π,b

Q

Q,brok

+ (cid:3)µ,π,b
Q,brok

(5.1.42)

(5.1.43)

(5.1.44)

with similar equalities and inequalities for (cid:52) and E. Here Cbrok (Q) denotes the set of broken
children, i.e. those Q(cid:48) ∈ C (Q) for which bQ(cid:48)
(cid:54)= 1Q(cid:48)bQ, and more generally and typically,

127

Cbrok (Q) = C (Q) ∩ A where A is a collection of stopping cubes that includes the broken
children and satisfies a σ-Carleson condition and πQ is the dyadic father of Q.

Define another modified dual martingale difference by

(cid:3)σ,(cid:91),b

I

f ≡ (cid:3)σ,b

I f − (cid:88)

I(cid:48)∈Cbrok(I)

Fσ,b
I(cid:48) f =

 (cid:88)

I(cid:48)∈Cnat(I)

 − Fσ,b

I f,

Fσ,b
I(cid:48) f

(5.1.45)

where we have removed the averages over broken children from (cid:3)σ,b
over I intact. On any child I(cid:48) of I, the function (cid:3)σ,(cid:91),b
and so we have

I f, but left the average
f is thus a constant multiple of bI,

I

(cid:19)
(cid:19)

f

(cid:3)σ,(cid:91),b

I

I

(cid:3)σ,(cid:91),b

(cid:98)(cid:3)σ,(cid:91),b

I

f = bI

I(cid:48)∈C(I)

(cid:88)
f ≡ (cid:88)
(cid:88)
1I(cid:48)
I(cid:48)∈Cnat(I)

I(cid:48)∈C(I)

=

1I(cid:48)Eσ
I(cid:48)

bI

(cid:18) 1
(cid:18) 1
1(cid:82)

I(cid:48) bI dµ

bI

(cid:20)

1I(cid:48) Eσ
I(cid:48)

(cid:88)

I(cid:48)∈C(I)

1I(cid:48)Eσ
I(cid:48)

= bI

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

I

(cid:17)

;

f

(5.1.46)

(cid:3)σ,(cid:91),b

,

I

f

(cid:90)
I(cid:48) f dµ− 1(cid:82)

I bI dµ

(cid:21)

f dµ

(cid:90)

I

− (cid:88)

1I(cid:48)
I(cid:48)∈Cbrok(I)

(cid:20)

(cid:90)

I

1(cid:82)

I bI dµ

(cid:21)

f dµ

Thus for I ∈ CA we have

(cid:3)σ,(cid:91),b

I

f = bA

I

I

f

f,

(cid:17)

(5.1.47)

1I(cid:48)Eσ
I(cid:48)

I(cid:48)∈C(I)

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

= bA(cid:98)(cid:3)σ,(cid:91),b

(cid:88)
(cid:17) satisfy the following telescoping property for all K ∈


A(cid:48)∈CA(A) A(cid:48)(cid:17) and L ∈ CA with K ⊂ L:
L(cid:98)Fσ
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
−Eσ
K(cid:98)Fσ
L(cid:98)Fσ
Lf
K f − Eσ

if K ∈ CA (A)
K ∈ CA
if

(5.1.48)

Eσ
IK

(cid:17)

Eσ

Lf

=

f

f

I

I

,

where the averages Eσ
I(cid:48)

(CA \ {A}) ∪(cid:16)(cid:83)
(cid:88)

I: πK⊂I⊂L

128

where(cid:98)Fσ

K

is defined in (5.1.37) above.

Finally, in analogy with the broken differences (cid:52)µ,π,b

Q,brok

and (cid:3)µ,π,b
Q,brok

introduced above,

we define

I,brok f ≡ (cid:88)

(cid:52)µ,(cid:91),b

I(cid:48)∈Cbrok(I)

I(cid:48) f and (cid:3)µ,(cid:91),b
Eσ,b

I,brokf ≡ (cid:88)

I(cid:48)∈Cbrok(I)

Fσ,b
I(cid:48) f ,

(5.1.49)

so that

(cid:52)µ,b
I = (cid:52)µ,(cid:91),b

I

+ (cid:52)µ,(cid:91),b

I,brok

and (cid:3)µ,b

I = (cid:3)µ,(cid:91),b

I

+ (cid:3)µ,(cid:91),b

I,brok .

(5.1.50)

These modified differences and the identities (5.1.47) and (5.1.48) play a useful role in the

analysis of the nearby and paraproduct forms.

Lemma 5.1.22. For dyadic cubes R and Q we have

 (cid:52)µ,b

Q

0

(cid:52)µ,b
R (cid:52)µ,b

Q =

if R = Q
if R (cid:54)= Q

.

For the reader’s convenience we now collect the various martingale and probability es-

timates that will be used in the proof that follows. First we summarize the martingale

identities and estimates that we will use in our proof. Suppose µ is a positive locally finite
Borel measure, and that b is a ∞-weakly µ-controlled accretive family. Then,
Martingale identities: Both of the following identities hold pointwise µ-almost every-

129

where, as well as in the sense of strong convergence in L2 (µ):

(cid:88)
(cid:88)

f =

f =

I∈D: I⊂I∞, (cid:96)(I)≥2−N

I∈D: I⊂I∞, (cid:96)(I)≥2−N

(cid:3)σ,b
I f + Fσ,b
I∞f,

(cid:52)σ,b
I f + Eσ,b
I∞f.

Frame estimates: Both of the following frame estimates hold:

(cid:107)f(cid:107)2

L2(µ)

≈ (cid:88)
≈ (cid:88)

Q∈D

Q∈D

(cid:26)(cid:13)(cid:13)(cid:13)(cid:3)µ,b
(cid:26)(cid:13)(cid:13)(cid:13)(cid:52)µ,b

Q f

Q f

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

L2(µ)

L2(µ)

(cid:13)(cid:13)(cid:13)(cid:53)µ,b
(cid:13)(cid:13)(cid:13)(cid:53)µ,b

Q f

Q f

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

+

+

L2(µ)

L2(µ)

(cid:27)
(cid:27)

Weak upper Riesz estimates: Define the pseudoprojections,

Ψµ,bB f ≡ (cid:88)
(cid:17)∗
f ≡ (cid:88)

I∈B

I∈B

(cid:16)

Ψµ,bB

I f,

(cid:3)µ,b

(cid:16)(cid:3)µ,b

(cid:17)∗

I

(cid:88)

I∈B

f =

(cid:52)µ,b
I f.

(5.1.51)

.

(5.1.52)

(cid:17)∗

:

Ψµ,bB

,

(5.1.53)

We have the ‘upper Riesz’ inequalities for pseudoprojections Ψµ,bB and(cid:16)
(cid:13)(cid:13)(cid:13)(cid:98)(cid:53)µ,b
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:13)(cid:16)(cid:98)(cid:53)µ,b
(cid:17)∗

(cid:13)(cid:13)(cid:13)(cid:3)µ,b
(cid:13)(cid:13)(cid:13)(cid:52)µ,b

(cid:13)(cid:13)(cid:13)Ψµ,bB f
(cid:17)∗

(cid:88)
(cid:88)

(cid:88)
(cid:88)

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

(cid:13)(cid:13)(cid:13)(cid:16)

≤ C

≤ C

Ψµ,bB

I f

I f

L2(µ)

L2(µ)

L2(µ)

I∈B

+

L2(µ)

I∈B

I f

f

L2(µ)

(cid:13)(cid:13)(cid:13)(cid:13)2

I∈B

+

I∈B

I

f

,

L2(µ)

for all f ∈ L2 (µ) and all subsets B of the grid D. Here the positive constant C and depends
only on the accretivity constants, and is independent of the subset B and the testing family
b. The Haar martingale differences (cid:52)µ,b
are independent of both the testing families and

Q

130

the grid, while the Carleson averaging operators (cid:53)µ
choice of broken children of Q.

Q

depend on the grid only through the

5.1.13 Monotonicity Lemma

As in virtually all proofs of a two weight T 1 theorem (see e.g. [27], [34] , [64] and/or [63]),

the key to starting an estimate for any of the forms we consider below, is the Monotonicity

Lemma and the Energy Lemma, to which we now turn.

In dimension n = 1 ([34], [27])

the Haar functions have opposite sign on their children, and this was exploited in a simple

but powerful monotonicity argument. In higher dimensions, this simple argument no longer

holds and that Monotonicity Lemma is replaced with the Lacey-Wick formulation of the

Monotonicity Lemma (see [37], and also [63]) involving the smaller Poisson operator. As the

martingale differences with test functions bQ here are no longer of one sign on children, we
will adapt the Lacey-Wick formulation of the Monotonicity Lemma to the operator T α and

the dual martingale differences(cid:110)(cid:3)ω,b∗

(cid:111)

, bearing in mind that the operators (cid:3)ω,b∗

J

are

J

J∈G

no longer projections, which results in only a one-sided estimate with additional terms on
the right hand side. It is here that we need the crucial property that the Range of (cid:3)ω,b∗

is

J

orthogonal to constants,(cid:82)(cid:16)(cid:3)ω,b∗

J Ψ

(cid:17)

dσ =(cid:82)(cid:16)(cid:52)σ,b∗

J

(cid:17)

1

Ψdω =(cid:82) (0) Ψdω = 0.

We will also need the smaller Poisson integral used in the Lacey-Wick formulation of the

Monotonicity Lemma,

1+δ (J, µ) ≡
Pα

(cid:90)

|J| 1+δ

n

(|J| + |y − cJ|)n+1+δ−α

dµ (y) ,

which is discussed in more detail below.

Lemma 5.1.23 (Monotonicity Lemma). Suppose that I and J are cubes in Rn such that

131

J ⊂ γJ ⊂ I for some γ > 1, and that µ is a signed measure on Rn supported outside I.
Let 0 < δ < 1 and let Ψ ∈ L2 (ω). Finally suppose that T α is a standard fractional singular
integral on Rn with 0 ≤ α < 1, and suppose that b∗ is an ∞-weakly µ-controlled accretive
family on Rn. Then we have the estimate

(cid:12)(cid:12)(cid:12)(cid:68)

T αµ, (cid:3)ω,b∗
J Ψ

(cid:69)

ω

(cid:12)(cid:12)(cid:12) (cid:46) Cb∗CCZ Φα (J,|µ|)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J Ψ

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

,

(5.1.54)

where

Φα (J,|µ|) ≡ Pα (J,|µ|)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

J Ψ

x

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

L2(µ)

|J|

≡ (cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
≡ (cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

J Ψ

x

J

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

L2(ω)

L2(µ)

+

x

+

L2(ω)

(cid:13)(cid:13)(cid:13)♠
(cid:88)
(cid:88)
J(cid:48)∈Cbrok(J)
J(cid:48)∈Cbrok(J)

+ inf
z∈R

|J|

(cid:107)x − mJ(cid:107)L2(1J ω) ,
Eω

1+δ (J,|µ|)
Pα
(cid:16)
J(cid:48) |x − z|(cid:17)2
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω
(cid:104)
J(cid:48) |Ψ|(cid:105)2
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

Eω

,

.

All of the implied constants above depend only on γ > 1, 0 < δ < 1 and 0 < α < 1.

1J(cid:48) defined in (5.1.39), we can rewrite the expressions

Using (cid:53)ω

J h =

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

(cid:13)(cid:13)(cid:13)♠2

x

L2(ω)

Eω

(cid:88)
and(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
J(cid:48)∈Cbrok(J)
J Ψ
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

(cid:16)
J(cid:48) |h|(cid:17)
(cid:13)(cid:13)(cid:13)(cid:70)2
≡ (cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
≡ (cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J Ψ

L2(ω)

L2(µ)

as

x

J

J

L2(µ)

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

x

L2(ω)

J Ψ

L2(µ)

(cid:13)(cid:13)(cid:53)ω
J (x − z)(cid:13)(cid:13)2
+(cid:13)(cid:13)(cid:53)ω
J Ψ(cid:13)(cid:13)2

+ inf
z∈R

L2(ω) .

L2(ω) ,

132

Proof. Using (cid:3)ω,b∗

J

(cid:12)(cid:12)(cid:12)(cid:68)

T αµ, (cid:3)ω,b∗
J Ψ

ω

(cid:3)ω,π,b∗

J

= (cid:3)ω,π,b∗
(cid:69)

J

(cid:12)(cid:12)(cid:12) =
(cid:12)(cid:12)(cid:12)(cid:68)
≤ (cid:12)(cid:12)(cid:12)(cid:68)

J,brok

+ (cid:3)ω,π,b∗
(cid:16)(cid:3)ω,π,b∗
T αµ,
T αµ, (cid:3)ω,π,b∗

J

J

, we write

(cid:3)ω,π,b∗
(cid:3)ω,π,b∗

J

Ψ

J

+ (cid:3)ω,π,b∗
(cid:69)

J,brok

(cid:12)(cid:12)(cid:12) +

(cid:12)(cid:12)(cid:12)(cid:68)

ω

(cid:12)(cid:12)(cid:12)

(cid:69)

(cid:17)
T αµ, (cid:3)ω,π,b∗
J,brok Ψ

Ψ

ω

(cid:69)

(cid:12)(cid:12)(cid:12)

ω

Since(cid:68)

(cid:69)

h

ω

1, (cid:3)ω,π,b∗

J

≡ I + II.

= 0, we use mJ =

(cid:90)

1
|J|ω

xdω (x) to obtain

J

(cid:90)
(cid:90) (cid:104)∇(Kα)T (θ (x, mJ ) , y) · (x − mJ )

[(Kα) (x, y) − (Kα) (mJ , y)] dµ (y)

(cid:105)

dµ (y)

T αµ (x) − T αµ (mJ ) =

=

for some θ (x, mJ ) ∈ J to obtain

I =

=

≤

J

J

(cid:12)(cid:12)(cid:12)(cid:12)

Ψ (x) dω (x)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)
[T αµ (x) − T αµ (mJ )] (cid:3)ω,π,b∗
(cid:3)ω,π,b∗
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:26)(cid:90)
(cid:27)
· (x − mJ ) (cid:3)ω,π,b∗
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:26)(cid:90)
(cid:27)
∇(Kα)T (θ (x, mJ )) dµ (y)
· (x − mJ ) (cid:3)ω,π,b∗
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
∇(Kα)T (mJ , y) dµ (y)
(cid:90)(cid:26)(cid:90) (cid:104)∇(Kα)T (θ (x, mJ ) , y)−∇(Kα)T (mJ , y)
(cid:105)
·(x − mJ ) (cid:3)ω,π,b∗

(cid:3)ω,π,b∗

Ψ (x) dω (x)

+

J

J

J

J

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

J

(cid:3)ω,π,b∗
(cid:3)ω,π,b∗
(cid:27)

J

dµ (y)

(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)

Ψ (x) dω (x)

Ψ (x) dω (x)

≡ I1 + I2

133

Now we estimate

I1 =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:20)(cid:90)
(cid:90) (cid:90)
∇(Kα) (mJ , y) dµ (y)
≤ n
(cid:46) n · CCZ

(cid:21)T ·
(cid:90)
(x − mJ ) (cid:3)ω,π,b∗
(cid:12)(cid:12)(cid:12)(cid:52)ω,π,b∗
(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:3)ω,π,b∗
(cid:13)(cid:13)(cid:13)(cid:52)ω,π,b∗
(cid:13)(cid:13)(cid:13)(cid:3)ω,π,b∗
(cid:13)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)L2(ω)

|∇(Kα) (mJ , y)| d|µ| (y)
Pα (J,|µ|)

Ψ

x

x

J

J

J

J

J

|J| 1

n

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

Ψ (x) dω (x)

(cid:3)ω,π,b∗

J

Ψ (x)

(cid:12)(cid:12)(cid:12) dω (x)

and

I2 (cid:46) CCZ

(cid:46) CCZ

(cid:46) CCZ

n

1+δ (J,|µ|)
Pα
|J| 1
1+δ (J,|µ|)
Pα
1+δ (J,|µ|)
Pα

|J|

|J|

(cid:90)
|x − mJ|(cid:12)(cid:12)(cid:12)(cid:3)ω,π,b∗
(cid:115)(cid:90)

J

|x − mJ|2 dω (x)

J

(cid:107)x − mJ(cid:107)L2(1J ω)

J

Ψ (x)

(cid:3)ω,π,b∗

(cid:12)(cid:12)(cid:12) dω (x)
(cid:13)(cid:13)(cid:13)(cid:3)ω,π,b∗
(cid:13)(cid:13)(cid:13)(cid:3)ω,π,b∗
(cid:13)(cid:13)(cid:13)L2(ω)

(cid:3)ω,π,b∗

Ψ

Ψ

J

J

J

.

(cid:13)(cid:13)(cid:13)L2(ω)

For term II we fix z ∈ J for the moment. Then since

(cid:68)

(cid:69)

(cid:68)

1, (cid:3)ω,b∗

J,brokh

=

ω

1, (cid:3)ω,b∗

J

h − (cid:3)ω,π,b∗

J

h

= 0

ω

(cid:69)

we have

II =

=

T αµ, (cid:3)ω,b∗

(cid:12)(cid:12)(cid:12)(cid:68)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:26)(cid:90)

J,brokΨ
∇(Kα)T (θ (x, z) , y) dµ (y)

ω

(cid:69)
(cid:90)

(cid:12)(cid:12)(cid:12)
(cid:27)
|x − z| ·(cid:12)(cid:12)(cid:12)(cid:3)ω,π,b∗
(cid:88)
J(cid:48)∈Cbrok(J)

(cid:12)(cid:12)(cid:12)dω(x)
(cid:90)
J,brok Ψ (x)
J(cid:48) |x − z| · 1J(cid:48)Eω

≤ CCZ

≤ CCZ

Pα (J,|µ|)

|J| 1

n

Pα (J,|µ|)

|J| 1

n

J(cid:48) |Ψ| dω(x)

· (x − z) (cid:3)ω,π,b∗

J,brok Ψ (x) dω (x)

(cid:12)(cid:12)(cid:12)(cid:12)

134

having used the reverse Hölder control of children (5.1.23) to obtain

(cid:12)(cid:12)(cid:12)(cid:3)ω,b∗

J,brokΨ

(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)

J(cid:48)∈Cbrok(JQ)

(cid:18)
Fω,bJ(cid:48)
J(cid:48) − Fω,bJ
J(cid:48)

(cid:90)
J(cid:48) |x − z| · 1J(cid:48)Eω

J(cid:48) |Ψ| dω(x) =

(cid:90)

and since

we get

1J(cid:48)Eω

J(cid:48) |Ψ| ,

J(cid:48)∈Cbrok(J)

J(cid:48) |Ψ|dω(x)

dω(x)

Ψ

(cid:19)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:88)
1J(cid:48)(cid:82)
|x − z|(cid:112)|J(cid:48)|ω
(cid:112)|J(cid:48)|ω
J(cid:48) |x − z|(cid:17)2(cid:118)(cid:117)(cid:117)(cid:116) (cid:88)

J(cid:48)

II ≤ CCZ

Pα (J,|µ|)

|J| 1

n

(cid:118)(cid:117)(cid:117)(cid:116) (cid:88)

J(cid:48)∈Cbrok(J)

(cid:16)

|J(cid:48)|ω

Eω

(cid:104)

J(cid:48) |Ψ|(cid:105)2

.

Eω

|J(cid:48)|ω

J(cid:48)∈Cbrok(J)

Combining the estimates for terms I and II, we obtain

(cid:12)(cid:12)(cid:12)(cid:68)

T αµ, (cid:3)ω,b∗
J Ψ
Pα (J,|µ|)

(cid:46) CCZ

+ CCZ

+ CCZ

n

|J| 1
1+δ (J,|µ|)
Pα
|J| 1

n

Pα (J,|µ|)

|J| 1

n

ω

(cid:12)(cid:12)(cid:12)
(cid:69)
(cid:13)(cid:13)(cid:13)(cid:52)ω,π,b∗
(cid:13)(cid:13)(cid:13)L2(ω)
(cid:118)(cid:117)(cid:117)(cid:116) (cid:88)

(cid:107)x − mJ(cid:107)L2(1J ω)
|J(cid:48)|ω

x

J

J

inf
z∈J

J(cid:48)∈Cbrok(J)

Ψ

(cid:13)(cid:13)(cid:13)(cid:3)ω,π,b∗
(cid:13)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)(cid:3)ω,π,b∗
(cid:13)(cid:13)(cid:13)L2(ω)
J(cid:48) |x − z2|(cid:17)2(cid:118)(cid:117)(cid:117)(cid:116) (cid:88)
(cid:16)

Eω

Ψ

J

(cid:104)

J(cid:48) |Ψ|(cid:105)2

Eω

|J(cid:48)|ω

J(cid:48)∈Cbrok(J)

and then noting that the infimum over z ∈ R is achieved for z ∈ J, and using the triangle
inequality on (cid:3)ω,π,b∗

we get (5.1.54).

− (cid:3)ω,π,b∗

= (cid:3)ω,b∗

J

J

J,brok

The right hand side of (5.1.54) in the Monotonicity Lemma will be typically estimated

135

in what follows using the frame inequalities for any cube K,

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

J Ψ

x

(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)♠2

L2(ω)

L2(ω)

(cid:88)
(cid:88)

J⊂K

J⊂K

(cid:46) (cid:107)Ψ(cid:107)2

L2(ω)

,

(cid:90)

K

(cid:46)

|x − mK|2 dω (x) ,

together with these inequalities for the square function expressions. To see the last one,
write x = (x1, . . . , xn) and note that for J ⊂ K,
(cid:90)
n(cid:88)
(cid:90)
≤ n(cid:88)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

|xi − mK i|2 dω =||x − mk||2

(cid:12)(cid:12)(cid:12)(cid:52)ω,b∗

(cid:13)(cid:13)(cid:13)♠2

(cid:12)(cid:12)(cid:12)(cid:52)ω,b∗

J

(cid:12)(cid:12)(cid:12)2

(cid:12)(cid:12)(cid:12)2

L2(ω)

x

dω =

J

i=1

xi

dω

(cid:90)

=

x

J

J

J

L2(1K ω)

using the one-variable result from [69].

K

i=1

Lemma 5.1.24. For any cube K we have

(cid:88)

J⊂K

inf
z∈R

(cid:88)
(cid:88)
J(cid:48)∈Cbrok(J)
J(cid:48)∈Cbrok(J)

(cid:104)
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω
(cid:16)
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

J(cid:48) |Ψ| (x)
Eω

(cid:105)2 (cid:46)
J(cid:48) |x − z|(cid:17)2 (cid:46)

Eω

and (cid:88)

J⊂K

(cid:90)
(cid:90)

K

K

|Ψ (x)|2 dω (x) ,

(5.1.55)

|x − mK|2 dω (x) .

Proof. The first inequality in (5.1.55) is just the Carleson embedding theorem since the cubes

(cid:8)J(cid:48) ∈ Cbrok (J) : J ⊂ K(cid:9) satisfy an ω-Carleson condition, and the second inequality in

(5.1.55) follows by choosing z = mK to obtain

(cid:88)
J(cid:48)∈Cbrok(J)

inf
z∈R

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:16)

J(cid:48) |x − z|(cid:17)2 ≤ (cid:88)

Eω

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:16)

J(cid:48) |x − mK|(cid:17)2

Eω

,

J(cid:48)∈Cbrok(J)

136

and then applying the Carleson embedding theorem again:

(cid:88)

J⊂K

(cid:88)
J(cid:48)∈Cbrok(J)

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:16)

J(cid:48) |x − mK|(cid:17)2 (cid:46)

Eω

(cid:90)

|x − mK|2 dω (x) .

K

5.1.13.1 The smaller Poisson integral

The expressions

1+δ (J,|µ|)
Pα

|J|

inf
z∈R

(cid:107)x − z(cid:107)L2(1J ω)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J Ψ

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

are typically easier to sum due to the small Poisson operator Pα

1+δ (J,|µ|). To illlustrate, we
show here one way in which we can exploit the additional decay in the Poisson integral Pα
.
Suppose that J is good in I with (cid:96) (J) = 2−s(cid:96) (I) (see Definition 5.2.5 below for ‘goodness’).
We then compute

1+δ

(cid:16)

(cid:17)

Pα

1+δ

J, 1A\I σ
|J| 1

n

|y − cJ|n+1+δ−α

n

A\I

|J| δ

(cid:90)
 |J| 1
(cid:90)
 |J| 1
δ

A\I

dist (cJ , Ic)

n

n

dist (cJ , Ic)

δ
Pα(cid:16)

≈

≤

(cid:46)

dσ (y)

1

(cid:17)

|y − cJ|n+1−α dσ (y)
J, 1A\I σ
|J| 1

n

,

and use the goodness inequality,

dist (cJ , Ic) ≥ 2(cid:96) (I)1−ε (cid:96) (J)ε ≥ 2 · 2s(1−ε)(cid:96) (J) ,

137

to conclude that

Pα

1+δ

Now we can estimate

(cid:16)

J, 1A\I σ
|J| 1

n

(cid:17)

 (cid:46) 2−sδ(1−ε)

Pα(cid:16)

(cid:17)

J, 1A\I σ
|J| 1

n

(5.1.56)

J⊂K: J good in K

(cid:88)
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) (cid:88)
Pα

J⊂K

J good in K

inf
z∈R

|J| 1

n

1+δ (J, 1Kc |µ|)

|J| 1

n

2

1+δ (J, 1Kc |µ|)
Pα

(cid:107)x − z(cid:107)L2(1J ω)

z∈R(cid:107)x − z(cid:107)2

inf

L2(1J ω)

J Ψ

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:88)

J⊂K

(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

L2(ω)

J good in K

(cid:13)(cid:13)(cid:13)(cid:70)2

J Ψ

L2(ω)

≤

where

(cid:88)

inf

|J|

(cid:33)2

1+δ (J, 1Kc |µ|)

|J|
1+δ (J, 1Kc |µ|)

(cid:32)Pα
z∈R(cid:107)x − z(cid:107)2
(cid:32)Pα
(cid:33)2
2−sδ(1−ε) Pα (J, 1Kcσ)
2 ∞(cid:88)
2

J⊂K: J good in K

(cid:96)(J)=2−s(cid:96)(I)

2

z∈R(cid:107)x − z(cid:107)2

(cid:88)

L2(1K ω)

|J| 1

s=0

inf

n

,

J⊂K: J good in K

(cid:96)(J)=2−s(cid:96)(I)

s=0

J⊂K: J good in K

∞(cid:88)
(cid:88)
∞(cid:88)
(cid:88)
Pα (K, 1Kcσ)
Pα (K, 1Kcσ)

|K| 1

s=0

n

|K| 1

n

J⊂K: J good in K

(cid:96)(J)=2−s(cid:96)(I)

=

≤

≤

(cid:46)

L2(1J ω)

z∈R(cid:107)x − z(cid:107)2

inf

L2(1J ω)

z∈R(cid:107)x − z(cid:107)2

inf

L2(1J ω)

2−2sδ(1−ε) inf

z∈R(cid:107)x − z(cid:107)2

L2(1K ω)

and where we have used (5.5.10), which gives in particular

Pα(J, µ1Ic) (cid:46)

Pα(I, µ1Ic).

(cid:18) (cid:96) (J)

(cid:19)1−ε(n+1−α)

(cid:96) (I)

138

for J ⊂ I and d (J, ∂I) > 2(cid:96) (J)ε (cid:96) (I)1−ε. We will use such arguments repeatedly in the
sequel.

Armed with the Monotonicity Lemma and the lower frame inequality

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

I

(cid:13)(cid:13)(cid:13)(cid:70)2

g

L2(µ)

(cid:88)

I∈D

(cid:46) (cid:107)g(cid:107)2

L2(ω)

,

we can obtain a b∗-analogue of the Energy Lemma as in [64] and/or [63].

5.1.13.2 The Energy Lemma

Suppose now we are given a subset H of the dyadic grid G. Due to the failure of both mar-
tingale and dual martingale pseudoprojections Qω,b∗
H g (see below for definition)
when the children ‘break’, it is

to satisfy inequalities of the form(cid:13)(cid:13)(cid:13)Pω,b∗
convenient to define the ‘square function norms’(cid:13)(cid:13)(cid:13)Qω,b∗

and(cid:13)(cid:13)(cid:13)Pω,b∗

(cid:13)(cid:13)(cid:13)L2(ω)

(cid:13)(cid:13)(cid:13)(cid:70)

H g

H g

of the

L2(ω)

H x and Pω,b∗
(cid:13)(cid:13)(cid:13)♠
(cid:46) (cid:107)g(cid:107)L2(ω)
H x
(cid:88)

L2(ω)

pseudoprojections

Qω,b∗
H x =

(cid:88)

J∈H

(cid:52)ω,b∗

J

x and Pω,b∗

H g =

(cid:3)ω,b∗

J

g ,

J∈H

by

H x

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)Pω,b∗

H g

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

L2(ω)

J∈H

≡ (cid:88)
(cid:88)
≡ (cid:88)
(cid:88)

=

J∈H

J∈H

=

J∈H

L2(ω)

L2(ω)

J

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

J

J

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)2

x

x

g

g

L2(ω)

L2(ω)

(cid:88)

J∈H

(cid:88)

J∈H

+

+

Eω

(cid:16)

inf
z∈R

(cid:88)
J(cid:48)∈Cbrok(J)
(cid:88)
J(cid:48)∈Cbrok(J)

J(cid:48) |x − z|(cid:17)2
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω
(cid:104)
J(cid:48) |g|(cid:105)2
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

Eω

139

for any subset H ⊂ G. The average Eω
i.e. Eω

(cid:82) |x − z| dω (x), and it is important that the infimum infz∈R is taken

J |x − z| above is taken with respect to the variable x,

inside the sum(cid:80)
J |x − z| = 1|J|ω
J∈H.

Note that we are defining here square function expressions related to pseudoprojections,

cause confusion, and it provides a useful way of bookkeeping the sums of squares of norms of

H x and Pω,b∗

H g, but also on the particular
g. This slight abuse of notation should not

, along with

J

J

J∈H (cid:52)ω,b∗

x and(cid:80)

which depend not only on the functions Qω,b∗
representations(cid:80)
J∈H (cid:3)ω,b∗
martingale and dual martingale differences(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:88)
(cid:88)

(cid:13)(cid:13)∇ω
J (x − z)(cid:13)(cid:13)2
(cid:88)
(cid:13)(cid:13)∇ω
J Ψ(cid:13)(cid:13)2

(cid:88)

L2(ω) =

L2(ω) =

inf
z∈R

inf
z∈R

J∈H

J∈H

J

J∈H

J∈H

J

g

x

L2(ω)

L2(ω)

(cid:13)(cid:13)(cid:13)2
(cid:88)
(cid:88)
J(cid:48)∈Cbrok(J)
J(cid:48)∈Cbrok(J)

and(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)2
J(cid:48) |x − z|(cid:17)2
(cid:16)
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω
(cid:104)
J(cid:48) |Ψ|(cid:105)2
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

Eω

Eω

.

the norms of the associated Carleson square function expressions

Note also that the upper weak Riesz inequalities yield the inequalities

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

J

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

x

g

L2(ω)

L2(ω)

≤(cid:13)(cid:13)(cid:13)Qω,b∗
≤(cid:13)(cid:13)(cid:13)Pω,b∗

H g

H x

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

L2(ω)

J∈H

L2(ω)

H g

H x

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:46) (cid:88)
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:46) (cid:88)
(cid:13)(cid:13)(cid:13)♠2
We will exclusively use(cid:13)(cid:13)(cid:13)Qω,b∗
and(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)Pσ,b∗
H g
H x = Qω,b∗

Finally, note that Qω,b∗

(cid:13)(cid:13)(cid:13)(cid:70)2

H x

H f

L2(ω)

L2(σ)

J∈H

in connection with energy terms, and use

L2(ω)
in connection with functions f ∈ L2 (σ) and g ∈ L2 (ω).

L2(ω)
H (x − m) for any constant m.

140

Recall that

Φα (J, ν) ≡ Pα (J, ν)

|J| 1

n

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

x

(cid:13)(cid:13)(cid:13)♠

L2(ω)

+

Pα
1+δ (J, ν)
|J| 1

n

(cid:107)x − mJ(cid:107)L2(1J ω) .

Lemma 5.1.25 (Energy Lemma). Let J be a cube in G. Let ΨJ be an L2 (ω) function
supported in J with vanishing ω-mean, and let H ⊂ G be such that J(cid:48) ⊂ J for every J(cid:48) ∈ H.
Let ν be a positive measure supported in R\γJ with γ > 1, and for each J(cid:48) ∈ H, let

dνJ(cid:48) = ϕJ(cid:48)dν with(cid:12)(cid:12)ϕJ(cid:48)(cid:12)(cid:12) ≤ 1. Suppose that b∗ is an ∞-weakly µ-controlled accretive family
we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

on Rn. Let T α be a standard α-fractional singular integral operator with 0 ≤ α < 1. Then

J(cid:48) ΨJ

L2(µ)

J(cid:48)∈H

ω

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) Cγ
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J(cid:48) ΨJ

Φα(cid:0)J(cid:48), ν(cid:1)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:88)
(cid:13)(cid:13)(cid:13)(cid:70)2

L2(µ)

J(cid:48)∈H

J(cid:48) ΨJ

(cid:68)
(cid:69)
(cid:88)
T α(cid:0)νJ(cid:48)(cid:1) , (cid:3)ω,b∗
(cid:118)(cid:117)(cid:117)(cid:116)(cid:88)
(cid:115)(cid:88)
Pα (J, ν)
(cid:13)(cid:13)(cid:13)♠

(cid:13)(cid:13)(cid:13)Qω,b∗

Φα (J(cid:48), ν)2

J(cid:48)∈H

H x

J(cid:48)∈H

L2(ω)

|J|

+

(cid:46) Cγ

(cid:46)

(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)Pω,b∗

H ΨJ

(cid:13)(cid:13)(cid:13)(cid:70)

L2(µ)

(cid:70)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

L2(µ)

Pα
1+δ (J, ν)
|J| 1

n

(cid:107)x − mJ(cid:107)L2(1J ω)

and in particular the ‘energy’ estimate

|(cid:104)T αϕν, ΨJ(cid:105)ω|

Pα (J, ν)

(cid:13)(cid:13)(cid:13)Qω,b∗

J

|J| 1

n

≤Cγ

(cid:13)(cid:13)(cid:13)♠

+
L2(ω)

x

Pα
1+δ (J, ν)
|J| 1

n

(cid:107)x − mJ(cid:107)L2(1J ω)

(cid:3)ω,b∗
J(cid:48) ΨJ

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

(cid:88)

J(cid:48)⊂J

141

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

(cid:88)

J(cid:48)⊂J

where

(cid:3)ω,b∗
J(cid:48) ΨJ

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

(cid:70)

L2(µ)

(cid:46) (cid:107)ΨJ(cid:107)L2(µ), and the ‘pivotal’ bound

|(cid:104)T α (ϕν) , ΨJ(cid:105)ω| (cid:46) CγPα (J,|ν|)

(cid:113)|J|ω (cid:107)ΨJ(cid:107)L2(ω) ,

for any function ϕ with |ϕ| ≤ 1.

Proof. Using the Monotonicity Lemma 5.1.23, followed by(cid:12)(cid:12)νJ(cid:48)(cid:12)(cid:12) ≤ ν, the Poisson equivalence

Pα(cid:0)J(cid:48), ν(cid:1)

|J(cid:48)| 1

n

≈ Pα (J, ν)

|J| 1

n

J(cid:48) ⊂ J ⊂ γJ,

suppν ∩ γJ = ∅,

,

(5.1.57)

and the weak frame inequalities for dual martingale differences, we have

(cid:46)

ω

J(cid:48)∈H

J(cid:48)∈H

J(cid:48) ΨJ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:88)
Φα(cid:0)J(cid:48),|µ|(cid:1)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:68)
(cid:69)
(cid:88)
T α(cid:0)νJ(cid:48)(cid:1) , (cid:3)ω,b∗
 1
(cid:88)
2(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
2(cid:88)
Pα(cid:0)J(cid:48), ν(cid:1)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)♠2
 1
(cid:88)
J(cid:48)∈H
2(cid:13)(cid:13)x − mJ(cid:48)(cid:13)(cid:13)2
2(cid:88)
Pα
(cid:0)J(cid:48),|µ|(cid:1)
(cid:17)
L2(cid:16)
(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)Qω,b∗

(cid:107)ΨJ(cid:107)L2(ω)+

1J(cid:48) ω
Pα
1+δ(cid:48) (J, ν)
|J| 1

J(cid:48) x

|J(cid:48)| 1

|J(cid:48)| 1

J(cid:48)∈H

J(cid:48)∈H

H x

1

γδ(cid:48)

L2(ω)

L2(ω)

+

n

1+δ

n

(cid:46)Pα (J, ν)

|J| 1

n

L2(µ)

(cid:13)(cid:13)(cid:13)(cid:70)
 1
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J(cid:48) ΨJ

L2(ω)

2

J(cid:48) ΨJ

J(cid:48) ΨJ

J(cid:48)∈H

 1

2

(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

(cid:107)x − mJ(cid:107)L2(1J ω)(cid:107)ΨJ(cid:107)L2(ω) .

n

142

(5.1.58)

n + |y − cJ| and

The last inequality follows from the following calculation using Haar projections (cid:52)ω

:

K

n

n

=

=

1+δ

1+δ

1J(cid:48) ω

J(cid:48)∈H

J(cid:48)∈H

|J(cid:48)| 1

|J(cid:48)| 1

J(cid:48)(cid:48)⊂J(cid:48)

(cid:88)

Pα

2(cid:13)(cid:13)x − mJ(cid:48)(cid:13)(cid:13)2
(cid:0)J(cid:48), ν(cid:1)
L2(cid:16)
(cid:17)
2 (cid:88)
Pα
(cid:0)J(cid:48), ν(cid:1)
(cid:13)(cid:13)(cid:13)(cid:52)ω
(cid:88)

Pα
(cid:0)J(cid:48), ν(cid:1)
(cid:88)
(cid:88)
2(cid:13)(cid:13)(cid:13)(cid:52)ω
Pα
1+δ(cid:48)(cid:0)J(cid:48)(cid:48), ν(cid:1)
(cid:88)
2 (cid:88)
Pα
(cid:13)(cid:13)(cid:13)(cid:52)ω
which in turn follows from (recalling δ = 2δ(cid:48) and (cid:12)(cid:12)J(cid:48)(cid:12)(cid:12) 1
|J|+|y−cJ| ≤ 1

1+δ(cid:48) (J, ν)
|J| 1

for y ∈ Rn\γJ)

≤ 1
γ2δ(cid:48)

(cid:46) 1
γ2δ(cid:48)

J(cid:48): J(cid:48)(cid:48)⊂J(cid:48)⊂J

|J(cid:48)(cid:48)| 1

J(cid:48)(cid:48)⊂J

J(cid:48)(cid:48)⊂J

J(cid:48)(cid:48)⊂J

|J(cid:48)| 1

|J|

1+δ

n

n

n

γ

J(cid:48)(cid:48)x

J(cid:48)(cid:48)x

L2(ω)

L2(ω)

J(cid:48)(cid:48)x

(cid:13)(cid:13)(cid:13)2

(cid:13)(cid:13)(cid:13)(cid:52)ω

(cid:13)(cid:13)(cid:13)2
2
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2
n +(cid:12)(cid:12)y − cJ(cid:48)(cid:12)(cid:12) ≈ |J| 1

L2(ω)

L2(ω)

,

J(cid:48)(cid:48)x

2
(cid:18)

=

J(cid:48): J(cid:48)(cid:48)⊂J(cid:48)⊂J

J(cid:48): J(cid:48)(cid:48)⊂J(cid:48)⊂J

(cid:88)
(cid:88)
(cid:46) (cid:88)

J(cid:48): J(cid:48)(cid:48)⊂J(cid:48)⊂J

1

γ2δ(cid:48)

 (cid:88)

n

n

1+δ

|J(cid:48)| 1

Pα
(cid:0)J(cid:48), ν(cid:1)
(cid:32)(cid:90)
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12) 2δ
(cid:32)(cid:90)
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12) 2δ
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12) 2δ

|J| 2δ

n

n

n

=

1

γ2δ(cid:48)

J(cid:48): J(cid:48)(cid:48)⊂J(cid:48)⊂J

|J| 2δ

n

Rn\γJ

|J(cid:48)| 1

1

n +(cid:12)(cid:12)y − cJ(cid:48)(cid:12)(cid:12)(cid:19)n+1+δ−α
(cid:19)n+1+δ(cid:48)−α
Pα

|J| δ(cid:48)
n + |y − cJ|

2

n

(cid:46) 1
γ2δ(cid:48)

1+δ(cid:48) (J, ν)
|J| 1

n

|J| 1

Rn\γJ

(cid:18)
Pα

dν (y)

dν (y)

1+δ(cid:48) (J, ν)
|J| 1

n

.

(cid:33)2
(cid:33)2
2

143

Finally we obtain the ‘energy’ estimate from the equality

(cid:88)

J(cid:48)⊂J

ΨJ =

(cid:3)ω,b∗
J(cid:48) ΨJ ,

(since ΨJ has vanishing ω-mean),

and we obtain the ‘pivotal’ bound from the inequality

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J(cid:48)(cid:48) x

(cid:13)(cid:13)(cid:13)♠2

(cid:88)

J(cid:48)(cid:48)⊂J

(cid:46) (cid:107)(x − mJ )(cid:107)2

L2(1J ω)

≤ |J|2 |J|ω .

L2(ω)

5.1.14 Organization of the proof

We adapt the proof of the main theorem in [66], but beginning instead with the decomposition

of Hytönen and Martikainen [24], to obtain the norm inequality

(cid:113)

NT α (cid:46) Tb

T α + Tb∗

T α +

2 + Eα
Aα
2

under the apriori assumption NT α < ∞, which is achieved by considering one of the trun-
defined in (5.1.3) above. This will be carried out in the next four sections of
cations T α

σ,δ,R

this paper. In the next section we consider the various form splittings and reduce matters to

the disjoint form, the nearby form and the main below form. Then these latter three forms

are taken up in the subsequent three sections, using material from the appendices.

A major source of difficulty will arise in the infusion of goodness for the cubes J into
the below form where the sum is taken over all pairs (I, J) such that (cid:96) (J) ≤ (cid:96) (I). We
will infuse goodness in a weak way pioneered by Hytönen and Martikainen in a one weight

setting. This weak form of goodness is then exploited in all subsequent constructions by

144

typically replacing J by J

(cid:122) in defining relations, where J

(cid:122) is the smallest cube K for which

J is good w.r.t. K and beyond.

Another source of difficulty arises in the treatment of the nearby form in the setting of

two weights. The one weight proofs in [24] and [29] relied strongly on a property peculiar

to the one weight setting - namely the fact already pointed out in Remark 5.1.6 above
that both of the Poisson integrals are bounded, namely Pα (Q, µ) (cid:46) 1 and Pα (Q, µ) (cid:46) 1.
We will circumvent this difficulty by combining a recursive energy argument with the full

testing conditions assumed for the original testing functions borig

Q

, before these conditions

were suppressed by corona constructions that delivered only weak testing conditions for the

new testing functions bQ.

Of particular importance will be a result proved in Appendices A that follow from known

work with some new twists. We show that the functional energy for an arbitrary pair of

grids is controlled by the Muckenhoupt and energy side conditions. The somewhat lengthy

proof of this latter assertion is similar to the corresponding proof in the T 1 setting - see e.g.

[66] - but requires a different decomposition of the stopping cubes into ‘Whitney cubes’ in

order to accomodate the weaker notion of goodness used here.

5.2 Form splittings

Notation 5.2.1. Fix grids D and G. We will use D to denote the grid associated with
f ∈ L2 (σ), and we will use G to denote the grid associated with g ∈ L2 (ω).

Now we turn to the probability estimates for martingale differences and halos that we
for all 1 ≤ i ≤ n, the λ-halo of J is

will use. Recall that given −→

λ = (λ1, ..., λn), 0 < λi < 1
2

145

defined to be

J ≡(cid:16)

1 +

(cid:17)

J\(cid:16)
1 − −→

λ

(cid:17)

−→
λ

J.

∂−→
λ

Suppose µ is a positive locally finite Borel measure, and that b is a p-weakly µ-controlled

accretive family for some p > 2. Then the following probability estimate holds.

g ≡(cid:80)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2

L2(µ)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Ψµ,b∗

GD
k−bad

g

ED

Ω

Bad cube probability estimates. Suppose that D and G are independent random dyadic
grids. With Ψµ,b∗
g equal to the pseudoprojection of g onto k-bad
GD
k−bad
G-cubes, we have

(cid:3)µ,b∗

J∈GD

k−bad

J

 (cid:46) ED

Ω

 (cid:88)

J∈GD
≤ Ce−kε (cid:107)g(cid:107)2

k−bad

L2(µ)

(cid:20)(cid:13)(cid:13)(cid:13)(cid:3)µ,b∗

J,G g

(cid:13)(cid:13)(cid:13)2

L2(µ)

(cid:13)(cid:13)(cid:13)∇µ

J,Gg

+

(cid:13)(cid:13)(cid:13)2

L2(µ)

(cid:21)

,

(5.2.1)

where the first inequality is the ‘weak upper half Riesz’ inequality from Appendix A of
[69] for the pseudoprojection Ψµ,b∗
GD
k−bad
inequality in (5.2.10) below.

, and the second inequality is proved using the frame

Halo probability estimates. Suppose that D and G are independent random grids.
Using the parameterization by translations of grids and taking the average over certain
translates τ + D of the grid D we have

(cid:88)
(cid:88)

ED

Ω

EG

Ω

I(cid:48)∈D: (cid:96)(I(cid:48))≈(cid:96)(J(cid:48))

J(cid:48)∈G: (cid:96)(J(cid:48))≈(cid:96)(I(cid:48))

(cid:90)
J(cid:48)∩∂δI(cid:48) dω (cid:46) δ
(cid:90)
I(cid:48)∩∂δJ(cid:48) dσ (cid:46) δ

(cid:90)
(cid:90)
J(cid:48) dω,
I(cid:48) dσ,

J(cid:48) ∈ C (J) , J ∈ G,

(5.2.2)

I(cid:48) ∈ C (I) , I ∈ D,

146

Ω

Ω

and EG

and where the expectations ED
are taken over grids D and G respectively. Indeed,
it is geometrically evident that for any fixed pair of side lengths (cid:96)1 ≈ (cid:96)2, the average of the

of the set J(cid:48) ∩ ∂δI(cid:48), as a cube I(cid:48) ∈ D with side length (cid:96)(cid:0)I(cid:48)(cid:1) = (cid:96)1 is

measure (cid:12)(cid:12)J(cid:48) ∩ ∂δI(cid:48)(cid:12)(cid:12)ω
translated across a cube J(cid:48) ∈ G of side length (cid:96)(cid:0)J(cid:48)(cid:1) = (cid:96)2, is at most C(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω
(cid:9)
In the σ-iterated corona construction we redefined the family b =(cid:8)bQ

observation it is now easy to see that (5.2.2) holds.

. Using this

Q

are given in terms of the original functions borig

, and of course we then dropped the superscript new. We continue to refer to the

new functions bnew
Q ∈ Cσ
triple stopping cubes A as ‘breaking’ cubes even if bA happens to equal 1AbπA. The results
of Appendix A of [69] apply with this more inclusive definition of ‘breaking’ cubes, and

by bnew

Q

A

A

Q∈D so that the
for
Q = 1Qborig

the associated definition of ‘broken’ children, since only the Carleson condition on stopping

cubes is relevant here.

This and Proposition 5.1.19 give us the triple corona decomposition of f = (cid:80)
where the pseudoprojection PσCA

is defined as:

A∈A

f,

PσCA

(cid:88)

I∈CA

PσCA

f =

(cid:3)µ,b
I f

We now record the main facts proved above for the triple corona.

Lemma 5.2.2. Let f ∈ L2(σ). We have

f =

(cid:88)

A∈A

PσCA

f

both in the sense of norm convergence in L2 (σ) and pointwise σ-a.e. The corona tops A and
stopping bounds {αA (A)}A∈A satisfy properties (1), (2), (3) and (4) in Definition 5.1.18,

147

hence constitute stopping data for f. Moreover, b = {bI}I∈D is a ∞-weakly σ-controlled
accretive family on D with corona tops A ⊂ D, where bI = 1I bA for all I ∈ CA, and the
weak corona forward testing condition holds uniformly in coronas, i.e.

(cid:90)

I

1
|I|σ

σ bA|2 dσ ≤ C,
|T α

I ∈ Cσ
A .

Similar statements hold for g ∈ L2(ω).
We have defined corona decompositions of f and g in the σ-iterated triple corona con-

struction above, but in order to start these corona decompositions for f and g respectively
within the dyadic grids D and G, we need to first restrict f and g to be supported in a
large common cube Q∞. Then we cover Q∞ with 2n pairwise disjoint cubes I∞ ∈ D with
(cid:96) (I∞) = (cid:96) (Q∞), and similarly cover Q∞ with 2n pairwise disjoint cubes J∞ ∈ G with
(cid:96) (J∞) = (cid:96) (Q∞). We can now use the broken martingale decompositions, together with
random surgery, to reduce matters to consideration of the four forms

(cid:88)

(cid:88)

I∈D: I⊂I∞

J∈G: J⊂J∞

(cid:90) (cid:16)

(cid:17) (cid:3)ω,b∗

J

gdω,

σ (cid:3)σ,b
T α
I f

with I∞ and J∞ as above, and where we can then use the cubes I∞ and J∞ as the starting
cubes in our corona constructions below. Indeed, the identities in [24, Lemma 3.5]), give

(cid:88)
(cid:88)

I∈D: I⊂I∞, (cid:96)(I)≥2−N

J∈G: J⊂J∞, (cid:96)(J)≥2−N

(cid:3)σ,b
I f + Fσ,b
I∞f,
g + Fω,b∗
(cid:3)ω,b∗
J∞ g,

J

f =

g =

148

(cid:88)

which can then be used to write the bilinear form(cid:82) (Tσf ) gdω as a sum of the forms
(cid:17) Fω,b∗
 (5.2.3)

(cid:17) (cid:3)ω,b∗
(cid:17) (cid:3)ω,b∗

(cid:88)
(cid:88)

(cid:90) (cid:16)
(cid:90) (cid:16)

(cid:88)
(cid:90) (cid:16)

(cid:17) Fω,b∗

2n+1pairs
(I∞,J∞)

σ (cid:3)σ,b
T α
I f

σ (cid:3)σ,b
T α
I f

σ Fσ,b
T α
I∞f

σ Fσ,b
T α
I∞f

(cid:90) (cid:16)



(cid:88)

I∈D
I⊂I∞

J∈G
J⊂J∞

I∈D
I⊂I∞

J∞ gdω

J∞ gdω

J

J

gdω +

gdω +

+

J∈G: J⊂J∞

taken over the 2n+1 pairs of cubes (I∞, J∞) above. The second, third and fourth sums in
(5.2.3) can be controlled using testing and random surgery. For example, for the second sum

we have

(cid:90)(cid:16)

σ (cid:3)σ,b
T α
I f

(cid:90)

I∞∩J∞

J∞ gdω

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
 (cid:88)
(cid:17) Fω,b∗
 (cid:88)
 T α,∗
(cid:16)Fω,b∗
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
 T α,∗
 (cid:88)
(cid:16)Fω,b∗
(cid:17)

I∈D: I⊂I∞

I∈D: I⊂I∞

(cid:3)σ,b
I f

(cid:3)σ,b
I f

J∞ g

dσ

ω

ω

J∞ g

I∈D: I⊂I∞

(cid:3)σ,b
I f

(cid:17)

dσ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

I∞∩((1+δ)J∞\J∞)

I∈D: I⊂I∞

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:90)

+

+

I∞\(1+δ)J∞
≡ A1 + A2 + A3

 T α,∗

ω

(cid:17)

(cid:16)Fω,b∗

J∞ g

dσ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

So we are left with bounding A1, A2, A3. We have

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)

A1 ≤

(cid:3)σ,b
I f

dσ

I∞

I∈D: I⊂I∞

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

 1
2(cid:18)(cid:90)

J∞

(cid:12)(cid:12)(cid:12)T α,∗

ω

(cid:16)Fω,b∗

J∞ g

(cid:17)(cid:12)(cid:12)(cid:12)2

(cid:19) 1

2

dσ

149

and since Fω,b∗

J∞ g = b∗
J∞
(cid:88)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

Eω
J∞ g
J∞ b∗
J∞

Eω

is b∗

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(σ)

A1 ≤

(cid:3)σ,b
I f

I∈D: I⊂I∞
Tα,∗ (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)

(cid:46) Tb*

J∞ times an ‘accretive’ average of g on J∞, we get
(cid:18)(cid:90)

(cid:19)1

ω (1J∞b∗

J∞)

dσ

2 |Eω

J∞g| ·

(cid:12)(cid:12)(cid:12)T α,∗

(cid:12)(cid:12)(cid:12)2

1

cb∗|J∞|ω

J∞

where in the last inequality we used the frame estimates (5.1.51) and the dual testing con-
dition on b∗

J∞.

For A2 we use expectation on the grid G.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

I∈D: I⊂I∞

(cid:3)σ,b
I f

dσ

≤ EG

I∞∩[(1+δ)J∞\J∞]

I∞∩[(1+δ)J∞\J∞]

EGA2 ≤ EG(cid:90)
(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:88)
EG(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:88)
Cδ
 1
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
2(cid:18)
(cid:90)
≤(cid:112)CδNT α (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)

I∞∩[(1+δ)J∞\J∞]

I∈D: I⊂I∞

I∈D: I⊂I∞

I∈D: I⊂I∞

(cid:88)

(cid:3)σ,b
I f

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

I∞

≤

≤

dσ

(cid:3)σ,b
I f

NT α

(cid:17)(cid:12)(cid:12)(cid:12) dσ
(cid:19)1
(cid:17)(cid:12)(cid:12)(cid:12)2
(cid:19) 1

dσ

2

2

J∞ g

ω

(cid:3)σ,b
I f

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)T α,∗
(cid:16)Fω,b∗
1
2(cid:18)(cid:90) (cid:12)(cid:12)(cid:12)T α,∗
(cid:16)Fω,b∗
 1
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
2(cid:18)
(cid:90)
(cid:19) 1
(cid:90)

NT α

dσ

ω

J∞ g

2

|g|2 dω

|g|2 dω

Finally for A3 we use lemma 5.4.3 since dist(I∞\(1 + δ)J∞, J∞) ≈ δ(cid:96)(J∞) to get

A3 (cid:46)(cid:113)

2 δα−n (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .
Aα

150

Altogether we get

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)

(cid:90) (cid:16)

I∈D
I⊂I∞

EG

Ω

(cid:17) Fω,b∗

J∞ gdω

σ (cid:3)σ,b
T α
I f

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:46)(cid:16)

Tb
T α +

(cid:113)

(cid:17)(cid:107)f(cid:107)L2(σ)(cid:107)g(cid:107)L2(ω)

2 δα−n+δNT α
Aα

Similarly we deal with the third and fourth sum of (5.2.3). We are left to deal with the first

sum in (5.2.3).

5.2.1 The Hytönen-Martikainen decomposition and weak goodness

Now we turn to the various splittings of forms, beginning with the two weight analogue
of the decomposition of Hytönen and Martikainen [24]. Let b (respectively b∗) be a ∞-
weakly σ-controlled (respectively ω-controlled) accretive family. Fix the stopping data A
and {αA (A)}A∈A and dual martingale differences (cid:3)σ,b
constructed above with the triple
iterated coronas, as well as the corresponding data for g. We are left with the estimation of

I

the bilinear form(cid:82) (Tσf ) gdω to that of the sum

(cid:90) (cid:16)

(cid:88)

(cid:88)

I∈D

J∈G

σ (cid:3)σ,b
T α
I f

gdω,

J

(cid:17) (cid:3)ω,b∗


(cid:90) (cid:16)

151

We split the form (cid:104)T α

σ f, g(cid:105)ω

(cid:90)

(Tσf ) gdω =



(cid:88)

(cid:88)

+

I∈D: J∈G
(cid:96)(J)≤(cid:96)(I)

I∈D: J∈G
(cid:96)(J)>(cid:96)(I)

≡ Θ(f, g) + Θ∗(f, g)

into the sum of two essentially symmetric forms by cube size,

(cid:17) (cid:3)ω,b∗

J

σ (cid:3)σ,b
T α
I f

gdω,

(5.2.4)

and focus on the first sum,

Θ (f, g) =

(cid:88)

(cid:68)

I∈D and J∈G: (cid:96)(J)≤(cid:96)(I)

σ (cid:3)σ,b
T α

I f, (cid:3)ω,b∗

J

(cid:69)

,

ω

since the second sum is handled dually, but is easier due to the missing diagonal. Before

introducing goodness into the sum, we follow [24] and split the form Θ (f, g) into 3 pieces:

(cid:88)

(cid:88)

J∈G: (cid:96)(J)≤(cid:96)(I)

+
+
J∈G: (cid:96)(J)≤2−r(cid:96)(I)
J∈G: 2−r(cid:96)(I)<(cid:96)(J)≤(cid:96)(I)
d(J,I)>2(cid:96)(J)ε(cid:96)(I)1−ε
d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε
d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε

≡ Θ1 (f, g) + Θ2 (f, g) + Θ3 (f, g) ,



(cid:88)

I∈D

(cid:88)

(cid:69)
I f, (cid:3)ω,b∗

J

ω

σ (cid:3)σ,b
T α

(cid:68)



where ε > 0 will be chosen to satisfy 0 < ε <

later. Now the disjoint form Θ1 (f, g)
can be handled by ‘long-range’ and ‘short-range’ arguments which we give in a section below,

n+1−α

1

and the nearby form Θ3 (f, g) will be handled using surgery methods and a new recursive
argument involving energy conditions and the ‘original’ testing functions discarded in the

corona construction. The remaining form Θ2(f, g) will be treated further in this section after
introducing weak goodness.

5.2.1.1 Good cubes with ‘body’

. We begin with the weaker extension of goodness introduced in [24], except that we will

make it a bit stronger by replacing the skeleton ‘skelK’ of a cube K, as used in [24], by a

larger collection of points ‘bodyK’, which we call the dyadic body of K. This modification

will prove useful in establishing the Straddling Lemma in the treatment of the stopping form
in Section 5.6 below. Let P denote the collection of all cubes in Rn. The content of the

152

next four definitions is inspired by, or sometimes identical with, that already appearing in

the work of Nazarov, Treil and Volberg in [41] and [43].

Definition 5.2.3. Given a dyadic cube K ∈ Rn, we define W (K) to be the Whitney cubes
in K. Namely, S ∈ W (K) if:

• 3S ⊂ K.

• S(cid:48) ∩ S (cid:54)= ∅ and 3S(cid:48) ⊂ K imply S(cid:48) ⊂ S.

Definition 5.2.4. We define the dyadic body ‘bodyK’ of a dyadic cube K ∈ Rn by

(cid:91)

∂S

S∈W (K)

bodyK =

where ∂S is the boundary of S.

Definition 5.2.5. Let 0 < 
 < 1. For dyadic cubes J, K ∈ Rn with (cid:96)(J) ≤ (cid:96)(K) we define
J to be 
−good in K if

dist(J, bodyK) > 2(cid:96)(J)
(cid:96)(K)1−


(5.2.5)

and we say it is 
−bad in K if (5.2.5) fails.

Definition 5.2.6. Let D and G be two dyadic grids in Rn. Define GD
(k,
)−good to consist
of those cubes J ∈ G such that J is 
−good inside every cube K ∈ D with K ∩ J (cid:54)= ∅ and
(cid:96)(K) ≥ 2k(cid:96)(J).

5.2.1.2 Grid probability

As pointed out on page 14 of [24] by Hytönen and Martikainen, there are subtle difficulties

associated in using dual martingale decompositions of functions which depend on the entire

153

dyadic grid, rather than on just the local cube in the grid. We will proceed at first in

the spirit of [24], and the goodness that we will infuse below into the main ‘below’ form

B(cid:98)r (f, g) will be the Hytönen-Martikainen ‘weak’ version of NTV goodness, but using the
body ‘bodyI’ of a cube rather than its skeleton ‘skelI’: every pair (I, J) ∈ D × G that arises
in the form B(cid:98)r (f, g) will satisfy J ∈ GD

where (cid:96) (I) = 2k(cid:96) (J).

(k,ε)−good

Now we return to the martingale differences (cid:3)σ,b

with controlled families b
and b∗ in Rn. When we want to emphasize that the grid in use is D or G, we will denote
the martingale difference by (cid:3)σ,b
J,G . Recall Definition 5.2.5 for the

I,D, and similarly for (cid:3)ω,b∗

J

I

and (cid:3)ω,b∗

meaning of when an cube J is ε-bad with respect to another cube K.

Definition 5.2.7. We say that J ∈ P is k-bad in a grid D if there is a cube K ∈ D

with (cid:96) (K) = 2k(cid:96) (J) such that J is ε-bad with respect to K (context should eliminate any
ambiguity between the different use of k-bad when k ∈ N and ε-bad when 0 < ε < 1
2).

Following [69] we know that in one dimension for an interval J and grids D0

(cid:90)

Ω

D0
Ω (D0 : J is k-bad in D0) ≡

P

Thus we conclude:

1{D0: J is k-bad in D0}dµΩ (D0) ≤ Cεk2−εk.

(5.2.6)

D0
Ω (D0 : J is k-good in D0) ≥ 1 − Cεk2−εk.

P

(5.2.7)

Now for a cube J to be good in our n-dimensional setting, it needs to be good in each side.

So, we conclude that

P D
Ω (D : J is k-good in D) ≥ (1 − Cεk2−εk)n.

(5.2.8)

154

and therefore a cube is bad with probability bounded by:

P D
Ω (D : J is k-bad in D) ≤ 1 − (1 − Cεk2−εk)n.

(5.2.9)

Then we obtain from (5.2.9), using the lower frame inequality, the expectation estimate

(cid:21)

dµΩ (D)

L2(ω)

J,Gg

J,G g

+

L2(ω)

k−bad

(cid:90)
(cid:88)

Ω

J∈GD

(cid:20)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)2

(cid:88)
(cid:20)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

(cid:13)(cid:13)(cid:13)(cid:53)ω
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2
(cid:21)(cid:90)
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:53)ω
(cid:20)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)2
(cid:88)
L2(ω)
≤ (1 − (1 − Cεk2−εk)n)
J∈G
≤ (1 − (1 − Cεk2−εk)n)(cid:107)g(cid:107)2

J,G g

J,G g

L2(ω)

Ω

+

J,Gg

J∈G

=

L2(ω)

,

1{D: J is k-bad in D}dµΩ (D)

(cid:13)(cid:13)(cid:13)(cid:53)ω

(cid:13)(cid:13)(cid:13)2

+

L2(ω)

J,Gg

L2(ω)

(cid:21)

J,G denotes the ‘broken’ Carleson averaging operator in (5.1.39) that depends on

where (cid:53)ω
the broken children in the grid G. Altogether then it follows easily that

 (cid:88)
J∈(cid:83)∞

ED

Ω

(cid:96)=k GD

(cid:96)−bad

(cid:20)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J,G g

(cid:13)(cid:13)(cid:13)2

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:53)ω

J,Gg

+

(cid:13)(cid:13)(cid:13)2

L2(ω)

(cid:21)≤ (1−(1−Cεk2−εk)n)(cid:107)g(cid:107)2

L2(ω)

(5.2.10)

for some large positive constant C.

From such inequalities summed for k ≥ r, it can be concluded as in [43] that there is an
so that the following holds. Let T : L2(σ) →
L2(ω) be a bounded linear operator. We then have the following traditional inequality for

absolute choice of r depending on 0 < ε < 1
2

155

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:42) (cid:88)

I,J∈DG

r−good

(cid:16)(cid:3)σ,b

I,Df

(cid:17)

T

f, (cid:3)ω,b∗
J,D g

(cid:43)

ω

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.2.11)

two random grids in the case that b is an ∞-weakly µ-controlled accretive family:

(cid:107)T(cid:107) ≤ 2

(cid:107)f(cid:107)

sup
L2(σ)

(cid:107)g(cid:107)

=1

sup
L2(ω)

=1

EΩEΩ(cid:48)

However, this traditional method of introducing goodness is flawed here in the general

setting of dual martingale differences, since these differences are no longer orthogonal pro-

jections, and as emphasized in [24], we cannot simply add back in bad cubes whenever we

want telescoping identities to hold - but these are needed in order to control the right hand

side of (5.2.11). In fact, in the analysis of the form Θ (f, g) above, it is necessary to have

goodness for the cubes J and telescoping for the cubes I. On the other hand, in the analysis
of the form Θ∗ (f, g) above, it is necessary to have just the opposite - namely goodness for
the cubes I and telescoping for the cubes J.

Thus, because in this unfortunate set of circumstances we can no longer ‘add back in’

bad cubes to achieve telescoping, we are prevented from introducing goodness in the full

sum (5.2.4) over all I and J, prior to splitting according to side lengths of I and J. Thus

the infusion of goodness must come after the splitting by side length, but one must work

much harder to introduce goodness directly into the form Θ (f, g) after we have restricted

the sum to cubes J that have smaller side length than I. This is accomplished in the

next subsubsection using the weaker form of NTV goodness introduced by Hytönen and

Martikainen in [24] (that permits certain additional pairs (I, J) in the good forms where
(cid:96) (J) ≤ 2−r(cid:96) (I) and yet J is bad in the traditional sense), and that will prevail later in
the treatment of the far below forms T1
f arbelow (f, g), and of the local forms BA(cid:98)r (f, g) (see
Subsection 5.7) where the need for using the ‘body’ of a cube will become apparent in dealing

156

with the stopping form, and also in the treatment of the functional energy in Appendix .

5.2.1.3 Weak goodness

Let D and G be dyadic grids. It remains to estimate the form Θ2 (f, g) which, following [24],
we will split into a ‘bad’ part and a ‘good’ part. For this we introduce our main definition

associated with the above modification of the weak goodness of Hytönen and Martikainen,

namely the definition of the cube R

(cid:122) in a grid D, given an arbitrary cube R ∈ P.

(cid:122) be the smallest (if any
Definition 5.2.8. Let D be a dyadic grid. Given R ∈ P, let R
such exist) D-dyadic supercube Q of R such that R is good inside all D-dyadic supercubes
(cid:122) will not exist if there is no D-dyadic cube Q containing R in which
(cid:122) exists, let

K of Q. Of course R
R is good. For cubes R, Q ∈ P let κ (Q, R) = log2
κ (R) ≡ κ

(cid:96)(R). For R ∈ P for which R

(cid:17)

(cid:16)

(cid:96)(Q)

(cid:122)

R

, R

.

Note that we typically suppress the dependence of R

(cid:122) on the grid D, since the grid
(cid:122) exists, we thus have that R is good inside all

(cid:122)(cid:17). Note in particular the monotonicity

(cid:16)

R

is usually understood from context. If R
D-dyadic supercubes K of R with (cid:96) (K) ≥ (cid:96)
property for J(cid:48), J ∈ P:

J(cid:48) ⊂ J =⇒(cid:0)J(cid:48)(cid:1)(cid:122) ⊂ J

(cid:122)

.

157

Here now is the decomposition:

Θ2 (f, g) =

J∈G: J

(cid:122)(cid:54)(cid:36)I, (cid:96)(J)≤2−r(cid:96)(I)

d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε

(cid:88)
(cid:88)

(cid:90) (cid:16)
(cid:90) (cid:16)

gdω

(cid:17) (cid:3)ω,b∗
(cid:17) (cid:3)ω,b∗

J

J

gdω

σ (cid:3)σ,b
T α
I f

σ (cid:3)σ,b
T α
I f

(cid:88)
(cid:88)

I∈D

+

I∈D

J∈G: J

(cid:122)(cid:36)I, (cid:96)(J)≤2−r(cid:96)(I)

d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε
(f, g) ,

(f, g) + Θgood

2

≡ Θbad

2

and where if J

(cid:122) fails to exist, we assume by convention that J

(cid:122) (cid:54)(cid:36) I, i.e. J
contained in I, so that the pair (I, J) is then included in the bad form Θbad

(cid:122) is not strictly
(f, g). We will

2

in fact estimate a larger quantity corresponding to the bad form, namely

(f, g) ≡ (cid:88)

I∈D

Θbad(cid:92)

2

(cid:88)

J∈G: J

(cid:122)(cid:54)(cid:36)I, (cid:96)(J)≤2−r(cid:96)(I)

d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:16)

σ (cid:3)σ,b
T α
I f

(cid:17) (cid:3)ω,b∗

J

gdω

(cid:12)(cid:12)(cid:12)(cid:12)

(5.2.12)

with absolute value signs inside the sum.

Remark 5.2.9. We now make some general comments on where we now stand and where

we are going.

1. In the first sum Θbad

2

(f, g) above, we are roughly keeping the pairs of cubes (I, J) such

that J is bad with respect to some ‘nearby’ cube having side length larger than that of

I.

2. We have defined energy and dual energy conditions that are independent of the test-

(cid:18)(cid:12)(cid:12)(cid:12) x−x(cid:48)

(cid:96)(J)

(cid:12)(cid:12)(cid:12)2(cid:19)

does not involve

ing families (because the definition of E (J, ω) = Eω,x

J

Eω,x(cid:48)

J

158

pseudoprojections (cid:3)ω,b∗
volve the dual martingale pseudoprojections (cid:3)ω,b∗
J,D .

J,D ), but the functional energy condition defined below does in-

3. Using the notion of weak goodness above, we will be able to eliminate all pairs of cubes

with J bad in I, which then permits control of the short range form in Section 5.3 and

the neighbour form in Section 5.5 provided 0 < ε <

n+1−α. Defining shifted coronas in
(cid:122) will then allow existing arguments to prove the Intertwining Proposition

terms of J

1

and obtain control of the functional energy in Appendix , as well as permitting control

of the stopping form in Section 5.6, but all of this with some new twists, for example

the introduction of a top/down ‘indented corona’ in the analysis of the stopping form.

4. The nearby form Θ3 (f, g) is handled in Section 5.4 using the energy condition assump-

tion along with the original testing functions borig

Q discarded during the construction of

the testing/accretive corona.

(k,ε)−good

k−good = GD

These remarks will become clear in this and the following sections. Recall that we earlier
defined in Definition 5.2.6, the set GD
to consist of those J ∈ G such that
J is ε − good inside every cube K ∈ D with K ∩ J (cid:54)= ∅ that lies at least k levels ‘above’ J,
i.e. (cid:96) (K) ≥ 2k(cid:96) (J). We now define an analogous notion of GD

k−bad
Definition 5.2.10. Let ε > 0. Define the set GD
(k,ε)−bad to consist of all J ∈ G
such that there is a D-cube K with sidelength (cid:96) (K) = 2k(cid:96) (J) for which J is ε − bad with

k−bad = GD

.

respect to K.

Note that for grids D and G, the complement of GD

k−good

is the union of GD

(cid:96)−bad

for (cid:96) ≥ k,

i.e.

G \ GD

k−good =

(cid:91)

(cid:96)≥k

159

GD
(cid:96)−bad .

Now assume ε > 0. We then have the following important property, namely for all cubes R,
and all k ≥ r (where the goodness parameter r will be fixed given ε > 0 in (5.2.16) below):

(cid:110)
Q : κ (Q, R) = k and d (R, Q) ≤ 2(cid:96) (R)ε (cid:96) (Q)1−ε(cid:111) (cid:46) 1.

(5.2.13)

#

As in [24], set

GD
bad,n ≡ {J ∈ G : J is ε − bad with respect to some K ∈ D with (cid:96) (K) ≥ n} .

We will now use the set equality

(cid:110)
(cid:110)
J ∈ G : J
R ∈ GD

(cid:122) (cid:54)⊂ I, (cid:96) (J) ≤ 2−r(cid:96) (I) , d (J, I) ≤ 2(cid:96) (J)ε (cid:96) (I)1−ε(cid:111)
bad,(cid:96)(Q) : r ≤ κ (Q, R) < κ (R) , d (R, Q) ≤ 2(cid:96) (R)ε (cid:96) (Q)1−ε(cid:111)

=

(5.2.14)

,

which the careful reader can prove by painstakingly verifying both containments.

Assuming only that b is 2-weakly µ-controlled accretive, and following the proof in [24],
we use (5.2.14) to show that for any fixed grids D and G, and any bounded linear operator
σ we have the following inequality for the form Θbad(cid:92),strict
T α

(f, g), defined to be Θbad(cid:92)

(f, g)

2

2

160

as in (5.2.12) with the pairs (I, J) removed when J

(cid:122)

= I. We use εQ,R = ±1 to obtain

Θbad(cid:92),strict

2

(f, g) =

(cid:88)

Q∈D

R∈GD

(cid:88)

(cid:88)
(cid:68)

: r≤κ(Q,R)<κ(R)

bad,(cid:96)(Q)

d(R,Q)≤2(cid:96)(R)ε(cid:96)(Q)1−ε

(cid:16)(cid:3)σ,b

Q,Df

εQ,R

T α
σ

(cid:69)(cid:12)(cid:12)(cid:12)

T α
σ

Q,Df

, (cid:3)ω,b∗
R,G g

(cid:12)(cid:12)(cid:12)(cid:68)
(cid:17)

(cid:17)

(cid:16)(cid:3)σ,b
(cid:69)

R∈GD

: r≤κ(Q,R)<κ(R)

bad,(cid:96)(Q)

d(R,Q)≤2(cid:96)(R)ε(cid:96)(Q)1−ε

(cid:16)(cid:3)σ,b

Q,Df

(cid:17)

,

(cid:88)

R∈GD

: r≤κ(Q,R)<κ(R)

bad,(cid:96)(Q)

d(R,Q)≤2(cid:96)(R)ε(cid:96)(Q)1−ε

, (cid:3)ω,b∗
R,G g

(cid:43)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

εQ,R(cid:3)ω,b∗
R,G g

(cid:88)

Q∈D

=

≤ (cid:88)

Q∈D

≤ NT α

≤ NT α

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:42)

T α
σ

(cid:88)

Q∈D

(cid:88)

Q∈D

(cid:13)(cid:13)(cid:13)(cid:3)σ,b

Q,Df

(cid:13)(cid:13)(cid:13)L2(σ)

(cid:13)(cid:13)(cid:13)(cid:3)σ,b

Q,Df

(cid:13)(cid:13)(cid:13)L2(σ)

∞(cid:88)

k=r

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

εQ,R(cid:3)ω,b∗
R,G g

R∈GD

: r≤κ(Q,R)<κ(R)

bad,(cid:96)(Q)

d(R,Q)≤2(cid:96)(R)ε(cid:96)(Q)1−ε

(cid:88)

(cid:88)

εQ,R(cid:3)ω,b∗
R,G g

,

R∈GD

:k=κ(Q,R)<κ(R)

bad,(cid:96)(Q)

d(R,Q)≤2(cid:96)(R)ε(cid:96)(Q)1−ε

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(ω)

161

by Minkowski’s inequality, and we continue with

≤ 2NT α
(cid:88)



(cid:88)

Q∈D

∞(cid:88)

k=r

L2(σ)

Q,Df

(cid:13)(cid:13)(cid:13)2

(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:88)
 (cid:88)

R∈GD

∞(cid:88)

k=r

2

·

 1
(cid:18)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:18)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

bad,2k(cid:96)(R)

Q∈D

R∈GD

: k=κ(Q,R)<κ(R)

bad,(cid:96)(Q)

d(R,Q)≤2(cid:96)(R)ε(cid:96)(Q)1−ε

R,G g

L2(ω)

R,Gg

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:53)ω
(cid:13)(cid:13)(cid:13)(cid:53)ω

+

+

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2


(cid:19)
(cid:19)

1
2

1
2

,

(cid:46) NT α (cid:107)f(cid:107)L2(σ)

R,G g

L2(ω)

R,Gg

L2(ω)

R,G denotes the ‘broken’ Carleson averaging operator in (5.1.39) that depends on

where (cid:53)ω
the grid G, and

1. the penultimate inequality uses Cauchy-Schwarz in Q and the weak upper Riesz in-
R,G , once for the sum when

equalities (5.1.53) for

εQ,R(cid:3)ω,b∗

(cid:88)

R∈GD

: k=κ(Q,R)<κ(R)

bad,(cid:96)(Q)

d(R,Q)≤2(cid:96)(R)ε(cid:96)(Q)1−ε

εQ,R = 1, and again for the sum when εQ,R = −1. However, we note that since
the sum in R is pigeonholed by k = κ (Q, R), the R’s are pairwise disjoint cubes and
the pseudoprojections (cid:3)ω,b∗
R,G g are pairwise orthogonal. Thus we could instead apply
Cauchy-Schwarz first in R, and then in Q as was done in [24], but we must still apply

weak upper Riesz inequalities as above.

2. and the final inequality uses the frame inequality (5.1.51) together with (5.2.13), namely
the fact that there are at most C cubes Q such that κ (Q, R) ≥ r is fixed and d (R, Q) ≤
2(cid:96) (R)ε (cid:96) (Q)1−ε.

162

Now it is easy to verify that we have the same inequality for the pairs(cid:16)

(cid:17) that were

(cid:122)

J

, J

removed, and then we take grid expectations and use the probability estimate (5.2.10) to
obtain for ε(cid:48) = 1

Θbad(cid:92)

(f, g)

2 ε that ED

Ω

2

(cid:16)

(cid:17) is bounded by
 (cid:88)
∞(cid:88)
ED
(cid:88)
∞(cid:88)

(cid:18)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:18)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

bad,2k(cid:96)(R)
(1 − (C12−εk)n)(cid:107)g(cid:107)2

bad,2k(cid:96)(R)

R∈GD

R∈GD

(cid:16)

k=r

Ω

R,G g

R,G g

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

L2(ω)

(cid:17) 1

2

L2(ω)

(5.2.15)

L2(ω)

R,Gg

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:53)ω
(cid:13)(cid:13)(cid:13)(cid:53)ω

+

+

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

R,G

L2(ω)

1
2

(cid:19)
(cid:19)

1
2

≤ ED

Ω NT α (cid:107)f(cid:107)L2(σ)
∞(cid:88)

≤ NT α (cid:107)f(cid:107)L2(σ)

k=r

(cid:46) 2

2 ε(cid:48)rNT α (cid:107)f(cid:107)L2(σ)
− 1

≤ Cgood2

− 1
2 εrNT α (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .

k=r

Clearly we can now fix r sufficiently large depending on ε > 0 so that

− 1

2 εr <

Cgood2

1
100

,

(5.2.16)

and then the final term above, namely Cgood2
at the end of the proof in Subsection 5.7. Note that (5.2.16) fixes our choice of the parameter

, can be absorbed

− 1
2 εrNT α (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)

r for any given ε > 0. Later we will choose 0 < ε < 1
goodness that we will exploit in the local forms BA(cid:98)r (f, g) treated below in Section 5.5.

. It is this type of weak

n+1−α

1

2 ≤

163

We are now left with the following ‘good’ form to control:

(cid:17) (cid:3)ω,b∗

J

gdω.

σ (cid:3)σ,b
T α
I f

Θgood

2

(f, g) =

(cid:88)

I∈D

(cid:88)

J

(cid:122)(cid:36)I: (cid:96)(J)≤2−r(cid:96)(I)
d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε

(cid:90) (cid:16)

The first thing we observe regarding this form is that the cubes J which arise in the sum
(cid:122) (cid:36) I. Then in the remainder of the

(f, g) must lie entirely inside I since J ⊂ J

for Θgood

2

paper, we proceed to analyze

Θgood

2

(f, g) =

(cid:88)

(cid:88)

I∈D

J(cid:122)(cid:36)I: (cid:96)(J)≤2−r(cid:96)(I)

(cid:90) (cid:16)

(cid:17) (cid:3)ω,b∗

J

gdω,

σ (cid:3)σ,b
T α
I f

(5.2.17)

in the same way we analyzed the below term B(cid:98)r (f, g) in [63]; namely, by implementing the
canonical corona splitting and the decomposition into paraproduct, neighbour and stopping

forms, but now with an additional broken form. We have (κ, ε)-goodness available for all
(f, g), and moreover, the cubes I ∈ D arising in
the cubes J ∈ G arising in the form Θgood
(f, g) for a fixed J are tree-connected, so that telescoping identities hold for
the form Θgood

2

2

these cubes I. This will prove decisive in the following three sections of the paper.

The forms Θ1 (f, g) and Θ3 (f, g) are analogous to the disjoint and nearby forms B∩ (f, g)
In the next two sections, we control the disjoint form

and B/ (f, g) in [63] respectively.
Θ1 (f, g) in essentially the same way that the disjoint form B∩ (f, g) was treated in [63] and
in earlier papers of many authors beginning with Nazarov, Treil and Volberg (see e.g. [75]),

and we control the nearby form Θ3 (f, g) using the probabilistic surgery of Hytönen and
Martikainen building on that of NTV, together with a new deterministic surgery involving

the energy condition and the original testing functions. But first we recall, in the follow-

ing subsection, the characterization of boundedness of one-dimensional forms supported on

164

disjoint cubes [22].

5.3 Disjoint form

which can be rewritten as

Θ1 (f, g) =

(cid:88)



(cid:88)

I∈D

Here we control the disjoint form Θ1 (f, g) by further decomposing it as follows:

(cid:88)

(cid:88)

I∈D

J∈G: (cid:96)(J)≤(cid:96)(I)

d(J,I)>2(cid:96)(J)ε(cid:96)(I)1−ε

(cid:90) (cid:16)

(cid:17) (cid:3)ω,b∗

J

gdω

Tσ(cid:3)σ,b
I f

(cid:88)

(cid:90)(cid:16)

(cid:17) (cid:3)ω,b∗

J

gdω

Tσ(cid:3)σ,b
I f



J∈G: (cid:96)(J)≤(cid:96)(I)

+
(cid:96)(I)≥d(J,I)>2(cid:96)(J)ε(cid:96)(I)1−ε
d(J,I)>max((cid:96)(I),2(cid:96)(J)ε(cid:96)(I)1−ε)
(f, g) ,

J∈G: (cid:96)(J)≤(cid:96)(I)

(f, g) + Θshort

1

≡ Θlong

1

where Θlong

1

(f, g) is a ‘long range’ form in which J is far from I, and where Θshort

1

(f, g) is a

short range form. It should be noted that the goodness plays no role in treating the disjoint

form.

5.3.1 Long range form

Lemma 5.3.1. We have

(cid:88)

(cid:88)

I∈D

J∈G: (cid:96)(J)≤(cid:96)(I)

d(J,I)>(cid:96)(I)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:16)

(cid:17) (cid:3)ω,b∗

J

gdω

Tσ(cid:3)σ,b
I f

(cid:12)(cid:12)(cid:12)(cid:12) (cid:46)(cid:113)

2 (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)
Aα

165

Proof. Since J and I are separated by at least max{(cid:96) (J) , (cid:96) (I)}, we have the inequality

(cid:12)(cid:12)(cid:12) dσ (y)

(cid:12)(cid:12)(cid:12)(cid:3)σ,b
(cid:96) (J)(cid:112)|I|σ

I f (y)

d (I, J)n+1−α ,

Pα(cid:16)

|y − cJ|n+1−α
I f

I

J,

(cid:96) (J)

I f

I f

(cid:12)(cid:12)(cid:12) σ

(cid:12)(cid:12)(cid:12)(cid:3)σ,b

(cid:90)
(cid:17) ≈
(cid:46) (cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:113)|I|σ
(cid:12)(cid:12)(cid:12) dσ (y) ≤(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:88)
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)L2(ω)
(cid:113)|J|ω;
(cid:113)|I|σ

J : (cid:96)(J)≤(cid:96)(I)
d(I,J)≥(cid:96)(I)

I f

I f

d (I, J)n+1−α

(cid:96) (J)

g

J

J

A (I, J) ;

(cid:13)(cid:13)(cid:13)L2(ω)

g

since

I

(cid:90)

I f (y)

(cid:12)(cid:12)(cid:12)(cid:3)σ,b
A (f, g) (cid:46) (cid:88)
≡(cid:88)

I∈D

(I,J)∈P
with A (I, J) ≡

of the conclusion of Lemma 5.3.1, we have using first the Energy Lemma,

. Thus if A (f, g) denotes the left hand side

(cid:113)|I|σ

(cid:113)|J|ω

(cid:96) (J)

d (I, J)n+1−α

and P ≡ {(I, J) ∈ D × G : (cid:96) (J) ≤ (cid:96) (I) and d (I, J) ≥ (cid:96) (I)} .

Now let DN ≡(cid:110)

K ∈ D : (cid:96) (K) = 2N(cid:111) for each N ∈ Z. For N ∈ Z and s ∈ Z+, we further

decompose A (f, g) by pigeonholing the sidelengths of I and J by 2N and 2N−s respectively:

(cid:88)
∞(cid:88)
N (f, g) ≡ (cid:88)

A (f, g) =

s=0

N∈Z

As

(I,J)∈Ps
N

N (f, g) ;

As

(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

(cid:13)(cid:13)(cid:13)L2(σ)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

(cid:13)(cid:13)(cid:13)L2(ω)

g

A (I, J)

where Ps

N ≡ {(I, J) ∈ DN × GN−s : d (I, J) ≥ (cid:96) (I)} .

M = (cid:80)

Now let Pσ

Span(cid:110)(cid:3)σ,b

(cid:111)

K

K∈DM

(cid:3)σ,b
K

denote the dual martingale pseudoprojection onto

K∈DM
. Since the cubes K in DM are pairwise disjoint, the pseudoprojections

166

(cid:3)σ,b
K

claim that

are mutually orthogonal, which means that(cid:13)(cid:13)Pσ
M f(cid:13)(cid:13)2
L2(σ) = (cid:80)
N (f, g)(cid:12)(cid:12) ≤ C2−s(cid:113)
(cid:13)(cid:13)Pω
N−sg(cid:13)(cid:13)(cid:70)
(cid:12)(cid:12)As

(cid:13)(cid:13)Pσ
N f(cid:13)(cid:13)(cid:70)

Aα
2

,

L2(ω)

L2(σ)

K∈DM

(cid:13)(cid:13)(cid:13)(cid:3)σ,b

K f

(cid:13)(cid:13)(cid:13)2

L2(σ)

. We

for s ≥ 0 and N ∈ Z.

(5.3.1)

With this proved, we can then obtain

A (f, g) =

As

s=0

∞(cid:88)
(cid:113)
(cid:113)
(cid:113)

N∈Z

Aα
2

(cid:88)
∞(cid:88)
∞(cid:88)
∞(cid:88)

s=0

s=0

Aα
2

Aα
2

s=0

≤ C

≤ C

≤ C

N (f, g) =

s=0

As

N∈Z

(cid:88)
∞(cid:88)
2−s (cid:88)
(cid:13)(cid:13)Pσ
N f(cid:13)(cid:13)(cid:70)
(cid:88)
(cid:13)(cid:13)Pσ
N f(cid:13)(cid:13)(cid:70)2

2−s

N∈Z

L2(σ)

N∈Z

N (f, g)

(cid:13)(cid:13)Pω
N−sg(cid:13)(cid:13)(cid:70)
2(cid:88)
 1
(cid:113)

L2(σ)

2−s (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) = C

 1

2

L2(ω)

(cid:13)(cid:13)Pω
N−sg(cid:13)(cid:13)(cid:70)2

L2(ω)

N∈Z
2 (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .
Aα

To prove (5.3.1), we pigeonhole the distance between I and J:

N,(cid:96)

g

J

(cid:96)=0

As

As

As

I f

(I,J)∈Ps

N,(cid:96) (f, g) ;

N (f, g) =

where Ps

(cid:13)(cid:13)(cid:13)(cid:3)σ,b

(cid:13)(cid:13)(cid:13)L2(σ)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

∞(cid:88)
N,(cid:96) (f, g) ≡ (cid:88)
N,(cid:96) ≡ (cid:110)
If we define H(cid:16)
(cid:17) to be the bilinear form on (cid:96)2 × (cid:96)2 with matrix [A (I, J)](I,J)∈Ps
then it remains to show that the norm (cid:13)(cid:13)(cid:13)H(cid:16)
(cid:17) on the sequence
space (cid:96)2 is bounded by C2−s−(cid:96)(cid:112)Aα
(cid:13)(cid:13)(cid:13)H(cid:16)
(cid:17)(cid:13)(cid:13)(cid:13)(cid:96)2→(cid:96)2
(cid:17) on the sequence
of the bilinear form H(cid:16)

(cid:13)(cid:13)(cid:13)L2(ω)
(I, J) ∈ DN × GN−s : d (I, J) ≈ 2N +(cid:96)(cid:111)
of H(cid:16)
(cid:17)tr H(cid:16)

(cid:17)(cid:13)(cid:13)(cid:13)(cid:96)2→(cid:96)2
(cid:17) ≡ H(cid:16)

. In turn, this is equivalent to showing that the norm

A (I, J)

Bs

Bs

As

As

As

As

As

N,(cid:96)

,

N,(cid:96)

N,(cid:96)

N,(cid:96)

N,(cid:96)

N,(cid:96)

N,(cid:96)

2

.

N,(cid:96)

167

space (cid:96)2 is bounded by C22−2s−2(cid:96)Aα

kernel(cid:104)

Bs

N,(cid:96)

(cid:0)J, J(cid:48)(cid:1)(cid:105)
(cid:0)J, J(cid:48)(cid:1) ≡

Bs

N,(cid:96)

J,J(cid:48)∈DN−s

having entries:

(cid:88)

I∈DN : d(I,J)≈d(I,J(cid:48))≈2N +(cid:96)

. Here H(cid:16)

Bs

N,(cid:96)

(cid:17) is the quadratic form with matrix

2

A (I, J) A(cid:0)I, J(cid:48)(cid:1) ,

for J, J(cid:48) ∈ GN−s.

We are reduced to showing the bilinear form inequality,

≤ C2−2s−2(cid:96)Aα

2

for s ≥ 0, (cid:96) ≥ 0 and N ∈ Z.

(cid:13)(cid:13)(cid:13)H(cid:16)

Bs

N,(cid:96)

(cid:17)(cid:13)(cid:13)(cid:13)(cid:96)2→(cid:96)2
(cid:0)J, J(cid:48)(cid:1):

N,(cid:96)

We begin by computing Bs

(cid:0)J, J(cid:48)(cid:1) =

Bs

N,(cid:96)

d(I,J)≈d

=

d(I,J)≈d

(cid:96)(cid:0)J(cid:48)(cid:1)

(cid:96) (J)

d (I, J)n+1−α

d (I, J(cid:48))n+1−α

(cid:113)|I|σ

(cid:113)|J|ω

(cid:113)|J(cid:48)|ω
(cid:113)|I|σ
(cid:88)
(cid:16)
I,J(cid:48)(cid:17)≈2N +(cid:96)
I∈DN
(cid:113)|J(cid:48)|ω.
d (I, J)n+1−α d (I, J(cid:48))n+1−α · (cid:96) (J) (cid:96)(cid:0)J(cid:48)(cid:1)(cid:113)|J|ω
(cid:88)
(cid:16)
I,J(cid:48)(cid:17)≈2N +(cid:96)
I∈DN
(cid:13)(cid:13)(cid:13)Bs

(cid:46) 2−2s−2(cid:96)Aα
2 ,

(cid:13)(cid:13)(cid:13)(cid:96)2→(cid:96)2

|I|σ

N,(cid:96)

(5.3.2)

Now we show that

β (K) =

1√|K|ω

by applying the proof of Schur’s lemma. Fix (cid:96) ≥ 0 and s ≥ 0. Choose the Schur function
. Fix J ∈ DN−s. We now group those I ∈ DN with d (I, J) ≈ 2N +(cid:96) into
finitely many groups G1, ...GC for which the union of the I in each group is contained in a
I for 1 ≤ k ≤ C (note that I∗
cube of side length roughly 1

1002N +(cid:96) , and we set I∗

k ≡ (cid:83)

k

I∈Gk

168

is not a cube). We then have

(cid:0)J, J(cid:48)(cid:1)

Bs

N,(cid:96)

β (J)
β (J(cid:48))

Bs

N,(cid:96)

β (J)
β (J(cid:48))

(cid:88)
(cid:88)
J(cid:48)∈GN−s
(cid:16)
(cid:17)≤ 1
J(cid:48)∈GN−s

d

=

J(cid:48),J
≡ A + B,

100 2N +(cid:96)+2

(cid:0)J, J(cid:48)(cid:1) +

(cid:88)
(cid:17)
J(cid:48)∈GN−s
> 1
100 2N +(cid:96)+2

J(cid:48),J

(cid:16)

d

(cid:0)J, J(cid:48)(cid:1)

β (J)
β (J(cid:48))

Bs

N,(cid:96)

where

A (cid:46)

=

=



|I|σ

100 2N +(cid:96)+2

(cid:88)
J,J(cid:48)(cid:17)≤ 1
J(cid:48)∈GN−s
(cid:88)
J,J(cid:48)(cid:17)≤ 1
J(cid:48)∈GN−s

(cid:16)

d

(cid:16)

d

100 2N +(cid:96)+2

22(N−s)

22((cid:96)+N )(n+1−α)

(cid:88)


 C(cid:88)
C(cid:88)
(cid:12)(cid:12)σ
(cid:12)(cid:12)I∗

(cid:16)

k=1

k

d

I∈DN

k

k=1

d(I,J)≈2N +(cid:96)


(cid:12)(cid:12)I∗
(cid:12)(cid:12)σ
(cid:88)
J,J(cid:48)(cid:17)≤ 1
J(cid:48)∈GN−s
(cid:12)(cid:12)(cid:12) 1

(cid:46) 2−2s−2(cid:96)

C(cid:88)

k=1

2((cid:96)+N )(n−α)

100 2N +(cid:96)+2
1002N +(cid:96)+2J
2((cid:96)+N )(n−α)

(cid:46) 2−2s−2(cid:96)Aα
2 ,

22(N−s)

22((cid:96)+N )(n+1−α)

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

22(N−s)

22((cid:96)+N )(n+1−α)

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:12)(cid:12)σ

k

(cid:12)(cid:12)I∗
(cid:12)(cid:12)(cid:12)ω

is contained in a cube ˜I∗

since I∗
only on dimension, and ˜I∗

such that |I∗

k|, with an implied constant depending
1002N +(cid:96)+2J are well separated. If we let Qk be the smallest

k| ≈ | ˜I∗

k

k

1

,

k

169

cube containing the set

Ek ≡

we then have

(cid:91)
(cid:16)
k ,J(cid:48)(cid:17)≈2N +(cid:96)
J,J(cid:48)(cid:17)
J(cid:48)∈DN−s: d
I∗
> 1
100 2N +(cid:96)+2

(cid:16)

d

B (cid:46)

(cid:46)

(cid:46)

(cid:88)
J,J(cid:48)(cid:17)
J(cid:48)∈DN−s
> 1
100 2N +(cid:96)+2
(cid:88)
J,J(cid:48)(cid:17)
J(cid:48)∈DN−s
> 1
100 2N +(cid:96)+2
22(N−s)

(cid:16)

d

(cid:16)

d

22((cid:96)+N )(n+1−α)

d



C(cid:88)
(cid:12)(cid:12)I∗
(cid:12)(cid:12)σ
(cid:12)(cid:12)I∗

k=1

k

k

C(cid:88)

k=1

(cid:88)

|I|σ



(cid:12)(cid:12)σ

(cid:16)

I∈DN

I,J(cid:48)(cid:17)≈d(I,J)≈2N +(cid:96)
(cid:12)(cid:12)I∗

(cid:88)
(cid:16)
,J(cid:48)(cid:17)≈2N +(cid:96)
(cid:12)(cid:12)σ |Ek|ω

I∗

k

k

k: d

J(cid:48)



22(N−s)

22((cid:96)+N )(n+1−α)

22(N−s)

22((cid:96)+N )(n+1−α)

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:46) 2−2s−2(cid:96)

2((cid:96)+N )(n−α)

|Qk|ω

2((cid:96)+N )(n−α)

(cid:46) 2−2s−2(cid:96)Aα
2 ,

is contained in a cube ˜I∗

since I∗
only on dimension, and ˜I∗

such that |I∗

k|, with an implied constant depending
1002N +(cid:96)+2J are well separated. Thus we can now apply Schur’s
, 1

k| ≈ | ˜I∗

k

k

k

170

(cid:88)

J(cid:48)

aJ bJ(cid:48)Bs

N,(cid:96)

J

(aJ )2 =

argument with(cid:88)
(cid:88)
J,J(cid:48)∈GN−s
(aJ β (J))2(cid:88)
(cid:88)
(cid:88)

≤(cid:88)
(cid:88)

(cid:46) 2−2s−2(cid:96)Aα

(aJ )2

J(cid:48)

=

J

J

2

(cid:0)bJ(cid:48)(cid:1)2 = 1 to obtain
(cid:88)
aJ β (J) bJ(cid:48)β(cid:0)J(cid:48)(cid:1) Bs
(cid:0)J, J(cid:48)(cid:1) =
(cid:0)J, J(cid:48)(cid:1)
(cid:0)J, J(cid:48)(cid:1)
J,J(cid:48)∈GN−s
(cid:0)bJ(cid:48)β(cid:0)J(cid:48)(cid:1)(cid:1)2(cid:88)
(cid:88)
(cid:0)J, J(cid:48)(cid:1)+
(cid:40)(cid:88)
(cid:88)
(cid:0)bJ(cid:48)(cid:1)2
 = 21−2s−2(cid:96)Aα
(cid:88)
(cid:0)bJ(cid:48)(cid:1)2

Bs
β (J) β (J(cid:48))

β(cid:0)J(cid:48)(cid:1)

β (J)

Bs

Bs

J(cid:48)

J(cid:48)

N,(cid:96)

N,(cid:96)

N,(cid:96)

N,(cid:96)

2 .

+

N,(cid:96)

Bs
β (J) β (J(cid:48))

J

J

J(cid:48)
β (J)
β (J(cid:48))

(aJ )2 +

J

J(cid:48)

(cid:0)J, J(cid:48)(cid:1)

β (J) β (J(cid:48))

(cid:0)J, J(cid:48)(cid:1)(cid:41)

This completes the proof of (5.3.2). We can now sum in (cid:96) to get (5.3.1) and we are done.

This completes our proof of the long range estimate

A (f, g) (cid:46)(cid:113)

2 (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .
Aα

5.3.2 Short range form

The form Θshort

1

(f, g) is handled by the following lemma.

Lemma 5.3.2. We have

(cid:88)

(cid:88)

I∈D

J∈G: (cid:96)(J)≤2−ρ(cid:96)(I)

(cid:96)(I)≥d(J,I)>2(cid:96)(J)ε(cid:96)(I)1−ε

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:16)

(cid:17) (cid:3)ω,b∗

J

gdω

Tσ(cid:3)σ,b
I f

(cid:12)(cid:12)(cid:12)(cid:12) (cid:46)(cid:113)

2 (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)
Aα

171

Proof. The pairs (I, J) that occur in the sum above satisfy J ⊂ 4I\I, so we consider

P ≡(cid:110)

(cid:111)
(I, J)∈D×G : (cid:96) (J)≤ 2−ρ(cid:96) (I) , (cid:96) (I)≥ d (J, I) > 2(cid:96) (J)ε (cid:96) (I)1−ε , J ⊂ 4I\I

For (I, J) ∈ P, the ‘pivotal’ estimate from the Energy Lemma 5.1.25 gives

(cid:12)(cid:12)(cid:12)(cid:68)

T α
σ

(cid:16)(cid:3)σ,b

I f

(cid:17)

, (cid:3)ω,b∗

J

g

(cid:69)

ω

(cid:12)(cid:12)(cid:12) (cid:46)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

(cid:13)(cid:13)(cid:13)L2(ω)

g

Pα(cid:0)J,(cid:12)(cid:12)(cid:52)σ

I f(cid:12)(cid:12) σ(cid:1)(cid:113)|J|ω .

Now we pigeonhole the lengths of I and J and the distance between them by defining

N,d ≡(cid:110)

Ps

(I, J) ∈ P : (cid:96) (I) = 2N , (cid:96) (J) = 2N−s, 2d−1 ≤ d (I, J) ≤ 2d, J ⊂ 4I\I

.

(cid:111)

Note that the closest a cube J can come to I is determined by:

2d ≥ 2(cid:96) (I)1−ε (cid:96) (J)ε = 21+N (1−ε)2(N−s)ε = 21+N−εs;
which implies N − εs + 1 ≤ d ≤ N.

Thus we have

T α
σ

(I,J)∈P

(cid:12)(cid:12)(cid:12)(cid:68)
(cid:17)
(cid:16)(cid:3)σ,b
(cid:88)
(cid:13)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:46) (cid:88)
N(cid:88)
(cid:88)
∞(cid:88)

(I,J)∈P

I f

g

J

=

J

, (cid:3)ω,b∗
Pα(cid:16)
(cid:88)

J,

g

ω

(cid:12)(cid:12)(cid:12)
(cid:69)
(cid:12)(cid:12)(cid:12) σ
(cid:12)(cid:12)(cid:12)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

I f

J

s=0

N∈Z

d=N−εs+1

(I,J)∈Ps

N,d

(cid:17)(cid:113)|J|ω
(cid:13)(cid:13)(cid:13)L2(ω)

g

Pα(cid:16)

J,

(cid:12)(cid:12)(cid:12)(cid:3)σ,b

I f

(cid:12)(cid:12)(cid:12) σ

(cid:17)(cid:113)|J|ω.

172

Now we use

Pα(cid:16)

(cid:12)(cid:12)(cid:12)(cid:3)σ,b

I f

(cid:12)(cid:12)(cid:12) σ

(cid:17)

J,

=

(cid:46)

(cid:90)

I

((cid:96) (J) + |y − cJ|)n+1−α
2N−s

(cid:96) (J)

(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

(cid:13)(cid:13)(cid:13)L2(σ)

2d(n+1−α)

(cid:12)(cid:12)(cid:12) dσ (y)

(cid:12)(cid:12)(cid:12)(cid:3)σ,b
(cid:113)|I|σ

I f (y)

and apply Cauchy-Schwarz in J and use J ⊂ 4I\I to get

(cid:69)

g

(cid:12)(cid:12)(cid:12)(cid:68)
(cid:88)
(cid:88)
∞(cid:88)

(I,J)∈P

T α
σ

(cid:17)

(cid:16)(cid:3)σ,b
N(cid:88)

I f

(cid:46)

J

, (cid:3)ω,b∗
(cid:88)

s=0

N∈Z

d=N−εs−1

I∈DN

2N−s2N (n−α)
2d(n+1−α)

∞(cid:88)
∞(cid:88)

s=0

2N−s2N (n−α)
2(N−εs)(n+1−α)

(cid:88)
2−s[1−ε(n+1−α)](cid:113)

N∈Z

(cid:88)

Aα
2

I∈DN

2 (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)
Aα

(cid:46) (1 + εs)

(cid:46) (1 + εs)

s=ρ

ω

(cid:12)(cid:12)(cid:12)
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)
(cid:113)

·

J∈GN−s

J⊂4I\I and d(I,J)≈2d

I f

(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:88)
(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

J

g

2N (n−α)

(cid:112)|I|σ
(cid:112)|4I\I|ω
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:88)
(cid:46)(cid:113)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

L2(ω)

J

·

(cid:13)(cid:13)(cid:13)2

g

L2(ω)

J∈GN−s
J⊂4I\I
2 (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)
Aα

1 (cid:46) 1 + εs, and in the last line

where in the third line above we have used
2N−s2N (n−α)
= 2−s[1−ε(n+1−α)] followed by Cauchy-Schwarz in I and N, using that
2(N−εs)(n+1−α)
we have bounded overlap, depending only on dimension and the goodness constant in the
quadruples of I for I ∈ DN . More precisely, if we define fk ≡ Ψσ,bDk

f = (cid:80)

(cid:3)σ,b
I f and

d=N−εs−1

I∈Dk

N(cid:88)

173

g = (cid:80)

J∈Gk

(cid:3)ω,b∗

J

gk ≡ Ψσ,b∗
Gk
(cid:88)

N∈Z

(cid:107)fN(cid:107)L2(σ) (cid:107)gN−s(cid:107)L2(ω) ≤

g, then we have the quasi-orthogonality inequality

(cid:88)

(cid:107)fN(cid:107)2

N∈Z

L2(σ)
(cid:46) (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .

2(cid:88)
 1

N∈Z

 1

2

(cid:107)gN−s(cid:107)2

L2(ω)

We have assumed that

0 < ε <

1

n + 1 − α

(5.3.3)

in the calculations above, and this completes the proof of Lemma 5.3.2.

5.4 Nearby form

We dominate the nearby form Θ3(f, g) by

|Θ3 (f, g)| ≤ (cid:88)

I∈D

(cid:88)

J∈G: 2−rn|I|<|J|≤|I|
d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:16)

σ (cid:3)σ,b
T α
I f

(cid:17) (cid:3)ω,b∗

J

gdω

(cid:12)(cid:12)(cid:12)(cid:12) ,

and prove the following proposition that controls the expectation, over two independent

grids, of the nearby form Θ3 (f, g). It should be noted that weak goodness plays no role in
treating the nearby form. Note also that in various steps we will use a small δ > 0. In all

those different instances δ is free of any dependence. Our goal is the following proposition.

Proposition 5.4.1. Suppose T α is a standard fractional singular integral with 0 ≤ α < n.
Let θ ∈ (0, 1) be sufficiently small depending only on α, n. Then there is a constant Cθ such
that for f ∈ L2 (σ) and g ∈ L2 (ω), and dual martingale differences (cid:3)σ,b
with

and (cid:3)ω,b∗

I

J

174

∞-weakly accretive families of test functions b and b∗, we have

(cid:17)

(cid:69)

(cid:12)(cid:12)(cid:12)

ω

, (cid:3)ω,b∗

J

g

(5.4.1)

(cid:88)

I∈D

Ω EG
ED

Ω

(cid:12)(cid:12)(cid:12)(cid:68)

(cid:16)(cid:3)σ,b
(cid:88)
(cid:17)(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .

J∈G: 2−rn|I|<|J|≤|I|
d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε
√
θNT α

I f

T α
σ

(cid:46) (cid:16)

CθNT Vα +

The following diagram is a sketch of the proof of proposition (5.4.1).

175

Figure 5.4.1: Nearby form diagram

176

Before we proceed any further let us mention that we will repeatedly use the inequality

Lemma 5.4.2. For f ∈ L2(σ) and I ∈CA(A) we have

Proof. Let I(cid:48) ∈ CD (I) ∩ CA (A). Since I(cid:48) ∈ CA (A), from the corona construction we have

(5.4.2)

(cid:13)(cid:13)(cid:13)L2(σ)

(cid:46)(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

(cid:13)(cid:13)(cid:13)(cid:70)

L2(σ)

.

f

(5.4.3)

(cid:13)(cid:13)(cid:13)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:13)(cid:13)(cid:13)L2(σ)

(cid:46)(cid:13)(cid:13)(cid:13)(cid:3)σ,b

L2(σ)

I f

(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:98)(cid:3)σ,(cid:91),b
(cid:12)(cid:12)(cid:12)(cid:12) > γ.

I

(cid:12)(cid:12)(cid:12)(cid:12) 1

|I(cid:48)|σ

(cid:90)
I(cid:48) bAdσ

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12) 1

|S|σ

bAdσ

S

(cid:12)(cid:12)(cid:12)(cid:12) < γ2.

Now let {I(cid:48)

j}j∈N be the collection of maximal subcubes S of I(cid:48) such that

(cid:91)

j

I(cid:48)

j

Let E =

which together with (5.4.3) gives

. We then have

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

E

bAdσ

γ(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ <

j

j

I(cid:48)

j

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

bAdσ

(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:88)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < γ2(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12) +
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12) =
(cid:115)(cid:90)
≤ γ2(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ +
(cid:12)(cid:12)I(cid:48) \ E(cid:12)(cid:12)σ ,
≤ γ2(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ + Cb

I(cid:48) bAdσ

I(cid:48)\E

bAdσ

E

|bA|2 dσ

j

≤ γ2(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)I(cid:48)
(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:90)
(cid:113)|I(cid:48) \ E|σ

I(cid:48)\E

bAdσ

where in the last inequality we used the ∞-accretivity of bA. Rearranging the inequality

177

yields successively

γ (1 − γ)(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ ≤ Cb

(cid:12)(cid:12)I(cid:48) \ E(cid:12)(cid:12)σ ;
(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ ≤ (cid:12)(cid:12)I(cid:48) \ E(cid:12)(cid:12)σ ,

γ (1 − γ)

Cb

which in turn gives

(cid:12)(cid:12)(cid:12)I(cid:48)

j

(cid:88)

j

(cid:12)(cid:12)(cid:12)σ

= (cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ −(cid:12)(cid:12)I(cid:48) \ E(cid:12)(cid:12)σ
≤ (cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ − γ (1 − γ)

Cb

(cid:18)

(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ =

1 − γ (1 − γ)

Cb

(cid:19)(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ ≡ β(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ

(5.4.4)

f = cI(cid:48) we can

where 0 < β < 1 since 1 ≤ Cb. This implies

(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ ≤ 1
Having that in hand and the fact that (cid:98)(cid:3)σ,(cid:91),b
(cid:90)

now calculate:

I

(cid:13)(cid:13)(cid:13)1I(cid:48)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:13)(cid:13)(cid:13)2

L2(σ)

=

≤

I

I(cid:48)
|I(cid:48) \ E|σ

1

=

1
γ4
≤ 1
γ4
≤ 1
γ4

1 − β

I

(cid:12)(cid:12)I(cid:48) \ E(cid:12)(cid:12)σ
f is constant on I(cid:48), say 1I(cid:48)(cid:98)(cid:3)σ,(cid:91),b
(cid:12)(cid:12)(cid:12)(cid:98)(cid:3)σ,(cid:91),b
(cid:90)
(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ
(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ
|I(cid:48) \ E|σ
(cid:90)
|I(cid:48) \ E|σ
1
1 − β
I(cid:48)

(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)cI(cid:48)(cid:12)(cid:12)2
dσ =(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ
(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ
(cid:12)(cid:12)cI(cid:48)(cid:12)(cid:12)2
(cid:90)
|bA|2(cid:12)(cid:12)cI(cid:48)(cid:12)(cid:12)2 dσ
(cid:90)
(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)bA(cid:98)(cid:3)σ,(cid:91),b
(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)bA(cid:98)(cid:3)σ,(cid:91),b

|bA|2 dσ

I(cid:48)\E
γ4

I(cid:48)\E

dσ,

I(cid:48)

dσ

f

f

f

I

I

178

and thus for I(cid:48) ∈ CA we obtain
(cid:90)

(cid:12)(cid:12)(cid:12)(cid:98)(cid:3)σ,(cid:91),b

I

(cid:12)(cid:12)(cid:12)2

f

dσ (cid:46)

(cid:90)

I(cid:48)

(cid:12)(cid:12)(cid:12)bA(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:12)(cid:12)(cid:12)2

dσ,

I(cid:48)

(cid:13)(cid:13)(cid:13)2

f

L2(σ)

≤(cid:13)(cid:13)(cid:13)bA(cid:98)(cid:3)σ,(cid:91),b

I

(cid:13)(cid:13)(cid:13)2

L2(σ)

.

f

which in turn gives, after summing over all I(cid:48) ∈ CD (I) ∩ CA (A),

I

f

L2(σ)

(cid:13)(cid:13)(cid:13)2

(cid:88)

(cid:13)(cid:13)(cid:13)1I(cid:48)(cid:98)(cid:3)σ,(cid:91),b

(cid:46)(cid:13)(cid:13)(cid:13)1I bA(cid:98)(cid:3)σ,(cid:91),b
I(cid:48)∈CD(I)∩CA(A)
Now if I(cid:48) ∈ CD (I) ∩ A, from the definition of (cid:98)∇µ
(cid:13)(cid:13)(cid:13)2

(cid:13)(cid:13)(cid:13)1I(cid:48)(cid:98)(cid:3)σ,(cid:91),b

(cid:88)
I(cid:48)∈CD(I)∩A

L2(σ)

f

I

I

Qf in (5.1.39),

(cid:46)(cid:13)(cid:13)(cid:13)(cid:98)∇σ

I f

(cid:13)(cid:13)(cid:13)2

.

L2(σ)

Now we are ready to prove (5.4.2). As bA = bI and

(cid:13)(cid:13)(cid:13)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:13)(cid:13)(cid:13)2

L2(σ)

=

(cid:88)
(cid:46) (cid:13)(cid:13)(cid:13)bI(cid:98)(cid:3)σ,(cid:91),b

I

I(cid:48)∈CD(I)∩CA(A)

(cid:13)(cid:13)(cid:13)2

f

(cid:13)(cid:13)(cid:13)1I(cid:48)(cid:98)(cid:3)σ,(cid:91),b
(cid:13)(cid:13)(cid:13)(cid:98)∇σ

+

I

L2(σ)

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

f

L2(σ)

I f

L2(σ)

we obtain

(cid:13)(cid:13)(cid:13)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:13)(cid:13)(cid:13)L2(σ)

(cid:46)(cid:13)(cid:13)(cid:13)bI(cid:98)(cid:3)σ,(cid:91),b
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)(cid:98)∇σ
(cid:13)(cid:13)(cid:13)(cid:3)σ,(cid:91),b
(cid:13)(cid:13)(cid:13)L2(σ)
≤(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

+

+

f

I

I f

(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)L2(σ)

=

I,brokenf

179

(cid:88)
I(cid:48)∈CD(I)∩A

+

(cid:13)(cid:13)(cid:13)1I(cid:48)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:13)(cid:13)(cid:13)2

L2(σ)

(cid:13)(cid:13)(cid:13)(cid:3)σ,(cid:91),b
(cid:13)(cid:13)(cid:13)(cid:98)∇σ

+

I

I f

f

(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)L2(σ)

(cid:13)(cid:13)(cid:13)(cid:98)∇σ
(cid:46)(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

+

(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)(cid:70)

I f

L2(σ)

.

Now from quasiorthogonality and (5.4.2) we get,

(cid:88)

(cid:88)

J∈G

J(cid:48)∈C(J)

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:12)(cid:12)(cid:12)Eω

J(cid:48)

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

J

g

(cid:17)(cid:12)(cid:12)(cid:12)2 (cid:46) (cid:88)
(cid:46) (cid:88)

J∈G

J∈G

(cid:13)(cid:13)(cid:13)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:18)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

J

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

g

g

L2(ω)

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:3)ω,(cid:91),b∗
(cid:13)(cid:13)(cid:13)2
(cid:46) (cid:88)
(cid:19)
+(cid:13)(cid:13)∇ω
J g(cid:13)(cid:13)2

J∈G

L2(ω)

g

J

L2(ω)

(cid:46) (cid:107)g(cid:107)2

L2(ω)

.

We also need the following lemma, that controls the above inner product for cubes of

positive distance.

Lemma 5.4.3. Given the ∞-weakly accretive families of test functions b and b∗ and cubes
Q, R ⊂ Rn, we have

R1R\(1+δ)Q(cid:105)ω| (cid:46) δα−n(cid:113)

Aα
2

(cid:112)|Q|σ

(cid:112)|R|ω

|(cid:104)T α

σ (bQ1Q), b∗

(5.4.5)

where the implied constant depends on the accretivity constants of the families b, b∗ and the

(cid:12)(cid:12)(cid:12)

T α
σ

(cid:69)

dimension n.

R1R\(1+δ)Q

Proof. We have that(cid:12)(cid:12)(cid:12)(cid:68)
(cid:0)bQ1Q
(cid:1) , b∗
(cid:12)(cid:12)T α
(cid:1)(cid:12)(cid:12)(cid:12)(cid:12)b∗
(cid:12)(cid:12) dω
(cid:0)bQ1Q
2(cid:32)(cid:90)
(cid:33) 1
(cid:1)(cid:12)(cid:12)2 dω
(cid:12)(cid:12)T α
(cid:0)bQ1Q
(cid:18)(cid:90)
|x − y|α−n(cid:12)(cid:12)bQ (y)(cid:12)(cid:12) dσ (y)

(cid:90)
(cid:32)(cid:90)
(cid:32)(cid:90)

R\(1+δ)Q

R\(1+δ)Q

≤

≤

(cid:46)

R

ω

σ

σ

R\(1+δ)Q

Rn\(1+δ)Q

Q

(cid:33) 1
(cid:12)(cid:12)2 dω
(cid:33) 1
2(cid:18)(cid:90)

2

(cid:12)(cid:12)b∗
(cid:19)2

R

dω (x)

(cid:19) 1

2

(cid:12)(cid:12)b∗

R

(cid:12)(cid:12)2 dω

R

180

Rn\(1+δ)Q

(cid:32)(cid:90)

(cid:46)

(cid:46) δα−n

(cid:32)(cid:90)
(cid:32)(cid:90)
≤ δα−n(cid:113)

(cid:46) δα−n

(cid:18)(cid:90)

Q

(cid:0)δ(cid:12)(cid:12)x − cQ
(cid:12)(cid:12)x − cQ
(cid:12)(cid:12)x − cQ
(cid:113)|R|ω

Rn\(1+δ)Q

Rn\(1+δ)Q

(cid:113)|Q|σ

Aα
2

(cid:12)(cid:12)(cid:1)α−n(cid:12)(cid:12)bQ (y)(cid:12)(cid:12) dσ (y)
(cid:33) 1
2(cid:18)(cid:90)
(cid:12)(cid:12)2(α−n) dω (x)
(cid:33) 1
(cid:12)(cid:12)2(α−n) dω (x)
2 |Q|σ

Q

dω (x)

(cid:33) 1
(cid:19)2
2(cid:113)|R|ω
(cid:19)(cid:113)|R|ω
(cid:12)(cid:12)bQ (y)(cid:12)(cid:12) dσ (y)
(cid:113)|R|ω

since(cid:32)(cid:90)

(cid:12)(cid:12)x − cQ

Rn\(1+δ)Q

(cid:33)
(cid:12)(cid:12)2(α−n) dω (x)

(cid:90)

|Q|σ=

 |Q| 1
(cid:12)(cid:12)x − cQ

n

n−α

(cid:12)(cid:12)2

 |Q|σ

|Q|1− α

n

dω (x)

Rn\(1+δ)Q
(cid:46) Pα (Q, ω)

|Q|σ
|Q|1− α

n

≤ Aα,∗
2 .

As usual, we continue to write the independent grids for f and g as D and G respectively.

Write the dual martingale averages (cid:3)σ,b

g as linear combinations

(cid:3)σ,b

I f = bI

(cid:3)ω,b∗

J

g = b∗

J

(cid:88)
(cid:88)
I(cid:48)∈Cnat(I)
J(cid:48)∈Cnat(J)

1I(cid:48) Eσ
I(cid:48)

1J(cid:48) Eω
J(cid:48)

J

I f and (cid:3)ω,b∗
(cid:16)(cid:98)(cid:3)σ,b
(cid:17)
(cid:16)(cid:98)(cid:3)ω,b∗

(cid:88)
+
(cid:17)
(cid:88)
I(cid:48)∈Cbrok(I)
b∗
+
J(cid:48)∈Cbrok(J)

I f

g

J

I(cid:48)

bI(cid:48) 1I(cid:48)(cid:98)Fσ,bI(cid:48)
J(cid:48) 1J(cid:48)(cid:98)Fω,b∗

J(cid:48)

(cid:88)
(cid:88)
I(cid:48)∈Cbrok(I)
J(cid:48)∈Cbrok(J)

1I(cid:48)(cid:98)Fσ,bI
1J(cid:48)(cid:98)Fω,b∗

f,

J

J

I

J

f − bI

J(cid:48)

g − b∗

g,

of the appropriate function b times the indicators of their children, denoted I(cid:48) and J(cid:48) respec-
tively. We will regroup the terms as needed below.

On the natural child I(cid:48), the expression (cid:98)(cid:3)σ,b
(cid:3)σ,b
I f simply denotes the dual mar-
tingale average with bI removed, so that we need not assume |bI| is bounded below in order

I f = 1
bI

181

(cid:3)σ,b
I f. Similar comments apply to the expressions

(cid:98)Fσ,b

to make sense of 1
bI
I(cid:48)
bI(cid:48) Fσ,b
I(cid:48)

f = 1

I(cid:48)

I(cid:48)

f and(cid:98)Fσ,bI

I

f = 1
bI

Fσ,bI
I

f. Now if we set

N (I) = {J ∈ G : 2−rn|I| < |J| ≤ |I|, d (J, I) ≤ 2(cid:96)(J)ε(cid:96)(I)1−ε}

for the cubes or similar size to I, the left hand side of (5.4.1) is bounded by

I + II ≡ (cid:88)
(cid:88)

I∈D

+

(cid:88)
(cid:88)

J∈N (I)

(1+δ)I∩J=∅

I∈D

J∈N (I)

(1+δ)I∩J(cid:54)=∅

(cid:69)

g

ω

(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:68)

T α
σ

I f

(cid:16)(cid:3)σ,b
(cid:17)
(cid:12)(cid:12)(cid:12)(cid:68)
(cid:16)(cid:3)σ,b

T α
σ

I f

J

, (cid:3)ω,b∗
(cid:17)

, (cid:3)ω,b∗

J

g

(5.4.6)

(cid:69)

(cid:12)(cid:12)(cid:12)

ω

When working in higher dimensions, run the proof pretending you have Hytönen’s es-

timate (which is of course not true due to the result in chapter 4). Then wherever we

were supposed to use Hytönen, we use the delta separation trick. The δ-separated part is

easily seen to be bounded by the Muckenhoupt conditions, and the δ-close part will get a
√

δ estimate. But δ can be chosen at the end, is independent of everything else (it is the

Hytönen-delta, not related to anything else in the proof). So, provided the proof only deals

with finite estimates and finitely many constructions (like the Cantor set construction, that

√

δ terms will be absorbable at the end. Here are

only does finitely many iterations), those

the details:

5.4.1 The case of δ-separated cubes.

In this subsection we are estimating I in (5.4.6) by using Lemma 5.4.3.

Definition 5.4.4. We say that the cubes J and I are δ-separated, where δ > 0, if J ∩ (1 +

182

δ)I = ∅.

For the first sum in (5.4.6) we have, following the proof of Lemma 5.4.3, the satisfactory

(cid:12)(cid:12)(cid:12)(cid:68)

T α
σ

(cid:16)(cid:3)σ,b

I f

(cid:17)

, (cid:3)ω,b∗

J

g

(cid:69)

ω

(cid:12)(cid:12)(cid:12) (cid:46) δα−n(cid:113)

Aα
2

(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

(cid:13)(cid:13)(cid:13)L2(σ)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

(cid:13)(cid:13)(cid:13)L2(ω)

g

.

estimate

Indeed,

≤

T α
σ

(cid:12)(cid:12)(cid:12)(cid:68)
(cid:90)
(cid:32)(cid:90)

J\(1+δ)I

σ

J\(1+δ)I

σ

J

g

J

I f

I f

g

ω

(cid:12)(cid:12)(cid:12)
(cid:16)(cid:3)σ,b
(cid:17)
(cid:69)
, (cid:3)ω,b∗
(cid:12)(cid:12)(cid:12)T α
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:3)ω,b∗
(cid:12)(cid:12)(cid:12) dω
(cid:16)(cid:3)σ,b
(cid:33) 1
(cid:19) 1
2(cid:18)(cid:90)
(cid:17)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)T α
(cid:12)(cid:12)(cid:12)(cid:3)ω,b∗
(cid:12)(cid:12)(cid:12)2
(cid:16)(cid:3)σ,b
(cid:33) 1
2(cid:18)(cid:90)
(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:3)ω,b∗
(cid:12)(cid:12)(cid:12)(cid:3)σ,b
(cid:12)(cid:12)(cid:12) dσ (y)
(cid:33) 1
2(cid:113)|I|σ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:3)ω,b∗
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(σ)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:3)σ,b
|x − cI|2(α−n) dω (x)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)

|x − cI|2(α−n) dω (x)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:3)ω,b∗

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:3)σ,b

Rn\(1+δ)I
Aα
2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(σ)

Rn\(1+δ)I

I f

g

dω

I f

I f

g

J

J

J

I f

I

dω

2

≤

(cid:46) δα−n

(cid:32)(cid:90)
(cid:32)(cid:90)
≤ δα−n(cid:113)

(cid:46) δα−n

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)

g

g

J

J

183

So combining all the above we get for the δ-separated cubes that

I∈D

I ≤ (cid:88)
≤δα−n(cid:113)
(cid:46)δα−n(cid:113)

J∈N (I)

(1+δ)I∩J=∅

(cid:88)
(cid:88)

I∈D

Aα
2

δα−n(cid:113)
(cid:88)

J∈N (I)

I f

(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)2

L2(σ)

J

g

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)L2(ω)
2(cid:88)

(cid:88)

I∈D

1

J∈N (I)

(1+δ)I∩J=∅

(5.4.7)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

g

(cid:13)(cid:13)(cid:13)2

L2(ω)

1
2



Aα
2

(1+δ)I∩J=∅
2||f||L2(σ)||g||L2(ω)
Aα

where the implied constant in the last line depends only on the goodness parameter r and

the finite repetition of I and J in each sum respectively.

5.4.2 The case of δ-close cubes.

Now we turn to the second sum in (5.4.6) which we will bound by using random surgery and

expectation.

Definition 5.4.5. We say that the cubes J and I are δ-close, if J ∩ (1 + δ)I (cid:54)= ∅.

We have

(cid:68)

T α
σ

(cid:16)(cid:3)σ,b

I f

(cid:17)

, (cid:3)ω,b∗

J

g

(cid:69)

=

ω

(cid:68)

I

f

(cid:17)
(cid:16)(cid:3)σ,(cid:91),b
(cid:16)(cid:3)σ,(cid:91),b
(cid:16)(cid:3)σ,(cid:91),b
(cid:16)(cid:3)σ,(cid:91),b

I

T α
σ

T α
σ

T α
σ

f

J

ω

g

(cid:69)
, (cid:3)ω,(cid:91),b∗
(cid:17)
, (cid:3)ω,(cid:91),b∗
(cid:17)
J,brokg
, (cid:3)ω,(cid:91),b∗
(cid:17)
J,brokg
, (cid:3)ω,(cid:91),b∗

(cid:69)
(cid:69)
(cid:69)

g

ω

J

I,brokf

I,brokf

T α
σ

(cid:68)
(cid:68)
(cid:68)

+

+

+

.

ω

(5.4.8)

ω

184

The estimation of the latter three inner products, i.e. those in which a broken operator

(cid:3)σ,(cid:91),b
I,brok

or (cid:3)ω,(cid:91),b∗

J,brok

arises, is simpler, but still requires the use of random surgery in order to

avoid the full testing condition that was available in one dimension. Indeed, recall that

(cid:3)σ,(cid:91),b

I,brokf =
(cid:3)ω,(cid:91),b∗
J,brokg =

(cid:88)
(cid:88)
I(cid:48)∈Cbrok(I)
J(cid:48)∈Cbrok(J)

(cid:16)

(cid:88)
(cid:88)
I(cid:48)∈Cbrok(I)
J(cid:48)∈Cbrok(J)

Eσ

(cid:17)
I(cid:48)(cid:98)Fσ,b
(cid:16)
J(cid:48)(cid:98)Fω,b∗

I(cid:48) f

Eω

J(cid:48)

bI(cid:48)

(cid:17)

g

b∗
J(cid:48)

Fσ,b
I(cid:48) f =
Fω,b∗
J(cid:48)

g =

so that if at least one broken difference appears in the inner product, as is the case for the

latter three inner products in (5.4.8), we need to use random surgery to get the necessary

bound. For example, the fourth term satisfies

(cid:12)(cid:12)(cid:12)(cid:68)

T α
σ

(cid:16)(cid:3)σ,(cid:91),b

I,brokf

(cid:17)

, (cid:3)ω,(cid:91),b∗

J

g

(cid:69)

(cid:12)(cid:12)(cid:12) =

ω

(cid:16)

(cid:88)
I(cid:48)∈Cbrok(I)

I(cid:48)(cid:98)Fσ,b

I(cid:48) f

Eσ

(cid:17)(cid:68)

σ bI(cid:48), (cid:3)ω,(cid:91),b∗

J

T α

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

and since

(cid:68)

σ bI(cid:48), (cid:3)ω,(cid:91),b∗

J

T α

(cid:69)

g

=

ω

(cid:68)

(cid:68)

1I(cid:48)∩J T α

σ bI(cid:48), (cid:3)ω,(cid:91),b∗
1(J\I(cid:48))∩(1+δ)I(cid:48)T α

J

+

(cid:69)
(cid:68)
σ bI(cid:48), (cid:3)ω,(cid:91),b∗

+

g

ω

J

(cid:69)

g

ω

1J\(1+δ)I(cid:48)T α

σ bI(cid:48), (cid:3)ω,(cid:91),b∗

J

(cid:69)

g

ω

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:69)

g

ω

≡ A(f, g) + B(f, g) + C(f, g)

185

we have

Eσ

(cid:88)
I(cid:48)∈Cbrok(I)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:16)
≤ Cb,b∗ (cid:88)
(cid:13)(cid:13)∇σ
I f(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)(cid:70)

I(cid:48)∈Cbrok(I)

≤ Tb
T α

(cid:46) Tb
T α

I f

L2(σ)

J

J

g

T α

I(cid:48) f

I(cid:48) f

A(f, g)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:113)|I(cid:48)|σ
(cid:12)(cid:12)(cid:12) Tb
(cid:13)(cid:13)(cid:13)(cid:3)ω,(cid:91),b∗
(cid:18)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:70)

(cid:17)
I(cid:48)(cid:98)Fσ,b
(cid:12)(cid:12)(cid:12)Eσ
I(cid:48)(cid:98)Fσ,b
(cid:32) (cid:88)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)≤ (cid:88)
(cid:12)(cid:12)(cid:12)Eσ
(cid:12)(cid:12)(cid:12) δα−n(cid:113)
I(cid:48)(cid:98)Fσ,b
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:3)σ,b
≤ δα−n(cid:113)

I(cid:48)∈Cbrok(I)
g

I(cid:48)∈Cbrok(I)

I(cid:48) f

L2(ω)

L2(ω)

I f

Aα
2

+

J

L2(σ)

Next by Lemma 5.4.3,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:16)
(cid:88)
I(cid:48)∈Cbrok(I)

I(cid:48)(cid:98)Fσ,b

I(cid:48) f

Eσ

(cid:17)

B(f, g)

g

(cid:13)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)(cid:3)ω,(cid:91),b∗

J,brokg

(cid:19)(cid:33) 1

2

(cid:13)(cid:13)(cid:13)2

L2(ω)

Aα
2

(cid:113)|I(cid:48)|σ
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

g

J

(cid:13)(cid:13)(cid:13)L2(ω)

g

(cid:13)(cid:13)(cid:13)(cid:3)ω,(cid:91),b∗
(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

J

186

Finally, using Cauchy-Schwarz, the norm inequality and accretivity we get

I∈D

(cid:88)

J∈N (I)
I∩J(cid:54)=∅

≤ CbNT α

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:88)
(cid:88)
(cid:18) (cid:88)
(cid:88)

(cid:88)
I(cid:48)∈Cbrok(I)
(cid:88)
(cid:104)
Eω
J(cid:48)
(cid:18)(cid:88)
≤ Cb,r,nNT α||f||L2(σ) ·
(cid:88)
I(cid:48)∈Cbrok(I)

J∈N (I)
I∩J(cid:54)=∅

J(cid:48)∈C(J)

Eσ

I(cid:48) f

C(f, g)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:17)
(cid:16)
I(cid:48)(cid:98)Fσ,b
(cid:12)(cid:12)(cid:12)(cid:112)|I(cid:48)|σ ·
(cid:12)(cid:12)(cid:12)Eσ
(cid:88)
I(cid:48)(cid:98)Fσ,b
I(cid:48)∈Cbrok(I)
(J\I(cid:48)) ∩ (1 + δ)I(cid:48)(cid:17) ∩ J(cid:48)(cid:12)(cid:12)(cid:12)ω
(cid:17)(cid:105)2(cid:12)(cid:12)(cid:12)(cid:16)
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:104)
(cid:88)

(cid:17)(cid:105)2(cid:12)(cid:12)(cid:12)(cid:16)

I(cid:48) f

Eω
J(cid:48)

J(cid:48)∈C(J)

I∈D

·

·

g

J

g

J

I∈D

J∈N (I)
I∩J(cid:54)=∅

(cid:19) 1

2

(J\I(cid:48)) ∩ (1 + δ)I(cid:48)(cid:17) ∩ J(cid:48)(cid:12)(cid:12)(cid:12)ω

(cid:19) 1

2

.

Now, it is geometrically evident that for the Lebesque measure we have

(cid:12)(cid:12)(cid:12)(cid:16)

(J\I(cid:48)) ∩ (1 + δ)I(cid:48)(cid:17) ∩ J(cid:48)(cid:12)(cid:12)(cid:12) (cid:46) δ|J(cid:48)|.

Taking averages over the grid D we get the same inequality for the ω measure:

(cid:12)(cid:12)(cid:12)(cid:16)

(J\I(cid:48)) ∩ (1 + δ)I(cid:48)(cid:17) ∩ J(cid:48)(cid:12)(cid:12)(cid:12)ω

(cid:46) δ(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω .

ED

Ω

Thus, if we fix J(cid:48), there are only finitely many I(cid:48) involved that contribute (are non-zero),
and then the expectation in D can "go through" the sum in I(cid:48) to get the estimate

(cid:88)

I∈D

ED

Ω

(cid:88)

J∈N (I)
I∩J(cid:54)=∅

(cid:16)

(cid:88)
I(cid:48)∈Cbrok(I)

(cid:17)

I(cid:48)(cid:98)Fσ,b

I(cid:48) f

Eσ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cb,r,n

C(f, g)

√
δNT α||f||L2(σ)||g||L2(ω).

187

The constant Cb,r,n depends on the accretivity constant of the family b, the dimension n
and the finite repetition of the intervals J(cid:48) appearing in the sum.

The third term in (5.4.8) is handled similarly if we change to(cid:68)(cid:3)σ,(cid:91),b

(cid:16)(cid:3)ω,(cid:91),b∗

(cid:17)(cid:69)

,

σ
the dual operator. For the second term in (5.4.8) the proof is somewhat different: it does

f, T α,∗

ω

J,brokg

I

not use probability, it is easier because the terms involving g can be estimated as the terms

involving f in the proof just done for the fourth term, and then using Carleson estimates.

So combining the above we get the following

(cid:17)

T α
σ

(cid:12)(cid:12)(cid:12)(cid:68)
(cid:16)(cid:3)σ,b
(cid:16)(cid:3)σ,(cid:91),b
(cid:17)

f

I

I f

J

g

, (cid:3)ω,b∗
(cid:69)

, (cid:3)ω,(cid:91),b∗

g

J

ED

Ω

≤ (cid:88)

I∈D

(cid:88)

(cid:88)

I∈D

J∈N (I)

(1+δ)I∩J(cid:54)=∅
T α
σ

(cid:12)(cid:12)(cid:12)(cid:68)

(cid:88)
(cid:16)

J∈N (I)

(1+δ)I∩J(cid:54)=∅
√

+

Cb,r,n

ω

(cid:12)(cid:12)(cid:12)

(cid:69)
(cid:12)(cid:12)(cid:12)
(cid:17)||f||L2(σ)||g||L2(ω)
(cid:16)(cid:3)σ,(cid:91),b

(cid:17)

ω

(5.4.9)

(cid:69)

ω

on the

δNT α + (δα−n + 1)NT Vα

Thus it remains to consider the first inner product (cid:68)

, (cid:3)ω,(cid:91),b∗
right hand side of (5.4.9), which we call the problematic term, and write it as

T α
σ

f

J

I

g

P (I, J) ≡ (cid:68)

T α
σ

(cid:17)

f

I

(cid:16)(cid:3)σ,(cid:91),b
(cid:68)
(cid:88)
(cid:88)

I(cid:48)∈C(I),J(cid:48)∈C(J)
Eσ
I(cid:48)
I(cid:48)∈C(I),J(cid:48)∈C(J)

T α
σ

J

(cid:69)
, (cid:3)ω,(cid:91),b∗
(cid:17)
(cid:16)
g
1I(cid:48)(cid:3)σ,(cid:91),b
(cid:17)(cid:10)T α
(cid:16)(cid:98)(cid:3)σ,(cid:91),b

f

f

ω

I

σ

I

=

=

(cid:69)
, 1J(cid:48)(cid:3)ω,(cid:91),b∗
(cid:1) , 1J(cid:48)b∗
(cid:0)1I(cid:48)bI

g

J

J

ω

(cid:11)

ω Eω
J(cid:48)

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

J

g

(cid:17)

.(5.4.10)

It now remains to show that

(cid:88)

(cid:88)

I∈D

J∈N (I)

Ω EG
ED

Ω

|P (I, J)| (cid:46)(cid:16)

CθNT Vα +

√
θNT α

(cid:17)(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .

(5.4.11)

188

Suppose now that I ∈ CA for A ∈ A, and that J ∈ CB for B ∈ B. Then the inner

product in the third line of (5.4.10) becomes

(cid:10)T α

σ

J 1J(cid:48)(cid:11)
(cid:0)bI 1I(cid:48)(cid:1) , b∗

ω =(cid:10)T α

σ

B1J(cid:48)(cid:11)
(cid:0)bA1I(cid:48)(cid:1) , b∗

ω ,

and we will write this inner product in either form, depending on context. We also introduce

the following notation:

P(I,J) (E, F ) ≡(cid:10)T α

σ (bI 1E) , b∗

J 1F

(cid:11)

ω ,

for any sets E and F,

so that

(cid:88)

I(cid:48)∈C(I) and J(cid:48)∈C(J)

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:17)

Eσ
I(cid:48)

(cid:0)I(cid:48), J(cid:48)(cid:1) Eω

J(cid:48)

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

J

(cid:17)

.

g

P(I,J)

P (I, J) =

The first thing we do is reduce matters to showing inequality (5.4.11) in the case that

(cid:0)I(cid:48), J(cid:48)(cid:1) is replaced by

P(I,J)

(cid:0)I(cid:48) ∩ J(cid:48), I(cid:48) ∩ J(cid:48)(cid:1)

P(I,J)

in the terms P (I, J) appearing in (5.4.11). To see this, write(cid:10)T α
(cid:68)
+(cid:10)T α

(cid:68)

(cid:69)

(cid:69)

(cid:17)

(cid:16)

+

σ

T α
σ

bI 1I(cid:48)\J(cid:48)

, b∗
J 1J(cid:48)

J 1J(cid:48)\I(cid:48)

T α
σ

J 1J(cid:48)(cid:11)
(cid:0)bI 1I(cid:48)(cid:1) , b∗
J 1I(cid:48)∩J(cid:48)(cid:11)
(cid:0)bI 1I(cid:48)∩J(cid:48)(cid:1) , b∗

as

ω

ω

ω

σ

ω

Set

(cid:68)

T α
σ

II =

I =

(cid:0)bI 1I(cid:48)∩J(cid:48)(cid:1) , b∗

J 1J(cid:48)\I(cid:48)

ω

(cid:69)
(cid:0)bI 1I(cid:48)∩J(cid:48)(cid:1) , b∗
and III =(cid:10)T α

, b∗
J 1J(cid:48)

ω

σ

J 1I(cid:48)∩J(cid:48)(cid:11)

ω

(cid:0)bI 1I(cid:48)∩J(cid:48)(cid:1) , b∗
(cid:17)
(cid:16)
(cid:68)
(cid:69)

bI 1I(cid:48)\J(cid:48)

T α
σ

189

For the first one, we have

(cid:17)

EG

Ω

Ω

(cid:17)

2

f

I

2

T α
σ

ω

(cid:16)

T α
σ

NT α

I(cid:48)

(cid:69)

ω

, b∗
J 1J(cid:48)

, b∗
J 1J(cid:48)

(cid:17) 1

2 ·

I∈D

J∈N (I)

bI 1I(cid:48)\(1+δ)J(cid:48)

I(cid:48)∈C(I)

J(cid:48)∈C(J)

bI 1(I(cid:48)\J(cid:48))∩(1+δ)J(cid:48)

|bI|2dσ
(I(cid:48)\J(cid:48))∩(1+δ)J(cid:48)

Summing all the terms for I2 and using Lemma 5.4.2, we have

(cid:12)(cid:12)(cid:12) ≡ I1 + I2
(cid:12)(cid:12)(cid:12) +
(cid:12)(cid:12)(cid:12)(cid:68)
I ≤(cid:12)(cid:12)(cid:12)(cid:68)
(cid:16)
(cid:69)
(cid:112)|I(cid:48)|σ
(cid:112)|J(cid:48)|ω and for I2 we need to use random surgery.
Using Lemma 5.4.3, I1 (cid:46) δα−n(cid:112)Aα
(cid:17)(cid:12)(cid:12)(cid:12)(cid:16)(cid:90)
(cid:12)(cid:12)(cid:12)Eσ
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:88)
(cid:88)
(cid:88)
(cid:88)
(cid:17)(cid:12)(cid:12)(cid:12)(cid:16)(cid:90)
·(cid:12)(cid:12)(cid:12) Eω
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:17) 1
(cid:12)(cid:12)(cid:12)Eσ
(cid:17)(cid:12)(cid:12)(cid:12) ·
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
J(cid:48) |bJ|2dω
(cid:88)
(cid:88)
(cid:88)
(cid:88)
(cid:12)(cid:12)(cid:12) Eω
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12) 1
·(cid:12)(cid:12)(cid:12)(I(cid:48)\J(cid:48)) ∩ (1 + δ)J(cid:48)(cid:12)(cid:12)(cid:12) 1
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:33)1
2(cid:18)(cid:88)(cid:104)
(cid:19)1
(cid:17)(cid:105)2(cid:12)(cid:12)(cid:12)(I(cid:48)\J(cid:48)) ∩ (1 + δ)J(cid:48)(cid:12)(cid:12)(cid:12)σ
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)(cid:105)2|J(cid:48)|ω
(cid:18)(cid:88)
(cid:19)1
(cid:12)(cid:12)(cid:12)(I(cid:48)\J(cid:48)) ∩ (1 + δ)J(cid:48)(cid:12)(cid:12)(cid:12)σ
(cid:104)
(cid:17)(cid:105)2
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:88)
(cid:88)
(cid:88)
(cid:18)(cid:88)
(cid:19) 1
(cid:104)
(cid:17)(cid:105)2
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:88)
√
δ||f||L2(σ)||g||L2(ω)

≤ NT αCn,r||g||L2(ω)
≤ NT αCn,r

≤ NT αCn,r||g||L2(ω)

J(cid:48)
(cid:46) NT αEG

I(cid:48)∈C(I)

J(cid:48)∈C(J)

≤ NT αEG

Ω

(cid:32)(cid:88)(cid:104)

I∈D

J∈N (I)

δ|I(cid:48)|σ

2

(5.4.12)

I(cid:48)

g

J(cid:48)

J

I(cid:48)

I(cid:48)

Eσ
I(cid:48)

Eσ
I(cid:48)

g

J

g

J

J(cid:48)

J

Eσ
I(cid:48)

f

I

f

f

I

I

f

I

2
ω

2
σ

I

I

Eω
J(cid:48)

EG

Ω

2

2

Similarly, we get the bound for II.

We are left then with III where we are integrating over I(cid:48) ∩ J(cid:48). We have to overcome two
difficulties at this step. First, I(cid:48) ∩ J(cid:48) is not necessarily a cube, so we cannot apply any of
the testing conditions available. Second, I(cid:48) ∩ J(cid:48), even if it is a cube, does not need to belong
in either of the grids D or G. We would like to split I(cid:48) ∩ J(cid:48) in smaller cubes of the grid G.

190

The problem is that the boundary of I(cid:48) ∩ J(cid:48) does not necessarily align with the grid G. To
deal with this, we cut a slice around I(cid:48) ∩ J(cid:48) so that what is left inside can be split in cubes
of the grid G. This small slice will be bounded using once again random surgery. While for
the remaining cubes, we will use a more involved random surgery technique along with the
A2 and testing condition.

Here are the details: Let η0 = 2−m for m large enough. For any cube L we define the

−→η1-halo for −→η1 = (η1

1, . . . , ηn

1 ) by

L = (1 + −→η1)L − (1 − −→η1)L

∂−→η1

where (1 + −→η1)L means a dilation of each coordinate of L according to the corresponding
coordinates of 1 +−→η1. Choose the coordinates of −→η1 such that η0
1 < η0 for all 1 ≤ i ≤ n
and such that if

2 ≤ ηi

(cid:20)(cid:16)

I(cid:48) ∩ J(cid:48) =

I(cid:48)\∂−→η1

I(cid:48)(cid:17) ∩ J(cid:48)(cid:21) ·∪

(cid:20)(cid:16)

I(cid:48) ∩ I(cid:48)(cid:17) ∩ J(cid:48)(cid:21)

∂−→η1

≡ M

·∪ L

(5.4.13)

then M consists of B (cid:46) 2n·m cubes Ks ∈ G with (cid:96)(Ks) ≥ 2−m−1(cid:96)(J(cid:48)). Note that either M
or L might be empty depending on where J(cid:48) is located, but this is not a problem. Thus

(cid:10)T α

σ

(cid:0)bI 1I(cid:48)∩J(cid:48)(cid:1) , b∗

J 1I(cid:48)∩J(cid:48)(cid:11)

ω =(cid:10)T α

(cid:11)
ω +(cid:10)T α

(cid:11)

σ (bI 1M ) , b∗

+(cid:10)T α
J 1L
σ (bI 1L) , b∗

σ (bI 1L) , b∗
(cid:11)

ω +(cid:10)T α

ω

J 1M
σ (bI 1M ) , b∗

J 1L

J 1M

(cid:11)

ω

The first two can be estimated using Lemma 5.4.3 and a random surgery. It is important to
mention here that the averages will be taken on the grid D, so that we do not have common

191

intersection among the different translations of the halo. Indeed,

(cid:10)T α
σ (bI 1M ) , b∗

J 1L

(cid:11)

(cid:68)
≡ A1 + A2

ω =

σ (bI 1M ) , b∗
T α

J 1L\(1+δ)M

σ (bI 1M ) , b∗
T α

J 1L∩(1+δ)M

(cid:68)

(cid:69)

+

ω

(cid:69)

ω

and

(cid:10)T α
σ (bI 1L) , b∗

J 1M

(cid:11)

ω =

σ (bI 1L) , b∗
T α

J 1M\(1+δ)L

(cid:68)
≡ A3 + A4

(cid:68)

(cid:69)

+

ω

σ (bI 1L) , b∗
T α

J 1M∩(1+δ)L

(cid:69)

ω

the left hand side of (5.4.11) we get by using Cauchy-Schwarz that

The first terms on the right hand side of both displays, A1 and A3, are bounded, by applying
the proof of Lemma 5.4.3 for M and L and using the fact that M consists of B (cid:46) 2nm cubes.

(cid:112)|J(cid:48)|ω, which when plugged into
(cid:112)|I(cid:48)|σ
The bound is a constant multiple of 2nδα−n(cid:112)Aα
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(A1 + A3)
(cid:12)(cid:12)(cid:12)(cid:12) Eω
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:17)(cid:12)(cid:12)(cid:12)δα−n(cid:113)
(cid:112)|J(cid:48)|ω
(cid:112)|I(cid:48)|σ

(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:16)(cid:98)(cid:3)σ,(cid:91),b

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

(cid:88)
(cid:88)

I(cid:48)∈C(I)
J(cid:48)∈C(J)

I(cid:48)

(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:12)(cid:12)(cid:12)Eσ

I(cid:48)

J∈N (I)

g

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)Eω

J(cid:48)

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)

(5.4.14)

f

f

Aα
2

g

J

J(cid:48)

J

2

I

I∈D

(cid:88)
(cid:88)
(cid:46) (cid:88)
(cid:88)
(cid:46) δα−n(cid:113)

I∈D

J∈N (I)

I

I(cid:48)∈C(I)
J(cid:48)∈C(J)
2||f||L2(σ)||g||L2(ω)
Aα

192

For A2 (and similarly for A4), we have

(cid:88)

(cid:88)

ED

Ω

I∈D

J∈N (I)

I(cid:48)∈C(I),J(cid:48)∈C(J)

≤ NT αCbED

Ω

I(cid:48)
I(cid:48)∈C(I)&J(cid:48)∈C(J)

J∈N (I)

≤ NT αCb,b∗,r,n

≤ NT αCb,b∗,r,n

I(cid:48)∈C(I)&J(cid:48)∈C(J)

(cid:18)
J∈N (I)
ED

Ω

·

I(cid:48)∈C(I)&J(cid:48)∈C(J)
J∈N (I)
√
δ||f||L2(σ)||g||L2(ω)

(5.4.15)

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)

g

·

(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)b∗
(cid:12)(cid:12)(cid:12)Eσ
(cid:88)

J 1L∩(1+δ)M )

g

J

·

f

I

f

I

I(cid:48)

J(cid:48)

σ (bI 1M )

(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)T α
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ·
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)
(cid:12)(cid:12)(cid:12)(cid:12) Eω
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:12)(cid:12)(cid:12)(cid:12) Eω
(cid:12)(cid:12)(cid:12)L ∩ (1 + δ)M
(cid:12)(cid:12)(cid:12) 1
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)M
(cid:12)(cid:12)(cid:12) 1
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)2|M|σ
(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)2|L ∩ (1 + δ)M|ω
(cid:12)(cid:12)(cid:12)(cid:12) Eω
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

(cid:88)

(cid:19)1

J(cid:48)

J(cid:48)

I(cid:48)

2
ω

2
σ

1
2

f

I

·

g

J

J

2

(cid:88)



√
δ, not √

by noting that (1 + δ)M ∩ L is a halo of width δ, much smaller than η0 (so as to get the
estimate by
η0 is easy to obtain (as L already has
√
δ will be crucially

width η0) and is sufficient for the purposes of this term, the estimate of
used later in (5.4.19) to kill the B term. Note also that we can take the averages over all

η0). Although an estimate of √

directions, so that we avoid common intersection along the different translations. Notice that

L, M are "moving" together. This is not a problem since by "moving" they cover different
parts of the cube J(cid:48).

Thus we only need to estimate (cid:10)T α

σ (bI 1L) , b∗

J 1L

σ (bI 1M ) , b∗

J 1M

. Applying

ω

(cid:11)

ω +(cid:10)T α

(cid:11)

193

one more time random surgery to the first term we get that

(cid:88)

(cid:88)

I∈D

J∈N (I)

Ω EG
ED

Ω

(cid:46) EG

ΩNT α (cid:107)f(cid:107)L2(σ) ED

Ω

I(cid:48)∈C(I)
J(cid:48)∈C(J)

I

f

I(cid:48)

(cid:12)(cid:12)(cid:12)Eσ
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17) (cid:10)T α
(cid:88)
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)
(cid:32)(cid:90)
(cid:88)
(cid:88)
(cid:88)
∂η1I(cid:48)∩J(cid:48)

J∈N (I)

I∈D

I(cid:48)∈C(I)
J(cid:48)∈C(J)

σ (bI 1L) , b∗

J 1L

(cid:12)(cid:12)b∗

J

(cid:12)(cid:12)2 dω

ω Eω
J(cid:48)

(cid:11)
(cid:33)(cid:12)(cid:12)(cid:12)Eω

J(cid:48)

J

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

J

g

(cid:17)(cid:12)(cid:12)(cid:12)
(cid:17)(cid:12)(cid:12)(cid:12)2

g

using (5.4.2) and the frame inequalities again. Then using Cauchy-Schwarz on the expecta-
tion ED

, this is dominated by

Ω



(cid:88)

(cid:12)(cid:12)(cid:12)∂−→η1

I(cid:48) ∩ J(cid:48)(cid:12)(cid:12)(cid:12)ω

(cid:12)(cid:12)(cid:12)Eω

J(cid:48)

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

J

(cid:17)(cid:12)(cid:12)(cid:12)2

g

I∈D: 2−rn|I|<|J|≤|I|
d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε

I(cid:48)∈C(I)

EG
ΩNT α(cid:107)f(cid:107)L2(σ)

ED

Ω

(cid:88)

J(cid:48)∈C(J)

J∈G

(cid:88)

(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)
(cid:118)(cid:117)(cid:117)(cid:117)(cid:116)(cid:88)

(cid:88)

ED

Ω

(cid:88)

(cid:12)(cid:12)∂η0I(cid:48) ∩ J(cid:48)(cid:12)(cid:12)ω

(cid:12)(cid:12)(cid:12)Eω

J(cid:48)

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

J

(cid:17)(cid:12)(cid:12)(cid:12)2

g

(cid:46) EG
(cid:46) √

ΩNT α (cid:107)f(cid:107)L2(σ)
2r
J(cid:48)∈C(J)
√
η0NT α (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) ≤

J∈G

I(cid:48)∈D:|J(cid:48)|≤|I(cid:48)|≤2r|J(cid:48)|
λNT α (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)

where in the last line we have used ηi

1 ≤ η0, and then
(cid:88)

(cid:12)(cid:12)∂η0I(cid:48) ∩ J(cid:48)(cid:12)(cid:12)ω

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:46) η0

ED

Ω

I(cid:48)∈D:|J(cid:48)|≤|I(cid:48)|≤2r|J(cid:48)|

194

This leaves us to estimate the term(cid:10)T α

as long as we choose η0 (cid:28) 2−r.
·(cid:91)

use the decomposition M =

1≤s≤B

(cid:10)T α
σ (bI 1M ) , b∗

J 1M

(cid:11)

ω =

which can be rewritten as

σ (bI 1M ) , b∗

J 1M

(cid:11)

ω

. It is at this point that we will

Ks constructed above. We have

B(cid:88)

(cid:68)

s,s(cid:48)=1

(cid:0)bI 1Ks

T α
σ

(cid:1) , b∗

J 1Ks(cid:48)

(cid:69)

ω

(cid:10)T α

σ

(cid:0)bI 1Ks

(cid:1) , b∗

J 1Ks

(cid:11)

ω +

B(cid:88)

s=1

(cid:18)(cid:88)(cid:88)

Ks ∼
Sep

Ks(cid:48)

(cid:88)(cid:88)

Ks ∼
Adj

Ks(cid:48)

+

(cid:19)(cid:68)

(cid:0)bI 1Ks

T α
σ

(cid:1) , b∗

J 1Ks(cid:48)

(cid:69)

ω

(5.4.16)

Sep

Ks(cid:48) the separated cubes, i.e. 3Ks ∩ Ks(cid:48) = ∅, while by Ks ∼

where we call Ks ∼
Ks(cid:48) are
estimated directly by(cid:112)Aα
the adjacent cubes, i.e. Ks ∩ Ks(cid:48) = ∅ and Ks ∩ Ks(cid:48) (cid:54)= ∅. The separated terms sum can be
(cid:68)
(cid:69)

. Indeed, as in the proof of Lemma 5.4.3,

Adj

2

(cid:0)bI 1Ks

(cid:1) , b∗

T α
σ

J 1Ks(cid:48)

ω

(cid:46)

(cid:32)(cid:90)
(cid:32)(cid:90)
(cid:46)(cid:113)

(cid:46)

Ks(cid:48)

Ks

(cid:19)2
|x − y|α−n |bI (y)| dσ (y)
(cid:33) 1
(cid:12)(cid:12)2α−2n dω(x)
2 |Ks|σ

(cid:18)(cid:90)
(cid:12)(cid:12)x − xKs
(cid:113)|Ks(cid:48)|ω
(cid:112)|Ks|σ

Rn\Ks
Aα
2

dω (x)

(cid:33) 1
2(cid:113)(cid:12)(cid:12)Ks(cid:48)(cid:12)(cid:12)ω
(cid:113)(cid:12)(cid:12)Ks(cid:48)(cid:12)(cid:12)ω

thus,

(cid:68)

T α
σ

(cid:0)bI 1Ks

(cid:69)

(cid:1) , b∗

J 1Ks(cid:48)

≤ Cb

ω

(cid:88)(cid:88)

Ks ∼
Sep

Ks(cid:48)

195

(cid:88)(cid:88)

(cid:113)

Ks ∼
Sep

Ks(cid:48)

Aα
2

(cid:112)|Ks|σ

(cid:113)|Ks(cid:48)|ω (5.4.17)

which plugged into (5.4.10) appropriately, we get the bound B(cid:112)Aα

2

To deal with the adjacent cubes term in (5.4.16), we write

(cid:112)|I(cid:48)|σ
(cid:16)

(cid:112)|J(cid:48)|ω.
(cid:17)(cid:69)

b∗
J 1Ks(cid:48)

σ

bI 1Ks, T α,∗

ω

(cid:1) , b∗
(cid:0)bI 1Ks
s(cid:48) , T α,∗

ω

T α
σ

Ks(cid:48)

bI 1Ks∩(1+δ)K

(cid:68)

(cid:68)

Ks ∼
Adj

(cid:88)(cid:88)
(cid:68)
(cid:88)(cid:88)

Ks(cid:48)

Ks ∼
Adj

Ks(cid:48)

bI 1Ks\(1+δ)K

(cid:69)

(cid:68)

(cid:88)(cid:88)
(cid:17)(cid:69)

Ks ∼
Adj

=

Ks(cid:48)

σ

b∗
s(cid:48)
J 1K

(cid:17)(cid:69)

σ

ω

J 1Ks(cid:48)

(cid:16)
b∗
J 1Ks(cid:48)
s(cid:48) , T α,∗

(cid:16)

ω

(cid:88)(cid:88)

Ks ∼
Adj

=

+

≡ ∼

I +

∼
II

∼
II we use Lemma 5.4.3 to get

For

∼

II (cid:46) δα−n(cid:113)
(cid:113)

(cid:46) δα−nB

Aα
2

 B(cid:88)
(cid:112)|I(cid:48)|σ

s=1

2

 1
(cid:112)|J(cid:48)|ω

|Ks|σ

Aα
2

 B(cid:88)

s=1

(cid:88)

s(cid:48)≥s

2 1

2

(cid:12)(cid:12)(cid:12)Ks(cid:48)

(cid:12)(cid:12)(cid:12) 1

2
ω

(5.4.18)

while summing

∼
I over

T = {I ∈ D, J ∈ N (I), I(cid:48) ∈ Cnat(I), J(cid:48) ∈ Cnat(J)}

196

and using Cauchy-Schwarz, accretivity, taking averages and using Jensen, we get

EG

Ω

(cid:46) EG

Ω

Ω

·

(cid:46) NT αEG
(cid:18) B(cid:88)
(cid:46) NT αEG
(cid:18) B(cid:88)

s=1

Ω

·

(cid:46) NT α

f

I(cid:48)

I(cid:48)

T

T

g

J

g

J

f

f

T

T

f

I

I

I(cid:48)

s=1

(cid:17)

K

s(cid:48)

Eω
J(cid:48)

Eω
J(cid:48)

Eω
J(cid:48)

NT α
Ks(cid:48)

Ks ∼
Adj

Ks ∼
Adj

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88)(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:68)
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)
(cid:88)
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88)(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)
(cid:88)
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) B(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:88)
(cid:17)2(cid:19) 1
(cid:113)|Ks ∩ (1 + δ)Ks(cid:48)|σ
(cid:16)(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:112)|J(cid:48)|ω ·
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:88)
(cid:17)(cid:19) 1
(cid:16)(cid:88)
|Ks ∩ (1 + δ)Ks(cid:48)|σ ·(cid:88)

(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:88)
(cid:32)(cid:88)
I(cid:48)∈Cnat(I)
(cid:12)(cid:12)(cid:12)(cid:12)Eσ
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:16)(cid:98)(cid:3)σ,(cid:91),b

B (cid:107)g(cid:107)L2(ω)

2nδ|I(cid:48)|σ

s≤s(cid:48)

s≤s(cid:48)

s≤s(cid:48)

Eω
J(cid:48)

I∈D

(cid:17)

√

√

s=1

I

I

f

I

f

I

g

g

J

J

2

1

I(cid:48)

I(cid:48)

I(cid:48)

2

(cid:88)
B(cid:88)
EG
J∈N (I)
(cid:33) 1
J(cid:48)∈Cnat(J)

s=1

Ω

2

(cid:46) NT α
(cid:46) NT α22n

B (cid:107)g(cid:107)L2(ω)
√

√

B

T

δ (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)

(5.4.19)

b∗
J 1Ks(cid:48)

(cid:17)(cid:69)
(cid:113)|Ks(cid:48)|ω

σ

ω

(cid:16)
bI 1Ks∩(1+δ)Ks(cid:48) , T α,∗
(cid:113)|Ks ∩ (1 + δ)Ks(cid:48)|σ
(cid:17) 1

|Ks(cid:48)|ω

2 ·

1
2



|Ks∩(1+δ)Ks(cid:48)|σ

(cid:88)

s≤s(cid:48)

because there are up to 2n adjacent cubes Ks(cid:48) for a given Ks. The implied constant depends
on r of the nearby form. Note that δ is independent of B or r and will later be chosen small

enough so that the terms containing the norm inequality constant will be absorbed.

197

Thus now we are left only with the first term of (5.4.16), i.e. we need to estimate

(cid:10)T α

σ

(cid:0)bI 1Ks

(cid:1) , b∗

J 1Ks

(cid:11)

ω

B(cid:88)

s=1

Before proceeding further it will prove convenient to introduce some additional notation,

namely we will write the energy estimate in the second display of the Energy Lemma as

(cid:90)

|(cid:104)T αν, ΨJ(cid:105)ω| (cid:46) Cγ,δ Pα

δ Qω (J, υ) (cid:107)ΨJ(cid:107)L2(µ)

if

ΨJ dω = 0 and γJ∩suppν = ∅ (5.4.20)

where

δ Qω (J, υ) ≡ Pα (J, ν)
Pα

|J|

(cid:13)(cid:13)(cid:13)Qω,b∗

J

x

(cid:13)(cid:13)(cid:13)♠

L2(ω)

Pα
1+δ (J, ν)

|J|

+

(cid:107)x − mJ(cid:107)L2(1J ω) .

The use of the compact notation Pα

δ Qω (J, υ) to denote the complicated expression on the

right hand side will considerably reduce the size of many subsequent displays.

We now consider the inner product(cid:10)T α

(cid:11)

σ (bA1K ) , b∗

B1K

ω

and estimate the case when

K ∈ G, K ⊂ I(cid:48) ∩ J(cid:48), I(cid:48) ∈ C (I) , J(cid:48) ∈ C (J) , I ∈ CA

A , J ∈ CB

B, (cid:96)(K) = 2−m−1(cid:96)(J(cid:48)).

For subsets E, F ⊂ A ∩ B and cubes K ⊂ A ∩ B we define

{E, F} ≡(cid:10)T α

σ (bA1E) , b∗

B1F

(cid:11)

ω ,

(5.4.21)

and Kin the 2n grandchildren of K that do not intersect the boundary of K while Kout the

198

rest 4n − 2n grandchildren of K that intersect its boundary i.e.

(cid:110)
K(cid:48)(cid:48) ∈ C(2) (K) : ∂K(cid:48)(cid:48) ∩ ∂K = ∅(cid:111)
K(cid:48)(cid:48) ∈ C(2) (K) : ∂K(cid:48)(cid:48) ∩ ∂K (cid:54)= ∅(cid:111)
(cid:110)

Kin =

Kout =

We can write

{K, K} = {A, Kin} − {A\K, Kin} + {Kout, Kout} + {Kin, Kout} .

(5.4.22)

Note that the first two terms on the right hand side of (5.4.22) decompose the inner product
{K, Kin}, which ‘includes’ one of the difficult symmetric inner product {Kin, Kin}, and
where the other difficult symmetric inner products are contained in {Kout, Kout}, which can
be handled recursively. Thus the difficult symmetric inner products are ultimately controlled
by testing on the cube A to handle the ‘paraproduct’ term {A, Kin}, and by using the
,
J∈G
discarded in the corona constructions above, to handle the ‘stopping’ term {A\K, Kin}.
are the testing functions obtained
More precisely, these original testing functions b

energy condition and a trick that resurrects the original testing functions (cid:110)

∗,orig
J

(cid:111)

b

∗,orig
J

after reducing matters to the case of bounded testing functions.

The first term on the right side of (5.4.22) satisfies

|{A, Kin}| =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:90)
≤ (cid:13)(cid:13)b∗

B

Kin

σ bA) b∗
(T α

(cid:13)(cid:13)(cid:13)1Kin

(cid:13)(cid:13)∞

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:13)(cid:13)(cid:13)1Kin
(cid:13)(cid:13)(cid:13)L2(ω)

(cid:13)(cid:13)(cid:13)L2(ω)
(cid:113)|Kin|ω .

T α
σ bA

Bdω

T α
σ bA

(cid:13)(cid:13)(cid:13)1Kin

b∗

B

(cid:13)(cid:13)(cid:13)L2(ω)

(5.4.23)

199

We now turn to the term {A\K, Kin}. Decompose 1Kin
2n(cid:88)

2n(cid:88)

(cid:90)

 +

b∗
Bdω

b∗
B =

1Kin

1

K(cid:96)
in

(cid:96)=1

(cid:96)=1

in

K(cid:96)
in

(cid:12)(cid:12)(cid:12)ω

b∗
B − 1(cid:12)(cid:12)(cid:12)K(cid:96)
b∗
B − 1(cid:12)(cid:12)(cid:12)K(cid:96)

K(cid:96)
in

1

in

≡ 2n(cid:88)

(cid:96)=1

k∗
Kin

(cid:90)

(cid:12)(cid:12)(cid:12)ω

b∗
Bdω

K(cid:96)
in

and then apply the Energy Lemma to the function

(cid:90)

K(cid:96)
in

(cid:12)(cid:12)(cid:12)ω

b∗
Bdω,

b∗

B

as

in

1

K(cid:96)
in

1(cid:12)(cid:12)(cid:12)K(cid:96)
 ≡ 2n(cid:88)

j=1

∗,j
k
Kin

= 0 unless K(cid:48) is a dyadic subcube of K that is contained
(Furthermore, we could even replace grandchildren by m-grandchildren in this
= 0 unless K(cid:48) is a dyadic m-grandchild of K that is

Kin

which does indeed satisfy (cid:3)ω,b∗
K(cid:48) k∗
in Kin.
argument in order that (cid:3)ω,b∗
K(cid:48) k∗
contained in Kin, but we will not need this.) We obtain
(cid:17)
(cid:42)

bA1A\K

, 1Kin

(cid:68)

(cid:69)

(cid:16)

(cid:16)

T α
σ

Kin

b∗

=

B

ω

(cid:69)

ω

, k∗

Kin

(cid:90)

(cid:12)(cid:12)(cid:12)ω

in

(cid:43)

ω

b∗
Bdω

(5.4.24)

(cid:16)

+

T α
σ

bA1A\K

and

(cid:16)

(cid:12)(cid:12)(cid:12)(cid:68)

T α
σ

bA1A\K

(cid:17)

(cid:69)

ω

, k∗

Kin

(cid:12)(cid:12)(cid:12) ≤ 2n(cid:88)

∗,(cid:96)
Kin

, k

(cid:17)(cid:13)(cid:13)(cid:13)k∗

ω

(cid:96)=1

≤ Cη0,n

(cid:13)(cid:13)(cid:13)L2(ω)
Kin ∩ (A\K) = ∅ , and where we have written(cid:110)

depends on the constant Cγ in the statement of the Monotonicity
since

in, 1A\K σ

(cid:111)2n

Kin

Pα

(cid:96)=1

1

K(cid:96)
in

1−η0

(cid:96)=1

where the constant Cη0
Lemma with γ = 1
1−η0

(cid:68)
2n(cid:88)

T α
σ

,

(cid:96)=1

(cid:17)

T α
σ

(cid:12)(cid:12)(cid:12)(cid:68)
(cid:16)
 2n(cid:88)

1

K(cid:96)
in

bA1A\K

(cid:17)
 1(cid:12)(cid:12)(cid:12)K(cid:96)
(cid:17)
δ Qω(cid:16)

K(cid:96)

bA1A\K

K(cid:96)
in

(cid:69)

(cid:12)(cid:12)(cid:12)

200

(cid:88)

in the Energy Lemma can be taken to be pseudo-

with K(cid:96)
in

denoting the innner grandchildren of K.

Thus we see that Pω,b∗
H

and Qω,b∗
(cid:88)
H

Kin

(cid:3)ω,b∗

and Qω,b∗

projection onto Kin, i.e. Pω,b∗
see below that the cubes Kin that arise in subsequent arguments will be pairwise disjoint.
Furthermore, the energy condition will be used to control these full pseudoprojections Pω,b∗
when taken over pairwise disjoint decompositions of cubes by subcubes of the form Kin.

=
J∈G: J⊂Kin

=
J∈G: J⊂Kin

, and we will

(cid:52)ω,b∗

Kin

Kin

J

J

However, the second line of (5.4.24) remains problematic because we cannot use any type
does not necessarily belong to CB, and this is our point

with b∗

since K(cid:96)
in

of testing in K(cid:96)
in

B

in which we exploit the original testing functions b

∗,orig
K(cid:96)
in

.

5.4.2.1 Return to the original testing functions

From the discussion above, we recall the identity (5.4.24) and the estimate (5.4.25). We also

have the analogous identity and estimate with b

∗,orig
K(cid:96)
in

in place of 1Kin

b∗

B

:

(cid:28)

T α
σ

(cid:16)

(cid:17)
(cid:42)

bA1A\K

(cid:29)

ω

(cid:16)

, b

∗,orig
K(cid:96)
in

(cid:16)

(cid:42)

=

T α
σ

bA1A\K

(cid:17)

, 1

K(cid:96)
in

b
(cid:17)

∗,orig
K(cid:96)
in

(cid:90)
− 1(cid:12)(cid:12)(cid:12)K(cid:96)
(cid:12)(cid:12)(cid:12)ω
 1(cid:12)(cid:12)(cid:12)K(cid:96)
(cid:90)
(cid:12)(cid:12)(cid:12)ω

in

in

b

∗,orig
K(cid:96)
in

dω

K(cid:96)
in

b

∗,orig
K(cid:96)
in

dω

K(cid:96)
in

(cid:43)
(cid:43)

ω

ω

+

T α
σ

bA1A\K

, 1

K(cid:96)
in

(5.4.25)

201

and

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

T α
σ

(cid:42)
(cid:16)
δ Qω(cid:16)

(cid:17)

bA1A\K

, 1

(cid:46) Pα

K(cid:96)

in, 1A\K σ

b
(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)1

K(cid:96)
in

K(cid:96)
in

(cid:90)
(cid:12)(cid:12)(cid:12)ω
− 1(cid:12)(cid:12)(cid:12)K(cid:96)
− 1(cid:12)(cid:12)(cid:12)K(cid:96)

in

∗,orig
K(cid:96)
in

in

∗,orig
K(cid:96)
in

b

K(cid:96)
in

(cid:90)

(cid:12)(cid:12)(cid:12)ω

(cid:43)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(ω)

ω

b

∗,orig
K(cid:96)
in

dω

b

∗,orig
K(cid:96)
in

dω

K(cid:96)
in

(5.4.26)

for 1 ≤ (cid:96) ≤ 2n, where the implied constants depend on L∞ norms of testing functions and
the constant in the Energy Lemma. Using the notation

(cid:110)

Kout, K(cid:96)
in

(cid:28)

(cid:111)orig ≡

(cid:29)

T α
σ bA1Kout

, b

∗,orig
K(cid:96)
in

ω

for 1 ≤ (cid:96) ≤ 2n.

note that

(cid:96)=1

{A\K, Kin} +

2n(cid:88)
= {A\K, Kin} − 2n(cid:88)
(cid:82)
(cid:82)

2n(cid:88)

(cid:32)

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω

(cid:96)=1

(cid:96)=1

+

≡ B + C

K(cid:96)
in

(cid:32)
(cid:32)

(cid:82)
(cid:82)

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω
1
|K(cid:96)
in|ω
1
in|ω
|K(cid:96)
b∗
Bdω
K(cid:96)
in
∗,orig
B

dω

b

(cid:33)(cid:110)
(cid:33)(cid:28)

K(cid:96)
in

(cid:82)
(cid:82)
(cid:33)(cid:20)(cid:28)

K(cid:96)
in

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

dω

dω

(cid:111)orig
(cid:17)

Kout, K(cid:96)
in

(cid:16)

T α
σ

bA1A\K

(cid:29)

−(cid:68)

ω

T α
σ (bA1A) , b

∗,orig
K(cid:96)
in

bA1Kin

, T α,∗
ω b

∗,orig
K(cid:96)
in

(cid:29)

ω

, b

∗,orig
K(cid:96)
in

(cid:21)

(cid:69)

σ

Now for B, using Energy Lemma to the function

(cid:32)

Ψ(cid:96)

J =

(cid:82)

(cid:82)

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

dω

K(cid:96)
in

 1(cid:12)(cid:12)(cid:12)K(cid:96)

in

(cid:12)(cid:12)(cid:12)ω

(cid:90)

K(cid:96)
in

 1

b∗
Bdω

K(cid:96)
in

(cid:33)

b

∗,orig
K(cid:96)
in

−

202

(cid:16)

(cid:17)

, 1

(cid:69)

K(cid:96)
in

ω

(cid:12)(cid:12)(cid:12)(cid:12)

bA1A\K

for 1 ≤ (cid:96) ≤ 2n we have

b∗

B

, 1Kin

(cid:12)(cid:12)(cid:12)(cid:12)(cid:68)

+O

T α
σ

bA1A\K

(cid:16)
(cid:17)
(cid:20) 2n(cid:88)
(cid:18)Pα(K(cid:96)
(cid:20) 2n(cid:88)
(cid:18)Pα
 2n(cid:88)
δ Qω(cid:16)

K(cid:96)

Pα

+O

(cid:96)=1

(cid:96)=1

(cid:96)=1

|B|=

(cid:46)

in1A\K σ)
in|
|K(cid:96)
1+δ(K(cid:96)
in1A\K σ)
in|
|K(cid:96)

in, 1A\K σ

(cid:12)(cid:12)(cid:12)ω

T α
σ

(cid:90)

K(cid:96)
in

b∗
Bdω

(cid:68)
(cid:19)(cid:21)(cid:112)|Kin|ω
(cid:19)(cid:21)(cid:112)|Kin|ω
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)

L2(ω)

K(cid:96)
in

(cid:69)

in

ω

(cid:96)=1

 1(cid:12)(cid:12)(cid:12)K(cid:96)
− 2n(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Qω,b∗
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)♠
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x − m
(cid:17)(cid:113)|Kin|ω

K(cid:96)
in

x

having used the triangle inequality to get

(cid:12)(cid:12)(cid:12)(cid:12)Ψ(cid:96)

J

(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)

(cid:46)

(cid:82)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:82)

b∗
Bdω
K(cid:96)
in
∗,orig
b
B

K(cid:96)
in

(cid:113)|K(cid:96)
in|ω (cid:46)(cid:112)|Kin|ω,

1 ≤ (cid:96) ≤ 2n

in|ω +

(cid:12)(cid:12)(cid:12)ω
(cid:32)

in

(cid:90)

K(cid:96)
in

b∗
Bdω

b∗
B − 1
|K(cid:48)(cid:48)
l |ω

(cid:69)

(cid:12)(cid:12)(cid:12)(cid:12)

ω

, 1

K(cid:96)
in

(cid:68)
(cid:90)

k(cid:48)(cid:48)

(cid:96)

(cid:16)

T α
σ

b∗
Bdω

bA1A\K

(cid:17)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)

(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

dω

(cid:113)|K(cid:96)
 1(cid:12)(cid:12)(cid:12)K(cid:96)
− 2n(cid:88)
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1K(cid:48)(cid:48)
2n(cid:88)
(cid:17)(cid:113)|Kin|ω

(cid:96)=1

(cid:96)=1

(cid:96)

and

(cid:12)(cid:12)(cid:12)(cid:12)(cid:68)

(cid:46)

(cid:46)

T α
σ

bA1A\K

(cid:16)
 2n(cid:88)
 2n(cid:88)

(cid:96)=1

(cid:17)
δ Qω(cid:16)
δ Qω(cid:16)

(cid:69)

ω

b∗

B

, 1Kin

K(cid:96)

in, 1A\K σ

K(cid:96)

in, 1A\K σ

Pα

Pα

(cid:96)=1

203

where in the last inequality we used accretivity and triangle inequality. We turn our attention

in term C. We have that

(cid:29)

ω

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

∗,orig
K(cid:96)
in

(cid:33)(cid:28)
(cid:12)(cid:12)(cid:12)(cid:12)b

T α
σ (bA1A) , b

(cid:12)(cid:12)(cid:12)(cid:12)2

∗,orig
K(cid:48)(cid:48)
(cid:96)

dω

(cid:69)

σ

, T α,∗
ω b

∗,orig
K(cid:96)
in

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≡ I + II + III
(cid:69)

T α,∗
ω b

∗,orig
K(cid:96)
in

σ

bA1Kin

, 1

K(cid:96)
in

Kin\(1+δ)K(cid:96)

in

, T α,∗
ω b

∗,orig
K(cid:96)
in

(cid:69)

σ

bA1

(cid:69)

σ

, T α,∗
ω b

∗,orig
K(cid:96)
in

(Kin\K(cid:96)

in)∩(1+δ)K(cid:96)

in

Also,

where

(cid:46)

(cid:46)

(cid:96)=1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 2n(cid:88)
(cid:32)
(cid:115)(cid:90)
2n(cid:88)
(cid:115)(cid:90)

(cid:96)=1

Kin

(cid:82)

(cid:82)

(cid:82)

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω
|T α
σ bA|2 dω

K(cid:96)
in

K(cid:48)(cid:48)

(cid:96)

|T α
σ bA|2 dω

K(cid:96)
in

dω

(cid:118)(cid:117)(cid:117)(cid:116)(cid:90)
(cid:113)|Kin|ω
(cid:33)(cid:68)

bA1Kin

(cid:32)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 2n(cid:88)

(cid:96)=1

(cid:82)

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

dω

K(cid:96)
in

(cid:96)=1

I =

(cid:32)
2n(cid:88)
(cid:32)
2n(cid:88)
(cid:32)
2n(cid:88)

(cid:96)=1

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω

(cid:82)
(cid:82)

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω

(cid:82)
(cid:82)

II =

(cid:82)

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

K(cid:96)
in

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

K(cid:96)
in

dω

(cid:33)(cid:68)
(cid:33)(cid:68)
(cid:33)(cid:68)

dω

bA1

III =

(cid:96)=1

(cid:82)

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

dω

K(cid:96)
in

The first term I is bounded using the dual testing condition. Indeed,

I ≤(cid:12)(cid:12)(cid:12)(cid:12)bA1Kin

(cid:12)(cid:12)(cid:12)(cid:12)L2(σ)

2n(cid:88)

(cid:96)=1

(cid:113)|K(cid:96)
in|ω ≤ 2nT∗Cb∗(cid:12)(cid:12)(cid:12)(cid:12)bA1Kin

(cid:12)(cid:12)(cid:12)(cid:12)L2(σ)

(cid:112)|Kin|ω

T∗Cb∗

204

The second term II is bounded using Lemma 5.4.3. Indeed,

δα−n(cid:113)
II ≤ 2n(cid:88)
≤ 2nδα−n(cid:113)

(cid:96)=1

(cid:113)|Kin\(1 + δ)K(cid:96)
(cid:112)|Kin|σ
(cid:112)|Kin|ω

Aα
2

Aα
2

(cid:113)|K(cid:96)

in|ω

in|σ

Finally,

III ≤ 2n(cid:88)

σ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)T α
(cid:0)bA1
(cid:18) 2n(cid:88)
(cid:112)CbCb∗
≡ (cid:112)CbCb∗ · ∆(K)

≤ NT α

(cid:96)=1

(cid:96)=1

(Kin\K(cid:96)

(cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)b

in

in)∩(1+δ)K(cid:96)

(cid:12)(cid:12)(cid:12)(Kin\K(cid:96)

in) ∩ (1 + δ)K(cid:96)

in

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)
(cid:19) 1
2(cid:112)|Kin|ω

∗,orig
K(cid:96)
in

(cid:12)(cid:12)(cid:12)σ

where we have defined

∆ (K) = NT α

(cid:18) 2n(cid:88)

(cid:96)=1

(cid:12)(cid:12)(cid:12)(Kin\K(cid:96)

in) ∩ (1 + δ)K(cid:96)

in

(cid:19) 1
2(cid:112)|Kin|ω

(cid:12)(cid:12)(cid:12)σ

This last term will be iterated and a final random surgery will give us the desired bound.

5.4.2.2 A finite iteration and a final random surgery.

Letting

ΦA,B(Kin) =

T α
σ

(cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(ω)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1Kin
(cid:0)bA
δ Qω(cid:16)
2n(cid:88)
Tα + Tα,∗ + δα−n(cid:113)
(cid:16)

(cid:113)|Kin|ω
(cid:17)(cid:113)|Kin|ω
(cid:17)(cid:113)|Kin|σ

in, 1A\K σ

K(cid:96)

Pα

(cid:96)=1

Aα
2

+

+

(5.4.27)

(cid:113)|Kin|ω

205

and simplifying more our notation

{Kout, Kin}orig ≡ 2n(cid:88)

(cid:96)=1

(cid:32)

(cid:82)

(cid:82)

1
|K(cid:96)
1
|K(cid:96)

in|ω
in|ω

b∗
Bdω
K(cid:96)
in
∗,orig
B

b

dω

K(cid:96)
in

(cid:33)(cid:110)

Kout, K(cid:96)
in

(cid:111)orig

we have so far that (5.4.22) is written as

{K, K} = {Kout, Kin}orig + {Kout, Kout} + {Kin, Kout} + O(cid:0)ΦA,B(Kin) + ∆(K)(cid:1)

Now

{Kout, Kout} =

(cid:88)

(cid:96)

(cid:88)

(cid:88)

{K(cid:96)

out, K(cid:96)

out} +

{K(cid:96)

out, Km

out} +

{K(cid:96)

out}
out, Km

m(cid:54)=(cid:96)
out∩K(cid:96)
Km

out=∅

m(cid:54)=(cid:96)
out∩K(cid:96)
Km

out(cid:54)=∅

where K(cid:96)

out, 1 ≤ (cid:96) ≤ 4n − 2n, are the outer grandchildren of K. For the second sum above,

we get

(cid:12)(cid:12)(cid:12)(cid:12) (cid:88)

m(cid:54)=(cid:96)
out∩K(cid:96)
Km

out=∅

{K(cid:96)

out}
out, Km

(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:113)
(cid:46) (cid:113)

(cid:113)|K(cid:96)
(cid:88)
(cid:112)|Kout|σ

(cid:88)
(cid:112)|Kout|ω

out|σ
m(cid:54)=(cid:96)
out∩K(cid:96)
Km

(cid:96)

Aα
2

Aα
2

out=∅

(cid:113)|Km

out|ω

where the implied constant depends on dimension and the accretivity of functions involved

and since dist(K(cid:96)

out, Km

out) ≥ (cid:96)(K(cid:96)

out) there is no δ. For the third sum, we need to use random

206

surgery again. Using Lemma 5.4.3,

out, Km

out}| =

(cid:19)

T α
σ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:28)
(cid:18)
(cid:19)
(cid:113)|Km

out

B

b∗

(cid:29)
(cid:12)(cid:12)(cid:12)(cid:12) +
(cid:12)(cid:12)(cid:12)(cid:12)(cid:28)
(cid:29)
(cid:113)|Km

ω
T α
σ
out|ω

(cid:12)(cid:12)(cid:12)(cid:12)
(cid:18)
(cid:113)|K(cid:96)

out

K(cid:96)

bA1

, 1Km
out
b∗
out|ω + NT α

, 1Km
out

B

ω

bA1

out\(1+δ)Km
K(cid:96)

(cid:113)|K(cid:96)

out|σ

Aα
2

|{K(cid:96)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:28)
(cid:18)
≤ δα−n(cid:113)

T α
σ

≤

(cid:19)

b∗

B

, 1Km
out

(cid:29)

(cid:12)(cid:12)(cid:12)(cid:12)

ω

bA1
out∩(1+δ)Km
K(cid:96)
out ∩ (1 + δ)Km

out|σ

out

Thus, summing

m(cid:54)=(cid:96)
(cid:96)
out∩K(cid:96)
Km

(cid:88)
(cid:46) δα−n(cid:113)
≤δα−n(cid:113)

Aα
2

out(cid:54)=∅

(cid:88)
(cid:112)|Kout|σ
(cid:112)|Kout|σ

Aα
2

|{K(cid:96)

out, Km

out}|

m(cid:54)=(cid:96)
(cid:96)
out∩K(cid:96)
Km

(cid:88)
(cid:88)
(cid:112)|Kout|ω + NT α
(cid:18)(cid:88)
(cid:88)
(cid:112)|Kout|ω + NT α
(cid:18)(cid:88)
(cid:88)

m(cid:54)=(cid:96)

(cid:96)

out(cid:54)=∅
|K(cid:96)

E(K) = NT α

(cid:96)

m(cid:54)=(cid:96)

|K(cid:96)

out ∩ (1 + δ)Km

out|σ

(5.4.28)

(cid:113)|K(cid:96)

out ∩ (1 + δ)Km

out|σ

(cid:113)|Km

out|ω

(cid:19)1
2(cid:112)|Kout|ω

out ∩ (1 + δ)Km

out|σ

(cid:19) 1
2(cid:112)|Kout|ω

Let

We will iterate this term below and we will the necessary bound. We now turn to {Kin, Kout}
and we have

|{Kin, Kout}|

(cid:12)(cid:12)(cid:12)(cid:12)(cid:68)
(cid:16)
(cid:46) δα−n(cid:113)

T α
σ

≤

(cid:112)|Kout|σ

bA1Kout\(1+δ)Kin
Aα
2

(cid:12)(cid:12)(cid:12)(cid:12) +
(cid:12)(cid:12)(cid:12)(cid:12)(cid:68)
(cid:17)
(cid:69)
(cid:112)|Kin|ω + NT α
(cid:112)|Kin|ω

, 1Kin

b∗

B

ω

(cid:16)
(cid:17)
(cid:112)|Kout ∩ (1 + δ)Kin|σ

bA1Kout∩(1+δ)Kin

T α
σ

(cid:69)

ω

(cid:12)(cid:12)(cid:12)(cid:12)

b∗

B

, 1Kin

207

and similarly |{Kout, Kin}orig | is bounded by

(cid:46) δα−n(cid:113)

(cid:112)|Kout|σ

Aα
2

(cid:112)|Kin|ω
(cid:112)|Kin|ω + NT α
(cid:112)|Kout ∩ (1 + δ)Kin|σ

(cid:112)|Kout ∩ (1 + δ)Kin|σ
(cid:112)|Kin|ω

F(K) = NT α

Let

Using the bounds we found above we have from (5.4.22),

|{K, K}| (cid:46)

4n−2n(cid:88)

(cid:96)=1

|{K(cid:96)

out, K(cid:96)

out}| + O(cid:0)ΦA,B(Kin)(cid:1)
(cid:113)

+∆(K) + E(K) + F(K) + Cδ,η0,b,b∗

Aα
2

(cid:112)|K|σ

(cid:112)|K|ω

Iterating the first term above a finite number of times, using again the norm inequality and
a final random surgery we get the bound we need. Indeed, for ν ∈ N

|{K, K}| ≤ (cid:88)

M∈Mν

|{M, M}| + O

(cid:113)

+Cδ,η0,b,b∗

Aα
2

 (cid:88)
(cid:104)
(cid:113)|M|σ
(cid:88)

M∈M∗

ν

(cid:105)
(cid:113)|M|ω

ΦA,B (Min)

M∈M∗

ν



+ ∆(M ) + E(M ) + F(M )

≡ A (K) + B (K) + C (K) = A(I(cid:48),J(cid:48)) (K) + B(I(cid:48),J(cid:48)) (K) + C(I(cid:48),J(cid:48)) (K) ,(5.4.29)

208

where the collections of cubes Mν = Mν (K) and M∗
by

ν = M∗

ν (K) are defined recursively

(cid:110)

M (cid:96)

out

(cid:111)

,

k ≥ 0,

M0 ≡ {K} ,

Mk+1 ≡ (cid:91)
ν ≡ ν(cid:91)

M∗

M∈Mk

Mk .

k=0

We will include the subscript (cid:0)I(cid:48), J(cid:48)(cid:1) in the notation when we want to indicate the pair
(cid:0)I(cid:48), J(cid:48)(cid:1) that are defined after (5.4.13). Now the term C (K) can be estimated by
(cid:113)|K|ω

(cid:113)|M|ω ≤ νCδ,η0,b,b∗

C (K) = Cδ,η0,b,b∗

(cid:113)|M|σ

(cid:113)|K|σ

(cid:88)

(5.4.30)

(cid:113)

(cid:113)

Aα
2

Aα
2

M∈M∗

ν

where ν is chosen below depending on η0. For the first term A (K), we will apply the norm
inequality and use probability, namely

 (cid:88)

M∈Mν

EG

Ω

 ≤ ε(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ ,

|M|σ

209

|A (K)| ≤ (cid:112)CbCb∗NT α
≤ (cid:112)CbCb∗NT α
≤ (cid:112)CbCb∗NT α

M∈Mν

(cid:88)
(cid:115) (cid:88)
(cid:115) (cid:88)

M∈Mν

M∈Mν

(cid:113)|M|σ

|M|σ

|M|σ

(cid:113)|M|ω
(cid:115) (cid:88)
(cid:113)|K|ω,

M∈Mν

|M|ω

where(cid:112)CbCb∗ is an upper bound for the testing functions involved, followed by

for a sufficiently small ε > 0, where roughly speaking, we use the fact that the cubes M ∈ Mν
depend on the grid G and form a relatively small proportion of I(cid:48), which captures only a
as the grid is translated relative to the grid D that

small amount of the total mass (cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ

contains I(cid:48).

Here are the details. Recall that the cubes K are taken from the set of consecutive cubes
4nν (cid:96) (Ki), and that

that lie in I(cid:48) ∩ J(cid:48), that the cubes M ∈ Mν (Ki) have length 1

{Ki}B
there are (4n − 2n)ν such cubes in Mν (Ki) for each i. Thus we have

i=1

(cid:18)4n − 2n

4n

and

1

|M| =

(cid:88)

M∈Mν (K)

M∈Mν (K)

4nν |K|

4nν |K| = (4n − 2n)ν 1

(cid:88)
(cid:19)ν → 0 as ν → ∞, which implies
 B(cid:88)
 ≤ B
(cid:17)ν ≤ ε. Then we have by Cauchy-Schwarz
(cid:16) 4n−2n

(cid:19)ν(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ ≤ ε(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ

(cid:18)4n − 2n

M∈Mν(Ki)

(cid:88)

|M|σ

EG

4n

i=1

Ω

where we have used that the variable B is at most 2nm and where the final inequality holds

if ν is chosen large enough such that B

applied first to

M∈Mν(Ki)

i=1

(cid:88)

B(cid:88)
 B(cid:88)
≤ (cid:112)CbCb∗NT α
≤ (cid:112)CbCb∗NT α

EG

i=1

Ω

|A (Ki)|

4n
and then to EG

,

Ω

Ω

(cid:112)CbCb∗NT α
(cid:88)

 ≤ EG
(cid:118)(cid:117)(cid:117)(cid:117)(cid:116)EG
B(cid:88)
(cid:113)|J(cid:48)|ω =(cid:112)CbCb∗
(cid:113)
ε|I(cid:48)|σ

M∈Mν(Ki)

|M|σ

i=1

Ω

(cid:118)(cid:117)(cid:117)(cid:117)(cid:116) B(cid:88)
(cid:113)|J(cid:48)|ω

i=1

√

εNT α

(cid:88)

(cid:113)|J(cid:48)|ω

|M|σ

(5.4.31)

M∈Mν(Ki)

(cid:113)|I(cid:48)|σ

(cid:113)|J(cid:48)|ω,

210

as required.

(cid:88)

Now we turn to summing up the remaining terms

ΦA,B (Min) + ∆(M ) + E(M ) + F(M ) above. In the case when the cube

ν

M∈M∗

B (K) = C
I(cid:48) is a natural child of I, i.e. I(cid:48) ∈ Cnat (I) so that I(cid:48) ∈ CA
(cid:88)

(cid:88)

(cid:90)

A

σ bA|2 dω ≤
|T α

(cid:13)(cid:13)(cid:13)1Min

T α
σ bA

=
L2(ω)
M∈M∗

Min

ν (K)

M∈M∗

ν (K)

(cid:13)(cid:13)(cid:13)2

, we have

(cid:90)
σ bA|2 dω (cid:46)(cid:16)
I(cid:48) |T α

(cid:17)2(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ

Tb
T α

by the weak testing condition for I(cid:48) in the corona CA. Also,

(cid:88)

|Min|ω ≤ |K|ω ≤(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

M∈M∗

ν (K)

because of the crucial fact that the cubes {Min}M∈M∗
composition of K ⊂ I(cid:48) ∩ J(cid:48) (for any ν ≥ 1). Of course, this implies

ν (K)

form a pairwise disjoint subde-

 (cid:88)

M∈M∗

ν (K)

(TT α,∗ + Aα

2 )2 |Min|σ

2 (cid:88)
1

M∈M∗

|Min|ω

ν (K)

1
(cid:46)(cid:16)

2

(cid:17)(cid:113)|I(cid:48)|σ |J(cid:48)|ω

TT α,∗ + Aα

2

and using the definition of Pα

δ Qω (J, υ) in (5.4.2),

(cid:88)
(cid:46) (cid:88)

M∈M∗

ν (K)

M∈M∗

ν (K)
2 + Aα

(cid:46) (E α

(cid:96)=1

Pα

M (cid:96)

δ Qω(cid:16)
2n(cid:88)
Pα(cid:16)
2n(cid:88)
(cid:12)(cid:12)(cid:12)M (cid:96)
2 )(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ

M (cid:96)

(cid:96)=1

(cid:12)(cid:12)(cid:12)

in

(cid:17)2
2(cid:13)(cid:13)(cid:13)(cid:13)x − m

(cid:17)

in, 1A\K σ

in, 1Aσ

(cid:13)(cid:13)(cid:13)(cid:13)2

L2

M (cid:96)
in

(cid:32)

(cid:33)

ω

1
M (cid:96)
in

upon using the stopping energy condition for I(cid:48) in the corona CA, i.e. the failure of (5.1.28),

211

in the corona CA with the subdecomposition
·(cid:91)

I(cid:48) ⊃

M∈M∗

ν (K)

(cid:96)=1

2n(cid:91)

M (cid:96)
in

Combining these four bounds together with the definition of ΦA,B in (5.4.27), after applying

Cauchy-Schwarz, gives

(cid:88)

M∈M∗

ν (K)

ΦA,B (Min) (cid:46) δα−n · NT Vα

(cid:113)|I(cid:48)|σ |J(cid:48)|ω

In particular then, if we now sum over natural children I(cid:48) of I ∈ CA and the associated

children J(cid:48) of J ∈ N (I), where

N (I) ≡(cid:110)

J ∈ G : 2−r(cid:96) (I) < (cid:96) (J) ≤ (cid:96) (I) and d (J, I) ≤ 2(cid:96) (J)ε (cid:96) (I)1−ε(cid:111)

.

we obtain the following corona estimate, using the collection of K that is defined after

212

(5.4.13),

(5.4.32)

J

g

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:12)(cid:12)(cid:12)Eω

(cid:17)(cid:12)(cid:12)(cid:12)
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

J(cid:48)

J

(cid:17)(cid:12)(cid:12)(cid:12)

g

I

I

f

f

I(cid:48)

J(cid:48)

(cid:12)(cid:12)(cid:12)Eσ

(cid:88)

I∈CA
J∈N (I)

I∈CA
J∈N (I)

(cid:88)
I(cid:48),J(cid:48)(cid:17)
K∈K(cid:16)
I(cid:48)∈Cnat(I)&J(cid:48)∈C(J)
(cid:88)
(cid:46) δα−n · B · NT Vα
(cid:88)
(cid:88)
(cid:13)(cid:13)(cid:13)PσCA

(cid:46) δα−n · B · NT Vα

(cid:46) δα−n · B · NT Vα

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Eω
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)B(I(cid:48),J(cid:48)) (K)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)(cid:12)(cid:12)(cid:12) (cid:113)|I(cid:48)|σ |J(cid:48)|ω
(cid:12)(cid:12)(cid:12)Eσ
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:88)
I(cid:48)
I(cid:48)∈Cnat(I)
1
J(cid:48)∈C(J)
(cid:12)(cid:12)(cid:12)Eσ
(cid:17)(cid:12)(cid:12)(cid:12)2
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:88)
(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ
I(cid:48)∈Cnat(I)
(cid:12)(cid:12)(cid:12)Eω
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:88)
(cid:88)
(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω
(cid:13)(cid:13)(cid:13)(cid:70)

CG,nearby

J(cid:48)∈C(J)

J∈N (I)

I∈CA

I∈CA

L2(σ)

J(cid:48)

I(cid:48)

f

f

2

·

·

g

A

L2(σ)

J

I

 1

2

(cid:17)(cid:12)(cid:12)(cid:12)2

g

A

= (cid:83)
where CG,nearby
(cid:88)
(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ
I(cid:48)∈Cnat(I)

(cid:88)

I∈CA

I∈CA

N (I), and the final line uses (5.4.2) to obtain

(cid:12)(cid:12)(cid:12)Eσ

I(cid:48)

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:17)(cid:12)(cid:12)(cid:12)2

=

(cid:88)
(cid:46) (cid:88)

I∈CA

I∈CA

(cid:13)(cid:13)(cid:13)(cid:98)(cid:3)σ,(cid:91),b
(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

I

f

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

L2(σ)

≤(cid:13)(cid:13)(cid:13)PσCA

(cid:13)(cid:13)(cid:13)(cid:70)2

L2(σ)

f

L2(σ)

and similarly for the sum in J and J(cid:48), once we note that given J ∈ CG,nearby
boundedly many I ∈ CA for which J ∈ N (I).

A

, there are only

In order to deal with this sum in the case when the child I(cid:48) is broken, we must take
the estimate one step further and sum over those broken cubes I(cid:48) whose parents belong to

the corona CA, i.e. (cid:8)I(cid:48) ∈ D : I(cid:48) ∈ Cbrok (I) for some I ∈ CA

(cid:9). Of course this collection is

213

precisely the set of A -children of A, i.e.

(cid:8)I(cid:48) ∈ D : I(cid:48) ∈ Cbrok (I) for some I ∈ CA

(cid:9) = CA (A) .

(5.4.33)

To obtain the same corona estimate when summing over broken I(cid:48), we will exploit the
fact that the cubes A(cid:48) ∈ CA (A) are pairwise disjoint. But first we note that when I(cid:48) is a
broken child, neither weak testing nor stopping energy is available. But if we sum over such
broken I(cid:48), and use (5.4.33) to see that the broken children are pairwise disjoint, we obtain
ν (K):
the following estimate where for convenience we use the notation

M∗

K∈K(I(cid:48),J(cid:48))

(cid:17)(cid:12)(cid:12)(cid:12)
(cid:17)(cid:12)(cid:12)(cid:12) ·

g

1/2

|Min|σ

J(cid:48)

J

J(cid:48)

g

J

I(cid:48)

f

I

I(cid:48)

f

I

(cid:88)

I∈CA
J∈N (I)

I∈CA
J∈N (I)

 (cid:88)

(cid:83)
∼Mν ≡
(cid:12)(cid:12)(cid:12)Eσ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Eω
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)B(I(cid:48),J(cid:48)) (K)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:88)
K∈K(cid:16)
I(cid:48),J(cid:48)(cid:17)
I(cid:48)∈Cbrok(I)&J(cid:48)∈C(J)
(cid:17)(cid:12)(cid:12)(cid:12)(cid:113)|J(cid:48)|ω
(cid:12)(cid:12)(cid:12)Eω
(cid:12)(cid:12)(cid:12)Eσ
(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:88)
(cid:88)
(cid:46)δα−n · B · NT Vα
I(cid:48)∈Cbrok(I)
J(cid:48)∈C(J)
(cid:17)2
δ Qω(cid:16)
(cid:88)
(cid:88)
2n(cid:88)
(cid:40)(cid:13)(cid:13)(cid:13)1Min
(cid:88)
(cid:41)(cid:33) 1
2 ·
(cid:88)
I∈CA: J∈N (I)
I(cid:48)∈Cbrok(I)

∼
Mν
(cid:46)Bδα−nNT Vα
δ Qω(cid:16)
(cid:88)
(cid:18) 1
(cid:90)

(cid:88)
(cid:17)2 |Min|σ
(cid:88)
J∈CG,nearby
J(cid:48)∈C(J)

(cid:13)(cid:13)(cid:13)1Min
(cid:13)(cid:13)(cid:13)2
(cid:32) (cid:88)

T α
σ bA

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

I∈CA

I(cid:48)∈Cbrok(I)

(cid:12)(cid:12)(cid:12)Eω

J(cid:48)

1
2



J∈N (I)
J(cid:48)∈C(J)

(cid:19)



+
M∈

∼
Mν

+
M∈

∼
Mν

(cid:17)(cid:12)(cid:12)(cid:12)2

g

∼
Mν

M∈

M (cid:96)

in, 1Aσ

|A|σ

A

+

L2(ω)

Pα

M (cid:96)

in, 1Aσ

·

·

Pα

(cid:96)=1

(cid:13)(cid:13)(cid:13)2

T α
σ bA

|f| dσ

A

M∈

L2(ω)

(cid:96)=1

J

214

which gives that

(cid:88)

I∈CA
J∈N (I)

(cid:46) NT Vα

(cid:88)
I(cid:48),J(cid:48)(cid:17)
K∈K(cid:16)
I(cid:48)∈Cbrok(I)&J(cid:48)∈C(J)
(cid:115)
(cid:18) 1
(cid:90)
|A|σ

|A|σ

A

|f| dσ

because

(cid:12)(cid:12)(cid:12)Eσ

I(cid:48)

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:17)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)

1(cid:82)

I bI dσ

(cid:16)(cid:98)(cid:3)ω,(cid:91),b∗

J

g

(5.4.34)

(cid:17)(cid:12)(cid:12)(cid:12)

J(cid:48)

(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)Eω
(cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)B(I(cid:48),J(cid:48)) (K)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

g

L2(σ)

I

f

I(cid:48)

(cid:12)(cid:12)(cid:12)Eσ
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:19)2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω
(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) 1

(cid:90)

f dσ

A

I

|I|σ

CG,nearby

(cid:90)

I

|f| dσ (cid:46) 1
|A|σ

(cid:90)

A

|f| dσ

if I(cid:48) ∈ Cbrok (I) and I ∈ CA, and because

(cid:88)

J∈N (I)
J(cid:48)∈C(J)

(cid:88)
(cid:46) (cid:16)
I(cid:48)∈Cbrok(I)

I∈CA

Tb
T α + Eα

2 + 1

(cid:13)(cid:13)(cid:13)1Min
(cid:88)
(cid:17)2 |A|σ

∼
Mν

M∈

(cid:13)(cid:13)(cid:13)2

L2(ω)

δ Qω(cid:16)

Pα

2n(cid:88)

(cid:96)=1

+

T α
σ bA

(cid:17)2

M (cid:96)

in, 1Aσ

(5.4.35)



+ |Min|σ

Indeed, in this last inequality (5.4.35), we have used first the testing condition,

(cid:88)
I(cid:48)∈Cbrok(I)

I∈CA

(cid:88)

(cid:88)

J∈N (I)
J(cid:48)∈C(J)

∼
Mν

M∈

(cid:13)(cid:13)(cid:13)1Min

(cid:13)(cid:13)(cid:13)2

T α
σ bA

L2(ω)

(cid:88)
≤ Tb
T α
(cid:88)
I(cid:48)∈Cbrok(I)
(cid:46) Tb
T α
I(cid:48)∈Cbrok(I)

I∈CA

I∈CA

(cid:88)

|I|σ

J∈N (I)
J(cid:48)∈C(J)
|I|σ ≤ Tb

T α|A|σ

215

where in the first inequality we used the fact that the Min that appear are all disjoint and
form a subdecomposition of I(cid:48) ⊂ I and then used testing. On the second inequality we used
the bounded overlap of J for any given I, since we are in the case of nearby cubes, and we
get the last inequality because the I ∈ CA, which have a broken child I(cid:48), are disjoint and
form a subdecomposition of A. The same argument can be applied for the second sum of
(5.4.35) upon using the energy condition for all I ∈ CA which have a broken child I(cid:48) and
using the finite repetition again since we are in the nearby form.

The inequality (5.4.34) is a suitable estimate since

(cid:115)
|A|σ

(cid:18) 1

(cid:90)

|A|σ

A

(cid:88)

A∈A

(cid:19)2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω

|f| dσ

CG,nearby

g

A

L2(σ)

(cid:46) (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(σ)

by quasiorthogonality and the frame inequalities (5.1.40) and (5.1.51), together with the

bounded overlap of the ‘nearby’ coronas (cid:110)CG,nearby

A

A∈A

. We are left with estimating

∆, E, F that we get after the iteration.

Let us first deal with ∆. By Kj
i,(cid:96)

we mean a grandchild of a cube Kj
i

and Kj
i

comes

from Ki after having iterated j times, so Kj
i,(cid:96)

is a (2j + 2)-child of Ki. We have

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)
(cid:111)

B(cid:88)

ν(cid:88)

i=1

j=1

(cid:96)=1

i,(cid:96))

∆(Kj

4n−2n(cid:88)
(cid:18) 2n(cid:88)
B(cid:88)
ν(cid:88)
4n−2n(cid:88)
(cid:18) B(cid:88)
2n(cid:88)
4n−2n(cid:88)
ν(cid:88)

(cid:96)=1

i=1

j=1

q=1

i=1

j=1

(cid:96)=1

q=1

≤ NT αCb,b∗,ν

≤ NT αCb,b∗,ν

(cid:12)(cid:12)(cid:12)(cid:0)Kj
(cid:12)(cid:12)(cid:12)(cid:0)Kj

i,(cid:96),in\Kj,q

i,(cid:96),in

i,(cid:96),in\Kj,q

i,(cid:96),in

(cid:1) ∩ (1 + δ)Kj,q
(cid:1) ∩ (1 + δ)Kj,q

i,(cid:96),in

i,(cid:96),in

(cid:19)1
2(cid:113)|Kj
(cid:12)(cid:12)(cid:12)σ
(cid:19) 1
(cid:12)(cid:12)(cid:12)σ
2(cid:112)|J(cid:48)|ω

i,(cid:96)|ω

where Kj,q

i,(cid:96),in

is one of the inner grandchildren of Kj

i,(cid:96),in

. Now fixing q = q0 and taking

216

averages over the grid G we get

B(cid:88)

ν(cid:88)

4n−2n(cid:88)

i=1

j=1

(cid:96)=1

EG

Ω

(cid:12)(cid:12)(cid:12)(cid:0)Kj

i,(cid:96),in\Kj,q

i,(cid:96),in

(cid:1) ∩ (1 + δ)Kj,q

i,(cid:96),in

(cid:12)(cid:12)(cid:12)σ

≤ Cnδ|I|σ

the constant depends on dimension since for the same i, j we can have intersection as (cid:96)

moves. Adding the different q we get finally

B(cid:88)

ν(cid:88)

4n−2n(cid:88)

i=1

j=1

(cid:96)=1

EG

Ω

∆(Kj

i,(cid:96)) ≤ NT αCb,b∗,ν,n

δ(cid:112)|I(cid:48)|σ

(cid:112)|J(cid:48)|ω.

√

(5.4.36)

For F we get,

B(cid:88)

ν(cid:88)

4n−2n(cid:88)

i=1

j=1

(cid:96)=1

F(Kj

i,(cid:96)) ≤ NT αCb,b∗

(cid:18) B(cid:88)

ν(cid:88)

4n−2n(cid:88)

i=1

j=1

(cid:96)=1

(cid:12)(cid:12)(cid:12)Kj

i,(cid:96),out ∩ (1 + δ)Kj

i,(cid:96),in

(cid:19) 1
2(cid:112)|J(cid:48)|ω

(cid:12)(cid:12)(cid:12)σ

and again averaging over grids G, we get the bound

B(cid:88)

ν(cid:88)

4n−2n(cid:88)

EG

Ω

F(Kj

i,(cid:96)) ≤ NT αCb,b∗

δ(cid:112)|I(cid:48)|σ

(cid:112)|J(cid:48)|ω

√

(5.4.37)

i=1

j=1

(cid:96)=1

Note here that upon choosing δ small enough there is no repetition in the different terms

217

that arise. Finally, for E, we have

B(cid:88)

i=1

≤ NT α

≤ NT α

i,(cid:96))

j=1

(cid:96)=1

E(Kj

ν(cid:88)
4n−2n(cid:88)
B(cid:88)
ν(cid:88)
4n−2n(cid:88)
(cid:18) B(cid:88)
4n−2n(cid:88)
ν(cid:88)

j=1

(cid:96)=1

i=1

i=1

j=1

(cid:96)=1

(5.4.38)

r>q

q=1

i,(cid:96),out ∩ (1 + δ)Kj,r

(cid:18)(cid:88)
(cid:12)(cid:12)(cid:12)Kj,q
4n−2n(cid:88)
(cid:12)(cid:12)(cid:12)Kj,q
(cid:88)
4n−2n(cid:88)
i,(cid:96),out ∩ (1 + δ)Kj,r
(cid:18) B(cid:88)
(cid:88)
4n−2n(cid:88)
4n−2n(cid:88)
ν(cid:88)
(cid:12)(cid:12)(cid:12)Kj,q
(cid:88)
4n−2n(cid:88)

4n−2n(cid:88)

q=1

q=1

j=1

r>q

r>q

(cid:96)=1

i=1

·

2 ·

i,(cid:96),out

(cid:12)(cid:12)(cid:12)ω

2(cid:114)(cid:12)(cid:12)(cid:12)Kj
(cid:19) 1
(cid:12)(cid:12)(cid:12)σ
(cid:19) 1
(cid:12)(cid:12)(cid:12)σ
(cid:19) 1
(cid:12)(cid:12)(cid:12)Kj
(cid:12)(cid:12)(cid:12)ω
(cid:19) 1
(cid:12)(cid:12)(cid:12)σ
2(cid:112)|J(cid:48)|ω

i,(cid:96),out

i,(cid:96),out

2

i,(cid:96),out ∩ (1 + δ)Kj,r

i,(cid:96),out

i,(cid:96),out

j=1

(cid:96)=1

q=1

r>q

ν(cid:88)

≤ NT α · Cn,ν

(cid:18) B(cid:88)

i=1

Taking averages,

B(cid:88)

ν(cid:88)

4n−2n(cid:88)

i=1

j=1

(cid:96)=1

EG

Ω

E(Kj

i,(cid:96)) ≤ NT α · Cn,ν

δ(cid:112)|I(cid:48)|σ

√

(cid:112)|J(cid:48)|ω

The constant Cn,ν comes from the intersection of the sets Kj

i,(cid:96),out

.

Recall that after splitting in the cases of δ-seperated and δ-close cubes, we got the bound

(5.4.7) in the separated case and after an initial application of random surgery, we reduced

the proof of Proposition 5.4.1 to establishing inequality (5.4.11). Then using the bounds in

(5.4.12), (5.4.14), (5.4.15), (5.4.16), (5.4.17), (5.4.18) we reduced P (I, J) to getting a bound
for {K, K} in the notation used in (5.4.21). Then using the estimates in (5.4.30), (5.4.31),
(5.4.32) and (5.4.34) together with (5.4.29), (5.4.36), (5.4.37) and (5.4.38) establishes prob-
abilistic control of the sum of all the inner products {K, K} taken over appropriate cubes
K, yielding (5.4.11) as required if we choose ε, λ, η0 and δ sufficiently small. And combining

218

all the above bounds we proved proposition 5.4.1, namely we got the bound

(cid:88)

I∈D

Ω EG
ED

Ω

(cid:88)

J∈G: 2−rn|I|<|J|≤|I|
d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε

(cid:12)(cid:12)(cid:12)(cid:68)

T α
σ

(cid:16)

(cid:16)(cid:3)σ,b

I f

(cid:17)

, (cid:3)ω,b∗

J

g

CθNT Vα +

√
θNT α

(cid:69)

(cid:12)(cid:12)(cid:12)
(cid:17)

ω

(cid:46)

(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)

(cid:17) (cid:3)ω,b∗

J

gdω.

5.5 Main below form

Now we turn to controlling the main below form (5.2.17),

Θgood

2

(f, g) =

(cid:88)

(cid:88)

I∈D

J(cid:122)(cid:36)I: (cid:96)(J)≤2−ρ(cid:96)(I)

(cid:90) (cid:16)

Tσ(cid:3)σ,b
I f

To control Θgood

2

(f, g) ≡ B(cid:98)ρ (f, g) we first perform the canonical corona splitting of

B(cid:98)ρ (f, g) into a diagonal form and a far below form, namely Tdiagonal (f, g) and
Tf arbelow (f, g) as in [63]. This canonical splitting of the form B(cid:98)ρ (f, g) involves the corona
acting
pseudoprojections Pσ,b
CD
A
on g, where B is a stopping cube in A. The stopping cubes B constructed relative to
g ∈ L2 (ω) play no role in the analysis here, except to guarantee that the frame and weak
is defined

Riesz inequalities hold for g and(cid:110)(cid:3)ω,b∗

acting on f and the shifted corona pseudoprojections Pω,b∗

CG,shif t

(cid:111)

B

g

. Here the shifted corona CG,shif t

J∈G
. Recall that the parameters τ and ρ are fixed

B

J

(cid:122) ∈ CD

B

to include those cubes J ∈ G such J
to satisfy

τ > r and ρ > r + τ,

where r is the goodness parameter already fixed in (5.2.16).

219

Definition 5.5.1. For B ∈ A we define the shifted G-corona by

(cid:110)

CG,shif t

B

=

J ∈ G : J

(cid:122) ∈ CD

B

(cid:111)

.

B

We will use repeatedly the fact that the shifted coronas CG,shif t

are pairwise disjoint in

B:

(cid:88)

B∈A

(J) ≤ 1,

J ∈ D.

1CG,shif t

B

(5.5.1)

The forms B(cid:98)ρ,ε (f, g) are no longer linear in f and g as the ‘cut’ is determined by the
coronas CD
, which depend on f as well as the measures σ and ω. However,
if the coronas are held fixed, then the forms can be considered bilinear in f and g. It is

and CG,shif t

B

A

convenient at this point to introduce the following shorthand notation:

(cid:42)

(cid:32)

(cid:33)

(cid:43)(cid:98)ρ,ε

T α
σ

f

Pσ,b
CD
A

, Pω,b∗
CG,shif t

B

g

ω

≡
I∈CD

(cid:88)

A and J∈CG,shif t
(cid:96)(J)≤2−ρ(cid:96)(I)

B

(cid:122)(cid:36)I

: J

(cid:68)

T α
σ

(cid:16)(cid:3)σ,b

I f

(cid:17)

(cid:69)

.

ω

, (cid:3)ω,b∗

J

g

(5.5.2)

Caution One must not assume, from the notation on the left hand side above, that the

(cid:16)

(cid:17) is simply integrated against the function Pω

function T α
σ

PσCA

f

CG,shif t
(cid:122) ∈ CD

B

g. Indeed, the
(cid:122) (cid:36) I

and J

B

sum on the right hand side is taken over pairs (I, J) such that J
and (cid:96) (J) ≤ 2−ρ(cid:96) (I).

5.5.1 The canonical splitting and local below forms

We then have the canonical splitting determined by the coronas CD
for A ∈ A (the stopping
times B play no explicit role in the canonical splitting of the below form, other than to

A

220

guarantee the weak Riesz inequalities for the dual martingale pseudoprojections (cid:3)ω,b∗

J

)

=

=

B(cid:98)ρ,ε (f, g)

(cid:42)

T α
σ

(cid:16)
(cid:42)

T α
σ

A,B∈A

(cid:88)
(cid:42)
(cid:88)
(cid:88)

A∈A

+

A,B∈A
B(cid:37)A

(cid:16)

f

Pσ,bCA
(cid:17)

f

Pσ,bCA
(cid:16)

T α
σ

(cid:17)

, Pω,b∗
CG,shif t

B

g

, Pω,b∗
CG,shif t
(cid:17)

A

g

(cid:43)(cid:98)ρ,ε
(cid:43)(cid:98)ρ,ε
(cid:43)(cid:98)ρ,ε

+

ω

ω

Pσ,bCA

f

, Pω,b∗
CG,shif t

B

g

ω

T α
σ

(cid:16)
(cid:42)

(cid:42)
(cid:88)
(cid:88)

A,B∈A
B(cid:36)A

+
A,B∈A
A∩B=∅

≡ Tdiagonal (f, g) + Tf arbelow (f, g) + Tf arabove (f, g) + Tdisjoint (f, g) .

(5.5.3)

(cid:17)

f

(cid:43)(cid:98)ρ,ε
(cid:43)(cid:98)ρ,ε

ω

Pσ,bCA
(cid:16)

g

, Pω,b∗
CG,shif t
(cid:17)

B

T α
σ

Pσ,bCA

f

, Pω,b∗
CG,shif t

B

g

ω

Now the final two terms Tf arabove (f, g) and Tdisjoint (f, g) each vanish since there are no
pairs (I, J) ∈ CD
(cid:122) (cid:36) I and (ii) either B (cid:36) A or B ∩ A = ∅. The
far below form Tf arbelow (f, g) requires functional energy, which we discuss in a moment.

A × CG,shif t

with both (i) J

B

Next we follow this splitting by a further decomposition of the diagonal form into local

below forms BA(cid:98)ρ (f, g) given by the individual corona pieces

(cid:42)

(cid:16)

(cid:17)

BA(cid:98)ρ,ε (f, g) ≡

T α
σ

Pσ,bCA

f

(cid:43)(cid:98)ρ,ε

ω

, Pω,b∗
CG,shif t

A

g

and prove the following estimate:

(cid:12)(cid:12)(cid:12)BA(cid:98)ρ,ε (f, g)

(cid:12)(cid:12)(cid:12) (cid:46) NT Vα

(cid:18)

αA (A)

(cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A

CG,shif t

(cid:113)|A|σ +

(cid:13)(cid:13)(cid:13)Pσ,bCA

f

(cid:13)(cid:13)(cid:13)(cid:70)

L2(σ)

221

(5.5.4)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

g

.

This reduces matters to the local forms since we then have from Cauchy-Schwarz that

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b

CD

A

f

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2

L2(σ)

 1

2

·

(cid:12)(cid:12)(cid:12)BA(cid:98)ρ,ε (f, g)

(cid:88)

A∈A

(cid:12)(cid:12)(cid:12) (cid:46) NT Vα
(cid:88)

·

(cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A∈A

αA (A)2 |A|σ +

 1

2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2

CG,shif t

g

A∈A

L2(ω)
(cid:46) NT Vα (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .

A

by the lower frame inequalities

(cid:13)(cid:13)(cid:13)Pσ,bCA

f

(cid:13)(cid:13)(cid:13)(cid:70)2

L2(σ)

(cid:88)

A∈A

(cid:46) (cid:107)f(cid:107)2

L2(σ)

and (cid:88)

A∈A

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

CG,shif t

using also quasi-orthogonality (cid:80)
(cid:46) (cid:107)f(cid:107)2
the pairwise disjointedness of the shifted coronas CG,shif t

αA (f )2 |A|σ

A∈A

A

:

L2(σ)

A

(cid:88)

A∈A

1CG,shif t

A

≤ 1D.

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

g

(cid:46) (cid:107)g(cid:107)2

L2(ω)

in the stopping cubes A, and

From now on we will often write CA in place of CD

A

when no confusion is possible.

Finally, the local forms BA(cid:98)ρ,ε (f, g) are decomposed into stopping BA
paraproduct (f, g) and neighbour BA

stop (f, g), paraprod-
neighbour (f, g) forms. The paraproduct and neighbour
terms are handled as in [63], which in turn follows the treatment originating in [43], and this

uct BA

leaves only the stopping form BA

stop (f, g) to be bounded, which we treat last by adapting

the bottom/up stopping time and recursion of M. Lacey in [27].

However, in order to obtain the required bounds of the above forms into which the below

form B(cid:98)ρ (f, g) was decomposed, we need functional energy. Recall that the vector-valued

222

function b in the accretive coronas ‘breaks’ only at a collection of cubes satisfying a Carleson
condition. We define M(r,ε)−deep (F ) to consist of the maximal r-deeply embedded dyadic
G-subcubes of a D-cube F - see (.0.7) in Appendix for more detail.

Definition 5.5.2. Let Fα = Fα (D,G) be the smallest constant in the ‘functional energy’
inequality below, holding for all h ∈ L2 (σ) and all σ-Carleson collections F ⊂ D with

Carleson norm CF bounded by a fixed constant C:

(cid:88)

(cid:88)

F∈F

M∈M(r,1)−deep,D(F )

Pα (M, hσ)

|M| 1

n

2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b

F

CG,shif t

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

;M

≤ Fα(cid:107)h(cid:107)L2(σ) ,

(5.5.5)

The main ingredient used in reducing control of the below form B(cid:98)ρ (f, g) to control of
stop (f, g), is the Intertwining
the functional energy Fα constant and the stopping form BA
Proposition from [63]. The control of the functional energy condition by the energy and

Muckenhoupt conditions must also be adapted in light of the p-weakly accretive function b

that only ‘breaks’ at a collection of cubes satisfying a Carleson condition, but this poses no

real difficulties. The fact that the usual Haar bases are orthonormal is here replaced by the

weaker condition that the corresponding broken Haar ‘bases’ are merely frames satisfying

certain lower and weak upper Riesz inequalities, but again this poses no real difference in

the arguments. Finally, the fact that goodness for J has been replaced with weak goodness,

namely J

(cid:122) (cid:36) I, again forces no real change in the arguments.

We then use the paraproduct / neighbour / stopping splitting mentioned above to reduce

223

boundedness of BA(cid:98)ρ,ε (f, g) to boundedness of the associated stopping form

stop (f, g) ≡ (cid:88)

BA

I∈CA

(cid:16)

(cid:88)
J∈CG,shif t

(cid:122)(cid:36)I

: J

A

(cid:96)(J)≤2−ρ(cid:96)(I)

(cid:17) (cid:68)

(cid:3)σ,b
I f

Eσ
IJ

T α
σ 1A\IJ

bA, (cid:3)ω,b∗

J

g

(cid:69)

ω

(5.5.6)

, the dual martingale support of f is contained in the corona Cσ

where f is supported in the cube A and its expectations Eσ
for I ∈ Cσ
martingale support of g is contained in CG,shif t
contains J.

I |f| are bounded by αA (A)
, and the dual
, and where IJ is the D-child of I that

A

A

A

5.5.2 Diagonal and far below forms

Now we turn to the diagonal and the far below terms Tdiagonal (f, g) and Tf arbelow (f, g),
where in [63] the far below terms were bounded using the Intertwining Proposition and the

control of functional energy condition by the energy conditions, but of course under the

restriction there that the cubes J were good. Here we write

Tf arbelow (f, g) =

=

(cid:88)
(cid:42)

A,B∈A
B(cid:36)A

(cid:88)
(cid:88)
− (cid:88)

B∈A

J

B

I∈CA and J∈CG,shif t
(cid:17)
(cid:16)(cid:3)σ,b
(cid:88)
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)
(cid:16)(cid:3)σ,b
(cid:88)

(cid:42)

I f

T α
σ

T α
σ

I f

I∈D: B(cid:36)I

B∈A

I∈D: B(cid:36)I

(5.5.7)

(cid:17)(cid:69)

g

ω

(cid:43)

(cid:16)(cid:3)ω,b∗
(cid:43)

J

ω

(cid:3)ω,b∗

J

(cid:68)

,

T α
σ

(cid:17)

I f

(cid:16)(cid:3)σ,b
(cid:88)
J∈CG,shif t
(cid:17)
(cid:88)
J∈CG,shif t
(cid:96)(J)>2−r(cid:96)(I)

B

B

J

,

,

(cid:3)ω,b∗

g

g

ω

= T1

f arbelow (f, g) − T2

f arbelow (f, g) .

224

since if I ∈ CA and J ∈ CG,shif t
First, we note that expectation of the second sum T2

(cid:122) (cid:36) I and B (cid:36) A, then we must have B (cid:36) I.
f arbelow (f, g) is controlled by (5.4.1) in

, with J

B

Proposition 5.4.1 , i.e.

Ω EG
ED

Ω

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)
(cid:88)

B∈A

I∈D

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:43)
(cid:69)

ω

(cid:12)(cid:12)(cid:12)

ω

(cid:3)ω,b∗

J

g

, (cid:3)ω,b∗

J

g

(cid:88)

I∈D: B(cid:36)I

(cid:42)

,

T α
σ

I f

(cid:16)(cid:3)σ,b
(cid:17)
(cid:12)(cid:12)(cid:12)(cid:68)

(cid:88)
J∈CG,shif t
(cid:16)(cid:3)σ,b
(cid:17)
(cid:88)
(cid:96)(J)>2−r(cid:96)(I)
(cid:17)(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .

I f

T α
σ

B

(cid:46) ED

Ω EG

Ω

(cid:46) (cid:16)

J∈G: 2−r(cid:96)(I)<(cid:96)(J)≤(cid:96)(I)
d(J,I)≤2(cid:96)(J)ε(cid:96)(I)1−ε
√

CθNT Vα +

θNT α

The form T1

f arbelow (f, g) can be written as

T1

f arbelow (f, g) =

(cid:88)
where gB ≡ (cid:88)

B∈A

(cid:88)

I∈D: B(cid:36)I

(cid:3)ω,b∗

J

J∈CG,shif t

B

(cid:17)

(cid:69)

;

ω

, gB

(cid:68)

(cid:16)(cid:3)σ,b
T α
I f
σ
g = Pω,b∗

CG,shif t

g

F

and the Intertwining Proposition 5.5.7 can now be applied to this latter form to show that it
2 +E α
is bounded by NT Vα+Fα. Then Proposition .0.1 can be applied to show that Fα (cid:46) Aα
which completes the proof that

2

,

(cid:12)(cid:12)Tf arbelow (f, g)(cid:12)(cid:12) (cid:46) NT Vα (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .

(5.5.8)

5.5.3

Intertwining Proposition

First we adapt the relevant definitions and theorems from [63].

225

Definition 5.5.3. A collection F of dyadic cubes is σ-Carleson if

(cid:88)

F∈F: F⊂S

|F|σ ≤ CF |S|σ ,

S ∈ F.

The constant CF is referred to as the Carleson norm of F.

Definition 5.5.4. Let F be a collection of dyadic cubes in a grid D. Then for F ∈ F, we
define the shifted corona CG,shif t

in analogy with Definition 5.5.1 by

F

(cid:110)

CG,shif t

F

=

J ∈ G : J

(cid:122) ∈ CF

(cid:111)

.

F

Note that the collections CG,shif t

are pairwise disjoint in F . Let CF (F ) denote the set of
F-children of F . Given any collection H ⊂ G of cubes, a family b∗ of dual testing functions,
and an arbitrary cube K ∈ P, we define the corresponding dual pseudoprojection Pω,b∗
H and
its localization Pω,b∗
H;K

to K by

Qω,b∗
H =

(cid:52)ω,b∗

H

and Qω,b∗
H;K =

(cid:52)ω,b∗

H

.

(5.5.9)

(cid:88)

H∈H

(cid:88)

H∈H: H⊂K

Recall from Definition 5.5.2 that Fα = Fα (D,G) = Fb∗
i.e.

α (D,G) is the best constant in (5.5.5),

(cid:88)

(cid:88)

F∈F

M∈M(r,1)−deep,D(F )

Pα (M, hσ)

|M| 1

n

2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b

F

CG,shif t

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

;M

≤ Fα(cid:107)h(cid:107)L2(σ) .

Remark 5.5.5. If in (5.5.5), we take h = 1I and F to be the trivial Carleson collec-
tion {Ir}∞
r=1 where the cubes Ir are pairwise disjoint in I, then we obtain the deep energy

226

condition in Definition .0.4, but with Pω,b∗

pseudoprojection Pweakgood,ω

J

CG,shif t
is larger than Pω,b∗

F

;M

in place of Pweakgood,ω

J

. However, the

CG,shif t

, and so we just miss obtaining the

;J

F

deep energy condition as a consequence of the functional energy condition. Nevertheless, this

near miss with h = 1I explains the terminology ‘functional’ energy.

We will need the following ‘indicator’ version of the estimates proved above for the disjoint

form.

Lemma 5.5.6. Suppose T α is a standard fractional singular integral with 0 ≤ α < 1, that
ρ > r, that f ∈ L2 (σ) and g ∈ L2 (ω), that F ⊂ Dσ and G ⊂ Dω are σ-Carleson and

ω-Carleson collections respectively, i.e.,

(cid:88)

(cid:12)(cid:12)F(cid:48)(cid:12)(cid:12)σ

(cid:12)(cid:12)G(cid:48)(cid:12)(cid:12)ω
that there are numerical sequences {αF (F )}F∈F and(cid:8)βG (G)(cid:9)

F ∈ F, and (cid:88)

F(cid:48)∈F: F(cid:48)⊂F

G(cid:48)∈G: G(cid:48)⊂G

(cid:46) |F|σ ,

(cid:46) |G|ω ,

G ∈ G,

G∈G such that

αF (F )2 |F|σ ≤ (cid:107)f(cid:107)2

L2(σ)

βG (G)2 |G|σ ≤ (cid:107)g(cid:107)2

L2(σ)

and (cid:88)

G∈G

(cid:88)

F∈F

Then

,

(cid:12)(cid:12)(cid:12)

(5.5.10)

(5.5.11)

(cid:12)(cid:12)(cid:12)

ω

(cid:88)
(cid:88)

F∈F

+

(cid:88)
(cid:88)

J∈G: (cid:96)(J)≤(cid:96)(F )

d(J,F )>2(cid:96)(J)ε(cid:96)(F )1−ε

(cid:12)(cid:12)(cid:12)(cid:68)
σ (1F αF (F )) , (cid:3)ω,b∗
T α
(cid:12)(cid:12)(cid:12)(cid:68)
(cid:16)(cid:3)σ,b

I f

(cid:17)

T α
σ

J

, 1GβG (G)

(cid:69)
(cid:69)

ω

g

(cid:46) (cid:113)

G∈G

I∈D:(cid:96)(I)≤(cid:96)(G)

d(I,G)>2(cid:96)(I)ε(cid:96)(G)1−ε
2 (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .
Aα

227

The proof of this lemma is similar to those of Lemmas 5.3.1 and 5.3.2 in Section 5.3

above, using the square function inequalities for (cid:3)σ,b

I

, ∇σ

I,F and (cid:3)ω,b∗

J

, ∇ω

J,G.

Proposition 5.5.7 (The Intertwining Proposition). Let D and G be grids, and suppose that
b and b∗ are ∞-weakly σ-accretive families of cubes in D and G respectively. Suppose that
F ⊂ D is σ-Carleson and that the F-coronas

CF ≡(cid:8)I ∈ D : I ⊂ F but I (cid:54)⊂ F(cid:48) for F(cid:48) ∈ CF (F )(cid:9)

satisfy

Then

I |f| (cid:46) Eσ
Eσ

F |f| and bI = 1I bF ,

for all I ∈ CF , F ∈ F.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

ED

Ω

(cid:88)
(cid:113)

F∈F

Fα +Tb

T α +

(cid:16)

(cid:42)

(cid:88)

σ (cid:3)σ,b
T α

I: I(cid:37)F
2 δα−n+δNT α
Aα

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46)
(cid:43)
I f, Pω,b∗
(cid:17)(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) ,

CG,shif t

g

F

ω

where the implied constant depends on the σ-Carleson norm CF of the family F.

Proof. We write the sum on the left hand side of the display above as

σ (cid:3)σ,b
T α

I f, Pω
F

CG,shif t

(cid:43)

g

=

ω

=

(cid:88)
(cid:88)

F∈F

F∈F

(cid:42)
(cid:10)T α

σ

T α
σ

 (cid:88)
(cid:0)f∗
(cid:1) , gF

I: I(cid:37)F

F

(cid:11)

(cid:3)σ,b
I f

ω ;

 , Pω

CG,shif t

F

(cid:43)

g

ω

(cid:42)

(cid:88)

(cid:88)

F∈F

I: I(cid:37)F

where f∗

F ≡ (cid:88)

I: I(cid:37)F

Note that gF is supported in F . By the telescoping identity for (cid:3)σ,b

, the function f∗

F

I

I f and gF ≡ Pω
(cid:3)σ,b

CG,shif t

F

g.

228

satisfies

1F f∗

F =

(cid:88)

I: I∞⊃I(cid:37)F

(cid:3)σ,b
I f = Fσ,b

F f − 1F Fσ,b

I∞f = bF

Eσ
F f
Eσ
F bF

− 1F bI∞

Eσ
I∞f
Eσ
I∞bI∞

.

where I∞ is the starting cube for corona constructions in D. However, we cannot apply the
testing condition to the function 1F bI∞, and since Eσ
I∞f does not vanish in general, we will
instead add and subtract the term Fσ,b

(cid:88)

F∈F

(cid:10)T α

σ

(cid:0)f∗

F

(cid:1) , gF

(cid:11)

ω =

=

(cid:43)
 , Pω

g

ω

CG,shif t

F

 , Pω
(cid:43)

(cid:3)σ,b
I f

CG,shif t

F

(5.5.12)

(cid:43)

g

ω

(cid:3)σ,b
I f

(cid:88)

I: I∞⊃I(cid:37)F

, Pω
F

CG,shif t

g

,

ω

I: I∞⊃I(cid:37)F

I∞f to get
(cid:42)
(cid:88)
(cid:42)
(cid:88)
− (cid:88)

(cid:42)

F∈F

F∈F

T α
σ

T α
σ

 (cid:88)
Fσ,b
(cid:16)Fσ,b
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:46)(cid:16)

Tb
T α +

I∞f

T α
σ

(cid:43)

F∈F

I∞f +

(cid:17)

(cid:113)

(cid:90)

J∞\(1+δ)I∞

(cid:35)(cid:88)

F∈F

Pω
CG,shif t
F

g

 T α

σ

(cid:17)

(cid:16)Fσ,b

I∞f

dω

229

where the second sum on the right hand side of the identity satisfies

(cid:17)

(cid:16)Fσ,b

I∞f

T α
σ

, Pω
F

CG,shif t

g

ω

(cid:17)(cid:107)f(cid:107)L2(σ)(cid:107)g(cid:107)L2(ω)

2 δα−n+δNT α
Aα

ED

Ω

Indeed, as

(cid:42)
(cid:88)

F∈F

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:42)
(cid:88)
(cid:34)(cid:90)

F∈F

I∞f

T α
σ

(cid:16)Fσ,b
(cid:90)
+
=
≡ A1 + A2 + A3

I∞∩J∞

(cid:17)

(cid:43)

, Pω
F

CG,shif t

g

ω

J∞∩((1+δ)I∞\I∞)

+

by Cauchy-Schwarz and Riesz inequalities, the term A1 is controlled by testing, the term A3
by Muckenhoupt’s condition using lemma 5.4.3 and finally

ED
Ω A2 ≤

(cid:90)

(cid:88)

Cδ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
≤ (cid:112)CδNT α (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .

Pω
CG,shif t
F

F∈F

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

I∞

 1
2(cid:18)

g

dω

NT α

(cid:90)

(cid:19) 1

2

|f|2 dσ

The advantage now is that with

fF ≡ Fσ,b

I∞f + f∗

F = Fσ,b

I∞f +

(cid:88)

I: I∞⊃I(cid:37)F

(cid:3)σ,b
I f

then in the first term on the right hand side of (5.5.12), the telescoping identity gives

Fσ,b

(cid:88)

I: I∞⊃I(cid:37)F

(cid:3)σ,b
I f

 = Fσ,b

F f = bF

Eσ
F f
Eσ
F bF

,

1F fF = 1F

I∞f +

which shows that fF is a controlled constant times bF on F .

The cubes I occurring in this sum are linearly and consecutively ordered by inclusion,

along with the cubes F(cid:48) ∈ F that contain F . More precisely we can write

F ≡ F0 (cid:36) F1 (cid:36) F2 (cid:36) ... (cid:36) Fn (cid:36) Fn+1 (cid:36) ...FN = I∞

where Fm = πmF F for all m ≥ 1. We can also write

F = F0 ≡ I0 (cid:36) I1 (cid:36) I2 (cid:36) ... (cid:36) Ik (cid:36) Ik+1 (cid:36) ... (cid:36) IK = FN = I∞

230

where Ik = πkDF for all k ≥ 1. There is a (unique) subsequence {km}N

m=1

such that

Fm = Ikm,

1 ≤ m ≤ N.

Then we have

fF (x)≡ Fσ,b

I∞f (x) +

K(cid:88)

(cid:96)=1

(cid:3)σ,b
I(cid:96)

f (x)

and gF ≡ (cid:88)

J∈CG,shif t

F

(cid:3)ω,b∗

J

g.

Assume now that km ≤ k < km+1. We denote by θ (I) the 2n − 1 siblings of I,
i.e. ˜I ∈ θ (I) implies ˜I ∈ CD (πDI)\{I}. There are two cases to consider here:

˜Ik /∈ F and ˜Ik ∈ F.

We first note that in either case, using a telescoping sum, we compute that for

x ∈ ˜Ik ⊂ Fm+1\Fm,

we have the formula

fF (x) = Fσ,b

I∞f (x) +

K(cid:88)

(cid:96)=k+1

(cid:3)σ,b
I(cid:96)

= Fσ,b
˜Ik
= Fσ,b
˜Ik

f (x) − Fσ,b
Ik+1

f (x) +

f (x) .

(cid:96)=k+1

231

f (x)

K−1(cid:88)

(cid:16)Fσ,b

I(cid:96)

f (x) − Fσ,b
I(cid:96)+1

f (x)

(cid:17)

+ Fσ,b

I∞f (x)

Now fix x ∈ ˜Ik. If ˜Ik /∈ F, then ˜Ik ∈ CFm+1

, and we have

(cid:12)(cid:12)(cid:12)(cid:12)Fσ,b

˜Ik

(cid:12)(cid:12)(cid:12)(cid:12) (cid:46)(cid:12)(cid:12)(cid:12)b ˜Ik

(cid:12)(cid:12)(cid:12)

|fF (x)| =

f (x)

(x)

|f|

(cid:12)(cid:12)(cid:12)(cid:12)Eσ

Eσ
˜Ik
bθ(Ik)

˜Ik

(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) Eσ

Fm+1

|f| ,

(5.5.13)

since the testing functions b ˜Ik
hypothesis. On the other hand, if ˜Ik ∈ F, then Ik+1 ∈ CFm+1

are bounded and accretive, and Eσ
˜Ik

and we have

|f| (cid:46) Eσ

Fm+1

|f| by

(cid:12)(cid:12)(cid:12)(cid:12)Fσ,b

˜Ik

(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) Eσ

˜Ik

|fF (x)| =

f (x)

|f| .

Note that F c =

θ (Ik). Now we write

·(cid:91)

k≥0

fF = ϕF + ψF ,

and ψF = fF − ϕF ;

(cid:88)

ϕF ≡(cid:88)
(cid:88)

k≥0

F∈F

Fσ,b
˜Ik

f

˜Ik∈θ(Ik)
˜Ik∈F
(cid:104)T α
σ fF , gF(cid:105)ω =

(cid:88)

F∈F

(cid:104)T α
σ ϕF , gF(cid:105)ω +

(cid:88)

F∈F

(cid:104)T α
σ ψF , gF(cid:105)ω ,

and note that ϕF = 0 on F , and ψF = bF
using ˜Ik ∈ F to the first sum above since J ∈ CG,shif t

on F . We can apply the first line in (5.5.11)
(cid:122) ⊂ F ⊂ Ik, which

implies J ⊂ J

F

Eσ
F f
Eσ
F bF

232

implies that d(J, ˜Ik) > 2(cid:96) (J)ε (cid:96)( ˜Ik)1−ε. Thus we obtain after substituting F(cid:48) for ˜Ik below,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)

F∈F

(cid:104)T α
σ ϕF , gF(cid:105)ω

 , (cid:3)ω,b∗

J

, (cid:3)ω,b∗

J

g

, (cid:3)ω,b∗

J

g

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

g

(cid:43)
(cid:29)
(cid:69)

ω

ω

ω

(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)

f

f

˜Ik∈θ(Ik)
˜Ik∈F
T α
σ

(cid:88)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:28)
(cid:12)(cid:12)(cid:12)(cid:68)

T α
σ

Fσ,b
˜Ik

Fσ,b
˜Ik

(cid:18)
(cid:16)Fσ,b

F(cid:48) f

(cid:19)
(cid:17)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

F∈F

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =
(cid:88)
≤ (cid:88)
≤ (cid:88)
(cid:46) (cid:113)

F∈F

F(cid:48)∈F



(cid:88)
(cid:88)

k≥0

T α
σ

(cid:42)

F

(cid:88)
J∈CG,shif t
(cid:88)
(cid:88)
J∈CG,shif t
(cid:88)

k≥0

F

˜Ik∈θ(Ik)
˜Ik∈F
(cid:16)
F(cid:48)(cid:17)
(cid:16)
F(cid:48)(cid:17)1−ε
(cid:16)
J,F(cid:48)(cid:17)
2 (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) .
Aα

J∈G: (cid:96)(J)≤(cid:96)
>2(cid:96)(J)ε(cid:96)

d

Turning to the second sum, we note that for km ≤ k < km+1 and x ∈ ˜Ik with ˜Ik /∈ F,

we have

|ψF (x)| (cid:46)(cid:12)(cid:12)(cid:12)b ˜Ik

(cid:12)(cid:12)(cid:12) Eσ

|f| 1 ˜Ik

(x) (cid:46) αF (Fm+1) 1 ˜Ik

˜Ik

Note that for σ-almost all x ∈ I∞ there exists a unique F ∈ F such that x ∈ F\ (cid:91)

F(cid:48)
since the family F is a Carleson family. Also from the stopping criteria we have αF (F ) ≤
αF (F(cid:48)) for F(cid:48) ⊂ F . Hence we get the following inequality for x /∈ F ,

F(cid:48)∈CF (F )

(x)

|ψF (x)| (cid:46) Φ (x) 1F c (x) ,

(5.5.14)

where we have defined

Φ ≡ (cid:88)

F∈F

αF (F ) 1F\∪CF (F ) .

233

Now we write

(cid:88)

F∈F

σ ψF , gF(cid:105)ω =
(cid:104)T α

(cid:88)

F∈F

(cid:104)T α
σ (1F ψF ) , gF(cid:105)ω +

(cid:88)

F∈F

σ (1F cψF ) , gF(cid:105)ω ≡ I + II.
(cid:104)T α

Then by cube testing,

|(cid:104)T α

σ (bF 1F ) , gF(cid:105)ω| = |(cid:104)1F T α

σ (bF 1F ) , gF(cid:105)ω| (cid:46) TT α

(cid:113)|F|σ (cid:107)gF(cid:107)(cid:70)

L2(ω)

,

and so quasi-orthogonality, together with the fact that on F , ψF = bF

c =

Eσ
F f
Eσ
F bF

times bF , where |c| is bounded by αF (F ), give

Eσ
F f
Eσ
F bF

is a constant

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:88)

F∈F

(cid:104)T α
σ (1F cbF ) , gF(cid:105)ω

|I| =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:88)
(cid:46) (cid:88)

F∈F

F∈F

αF (F )

αF (F ) TT α

σ bF , gF(cid:105)ω

(cid:12)(cid:12)(cid:12)(cid:104)T α
(cid:12)(cid:12)(cid:12)
(cid:113)|F|σ (cid:107)gF(cid:107)(cid:70)
 1
(cid:88)

(cid:107)gF(cid:107)(cid:70)2
L2(ω)

2

F∈F

L2(ω)

(cid:46) TT α (cid:107)f(cid:107)L2(σ)

Now 1F cψF is supported outside F , and each J in the dual martingale support CG,shif t
of gF = Pω
g is in particular good in the cube F , and as a consequence, each such cube
F
J as above is contained in some cube M for M ∈ W (F ). This containment will be used in
the analysis of the term IIG below.

CG,shif t

F

(cid:17)-deeply embedded in F , i.e. J (cid:98)(cid:104) 3

In addition, each J in the dual martingale support CG,shif t
(cid:105)
(cid:16)(cid:104) 3
of gF = Pω
F
F the definition of CG,shif t
each such cube J as above is contained in some cube M for M ∈ M(cid:16)(cid:104) 3
(cid:105)

. As a consequence,

CG,shif t
(cid:17)−deep,D (F ). This

g is

(cid:105)

, ε

,ε

,ε

ε

F

F

ε

ε

containment will be used in the analysis of the term IIB below.

234

Notation 5.5.8. Define ρ ≡(cid:104) 3

ε

(cid:105)

, so that for every J ∈ CG,shif t

F

, there is

M ∈ M(ρ,ε)−deep,G (F ) such that J ⊂ M.

(cid:105)

,ε

ε

|II| =

The collections W (F ) and M(ρ,ε)−deep,G (F ) used here, and in the display below, are
defined in (.0.7) in Appendix. Finally, since the cubes M ∈ W (F ), as well as the cubes
M ∈ M(cid:16)(cid:104) 3
(cid:17)−deep,G (F ), satisfy 3M ⊂ F , we can apply (5.1.54) in the Monotonicity
Lemma 5.1.23 using (5.5.14) with µ = 1F cψF and J(cid:48) in place of J there, to obtain
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:69)
(cid:88)
σ (1F cψF ) , gF(cid:105)ω
(cid:104)T α
(cid:88)
(cid:46) (cid:88)
J(cid:48)∈CG,shif t
(cid:88)
(cid:88)
J(cid:48)∈CG,shif t
(cid:88)
(cid:46) (cid:88)
(cid:88)
(cid:88)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =
(cid:68)
(cid:88)
(cid:88)
σ (1F cψF ) , (cid:3)ω,b∗
T α
J(cid:48)
Pα(cid:0)J(cid:48), 1F c|ψF|σ(cid:1)
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
J(cid:48)∈CG,shif t
(cid:0)J(cid:48), 1F c|ψF|σ(cid:1)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)x − mJ(cid:48)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)gF ;M
(cid:0)J(cid:48), 1F c|ψF|σ(cid:1)
(cid:13)(cid:13)x − mJ(cid:48)(cid:13)(cid:13)
L2(cid:16)

(cid:17)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

Pα (M, 1F cΦσ)

(cid:13)(cid:13)(cid:13)(cid:70)

(cid:13)(cid:13)(cid:13)(cid:70)

CG,shif t

M∈W(F )

J(cid:48) x

|J(cid:48)| 1

n

|J(cid:48)| 1

n

Pα

1+δ

g

ω

J(cid:48)

g

J(cid:48)

g

J(cid:48)

g

+

F∈F

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

F ;M

L2(ω)

♠

x

F∈F

L2(ω)

L2(ω)

F

F

F∈F

F∈F

F∈F

L2(ω)

L2(ω)

F

1J(cid:48) ω

L2(ω)

n

|M| 1

(cid:88)
Pα
J(cid:48)∈CG,shif t

F ;J

1+δ

|J(cid:48)| 1

n

+

F∈F

J∈Mdeep
≡ IIG + IIB .

(ρ,ε),G (F )

(cid:88)
J(cid:48)∈CG,shif t

F

(cid:3)ω,b∗
J(cid:48)

g.

: J(cid:48)⊂M

where gF ;M denotes the pseudoprojection gF ;M =

Note: We could also bound IIG by using the decomposition M(ρ,ε)−deep,G (F ) of F into
certain maximal G-cubes, but the ‘smaller’ choice W (F ) of D-cubes is needed for IIG in
order to bound it by the corresponding functional energy constant Fα, which can then be
controlled by the energy and Muckenhoupt constants in Appendix .

235

Then from Cauchy-Schwarz, the functional energy condition, and

≤ (cid:88)

F∈F

(cid:107)Φ(cid:107)2

L2(σ)

αF (F )2 |F|σ

(cid:46) (cid:107)f(cid:107)2

L2(σ)

,

we obtain

|IIG|≤

(cid:88)

F∈F

(cid:88)

M∈W(F )

(cid:18)Pα (M, 1F cΦσ)
(cid:88)

(cid:107)gF(cid:107)(cid:70)2
L2(ω)

|M|

F∈F

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:19)2
 1

2

1
2(cid:88)

♠2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

(cid:88)

(cid:13)(cid:13)gF ;M

(cid:13)(cid:13)(cid:70)2

L2(ω)

1

2

x

CG,shif t

F ;M

L2(ω)

F∈F

M∈W(F )

(cid:46) Fα (cid:107)Φ(cid:107)L2(σ)

(cid:46) Fα (cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω) ,

F ;M

jointly in F and M, which in turn

by the pairwise disjointedness of the coronas CG,shif t
follows from the pairwise disjointedness (5.5.1) of the shifted coronas CG,shif t
with the pairwise disjointedness of the cubes M. Thus we obtain the pairwise disjointedness
and Qω,b∗
of both of the pseudoprojections Pω,b∗
In term IIB the quantities (cid:13)(cid:13)x − mJ(cid:48)(cid:13)(cid:13)2
CG,shif t
L2(cid:16)

CG,shif t
(cid:17) are no longer additive except when the

jointly in F and M.

in F , together

F ;M

F ;M

F

cubes J(cid:48) are pairwise disjoint. As a result we will use (5.1.58) in the form,

1J(cid:48) ω

Pα

(cid:88)

J(cid:48)⊂J

(cid:0)J(cid:48), ν(cid:1)

2(cid:13)(cid:13)x − mJ(cid:48)(cid:13)(cid:13)2
L2(cid:16)

1J(cid:48)

1+δ

|J(cid:48)| 1

n

Pα
(cid:17) (cid:46) 1
Pα
γ2δ(cid:48)

(cid:46)

1+δ(cid:48) (J, ν)
|J| 1
1+δ(cid:48) (J, ν)
|J| 1

2

n

n

2 (cid:88)

J(cid:48)(cid:48)⊂J

(5.5.15)

(cid:13)(cid:13)(cid:13)(cid:52)ω

J(cid:48)(cid:48)x

(cid:13)(cid:13)(cid:13)2

L2

(cid:107)x − mJ(cid:107)2

,

L2(1J )

and exploit the decay in the Poisson integral Pα

1+δ(cid:48) along with weak goodness of the cubes
J. As a consequence we will be able to bound IIB directly by the strong energy condition

236

(5.1.8), without having to invoke the more difficult functional energy condition. For the
decay we compute that for J ∈ M(ρ,ε)−deep,G (F )

1+δ(cid:48) (J, 1F c|ψF|σ)
Pα

|J| 1

n

≈

≤

(cid:46)

F c

(cid:90)
∞(cid:88)
∞(cid:88)

t=0

t=0

|J| δ(cid:48)

n

δ(cid:48)

|ψF| (y) dσ

|y − cJ|n+1+δ(cid:48)−α
(cid:90)



dist(cid:0)cJ ,(cid:0)πtF F(cid:1)c(cid:1)

πt+1F F\πtF F

|J| 1

n

n

|J| 1

dist(cid:0)cJ ,(cid:0)πtF F(cid:1)c(cid:1)
(cid:18)
δ(cid:48)

Pα

J, 1

|ψF| (y)

(cid:19)
|y − cJ|n+1−α dσ

|ψF|σ

,

πt+1F F\πtF F

|J| 1

n

and then use the weak goodness inequality and the fact that J ⊂ F

(cid:16)

(cid:16)

dist

cJ ,

πtF F

(cid:17)c(cid:17) ≥ 2(cid:96)

(cid:16)

πtF F

(cid:17)1−ε

(cid:96) (J)ε ≥ 2 · 2t(1−ε)(cid:96) (F )1−ε (cid:96) (J)ε ≥ 2t(1−ε)+1(cid:96) (J) ,

to conclude that

Pα

1+δ(cid:48) (J, 1F c|ψF|σ)

|J| 1

n

2

(cid:46)

 ∞(cid:88)
∞(cid:88)

t=0

t=0

(cid:46)

2−tδ(cid:48)(1−ε)

2−tδ(cid:48)(1−ε)

Pα

(cid:18)
Pα

J, 1

(cid:18)

(cid:19)

|ψF|σ

πt+1F F\πtF F

|J| 1

n

|ψF|σ

J, 1

πt+1F F\πtF F

|J| 1

n

(5.5.16)

2
(cid:19)
2

.

where in the last inequality we used the Cauchy-Schwarz inequality. Now we again apply

237

Cauchy-Schwarz and (5.5.16) to obtain

IIB =

≤

F∈F

F∈F

J∈Mdeep

(cid:88)

(cid:88)

(ρ,ε),G (F )

(cid:88)
(cid:88)
(cid:88)
1
(cid:107)gF(cid:107)(cid:70)2
(cid:88)
L2(ω)
(cid:88)
≡ (cid:112)IIenergy (cid:107)g(cid:107)L2(ω) ,

(ρ,ε),G (F )

J∈Mdeep

F∈F

F∈F

≤

2

J∈M(ρ,ε)−deep,G (F )

1+δ

(cid:88)
Pα
J(cid:48)∈CG,shif t
(cid:88)
J(cid:48)∈CG,shif t

F ;J

F ;J

|J(cid:48)| 1

(cid:0)J(cid:48), 1F c|ψF|σ(cid:1)
Pα

(cid:17)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)x − mJ(cid:48)(cid:13)(cid:13)
L2(cid:16)
2(cid:13)(cid:13)x − mJ(cid:48)(cid:13)(cid:13)2
(cid:0)J(cid:48), 1F c|ψF|σ(cid:1)
L2(cid:16)

1J(cid:48) ω

J(cid:48)

1+δ

n

|J(cid:48)| 1

n

1J(cid:48) ω

g

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:70)


(cid:17)

1
2

Pα

1+δ(cid:48) (J, 1F c|ψF|σ)

|J| 1

n

2

(cid:107)x − mJ(cid:107)2

L2(1J ω)

1

2

(cid:107)g(cid:107)L2(ω)

and it remains to estimate IIenergy. From (5.5.16) and the strong energy condition (5.1.8),

238

we have

J∈M(ρ,ε)−deep,G (F )

(cid:88)

F∈F

F∈F

(ρ,ε),G (F )

IIenergy =

J∈Mdeep

(cid:88)
≤(cid:88)
(cid:88)
∞(cid:88)
2−tδ(cid:48)(1−ε)(cid:88)
∞(cid:88)
2−tδ(cid:48)(1−ε)(cid:88)
∞(cid:88)
2−tδ(cid:48)(1−ε) (cid:88)

G∈F

G∈F

(cid:46)

(cid:46)

t=0

t=0

=

t=0

G∈F

∞(cid:88)

t=0

2−tδ(cid:48)(1−ε)
(cid:88)
αF (G)2(cid:88)

(t+1)F

F∈C

(G)

F∈C

(t+1)F
αF (G)2 (E α

2

|ψF|σ

n

|J| 1

Pα
1+δ(cid:48) (J, 1F c|ψF|σ)
(cid:18)
Pα
(cid:88)

πt+1F F\πtF F

|J| 1

J, 1

J, 1

n

G\πtF F
|J| 1

n

(cid:18)
Pα
Pα

(cid:18)

J, 1

G\πtF F
|J| 1

n

2
(cid:19)
2
|ψF|σ
(cid:19)
2

σ

J∈Mdeep

(ρ,ε),G (F )

(cid:88)

(G)

J∈Mdeep
(ρ,ε)

(F )

2 )2 |G|σ

(cid:46) (E α

2 )2 (cid:107)f(cid:107)2

L2(σ)

.

(cid:107)x − mJ(cid:107)2
(cid:19)

L2(1J ω)

(cid:107)x − mJ(cid:107)2

L2(1J ω)

(cid:107)x − mJ(cid:107)2

L2(1J ω)

(cid:107)x − mJ(cid:107)2

L2(1J ω)

This completes the proof of the Intertwining Proposition 5.5.7.

The task of controlling functional energy is taken up in Appendix below.

5.5.4 Paraproduct, neighbour and broken forms

In this subsection we reduce boundedness of the local below form BA(cid:98)r,ε (f, g) defined in
(5.5.4) to boundedness of the associated stopping form

stop (f, g) ≡
BA

(cid:88)

A and J∈CG,shif t
I∈CD
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)

A

J

(cid:16)

(cid:17)(cid:68)

(cid:16)

(cid:17)

T α
σ

1A\IJ

bA

, (cid:3)ω,b∗

J

g

(5.5.17)

,

ω

(cid:69)

IJ(cid:98)(cid:3)σ,(cid:91),b

I

Eσ

f

239

where the modified difference (cid:98)(cid:3)σ,(cid:91),b

I

must be carefully chosen in order to control the corre-

sponding paraproduct form below. Indeed, below we will decompose

BA(cid:98)r,ε (f, g) = BA

paraproduct (f, g) − BA

stop (f, g) + BA

neighbour (f, g) + BA

brok (f, g) ,

and we will show that

(cid:12)(cid:12)(cid:12)BA(cid:98)r,ε (f, g) + BA

(cid:88)

A∈A

stop (f, g)

(cid:12)(cid:12)(cid:12) (cid:46)(cid:16)

Tb

T α +

(cid:113)

(cid:17)(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)

Aα
2

and the bound of BA

Note that the modified dual martingale differences (cid:3)σ,(cid:91),b

stop (f, g) will be the main subject of the next section.
,

and (cid:98)(cid:3)σ,(cid:91),b
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)
= bA(cid:98)(cid:3)σ,(cid:91),b
 (cid:83)
satisfy the following telescoping property for all K ∈ (cid:0)CA\{A}(cid:1) ∪

I f − (cid:88)

Fσ,b
I(cid:48) f = bA

(cid:3)σ,(cid:91),b

f ≡ (cid:3)σ,b

I(cid:48)∈Cbrok(I)

(cid:88)

I(cid:48)∈C(I)

1I(cid:48)Eσ
I(cid:48)

I

I

I

f

I

f,

I

 and

A(cid:48)

A(cid:48)∈CA(A)

L ∈ CA with K ⊂ L:
(cid:88)

Eσ
I

I: πK⊂I⊂L

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:17)

=



L(cid:98)Fσ,b
−Eσ
K(cid:98)Fσ,b
L(cid:98)Fσ,b
L f
K f − Eσ

L f

Eσ

if K ∈ CA (A)
K ∈ CA
if

.

Fix I ∈ CA for the moment. We will use

1I = 1IJ

1 ˜I ,

(cid:88)
˜I∈θ(IJ )
= 1A − 1A\IJ
,

+

1IJ

240

where θ (IJ ) denotes the 2n − 1 D-children of I other than the child IJ that contains J. We
begin with the splitting

(cid:68)
(cid:68)
(cid:68)

g

(cid:69)
I f, (cid:3)ω,b∗
(cid:17)
J
, (cid:3)ω,b∗
(cid:3)σ,b
1IJ
I f
(cid:17)

(cid:16)
σ (cid:3)σ,b
T α
(cid:16)

T α
σ

ω

J

T α
σ

(cid:3)σ,(cid:91),b

f

I

1IJ

, (cid:3)ω,b∗

J

g

=

=

≡ I + II + III .

(cid:17)

ω

+

(cid:69)
(cid:69)
(cid:88)

g

ω

˜I∈θ(IJ )

(cid:88)
(cid:42)
(cid:68)

T α
σ

T α
σ

(cid:16)
(cid:68)
1IJ
(cid:16)

1 ˜I

+

T α
σ

+
˜I∈θ(IJ )

(cid:3)σ,b
I f

1 ˜I

, (cid:3)ω,b∗

J

(cid:88)
(cid:17)
I(cid:48)∈Cbrok(I)
, (cid:3)ω,b∗
(cid:3)σ,b
I f

J

Fσ,b
I(cid:48) f

(cid:69)

g

ω

g

(cid:69)
 , (cid:3)ω,b∗

ω

J

(cid:43)

g

ω

From (5.1.47) we have

(cid:68)

I =

T α
σ

= Eσ
IJ

= Eσ
IJ

1IJ

(cid:16)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:16)(cid:98)(cid:3)σ,(cid:91),b

I

I

(cid:3)σ,(cid:91),b

f

I

(cid:17)(cid:68)
(cid:17)(cid:68)

f

f

J

g

(cid:17)

(cid:69)
, (cid:3)ω,b∗
(cid:16)
(cid:17)
T α
σ
σ bA, (cid:3)ω,b∗
T α

1IJ

bA

J

ω

=

(cid:68)
T α
σ
, (cid:3)ω,b∗
(cid:69)
g
− Eσ
IJ

J

ω

g

ω

I

J

g

f

bA

, (cid:3)ω,b∗

(cid:17)(cid:105)
(cid:16)

(cid:104)
(cid:16)
(cid:69)
1IJ(cid:98)(cid:3)σ,(cid:91),b
(cid:69)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)
(cid:17)(cid:68)
on IJ, we can define(cid:98)Fσ,b

1A\IJ

T α
σ

bA

f

ω

I

(cid:69)

g

ω

, (cid:3)ω,b∗

J

f ≡ 1
bIJ

Fσ,b
IJ

f

IJ

Since the function Fσ,b
IJ

f is a constant multiple of bIJ

and then

(cid:42)

II =

T α
σ

1IJ

, (cid:3)ω,b∗

J

(cid:88)
I(cid:48)∈Cbrok(I)

Fσ,b
I(cid:48) f

(cid:43)

g

= 1CA(A)(IJ ) Eσ
IJ

ω

(cid:16)(cid:98)Fσ,b

IJ

f

(cid:17)(cid:68)

T α
σ bIJ

(cid:69)

ω

, (cid:3)ω,b∗

J

g

where the presence of the indicator function 1CA(A) (IJ ) simply means that term II vanishes

241

unless IJ is an A-child of A. We now write these terms as

(cid:68)

σ (cid:3)σ,b
T α

I f, (cid:3)ω,b∗

J

g

(cid:69)

ω

= Eσ
IJ
−Eσ
IJ

I

J

g

f

(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)(cid:68)
(cid:69)
σ bA, (cid:3)ω,b∗
(cid:16)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)
(cid:17)(cid:68)
T α
(cid:17)
(cid:68)
(cid:16)
(cid:88)
T α
1A\IJ
bA
σ
, (cid:3)ω,b∗
(cid:3)σ,b
(cid:16)(cid:98)Fσ,b
(cid:17) (cid:68)
I f

˜I∈θ(IJ )
+1{IJ∈CA(A)} Eσ
IJ

T α
σ

1 ˜I

+

IJ

f

f

g

ω

J

I

(cid:69)

ω

, (cid:3)ω,b∗
(cid:69)

J

g

ω

, (cid:3)ω,b∗

J

g

T α
σ bIJ

(cid:69)

,

ω

where the four lines are respectively a paraproduct, stopping, neighbour and broken term.

The corresponding NTV splitting of BA(cid:98)r,ε (f, g) using (5.5.4) and (5.5.2) becomes

BA(cid:98)r,ε (f, g) =

T α
σ

PσCA

(cid:42)

(cid:16)

(cid:17)
(cid:88)

f

g

, Pω
A

(cid:68)
CG,shif t
T α
σ

(cid:43)(cid:98)r,ε
(cid:16)(cid:3)σ,b

ω

I f

=

J

= BA

I∈CA and J∈CG,shif t
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)
paraproduct (f, g) − BA

A

(cid:17)

(cid:69)

ω

, (cid:3)ω,b∗

J

g

stop (f, g) + BA

neighbour (f, g) + BA

brok (f, g) ,

where

paraproduct (f, g) ≡
BA

stop (f, g) ≡
BA

J

J

neighbour (f, g) ≡
BA

J

(cid:69)

g

ω

I

f

Eσ
IJ

(cid:17)(cid:68)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
I∈CA and J∈CG,shif t
(cid:17)(cid:68)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)
Eσ
IJ
I∈CA and J∈CG,shif t
(cid:16)
(cid:68)
(cid:88)
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)

(cid:88)
(cid:88)
(cid:88)

T α
σ

A

A

f

I

T α
σ

(cid:3)σ,b
I f

1 ˜I

J

σ bA, (cid:3)ω,b∗
T α
(cid:17)
(cid:16)
(cid:69)
(cid:17)

1A\IJ

, (cid:3)ω,b∗

bA

g

J

ω

(cid:69)

g

ω

, (cid:3)ω,b∗

J

I∈CA and J∈CG,shif t
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)

A

˜I∈θ(IJ )

242

correspond to the three original NTV forms associated with 1-testing, and where

(cid:17) (cid:68)

(cid:16)(cid:98)Fσ,b

IJ

f

(cid:69)

, (cid:3)ω,b∗

J

g

T α
σ bIJ

(5.5.18)

ω

(cid:88)

brok (f, g) ≡
BA

J

I∈CA and J∈CG,shif t
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)

A

1{IJ∈CA(A)} Eσ
IJ

"vanishes" since J

(cid:122) (cid:36) I and IJ ∈ CA (A) imply that J

(cid:122)

/∈ CG

A

, contradicting J ∈ CG,shif t

A

.

Remark 5.5.9. The inquisitive reader will note that the pairs (I, J) arising in the above

sum with J

(cid:122) (cid:36) I replaced by J

(cid:122)

= I are handled in the probabilistic estimate (5.2.15) for

the bad form Θbad(cid:92)

2

defined in (5.2.12).

5.5.4.1 The paraproduct form

The paraproduct form BA

paraproduct (f, g) is easily controlled by the testing condition for
T α together with weak Riesz inequalities for dual martingale differences. Indeed, recalling

the telescoping identity (5.1.48), and that the collection (cid:8)I ∈ CA: (cid:96) (J) ≤ 2−r(cid:96) (I)(cid:9) is tree

connected for all J ∈ CG,shif t

A

, we have

BA

paraproduct (f, g) =

J

J

A

f

I

g

ω

Eσ
IJ

(cid:17)(cid:68)

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

I∈CA and J∈CG,shif t
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)
(cid:69)
(cid:69)

(cid:88)

(cid:40)
I∈CA: J(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)
(cid:98)Fσ,b
1(cid:110)
J:I(cid:92)(J)J∈CA
A(cid:98)Fσ,b
(cid:111)Eσ

(cid:88)
(cid:111)Eσ
(cid:98)Fσ,b

σ bA, (cid:3)ω,b∗
T α

(cid:68)
(cid:68)
σ bA, (cid:3)ω,b∗
T α
(cid:40)
(cid:88)
1(cid:110)
J:I(cid:92)(J)J∈CA
J∈CG,shif t

f − Eσ

(cid:69)
σ bA, (cid:3)ω,b∗
T α
(cid:17)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:41)
A(cid:98)Fσ,b
(cid:43)
(cid:3)ω,b∗

f − Eσ

(cid:41)

A f

Eσ
IJ

I(cid:92)(J)J

I(cid:92)(J)J

I(cid:92)(J)J

I(cid:92)(J)J

g

g

ω

ω

J

J

g

J

f

I

A f

A

ω

(cid:88)
J∈CG,shif t
(cid:88)
(cid:42)
J∈CG,shif t

A

A

T α
σ bA,

=

=

=

243

I

I(cid:92)(J)J

(cid:122) (cid:36) I and (cid:96) (J) ≤ 2−r(cid:96) (I), and
denotes its child containing J. Note that by construction of the modified

where I(cid:92) (J) denotes the smallest cube I ∈ CA such that J
of course I(cid:92) (J)J
, the only time the average (cid:98)Fσ
difference operator (cid:3)σ,(cid:91),b
f appears in the above sum is
when I(cid:92) (J)J ∈ CA, since the case I(cid:92) (J)J ∈ A has been removed to the broken term. This is
reflected above with the inclusion of the indicator 1(cid:110)
(cid:111). It follows that we have
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) +
(cid:12)(cid:12)(cid:12)Eσ
A(cid:98)Fσ,b
(cid:32)
1(cid:110)
J:I(cid:92)(J)J∈CA

Thus from Cauchy-Schwarz, the upper weak Riesz inequalities for the pseudoprojections

the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1(cid:110)

g and the bound on the coefficients λJ ≡

(cid:3)ω,b∗
given by |λJ| (cid:46) αA (A), we have

(cid:12)(cid:12)(cid:12) (cid:46) Eσ

A |f| ≤ αA (A)

A(cid:98)Fσ,b

J:I(cid:92)(J)J∈CA

J:I(cid:92)(J)J∈CA

(cid:98)Fσ,b

(cid:98)Fσ,b

(cid:111)Eσ

(cid:111)Eσ

f − Eσ

I(cid:92)(J)J

I(cid:92)(J)J

I(cid:92)(J)J

A f

f

I(cid:92)(J)J

J

A f

(cid:33)

(cid:46) αA (A) (cid:107)1AT α

σ bA(cid:107)L2(ω)

≤ Tb

T α αA (A)

244

(cid:33)(cid:41)

A(cid:98)Fσ,b

A f

f − Eσ

(5.5.19)

(cid:43)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

ω

(cid:3)ω,b∗

J

g

(cid:12)(cid:12)(cid:12)BA

paraproduct (f, g)

(cid:12)(cid:12)(cid:12) =
(cid:88)
J∈CG,shif t

A

T α
σ bA,

(cid:42)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

≤ (cid:107)1AT α

σ bA(cid:107)L2(ω)

(cid:111)Eσ

(cid:98)Fσ,b

I(cid:92)(J)J

I(cid:92)(J)J

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

g

g

J

(cid:40)(cid:32)
1(cid:110)
J:I(cid:92)(J)J∈CA
(cid:88)
J∈CG,shif t

J

λJ(cid:3)ω,b∗
(cid:88)
J∈CG,shif t

A

A

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:113)|A|σ

CG,shif t

g

.

L2(ω)

5.5.4.2 The neighbour form

neighbour (f, g) is easily controlled by the Aα

Next, the neighbour form BA
pivotal estimate in Energy Lemma 5.1.25 and the fact that the cubes J ∈ CG,shif t
I and beyond when the pair (I, J) occurs in the sum. In particular, the information encoded
in the stopping tree A plays no role here, apart from appearing in the corona projections on
the right hand side of (5.5.25) below. We have

condition using the

are good in

A

2

(cid:88)

(cid:88)

(cid:68)

(cid:16)

(cid:17)

T α
σ

(cid:3)σ,b
I f

1 ˜I

, (cid:3)ω,b∗

J

g

(5.5.20)

(cid:69)

ω

BA

neighbour (f, g) =

I∈CA and J∈CG,shif t
(cid:122)(cid:36)I and (cid:96)(J)≤2−r(cid:96)(I)

A

J

˜I∈θ(IJ )

where we keep in mind that the pairs (I, J) ∈ D×G that arise in the sum for BA
satisfy the property that J

neighbour (f, g)
(cid:122) (cid:36) I, so that J is good with respect to all cubes K of size at
(cid:122), which includes I. Recall that IJ is the child of I that contains J, and that
least that of J
θ (IJ ) denotes its 2n − 1 siblings in I, i.e. θ (IJ ) = CD (I)\{IJ}. Fix (I, J) momentarily,
and an integer s ≥ r. Using (cid:3)σ,b
f is a constant
multiple of b ˜I

+ (cid:3)σ,(cid:91),b
I,brok
on the cube ˜I, we have the estimates

and the fact that (cid:3)σ,(cid:91),b

I = (cid:3)σ,(cid:91),b

I

I

(cid:12)(cid:12)(cid:12) ≤ Cb

(cid:12)(cid:12)(cid:12)Eσ
˜I(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:12)(cid:12)(cid:12) ,

(cid:12)(cid:12)(cid:12)1 ˜I
(cid:12)(cid:12)(cid:12)1 ˜I

(cid:3)σ,(cid:91),b

f

I

(cid:3)σ,(cid:91),b

I,brokf

b ˜I
|f| ,

I

f

Eσ

(cid:12)(cid:12)(cid:12)(cid:16)

(cid:12)(cid:12)(cid:12) =
(cid:17)
˜I(cid:98)(cid:3)σ,(cid:91),b
(cid:12)(cid:12)(cid:12) ≤ 1CA(A)( ˜I) Eσ
(cid:16)(cid:12)(cid:12)(cid:12)Eσ
(cid:12)(cid:12)(cid:12) ≤ C1 ˜I
˜I(cid:98)(cid:3)σ,(cid:91),b

˜I

f

I

and hence

(cid:12)(cid:12)(cid:12)(cid:3)σ,b

I f

1 ˜I

(cid:12)(cid:12)(cid:12) + 1CA(A)( ˜I) Eσ

˜I

|f|(cid:17)

,

(5.5.21)

245

which will be used below after an application of the Energy Lemma. We can write

neighbour (f, g) as
BA

(cid:88)
(cid:16) ˜I
(cid:17)1−ε
(κ(IJ ,J),ε)−good
>2(cid:96)(J)ε(cid:96)

I∈CA&J∈GD
(cid:17)
(cid:16)

d

J, ˜I

∩CG,shif t
and (cid:96)(J)≤2−r(cid:96)(I)

&J

A

(cid:122)(cid:36)I

(cid:16)

T α
σ

(cid:88)

(cid:68)

˜I∈θ(IJ )

(cid:17)

(cid:69)

ω

, (cid:3)ω,b∗

J

g

(cid:3)σ,b
I f

1 ˜I

where we have included the conditions

J ∈ GD

(κ(IJ ,J),ε)−good

and d(J, ˜I) > 2(cid:96) (J)ε (cid:96)( ˜I)1−ε

in the summation since they are already implied the remaining four conditions, and will be

used in estimates below.

We will also use the following fractional analogue of the Poisson inequality in [75].

Lemma 5.5.10. Suppose 0 ≤ α < 1 and J ⊂ I ⊂ K and that d (J, ∂I) > 2(cid:96) (J)ε(cid:96) (I)1−ε for

some 0 < ε <

1

n+1−α. Then for a positive Borel measure µ we have

(cid:18) (cid:96) (J)

(cid:19)1−ε(n+1−α)

Pα(I, µ1K\I ).

(5.5.22)

Pα(J, µ1K\I ) (cid:46)

Proof. We have

Pα(cid:16)

(cid:17) ≈
(cid:17) ∩ (K\I) (cid:54)= ∅ requires

J, µ1K\I

and(cid:16)

2kJ

(cid:96) (I)

∞(cid:88)

k=0

2−k

(cid:90)(cid:16)

(cid:17)∩(K\I)

2kJ

dµ,

(cid:12)(cid:12)2kJ(cid:12)(cid:12)1− α

1

n

d (J, K\I) ≤ c2k(cid:96) (J) ,

246

for some dimensional constant c > 0. Let k0 be the smallest such k. By our distance
assumption we must then have

or

2(cid:96) (J)ε (cid:96) (I)1−ε ≤ d (J, ∂I) ≤ c2k0(cid:96) (J) ,

(cid:18) (cid:96) (J)

(cid:19)1−ε

(cid:96) (I)

.

2−k0+1 ≤ c

Now let k1 be defined by 2k1 ≡ (cid:96)(I)
we have

(cid:96)(J)

. Then assuming k1 > k0 (the case k1 ≤ k0 is similar)

+

J, µ1K\I

Pα(cid:16)

 k1(cid:88)
(cid:17) ≈
∞(cid:88)
 1
(cid:90)(cid:16)
(cid:12)(cid:12)(cid:12)1− α
(cid:12)(cid:12)(cid:12)2k0J
(cid:18) (cid:96) (J)
(cid:19)(1−ε)(n+1−α)(cid:18) (cid:96) (I)

 2−k
(cid:17)∩(K\I)
(cid:19)n−α
Pα(cid:16)

(cid:90)(cid:16)
(cid:12)(cid:12)2kJ(cid:12)(cid:12)1− α
(cid:17)∩(K\I)
 + 2−k1Pα(cid:16)
(cid:17)
Pα(cid:16)

|I|1− α

2k1J

|I|1− α

n

k=k1

k=k0

2kJ

dµ

+

1

n

n

n

I, µ1K\I

(cid:96) (I)

(cid:96) (J)

(cid:96) (J)
(cid:96) (I)

dµ

(cid:17)

I, µ1K\I

(cid:17)

,

I, µ1K\I

(cid:46) 2−k0

(cid:46)

which is the inequality (5.5.22).

Now fix I0 = IJ , Iθ ∈ θ (IJ ) and assume that J (cid:98)r,ε I0. Let (cid:96)(J)

(cid:96)(I0) = 2−s in the pivotal

estimate from Energy Lemma 5.1.25 with J ⊂ I0 ⊂ I to obtain

σ

(cid:12)(cid:12)(cid:12)(cid:104)T α
(cid:16)
(cid:46) (cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:46) (cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

J

1Iθ

(cid:3)σ,b
I f

(cid:13)(cid:13)(cid:13)L2(ω)
(cid:13)(cid:13)(cid:13)L2(ω)

g

g

J

g(cid:105)ω

, (cid:3)ω,b∗

(cid:12)(cid:12)(cid:12)(cid:46)(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

(cid:13)(cid:13)(cid:13)L2(ω)
(cid:17)
(cid:113)|J|ω · 2−(1−ε(n+1−α))sPα(cid:16)
(cid:113)|J|ω · 2−(1−ε(n+1−α))sPα(cid:16)

g

J

(cid:113)|J|ωPα(cid:16)
(cid:12)(cid:12)(cid:12)(cid:3)σ,b

I0, 1Iθ

I0, 1Iθ

Eσ
Iθ

J, 1Iθ

(cid:17)

(cid:12)(cid:12)(cid:12) σ
(cid:17)

I f
f · σ

(cid:12)(cid:12)(cid:12)(cid:3)σ,b

I f

(cid:12)(cid:12)(cid:12) σ

(cid:17)

247

Here we are using (5.5.22) in the third line, which applies since J ⊂ I0, and we have used
(5.5.21) in the fourth line and the shorthand notation

f ≡(cid:12)(cid:12)(cid:12)Eσ

Iθ(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:12)(cid:12)(cid:12) + 1CA(A) (Iθ) Eσ

Iθ

|f|

Eσ
Iθ

where the cube I on the right hand side is determined uniquely by the cube Iθ ∈ θ (IJ ).

In the sum below, we keep the side lengths of the cubes J fixed at 2−s times that of I0,
and

and of course take J ⊂ I0. We also keep the underlying assumptions that J ∈ CG,shif t
that J ∈ GD
Matters will shortly be reduced to estimating the following term:

in mind without necessarily pointing to them in the notation.

(κ(IJ ,J),ε)−good

A

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

g

(cid:13)(cid:13)(cid:13)2

;(5.5.23)

L2(ω)

: 2s+1(cid:96)(J)=(cid:96)(I): J⊂I

(cid:88)
(cid:88)

I0∈CD(I)

(cid:88)

I∈CA

I0∈CD(I)

Λ(I, I0, Iθ, s)2 ≡

Λ(I, I0, Iθ, s)2 ≤

(cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

J∈CG,shif t

A

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A

CG,shif t

g

248

A(I, I0, Iθ, s) ≡

(cid:88)

J : 2s+1(cid:96)(J)=(cid:96)(I):J⊂I0
Eσ
Iθ

≤ 2−(1−ε(n+1−α))s(cid:16)
≤ 2−(1−ε(n+1−α))s(cid:16)
(cid:88)

Eσ
Iθ

(cid:113)|J|ω

(cid:3)σ,b
I f

1Iθ

(cid:16)

σ

(cid:12)(cid:12)(cid:12)(cid:104)T α
(cid:17)
(cid:17)

f

f

Pα(I0, 1Iθ

Pα(I0, 1Iθ

J

g(cid:105)ω

σ)
J:J⊂I0

(cid:12)(cid:12)(cid:12)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

(cid:17)
, (cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)L2(ω)
(cid:88)
(cid:113)|I0|ωΛ(I, I0, Iθ, s)
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

2s+1(cid:96)(J)=(cid:96)(I)

σ)

g

g

J

J

.

L2(ω)

where Λ(I, I0, Iθ, s)2 ≡

The last line follows upon using the Cauchy-Schwarz inequality and the fact that J ∈

A

J∈CG,shif t

: 2s+1(cid:96)(J)=(cid:96)(I): J⊂I0

CG,shif t

A

. We also note that since 2s+1(cid:96) (J) = (cid:96) (I),

Using (5.4.2) we obtain

(cid:12)(cid:12)(cid:12)Eσ

Iθ

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

I

(cid:17)(cid:12)(cid:12)(cid:12) ≤

f

(cid:114)

Eσ
Iθ

(cid:12)(cid:12)(cid:12)(cid:98)(cid:3)σ,(cid:91),b

I

f

(cid:12)(cid:12)(cid:12)2 (cid:46)(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

(cid:13)(cid:13)(cid:13)(cid:70)

L2(σ)

|Iθ|− 1

σ

2

(5.5.24)

and hence

Eσ
Iθ

f ≡ (cid:12)(cid:12)(cid:12)Eσ
Iθ(IJ )(cid:98)(cid:3)σ,(cid:91),b
(cid:32)(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)(cid:70)

I f

(cid:46)

I

f

L2(σ)

(cid:12)(cid:12)(cid:12) + 1CA(A) (Iθ) Eσ

|f|
+ 1CA(A) (Iθ) |Iθ| 1

Iθ

σ Eσ
2
Iθ

|Iθ|− 1

σ

2

(cid:33)

|f|

·

(cid:33)

|f|

σ Eσ
2
Iθ

+ 1CA(A) (Iθ) |Iθ| 1
(cid:113)|I0|ω
+ 1CA(A) (Iθ) |Iθ| 1

(cid:13)(cid:13)(cid:13)(cid:70)

σ)

L2(σ)

σ Eσ
2
Iθ

(cid:33)

|f|

Λ(I, I0, Iθ, s)

and thus A(I, I0, Iθ, s) is bounded by

(cid:32)(cid:13)(cid:13)(cid:13)(cid:3)σ,b

L2(σ)

(cid:13)(cid:13)(cid:13)(cid:70)
(cid:32)(cid:13)(cid:13)(cid:13)(cid:3)σ,b

I f

σ Pα(I0, 1Iθ

2−(1−ε(n+1−α))s

I f
·Λ(I, I0, Iθ, s)|Iθ|− 1
(cid:46)(cid:113)
2 2−(1−ε(n+1−α))s
Aα

2

since Pα(I0, 1Iθ

σ) (cid:46) |Iθ|σ
|Iθ|1− α

n

shows that

|Iθ|− 1

2

σ Pα(I0, 1Iθ

(cid:113)|I0|ω

(cid:46)

σ)

(cid:112)|Iθ|σ

(cid:112)|I0|ω

|Iθ|1− α

n

(cid:46)(cid:113)

Aα
2

where the implied constant depends on α and the dimension. An application of Cauchy-

249

(cid:13)(cid:13)(cid:13)(cid:70)2

(cid:88)

(cid:16)

Eσ
Iθ

|f|(cid:17)2 ·

|Iθ|σ

I f

L2(σ)

+
Iθ∈CA(A)

Schwarz to the sum over I using (5.5.23) then shows that

(cid:88)

(cid:88)

(cid:46)(cid:113)

I∈CA

I0,Iθ∈CD(I)

I0(cid:54)=Iθ

(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)

·

2 2−(1−ε(n+1−α))s
Aα
(cid:88)
(cid:88)

I∈CA

I0,Iθ∈CD(I)

I0(cid:54)=Iθ



A(I, I0, Iθ, s)

(cid:118)(cid:117)(cid:117)(cid:116)(cid:88)

I∈CA

(cid:13)(cid:13)(cid:13)(cid:3)σ,b


2

Λ(I, I0, Iθ, s)

(cid:118)(cid:117)(cid:117)(cid:116)(cid:13)(cid:13)(cid:13)PσCA

f

L2(σ)

(cid:13)(cid:13)(cid:13)(cid:70)2


2

(cid:46)(cid:113)

(cid:46)(cid:113)



(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)

·

2 2−(1−ε(n+1−α))s
Aα
(cid:88)
(cid:88)

I∈CA

I0∈CD(I)
I0(cid:54)=Iθ
2 2−(1−ε(n+1−α))s
Aα

(cid:88)
A(cid:48)∈CA(A)

+

(cid:16)

A(cid:48) |f|(cid:17)2 ·

Eσ

|A(cid:48)|σ

Λ(I, I0, Iθ, s)

(cid:107)PσCA

f(cid:107)(cid:70)

+

L2(σ)

(cid:118)(cid:117)(cid:117)(cid:116) (cid:88)

A(cid:48)∈CA(A)

(cid:16)

A(cid:48) |f|(cid:17)2

Eσ

|A(cid:48)|σ

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A

CG,shif t

This estimate is summable in s ≥ r since ε <

1

n+1−α

, and so the proof of

(cid:12)(cid:12)(cid:12)BA
(cid:46) (cid:113)

Aα
2

neighbour (f, g)

(cid:13)(cid:13)(cid:13)PσCA

f

(cid:12)(cid:12)(cid:12) ≤ (cid:88)
(cid:13)(cid:13)(cid:13)(cid:70)

(cid:88)
(cid:118)(cid:117)(cid:117)(cid:116) (cid:88)

I∈CA

L2(σ)

+

I0(cid:54)=Iθ

A(cid:48)∈CA(A)

I0 and Iθ∈CD(I)

∞(cid:88)

s=r

|A(cid:48)|σ αA (A(cid:48))2

A(I, I0, Iθ, s)



(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A

CG,shif t

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

g

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

g

(5.5.25)

250

is complete since Eσ

Now if we sum in A ∈ A the inequalities (5.5.19), (5.5.25) and (5.5.18) we get

(cid:12)(cid:12)(cid:12)

A∈A

A(cid:48) |f| (cid:46) αA(cid:0)A(cid:48)(cid:1).
(cid:12)(cid:12)(cid:12)BA(cid:98)r,ε (f, g) + BA
(cid:88)
(cid:17)(cid:118)(cid:117)(cid:117)(cid:117)(cid:116)(cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:113)
(cid:46) (cid:16)
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)(cid:88)
αA (A)2 |A|σ +
(cid:13)(cid:13)(cid:13)PσCA
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:113)
(cid:17)(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)
(cid:46) (cid:16)

CG,shif t

stop (f, g)

T α +

T α +

A∈A

A∈A

Aα
2

Tb

Aα
2

·

Tb

g

A

L2(ω)

f

+

L2(σ)

·

(cid:88)
A(cid:48)∈CA(A)



αA (A(cid:48))2 |A(cid:48)|σ

The stopping form is the subject of the following section.

5.6 The stopping form

Here we deal with the stopping form. We modify the adaptation of the argument of M. Lacey

in to apply in the setting of a T b theorem for an α-fractional Calderón-Zygmund operator T α

in Rn using the Monotonicity Lemma 5.1.23, the energy condition, and the weak goodness

of Hytönen and Martikainen [24]. We directly control the pairs (I, J) in the stopping form
according to the L -coronas (constructed from the ‘bottom up’ with stopping times involving
(cid:122) are associated. However, due to the fact that

the energies(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

) to which I and J

(cid:13)(cid:13)(cid:13)2

J

L2(ω)

the cubes I need no longer be good in any sense, we must introduce an additional top/down

‘indented’ corona construction on top of the bottom/up construction of M. Lacey, and in

connection with this we introduce a Substraddling Lemma. We then control the stopping
(cid:122) belong to the same L -corona, and by

form by absorbing the case when both I and J

251

using the Straddling and Substraddling Lemmas, together with the Orthogonality Lemma,
(cid:122) lie in different coronas, with a geometric gain coming

to control the case when I and J

from the separation of the coronas. This geometric gain is where the new ‘indented’ corona

is required.

Apart from this change, the remaining modifications are more cosmetic, such as

• the use of the weak goodness of Hytönen and Martikainen [24] for pairs (I, J) arising in

the stopping form, rather than goodness for all cubes J that was available in [27], [64],

[66] and [67]. For the most part definitions such as admissible collections are modified

to require J

(cid:122) ⊂ I;

• the pseudoprojections (cid:3)σ,b

I

, (cid:3)ω,b∗

J

are used in place of the orthogonal Haar projections,

and the frame and weak Riesz inequalities compensate for the lack of orthogonality.

Fix grids D and G. We will prove the bound

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

g

,

(5.6.1)

CG,shif t

where we recall that the nonstandard ‘norms’ are given by,

(cid:12)(cid:12)(cid:12)BA

stop (f, g)

(cid:12)(cid:12)(cid:12) (cid:46) NT Vα
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2

CG,shif t

L2(σ)

CD

L2(ω)

A

A

f

g

f

A

A

CD

L2(σ)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b
(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)2
≡ (cid:88)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
≡ (cid:88)

I∈CD

I f

A

J

J∈CG,shif t

A

L2(σ)

,

(cid:13)(cid:13)(cid:13)2

g

,

L2(ω)

252

and that the stopping form is given by

Eσ

(cid:16)

A and J∈CG,shif t
I∈CD
(cid:122)(cid:36)I and (cid:96)(J)≤2−ρ(cid:96)(I)

(cid:88)
(cid:88)
A and J∈CG,shif t
I: πI∈CD
(cid:122)(cid:36)I and (cid:96)(J)≤2−(ρ−1)(cid:96)(I)

A

A

stop (f, g) ≡
BA

=

J

J

T α
σ

(cid:17)(cid:68)
(cid:17)(cid:68)

I

f

IJ(cid:98)(cid:3)σ,(cid:91),b
(cid:16)
I(cid:98)(cid:3)σ,(cid:91),b

Eσ

πI f

(cid:16)

bA1A\IJ

(cid:17)

(cid:16)

bA1A\I

T α
σ

(cid:69)

g

J

, (cid:3)ω,b∗
(cid:17)

, (cid:3)ω,b∗

J

g

ω

(cid:69)

ω

where we have made the ‘change of dummy variable’ IJ → I for convenience in notation
(recall that the child of I that contains J is denoted IJ). Changing ρ − 1 to ρ we have:

BA

stop (f, g) =

(cid:88)
A and J∈CG,shif t
I: πI∈CD
(cid:122)(cid:36)I and (cid:96)(J)≤2−ρ(cid:96)(I)

A

J

(cid:16)

I(cid:98)(cid:3)σ,(cid:91),b

πI f

Eσ

(cid:17)(cid:68)

(cid:16)

T α
σ

bA1A\I

(cid:17)

(cid:69)

, (cid:3)ω,b∗

J

g

,

ω

For A ∈ A recall that we have defined the shifted G-corona by

≡(cid:110)

CG,shif t

A

J ∈ G : J

(cid:122) ∈ CD

A

(cid:111)

,

and also defined the restricted D-corona by

CD,restrict

A

≡ CA\{A} ≡ C(cid:48)
A.

Definition 5.6.1. Suppose that A ∈ A and that P ⊂ CD,restrict
collection of pairs P is A -admissible if

A

×CG,shif t

A

. We say that the

• (good and (ρ, ε)-deeply embedded) For every (I, J) ∈ P, and J

(cid:122) ⊂ I (cid:38) A.

• (tree-connected in the first component) if I1 ⊂ I2 and both (I1, J) ∈ P and (I2, J) ∈ P,

253

then (I, J) ∈ P for every I in the geodesic [I1, I2] = {I ∈ D : I1 ⊂ I ⊂ I2}.

From now on we often write CA and C(cid:48)

A

in place of CD

A

and CD,restrict

A

respectively when

there is no confusion. The basic example of an admissible collection of pairs is obtained from

the pairs of cubes summed in the stopping form BA

stop (f, g),

PA ≡(cid:110)

(I, J) : I ∈ C(cid:48)

and J ∈ GD

(ρ,ε)−good ∩ CG,shif t

A

A

(cid:111)

.

where J (cid:98)ρ,ε I

(5.6.2)

Definition 5.6.2. Suppose that A ∈ A and that P is an A -admissible collection of pairs.
Define the associated stopping form BA,P
I(cid:98)(cid:3)σ,(cid:91),b

stop (f, g) ≡ (cid:88)

, (cid:3)ω,b∗

(cid:17) (cid:68)

bA1A\I

stop by

BA,P

(cid:69)

(cid:16)

(cid:17)

(cid:16)

T α
σ

J

Eσ

πI f

g

.

ω

(I,J)∈P

Proposition 5.6.3. Suppose that A ∈ A and that P is an A-admissible collection of pairs.
Then the stopping form BA,P

stop satisfies the bound

(cid:12)(cid:12)(cid:12)BA,P

stop (f, g)

2 +

Aα
2

(cid:12)(cid:12)(cid:12) (cid:46)(cid:16)E α

(cid:113)

(cid:17)(cid:13)(cid:13)(cid:13)Pσ,bCA

(cid:13)(cid:13)(cid:13)(cid:70)

L2(σ)

f

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A

CG,shif t

g

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

(5.6.3)

With the above proposition in hand, we can complete the proof of (5.6.1) by summing

254

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
 1
2(cid:88)

CG,shif t

A

g

A∈A

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

L2(ω)

A

CG,shif t

 1

2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

g

A∈A

(cid:88)
(cid:46) (cid:88)
(cid:46) (cid:16)E α
(cid:46) (cid:16)E α

A∈A

2 +

2 +

f

L2(σ)

stop

Aα
2

2 +

(f, g)

(cid:12)(cid:12)(cid:12)(cid:12)BA,PA
(cid:12)(cid:12)(cid:12)(cid:12)
(cid:17)(cid:13)(cid:13)(cid:13)Pσ,bCA
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:113)
(cid:16)E α
(cid:17)(cid:88)
(cid:13)(cid:13)(cid:13)Pσ,bCA
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:113)
(cid:113)
(cid:17)(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)Pσ,bCA

A∈A

Aα
2

Aα
2

f

f

L2(σ)

by the lower Riesz inequality (cid:88)
(cid:88)
: (cid:88)
L2(σ)
A∈A
the shifted coronas CG,shif t

αA (f )2 |A|σ

(cid:46) (cid:107)f(cid:107)2

≤ 1D.
To prove Proposition 5.6.3, we begin by letting

1CG,shif t

A∈A

A

A

(cid:46) (cid:107)f(cid:107)2

L2(σ)

, quasi-orthogonality

L2(σ)

A∈A
in the stopping cubes A, and by the pairwise disjointedness of

over the stopping cubes A ∈ A with the choice PA of A-admissible pairs for each A:

Π1P ≡ (cid:110)
Π2P ≡ (cid:110)

I ∈ CD,restrict
J ∈ CG,shif t

: (I, J) ∈ P for some J ∈ CG,shif t
: (I, J) ∈ P for some I ∈ C(cid:48)

(cid:111)

A

A

,

A

A

(cid:111)

,

consist of the first and second components respectively of the pairs in P, and writing

(cid:68)
(cid:88)
(cid:88)

J∈Π2P

BA,P
stop (f, g) =
where ϕP

J ≡

σ ϕP
T α

J , (cid:3)ω,b∗

J

(cid:69)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b

g

ω

;

πI f

bAEσ
I

I∈C(cid:48)

A: (I,J)∈P

(cid:17)

1A\I

(since bI = bA for I ∈ CA).

By the tree-connected property of P, and the telescoping property of dual martingale differ-

255

ences, together with the bound αA (A) on the averages of f in the corona CA, we have

(cid:12)(cid:12)(cid:12)ϕP

J

(cid:12)(cid:12)(cid:12) (cid:46) αA (A) 1A\IP (J),

where IP (J) ≡(cid:84){I : (I, J) ∈ P} is the smallest cube I for which (I, J) ∈ P. It is important

to note that J is good with respect to IP (J) by our infusion of weak goodness above. Another
important property of these functions is the sublinearity:

(5.6.4)

(5.6.5)

to obtain

(cid:12)(cid:12)(cid:12) +

(cid:12)(cid:12)(cid:12)ϕ

P2

J

(cid:12)(cid:12)(cid:12) ,

P1

J

P = P1 ˙∪P2 .

J

(cid:12)(cid:12)(cid:12)ϕP
(cid:12)(cid:12)(cid:12) ≤(cid:12)(cid:12)(cid:12)ϕ
(cid:12)(cid:12)(cid:12) (cid:46) Pα(cid:16)

J g(cid:11)
Now apply the Monotonicity Lemma 5.1.23 to the inner product(cid:10)T α
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)♠
(cid:12)(cid:12)(cid:12)(cid:68)
(cid:17)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)L2(1J ω)
(cid:13)(cid:13)x − mω

J,|ϕJ| 1A\IP (J)σ
(cid:16)

|J| 1
J,|ϕJ| 1A\IP (J)σ

σ ϕJ , (cid:3)ω,b∗
T α

σ ϕJ , (cid:3)ω

L2(ω)

(cid:69)

(cid:17)

Pα

1+δ

+

x

g

g

ω

ω

n

J

J

J

J

J

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:70)

L2

g

|J| 1

n

Thus we have

(cid:12)(cid:12)(cid:12)BA,P

stop (f, g)

(cid:17)

Pα(cid:16)

J∈Π2P

J,|ϕJ| 1A\IP (J)σ
(cid:16)

(cid:12)(cid:12)(cid:12) ≤ (cid:88)
(cid:88)
J∈Π2P
stop,1,(cid:52)ω (f, g) + |B|A,P
≡ |B|A,P

|J| 1
J,|ϕJ| 1A\IP (J)σ

|J| 1

Pα

1+δ

+

n

n

stop,1+δ,Pω (f, g) ,

(5.6.6)

J

x

(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:17)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)L2(1J ω)
(cid:13)(cid:13)x − mω

L2(ω)

g

J

J

J

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:70)

L2

g

where we have dominated the stopping form by two sublinear stopping forms that involve the

256

Poisson integrals of order 1 and 1 + δ respectively, and where the smaller Poisson integral

is multiplied by the larger quantity (cid:13)(cid:13)x − mω

(cid:13)(cid:13)L2 (1J ω). This splitting turns out to

J

Pα

1+δ

be successful in separating the two energy terms from the right hand side of the Energy

Lemma, because of the two properties (5.6.4) and (5.6.5) above. It remains to show the two

inequalities:

stop,(cid:52)ω (f, g) (cid:46)(cid:16)E α

|B|A,P

2 +

(cid:113)

Aα
2

(cid:17)(cid:13)(cid:13)(cid:13)Pσ,b

π(Π1P)f

(cid:13)(cid:13)(cid:13)(cid:70)

L2(σ)

(cid:13)(cid:13)(cid:13)Pω,b∗

Π2P g

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

,

(5.6.7)

for f ∈ L2 (σ) satisfying where Eσ
{πDI : I ∈ Π1P}; and

stop,1+δ,Pω (f, g) (cid:46)(cid:16)E α

|B|A,P

2 +

Aα
2

I |f| ≤ αA (A) for all I ∈ CA; and where π (Π1P) ≡
(cid:113)

(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b

CD

A

f

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(σ)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(ω)

CG,shif t

g

(5.6.8)

where we only need the case P = PA in this latter inequality as there is no recursion involved
in treating this second sublinear form. We consider first the easier inequality (5.6.8) that

does not require recursion.

5.6.1 The bound for the second sublinear inequality

Now we turn to proving (5.6.8), i.e.

stop,1+δ,Pω (f, g) (cid:46)(cid:16)E α

|B|A,P

2 +

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A

CG,shif t

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(ω)

g

(cid:113)

Aα
2

(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b

CD

A

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(σ)

f

257

where since

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

|ϕJ| =

(cid:88)

I∈C(cid:48)

A: (I,J)∈P

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

πI f

(cid:17)

Eσ
I

bA1A\I

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88)

I∈C(cid:48)

A: (I,J)∈P

(cid:12)(cid:12)(cid:12)Eσ

I

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

πI f

(cid:17)

(cid:12)(cid:12)(cid:12) ,

bA 1A\I

the sublinear form |B|A,P
the ratio of side lengths of J and I:

stop,1+δ,Pω can be dominated and then decomposed by pigeonholing

=

J∈Π2P

|B|A,P
(cid:88)
≤ (cid:88)
∞(cid:88)

(I,J)∈P

≡

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

g

(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

L2(ω)

J

g

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

(cid:107)x − mJ(cid:107)L2(1J ω)

stop,1+δ (f, g)

(cid:16)
J,|ϕJ| 1A\IP (J)σ
(cid:16)(cid:3)σ,(cid:91),b
(cid:16)

|J| 1

(cid:12)(cid:12)(cid:12)Eσ

(cid:17)
(cid:17)(cid:12)(cid:12)(cid:12) 1A\I σ

πI f

J,

n

I

(cid:17)

Pα

1+δ

Pα

1+δ

(cid:107)x − mJ(cid:107)L2(1J ω)

|J| 1

n

|B|A,P;s

stop,1+δ (f, g) ;

s=0

We will now adapt the argument for the stopping term starting on page 42 of [32], where the

geometric gain from the assumed ‘Energy Hypothesis’ there will be replaced by a geometric

gain from the smaller Poisson integral Pα

1+δ

used here.

First, we exploit the additional decay in the Poisson integral Pα

1+δ

as follows. Suppose

258

that (I, J) ∈ P with (cid:96) (J) = 2−s(cid:96) (I). We then compute

(cid:16)

(cid:17)

Pα

1+δ

J,|bA| 1A\I σ
|J| 1

n

|y − cJ|n+1+δ−α

n

A\I

|J| δ

(cid:90)
 |J| 1
(cid:90)
δ
 |J| 1

A\I

dist (cJ , Ic)

n

n

dist (cJ , Ic)

δ
Pα(cid:16)

≈

≤

(cid:46)

|bA (y)| dσ (y)

1

|y − cJ|n+1−α |bA (y)| dσ (y)
J,|bA| 1A\I σ

(cid:17)

|J| 1

n

,

and using the goodness of J in I,

d (cJ , Ic) ≥ 2(cid:96) (I)1−ε (cid:96) (J)ε ≥ 2 · 2s(1−ε)(cid:96) (J) ,

to conclude, using accretivity, that

(cid:16)

Pα

1+δ

(cid:17)

 (cid:46) 2−sδ(1−ε)

Pα(cid:16)

(cid:17)

.

J, 1A\I σ
|J| 1

n

J,|bA|1A\I σ
|J| 1

n

We next claim that for s ≥ 0 an integer,

stop,1+δ,Pω (f, g) (cid:46) 2−sδ(1−ε) (cid:16)E α

|B|A,P;s

from which (5.6.8) follows upon summing in s ≥ 0. Now using both

(5.6.9)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(ω)

g

CG,shif t

f

A

A

Aα
2

CD

2 +

(cid:113)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(σ)
(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b
(cid:90)
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:12)(cid:12)(cid:12) dσ ≤(cid:13)(cid:13)(cid:13)(cid:3)σ,(cid:91),b
(cid:12)(cid:12)(cid:12)(cid:3)σ,(cid:91),b
(cid:18)(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:19)
(cid:13)(cid:13)(cid:13)2
+(cid:13)(cid:13)∇σ
πI f(cid:13)(cid:13)2

1(cid:112)|I|σ

πI f

πI f

πI f

L2(σ)

I

L2(σ)

,

≈ (cid:107)f(cid:107)2

L2(σ)

,

259

I

(cid:12)(cid:12)(cid:12)Eσ
(cid:13)(cid:13)(cid:13)(cid:3)σ,(cid:91),b

(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:13)(cid:13)(cid:13)2

πI f

πI f

L2(σ)

(cid:17)(cid:12)(cid:12)(cid:12) 1
(cid:46) (cid:88)

|I|σ

I∈D

(cid:88)

I∈D

we apply Cauchy-Schwarz in the I variable above to see that

stop,1+δ,Pω (f, g)

(cid:104)|B|A,P;s

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pσ,b
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L2(σ)

CD

A

f

(cid:46)

(cid:105)2

 1(cid:112)|I|σ

(cid:88)

I∈C(cid:48)

A

(cid:88)

Pα

1+δ

J: (I,J)∈P
(cid:96)(J)=2−s(cid:96)(I)

Using the frame inequality for (cid:3)ω,b∗
by

J



(cid:88)

I∈C(cid:48)

A

(cid:88)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

(cid:13)(cid:13)(cid:13)(cid:70)2

g

L2(ω)

J: (I,J)∈P
(cid:96)(J)=2−s(cid:96)(I)

(cid:16)

J, 1A\I σ
|J| 1

n

(cid:17)
(cid:107)x − mJ(cid:107)L2(1J ω)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

g

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

1
2

2


we can then estimate the sum inside the square brackets



(cid:88)

1
|I|σ
J: (I,J)∈P
(cid:96)(J)=2−s(cid:96)(I)

1+δ

(cid:16)
Pα
(cid:46) (cid:13)(cid:13)(cid:13)Pω,b∗

Π2P g

J, 1A\I σ
|J| 1

n

(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

(cid:17)

2

(cid:107)x − mJ(cid:107)2

L2(1J ω)

A (s)2 ,

where

Pα

1+δ

1
|I|σ

(cid:88)

J: (I,J)∈P
(cid:96)(J)=2−s(cid:96)(I)

A (s)2 ≡ sup
I∈C(cid:48)

A

(cid:16)

J, 1A\I σ
|J| 1

n

(cid:17)

2

(cid:107)x − mJ(cid:107)2

L2(1J ω)

Finally then we turn to the analysis of the supremum in last display. From the Poisson decay

(5.6.9) we have

A (s)2 (cid:46) sup
I∈C(cid:48)
A

1
|I|σ

(cid:46) 2−2sδ(1−ε)(cid:104)

2−2sδ(1−ε) (cid:88)
(cid:105)

2 )2 + Aα
2

(E α

,

J: (I,J)∈P
(cid:96)(J)=2−s(cid:96)(I)

260

Pα(cid:16)

(cid:17)

2

J, 1A\I σ
|J| 1

n

(cid:107)x − mJ(cid:107)2

L2(1J ω)

Indeed, from Definition 5.1.14, as (I, J) ∈ P , we have that I is not a stopping cube in A,
(cid:17)
and hence that (5.1.28) fails to hold, delivering the estimate above since J (cid:98)ρ,ε I good must
K,|bI|1A\I σ
.
|K| 1
n
are additive since the J(cid:48)s are pigeonholed by (cid:96) (J) = 2−s(cid:96) (I).

be contained in some K ∈ M(r,ε)−deep (I), and since Pα(cid:16)
The terms(cid:13)(cid:13)Pω
J x(cid:13)(cid:13)2

J,|bI|1A\I σ
|J| 1
n

≈ Pα(cid:16)

(cid:17)

L2(ω)

5.6.2 The bound for the first sublinear inequality
Now we turn to proving the more difficult inequality (5.6.7). Denote by NA,P
constant in

stop,(cid:52)ω the best

|B|A,P

stop,(cid:52)ω (f, g) ≤ NA,P

stop,(cid:52)ω

(cid:13)(cid:13)(cid:13)Pσ,b

(cid:13)(cid:13)(cid:13)(cid:70)

(cid:13)(cid:13)(cid:13)Pω,b∗

Π2P g

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

π(Π1P)f

L2(σ)

,

(5.6.10)

where f ∈ L2 (σ) satisfies Eσ
{πI : I ∈ Π1P}. We refer to NA,P
Inequality (5.6.7) follows once we have shown that NA,P

I |f| ≤ αA (A) for all I ∈ CA, and g ∈ L2 (ω) and π(Π1P) =
stop,(cid:52)ω as the restricted norm relative to the collection P.

2 +(cid:112)Aα

stop,(cid:52)ω (cid:46) E α

2

.

The following general result on mutually orthogonal admissible collections will prove very

useful in establishing (5.6.7). Given a set {Qm}∞
that the collections Qm are mutually orthogonal, if each collection Qm satisfies

m=0

of admissible collections for A, we say

(cid:8)Am,j × Bm,j

(cid:9)

∞(cid:91)

j=0

where the sets(cid:8)Am,j

(cid:9)

m,j

grids D and G:

Qm ⊂

(cid:9)

and(cid:8)Bm,j
∞(cid:88)

1Am,j

m,j=0

m,j

≤ 1D and

∞(cid:88)

m,j=0

1Bm,j

≤ 1G.

are each pairwise disjoint in their respective dyadic

Lemma 5.6.4. Suppose that {Qm}∞

m=0 is a set of admissible collections for A that are

261

mutually orthogonal. Then Q ≡ ∞(cid:83)
stop,(cid:52)ω (f, g) has its restricted norm NA,Q

m=0

|B|A,Q
norms NA,Qm

stop,(cid:52)ω:

Proof. If J ∈ Π2Qm, then ϕQ
is mutually orthogonal. Thus we have

stop,(cid:52)ω (f, g) =

|B|A,Q
(cid:88)
(cid:88)

m≥0

(cid:88)
J∈Π2Qm
|B|A,Qm

m≥0

=

=

stop,(cid:52)ω (f, g) ,

Qm is admissible, and the sublinear stopping form
stop,(cid:52)ω controlled by the supremum of the restricted

NA,Qm
stop,(cid:52)ω .

NA,Q
stop,(cid:52)ω ≤ sup
m≥0
Qm
J = ϕ
J

and IQ (J) = IQm (J), since the collection {Qm}∞

m=0

(cid:88)
(cid:12)(cid:12)(cid:12)ϕ

J∈Π2Q
Qm
J,
J

Pα(cid:16)

J

J,

(cid:12)(cid:12)(cid:12)ϕQ
Pα(cid:16)
(cid:12)(cid:12)(cid:12) 1A\IQm (J)σ

(cid:12)(cid:12)(cid:12) 1A\IQ(J)σ
(cid:17)
(cid:17)
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

|J| 1

n

J

x

|J| 1

n

J

x

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)♠

L2(ω)

(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

L2(ω)

g

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

g

J

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

|B|A,Q

and we can continue with the definition of (cid:98)NA,Qm
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)Pσ,b
stop,(cid:52)ω (f, g)≤ (cid:88)
(cid:33)(cid:118)(cid:117)(cid:117)(cid:116)(cid:88)
(cid:32)
(cid:13)(cid:13)(cid:13)Pσ,b
(cid:33)(cid:115)(cid:13)(cid:13)(cid:13)Pσ,b
(cid:32)

(cid:98)NA,Qm
(cid:98)NA,Qm
(cid:98)NA,Qm

π(Π1Qm)f

stop,(cid:52)ω

stop,(cid:52)ω

stop,(cid:52)ω

sup
m≥0

m≥0

m≥0

≤

≤

π(Π1Q)f

sup
m≥0

stop,(cid:52)ω and Cauchy-Schwarz to obtain

L2(σ)

π(Π1Qm)f

g

L2(ω)

Π2Qm

(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:118)(cid:117)(cid:117)(cid:116)(cid:88)
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:115)(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)2

m≥0

Π2Qg

L2(σ)

L2(ω)

.

(cid:13)(cid:13)(cid:13)(cid:70)2

L2(σ)

Π2Qm

(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

g

262

Now we turn to proving inequality (5.6.7) for the sublinear form |B|A,P

|B|A,P

J∈Π2P

stop,(cid:52)ω (f, g) ≡ (cid:88)
(cid:46) (cid:16)E α
(cid:88)

where ϕJ ≡

2 +

Pα(cid:16)
(cid:113)

Aα
2

|J|

(cid:17)(cid:13)(cid:13)(cid:13)Pσ,b
I(cid:98)(cid:3)σ,(cid:91),b

Eσ

πI f

(cid:16)

I∈C(cid:48)

A: (I,J)∈P

J,|ϕJ| 1A\IP (J)σ

stop,(cid:52)ω (f, g), i.e.

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

(cid:13)(cid:13)(cid:13)(cid:70)

g

L2(ω)

L2(ω)

(cid:17)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)Pω,b∗

J

Π2P g

x

(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:70)

(cid:13)(cid:13)(cid:13)(cid:70)

π(Π1P)f

L2(σ)

;

L2(ω)

(cid:17)

bA 1A\I

is supported in A\IP (J)

and IP (J) denotes the smallest cube I ∈ D for which (I, J) ∈ P. We recall the stopping
energy from (5.1.30),

Xα (CA)2 ≡ sup
I∈CA

1
|I|σ

sup

I⊃ ·∪Jr

∞(cid:88)

r=1

(cid:18)Pα (Jr, 1Aσ)

|Jr|

(cid:19)2(cid:13)(cid:13)x − mJr

(cid:13)(cid:13)2
L2(cid:16)

(cid:17) ,

1Jr ω

where the cubes Jr ∈ G are pairwise disjoint in I.

What now follows is an adaptation to our sublinear form |B|A,P

stop,(cid:52)ω of the arguments of
M. Lacey in [27], together with an additional ‘indented’ corona construction. We have the
following Poisson inequality for cubes B ⊂ A ⊂ I:

Pα(cid:16)

A, 1I\Aσ
|A| 1

n

(cid:17)

(cid:90)
(cid:90)

≈

(cid:46)

1

(|y − cA|)n+1−α dσ (y)

(|y − cB|)n+1−α dσ (y) ≈ Pα(cid:16)

1

(5.6.11)

(cid:17)

B, 1I\Aσ
|B| 1

n

I\A

I\A

where the implied constants depend on n, α.

Fix A ∈ A. Following [27] we will use a ‘decoupled’ modification of the stopping energy

263

Xα (CA) to define a ‘size functional’ of an A-admissible collection P. So suppose that P is
an A-admissible collection of pairs of cubes, and recall that Π1P and Π2P denote the cubes
in the first and second components of the pairs in P respectively.

Definition 5.6.5. For an A-admissible collection of pairs of cubes P, and a cube K ∈ Π1P,
define the projection of P ‘relative to K’ by

2 P ≡(cid:110)

ΠK

(cid:111)

J ∈ Π2P : J

(cid:122) ⊂ K

,

where we have suppressed dependence on A.

Definition 5.6.6. We will use as the ‘size testing collection’ of cubes for P the collection

Πbelow

1

P ≡ {K ∈ D : K ⊂ I for some I ∈ Π1P} ,

which consists of all cubes contained in a cube from Π1P.

Continuing to follow Lacey [27], we define two ‘size functionals’ of P as follows. Recall

that for a pseudoprojection QωH on x we have

(cid:13)(cid:13)(cid:13)Qω,b∗

H x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)2

x

J

J

L2(ω)

L2(ω)

(cid:88)
(cid:88)

J∈H

J∈H

=

=

(cid:88)
J(cid:48)∈Cbrok(J)

+ inf
z∈Rn

(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:16)

J(cid:48) |x − z|(cid:17)2

Eω



Definition 5.6.7. If P is A-admissible, define an initial size condition Sα,A

initsize (P) by

initsize (P)2 ≡
Sα,A

sup

K∈Πbelow

1

1
|K|σ

P

(cid:17)

2(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

2 P x
ΠK

(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

Pα(cid:16)

K, 1A\K σ
|K| 1

n

264

.

(5.6.12)

The following key fact is essential.

Key Fact #1:

If K ⊂ A and K /∈ CA, then ΠK

2 P = ∅ .

(5.6.13)

To see this, suppose that K ⊂ A and K /∈ CA. Then K ⊂ A(cid:48) for some A(cid:48) ∈ CA (A), and
so if there is J(cid:48) ∈ ΠK
, which
contradicts ΠK

2 P, then (cid:0)J(cid:48)(cid:1)(cid:122) ⊂ K ⊂ A(cid:48) , which implies that J(cid:48) /∈ CG,shif t

. We now observe from (5.6.13) that we may also write the

2 P ⊂ CG,shif t

A

A
initial size functional as

Sα,A
initsize (P)2 ≡

sup
K∈Πbelow

1

P∩C(cid:48)

A

1
|K|σ

Pα(cid:16)

(cid:17)

2(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

2 P x
ΠK

(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

K, 1A\K σ
|K| 1

n

(5.6.14)

.

However, we will also need to control certain pairs (I, J) ∈ P using testing cubes K
(2)D K. For
(cid:122), that will also play a crucial role in

(cid:122), namely those K ∈ CA such that K ⊂ J

this, we need a second key fact regarding the cubes J

which are strictly smaller than J

(cid:122) ⊂ π

one of the inner 2n grandchildren of J

controlling pairs in the indented corona below, and which is that J is always contained in
(cid:122). For M ∈ D, denote by M(cid:38) and M(cid:37) any of the
inner and outer respectively grandchildren of M and by MJ and M (cid:91) the child and grandchild
respectively of M that contains J, provided they exist.

Key Fact #2:

3J ⊂ J (cid:91) and J (cid:91) is an inner grandchild of J

(cid:122)

(5.6.15)

(cid:122)
To see this, suppose that the child J
J

of J

(cid:122) contains J (J

(cid:122)
J

exists because J is good in

265

(cid:122)). Then observe that J is by definition ε − bad in J

J

(cid:17) ≤ 2|J| ε

n

(cid:122)
J

(cid:12)(cid:12)(cid:12)J

, i.e.

(cid:12)(cid:12)(cid:12) 1−ε

n

(cid:122)
J

dist

(cid:122)
J, bodyJ
J

and so cannot lie in any of the 4n − 2n outermost grandchildren J
then

(cid:122)
(cid:37). Indeed, if J ⊂ J

(cid:122)
(cid:37),

(cid:16)

(cid:122)(cid:17)

(cid:16)

dist

J, bodyJ

= dist

(cid:16)

(cid:12)(cid:12)(cid:12)J
(cid:12)(cid:12)(cid:12) 1−ε
(cid:17) ≤ 2|J| ε
(cid:122)(cid:12)(cid:12)(cid:12) 1−ε
(cid:12)(cid:12)(cid:12)J

(cid:122)
J

n < 2|J| ε

n

n

n

n

(cid:122)
J, bodyJ
J

(cid:122)(cid:12)(cid:12)(cid:12) 1−ε

(cid:12)(cid:12)(cid:12)J

= 2ε |J| ε

n

contradicting the fact that J is ε − good in J
we get that J (cid:91) is an inner grandchild of J
interior of J (cid:91), thus permitting J to be ε − good in J
in J

(cid:122), (where the body of J

(cid:122). Thus we must have J ⊂ J (cid:91), and of course
(cid:122) does not intersect the
(cid:122)). Finally, the fact that J is ε − good

(cid:122) implies that 3J ⊂ J (cid:91).
This second key fact is what underlies the construction of the indented corona below, and

motivates the next definition of augmented projection, in which we allow cubes K satisfying
(2)D K, as well as K ∈ CA, to be tested over in the augmented size condition
J ⊂ K (cid:36) J
below.

(cid:122) ⊂ π

Definition 5.6.8. Suppose P is an A-admissible collection.

(1). For K ∈ Π1P, define the augmented projection of P relative to K by
(cid:111)

J ∈ Π2P : J ⊂ K and J

(cid:122) ⊂ π

P ≡(cid:110)

ΠK,aug

2

(2)D K

.

266

(2). Define the corresponding augmented size functional Sα,A

Pα(cid:16)

K, 1A\K σ
|K|

augsize (P) by
(cid:17)

2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

Π

K,aug
2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

P x

Sα,A
augsize (P)2 ≡ sup
K∈Πbelow

1

P∩C(cid:48)

A

1
|K|σ

P includes cubes J for which J ⊂ K (cid:36)
(2)D K, and hence J need not be ε−good inside K. Then by the second key fact (5.6.15),

We note that the augmented projection ΠK,aug
(cid:122) ⊂ π
and using that the boundaries of J

(cid:122), we have two consequences,

(cid:122)
(cid:38) lie in the body of J

J

2

K ∈(cid:110)

J

J , J (cid:91)(cid:111) and 3J ⊂ J (cid:91) ⊂ 3J (cid:91) ⊂ J

(cid:122)

(cid:122) for J ∈ ΠK,aug

2

P,

which will play an important role below.
The augmented size functional Sα,A

augsize (P) is a ‘decoupled’ form of the stopping energy
Xα (CA) restricted to P, in which the cubes J appearing in Xα (CA) no longer appear in the
augsize (P), and it plays a crucial role in Lacey’s argument in [27]. We
Poisson integral in Sα,A
note two essential properties of this definition of size functional:

1. Monotonicity of size: Sα,A

augsize (P) ≤ Sα,A

augsize (Q) if P ⊂ Q,

2. Control by energy and Muckenhoupt conditions: Sα,A

2 +(cid:112)Aα

2

.

augsize (P) (cid:46) E α
Q and ΠK

The monotonicity property follows from Πbelow

2 Q. The
control property is contained in the next lemma, which uses the stopping energy control for

2 P ⊂ ΠK

P ⊂ Πbelow

1

1

the form BA

stop (f, g) associated with A.

Lemma 5.6.9. If PA is as in (5.6.2) and P ⊂ PA, then

Sα,A
augsize (P) ≤ Xα (CA) (cid:46) E α
2 +

267

(cid:113)

Aα
2 .

Proof. We have

augsize (P)2 =
Sα,A

sup
K∈Πbelow

1

P∩C(cid:48)

A

(cid:46) sup
K∈C(cid:48)
A

1
|K|σ

Pα(cid:16)
Pα (K, 1Aσ)

1
|K|σ

|K| 1

n

K, 1A\K σ
|K| 1

n

2

(cid:17)

2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

2 P∪Π
ΠK

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

P x

K,aug
2

(cid:107)x − mK(cid:107)2

L2(1K ω)

≤ Xα (CA)2 ,

which is the first inequality in the statement of the lemma. The second inequality follows

from (5.1.31).

There is an important special circumstance, introduced by M. Lacey in [27], in which we

can bound our forms by the size functional, namely when the pairs all straddle a subpartition

of A, and we present this in the next subsection. In order to handle the fact that the cubes in

Πbelow

1

P ∩C(cid:48)

A

need no longer enjoy any goodness, we will need to formulate a Substraddling

Lemma to deal with this situation as well. See Remark on lack of usual goodness after

(5.6.41), where it is explained how this applies to the proof of (5.6.40). Then in the following

subsection, we use the bottom/up stopping time construction of M. Lacey, together with

an additional ‘indented’ top/down corona construction, to reduce control of the sublinear
stopping form |B|A,P
the Orthogonality Lemma, the Straddling Lemma and the Substraddling Lemma.

stop,(cid:52)ω (f, g) in inequality (5.6.7) to the three special cases addressed by

5.6.3 (cid:91)Straddling, Substraddling, Corona-Straddling Lemmas

We begin with the Corona-straddling Lemma in which the straddling collection is the set of
A-children of A, and applies to the ‘corona straddling’ subcollection of the initial admissible

268

collection PA - see (5.6.2). Define the ‘corona straddling’ collection PA

(cid:110)
(I, J) ∈ PA : J ⊂ A(cid:48) (cid:38) J

(cid:122) ⊂ π

cor by

(2)D A(cid:48)(cid:111)

.

(5.6.16)

cor ≡ (cid:91)

PA

A(cid:48)∈CA(A)

Note that PA
cor is an A-admissible collection that consists of just those pairs (I, J) for which
(cid:122) is either the D-parent or the D -grandparent of a stopping cube A(cid:48) ∈ CA (A). The bound
for the norm of the corresponding form is controlled by the energy condition.

J

Lemma 5.6.10. We have the sublinear form bound

cor

NA,PA
stop,(cid:52)ω ≤ CE α
2 .
(cid:16)

Proof. The key point here is our assumption that J ⊂ A(cid:48) (cid:38) J
which implies that in fact 3J ⊂ A(cid:48) since J ∩ body
(2)D A(cid:48). We start with

π

π

(2)D A(cid:48) for (I, J) ∈ PA
(cid:122) ⊂ π
cor,
= ∅ because J is ε − good in

L2(ω)

L2(ω)

x

(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)♠

J

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

g

J

g

(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

L2(ω)

|B|A,PA

cor

stop,(cid:52)ω (f, g) =

(cid:88)

Pα

J∈Π2PA

cor

(cid:88)
A(cid:48)∈CA(A)

=

(cid:88)

Pα

J∈Π2PA
3J⊂A(cid:48)

cor

where

PA
cor
ϕ
J

≡

(cid:18)

J,

(cid:12)(cid:12)(cid:12)(cid:12)ϕ
(cid:12)(cid:12)(cid:12)(cid:12)ϕ

(cid:18)

J,

PA
cor
J

|J|

(cid:12)(cid:12)(cid:12)(cid:12) 1A\IPA
(cid:12)(cid:12)(cid:12)(cid:12) 1A\IPA

cor

|J|

(J)σ

cor

(J)σ

PA
cor
J

(cid:88)

(2)D A(cid:48)(cid:17)
(cid:19)

(cid:19)

J

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:17)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b

πI f

J

x

bAEσ
I

1A\I .

I∈Π1PA

cor: (I,J)∈PA
cor

269

and we have

if A(cid:48) = J (cid:91)

if A(cid:48) = J

(cid:122)
J

(cid:122)
J

(cid:17)
(cid:17)

n

n

n

σ

σ

cor

cor

Pα

Pα

≈

Pα

(J)σ

(cid:19)

(cid:33)
(cid:33)

A(cid:48),1Aσ
|A(cid:48)| 1
A(cid:48),1Aσ
|A(cid:48)| 1



(cid:32)
A(cid:48),1A\IPA
(cid:32)
|A(cid:48)| 1
A(cid:48)
J ,1A\IPA

cor and J ⊂ A(cid:48) ∈ CA (A), then either A(cid:48) = J (cid:91) or A(cid:48) = J
If J ∈ Π2PA
(cid:18)
J, 1A\IPA
|J| 1

≤ Pα(cid:16)
(cid:46) Pα(cid:16)
(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) αA (A) 1A by (5.6.4), we can then bound |B|A,PA
(cid:13)(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)2
L2(cid:16)

(cid:88)
 (cid:88)
A(cid:48)∈CA(A)

(cid:12)(cid:12)(cid:12)(cid:12)ϕ

PA
cor
J

Since

≤ αA (A)

cor;A(cid:48)x

(cid:12)(cid:12)(cid:12)A(cid:48)

αA (A)

(cid:12)(cid:12)(cid:12) 1

L2(ω)

A(cid:48)

cor

n

n

J

A(cid:48)∈CA(A)

n

n

|A(cid:48)| 1

|A(cid:48)| 1

Π2PA

(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
Pα(cid:0)A(cid:48), 1Aσ(cid:1)
Pα(cid:0)A(cid:48), 1Aσ(cid:1)
2(cid:13)(cid:13)(cid:13)x − mσ
 (cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

A(cid:48)∈CA(A)

Π2PA

L2(ω)

Cshif t
A

L2(ω)

cor

·

g

g

(cid:113)|A|σ
(cid:113)|A|σ

≤ E α

2 αA (A)

≤ E α

2 αA (A)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

cor;A(cid:48)g

cor

stop,(cid:52)ω (f, g) by

Π2PA

(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
 1
(cid:17)
 1
(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

·

2

2

1A(cid:48) σ

Π2PA

cor;A(cid:48)g

where in the last line we have used the strong energy constant E α

2

in (5.1.8).

Definition 5.6.11. We say that an admissible collection of pairs P is reduced if it contains
no pairs from PA

cor, i.e.

P ∩ PA

cor = ∅.

270

Recall that in terms of J (cid:91) we rewrite

(cid:110)
(cid:110)

ΠK,aug

2

P =

=

(cid:122) ⊂ π
J ∈ Π2P : J ⊂ K and J
J ∈ Π2P : J ⊂ K and J (cid:91) ⊂ K

(cid:111)

(2)D K

(cid:111)

1

Q ∩ C(cid:48)

Definition 5.6.12. Given a reduced admissible collection of pairs Q for A, and a subpar-
tition S ⊂ Πbelow
A of pairwise disjoint cubes in A, we say that Q (cid:91) straddles S if
for every pair (I, J) ∈ Q there is S ∈ S ∩ [J, I] with J (cid:91) ⊂ S. To avoid trivialities, we
further assume that for every S ∈ S, there is at least one pair (I, J) ∈ Q with J (cid:91) ⊂ S ⊂ I.
Here [J, I] denotes the geodesic in the dyadic tree D that connects JD to I, where JD is the
minimal cube in D that contains J.

Definition 5.6.13. For any dyadic cube S ∈ D, define the Whitney collection W (S) to

consist of the maximal subcubes K of S whose triples 3K are contained in S. Then set
W∗ (S) ≡ W (S) ∪ {S}.

The following geometric proposition will prove useful in proving the (cid:91) Straddling Lemma

5.6.15 below. For S ∈ S, let QS ≡(cid:110)

(I, J) ∈ Q : J (cid:91) ⊂ S ⊂ I

(cid:111).

Proposition 5.6.14. Suppose Q is reduced admissible and (cid:91) straddles a subpartition S of
A. Fix S ∈ S. Define

(cid:88)

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

πI h

(cid:17)

bAEσ
I

1A\I ,

ϕQS

J

[h] ≡

I∈Π1QS : (I,J)∈QS

assume that h ∈ L2 (σ) is supported in the cube A, and that there is a cube H ∈ CA with

271

H ⊃ S such that

I |h| ≤ CEσ
Eσ

H |h| ,

for all I ∈ Πbelow

1

Q ∩ C(cid:48)

A with I ⊃ S.

Then

J∈Π2Q: J(cid:91)⊂S

(cid:46) αH (H)

S, 1A\Sσ
|S|

Pα(cid:16)

(cid:88)
Pα(cid:16)
(cid:88)

x

J

J,

(cid:17)

J [h]
|J|

(cid:12)(cid:12)(cid:12)ϕQ
(cid:12)(cid:12)(cid:12) 1A\IQ(J)σ
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)♠
(cid:17)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
Pα(cid:16)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

K, 1A\K σ
|K|

S,aug
2

S,aug
2

L2(ω)

Qx

Qx

(cid:17)

K,aug
2

Π

Π

Π

L2(ω)

Qg

J

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

K,aug
2

L2(ω)

Π

(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

Qg

L2(ω)

g

L2(ω)

+αH (H)

K∈W(S)

.

L2(ω)

The sum over Whitney cubes K ∈ W (S) is only required to bound the sum of those terms
on the left for which J (cid:91) ⊂ S(cid:48)(cid:48) for some S(cid:48)(cid:48) ∈ C(2)D (S).
Proof. Suppose first that J (cid:91) = S ∈ C(cid:48)
A
with αH (H) in place of αA (A), we have

(cid:122) ⊂ IQ (J) and using (5.6.4)

. Then 3S = 3J (cid:91) ⊂ J

Pα(cid:16)

J,

(cid:12)(cid:12)(cid:12)ϕQ

J

(cid:12)(cid:12)(cid:12) 1A\IQ(J)σ

(cid:17)

|J| 1

n

Pα(cid:16)
Pα(cid:16)

(cid:46) αH (H)

(cid:46) αH (H)

(cid:17)
(cid:17)

J, 1
|J| 1
S, 1
|S| 1

A\J(cid:122)σ
n
A\J(cid:122)σ
n

Pα(cid:16)

(cid:17)

.

S, 1A\Sσ
|S| 1

n

≤ αH (H)

272

Suppose next that J (cid:91) = S(cid:48) ∈ CD (S). Then 3S(cid:48) = 3J (cid:91) ⊂ J

Pα(cid:16)

J,

(cid:12)(cid:12)(cid:12)ϕQ

J

(cid:12)(cid:12)(cid:12) 1A\IQ(J)σ

(cid:17)

|J| 1

n

Pα(cid:16)
Pα(cid:16)
Pα(cid:16)

(cid:122) ⊂ IQ (J) and (5.6.4) give
(cid:17)
(cid:17)

Pα(cid:16)

(cid:17)

.

S, 1A\Sσ
|S| 1

n

≈ αH (H)

A\J(cid:122)σ
n
A\J(cid:122)σ
n

J, 1
|J| 1
S(cid:48), 1
|S(cid:48)| 1
S(cid:48), 1A\Sσ
|S(cid:48)| 1

(cid:17)

n

(cid:46) αH (H)

(cid:46) αH (H)

≤ αH (H)

Thus in these two cases, by Cauchy-Schwarz, the left hand side of our conclusion is bounded

by a multiple of

Pα(cid:16)

(cid:17)

S, 1A\Sσ
|S| 1

n

αH (H)



Pα(cid:16)

= αH (H)

1
2



(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

g

J

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

Π

x

L2(ω)

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠

Qx

S,aug
2

L2(ω)

1

2

(cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

J∈Π2Q
J(cid:91)⊂S

Π

S,aug
2

J

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

Qg

L2(ω)

(cid:88)

J∈Π2Q
J(cid:91)⊂S

(cid:17)

S, 1A\Sσ
|S| 1

n

Finally, suppose that J (cid:91) ⊂ S(cid:48)(cid:48) for some S(cid:48)(cid:48) ∈ C(2)D (S). Then J
#2 in (5.6.15) shows that 3J (cid:91) ⊂ J
J (cid:91) ⊂ K = K [J] for some K ∈ W (S) and so by (5.6.4) again,

(cid:122), so that 3J (cid:91) ⊂ J

Pα(cid:16)

J,

(cid:12)(cid:12)(cid:12)ϕQ

J

(cid:12)(cid:12)(cid:12) 1A\IQ(J)σ

(cid:17)

|J| 1

n

Pα(cid:16)
Pα(cid:16)

n

J, 1A\Sσ
|J| 1
K, 1A\Sσ
|K| 1

n

(cid:46) αH (H)

(cid:46) αH (H)

(cid:122) ⊂ S, and Key Fact
(cid:122) ⊂ S ⊂ IQ (J). Thus we have
(cid:17)
(cid:17)

(cid:17)

Pα(cid:16)

≤ αH (H)

K, 1A\K σ
|K| 1

n

.

Now we apply Cauchy-Schwarz again, but noting that J (cid:91) ⊂ K this time, to obtain that the

273

left hand side of our conclusion is bounded by a multiple of

(cid:17)



Pα(cid:16)

αH (H)

J

n

(cid:17)

K∈W(S)

(cid:88)

(cid:88)

(cid:88)

= αH (H)

J∈Π2Q
J(cid:91)⊂K

Pα(cid:16)

K, 1A\K σ
|K| 1

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
Recall the family of operators (cid:110)(cid:3)σ,π,b
(cid:111)
 − Fσ,b

I∈CA
is defined in (5.1.41), and satisfies

 (cid:88)

K, 1A\K σ
|K| 1

K∈W(S)

(cid:3)σ,π,b

f =

Π

A

f

n

I

I

Fσ,π,b
I(cid:48)

I

This completes the proof of Proposition 5.6.14.

difference (cid:3)σ,π,b

I f =

I(cid:48)∈C(I)

x

1


2
(cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗

J∈Π2Q
J(cid:91)⊂K

Π

L2(ω)

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠

Qx

g

J

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

Qg

1
2



(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

.

K,aug
2

L2(ω)

K,aug
2

L2(ω)

, where for I ∈ CA

A

, the dual martingale

(cid:88)

I(cid:48)∈C(I)

Fσ,bA
I(cid:48)

f − Fσ,bA

I

f .

Since (cid:3)σ,π,b

the superscript π is suppressed for convenience) shows that (cid:110)(cid:3)σ,π,b

is the transpose of (cid:52)σ,π,b

for I ∈ CA

A

I

I

(cid:111)

, the first line of Lemma 5.1.22 (where

I

I∈CA

A

is a family of

projections, and the second line of Lemma 5.1.22 shows it is an orthogonal family, i.e.

The orthogonal projections

I

0

K

=

 (cid:3)σ,π,b
π(Π1Q) ≡ (cid:88)

(cid:3)σ,π,b

I

(cid:3)σ,π,b

if

if

I = K
I (cid:54)= K

,

I, K ∈ CA
A .

(cid:88)

Pσ,π,b

(cid:3)σ,π,b
where π (Π1Q) ≡ {πDI : I ∈ Π1Q} and Π1Q ⊂ CA,(cid:48)
A ,

I∈π(Π1Q)

I∈Π1Q

(cid:3)σ,π,b

=

πI

I

,

274

thus satisfy the equalities

f = (cid:3)σ,π,b

πI Pσ,π,b

π(Π1Q)f and (cid:98)(cid:3)σ,π,b

πI

f = (cid:98)(cid:3)σ,π,b

πI Pσ,π,b

π(Π1Q)f

(5.6.17)

(cid:3)σ,π,b

πI

for I ∈ Π1Q ⊂ CArestrict
Haar projections in the proof of T 1 theorems.

A

, which will permit us to apply certain projection tricks used for

However, in our sublinear stopping form |B|A,Q

stop,(cid:52)ω, the dual martingale projections in

use in the function

ϕQS
J ≡

(cid:88)

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

πI f

(cid:17)

bAEσ
I

1A\I ,

given in Proposition 5.6.14 above, are the modified pseudoprojections(cid:110)(cid:98)(cid:3)σ,(cid:91),b

I∈Π1QS : (I,J)∈QS

πI

(cid:3)σ,(cid:91),b

πI

differs from the orthogonal projection (cid:3)σ,π,b

πI

for I ∈ Π1Q by

(5.6.18)

(cid:111)

I∈Π1Q

, where

πI f − (cid:3)σ,π,b
(cid:3)σ,(cid:91),b

πI

f


 (cid:88)
= − (cid:88)

=

I(cid:48)∈Cnat(πI)

Fσ,bA
I(cid:48)

f

Fσ,bA
I(cid:48)

f.

I(cid:48)∈Cbrok(πI)

 − Fσ,bA

πI f

 −



 (cid:88)

I(cid:48)∈C(πI)

 − Fσ,bA

πI f



Fσ,bA
I(cid:48)

f

But the "box support" Suppbox of this last expression (cid:88)

broken children of πI, Cbrok (πI), and is contained in the set

Fσ,bA
I(cid:48)

f consists of the

I(cid:48)∈Cbrok(πI)

(cid:8)I(cid:48)(cid:9)

(cid:91)

(cid:91)

I∈C(cid:48)

A

I(cid:48)∈CA(A)∩CD(πI)

275

i.e.

 (cid:88)

I(cid:48)∈Cbrok(πI)

Suppbox

 ⊂(cid:8)I(cid:48) ∈ CA (A) : I(cid:48) ∈ Cbrok (πI) for some I ∈ C(cid:48)
(cid:91)

(cid:91)

(cid:8)I(cid:48)(cid:9) .

A

I∈C(cid:48)

A

I(cid:48)∈CA(A)∩CD(πI)

(cid:9)

Fσ,bA
I(cid:48)

f

=

But I ∈ Π1QS ⊂ C(cid:48)

A

is a natural child of πI, and so

 (cid:88)

I(cid:48)∈Cbrok(πI)

 = ∅

Fσ,bA
I(cid:48)

f

I ∩ Suppbox

It now follows that we have

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

πI f

(cid:17)

(cid:16)(cid:98)(cid:3)σ,π,b

πI

f

(cid:17)

,

= Eσ
I

for I ∈ C(cid:48)

A

(5.6.19)

Eσ
I

Returning to (5.6.18), we have from (5.6.17) and (5.6.19) the identity

ϕQS

J

≡

=

(cid:88)
(cid:88)

I∈Π1QS : (I,J)∈QS

I∈Π1QS : (I,J)∈QS

bAEσ
I

bAEσ
I

(cid:16)(cid:98)(cid:3)σ,π,b
(cid:16)(cid:98)(cid:3)σ,π,b

πI

πI

(cid:17)
(cid:16)

f

1A\I

Pσ,π,b
π(Π1Q)f

(5.6.20)

(cid:17)(cid:17)

1A\I

which will play a critical role in proving the following (cid:91)Straddling and Substraddling lemmas.

The (cid:91)Straddling Lemma is an adaptation of Lemmas 3.19 and 3.16 in [27].

Lemma 5.6.15. Let Q be a reduced admissible collection of pairs for A, and suppose that
S ⊂ Πbelow
A is a subpartition of A such that Q (cid:91)straddles S. Then we have the

Q ∩ C(cid:48)

1

276

restricted sublinear norm bound

(cid:98)NA,Q
stop,(cid:52)ω ≤ Cr sup
S∈S

Sα,A;S
locsize (Q) ≤ CrSα,A

augsize (Q) ,

(5.6.21)

where Sα,A;S

locsize is an S-localized size condition with an S-hole given by

Sα,A;S
locsize (Q)2 ≡

sup

K∈W∗(S)∩C(cid:48)

A

1
|K|σ

Pα(cid:16)

K, 1A\Sσ
|K| 1

n

(cid:17)

2 (cid:88)

J∈Π

K,aug
2

Q

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(5.6.22)

Proof. We begin by using that the reduced collection Q (cid:91)straddles S to write

(cid:88)
(cid:18)

J∈Π2Q
Pα

J,

J

J,

(cid:12)(cid:12)(cid:12)ϕQ
Pα(cid:16)
(cid:12)(cid:12)(cid:12)(cid:12) 1A\IQ(J)σ
(cid:12)(cid:12)(cid:12)(cid:12)ϕQS

(cid:12)(cid:12)(cid:12) 1A\IQ(J)σ
(cid:17)
(cid:19)
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

|J| 1

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)♠

x

n

J

J

J

|J| 1

n

x

L2(ω)

(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)

g

J

J

L2(ω)

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

g

stop,(cid:52)ω (f, g) =

|B|A,Q
(cid:88)

=

(cid:88)

S∈S

J∈Π

S,aug
2

Q

where ϕQS

J

≡

(cid:88)

I∈Π1QS : (I,J)∈QS

(cid:16)(cid:98)(cid:3)σ,(cid:91),b

πI f

(cid:17)

bAEσ
I

1A\I .

At this point we invoke the identity (5.6.20),

ϕQS
J =

(cid:88)

I∈Π1QS : (I,J)∈QS

bAEσ
I

so that

(cid:16)(cid:98)(cid:3)σ,π,b

πI

(cid:16)

Pσ,π,b
π(Π1Q)f

(cid:17)(cid:17)

1A\I ,

|B|A,Q

stop,(cid:52)ω (f, g) = |B|A,Q

stop,(cid:52)ω (h, g) ,

where h ≡ Pσ,π,b

π(Π1Q)f .

277

stop,(cid:52)ω (h, g) with h = Pσ,π,b

We will treat the sublinear form |B|A,Q
π(Π1Q)f using a small variation
on the corresponding argument in Lacey [27]. Namely, we will apply a Calderón-Zygmund
stopping time decomposition to the function h = Pσ,π,b
π(Π1Q)f on the cube A with ‘obstacle’
S ∪ CA (A), to obtain stopping times H ⊂ CA with the property that for all H ∈ H\{A}
we have

H ∈ CA is not strictly contained in any cube from S,
H |h| > ΓEσ
Eσ
H(cid:48) |h| ≤ ΓEσ
Eσ

πHH |h| ,
πHH |h| for all H (cid:36) H(cid:48) ⊂ πHH with H(cid:48) ∈ CA.

More precisely, define generation 0 of H to consist of the single cube A. Having defined
generation n, let generation n + 1 consist of the union over all cubes M in generation n of
the maximal cubes M(cid:48) in CA that are contained in M with Eσ
M |h|, but are not
strictly contained in any cube S from S or contained in any cube A(cid:48) from CA (A) - thus the
construction stops at the obstacle S ∪ CA(A). Then H is the union of all generations n ≥ 0.

M(cid:48) |h| > ΓEσ

Denote by

H ≡(cid:8)H(cid:48) ∈ CA : H(cid:48) ⊂ H but H(cid:48) (cid:54)⊂ H(cid:48)(cid:48) for any H(cid:48)(cid:48) ∈ CH (H)(cid:9)

CH

the usual H-corona associated with the stopping cube H, but restricted to CA, and let
H |f| as is customary for a Calderón-Zygmund corona. Since these coronas CH
αH (H) = Eσ
are all contained in CA, we have the stopping energy from the A-corona CA at our disposal,

H

278

which is crucial for the argument. Furthermore, we denote by

QH ≡(cid:110)

(I, J) ∈ Q : J ∈ CH,(cid:91)shif t

H

, with CH,(cid:91)shif t

H

J ∈ Π2Q : J (cid:91) ∈ CH

H

(cid:111)

≡(cid:110)

(cid:111) (5.6.23)

the restriction of the pairs (I, J) in Q to those for which J lies in the flat shifted H-corona
CH,(cid:91)shif t
. Since the H-stopping cubes satisfy a σ-Carleson condition for Γ chosen large
H
enough, we have the quasiorthogonal inequality

(cid:88)

H∈H

αH (H)2 |H|σ

(cid:46) (cid:107)h(cid:107)2

L2(σ)

,

(5.6.24)

which below we will see reduces matters to proving inequality (5.6.21) for the family of
reduced admissible collections {QH}H∈H with constants independent of H:

(cid:98)N

A,QH
stop,(cid:52)ω ≤ Cr sup
S∈S

Sα,A;S
locsize (QH ) ≤ CrSα,A

augsize (QH ) ,

H ∈ H.

Given S ∈ S, define HS ∈ H to be the minimal cube in H that contains S, and then

define

HS ≡ {HS ∈ H : S ∈ S} .

Note that a given H ∈ HS may have many cubes S ∈ S such that H = HS, and we denote
the collection of these cubes by SH ≡ {S ∈ S : HS = H }. We will organize the straddling
cubes S as

(cid:91)

(cid:91)

S

H∈HS

S∈SH

S =

where each S ∈ S occurs exactly once in the union on the right hand side, i.e. the collections
{SH}H∈HS are pairwise disjoint.

279

We now momentarily fix H ∈ HS, and consider the reduced admissible collection QH,
so that its projection onto the second component Π2QH of QH is contained in the corona
CH,(cid:91)shif t
(cid:83)
. Then the collection QH (cid:91)straddles the set SH = {S ∈ S : HS = H }. Moreover,
QH =

and Π2QS

QH.

H = ΠS,aug

QS

H

H

2

S∈S: S⊂H

(cid:122) (cid:54)⊂ S. Since J

(cid:122) (cid:54)⊂ K, and hence 3J

(cid:122) shares a common part of the boundary with S (since if not, then 3J

Recall that a Whitney cube K was required in the right hand side of the conclusion of
Proposition 5.6.14 only in the case that J (cid:91) ⊂ S(cid:48)(cid:48) for some S(cid:48)(cid:48) ∈ C(2)D (S), which of course
(cid:122) ⊂ S. In this case we claim that K ∈ CA. Indeed, suppose in order to
implies 3J (cid:91) ⊂ J
(cid:122) ⊂ S, it
derive a contradiction, that K (cid:54)∈ CA. Then J
(cid:122) ⊂ S,
follows that J
a contradiction). Now Key Fact #2 in (5.6.15) implies that the inner grandchild containing
J, J (cid:91), is contained in K where K (cid:54)∈ CA. This then implies that the pair (I, J) belongs to
the corona straddling subcollection PA
cor, contradicting the assumption that Q is reduced.
Q∩C(cid:48)
and K ∈ W (S)∩C(cid:48)
and we can use Proposition (5.6.14)
stop,(cid:52)ω (f, g) by first summing over H ∈ HS and then over S ∈

Thus we have S ∈ Πbelow

with H = HS to bound |B|A,Q

SH. Indeed, QH (cid:91)straddles SH ≡ {S ∈ S : HS = H }, so that(cid:12)(cid:12)(cid:12)ϕ

(cid:12)(cid:12)(cid:12) (cid:46) αH (H) 1A\IQH

by (5.6.4), and so the sum over S ∈ SH of the first term on the right side of the conclusion

QH

(J)

A

1

A

J

280

of Proposition (5.6.14) is bounded by

αH (H)

S∈SH
≤ αH (H)

≤ αH (H)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

g

QH

n

Π

x

(cid:17)

S,aug
2

S, 1A\Sσ
|S| 1

Pα(cid:16)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠
(cid:113)|S|σ
(cid:88)
1(cid:112)|S|σ
Pα(cid:16)
(cid:17)
 sup
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
1(cid:112)|S|σ
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:113)|S|σ
· (cid:88)
(cid:41)(cid:113)|H|σ
(cid:40)
(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)

S, 1A\Sσ
|S| 1

Sα,A;S
locsize (QH )

S∈SH

S∈SH

S,aug
2

QH

QH

Π

Π

g

x

n

Π2QH

L2(ω)

sup
S∈SH

L2(ω)

S,aug
2

Π

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
 ·
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

QH

g

S,aug
2

L2(ω)

where ΠK,aug

2

QH is as in Definition 5.6.8, and the corresponding sum over S ∈ SH of the

second term is bounded by

αH (H)

S∈SH

(cid:88)

(cid:88)

K∈W(S)∩C(cid:48)

(cid:112)|K|σ(cid:112)|K|σ
locsize (QH )
Sα,A;S
(cid:46) αH (H) sup
(cid:41)
S∈SH
locsize (QH )
Sα,A;S

K, 1A\Sσ
|K| 1

Pα(cid:16)
(cid:88)
(cid:88)
(cid:113)|H|σ
(cid:13)(cid:13)(cid:13)Pω,b∗

(cid:17)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
 1
2(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)

K∈W(S)

αH (H)

|K|σ

K,aug
2

(cid:40)

QS
H

S∈S

≤

x

Π

A

Π2QHg

L2(ω)

n

sup
S∈SH

Π2QH

g

L2(ω)

K,aug
2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

Π

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

g

QS
H

Using the definition of |B|A,Q

stop,(cid:52)ω (f, g), we now sum the previous inequalities over the
cubes H ∈ HS to obtain the following string of inequalities (explained in detail after the

281

display)

αH (H)

H∈HS
αH (H)2 |H|σ

≤

(cid:26)

sup
S∈S

sup
S∈S

|B|A,Q

stop,(cid:52)ω (f, g) ≤

(cid:27) (cid:88)
Sα,A;S
locsize (Q)
(cid:27)(cid:115) (cid:88)
locsize (Q)
Sα,A;S
(cid:27)
locsize (Q)
Sα,A;S
(cid:27)(cid:13)(cid:13)(cid:13)Pσ,π,b
(cid:27)(cid:13)(cid:13)(cid:13)Pσ,b
where in the first line we have used Q = (cid:83)

Sα,A;S
locsize (Q)
Sα,A;S
locsize (Q)

H∈HS
(cid:107)h(cid:107)L2(σ)

(cid:26)
(cid:26)
(cid:26)
(cid:26)

π(Π1Q)f

π(Π1Q)f

sup
S∈S

sup
S∈S

sup
S∈S

≤

(cid:46)

(cid:46)

H∈HS

(cid:118)(cid:117)(cid:117)(cid:116) (cid:88)
(cid:13)(cid:13)(cid:13)L2(σ)
(cid:13)(cid:13)(cid:13)(cid:70)

L2(σ)

g

(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)(cid:70)2

L2(ω)

g

Π2QH

L2(ω)

Π2QH

H∈HS

(cid:113)|H|σ
(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:118)(cid:117)(cid:117)(cid:116) (cid:88)
(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)2
(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)Pω,b∗
(cid:13)(cid:13)(cid:13)(cid:70)
(cid:13)(cid:13)(cid:13)Pω,b∗

Π2QH

Π2Qg

Π2Qg

L2(ω)

L2(ω)

L2(ω)

g

QH, which follows from the fact that each J (cid:91)
is contained in a unique S ∈ S; in the third line we have used the quasiorthogonal inequality
(5.6.24); in the fourth line we have used that the sets Π2QH ⊂ CH,(cid:91)shif t
are pairwise disjoint
Π2QH. In the final line, we have used first the equality
in H and have union Π2Q =

H∈HS

·(cid:91)

H

(5.1.43), second the fact that the functions (cid:3)σ,π,b

I,brokf have pairwise disjoint supports, third
the upper weak Riesz inequality and fourth the estimate (5.1.44) - which relies on the reverse

H∈HS

282

Hölder property for children in Lemma 5.1.9 - to obtain

(cid:13)(cid:13)(cid:13)Pσ,π,b

π(Π1Q)f

(cid:13)(cid:13)(cid:13)2

L2(σ)

I∈π(Π1Q)

I∈π(Π1Q)

=

(cid:46)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:88)
(cid:46) (cid:13)(cid:13)(cid:13)Pσ,b
(cid:46) (cid:88)
(cid:46) (cid:13)(cid:13)(cid:13)Pσ,b

I∈π(Π1Q)
π(Π1Q)f

(cid:3)σ,b

I∈π(Π1Q)

I f − (cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:88)

I f

L2(σ)

L2(σ)

+

+

+

I∈π(Π1Q)

(cid:3)σ,b
I f

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:3)σ,b
(cid:13)(cid:13)(cid:13)(cid:70)2

L2(σ)

π(Π1Q)f

L2(σ)

L2(σ)

I,brokf

(cid:3)σ,π,b

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2
(cid:88)
(cid:13)(cid:13)(cid:13)(cid:3)σ,π,b
(cid:13)(cid:13)(cid:13)2
(cid:88)
(cid:13)(cid:13)(cid:53)σ
I f(cid:13)(cid:13)2

(cid:3)σ,π,b

I,brokf

I,brokf

L2(σ)

I∈π(Π1Q)

I∈π(Π1Q)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2

L2(σ)

L2(σ)

(5.6.25)

We now use the fact that the supremum in the definition of Sα,A;S

locsize (Q) is taken over

K ∈ W∗ (S) ∩ C(cid:48)

A

to conclude that

Sα,A;S
locsize (Q) ≤ Sα,A

augsize (Q) ,

sup
S∈S

and this completes the proof of Lemma 5.6.15.

In a similar fashion we can obtain the following Substraddling Lemma.

Definition 5.6.16. Given a reduced admissible collection of pairs Q for A, and a D-cube
L contained in A, we say that Q substraddles L if for every pair (I, J) ∈ Q there is
K ∈ W (L) ∩ C(cid:48)

A with J ⊂ K ⊂ 3K ⊂ I ⊂ L.

Lemma 5.6.17. Let L be a D-cube contained in A, and suppose that Q is an admissible

283

collection of pairs that substraddles L. Then we have the sublinear form bound

(cid:98)NA,Q
stop,(cid:52)ω ≤ CSα,A

augsize (Q) .

Proof. We will show that Q (cid:91)straddles the subset WL of Whitney cubes for L given by

WQ (L) ≡(cid:8)K ∈ W (L) ∩ C(cid:48)

A : J ⊂ K ⊂ 3K ⊂ I ⊂ L for some (I, J) ∈ Q(cid:9) .

Q ∩ C(cid:48)

A

1

Case 1: If π

It is clear that WQ (L) ⊂ Πbelow
is a subpartition of A. It remains to show that for
every pair (I, J) ∈ Q there is K ∈ WQ (L) ∩ [J, I] such that J (cid:91) ⊂ K. But our hypothesis
implies that there is K ∈ WQ (L) with J ⊂ K ⊂ 3K ⊂ I ⊂ L. We now consider two cases.
(3)D K ⊂ L, then since K is maximal Whitney cube, it is contained in an
(3)D K.
(cid:122). We thus have
(cid:122) which implies that J (cid:91) has the same endpoint as L, a

outer grandchild of π
Recall, from Key Fact #2 in (5.6.15), 3J ⊂ J (cid:91), an inner grandchild of J
(cid:122) ⊂ π
J
contradiction). This implies that J (cid:91) ⊂ K .

(1)D K has to share an endpoint with L. Then so does π

(2)D K (If not; π

(2)D K ⊂ J

(3)D K and π

Case 2: If π
(cid:122) ⊂ I = π

have J

(3)D K (cid:39) L , then K ⊂ 3K ⊂ I ⊂ L implies that I = L = π
(2)D K, which again gives J (cid:91) ⊂ K.

(2)D K. Thus we

Now that we know Q (cid:91)straddles the subset WQ (L), we can apply Lemma 5.6.15 to obtain

the required bound (cid:98)NA,Q

stop,(cid:52)ω ≤ CSα,A

augsize (Q).

5.6.4 The bottom/up stopping time argument of M. Lacey

Before introducing Lacey’s stopping times, we note that the Corona-straddling Lemma 5.6.10
allows us to remove the ‘corona straddling’ collection PA

cor of pairs of cubes in (5.6.16) from

284

the collection PA in (5.6.2 ) used to define the stopping form BA
PA\PA

cor is of course also A-admissible.

stop (f, g). The collection

We assume for the remainder of the proof that all admissible collections P are reduced,

i.e.

PA ∩ PA

cor = ∅, as well as P ∩ PA

cor = ∅ for all A-admissible P.

(5.6.26)

For a cube K ∈ D, we define

G [K] ≡ {J ∈ G : J ⊂ K}

J(cid:122)(cid:17)(cid:17)

J(cid:91)(cid:17)(cid:17),

c

(5.6.27)

(5.6.28)

to consist of all cubes J in the other grid G that are contained in K. For an A-admissible
collection P of pairs, define two atomic measures ωP and ω(cid:91)P in the upper half space Rn+1
by

+

J

x

and

L2(ω)

δ(cid:16)

J∈Π2P

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

ωP ≡ (cid:88)
ω(cid:91)P ≡ (cid:88)

(cid:13)(cid:13)(cid:13)♠2
(cid:16)
J(cid:122),(cid:96)
c
(cid:13)(cid:13)(cid:13)♠2
δ(cid:16)
Note that each cube J ∈ Π2P has its ‘energy’(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
to exactly one of the 2n points (cid:16)

J∈Π2P
where J (cid:91) is the inner grandchild of J

(cid:122) that contains J

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

J(cid:91) ,(cid:96)

(cid:16)

x

J

J

c

x

(cid:16)

L2(ω)

J(cid:91), 1
4 (cid:96)

(cid:122)(cid:17)(cid:17) in the upper half plane Rn+1

in the measure ω(cid:91)P assigned
since J is
(cid:122)
(cid:38), namely in J (cid:91), by Key Fact #2 in (5.6.15). Note also that the atomic
contained in one of J
measure ω(cid:91)P differs from the measure µ in (.0.20) in Appendix below - which is used there to
control the functional energy condition - in that here we bundle together all the J(cid:48)s having a
common J (cid:91). This is in order to rewrite the augmented size functional in terms of the measure

J

+

285

ω(cid:91)P. We can get away with this here, as opposed to in Appendix , due to the ‘smaller and
decoupled’ nature of the augmented size functional to which we will relate ω(cid:91)P.

Define the tent T (L) over a cube L to be the convex hull of the cube L×{0} and the point
(2)D K

(cid:111)
J (cid:91)(cid:17)(cid:17) ∈ T (K). We can now rewrite the augmented size functional

. Then for J ∈ Π2P we have J ∈ ΠK,aug

J ⊂ K and J

(cid:122) ⊂ π

P iff

(cid:110)

2

+

(cid:16)
(cL, (cid:96) (L)) ∈ Rn+1
iff J (cid:91) ⊂ K iff
of P in Definition 5.6.8 as

J(cid:91), (cid:96)

(cid:16)

c

Sα,A
augsize (P)2 ≡

sup
K∈Πbelow

1

P∩C(cid:48)

A

1
|K|σ

Pα(cid:16)

(cid:17)

2

K, 1A\K σ
|K| 1

n

ω(cid:91)P (T (K)) .

(5.6.29)

It will be convenient to write

Ψα (K;P)2 ≡

Pα(cid:16)

(cid:17)

2

K, 1A\K σ
|K| 1

n

ω(cid:91)P (T (K)) ,

so that we have simply

Sα,A
augsize (P)2 =

sup
K∈Πbelow

1

P∩C(cid:48)

A

Ψα (K;P)2

|K|σ

.

Remark 5.6.18. The functional ω(cid:91)P (T (K)) is increasing in K, while the functional

286

is ‘almost decreasing’ in K: if K0 ⊂ K then
(cid:17)

dσ (y)
|K| 1
n + |y − cK|
√
n)n+1−αdσ (y)

(cid:19)n+1−α
(cid:12)(cid:12)(cid:12)(cid:19)n+1−α
(cid:12)(cid:12)(cid:12)(cid:19)n+1−α = Cα,n

(cid:12)(cid:12)(cid:12)y − cK0
(cid:12)(cid:12)(cid:12)y − cK0

Cα,n dσ (y)

=

(cid:46)

A\K

A\K

(cid:18)
(cid:18)
(
|K0| 1
(cid:18)

(cid:90)
(cid:90)
(cid:90)
(cid:12)(cid:12)(cid:12) ≤ |K| + |y − cK| + 1

|K0| 1

A\K0

n +

n +

≤

Pα(cid:16)

(cid:17)

σ

K0, 1A\K0
|K0| 1

n

(cid:17)

K,1A\K σ
|K| 1
n

Pα(cid:16)
Pα(cid:16)

K, 1A\K σ
|K| 1

n

(cid:12)(cid:12)(cid:12)y − cK0

since |K0| +

2 diam (K) for y ∈ A\K.

Recall that if P is an admissible collection for a dyadic cube A, the corresponding sub-

linear form in (5.6.7) is given by

|B|A,P

Pα(cid:16)
stop,(cid:52)ω (f, g) ≡ (cid:88)
(cid:88)

J∈Π2P

where ϕP

J ≡

(cid:12)(cid:12)(cid:12)ϕP

J

J,

(cid:12)(cid:12)(cid:12) 1A\IP (J)σ
(cid:17)
(cid:16)(cid:98)(cid:3)σ,(cid:91),b
(cid:17)

πI f

n

|J| 1
bAEσ
I

I∈C(cid:48)

A: (I,J)∈P

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

x

(cid:13)(cid:13)(cid:13)♠

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:3)ω,b∗

J

g

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

;

1A\I .

In the notation for |B|A,P
clutter, we will often do so from now on when the dependence on α is inconsequential.

stop,(cid:52)ω, we are omitting dependence on the parameter α, and to avoid

Recall further that the ‘size testing collection’ of cubes Πbelow

P for the initial size testing
initsize (P) is the collection of all subcubes of cubes in Π1P, and moreover, by
. This latter set

functional Sα,A
Key Fact #1 in (5.6.13), that we can restrict the collection to Πbelow

P ∩ C(cid:48)

1

1

A

is used for the augmented size functional.

287

Assumption

We may assume that the corona CA is finite, and that each A-admissible collection P
is a finite collection, and hence so are Π1P, Πbelow
and Π2P, provided all of
the bounds we obtain are independent of the cardinality of these latter collections.

P ∩ C(cid:48)

A

1

Consider 0 < ε < 1, where ρ = 1 + ε will be chosen later in (5.6.37). Begin by defining

the collection L0 to consist of the minimal dyadic cubes K in Πbelow

1

P ∩ C(cid:48)

A

such that

Ψα (K;P)2

|K|σ

≥ εSα,A

augsize (P)2 .

where we recall that

Pα(cid:16)

(cid:17)

2

K, 1A\K σ
|K| 1

n

ω(cid:91)P (T (K)) .

Ψα (K;P)2 ≡

Note that such minimal cubes exist when 0 < ε < 1 because Sα,A
over K ∈ Πbelow

augsize (P)2 is the supremum
. A key property of the minimality requirement is that

of Ψα(K;P)2

P ∩ C(cid:48)

1

A

|K|σ

Ψα(cid:0)K(cid:48);P(cid:1)2

|K(cid:48)|σ

< εSα,A

augsize (P)2 ,

(5.6.30)

whenever there is K(cid:48) ∈ Πbelow

1

P ∩ C(cid:48)

A

with K(cid:48) (cid:38) K and K ∈ L0.

We now perform a stopping time argument ‘from the bottom up’ with respect to the

atomic measure ωP in the upper half space. This construction of a stopping time ‘from the
bottom up’, together with the subsequent applications of the Orthogonality Lemma and the

Straddling Lemma, comprise the key innovations in Lacey’s argument [27]. However, in our
P are no longer ‘good’ in any sense, and we must

situation the cubes I belonging to Πbelow

1

288

include an additional top/down stopping criterion in the next subsection to accommodate

this lack of ‘goodness’. The argument in [27] will apply to these special stopping cubes,

called ‘indented’ cubes, and the remaining cubes form towers with a common endpoint, that

are controlled using all three straddling lemmas.

We refer to L0 as the initial or level 0 generation of stopping cubes. Set

ρ = 1 + ε.

(5.6.31)

As in [64], [66] and [67], we follow Lacey [27] by recursively defining a finite sequence of
P ∩C(cid:48)
generations {Lm}m≥0
that contain a cube from some previous level L(cid:96), (cid:96) < m, such that

by letting Lm consist of the minimal dyadic cubes L in Πbelow

A

1

ω(cid:91)P (T (L)) ≥ ρω(cid:91)P

(5.6.32)



T(cid:0)L(cid:48)(cid:1)

(cid:91)
L(cid:48)∈m−1(cid:83)
L(cid:96): L(cid:48)⊂L

(cid:96)=0

 .

1

A

P ∩C(cid:48)

Since P is finite this recursion stops at some level M. We then let LM +1 consist of all the
that are not already in some Lm with m ≤ M. Thus LM +1
maximal cubes in Πbelow
P. We do not of course
will contain either none, some, or all of the maximal cubes in Πbelow
have (5.6.32) for A(cid:48) ∈ LM +1 in this case, but we do have that (5.6.32) fails for subcubes
K of A(cid:48) ∈ LM +1 that are not contained in any other L ∈ Lm with m ≤ M, and this is
sufficient for the arguments below.

1

We now decompose the collection of pairs (I, J) in P into collections P(cid:91)small and P(cid:91)big
according to the location of I and J (cid:91), but only after introducing below the indented corona
H. The collection P(cid:91)big will then essentially consist of those pairs (I, J) ∈ P for which there

289

L

L(cid:48) and I ∈ CH

are L(cid:48), L ∈ H with L(cid:48) (cid:38) L and such that J (cid:91) ∈ CH
. The collection P(cid:91)small will
consist of the remaining pairs (I, J) ∈ P for which there is L ∈ H such that J (cid:91), I ∈ CH
,
along with the pairs (I, J) ∈ P such that I ⊂ I0 for some I0 ∈ L0. This will cover all pairs
(I, J) in P ⊂ PA, since for such pairs, I ∈ C(cid:48)
, which in turn implies I ∈ CH
L(cid:48) for some L, L(cid:48) ∈ H. But a considerable amount of further analysis is required
and J (cid:91) ∈ CH
to prove (5.6.7).

and J ∈ CGshif t

A

A

L

L

First recall that L ≡ M +1(cid:83)

Lm is the tree of stopping ωP-energy cubes defined above. By
P ∩ C(cid:48)
.

the construction above, the maximal elements in L are the maximal cubes in Πbelow
For L ∈ L, denote by CL

the corona associated with L in the tree L,

m=0

A

1

L

L ≡(cid:8)K ∈ D : K ⊂ L and there is no L(cid:48) ∈ L with K ⊂ L(cid:48) (cid:36) L(cid:9) ,

CL

and define the (cid:91) shifted L-corona by

CL,(cid:91)shif t

L

≡ (cid:110)

J ∈ G : J (cid:91) ∈ CL

L

(cid:111)

.

Now the parameter m in Lm refers to the level at which the stopping construction was
performed, but for L ∈ Lm, the corona children L(cid:48) of L are not all necessarily in Lm−1, but
may be in Lm−t for t large.

290

At this point we introduce the notion of geometric depth d in the tree L by defining

G0 ≡ {L ∈ L : L is maximal} ,
G1 ≡ {L ∈ L : L is maximal wrt L (cid:36) L0 for some L0 ∈ G0} ,

(5.6.33)

...

Gd+1 ≡ {L ∈ L : L is maximal wrt L (cid:36) Ld for some Ld ∈ Gd} ,

...

We refer to Gd as the dth generation of cubes in the tree L, and say that the cubes in Gd are
at depth d in the tree L (the generations Gd here are not related to the grid G), and we write
dgeom (L) for the geometric depth of L. Thus the cubes in Gd are the stopping cubes in L
that are d levels in the geometric sense below the top level. While the geometric depth dgeom
is about to be superceded by the ‘indented’ depth dindent defined in the next subsection, we
will return to the geometric depth in order to iterate Lacey’s bottom/up stopping criterion

when proving the second line in (5.6.36) in Proposition 5.6.19 below.

5.6.5 The indented corona construction

Now we address the lack of goodness in Πbelow
top/down stopping time H over the collection L. Given the initial generation

A

1

. For this we introduce an additional

P ∩ C(cid:48)

H0 = {maximal L ∈ L} =

(cid:110)maximal I ∈ Πbelow

P(cid:111)

,

1

291

define subsequent generations Hk as follows. For k ≥ 1 and each H ∈ Hk−1, let

Hk (H) ≡ {maximal L ∈ L : 3L ⊂ H}

k=0

H∈Hk−1

Hk (H). Finally

consist of the next H-generation of L-cubes below H, and set Hk ≡ (cid:83)
set H ≡ ∞(cid:83)

Hk. We refer to the stopping cubes H ∈ H as indented stopping cubes since
3H ⊂ πHH for all H ∈ H at indented generation one or more, i.e. each successive such H
is ‘indented’ in its H-parent. This property of indentation is precisely what is required in
order to generate geometric decay in indented generations at the end of the proof. We refer
to k as the indented depth of the stopping cube H ∈ Hk, written k = dindent (H), which is a
refinement of the geometric depth dgeom introduced above. We will often revert to writing
the dummy variable for cubes in H as L instead of H. For L ∈ H define the H-corona CH
and H-(cid:91)shifted corona CH,(cid:91)shif t

by

L

L

L ≡ (cid:8)I ∈ D : I ⊂ L and I (cid:54)⊂ L(cid:48) for any L(cid:48) ∈ CH (L)(cid:9) ,
≡ (cid:110)

J ∈ G : J (cid:91) ∈ CH

(cid:111)

.

CH
CH,(cid:91)shif t

L

L

L

We will also need recourse to the coronas CH

restricted to cubes in L, i.e.

L ∩ L =(cid:8)T ∈ L : T ⊂ L and T (cid:54)⊂ L(cid:48) for any L(cid:48) ∈ H with L(cid:48) (cid:36) L(cid:9) .

CH
L (L) ≡ CH

and

T (L) ≡ CH,restrict

L

(L) = CH

L (L)\{L}

292

We emphasize the distinction ‘indented generation’ as this refers to the indented depth rather
than either the level of initial stopping construction of L, or the geometric depth. The point
of introducing the tree H of indented stopping cubes, is that the inclusion 3L ⊂ πHL for
all L ∈ H with dindent (L) ≥ 1 turns out to be an adequate substitute for the standard
‘goodness’ lost in the process of infusing the weak goodness of Hytönen and Martikainen in

Subsection 5.2.1 above.

5.6.5.1 Flat shifted coronas

We now define the (cid:91)shifted admissible collections of pairs P(cid:91)H

using the coronas

≡(cid:110)

CH,(cid:91)shif t

L

J ∈ Π2P : J (cid:91) ∈ CH

L

(cid:111) and CL,(cid:91)shif t

L

L,t

≡(cid:110)

(cid:111)

.

J ∈ Π2P : J (cid:91) ∈ CL

L

In these flat shifted H and L coronas, we have effectively shift the cubes J two levels ‘up’ by
requiring J (cid:91) ∈ CL
. We define

, but because P is admissible, we always have J

(cid:122) ∈ CA,restrict

L

L,t ≡ (cid:110)
(I, J) ∈ P : I ∈ CH
(cid:110)
(I, J) ∈ P : I ∈ CH

P(cid:91)H
P(cid:91)H
L,0 =

L

L , J ∈ CH,(cid:91)shif t

L(cid:48)

and J ∈ CH,(cid:91)shif t

L

A

(cid:111)
for some L(cid:48) ∈ Hdindent(L)+tL(cid:48) ⊂ L
(cid:111)

and

L,0

L,0 = P(cid:91)H−small
P(cid:91)H
P(cid:91)H−small

L,0

;

L,0

≡ (cid:110)
(I, J) ∈ P(cid:91)H
(cid:110)
(I, J) ∈ P(cid:91)H
≡ (cid:110)
(I, J) ∈ P(cid:91)H

˙∪P(cid:91)H−big
L,0 : there is no L(cid:48) ∈ T (L) with J (cid:91) ⊂ L(cid:48) ⊂ I
L,0 : I ∈ CL
(cid:111)
L,0 : there is L(cid:48) ∈ T (L) with J (cid:91) ⊂ L(cid:48) ⊂ I

L(cid:48)\(cid:8)L(cid:48)(cid:9) , J ∈ CL,(cid:91)shif t

L(cid:48)

=

,

P(cid:91)H−big

L,0

(cid:111)
for some L(cid:48) ∈ T (L)

(cid:111)

293

,

,

with one exception: if L ∈ H0 we set P(cid:91)H−small
≡ ∅ since in this case
L fails to satisfy (5.6.32) as pointed out above. Finally, for L ∈ H we further decompose
P(cid:91)H−small

and P(cid:91)H−big

≡ P(cid:91)H

as

L,0

L,0

L,0

L,0

P(cid:91)H−small

L,0

where P(cid:91)L−small

L(cid:48),0

·(cid:91)
L(cid:48)∈T (L)
(I, J) ∈ P : I ∈ CL

P(cid:91)L−small
L(cid:48),0

Then we set

=

≡ (cid:110)
(cid:91)
P(cid:91)small ≡ (cid:91)

P(cid:91)big ≡

L∈H

L∈L

P(cid:91)H−big

L,0

P(cid:91)L−small

L,0

L(cid:48)

L(cid:48)\(cid:8)L(cid:48)(cid:9) and J ∈ CL,(cid:91)shif t
(cid:91)(cid:91)

 ;

(cid:91)

P(cid:91)H

L,t

L∈H

t≥1

(cid:111)

(5.6.34)

L,0

are now even smaller than the regular coronas PL−small

We observed above that every pair (I, J) ∈ P is included in either Psmall or Pbig, and it
follows that every pair (I, J) ∈ P is thus included in either P(cid:91)small or P(cid:91)big, simply because
the pairs (I, J) have been shifted up by two dyadic levels in the cube J. Thus the coronas
P(cid:91)L−small
estimate (5.6.35) below to hold for the larger augmented size functional. On the other hand,
the coronas P(cid:91)H−big
lemmas above in order to obtain the estimates (5.6.36) below. More specifically, we will
see that stopping forms with pairs in P(cid:91)big will be estimated using the (cid:91) Straddling and
Substraddling Lemmas (Substraddling applies to part of P(cid:91)H−big
and (cid:91)Straddling applies to
the remaining part of P(cid:91)H−big
straddling collection PA

), and it is here that the removal of the corona-
cor is essential, while forms with pairs in P(cid:91)small will be absorbed.

are now bigger than before, requiring the stronger straddling

, which permits the

and to P(cid:91)H

and P(cid:91)H

L,0

L,0

L,0

L,0

L,t

L,t

294

5.6.6 Size estimates

restricted norm (cid:98)NA,P

Now we turn to proving the size estimates we need for these collections. Recall that the

stop,(cid:52)ω is the best constant in the inequality

stop,(cid:52)ω (f, g) ≤ (cid:98)NA,P

|B|A,P

stop,(cid:52)ω

(cid:13)(cid:13)(cid:13)Pσ,b

Π1P f

(cid:13)(cid:13)(cid:13)(cid:70)

L2(σ)

(cid:13)(cid:13)(cid:13)Pω,b∗

Π2P g

(cid:13)(cid:13)(cid:13)(cid:70)

L2(ω)

where f ∈ L2 (σ) satisfies Eσ

I |f| ≤ αA (A) for all I ∈ CA, and g ∈ L2 (ω).

Proposition 5.6.19. Suppose ρ in (5.6.31) is greater than 1, and P is a reduced admissible
collection of pairs for a dyadic cube A. Let P = P(cid:91)big ˙∪P(cid:91)small be the decomposition satisfying

above, i.e.

(cid:91)

L∈H

P =

(cid:91)(cid:91)

t≥1

(cid:91)

L∈H

 (cid:91) (cid:91)

L∈L

P(cid:91)H

L,t

P(cid:91)H−big

L,0



P(cid:91)L−small

L,0

Then all of these collections P(cid:91)L−small

L,0

, P(cid:91)H−big

L,0

and P(cid:91)H

L,t are reduced admissible, and we

have the estimate

Sα,A

augsize

(cid:16)P(cid:91)L−small

L,0

(cid:17)2 ≤ (ρ − 1)Sα,A

augsize (P)2 ,

L ∈ L

(5.6.35)

and the localized norm bounds,

(cid:98)N

L∈H
stop,(cid:52)ω

A, (cid:83)
(cid:98)N

L,0

P(cid:91)H−big
A, (cid:83)

P(cid:91)H

L,t

L∈H
stop,(cid:52)ω

≤ CSα,A

augsize (P) ,
− t
2Sα,A

augsize (P) ,

≤ Cρ

(5.6.36)

t ≥ 1.

Using this proposition on size estimates, we can finish the proof of (5.6.7), and hence the

295

proof of (5.6.1).

Proof. Recall that (cid:98)NA,P
(cid:110)P(cid:91)L−small

(cid:111)

L,0

L∈L

Corollary 5.6.20. The sublinear stopping form inequality (5.6.7) holds.

stop,(cid:52)ω is the best constant in the inequality (5.6.10). Since

is a mutually orthogonal family of A-admissible pairs, the Orthogonality

Lemma 5.6.4 implies that

(cid:98)N

A, (cid:83)

L∈L
stop,(cid:52)ω

P(cid:91)L−small

L,0

(cid:98)N

A,P(cid:91)L−small
L,0
stop,(cid:52)ω

≤ sup
L∈L

Using this, together with the decomposition of P and (5.6.36) above, we obtain

(cid:98)NA,P
stop,(cid:52)ω ≤ sup
L∈H

A, (cid:83)

L∈H
stop,(cid:52)ω

(cid:98)N
augsize (P) +

P(cid:91)H−big

L,0

M +1(cid:88)

t=1

A, (cid:83)

M +1(cid:88)
Sα,A

sup
L∈H

P(cid:91)H

L∈H
stop,(cid:52)ω

+(cid:98)N
(cid:98)N
(cid:98)N
augsize (P) + sup
L∈L

t=1

L,t

+

− t
2

ρ

(cid:46) Sα,A

P(cid:91)L−small

A, (cid:83)

L,0

L∈L
stop,(cid:52)ω
A,P(cid:91)L−small
L,0
stop,(cid:52)ω

Since the admissible collection PA in (5.6.2) that arises in the stopping form is finite, we
can define L to be the best constant in the inequality

(cid:98)NA,P
stop,(cid:52)ω ≤ LSα,A

augsize (P) for all A-admissible collections P.

Now choose P so that

(cid:98)NA,P
stop,(cid:52)ω
Sα,A
augsize (P)

>

1
2

L =

1
2

sup

Q is A-admissible

(cid:98)NA,Q
stop,(cid:52)ω
Sα,A
augsize (Q)

.

296

Then using

M +1(cid:88)

t=1

− t
2 ≤

ρ

1√
ρ − 1

we have

(cid:98)NA,P
stop,(cid:52)ω
Sα,A
augsize (P)

≤

L < 2

≤ C

1√
ρ − 1

L∈L L
+ C sup

C 1√

ρ−1Sα,A

augsize (P) + C sup
L∈L
(cid:17)
Sα,A
augsize (P)

(cid:16)P(cid:91)L−small

L,0

Sα,A

augsize

Sα,A
augsize (P)

(cid:98)N

A,P(cid:91)L−small
L,0
stop,(cid:52)ω

+ CL(cid:112)ρ − 1

≤ C

1√
ρ − 1

where we have used (5.6.35) in the last line. If we choose ρ > 1 so that

C(cid:112)ρ − 1 <

1
2

,

(5.6.37)

then we obtain L ≤ 2C 1√

ρ−1

. Together with Lemma 5.6.9, this yields

(cid:98)NA,P
stop,(cid:52)ω ≤ LSα,A

augsize (P) ≤ 2C

1√
ρ − 1

as desired, and completes the proof of inequality (5.6.7).

(cid:16)E α

(cid:113)

(cid:17)

2 +

Aα
2

Thus, in view of Conclusion 5.6.4, it remains only to prove Proposition 5.6.19 using the

Orthogonality and Straddling and Substraddling Lemmas above, and we now turn to this

task.

Proof of Proposition 5.6.19. We split the proof into three parts.

Proof of (5.6.35): To prove the inequality (5.6.35), suppose first that L /∈ LM +1. In
the case that L ∈ L0 is an initial generation cube, then from (5.6.30) and the fact that every

297

I ∈ P(cid:91)L−small

L,0

satisfies I (cid:36) L, we obtain that

(cid:16)P(cid:91)L−small

L,0

(cid:17)2

Sα,A

augsize

=

≤

Ψα(cid:16)
Ψα(cid:16)

A

(cid:17)2
K(cid:48);P(cid:91)L−small
(cid:17)2

|K(cid:48)|σ

L,0

K(cid:48);P(cid:91)L−small

L,0

|K(cid:48)|σ

K(cid:48)∈Πbelow

1

sup
P(cid:91)L−small

L,0

∩C(cid:48)

K(cid:48)∈Πbelow

sup
P∩C(cid:48)
augsize (P)2

1

≤ εSα,A

A: K(cid:48)(cid:38)L

Now suppose that L (cid:54)∈ L0 in addition to L /∈ LM +1. Pick a pair (I, J) ∈ P(cid:91)L−small
I is in the restricted corona CL,(cid:48)
is a finite collection, the definition of Sα,A
K ∈ Πbelow

and J is in the (cid:91)shifted corona CL,(cid:91)shif t

P(cid:91)L−small

so that

∩ C(cid:48)

augsize

L,0

L,0

L

1

L,0

A

L

L,0

. Then
. Since P(cid:91)L−small

(cid:17) shows that there is an cube
(cid:17)
2

ω(cid:91)P (T (K)) .

(cid:16)P(cid:91)L−small
Pα(cid:16)

K, 1A\K σ
|K| 1

n

(cid:16)P(cid:91)L−small

L,0

(cid:17)2

Sα,A

augsize

=

1
|K|σ

Note that K (cid:36) L by definition of P(cid:91)L−small

. Now let t be such that L ∈ Lt, and define

L,0

t(cid:48) = t(cid:48) (K) ≡ max(cid:8)s : there is L(cid:48) ∈ Ls with L(cid:48) ⊂ K(cid:9) ,

and note that 0 ≤ t(cid:48) < t. First, suppose that t(cid:48) = 0 so that K does not contain any L(cid:48) ∈ L.
Then it follows from the construction at level (cid:96) = 0 that

Pα(cid:16)

1
|K|σ

K, 1A\K σ
|K|

(cid:17)

2

ω(cid:91)P (T (K)) < εSα,A

augsize (P)2 ,

298

and hence from ρ = 1 + ε we obtain

(cid:16)P(cid:91)L−small

L,0

(cid:17)2

Sα,A

augsize

< εSα,A

augsize (P)2 = (ρ − 1)Sα,A

augsize (P)2 .

Now suppose that t(cid:48) ≥ 1. Then K fails the stopping condition ( 5.6.32) with m = t(cid:48) + 1,
contradicting our definition of t(cid:48), and so
since otherwise it would contain a cube L(cid:48)(cid:48) ∈ Lt(cid:48)+1

ω(cid:91)P (T (K)) < ρω(cid:91)P (V (K)) where V (K) ≡ (cid:91)
L(cid:48)∈ t(cid:48)(cid:83)

T(cid:0)L(cid:48)(cid:1) .

L(cid:96): L(cid:48)⊂K

(cid:96)=0

Now we use the crucial fact that the positive measure ω(cid:91)P is additive and finite to obtain

from this that

ω(cid:91)P (T (K)\V (K)) = ω(cid:91)P (T (K)) − ω(cid:91)P (V (K)) ≤ (ρ − 1) ω(cid:91)P (V (K)) .

(5.6.38)

Now recall that

Sα,A
augsize (Q)2 ≡

1

n

(cid:17)

1
|K|σ

(cid:48)
Q∩C
A

sup
K∈Πbelow

Pα(cid:16)

K, 1A\K σ
|K| 1

2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
the support (cid:16)
J(cid:91)(cid:17)(cid:17) is contained in the set T (K), but not in the set
 .

V (K) ≡(cid:91)T(cid:0)L(cid:48)(cid:1) : L(cid:48) ∈ t(cid:48)(cid:91)

L(cid:96) : L(cid:48) ⊂ K

P(cid:91)L−small

L,0

(cid:96)=0

We claim it follows that for each J ∈ ΠK,aug
atom δ(cid:16)

(cid:16)

2

c
J(cid:91) ,(cid:96)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2
J (cid:91)(cid:17)(cid:17) of the
(cid:16)

L2(ω)

.

Π

K,aug
2

Qx

c

J(cid:91), (cid:96)

299

Indeed, suppose in order to derive a contradiction, that (cid:16)
L(cid:48) ∈ L(cid:96) with 0 ≤ (cid:96) ≤ t(cid:48). Recall that L ∈ Lt with t(cid:48) < t so that L(cid:48) (cid:36) L. Thus(cid:16)
T(cid:0)L(cid:48)(cid:1) implies J (cid:91) ⊂ L(cid:48), which contradicts the fact that

J(cid:91), (cid:96)

(cid:16)

c

J ∈ ΠK

2 P(cid:91)L−small

L,0

⊂ Π2P(cid:91)L−small

L,0

=

L\{L} and J ∈ CL,(cid:91)shif t

(cid:110)
(I, J) ∈ P : I ∈ CL

c

(cid:16)

J(cid:91), (cid:96)

J (cid:91)(cid:17)(cid:17) ∈ T(cid:0)L(cid:48)(cid:1) for some
J (cid:91)(cid:17)(cid:17) ∈
(cid:111)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

P(cid:91)L−small

is

x

L

L,0

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

Π

K,aug
2

implies J (cid:91) ∈ CL

L

- because L(cid:48) /∈ CL

L

.

Thus from the definition of ω(cid:91)P in (5.6.28), the ‘energy’

at most the ω(cid:91)P-measure of T (K)\V (K). Using now

(cid:91)P(cid:91)L−small
ω

L,0

(T (K)) = ω

(cid:91)P(cid:91)L−small

L,0

(T (K)\V (K)) ≤ ω(cid:91)P (T (K)\V (K))

and (5.6.38), we then have

(cid:16)P(cid:91)L−small

(cid:17)2 ≤

L,0

Sα,A

augsize

K∈Πbelow

1

sup
P(cid:91)L−small

L,0

∩C(cid:48)

A

1
|K|σ

Pα(cid:16)

(cid:17)

2

ω(cid:91)P (T (K)\V (K))

n

K, 1A\K σ
|K| 1

Pα(cid:16)

(cid:17)

2

K, 1A\K σ
|K| 1

n

≤ (ρ − 1)

K∈Πbelow

1

sup
P(cid:91)L−small

L,0

∩C(cid:48)

A

1
|K|σ

ω(cid:91)P (V (K))

and we can continue with

(cid:16)P(cid:91)L−small

L,0

(cid:17)2 ≤ (ρ − 1)

Sα,A

augsize

sup
P∩C(cid:48)
K∈Πbelow
augsize (P)2 .

1

A

≤ (ρ − 1)Sα,A

Pα(cid:16)

1
|K|σ

K, 1A\K σ
|K| 1

n

(cid:17)

2

ω(cid:91)P (T (K))

300

In the remaining case where L ∈ LM +1 we can include L as a testing cube K and the

same reasoning applies. This completes the proof of (5.6.35).

To prove the other inequality (5.6.36) in Proposition 5.6.19, we will use the (cid:91) Straddling

and Substraddling Lemmas to bound the norm of certain ‘straddled’ stopping forms by
the augmented size functional Sα,A
‘mutually orthogonal’ stopping forms. Recall that

, and the Orthogonality Lemma to bound sums of

augsize

(cid:91)
≡ (cid:91)

L∈L

P(cid:91)H−big

L,0

L∈H
P(cid:91)H−big

L,0

,

(cid:91)(cid:91)

t≥1
Q(cid:91)H−big

1

P(cid:91)H

L,t

(cid:91)
≡ (cid:91)

L∈H

t≥1

 ≡ Q(cid:91)H−big

0

P(cid:91)H−big

,

t

(cid:91)Q(cid:91)H−big
≡ (cid:91)

1

;

P(cid:91)H−big

t

P(cid:91)H

L,t

L∈H

P(cid:91)big =

Q(cid:91)H−big

0

Proof of the second line in (5.6.36): We first turn to the collection

1

Q(cid:91)H−big
P(cid:91)H−big

t

(cid:91)
≡ (cid:91)

t≥1

=

(cid:91)
L∈H
P(cid:91)H
L,t ,

L∈L

(cid:91)

t≥1
t ≥ 1,

P(cid:91)H
L,t =

P(cid:91)H−big

;

t

where

P(cid:91)H
L,t =

(cid:110)
(I, J) ∈ P : I ∈ CH

L , J ∈ CH,(cid:91)shif t

L(cid:48)

for some L(cid:48) ∈ Hdindent(L)+t, L(cid:48) ⊂ L

(cid:111)

.

We now claim that the second line in (5.6.36) holds, i.e.

(cid:98)N

A,P(cid:91)H−big
stop,(cid:52)ω

t

≤ Cρ

− t
2Sα,A

augsize (P) ,

t ≥ 1,

(5.6.39)

which recovers the key geometric gain obtained by Lacey in [27], except that here we are

301

only gaining this decay relative to the indented subtree H of the tree L.

The case t = 1 can be handled with relative ease since decay is not relevant here. Indeed,
straddles the collection CH (L) of H -children of L, and so the localized (cid:91)Straddling

L,1

P(cid:91)H
Lemma 5.6.15 applies to give

(cid:98)N

A,P(cid:91)H
stop,(cid:52)ω ≤ CSα,A

L,1

augsize

(cid:16)P(cid:91)H

L,1

(cid:17) ≤ CSα,A

augsize (P) ,

and then the Orthogonality Lemma 5.6.4 applies to give

(cid:98)N

A,P(cid:91)H−big
stop,(cid:52)ω

1

≤ sup

L∈H N

A,P(cid:91)H
stop,(cid:52)ω ≤ CSα,A

L,1

augsize (P) ,

since (cid:110)P(cid:91)H

(cid:111)

L × CH,(cid:91)shif t
L(cid:48) ∈ Hk+1 for indented depth k = k (L). The case t = 2 is equally easy.

is mutually orthogonal as P(cid:91)H

L,1 ⊂ CH

L∈L

L(cid:48)

L,1

with L ∈ Hk and

Now we consider the case t ≥ 2, where it is essential to obtain geometric decay in t. We
are reduced by Conclusion 5.6.4.

with S = CH (L), so that for any (I, J) ∈ P(cid:91)H
. But this time we must use the stronger localized

, there

L,t

remind the reader that all of our admissible collections P(cid:91)H
We again apply Lemma 5.6.15 to P(cid:91)H
is H(cid:48) ∈ CH (L) with J (cid:91) ⊂ H(cid:48) (cid:36) I ∈ CH
L
bounds Sα,A;S
with an S-hole, that give

L,t

L,t

A,P(cid:91)H
stop,(cid:52)ω ≤ C

L,t

locsize

(cid:98)N
(cid:16)P(cid:91)H
Sα,A;H(cid:48)

locsize

L,t

(cid:17)2

Sα,A;H(cid:48)

locsize

(cid:17)

,

(cid:16)P(cid:91)H
Pα(cid:16)

L,t

K, 1A\H(cid:48)σ
|K| 1

n

sup

H(cid:48)∈CH(L)

= sup
K∈W∗(H(cid:48))∩C(cid:48)

A

1
|K|σ

t ≥ 0;

(cid:17)

2 (cid:88)

J∈Π

K,aug
2

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

P(cid:91)H

L,t

302

It remains to show that

(cid:88)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

(cid:13)(cid:13)(cid:13)♠2
for t ≥ 2, K ∈ W∗(cid:0)H(cid:48)(cid:1) ∩ C(cid:48)

K,aug
2

P(cid:91)H

J∈Π

L,t

x

J

L2(ω)

A, H(cid:48) ∈ CH (L)

≤ ρ−(t−2)ω(cid:91)P (T (K)) ,

(5.6.40)

so that we then have

Pα(cid:16)

1
|K|σ

≤ ρ−(t−2) 1
|K|σ

(cid:17)

2 (cid:88)
(cid:17)
2

J∈Π

K,aug
2

K, 1A\K σ
|K| 1

n

K, 1A\H(cid:48)σ
|K| 1

n

Pα(cid:16)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

P(cid:91)H

L,t

ω(cid:91)P (T (K)) ≤ ρ−(t−2)Sα,A

augsize (P)2

by (5.6.29), and hence conclude the required bound for N

A,P(cid:91)H
stop,(cid:52)ω, namely that

L,t

(cid:98)N

A,P(cid:91)H
L,t
stop,(cid:52)ω

≤ C sup

H(cid:48)∈CH(L)
(cid:113)

(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) 1

|K|σ

Pα(cid:16)

K, 1A\H(cid:48)σ
|K| 1

n

(cid:17)

2 (cid:88)

J∈Π

K,aug
2

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

P(cid:91)H

L,t

x

sup

K∈W∗(H(cid:48))∩C(cid:48)

A

(5.6.41)

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

≤ C

ρ−(t−2)Sα,A

augsize (P) = C(cid:48)ρ

− t
2Sα,A

augsize (P) .

in one of the Whitney cubes K ∈ W(cid:16)

Remark on lack of usual goodness: To prove (5.6.40), it is essential that the cubes
Hk+2 ∈ Hk+2 at the next indented level down from Hk+1 ∈ CH (L) are each contained
for some Hk+1 ∈ CH (L). And this is
the reason we introduced the indented corona - namely so that 3Hk+2 ⊂ Hk+1 for some

Hk+1(cid:17) ∩ C(cid:48)

A

303

Hk+1 ∈ CH (L), and hence Hk+2 ⊂ K for some K ∈ W(cid:16)

Hk+1(cid:17).

In the argument of

Lacey in [27], the corresponding cubes were good in the usual sense, and so the above triple

property was automatic.

So we begin by fixing K ∈ W∗(cid:16)

Hk+1(cid:17) ∩ C(cid:48)

A

above that each J ∈ ΠK,aug

2

P(cid:91)H

L,t

satisfies

with Hk+1 ∈ CH (L), and noting from the

J (cid:91) ⊂ Hk+t ⊂ Hk+t−1 ⊂ ... ⊂ Hk+2 ⊂ K

for Hk+j ∈ Hk+j uniquely determined by J (cid:91). Thus for t ≥ 2 we have

(cid:88)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

(cid:13)(cid:13)(cid:13)♠2

x

L2(ω)

J∈Π

K,aug
2

P(cid:91)H

L,t

=

(cid:88)
≤ (cid:88)

Hk+t∈Hk+t
Hk+t⊂K

(cid:88)
(cid:16)
(cid:16)

T

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
Hk+t(cid:17)(cid:17)

L,t

J

J∈Π

P(cid:91)H
K,aug
2
J(cid:91)⊂Hk+t

ω(cid:91)P

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

Hk+t∈Hk+t
Hk+t⊂K

In the case t = 2 we are done since the final sum above is at most ω(cid:91)P (T (K)).

Now suppose t ≥ 3. In order to obtain geometric gain in t, we will apply the stopping

criterion (5.6.32) in the following form,

ω(cid:91)P(cid:0)T(cid:0)L(cid:48)(cid:1)(cid:1) = ω(cid:91)P

(cid:88)
L(cid:48)∈CL(L0)

 (cid:91)

L(cid:48)∈CL(L0)

T(cid:0)L(cid:48)(cid:1) ≤ 1

ρ

ω(cid:91)P (T (L0)) ,

for all L0 ∈ L

(5.6.42)

where we have used the fact that the maximal cubes L(cid:48) in the collection

m−1(cid:91)

(cid:8)L(cid:48) ∈ L(cid:96) : L(cid:48) ⊂ L0

(cid:9)

(cid:96)=0

304

for L0 ∈ Lm (that appears in (5.6.32)) are precisely the L-children of L0 in the tree L (the
cubes L(cid:48) above are strictly contained in L0 since ρ > 1 in (5.6.32)), so that

(cid:91)

L(cid:48)∈Γ

L(cid:48) =

(cid:91)
L(cid:48)∈CL(L0)

L(cid:48) where Γ ≡ m−1(cid:91)

(cid:8)L(cid:48) ∈ L(cid:96) : L(cid:48) ⊂ L0

(cid:9) .

(cid:96)=0

(cid:16)

dgeom

L

+t−2 ∈ G

(cid:16)

Hk+2(cid:17)

In order to apply (5.6.42), we collect the pairwise disjoint cubes Hk+t ∈ Hk+t such
they

Hk+2(cid:17) is the geometric depth of Hk+2 in the tree L

that Hk+t ⊂ Hk+2 ⊂ K, into groups according to which cube Lk(cid:48)+t−2 ∈ Gk(cid:48)+t−2
are contained in, where k(cid:48) = dgeom
introduced in (5.6.33). It follows that each cube Hk+t ∈ Hk+t is contained in a unique cube
Hk+2(cid:17)
. Thus we obtain from the previous inequality
≤ (cid:88)
≤ (cid:88)

Hk+t(cid:17)(cid:17)
(cid:88)

Lk(cid:48)+t−2(cid:17)(cid:17)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

Hk+t∈Hk+t
Hk+t⊂K

J∈Π

K,aug
2

P(cid:91)H

L,t

(cid:13)(cid:13)(cid:13)♠2

(cid:88)

x

L2(ω)

(cid:16)

(cid:16)

(cid:16)

(cid:16)

ω(cid:91)P

T

dgeom

+t−2

that

(cid:16)

ω(cid:91)P

T

Lk(cid:48)+t−2∈Gk(cid:48)+t−2
Hk+2(cid:17)
(cid:16)
Lk(cid:48)+t−2⊂Hk+2
where k(cid:48)=dgeom

Hk+2∈Hk+2
Hk+2⊂K

305



and this last expression is equal to

(cid:88)

(cid:88)

Hk+2∈Hk+2
Hk+2⊂K

≤ (cid:88)

Hk+2∈Hk+2
Hk+2⊂K

Lk(cid:48)+t−3∈Gk(cid:48)+t−3
Hk+2(cid:17)
(cid:16)
k(cid:48)+t−3⊂Hk+2
(cid:88)
where k(cid:48)=dgeom

Lk(cid:48)+t−3∈Gk(cid:48)+t−3
Hk+2(cid:17)
(cid:16)
Lk(cid:48)+t−3⊂Hk+2
where k(cid:48)=dgeom

(cid:88)

ω(cid:91)P

Lk(cid:48)+t−2∈Gk(cid:48)+t−2
Hk+2(cid:17)
(cid:16)
Lk(cid:48)+t−2⊂Lk(cid:48)+t−3
Lk(cid:48)+t−3(cid:17)(cid:17)(cid:27)
(cid:16)
(cid:16)
where k(cid:48)=dgeom

T

ω(cid:91)P

(cid:26) 1

ρ

(cid:16)

T

(cid:16)

Lk(cid:48)+t−2(cid:17)(cid:17)



where in the last line we have used (5.6.42) with L0 = Lk(cid:48)+t−3 on the sum in braces. We
then continue (if necessary) with
Lk(cid:48)+t−3(cid:17)(cid:17)

(cid:88)

(cid:88)

(cid:88)

(cid:16)

(cid:16)

T

x

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

ω(cid:91)P

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

J∈Π

K,aug
2

P(cid:91)H

L,t

≤ 1
ρ

≤ 1
ρ2

Hk+2∈Hk+2
Hk+2⊂K

(cid:88)

Hk+2∈Hk+2
Hk+2⊂K

Lk(cid:48)+t−3∈Gk(cid:48)+t−3
Hk+2(cid:17)
(cid:16)
Lk(cid:48)+t−3⊂Hk+2
(cid:88)
where k(cid:48)=dgeom
Lk(cid:48)+t−4∈G
k(cid:48)+t−4
(cid:16)
Hk+2(cid:17)
Lk(cid:48)+t−4⊂Hk+2
where k(cid:48)=dgeom

ω(cid:91)P

(cid:16)

T

(cid:16)

Lk(cid:48)+t−4(cid:17)(cid:17)

...

≤

1
ρt−2

(cid:88)

Hk+2∈Hk+2
Hk+2⊂K

(cid:88)

ω(cid:91)P
Hk+2(cid:17)
Lk(cid:48)∈Gk(cid:48) : Lk(cid:48)⊂Hk+2
where k(cid:48)=dgeom

(cid:16)

(cid:16)

T

(cid:16)

Lk(cid:48)(cid:17)(cid:17)

306

Since Lk(cid:48) ⊂ Hk+2 implies Lk(cid:48)

= Hk+2, we now obtain

(cid:88)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J

(cid:13)(cid:13)(cid:13)♠2

x

L2(ω)

J∈Π

K,aug
2

P(cid:91)H

L,t

(cid:88)

(cid:16)

T

(cid:16)

Hk+2(cid:17)(cid:17)

ω(cid:91)P

1
ρt−2

≤

≤

1
ρt−2 ω(cid:91)P (T (K))

Hk+2∈Hk+2: Hk+2⊂K

which completes the proof of (5.6.40), and hence that of (5.6.41). Finally, an application of

the Orthogonality Lemma 5.6.4 proves (5.6.39).

Proof of the first line in (5.6.36): At last we turn to proving the first line in (5.6.36).

Recalling that T (L) = CH

L (L)\{L}, we consider the collection
(cid:91)
Q(cid:91)H−big
(cid:110)
L∈H
P(cid:91)H−big
(I, J) ∈ P(cid:91)H
(cid:110)
=
P(cid:91)H
(I, J) ∈ P : I ∈ CH
L,0 =

L J ∈ CH,(cid:91)shif t

P(cid:91)H−big

L,0

L,0

=

0

L,0 : there is L(cid:48) ∈ T (L) , J (cid:91) ⊂ L(cid:48) ⊂ I

for some L ∈ H(cid:111)

L

where

and

(cid:111)

, L ∈ H
, L ∈ H

(5.6.43)

and begin by claiming that

(cid:98)N

A,P(cid:91)H−big
L,0
stop,(cid:52)ω

≤ CSα,A

augsize

(cid:16)P(cid:91)H−big

L,0

(cid:17) ≤ CSα,A

L ∈ H.

augsize (P) ,
Lk,i(cid:111)
(cid:110)

To see this, we fix L ∈ H and order the cubes of T (L) =
, where 1 ≤ i ≤ nk
where L0 = L and L1,i are the maximal cubes in L0 and then Lk+1,i are the maximal cubes
inside a cube Lk,j of some previous generation. Then P(cid:91)H−big

can be decomposed as follows,

k,i

L,0

307

remembering that J (cid:91) ⊂ I ⊂ L for (I, J) ∈ P(cid:91)H−big

⊂ P(cid:91)H

L,0

:

L,0

P(cid:91)H−big

L,0

R(cid:91)L
R(cid:91)L

L

k,i
out,in

L

k,i
out,out

R(cid:91)L

L

k,i
in

L

L

=

=

k,i

R(cid:91)L

k,i
out,out

k,i
out,out

 ˙∪

(cid:40)
·(cid:91)
R(cid:91)L
 ·(cid:91)
≡ (cid:110)
(I, J) ∈ P(cid:91)H−big
≡ (cid:110)
(I, J) ∈ P(cid:91)H−big
≡ (cid:110)
(I, J) ∈ P(cid:91)H−big
(cid:110)
(I, J) ∈ P(cid:91)H−big

L,0

L,0

L,0

k,i

=

L,0

(cid:41)
 ˙∪

k,i
in

˙∪ R(cid:91)L

˙∪ R(cid:91)L

L

k,i
out,in

L

 ·(cid:91)

k,i

R(cid:91)L

L

k,i
out,in

 ·(cid:91)

k,i

R(cid:91)L

L

k,i
in

out,in

out,out

Lk−1,i

Lk−1,i

and J (cid:91) ⊂ Lk,i

: I ∈ CL
: I ∈ CL
and J (cid:91) ⊂ Lk,i
: I ∈ CL
and J (cid:91) ∈ CL
: I = Lk−1,i and J (cid:91) ∈ CL

Lk−1,i

Lk−1,i

Lk−1,i

 ;
(cid:111)
(cid:111)
and J (cid:91) ∩ Lk,i = ∅(cid:111)

,

,

and J (cid:91) ∩ Lk−1,i

out = ∅,

(cid:111)

,

where by Lk,i
in

we denote the union of the children of Lk,i that do not touch the boundary of

L, by Lk,i

out,in

the union of the grandchildren of Lk,i that do not touch the boundary of L while

out,out

their father does, and by Lk,i
where in the last line we have used the fact that if I, J (cid:91) ∈ CL
with J (cid:91) ⊂ L(cid:48) ⊂ I, then we must have I = Lk−1,i. All of the pairs (I, J) ∈ P(cid:91)H−big
included in either R(cid:91)L
boundary with L, which contradicts the fact that 3J (cid:91) ⊂ J

the grandchildren of Lk,i that touch the boundary of L and
and there is L(cid:48) ∈ T (L)
are
for some k, since if J (cid:91) ⊃ Lk,i , then J (cid:91) shares

(cid:122) ⊂ I ⊂ L.

or R(cid:91)L

We can easily deal with the ‘in’ collection Qin ≡ ·(cid:83)∞

by applying a trivial case

Lk−1,i

k,i
out,out

, R(cid:91)L

k,i
out,in

k,i
in

L,0

L

L

L

of the (cid:91)Straddling Lemma to R(cid:91)L
of the Orthogonality Lemma to Qin. More precisely, every pair (I, J) ∈ R(cid:91)L
J (cid:91) ⊂ Lk−1,i = I, so that the reduced admissible collection R(cid:91)L

k,i
in

with a single straddling cube, followed by an application

(cid:91)straddles the trivial

satisfies

k,i
in

L

L

k=1R(cid:91)L

L

k,i
in

L

k,i
in

308

Lk−1,i(cid:111), the singleton consisting of just the cube Lk−1,i. Then the inequality
(cid:110)

choice S =

(cid:98)N

A,R(cid:91)L
stop,(cid:52)ω ≤ CSα,A

k,i
in

L

augsize

(cid:32)

(cid:33)

,

k,i
in

L

R(cid:91)L
(cid:40)

(cid:41)

k,i

is mutually orthogonal

R(cid:91)L

L

k,i
in

follows from (cid:91)Straddling Lemma 5.6.15. The collection

since

·(cid:91)

Since

A,(cid:83)

R(cid:91)L

L

k,i
in

⊂ CL

Lk−1,i × CL,(cid:91)shif t
Lk−1,i

∞(cid:88)

nk(cid:88)

k=1

i=1

≤ 1 and

1CL

Lk−1,i

∞(cid:88)

nk(cid:88)

k=1

i=1

≤ 1.

1CL,(cid:91)shif t
Lk−1,i

 ·(cid:91)

k,i

R(cid:91)L

L

k,i
in

 is paired with a single

R(cid:91)L

L

k,i
in

k,i

is reduced and admissible (each J ∈ Π2

I, namely the top of the L-corona to which J (cid:91) belongs), the Orthogonality Lemma 5.6.4
applies to obtain the estimate

L

k,i
in

(cid:98)N

stop,(cid:52)ω

k,i R(cid:91)L

(cid:98)N
Now we turn to estimating the norm of the ‘out-in’ collection Qout,in ≡(cid:91)

A,R(cid:91)L
stop,(cid:52)ω ≤ C sup
1≤k
1≤i≤nk

≤ sup
1≤k
1≤i≤nk

≤ CSα,A

Sα,A

augsize

k,i
in

k,i
in

L

L

augsize

(cid:33)

(cid:32)
R(cid:91)L

(cid:16)P(cid:91)H−big

L,0

(cid:17)

(5.6.44)

R(cid:91)L

L

k,i
out,in

. First

k,i

since R(cid:91)L

if (I, J) ∈ R(cid:91)L

out,in ∈ CA,restrict

L

A

k,i
out,in

is reduced, i.e. doesn’t
we note that Lk,i
contain any pairs (I, J) with J (cid:91) ⊂ A(cid:48) for some A(cid:48) ∈ CA (A). Next we note that Qout,in
is admissible since if J ∈ Π2Qout,in, then J ∈ Π2R(cid:91)L
for a unique index (k, i), and of
course R(cid:91)L
is admissible, so that the cubes I that are paired with J are tree-connected.
Thus we can apply the Straddling Lemma 5.6.15 to the reduced admissible collection Qout,in

k,i
out,in

k,i
out,in

k,i
out,in

L

L

L

309

L(cid:48)∈Lk,i L(cid:48)(cid:17) ∩ CA,restrict
(cid:16)Qout,in(cid:17) ≤ CSα,A

A

to obtain the estimate

(cid:16)P(cid:91)H−big

L,0

(cid:17) (5.6.45)

augsize

(cid:83)

k,i

with the ‘straddling’ set S ≡(cid:16)(cid:83)
A,(cid:83)∞
(cid:98)N
= (cid:98)NA,Qout,in
k=1 R(cid:91)L

k,i
out,in

L

augsize

stop,(cid:52)ω

stop,(cid:52)ω ≤ CSα,A
A,(cid:83)
As for the remaining ‘out-out’ form |B|
R(cid:91)L
implies that either J (cid:91) = L(cid:48) (cid:36) J
an endpoint with L, or that J (cid:91) = L(cid:48)(cid:48) ∈ L(cid:48)
(I, J) ∈ R(cid:91)L

, then either J (cid:91) ⊂ L(cid:48) ∈ Lk,i

k,i
out,out

, then

out,out

L

L

k,i
out,out

k,i R(cid:91)L

k,i
out,out

L

(f, g),

(cid:122) or J

stop,(cid:52)ω
(cid:36) J

if the cube pair (I, J) ∈
. But J (cid:91) ⊂ L(cid:48) (cid:36) J
(cid:122)
(cid:122) ⊂ I ⊂ L, which is impossible since J (cid:91) cannot share
= Lk,i. So we conclude that if

(cid:122) ⊂ L(cid:48) ∈ Lk,i

and J

out,out

(cid:122)

in

either J

(cid:122) ⊂ Lk,i

out,out

(cid:122)

or {J

= Lk,i and J ⊂ Lk,i

out,out

}.

(5.6.46)

In either case in (5.6.46), there is a unique cube K [J] ∈ W (L) that contains J. It follows
that there are now two remaining cases:

,

L

A

Case 1: K [J] ∈ C(cid:48)
Case 2: K [J] ⊂ A(cid:48) (cid:36) I for some A(cid:48) ∈ CA (A).
However, since J (cid:91) ⊂ K[J], as K[J] is the maximal cube whose triple is contained in
is reduced, the pairs (I, J) in Case 2 lie in the ‘corona straddling’

L, and since R(cid:91)L
cor that was removed from all A-admissible collections in (5.6.26) of Conclusion
collection P A
5.6.4 above, and thus there are no pairs in Case 2 here. Thus we conclude that K [J] ∈ C(cid:48)
.
A
. To see this, suppose
for some k ≥ 1, 1 ≤ i ≤ nk. Then by (5.6.46) we have both that
implies that 3K [J] ⊂ Lk,i ⊂ I as

We now claim that 3K [J] ⊂ I for all pairs (I, J) ∈(cid:83)

that (I, J) ∈ R(cid:91)L
K [J] ⊂ Lk,i

and Lk,i (cid:36) I. But then K [J] ⊂ Lk,i

k,i R(cid:91)L
Lk

k,i
out,out

k,i
out,out

out,out

L

out,out

out,out

310

is admissible, since if J ∈ Π2Qout,out

claimed.

Now the ‘out-out’ collection Qout,out ≡(cid:91)
and Ij ∈ Π1Qout,out with(cid:0)Ij, J(cid:1) ∈ Qout,out for j = 1, 2, then Ij ∈ CL

k,i
out,out

R(cid:91)L

k,i

L

for some kj and
L
i and all of the cubes I ∈ [I1, I2] lie in one of the coronas CL
for k between k1 and k2.
⊂ Qout,out
And of course for those coronas we have J ∈ Lk,i
and we have proved the required connectedness. From the containment 3K [J] ⊂ I ⊂ L
, we now see that the reduced admissible collection Qout,out

for all (I, J) ∈ (cid:83)

. Thus (I, J) ∈ R(cid:91)L
Lk

Lk−1,i

kj−1,i

out,out

out,out

k,i R(cid:91)L

L

k,i
out,out

substraddles the cube L. Hence the Substraddling Lemma 5.6.17 yields the bound

A,(cid:83)

(cid:98)N

k,i R(cid:91)L

L

k,i
out,out

stop,(cid:52)ω

= (cid:98)NA,Qout,out

stop,(cid:52)ω ≤ CSα,A

augsize

(cid:16)Qout,out(cid:17) ≤ CSα,A

augsize

(cid:16)P(cid:91)H−big

L,0

(cid:17)

.

(5.6.47)

Combining the bounds (5.6.44), (5.6.45) and (5.6.47), we obtain (5.6.43).

Finally, we observe that the collections P(cid:91)H−big

L,0

themselves are mutually orthogonal,

namely

P(cid:91)H−big
(cid:88)

L,0

1CH

L

L∈H

⊂ CH

L × CH,(cid:91)shif t
≤ 1 and (cid:88)

L

,

1CH,(cid:91)shif t

L

L∈H

L ∈ H ,
≤ 1.

Thus an application of the Orthogonality Lemma 5.6.4 shows that

(cid:98)N

A,Q(cid:91)H−big
stop,(cid:52)ω

0

(cid:98)N

A,P(cid:91)H−big
L,0
stop,(cid:52)ω

≤ sup
L∈L

≤ CSα,A

augsize (P) .

Altogether, the proof of Proposition 5.6.19 is now complete.

This finishes the proofs of the inequalities (5.6.7) and (5.6.1).

311

5.7 Finishing the proof

At this point we have controlled, either directly or probabilistically, the norms of all of the

forms in our decompositions - namely the disjoint, nearby, far below, paraproduct, neighbour,

broken and stopping forms - in terms of the Muckenhoupt, energy and functional energy con-

ditions, along with an arbitrarily small multiple of the operator norm. Thus it only remains

to control the functional energy condition by the Muckenhoupt and energy conditions, since
σ f ) gdω = Θ (f, g) + Θ∗ (f, g) with the further decompositions above, we

will have shown that for any fixed tangent line truncation of the operator T α

σ we have

then, using (cid:82) (T α
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

σ f ) gdω

(T α

(cid:12)(cid:12)(cid:12)(cid:12) = ED

Ω EG

Ω

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

(T α

σ f ) gdω

3(cid:88)

(cid:12)(cid:12)(cid:12)(cid:12) ≤ED
≤(cid:0)CηNT Vα + ηNT α(cid:1)(cid:107)f(cid:107)L2(σ) (cid:107)g(cid:107)L2(ω)

(|Θi (f, g)| + |Θ∗

i (f, g)|)

Ω EG

Ω

i=1

for f ∈ L2 (σ) and g ∈ L2 (ω), for an arbitarily small positive constant η > 0, and a
correspondingly large finite constant Cη. Note that the testing constants TT α and TT α,∗
in NT Vα already include the supremum over all tangent line truncations of T α, while the
operator norm NT α on the left refers to a fixed tangent line truncation of T α. This gives

NT α =

(cid:107)f(cid:107)

sup
L2(σ)

(cid:107)g(cid:107)

=1

sup
L2(ω)

=1

(T α

σ f ) gdω

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12) ≤ CηNT Vα + ηNT α,

and since the truncated operators have finite operator norm NT α, we can absorb the term
ηNT α into the left hand side for η < 1 and obtain NT α ≤ C(cid:48)
ηNT Vα for each tangent line
truncation of T α. Taking the supremum over all such truncations of T α finishes the proof

of Theorem 5.1.5.

The task of controlling functional energy is taken up in Appendix, after first establish-

312

ing weak frame and weak Riesz inequalities for martingale and dual martingale differences
(except for the lower weak Riesz inequality for the martingale difference (cid:52)µ,b

).

Q

313

APPENDIX

314

Appendix

Control of functional energy

Now we arrive at one of the main propositions used in the proof of our theorem. This

result is proved independently of the main theorem, and only using the results on dual

martingale differences established in the previous appendix. The organization of the proof is

almost identical to that of the corresponding result in [64, pages 128-151], together with the

modifications in [66, pages 348-360] to accommodate common point masses, but we repeat

the organization here with modifications required for the use of two independent grids, and
(cid:122). Recall that the functional
α (D,G) in (5.5.5), 0 ≤ α < n, namely the best constant in the

the appearance of weak goodness entering through the cubes J
energy constant Fα = Fb∗
inequality (see (.0.7) below for the definition of W (F )),

(cid:88)

(cid:88)

F∈F

M∈W(F )

Pα (M, hσ)

|M| 1

n

2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

F

CG,shif t

x

;M

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

≤ Fα(cid:107)h(cid:107)L2(σ) ,

(.0.1)

depends on the grids D and G, the goodness parameter ε > 0 used in the definition
, and on the family of martingale differences
of J

(cid:111)
(cid:122) through the shifted corona CG,shif t

associated with x ∈ L2

F

(cid:110)(cid:52)ω,b∗
ferences(cid:110)(cid:3)σ,b

J∈G

J

(cid:111)

I

I∈D

loc (ω), but not on the family of dual martingale dif-
, since the function h ∈ L2 (σ) appearing in the definition of functional
I h. Finally, we emphasize that

energy is not decomposed as a sum of pseudoprojections (cid:3)σ,b

315

the pseudoprojection

Qω,b∗
CG,shif t

F

;M

≡

(cid:88)
J∈CG,shif t

F

(cid:52)ω,b∗

J

: J⊂M

here uses the shifted restricted corona in

CG,shif t

F

(cid:110)
; K ≡ (cid:110)
J ∈ G : J
J ∈ CG,shif t

=

F

CG,shif t

F

(cid:122) ∈ CD

F

,
: J ⊂ K

(cid:111)

,

(cid:111)

(.0.2)

(.0.3)

(cid:122) is defined using the body of a cube as in Definition 5.2.8, and where we have
2 P
(c.f. ΠK
(cid:122) ⊂ K). Moreover, recall from
in (5.1.39), that for any subset H of the grid

where J
defined here the ‘restriction’ CG,shif t
in Definition 5.6.5, which uses the stronger requirement J
Notation in 5.1.13.2 and the definition of (cid:53)ω
G,

; K to the cube K of the corona CG,shif t

F

F

J

so that we never need to consider the norm squared
jection Qω,b∗

CG,shif t

F

;M

x, something for which we have no lower Riesz inequality. Note moreover

(cid:13)(cid:13)(cid:13)Qω,b∗

H x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

≡ (cid:88)
(cid:88)

J∈H

=

J∈H

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:18)(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)2

x

x

J

J

L2(ω)

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:98)(cid:53)ω

J (x − z)

(cid:13)(cid:13)(cid:13)2

L2(ω)

(cid:19)

,

of the pseudopro-

+ inf
z∈R

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

F

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2

CG,shif t

x

;M

L2(ω)

316

that for J ∈ G and an arbitrary cube K, we have by the frame inequality in (5.1.51) ,

(cid:88)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:98)(cid:53)ω

J (x − z)

x

J

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

L2(ω)

L2(ω)

(cid:88)

J∈G: J⊂K

J∈G: J⊂K

inf
z∈R

(cid:46) (cid:13)(cid:13)x − mω
≤ (cid:88)

K

J∈G: J⊂K
(cid:46) (cid:107)(x − p)(cid:107)2

(cid:13)(cid:13)2
J {(x − p) 1K (x)}(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:98)(cid:53)ω

L2(1K ω) ,

p ∈ K,

,

L2(1K ω)

L2(ω)

(.0.4)

(.0.5)

where the second line follows from (5.1.40).

Important note If J ∈ CG,shif t

, then in particular J (cid:98)ρ,ε F with ρ =
above notation 5.5.8 , and so J ∩ M (cid:54)= ∅ for a unique M ∈ W (F ).

F

(cid:105) as mentioned

(cid:104) 3

ε

We will show that, uniformly in pairs of grids D and G, the functional energy constants
-
and the large energy constant Eα
2

Fα (D,G) in (5.5.5) are controlled by Aα
actually the proof shows that we have control by the Whitney plugged energy constant as

, Aα,punct

2

2

defined in (.0.16) below. More precisely this is our control of functional energy proposition.

Proposition .0.1. For all grids D and G, and ε > 0 sufficiently small, we have

Fb∗
α (D,G) (cid:46) Eα
2 +
Fb,∗
α (G,D) (cid:46) Eα,∗
2 +

(cid:113)Aα
(cid:113)Aα

2 +

(cid:113)Aα,∗
(cid:113)Aα,∗

2 +

2 +

2 +

(cid:113)
(cid:113)
Aα,punct
,
Aα,∗,punct

2

2

,

with implied constants independent of the grids D and G.

In order to prove this proposition, we first turn to recalling these more refined notions of

energy constants.

317

Various energy conditions

In this subsection we recall various refinements of the strong energy conditions appearing in

the main theorem above. Variants of this material already appear in earlier papers, but we

repeat it here both for convenience and in order to introduce some arguments we will use

repeatedly later on. These refinements represent the ‘weakest’ energy side conditions that

suffice for use in our proof, but despite this, we will usually use the large energy constant Eα
2
in estimates to avoid having to pay too much attention to which of the energy conditions we

need to use - leaving the determination of the weakest conditions in such situations to the

interested reader. We begin with the notion of ‘deeply embedded’. Recall that the goodness
parameter r ∈ N is determined by ε > 0 in (5.2.16), and that 0 < ε < 1

1

.

n+1 <

n+1−α

For arbitrary cubes in J, K ∈ P, we say that J is (ρ, ε)-deeply embedded in K, which we

write as J (cid:98)ρ,ε K, when J ⊂ K and both

(cid:96) (J) ≤ 2−ρ(cid:96) (K) ,

d (J, ∂K) ≥ 2(cid:96) (J)ε (cid:96) (K)1−ε .

(.0.6)

Note that we use the boundary of K for the definition of J (cid:98)ρ,ε K, rather than the skeleton
or body of K, which would result in a more restrictive notion of (ρ, ε)-deeply embedded. We
will use this notion for the purpose of grouping ε− good cubes into the following collections.
Fix grids D and G. For K ∈ D, define the collections,

M(ρ,ε)−deep,G (K) ≡ (cid:8)J ∈ G : J is maximal w.r.t J (cid:98)ρ,ε K(cid:9) ,
M(ρ,ε)−deep,D (K) ≡ (cid:8)M ∈ D : M is maximal w.r.t M (cid:98)ρ,ε K(cid:9) ,

W (K) ≡ {M ∈ D : M is maximal w.r.t 3M ⊂ K}

(.0.7)

318

where the first two consist of maximal (ρ, ε)-deeply embedded dyadic G-subcubes J, respec-
tively D-subcubes M, of a D-cube K, and the third consists of the maximal D-subcubes M
whose triples are contained in K.

Let γ > 1. Then the following bounded overlap property holds where

M(ρ,ε)−deep (K) can be taken to be either M(ρ,ε)−deep,G (K) or M(ρ,ε)−deep,D (K) or W (K)
throughout.

Lemma .0.2. Let 0 < ε ≤ 1 < γ ≤ 1 + 4 · 2ρ(1−ε). Then

(cid:88)

J∈M(ρ,ε)−deep(K)

1γJ ≤ β1

(cid:83)

J∈M(ρ,ε)−deep(K)



γJ

(.0.8)

holds for some positive constant β depending only on γ, ρ and ε. In addition γJ ⊂ K for all
J ∈ M(ρ,ε)−deep (K), and consequently
(cid:88)

(.0.9)

1γJ ≤ β1K .

J∈M(ρ,ε)−deep(K)

A similar result holds for W (K).

Proof. We suppose 0 < ε < 1 and leave the simpler case ε = 1 for the reader. To prove
(.0.8), we first note that there are at most 2n(ρ+1)−1
cubes J contained in K for which
(cid:96) (J) > 2−ρ(cid:96) (K). On the other hand, the maximal (ρ, ε)-deeply embedded subcubes J of K
also satisfy the comparability condition

2n−1

2(cid:96) (J)ε (cid:96) (K)1−ε ≤ d (J, ∂K) ≤ d (πJ, ∂K) + (cid:96) (J) ≤ 2 (2(cid:96) (J))ε (cid:96) (K)1−ε + (cid:96) (J)

≤ 4(cid:96) (J)ε (cid:96) (K)1−ε + (cid:96) (J) .

319

Now with 0 < ε < 1 and γ > 1 fixed, let y ∈ K. Then if y ∈ γJ, we have

2(cid:96) (J)ε (cid:96) (K)1−ε ≤ d (J, ∂K) ≤ γ(cid:96) (J) + d (γJ, ∂K)

Now assume that (cid:96)(J)

(cid:96)(K) ≤(cid:16) 1

γ

≤ γ(cid:96) (J) + d (y, ∂K) .

(cid:17) 1
1−ε . Then we have γ(cid:96) (J) ≤ (cid:96) (J)ε (cid:96) (K)1−ε and so

(cid:96) (J)ε (cid:96) (K)1−ε ≤ d (y, ∂K) .

But we also have

d (y, ∂K) ≤ γ(cid:96) (J) + d (J, ∂K) ≤ γ(cid:96) (J) + 4(cid:96) (J)ε (cid:96) (K)1−ε + (cid:96) (J) ≤ 6(cid:96) (J)ε (cid:96) (K)1−ε ,

and so altogether, under the assumption that (cid:96)(J)

d (y, ∂K) ≤ (cid:96) (J)ε (cid:96) (K)1−ε ≤ d (y, ∂K) ,

1
6

(cid:32)

i.e.

1
6

d (y, ∂K)
(cid:96) (K)1−ε

(cid:33) 1
ε ≤ (cid:96) (J) ≤

γ

(cid:96)(K) ≤(cid:16) 1
(cid:32)

(cid:17) 1

1−ε , we have

d (y, ∂K)
(cid:96) (K)1−ε

ε

,

(cid:33) 1
(cid:96)(K) ≤(cid:16) 1

γ

which shows that the number of J’s satisfying y ∈ γJ and (cid:96)(J)
On the other hand, the number of J’s contained in K satisfying y ∈ γJ and (cid:96)(J)
is at most C(cid:48) 1

1−ε (1 + log2 γ). This proves (.0.8) with

(cid:17) 1
(cid:17) 1
1−ε is at most C(cid:48) 1
.
ε
1−ε

(cid:16) 1

(cid:96)(K) >

γ

2n(ρ+1) − 1

2n − 1

β =

+ C(cid:48) 1
ε

+ C(cid:48)

1
1 − ε

(1 + log2 γ) .

320

In order to prove (.0.9) it suffices, by (.0.8), to prove γJ ⊂ K for all J ∈ M(ρ,ε)−deep (K).

But J ∈ M(ρ,ε)−deep (K) implies

2(cid:96) (J)ε (cid:96) (K)1−ε ≤ d (J, ∂K) = d (cJ , ∂K) − 1

2

(cid:96) (J) .

We wish to show γJ ⊂ K, which is implied by

(cid:96) (J) ≤ d (cJ , Kc) = d (J, ∂K) +

γ

1
2

1
2

(cid:96) (J) .

But we have

d (J, ∂K) +

1
2

(cid:96) (J) ≥ 2(cid:96) (J)ε (cid:96) (K)1−ε +

1
2

(cid:96) (J) ,

and so it suffices to show that

2(cid:96) (J)ε (cid:96) (K)1−ε +

(cid:96) (J) ≥ γ

1
2

1
2

(cid:96) (J) ,

which is equivalent to

γ − 1 ≤ 4(cid:96) (J)ε−1 (cid:96) (K)1−ε .

But the smallest that (cid:96) (J)ε−1 (cid:96) (K)1−ε can get for J ∈ M(ρ,ε)−deep (K) is 2ρ(1−ε) ≥ 1, and
so γ ≤ 1 + 4 · 2ρ(1−ε) implies γ − 1 ≤ 4(cid:96) (J)ε−1 (cid:96) (K)1−ε, which completes the proof.

The reader can easily verify the same argument works for the Whitney collection W (K).

Now we recall the notion of alternate dyadic cubes from [64], which we rename augmented

dyadic cubes here.

321

Definition .0.3. Given a dyadic grid D, the augmented dyadic grid AD consists of those
cubes I whose dyadic children I(cid:48) belong to the grid D.

Of course an augmented grid is not actually a grid because the nesting property fails,

but this terminology should cause no confusion. These augmented grids will be needed in

order to use the ‘prepare to puncture’ argument (introduced in [66]) at several places below.

Now we proceed to recall certain of the definitions of various energy conditions from [62]

and [64]. While these definitions are not explicitly used in the proof of functional energy,

some of the arguments we give to control them will be appealed to later, and so we take the

time to develop these definitions in detail.

Whitney energy conditions

The following definition of Whitney energy condition uses the Whitney decomposition
M(ρ,1)−deep,D (Ir) into D-dyadic cubes in which ε = 1, as well as the ‘large’ pseudoprojec-
tions

K ≡ (cid:88)

Qω,b∗

J∈G: J⊂K

(cid:52)ω,b∗

J

.

(.0.10)

Definition .0.4. Suppose σ and ω are locally finite positive Borel measures on Rn and fix
γ > 1. Then the Whitney energy condition constant E α,W hitney

is given by

2

Pα(cid:16)

M, 1I\γM σ
|M| 1

n

(cid:17)

2(cid:13)(cid:13)(cid:13)Qω,b∗

M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

,

(cid:16)E α,W hitney

2

(cid:17)2 ≡ supD,G sup

I= ˙∪Ir

1
|I|σ

∞(cid:88)

(cid:88)

r=1

M∈W(Ir)

where supD,G supI= ˙∪Ir is taken over

1. all dyadic grids D and G,

2. all D-dyadic cubes I,

322

3. and all partitions {Ir}N or ∞

r=1

of the cube I into D-dyadic subcubes Ir.

If the parameter γ > 1 above is chosen sufficiently close to 1, then the collection of

cubes {γM}M∈W(Ir)
E α,W hitney

2

is controlled by the strong energy constant E α

in (5.1.8),

2

has bounded overlap β by (.0.9), and the Whitney energy constant

E α,W hitney

2

(cid:46) E α
2 .

Indeed, to see this, fix a decomposition of a cube

·(cid:91)

I =

1≤r<∞

·(cid:91)

M∈W(Ir)

M

(.0.11)

(.0.12)

as in Definition .0.4. Then consider the subdecomposition

·(cid:91)

I ⊃

·(cid:91)

1≤r<∞

M∈W(Ir)

M

of the cube I given by the collection of cubes,

I ≡

We then have

∞(cid:88)

(cid:88)

r=1

M∈W(Ir)

(E α
2 )2 ≥ 1
|I|σ

·(cid:91)

1≤r<∞W (Ir) .

Pα (M, 1I σ)

|M| 1

n

2(cid:13)(cid:13)x − mω

M

(cid:13)(cid:13)2

L2(1M ω) .

323

Now Pα (M, 1I σ) ≥ Pα(cid:16)

(cid:17) and from (.0.4),
(cid:38)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)2

L2(1M ω)

M x

M, 1I\γM σ

(cid:13)(cid:13)x − mω

M

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

,

and combining these two inequalities, we obtain that

∞(cid:88)

(cid:88)

r=1

M∈W(Ir)

2 )2 ≥ c
(E α

1
|I|σ

Pα(cid:16)

M, 1I\γM σ
|M| 1

n

(cid:17)

2(cid:13)(cid:13)(cid:13)Qω,b∗

M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

.

Thus we conclude that

∞(cid:88)

(cid:88)

r=1

M∈W(Ir)

1
|I|σ

Pα(cid:16)

M, 1I\γM σ
|M| 1

n

(cid:17)

2(cid:13)(cid:13)(cid:13)Qω,b∗

M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

≤ C
c

β (E α

2 )2 ,

and taking the supremum over all decompositions (.0.12) as in Definition .0.4, we obtain

(.0.11).

There is a similar definition for the dual (backward) Whitney energy conditions that

simply interchanges σ and ω everywhere. These definitions of the Whitney energy conditions

depend on the choice of γ > 1.

Commentary on proofs We now introduce a number of results concerning partial plug-

ging of the hole for Whitney energy conditions.

Note that we can ‘partially’ plug the γ-hole in the Poisson integral Pα(cid:16)

(cid:17) for

J, 1I\γJ σ

E α,W hitney

2

using the offset Aα
2

condition and the bounded overlap property (.0.9). Indeed,

324

define

(cid:16)E α,W hitneypartial
∞(cid:88)
≡ supD,G sup
I= ˙∪Ir

1
|I|σ

2

r=1

(cid:17)2

(cid:88)

M∈W(Ir)

Pα(cid:16)

M, 1I\M σ
|M| 1

n

(.0.13)

(cid:17)

2(cid:13)(cid:13)(cid:13)Qω,b∗

M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

.

Recall from (.0.9) that

γM ⊂ Ir for all M ∈ W (Ir) provided γ ≤ 5.

At this point we need the following analogues of the ‘energy Aα
2

conditions’ from [66], which

we denote by Aα,energy

2

and Aα,∗,energy

2

, and define by

,

(.0.14)

(cid:13)(cid:13)(cid:13)Qω,b∗

Q

(cid:96)(Q)

x

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qσ,b

Q

|Q|1− α

n

|Q|σ
|Q|1− α

n

(cid:13)(cid:13)(cid:13)♠2

L2(σ)

.

L2(ω)

x

(cid:96)(Q)

|Q|1− α

n

Aα,energy

2

(σ, ω) ≡ sup
Q∈P

Aα,∗,energy

2

(σ, ω) ≡ sup
Q∈P

|Q|ω
|Q|1− α

n

325

Then if γ ≤ 5, we have

2

(cid:17)2

(cid:16)E α,W hitneypartial
∞(cid:88)
(cid:46) supD,G sup
I= ˙∪Ir
∞(cid:88)

1
|I|σ

r=1

(cid:88)
(cid:88)

M∈W(Ir)

1
|I|σ
(cid:17)2

+ supD,G sup
(cid:46) (cid:16)E α,W hitney
I= ˙∪Ir
(cid:17)2
(cid:46) (cid:16)E α,deep

2

2

r=1

M∈W(Ir)
1
|I|σ

+ supD,G sup
I= ˙∪Ir

+ βAα,energy

,

2

Pα(cid:16)
Pα(cid:16)
∞(cid:88)

M, 1I\γM σ
|M| 1

n

M, 1γM\M σ
|M| 1

n

(cid:88)

(cid:17)

2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:17)
2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2

M x

M x

L2(ω)

L2(ω)

Aα,energy

2

|γM|σ

(.0.15)

r=1

M∈W(Ir)

by (.0.9).

Plugged energy conditions

term Pα(cid:16)

M, 1I\M σ

We continue to recall some results from [66] and [67] that we will use repeatedly here.
and Aα,∗,punct
to control the plugged energy conditions, where the hole in the argument of the Poisson

For example, we will use the punctured Muckenhoupt conditions Aα,punct

2

2

(cid:17) in the partially plugged energy condition above, is replaced with the
(cid:17)2 ≡ supD,G sup

Pα (M, 1I σ)

2(cid:13)(cid:13)(cid:13)Qω,b∗

(cid:13)(cid:13)(cid:13)♠2

∞(cid:88)

(cid:88)

M x

1
|I|σ

L2(ω)

.

I= ˙∪Ir

|M| 1

n

r=1

M∈W(Ir)

(cid:16)E α,W hitneyplug

2

‘plugged’ term Pα (M, 1I σ), for example

By an argument similar to that in (.0.15), we obtain

E α,W hitneyplug

2

(cid:46) E α,W hitneypartial

2

+ Aα,energy

2

.

326

(.0.16)

(.0.17)

We first show that the punctured Muckenhoupt conditions Aα,punct

2

and Aα,∗,punct

2

control

respectively the ‘energy Aα
2

conditions’ in (.0.14). We will make reference to the proof of the

next lemma (for the T 1 theorem this is from [66, Lemma 3.2 on page 328.]) several times in

the sequel. We repeat the proof from [66, Lemma 3.2 on page 328.] but with modifications

to accommodate the differences that arise here in the setting of a local T b theorem. Recall
that P(σ,ω)

is defined below (5.1.6) above.

Lemma .0.5. For any positive locally finite Borel measures σ, ω we have

2

Aα,energy
Aα,∗,energy

2

(σ, ω) (cid:46) Aα,punct
(σ, ω) (cid:46) Aα,∗,punct

2

(σ, ω) ,

(σ, ω) .

2

(cid:16)

(cid:17) in (5.1.6). If ω

(cid:16)

Q, P(σ,ω)

(cid:17) ≥

Q, P(σ,ω)

Proof. Fix a cube Q ∈ D. Recall the definition of ω
2 |Q|ω

, then we trivially have

1

(cid:13)(cid:13)(cid:13)Qω,b∗

Q

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

(cid:96)(Q)

|Q|1− α

n

|Q|σ
|Q|1− α

n

(cid:46)

n

|Q|σ
|Q|ω
(cid:17)
(cid:16)
|Q|1− α
|Q|1− α
Q, P(σ,ω)
ω
|Q|1− α

n

n

≤ 2

|Q|σ
|Q|1− α

n

≤ 2Aα,punct

2

(σ, ω) .

On the other hand, if ω

(cid:16)

Q, P(σ,ω)

(cid:17)

< 1

2 |Q|ω

then there is a point p ∈ Q∩ P(σ,ω)

such that

ω ({p}) >

|Q|ω ,

1
2

and consequently, p is the largest ω-point mass in Q. Thus if we define (cid:101)ω = ω − ω ({p}) δp,

327

then we have

(cid:16)

Q, P(σ,ω)

(cid:17)

= |Q|(cid:101)ω .

ω

Now we observe from the construction of martingale differences that

(cid:52)(cid:101)ω,b∗

J

= (cid:52)ω,b∗

J

,

for all J ∈ D with p /∈ J.

So for each s ≥ 0 there is a unique cube Js ∈ D with (cid:96) (Js) = 2−s(cid:96) (Q) that contains the
point p. Now observe that, just as for the Haar projection, the one-dimensional projection
(cid:52)ω,b∗
is given by (cid:52)ω,b∗
Js
this cube we then have

for a unique up to ± unit vector hω,b∗

hω,b∗

hω,b∗

. For

(cid:68)

(cid:69)

f =

, f

Js

Js

Js

Js

ω

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

Js

(cid:13)(cid:13)(cid:13)2

x

L2(ω)

=

ω

=

Js

Js

, x

(cid:69)

, x − p

(cid:12)(cid:12)(cid:12)(cid:68)
(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:68)
(cid:12)(cid:12)(cid:12)2
(cid:69)
hω,b∗
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)
L2((cid:101)ω) ≤(cid:13)(cid:13)(cid:13)hω,b∗
≤ (cid:13)(cid:13)(cid:13)hω,b∗
(cid:13)(cid:13)1Js (x − p)(cid:13)(cid:13)2
≤ (cid:96) (Js)2 |Js|(cid:101)ω ≤ 2−2s(cid:96) (Q)2 |Q|(cid:101)ω ,

hω,b∗
hω,b∗
(x) (x − p) dω (x)
(cid:13)(cid:13)(cid:13)2
L2((cid:101)ω)

hω,b∗

Js

Js

Js

Js

Js

Js

=

=

ω

(cid:12)(cid:12)(cid:12)(cid:12)2
(x) (x − p) d(cid:101)ω (x)
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)1Js (x − p)(cid:13)(cid:13)2
L2((cid:101)ω)

L2(ω)

as well as

(cid:13)(cid:13)(cid:13)(cid:98)∇ω

(cid:13)(cid:13)(cid:13)2

inf
z∈R

Js (x − z)

L2(ω)

L2(cid:16)

(cid:46) (cid:107)(x − p)(cid:107)2
≤ 2−2s(cid:96) (Q)2 |Q|(cid:101)ω ,

1Js ω

L2(cid:16)
(cid:17) = (cid:107)(x − p)(cid:107)2

1Js(cid:101)ω

(cid:17) ≤ (cid:96) (Js)2 |Js|(cid:101)ω

328

(.0.18)



from (.0.4). Thus we can estimate

x

Q

1

1

L2(ω)

x

J

≤

=

(cid:96) (Q)

L2(ω)

(cid:96) (Q)2

(cid:96) (Q)2

+ inf
z∈R

J∈D: J⊂Q

(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

(cid:13)(cid:13)(cid:13)(cid:13)♠2
 (cid:88)
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)2
(cid:32) (cid:88)
(cid:13)(cid:13)(cid:13)(cid:52)(cid:101)ω,b∗
(cid:13)(cid:13)(cid:13)2
L2((cid:101)ω)
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:98)∇ω
Js (x − z)
(cid:32)(cid:13)(cid:13)(cid:13)Q(cid:101)ω,b∗
(cid:13)(cid:13)(cid:13)♠2
∞(cid:88)
2−2s(cid:96) (Q)2 |Q|(cid:101)ω
L2((cid:101)ω)
(cid:32)
(cid:33)
∞(cid:88)
(cid:96) (Q)2 |Q|(cid:101)ω +
2−2s(cid:96) (Q)2 |Q|(cid:101)ω
(cid:17)
(cid:16)
≤ 3|Q|(cid:101)ω = 3ω

J∈D: p /∈J⊂Q

(cid:13)(cid:13)(cid:13)(cid:98)∇ω
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
∞(cid:88)
(cid:33)
(cid:33)

s=0

Js

Q, P(σ,ω)

+ inf
z∈R

(cid:96) (Q)2

(cid:46)

(cid:46)

(cid:96) (Q)2

s=0

,

Q x

x

J

+

s=0

L2(ω)

1

1

+

Js (x − z)

L2(ω)

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

x

L2(ω)

and so

(cid:13)(cid:13)(cid:13)Qω,b∗

Q

x

(cid:96)(Q)

|Q|1− α

n

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(cid:17)

(cid:16)
Q, P(σ,ω)
|Q|1− α

n

|Q|σ
|Q|1− α

n

|Q|σ
|Q|1− α

n

(cid:46) 3ω

≤ 3Aα,punct

2

(σ, ω) .

Now take the supremum over Q ∈ D to obtain Aα,energy
inequality follows upon interchanging the measures σ and ω.

2

(σ, ω) (cid:46) Aα,punct

(σ, ω). The dual

2

We isolate a simple but key fact that will be used repeatedly in what follows:

(cid:88)

Q∈D: Q⊂P

(cid:96) (Q)2 |Q|µ

(cid:46) (cid:96) (P )2 |P|µ ,

for P ∈ D and µ a positive measure.

(.0.19)

Indeed, to see (.0.19), simply pigeonhole the length of Q relative to that of P and sum. The

329

next corollary follows immediately from Lemma .0.5, (.0.15) and (.0.17).

Corollary .0.6. Provided 1 < γ ≤ 5,

E α,W hitneyplug

2

(cid:46) E α,W hitneypartial

2

+ Aα,punct

2

(cid:46) E α,W hitney

2

+ Aα,punct

2

,

and similarly for the dual plugged energy condition.

Using Lemma .0.5 we can control the ‘plugged’ energy Aα

conditions:

Aα,∗,energyplug

2

(σ, ω) ≡ sup
Q∈P

Pα (Q, ω)

Lemma .0.7. We have

Aα,energyplug

2

(σ, ω) ≡ sup
Q∈P

L2(ω)

Pα (Q, σ) ,

(cid:13)(cid:13)(cid:13)Qω,b∗

Q

2

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qσ,b

Q

x

(cid:96)(Q)

|Q|1− α

n

(cid:13)(cid:13)(cid:13)♠2

x

(cid:96)(Q)

|Q|1− α

n

L2(σ)

.

2

Aα,energyplug
Aα,∗,energyplug

2

(σ, ω) (cid:46) Aα
(σ, ω) (cid:46) Aα,∗

2

2 (σ, ω) + Aα,energy

2

(σ, ω) ,

(σ, ω) + Aα,∗,energy

2

(σ, ω) .

330

Proof. We have

(cid:13)(cid:13)(cid:13)Qω,b∗

Q

x

(cid:96)(Q)

|Q|1− α

n

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

Pα (Q, σ) =

+

(cid:46)

Q

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)Qω,b∗

Q

(cid:96)(Q)

x

n

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)♠2
Pα(cid:16)

x

(cid:96)(Q)

L2(ω)

|Q|1− α

L2(ω)

|Q|1− α

n

|Q|ω
|Q|1− α
2 (σ, ω) + Aα,energy

Q, 1Qcσ

n

2

(cid:46) Aα

(cid:17)

Q, 1Qcσ

Pα(cid:16)
Pα(cid:0)Q, 1Qσ(cid:1)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:17)

Q

+

(σ, ω) .

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

(cid:96)(Q)

|Q|1− α

n

|Q|σ
|Q|1− α

n

The Poisson formulation

Recall from Definitions 5.2.8 and 5.5.1 that

(cid:110)

(cid:111)

,

(cid:122) ∈ CF

CG,shif t

F

=

J ∈ G : J

where F ∈ F is a stopping cube in the dyadic grid D. For convenience we repeat here the
main result of this section, Proposition .0.1.

Proposition .0.8. For all grids D and G, and ε > 0 sufficiently small, we have

Fb∗
α (D,G) (cid:46) Eα
2 +
Fb,∗
α (G,D) (cid:46) Eα,∗
2 +

(cid:113)Aα
(cid:113)Aα

2 +

(cid:113)Aα,∗
(cid:113)Aα,∗

2 +

2 +

2 +

(cid:113)
(cid:113)
Aα,punct
,
Aα,∗,punct

2

2

,

with implied constants independent of the grids D and G.

331

To prove Proposition .0.8, we fix grids D and G and a subgrid F of D as in (5.5.5), and

set

µ ≡ (cid:88)

F∈F

(cid:88)

M∈W(F )

(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

· δ(cM ,(cid:96)(M ))

and dµ (x, t) ≡ 1

t2 dµ (x, t) ,

(.0.20)

where W (F ) consists of the maximal D-subcubes of F whose triples are contained in F ,
denotes the Dirac unit mass at the point (cM , (cid:96) (M )) in the upper
and where δ(cM ,(cid:96)(M ))
. Here M ∈ D is a dyadic cube with center cM and side length (cid:96) (M ), and
half-space Rn+1
for any cube K ∈ P, the shorthand notation Pω,b∗
) is used for the localized
pseudoprojection Pω,b∗

(resp. Qω,b∗
F,K
F,K
) given in (5.5.9):

(resp. Qω,b∗

+

CG,shif t

F

;K

CG,shif t

F

;K

(.0.21)

(.0.22)

Pω,b∗
F,K ≡ Pω,b∗

CG,shif t

=

;K

F

resp. Qω,b∗

F,K ≡ Qω,b∗

CG,shif t

F

;K

=

(cid:88)

(cid:3)ω,b∗

J

 .

(cid:52)ω,b∗

J

F

J⊂K: J∈CG,shif t
(cid:88)

J⊂K: J∈CG,shif t

F

(cid:122) ⊂ F . Here J

We emphasize that all the subcubes J that arise in the projection Qω,b∗
are good inside
(cid:122) is defined in Definition 5.2.8 using the
the cubes F and beyond since J
body of a cube. Thus every J ∈ Qω,b∗
is contained in a unique M ∈ W (F ), so that
Qω,b∗
. We can replace x by x − c inside the projection for any choice of
F
c we wish; the projection is unchanged. More generally, δq denotes a Dirac unit mass at a
point q in the upper half-space Rn+1

M∈W(F )Qω,b∗

·(cid:83)

F,M

F,M

=

F

.

+

332

We will prove the two-weight inequality

(cid:18)

(cid:113)Aα

2 +

(cid:113)Aα,∗

2 +

(cid:113)

Aα,punct

2

(cid:19)

(cid:107)f(cid:107)L2(σ) ,

(.0.23)

(cid:107)Pα (f σ)(cid:107)

(cid:46)

L2(Rn+1

+ ,µ)

Eα

2 +

for all nonnegative f in L2 (σ), noting that F and f are not related here. Above, Pα(·)
denotes the α-fractional Poisson extension to the upper half-space Rn+1

,

+

(cid:90)

Pαρ (x, t) ≡

(cid:16)

t2 + |x − y|2(cid:17) n+1−α

t

2

dρ (y) ,

Rn

so that in particular

♠2

L2(R2

+,µ)

=

(cid:88)

(cid:88)

F∈F

M∈W(F )

(cid:107)Pα(f σ)(cid:107)2

Pα (f σ) (c(M ), (cid:96) (M ))2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
where(cid:101)Pαν (x, t) ≡
(cid:12)(cid:12)(cid:12)2 instead, and we will do this shifting often throughout the proof when it

and so (.0.23) proves the first line in Proposition .0.1 upon inspecting (5.5.5). Note also
that we can equivalently write (cid:107)Pα (f σ)(cid:107)
Pαν (x, t) is the renormalized Poisson operator. Here we have simply shifted the factor 1
t2

in µ to(cid:12)(cid:12)(cid:12)(cid:101)Pα (f σ)

F,M

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)L2(R2

(cid:13)(cid:13)(cid:13)(cid:101)Pα (f σ)

x
|M| 1

n

L2(R2

+,µ)

L2(ω)

+,µ)

=

1
t

,

is convenient to do so.

One version of the characterization of the two-weight inequality for fractional and Poisson
integrals in [55] was stated in terms of a fixed dyadic grid D of cubes in R with sides parallel
to the coordinate axes. Using this theorem for the two-weight Poisson inequality, but adapted

to the α-fractional Poisson integral Pα,1 we see that inequality (.0.23) requires checking these

two inequalities for dyadic cubes I ∈ D and boxes (cid:98)I = I × [0, (cid:96) (I)) in the upper half-space

1The proof for 0 ≤ α < 1 is essentially identical to that for α = 0 given in [55].

333

Rn+1

+

:

(cid:90)

R2
+

Pα (1I σ) (x, t)2 dµ (x, t) ≡ (cid:107)Pα (1I σ)(cid:107)2
2 )2 + Aα

(Eα

(cid:46) (cid:16)
µ)]2dσ(x) (cid:46)(cid:16)

(Eα

(cid:90)

[Qα(t1(cid:98)I

R

2 )2 + Aα

2 + Aα,punct

L2(µ)

2 + Aα,∗
(cid:17)(cid:90)
(cid:98)I

2

(cid:17)

σ(I) ,

(.0.24)

2 + Aα,punct

2

t2dµ(x, t),

(.0.25)

for all dyadic cubes I ∈ D, and where the dual Poisson operator Qα is given by

Qα(t1(cid:98)I

µ) (x) =

(cid:0)t2 + |x − y|2(cid:1) n+1−α

t2

2

dµ (y, t) .

(cid:90)
(cid:98)I

It is important to note that we can choose for D any fixed dyadic grid, the compensating
point being that the integrations on the left sides of (.0.24) and (.0.25) are taken over the

entire spaces R2
+

and R respectively2.

Poisson testing

We now turn to proving the Poisson testing conditions (.0.24) and (.0.25). Similar testing

conditions have been considered in [62], [64], [66] and [67], and the proofs there essentially

carry over to the situation here, but careful attention must now be paid to the changed

definition of functional energy and the weaker notion of goodness. We continue to circumvent

the difficulty of permitting common point masses here by using the energy Muckenhoupt

constants Aα,energy

2

constants Aα,punct

2

and Aα,∗,energy
and Aα,∗,punct

2

2

, which require control by the punctured Muckenhoupt

. The following elementary Poisson inequalities (see e.g.

2There is a gap in the proof of the Poisson inequality at the top of page 542 in [55]. However, this gap

can be fixed as in [70] or [31].

334

[75]) will be used extensively.

Lemma .0.9. Suppose that J, K, I are cubes in Rn, and that µ is a positive measure supported
in Rn \ I. If J ⊂ K ⊂ βK ⊂ I for some β > 1, then

while if J ⊂ βK, then

Pα (J, µ)

|J| 1

n

Pα (K, µ)

|K| 1

n

Proof. We have

(cid:90)

(cid:18)

Pα (J, µ)

|J| 1

n

=

1
|J| 1

n

where J ⊂ K ⊂ βK ⊂ I implies that

,

.

≈ Pα (K, µ)

|K| 1

n

(cid:46) Pα (J, µ)

|J| 1

n

|J| 1

n

(cid:19)n+1−α dµ (x) ,

|J| 1

n + |x − cJ|

|J| 1

n + |x − cJ| ≈ |K| 1

n + |x − cK| ,

x ∈ Rn \ I,

and where J ⊂ βK implies that

|J| 1

n + |x − cJ| (cid:46) |J| 1

n + |cK − cJ| + |x − cK| (cid:46) |K| 1

n + |x − cK| ,

x ∈ Rn.

Recall that in the case of the T 1 theorem in [64], where we assumed traditional goodness
in a single family of grids D, we had a strong bounded overlap property associated with the
projections Pω,b∗
defined there; namely, that for each cube I0 ∈ D, there were a bounded

F,J

335

number of cubes F ∈ F with the property that F (cid:37) I0 ⊃ J for some J ∈ M(ρ,ε)−deep (F )
with Pω,b∗
(cid:54)= 0 (see the first part of Lemma 10.4 in [64]). However, we no longer have
this strong bounded overlap property when ordinary goodness is replaced with the weak

F,J

goodness of Hytönen and Martikainen. Indeed, there may now be an unbounded number of
cubes F ∈ F with F (cid:37) I0 ⊃ J and Pω,b∗
(cid:54)= 0, simply because there can be J(cid:48) ∈ G with
both J(cid:48) ⊂ I0 and(cid:0)J(cid:48)(cid:1)(cid:122)
F,J
arbitrarily large.

What will save us in obtaining the following lemma is that the Whitney cubes M in
W (F ) that happen to lie in some I ∈ D with I ⊂ F have one of just two different forms:
if I shares an endpoint with F then the cubes M near that endpoint are the same as those
in W (I) - note that F has been replaced with I here - while otherwise there are a bounded
number of Whitney cubes M in I, and each such M has side length comparable to (cid:96) (I).

The next lemma will be used in bounding both of the local Poisson testing conditions.
Recall from Definition .0.3 that AD consists of all augmented D-dyadic cubes where K is

an augmented dyadic cube if it is a union of 2 D-dyadic cubes K(cid:48) with (cid:96)(cid:0)K(cid:48)(cid:1) = 1

2 (cid:96) (K).

Lemma .0.10. Let D and G and F ⊂ D be grids and let
above. For any augmented cube I ∈ AD define

(cid:88)

(cid:88)

F∈F: F (cid:37)I(cid:48) for some I(cid:48)∈C(I)

M∈W(F ): M⊂I

F,M

(cid:110)
(cid:111)
Qω,b∗
Pα (M, 1I σ)
(cid:17)|I|σ .

|M| 1

n

F∈F

M∈W(F )

be as in (.0.22)

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

.

(.0.26)

(.0.27)

B (I) ≡

Then

B (I) (cid:46)(cid:16)

(Eα

2 )2 + Aα,energy

2

Proof. We first prove the bound (.0.27) for B (I) ignoring for the moment the possible case
when M = I in the sum defining B (I). So suppose that I ∈ AD is an augmented D-dyadic

336

cube. Define

Λ∗ (I) ≡(cid:110)

Λ(cid:0)I(cid:48)(cid:1) ≡(cid:110)

(cid:111)
F,M x (cid:54)= 0

,

M (cid:36) I : M ∈ W (F ) for some F (cid:37) I(cid:48), I(cid:48) ∈ C (I) with Qω,b∗

and pigeonhole this collection as Λ∗ (I) = (cid:83)

Λ(cid:0)I(cid:48)(cid:1), where for each I(cid:48) ∈ C (I) we define

I(cid:48)∈C(I)

M ⊂ I(cid:48) : M ∈ W (F ) for some F (cid:37) I(cid:48) with Qω,b∗

Consider first the case when 3I(cid:48) ⊂ F , so that d(cid:0)I(cid:48), ∂F(cid:1) ≥ (cid:96)(cid:0)I(cid:48)(cid:1). Then if M ∈ W (F ) for

some F (cid:37) I(cid:48) we have (cid:96) (M ) = d (M, ∂F ), and if in addition M ⊂ I(cid:48), then M = I(cid:48). Consider
the sum over all F (cid:37) I(cid:48) = M:

(cid:111)
F,M x (cid:54)= 0

.

(cid:88)
2(cid:13)(cid:13)(cid:13)Qω,b∗

M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

Pα (M, 1I σ)
Pα (I, 1I σ)

2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2

|M| 1

F,M x

(cid:46)

x

n

I

L2(ω)

|I| 1

n

L2(ω)

BM (I) ≡

F∈F: F (cid:37)M for some M∈C(I)∩W(F )

Pα (M, 1I σ)

|M| 1
(cid:46) Aα,energy

n

2

|I|σ ,

≤

where we have used the definitions (.0.22) and (.0.10). Thus we have obtained the bound

(cid:88)

F∈F: F (cid:37)M for some M∈C(I)∩W(F )

Pα (M, 1I σ)

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

|M| 1

n

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(cid:46) Aα,energy

2

|I|σ .

Now we turn to the case 3I(cid:48) (cid:54)⊂ F , i.e. when ∂I(cid:48) ∩ ∂F consists of exactly one boundary
point. In this case, if both M ⊂ I(cid:48) and M ∈ W (F ) for some F (cid:37) I(cid:48), then we must have

either M ∈ W(cid:0)I(cid:48)(cid:1) or M ∈ C(cid:0)I(cid:48)(cid:1), since both M and I(cid:48) are then close to the same boundary

337

point in ∂F . Note that it is here that we use the Whitney decompositions to full advantage.

So again we can estimate

(cid:88)
(cid:88)

3I(cid:48)(cid:54)⊂F

F∈F: F (cid:37)I(cid:48) for some I(cid:48)∈C(I)

≤

M∈{W(I(cid:48))∪C(I(cid:48))}∩W(F )

(cid:88)
Pα (M, 1I σ)

M∈W(F ): M⊂I(cid:48)

Pα (M, 1I σ)
2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2

|M| 1

M x

n

|M| 1

n

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(cid:46) (Eα

2 )2 |I|σ .

L2(ω)

Finally, we consider the case M = I. In this case I ∈ D and so F (cid:37) I(cid:48) implies F ⊃ I

and we can estimate

(cid:88)

F∈F: F⊃I

Pα (I, 1I σ)

|I| 1

n

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,I x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

Pα (I, 1I σ)

2(cid:13)(cid:13)(cid:13)Qω,b∗

I

x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

≤

|I| 1
n
(cid:46) Aα,energy

2

|I|σ .

This completes the proof of Lemma .0.10.

The forward Poisson testing inequality

Fix I ∈ D. We split the integration on the left side of (.0.24) into a local and global piece:

(cid:90)

Rn+1

+

(cid:90)
(cid:98)I

Pα (1I σ)2 dµ =

Pα (1I σ)2 dµ +

+ \(cid:98)I
Pα (1I σ)2 dµ ≡ Local (I) + Global (I)

Rn+1

(cid:90)

338

where more explicitly,

(cid:90)
(cid:98)I
(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

F,M

i.e. µ ≡ (cid:88)

(cid:88)

F∈F

M∈W(F )

Local (I) ≡

[Pα (1I σ) (x, t)]2 dµ (x, t) ;

µ ≡ 1

t2 µ,

(.0.28)

(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

(cid:96) (M )

· δ(cM ,(cid:96)(M )),

where we recall Qω,b∗
decompositions, used in this subsection:

F,M

is defined in (.0.22) above. Here is a brief schematic diagram of the

Local (I)

↓

Localplug (I)

+

↓
↓

Localhole (I)

(cid:1)2

(cid:0)Eα

2

A

(cid:1)2 + Aα,energy

2

+

(cid:0)Eα

2

B

(cid:1)2 + Aα,energy

2

(cid:0)Eα

2

and

Global (I)

↓

A

Aα
2

+

(cid:0)Eα

2

B

(cid:1)2 + Aα

2 + Aα,energy

2

+ C
Aα,∗

2

+

D

Aα,∗
2 + Aα,punct

2

As in our earlier papers [59]-[67] that used a single family of random grids, we have the

useful equivalence that

(c (M ) , (cid:96) (M )) ∈(cid:98)I if and only if M ⊂ I,

(.0.29)

339

since M and I live in the common grid D. We thus have

Local (I) =

Pα (1I σ) (x, t)2 dµ (x, t)

(cid:90)
(cid:98)I
(cid:88)
(cid:88)

=

(cid:88)
≈ (cid:88)

F∈F

M∈W(F ): M⊂I

Pα (1I σ) (cM , (cid:96) (M ))2
Pα (M, 1I σ)2 (cid:107)Qω,b∗

F,M

F∈F

M∈W(F ): M⊂I

≈ Localplug (I) + Localhole (I) ,

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

F,M

x
|M| 1

n

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

♠2

L2(ω)

(cid:107)♠2
L2(ω)

x
|M| 1

n

The ‘plugged’ local sum Localplug (I) can be further decomposed into

where

F∈F

Localplug (I) ≡ (cid:88)
Localhole (I) ≡ (cid:88)


Localplug (I) =

F∈F

(cid:88)

(cid:88)



+
F∈F
F∈F
F⊂I
F (cid:37)I
= A + B.

(cid:88)
(cid:88)

M∈W(F )): M⊂I

M∈W(F ): M⊂I

Pα (M, 1F∩I σ)
(cid:17)
Pα(cid:16)

M, 1I\F σ
|M| 1

|M| 1

n

n

2(cid:13)(cid:13)(cid:13)Qω,b∗
2(cid:13)(cid:13)(cid:13)Qω,b∗

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)♠2

F,M x

L2(ω)

F,M x

L2(ω)

,

.

Pα (M, 1F∩I σ)

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

|M| 1

n

(cid:88)

M∈W(F )

M⊂I

340

Then an application of the Whitney plugged energy condition gives

A =

F∈F: F⊂I

M∈W(F )

(cid:88)
≤ (cid:88)
(cid:13)(cid:13)(cid:13)♠2

(cid:88)

2 +

Eα

(cid:18)
≤(cid:13)(cid:13)(cid:13)Qω,b∗

F∈F: F⊂I

n

|M| 1

Pα (M, 1F∩I σ)
(cid:19)2 |F|σ
(cid:113)
(cid:13)(cid:13)(cid:13)♠2

Aα,energy

L2(ω)

2

since(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

M x
a σ-Carleson measure estimate,

L2(ω)

2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:18)

F,M x

(cid:13)(cid:13)(cid:13)♠2
(cid:113)

L2(ω)

(cid:46)

Eα

2 +

Aα,energy

2

(cid:19)2 |I|σ ,

. We also used here that the stopping cubes F satisfy

(cid:88)

F∈F: F⊂F0

|F|σ

(cid:46) |F0|σ .

Lemma .0.10 applies to the remaining term B to obtain the bound

B (cid:46)(cid:16)

(cid:17)|I|σ .

(Eα

2 )2 + Aα,energy

2

Next we show the inequality with ‘holes’, where the support of σ is restricted to the

complement of the cube F .

Lemma .0.11. We have

Proof. Fix I ∈ D and define

Localhole (I) (cid:46) (Eα

2 )2 |I|σ .

(.0.30)

FI ≡ {F ∈ F : F ⊂ I} ∪ {I} ,

and denote by πF , for this proof only, the parent of F in the tree FI. Also denote by

341

d(cid:0)F, F(cid:48)(cid:1) ≡ dFI

(cid:0)F, F(cid:48)(cid:1) the distance from F to F(cid:48) in the tree FI, and denote by d (F ) ≡

(F, I) the distance of F from the root I. Since I \ F appears in the argument of the

dFI
Poisson integral, those F ∈ F \ FI do not contribute to the sum and so we estimate

(cid:88)

(cid:88)

F∈FI

M∈W(F ): M⊂I

Pα(cid:16)

M, 1I\F σ
|M| 1

n

(cid:17)

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

1

d(F(cid:48))2 ≤ C to obtain
Pα(cid:16)
d(cid:0)F(cid:48)(cid:1)
 ·

d (F(cid:48))2

d (F(cid:48))

F(cid:48)∈F: F⊂F(cid:48)(cid:36)I

1

(cid:17)

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

M, 1πF(cid:48)\F(cid:48)σ
|M| 1

n

S ≡ Localhole (I) =

by using(cid:80)

F(cid:48)∈F: F⊂F(cid:48)(cid:36)I

S =

M⊂I

M⊂I

F∈FI

F∈FI

M∈W(F )

M∈W(F )

(cid:88)

(cid:88)

(cid:88)

F(cid:48)∈F
F⊂F(cid:48)(cid:36)I

≤ (cid:88)


 (cid:88)

(cid:88)
Pα(cid:16)
(cid:88)
d(cid:0)F(cid:48)(cid:1)2
d(cid:0)F(cid:48)(cid:1)2 (cid:88)
(cid:88)
F(cid:48)∈FI
d(cid:0)F(cid:48)(cid:1)2 (cid:88)
(cid:88)
F(cid:48)∈FI
d(cid:0)F(cid:48)(cid:1)2 (cid:88)
(cid:46) (cid:88)

F(cid:48)∈F
F⊂F(cid:48)(cid:36)I

K∈W(F(cid:48))

F∈F
F⊂F(cid:48)

≤ C

= C

·

F(cid:48)∈FI

K∈W(F(cid:48))

(cid:13)(cid:13)(cid:13)♠2

F,M x

L2(ω)

(cid:13)(cid:13)(cid:13)Qω,b∗


(cid:17)
2
Pα(cid:16)
Pα(cid:16)
(cid:17)
2 (cid:88)

M, 1πF(cid:48)\F(cid:48)σ
|M| 1

n

M, 1πF(cid:48)\F(cid:48)σ
|M| 1

n

F∈F
F⊂F(cid:48)

M∈W(F )

M⊂I

F,M x

(cid:17)
2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:17)
2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)Qω,b∗

L2(ω)

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)♠2

F,M∩K x

L2(ω)

F,M∩K x

L2(ω)

(cid:88)

M, 1πF(cid:48)\F(cid:48)σ
|M| 1

n

M∈W(F ): M⊂I

(cid:88)

(cid:88)
(cid:88)
Pα(cid:16)

F∈F
F⊂F(cid:48)

M∈W(F )

M⊂I

K, 1πF(cid:48)\F(cid:48)σ
|K| 1

n

342

where in the fifth line we have used that each J(cid:48) appearing in Qω,b∗
Qω,b∗
F,M∩K
inequalities in Lemma .0.9.

F,M

since each M is contained in a unique K. We have also used there the Poisson

occurs in one of the

(cid:88)

(cid:88)

We now use the lower frame inequality applied to the function 1K

(cid:46)(cid:13)(cid:13)1K
(cid:0)x − mω
Since the collection FI satisfies a Carleson condition, namely(cid:80)

M∈W(F ): M⊂I

F∈F: F⊂F(cid:48)

(cid:13)(cid:13)(cid:13)Qω,b∗

(cid:13)(cid:13)(cid:13)♠2

F,M∩K x

L2(ω)

F∈FI

for all cubes I(cid:48), we have geometric decay in generations:

K

(cid:1) to obtain

(cid:0)x − mω
(cid:1)(cid:13)(cid:13)♠2
(cid:12)(cid:12)F ∩ I(cid:48)(cid:12)(cid:12)σ ≤ C(cid:12)(cid:12)I(cid:48)(cid:12)(cid:12)σ

L2(ω)

.

K

|F|σ

(cid:46) 2−δk |I|σ ,

k ≥ 0.

(.0.31)

1
2

(cid:12)(cid:12)F ∩ F(cid:48)(cid:12)(cid:12)σ <
(cid:12)(cid:12)F ∩ F(cid:48)(cid:12)(cid:12)σ ≥ m

(cid:12)(cid:12)F(cid:48)(cid:12)(cid:12)σ ,
(cid:12)(cid:12)F(cid:48)(cid:12)(cid:12)σ ,

1
2

(.0.32)

(cid:88)

F∈FI : d(F )=k

(cid:88)

Indeed, with m > 2C we have for each F(cid:48) ∈ FI,

F∈FI : F⊂F(cid:48) and d(F,F(cid:48))=m

since otherwise

(cid:88)

F∈FI : F⊂F(cid:48) and d(F,F(cid:48))≤m

a contradiction. Now iterate (.0.32) to obtain (.0.31).

343

Ak = k2 (cid:88)

(cid:88)

(cid:88)
F(cid:48)∈FI : d(F(cid:48))=k

K∈W(F(cid:48))

F(cid:48)(cid:48)∈FI : d(F(cid:48)(cid:48))=k−1

(cid:46) k2 (Eα

2 )2

Pα(cid:16)
(cid:12)(cid:12)F(cid:48)(cid:48)(cid:12)(cid:12)σ

(cid:17)

2(cid:13)(cid:13)1K

K, 1πF(cid:48)\F(cid:48)σ
|K| 1
n
(cid:46) (Eα

2 )2 k22−δk |I|σ ,

(cid:0)x − mω

K

(cid:1)(cid:13)(cid:13)♠2

L2(ω)

d(cid:0)F(cid:48)(cid:1)2 (cid:88)
k2 (cid:88)

K∈W(F(cid:48))

Pα(cid:16)
(cid:88)

F(cid:48)∈FI : d(F(cid:48))=k

K∈W(F(cid:48))

(cid:17)

2(cid:13)(cid:13)1K
(cid:17)

K, 1πF(cid:48)\F(cid:48)σ
|K| 1

n

Pα(cid:16)

K, 1πF(cid:48)\F(cid:48)σ
|K| 1

n

Thus we can write

S (cid:46) (cid:88)
F(cid:48)∈FI
∞(cid:88)
∞(cid:88)

k=1

=

≡

Ak ,

k=1

(cid:0)x − mω
2(cid:13)(cid:13)1K

(cid:1)(cid:13)(cid:13)♠2
(cid:0)x − mω

L2(ω)

K

K

(cid:1)(cid:13)(cid:13)♠2

L2(ω)

where Ak is defined at the end of the above display. Hence using the strong energy condition,

where we have applied the strong energy condition for each F(cid:48)(cid:48) ∈ FI with d(cid:0)F(cid:48)(cid:48)(cid:1) = k − 1

to obtain

(cid:88)

(cid:88)

F(cid:48)∈FI : πF(cid:48)=F(cid:48)(cid:48)

K∈W(F(cid:48))

Pα(cid:16)

K, 1F(cid:48)(cid:48)\F(cid:48)σ
|K| 1

n

(cid:17)

2(cid:13)(cid:13)1K

(cid:0)x − mω

K

(cid:1)(cid:13)(cid:13)♠2

L2(ω)

2 )2(cid:12)(cid:12)F(cid:48)(cid:48)(cid:12)(cid:12)σ .

≤ (Eα

(.0.33)

Finally then we obtain

∞(cid:88)

k=1

S (cid:46)

2 )2 k22−δk |I|σ
(Eα

(cid:46) (Eα

2 )2 |I|σ ,

which is (.0.30).

Altogether we have now proved the estimate Local (I)(cid:46)(cid:16)(cid:0)Eα

2

(cid:1)2+Aα,energy

2

(cid:17)|I|σ

when

344

I ∈ D, i.e. for every dyadic cube I ∈ D,

(cid:88)

Local (I) ≈ (cid:88)
(cid:46) (cid:16)

F∈F
(Eα

M∈W(F ): M⊂I
2 )2 + Aα,energy

2

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(.0.34)

Pα (M, 1I σ)
(cid:17)|I|σ ,

|M| 1

I ∈ D.

n

The augmented local estimate

For future use in the ‘prepare to puncture’ arguments below, we prove a strengthening of
the local estimate Local (I) to augmented cubes L ∈ AD.

Lemma .0.12. With notation as above and L ∈ AD an augmented cube, we have

Local (L) ≡ (cid:88)
(cid:46) (cid:16)

F∈F
(Eα

(cid:88)

M∈W(F ): M⊂L
2 )2 + Aα,energy

2

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

Pα (M, 1Lσ)
(cid:17)|L|σ , L ∈ AD.

|M| 1

n

(.0.35)

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

Proof. We prove (.0.35) by repeating the above proof of (.0.34) and noting the points requir-

ing change. First we decompose

Local (L) (cid:46) Localplug (L) + Localhole (L) + Localof f set (L)

where Localplug (L), Localhole (L) are analogous to Localplug (I) and Localhole (I) above,
and where Localof f set (L) is an additional term arising because L \ F need not be empty

345

when L ∩ F (cid:54)= ∅ and F is not contained in L:

(cid:88)

M∈W(F ): M⊂L

n

|M| 1

Pα (M, 1L∩F σ)
Pα(cid:16)
Pα(cid:16)

M, 1L\F σ
|M| 1

n

M, 1L\F σ
|M| 1

n

(cid:88)
(cid:88)

F∈F: F⊂L

M∈W(F ): M⊂L

F∈F: F(cid:54)⊂L

M∈W(F ): M⊂L

F∈F

Localplug (L) ≡ (cid:88)
Localhole (L) ≡ (cid:88)
Localof f set (L) ≡ (cid:88)


Localplug (L) =

We have

(cid:88)

F∈F: F⊂ some L(cid:48)∈C(L)

+

×

= A + B.

(cid:88)
Pα (M, 1F∩Lσ)

F∈F: F (cid:37) some L(cid:48)∈CD(L)

|M| 1

n

,

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)♠2

F,M x

L2(ω)

F,M x

L2(ω)

L2(ω)

F,M x

2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:17)
2(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:17)
2(cid:13)(cid:13)(cid:13)Qω,b∗

2(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:88)
(cid:13)(cid:13)(cid:13)♠2

L2(ω)

M∈W(F ): M⊂L

Term A satisfies

just as above using(cid:13)(cid:13)(cid:13)Qω

F,M x

(cid:13)(cid:13)(cid:13)2

A (cid:46)

satisfy a σ-Carleson measure estimate,

L2(ω)

L2(ω)

, and the fact that the stopping cubes F

(cid:19)2 |L|σ ,

Aα,energy

2

Eα

2 +

(cid:18)
(cid:113)
≤(cid:13)(cid:13)Qω
M x(cid:13)(cid:13)2
(cid:88)

F∈F: F⊂L

|F|σ

(cid:46) |L|σ .

346

Term B is handled directly by Lemma .0.10 with the augmented cube I = L to obtain

B (cid:46)(cid:16)

(cid:17)|L|σ .

(Eα

2 )2 + Aα,energy

2

To handle Localhole (L), we define

FL ≡ {F ∈ F : F ⊂ L} ∪ {L} ,

and follow along the proof there with only trivial changes. The analogue of (.0.33) is now

(cid:88)

(cid:88)

F(cid:48)∈FL: πF(cid:48)=F(cid:48)(cid:48)

K∈W(F(cid:48))

Pα(cid:16)

K, 1F(cid:48)(cid:48)\F(cid:48)σ
|K| 1

n

(cid:17)

2(cid:13)(cid:13)1K

(cid:0)x − mω

K

(cid:1)(cid:13)(cid:13)♠2

L2(ω)

2 )2(cid:12)(cid:12)F(cid:48)(cid:48)(cid:12)(cid:12)σ ,

≤ (Eα

the only change being that FL now appears in place of FI, so that the energy condition still
applies. We conclude that

Localhole (L) (cid:46) (Eα

2 )2 |L|σ .

Finally, the additional term Localof f set (L) is handled directly by Lemma .0.10, and this

completes the proof of the estimate (.0.35) in Lemma .0.12.

The global estimate

Now we turn to proving the following estimate for the global part of the first testing condition

(.0.24):

Global (I) =

(cid:90)

+ \(cid:98)I

Rn+1

Pα (1I σ)2 dµ (cid:46)(cid:16)

347

2 )2 + Aα,∗
(Eα

2 + Aα,punct

2

(cid:17)|I|σ .

We begin by decomposing the integral above into four pieces. We have from (.0.29):

(cid:90)



+ \(cid:98)I

Rn+1

Pα (1I σ)2 dµ

(cid:88)

=

=

+ \(cid:98)I
M : (cM ,(cid:96)(M ))∈Rn+1
(cid:88)

(cid:88)

+

+

M⊂3I\I

M∩3I=∅
(cid:96)(M )≤(cid:96)(I)

♠2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

n

L2(ω)

F,M

F∈F:

M∈W(F )

x
|M| 1

Pα (1I σ) (cM , (cid:96) (M ))2 (cid:88)
(cid:88)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
 Pα (1I σ) (cM , (cid:96) (M ))2 ·
(cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
· (cid:88)

M∩I=∅
(cid:96)(M )>(cid:96)(I)

x
|M| 1

F∈F:

M(cid:37)I

♠2

F,M

n

L2(ω)

+

M∈W(F )

= A + B + C + D.

We further decompose term A according to the length of M and its distance from I,

and then use the pairwise disjointedness of the projections Qω,b∗
(.0.22)) to obtain:
(cid:33)2

F,M

in F (see the definition in

|M|ω

∞(cid:88)

m=0

k=1

∞(cid:88)
∞(cid:88)
∞(cid:88)

m=0

(cid:88)
∞(cid:88)
∞(cid:88)

k=1

M⊂3k+1I\3kI
(cid:96)(M )=2−m(cid:96)(I)
|I|2 |I|σ

2−2m

2−2m

3−2k

m=0

k=1

2−m |I|

(cid:32)
d (M, I)n+1−α |I|σ
(cid:12)(cid:12)(cid:12)3k+1I \ 3kI
(cid:12)(cid:12)(cid:12)ω
(cid:12)(cid:12)3kI(cid:12)(cid:12)2(n+1−α)
(cid:12)(cid:12)(cid:12)3k+1I \ 3kI
(cid:12)(cid:12)(cid:12)3kI
(cid:12)(cid:12)(cid:12)ω

(cid:12)(cid:12)3kI(cid:12)(cid:12)2(1−α)

(cid:12)(cid:12)(cid:12)σ

|I|σ

A (cid:46)

(cid:46)

(cid:46)

|I|σ

(cid:46) Aα

2 |I|σ ,

where the offset Muckenhoupt constant Aα
2

applies because 3k+1I has only three times the

side length of 3kI.

348

For term B we first dispose of the nearby sum Bnearby that consists of the sum over
those M which satisfy in addition 2−ρ(cid:96) (I) ≤ (cid:96) (M ) ≤ (cid:96) (I). But it is a straightforward task
to bound Bnearby by CAα,energy
as there are at most 2ρ+1 such cubes M. To bound
Baway ≡ B − Bnearby, we further decompose the sum over F ∈ F according to whether or
not F ⊂ 3I \ I:

|I|σ

2

(cid:88)
(cid:88)

Pα (M, 1I σ)
Pα (M, 1I σ)

2 (cid:88)
2 (cid:88)

|M| 1

M∈W(F )

n

F∈F: F⊂3I\I

F,M x

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

L2(ω)

L2(ω)

F∈F: F(cid:54)⊂3I\I

M∈W(F )

Baway ≈

M⊂3I\I and (cid:96)(M )<2−ρ(cid:96)(I)

+

M⊂3I\I and (cid:96)(M )<2−ρ(cid:96)(I)

|M| 1

n

≡ B1

away + B2

away .

To estimate B1

away, let

J ∗ ≡ (cid:91)

F∈F
F⊂3I\I

(cid:91)

M∈W(F )

M⊂3I\I and (cid:96)(M )<2−ρ(cid:96)(I)

(cid:110)
J ∈ CG,shif t

F

(cid:111)

: J ⊂ M

(.0.36)

consist of all cubes J ∈ G for which the projection (cid:52)ω,b∗
Qω,b∗

away. In order to use J ∗ in the estimate for B1

in term B1

J

F,M

occurs in one of the projections

away we need the following

349

inequality. For any cube M ∈ W (F ) we have

(cid:18)Pα (M, 1I σ)

|M|

(cid:19)2(cid:13)(cid:13)(cid:13)Qω,b∗

F ;M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

|M|

(cid:18)Pα (M, 1I σ)
(cid:88)
J∈CG,shif t

F

: J⊂M

=

(cid:46)

(cid:19)2
(cid:88)
(cid:18)Pα (J, 1I σ)
J∈CG,shif t

F

|J|

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:19)2(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

: J⊂M

x

J

J

x

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(.0.37)

L2(ω)

since

Pα (M, 1I σ)

|M| 1

n

(cid:90)
(cid:90)

I

I

=

(cid:46)

1

((cid:96) (M ) + |x − cM|)n+1−α dσ (x)

1

((cid:96) (J) + |x − cJ|)n+1−α dσ (x) =

Pα (J, 1I σ)

|J| 1

n

for J ⊂ M because

(cid:96) (J) + |x − cJ| (cid:46) (cid:96) (M ) + |x − cM| ,

J ⊂ M and x ∈ Rn.

We now use (.0.37) to replace the sum over M ∈ W (F ) in B1

away, with a sum over J ∈ J ∗:

M⊂3I\I and (cid:96)(M )<2−ρ(cid:96)(I)

(cid:88)
(cid:88)
(cid:18)Pα (J, 1I σ)

|J|

n

|M| 1

F∈F: F⊂3I\I

2 (cid:88)
Pα (M, 1I σ)
Pα (J, 1I σ)
(cid:88)
(cid:88)
J∈CG,shif t
(cid:19)2 (cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

(cid:13)(cid:13)(cid:13)♠2

F∈F: F⊂3I\I

M∈W(F )

M∈W(F )

|J| 1

F
J⊂M

x

n

J

,

L2(ω)

F,M x

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qω,b∗
2(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x

J

L2(ω)

B1

away =

(cid:46)

M⊂3I\I&(cid:96)(M )<2−ρ(cid:96)(I)

(cid:46) (cid:88)

J∈J ∗

where the final line follows since for each J ∈ J ∗ there is a unique pair (F, M ) satisfying

350

the conditions in the second line.

We will now exploit the smallness of ε > 0 in the weak goodness condition by decomposing
the sum over J ∈ J ∗ according to the length of J, and then using the fractional Poisson
inequality (5.5.22) in Lemma 5.5.10 on the neighbour I(cid:48) of I containing J.
Indeed, for
J ⊂ I(cid:48) ⊂ R and I ⊂ R \ I(cid:48), we have

(cid:19)2−2(n+1−α)ε

(cid:18) (cid:96) (J)

(cid:96) (I)

Pα (J, 1I σ)2 (cid:46)

Pα (I, 1I σ)2 ,

J ∈ J ∗,

(.0.38)

where we have used that (cid:96)(cid:0)I(cid:48)(cid:1) = (cid:96) (I) and Pα(cid:0)I(cid:48), 1I σ(cid:1) ≈ Pα (I, 1I σ), and that the cubes

J ∈ J ∗ are good in I(cid:48) and beyond, and have side length at most 2−ρ(cid:96) (I), all because
(cid:122) ⊂ F ⊂ 3I \ I and we have already dealt with the term Bnearby. Moreover, we may also
J
assume here that the exponent 2 − 2 (n + 1 − α) ε is positive, i.e. ε <
, which is of
course implied by 0 < ε < 1
2
|J|2 |J|ω
CG,shif t

(cid:46)
, the pairwise disjointedness of the M ∈ W (F ), the uniqueness of F with J ∈
, and since F ⊂ 3I \ I in the sum over J ∈ J ∗, that

. We then obtain from (.0.38), the inequality(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

(cid:13)(cid:13)(cid:13)♠2

n+1−α

L2(ω)

x

J

1

F

B1

J∈J ∗

(cid:46)

away (cid:46) (cid:88)
∞(cid:88)
∞(cid:88)
∞(cid:88)

m=ρ

m=ρ

(cid:46)

(cid:46)

J

n

x

|J| 1

2 (cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:0)2−m(cid:1)2−2(n+1−α)ε

Pα (J, 1I σ)
(cid:88)
(cid:0)2−m(cid:1)2−2(n+1−α)ε
(cid:0)2−m(cid:1)2−2(n+1−α)ε |I|σ |3I \ I|ω

(cid:32) |I|σ

(cid:96)(J)=2−m(cid:96)(I)

|I|1− α

J∈J ∗

n

|3I|2(1− α
n )

L2(ω)

Pα (I, 1I σ)2 |J|ω

(cid:33)2 (cid:88)

|J|ω

J⊂3I\I

(cid:96)(J)=2−m(cid:96)(I)
|I|σ
(cid:46) Aα

2 |I|σ ,

m=ρ

351

since 2 − 2 (n + 1 − α) ε > 0.

To complete the bound for term B = Bnearby + B1

away, it remains to estimate
away in which we sum over F (cid:54)⊂ 3I\I. In this case F (cid:39) I(cid:48) for one of the two neighbours
term B2
I(cid:48) of I, and so we can apply Lemma .0.10, with I there replaced by the augmented cubes
I(cid:48) ∪ I, to obtain the estimate

away + B2

away (cid:46)(cid:16)

B2

(Eα

2 )2 + Aα,energy

2

(cid:17)|I|σ .

Next we turn to term D. The cubes M occurring here are included in the set of ancestors

Ak ≡ π

=

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

k=1

F∈F:

Ak∈W(F )

(k)D I of I, 1 ≤ k < ∞. Then D is equal to
∞(cid:88)
Pα (1I σ) (c (Ak) ,|Ak|)2 (cid:88)
∞(cid:88)
Pα (1I σ) (c (Ak) ,|Ak|)2 (cid:88)
Pα (1I σ) (c (Ak) ,|Ak|)2 (cid:88)
∞(cid:88)
Pα (1I σ) (c (Ak) ,|Ak|)2 (cid:88)

Ak∈W(F )

Ak∈W(F )

∞(cid:88)

F∈F:

F∈F:

k=1

k=1

+

+

k=1

Ak∈W(F )
≡ Ddisjoint + Ddescendent + Dancestor .

F∈F:

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

x
|Ak| 1

n

n

F

J(cid:48)

F,Ak

L2(ω)

x
|Ak| 1
(cid:88)
J(cid:48)∈CG,shif t
: J(cid:48)⊂Ak\I
(cid:88)
(cid:16)
J(cid:48)(cid:17)≤(cid:96)(I)
J(cid:48)∈CG,shif t
: J(cid:48)⊂Ak
J(cid:48)∩I(cid:54)=∅ and (cid:96)
(cid:88)
J(cid:48)(cid:17)
(cid:16)
J(cid:48)∈CG,shif t
: J(cid:48)⊂Ak
J(cid:48)∩I(cid:54)=∅ and (cid:96)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

>(cid:96)(I)

J(cid:48)

J(cid:48)

F

F

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

n

n

x
|Ak| 1

L2(ω)

x
|Ak| 1

L2(ω)

We thus have from the pairwise disjointedness of the projections Qω,b∗

F,Ak

in F once again,

352

Ddisjoint equals

F∈F:

Ak∈W(F )

k=1

∞(cid:88)

Pα (1I σ) (c (Ak) ,|Ak|)2 (cid:88)
(cid:32) |I|σ |Ak|
∞(cid:88)
(cid:40) |I|σ

|Ak \ I|ω =
(cid:41)
Pα (I, 1Icω)

|Ak|n+1−α

(cid:33)2

|I|σ

k=1

|I|1− α

n

(cid:46) Aα,∗

2

(cid:88)
J(cid:48)∈CG,shif t
(cid:40) |I|σ
J(cid:48)⊂Ak\I
∞(cid:88)

F

n

|I|1− α
|I|σ ,

(cid:46)

(cid:46)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J(cid:48)

:

x
|Ak| 1

n

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(cid:41)

n

|I|1− α
|Ak|2(n−α)

|Ak \ I|ω

|I|σ

k=1

since

∞(cid:88)

k=1

n

|I|1− α
|Ak|2(n−α)

|Ak \ I|ω =

=

(cid:46)

k=1

(cid:90) ∞(cid:88)
(cid:90) ∞(cid:88)

(cid:90)

Ic

k=1

(cid:20)

n

|I|1− α
|Ak|2(n−α)

1

22(n−α)k

|I| 1

n

|I| 1

n + d (x, I)

n

1Ak\I (x) dω (x)
|I|1− α
|I|2(n−α)
(cid:21)2



1Ak\I (x) dω (x)
n−α

dω (x) = Pα (I, 1Icω) ,

upon summing a geometric series with 2 (n − α) > 0.

353

The next term Ddescendent satisfies

Ddescendent (cid:46)

=

(cid:46)

(cid:32) |I|σ |Ak|

|Ak|n+1−α
2−2k(n+1−α)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

3I

x
|I| 1
n
|I|2(n−α)

n

x

3I

2k|I| 1

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2
(cid:33)2 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:19)2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:18) |I|σ
|I|σ
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

|I|n−α

L2(ω)

3I

2

x
|I| 1

n

∞(cid:88)
∞(cid:88)

k=1

k=1



|I|σ

L2(ω)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(cid:46) Aα,energy

|I|σ .

Lastly, for Dancestor we note that there are at most two cubes K1 and K2 in G having
side length (cid:96) (I) and such that Ki ∩ I (cid:54)= ∅. Then each J(cid:48) occurring in the sum in Dancestor
is of the form J(cid:48) = A(cid:96)
((cid:96))G Ki with J(cid:48) ⊂ Ak for some 1 ≤ (cid:96) ≤ k and i ∈ {1, 2}. Now we
write

i ≡ π

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

J(cid:48)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

L2(ω)

x
|Ak| 1

n

Dancestor =

(cid:46)

k=1

F∈F:

∞(cid:88)

Ak∈W(F )

Pα (1I σ) (c (Ak) ,|Ak|)2 (cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:32) |I|σ |Ak|
(cid:33)2 2(cid:88)
k(cid:88)
∞(cid:88)
(cid:33)2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
|Ak|n+1−α
(cid:32) |I|σ |Ak|
∞(cid:88)

|Ak|n+1−α

A(cid:96)
i

Ak

(cid:96)=1

i=1

k=1

k=1

x
|Ak| 1

n

≤ 2

(cid:88)
(cid:16)
J(cid:48)(cid:17)
J(cid:48)∈CG,shif t
: J(cid:48)⊂Ak
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2
J(cid:48)∩I(cid:54)=∅ and (cid:96)

x
|Ak| 1

L2(ω)

F

n

>(cid:96)(I)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2

.

L2(ω)

At this point we need a ‘prepare to puncture’ argument, as we will want to derive geometric

decay from(cid:13)(cid:13)(cid:13)Qω,b∗
using the Muckenhoupt energy constant. For this we define (cid:101)ω = ω − ω ({p}) δp where p is

by dominating it by the ‘nonenergy’ term(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)2(cid:12)(cid:12)J(cid:48)(cid:12)(cid:12)ω

(cid:13)(cid:13)(cid:13)♠2

, as well as

J(cid:48) x

L2(ω)

an atomic point in I for which

354

sup

ω ({p}) =
q∈P(σ,ω): q∈I
Then we have |I|(cid:101)ω = ω
|I|(cid:101)ω
|I|1− α

n

ω ({q}) . (If ω has no atomic point in common with σ in I set(cid:101)ω = ω.)
(cid:16)

(cid:17) and

I, P(σ,ω)

(cid:17)

(cid:16)
I, P(σ,ω)
|I|1− α

n

|I|σ
|I|1− α

n

≤ Aα,punct

2

.

|I|σ
|I|1− α

n

ω

=

A key observation, already noted in the proof of Lemma .0.5 above, is that

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗

K x

(cid:13)(cid:13)(cid:13)2

L2(ω)

=

if

if

p ∈ K
p /∈ K

≤ (cid:96) (K)2 |K|(cid:101)ω ,

(.0.39)

and so, as in the proof of (.0.18) in Lemma .0.5,

n

Ak

♠2

L2(ω)

L2(ω)



K x

x
|Ak| 1

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2
K (x − p)
L2((cid:101)ω)

(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:52)ω,b∗
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:33)2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:32) |I|σ |Ak|
(cid:32) |I|σ |Ak|
(cid:33)2
|Ak|(cid:101)ω
(cid:19)2 |Ak \ I|ω +
(cid:18) |I|σ
(cid:17)|I|σ ,
(cid:19)2 |Ak \ I|ω
(cid:18) |I|σ

|Ak|n+1−α

|Ak|n+1−α

|Ak|n−α

Ak

k=1

|Ak|n−α

2 + Aα,punct

2

355

Then we continue with

(cid:46)

k=1

∞(cid:88)
∞(cid:88)
∞(cid:88)
(cid:46) (cid:16)Aα,∗
where the inequality (cid:80)∞

k=1

k=1

=

the display estimating Ddisjoint.

(cid:46) |Ak|(cid:101)ω .
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)♠2
(cid:18)

L2(ω)

n

x
|Ak| 1
∞(cid:88)

k=1

(cid:19)2 |I|(cid:101)ω

|I|σ

2k(n−α) |I|n−α

(cid:46) Aα,∗

2

|I|σ

is already proved above in

Finally, for term C we will have to group the cubes M into blocks Bi. We first split the

sum according to whether or not I intersects the triple of M:

|M| 1

n

(cid:19)n+1−α |I|σ



2

·



(cid:88)

C ≈

+

M : I∩3M =∅
(cid:96)(M )>(cid:96)(I)

M : I⊂3M\M
(cid:96)(M )>(cid:96)(I)



(cid:18)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qωb∗

F,M

(cid:88)
· (cid:88)

F∈F:

M∈W(F )

|M| 1

n + d (M, I)

♠2

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

x
|M| 1

n

L2(ω)

= C1 + C2.

We first consider C1. Let M consist of the maximal dyadic cubes in the collection
{Q : 3Q ∩ I = ∅}, and then let {Bi}∞
be an enumeration of those Q ∈ M whose side
length is at least (cid:96) (I). Note in particular that 3Bi ∩ I = ∅. Now we further decompose
the sum in C1 by grouping the cubes M into the ‘Whitney’ cubes Bi, and then using the

i=1

356

pairwise disjointedness of the martingale supports of the pseudoprojections Qω,b∗

in F :

F,M

F,M x

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)♠2

L2(ω)

L2(ω)

2 (cid:88)

F∈F:

M∈W(F )

(cid:88)

F∈F:

M∈W(F )

|M| 2

n |M|ω


(cid:19)n+1−α |I|σ

2 (cid:88)

2 (cid:88)


|Bi| 2

2

M : M⊂Bi

M : M⊂Bi

(cid:19)n+1−α |I|σ

(cid:19)n+1−α |I|σ

(cid:19)n+1−α |I|σ

n |Bi|ω

1

M : M⊂Bi

|M| 1

n + d (M, I)



(cid:18)

∞(cid:88)

(cid:88)

i=1

i=1

∞(cid:88)

∞(cid:88)




∞(cid:88)
(cid:40) ∞(cid:88)

i=1

i=1

i=1

(cid:18)
|Bi| 1
(cid:18)
|Bi| 1
(cid:18)
|Bi| 1
|Bi|ω |I|σ
|Bi|2(n−α)

1

1

1

n + d (Bi, I)

n + d (Bi, I)

n + d (Bi, I)

(cid:41)

|I|σ ,

C1 ≤

(cid:46)

(cid:46)

(cid:46)

(cid:46)

Now since |Bi| ≈ d (x, I) for x ∈ Bi,

∞(cid:88)
|I|1−α
|Bi|2(n−α)
(cid:90)
∞(cid:88)
(cid:20)



i=1

Bi

i=1

n

|Bi|ω ≈ |I|σ
|I|1− α
(cid:21)2

|I| 1

n

n + d (x, I)

|I| 1

(cid:90)

∞(cid:88)


Bi

i=1
n−α

dω (x)

|I|1− α

n

d (x, I)2(n−α)

dω (x)

∞(cid:88)

i=1

|Bi|ω |I|σ
|Bi|2(n−α)

=

≈

≤

|I|σ
|I|1−α

n

|I|σ
|I|1− α
|I|σ
|I|1− α

n

Pα (I, 1Icω) ≤ Aα,∗
2 ,

357

we obtain

C1 (cid:46) Aα,∗

2

|I|σ .

Next we turn to estimating term C2 where the triple of M contains I but M itself does
not. Note that there are at most two such cubes M of a given side length. So with this in

mind, we sum over the cubes M according to their lengths to obtain



(cid:96)(M )=2m(cid:96)(I)

M :

m=1

I⊂3M\M

∞(cid:88)
(cid:88)
(cid:18) |I|σ
∞(cid:88)
(cid:40) |I|σ

m=1

|I|1− α

n

|2mI|n−α

n + dist (M, I)

n

|M| 1

|M| 1

(cid:18)
(cid:19)2|(5·2mI)\I|ω =
(cid:41)

(cid:19)n+1−α |I|σ
(cid:40) |I|σ
∞(cid:88)

m=1

n

|I|1− α
|I|σ ,

Pα (I, 1Icω)

|I|σ ≤ Aα,∗

2

C2 =

(cid:46)

(cid:46)



2 (cid:88)

F∈F:

M∈W(F )
|I|1− α

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Qω,b∗

F,M

♠2

L2(ω)

x
|M| 1

n

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:41)

n |(5 · 2mI) \ I|ω
|2mI|2(n−α)

|I|σ

since in analogy with the corresponding estimate above,

∞(cid:88)

m=1

|I|1− α

n |(5·2mI)\I|ω
|2mI|2(n−α)

=

(cid:90) ∞(cid:88)

m=1

|I|1− α

n

|2mI|2(n−α)

1(5·2mI)\I(x) dω(x) (cid:46) Pα (I, 1Icω) .

The backward Poisson testing inequality

The argument here follows the broad outline of the analogous argument in [64], but using

modifications from [66] that involve ‘prepare to puncture arguments’, using decompositions
W (F ) in place of (ρ, ε)-decompositions, and using pseudoprojections Qω,b∗
F,M x (see (.0.22) for
the definition). The final change here is that there is no splitting into local and global parts

as in [64] - instead, we follow the treatment in [63] in this regard.

358

Fix I ∈ D. It suffices to prove

Back

(y)

dσ(y)

(cid:16)(cid:98)I
(cid:17) ≡

(cid:46)

(cid:90)
(cid:26)

(cid:104)Qα(cid:16)
(cid:18)

Rn
Aα
2 +

t1(cid:98)I

µ

(cid:17)
(cid:113)

(cid:105)2

(cid:19)(cid:113)

Eα

2 +

Aα,energy

2

Aα,punct

2

(cid:27)(cid:90)
(cid:98)I

t2dµ(x, t).

(.0.40)

Note that for a ‘Poisson integral with holes’ and a measure µ built with Haar projections,

Hytönen obtained in [22] the simpler bound Aα
2

for a term analogous to, but significantly

smaller than, (.0.40). Using (.0.29) we see that the integral on the right hand side of (.0.40)

is

(cid:90)
(cid:98)I

t2dµ =

(cid:88)

(cid:88)

F∈F

M∈W(F ): M⊂I

where Qω,b∗

F,M

was defined in (.0.22).

(cid:107)Qω,b∗

F,M x(cid:107)♠2

L2(ω)

.

(.0.41)

We now compute using (.0.29) again that

Qα(cid:16)

(cid:17)

t1(cid:98)I

µ

(y) =

(cid:90)
(cid:16)
(cid:98)I
≈ (cid:88)

t2

t2 + |x − y|2(cid:17) n+1−α

2

(cid:88)

F∈F

M∈W(F ): M⊂I

dµ (x, t)

(.0.42)

(cid:107)Qω,b∗

F,M x(cid:107)♠2

L2(ω)

(|M| + |y − cM|)n+1−α ,

(cid:17) is
(cid:16)(cid:98)I

and then expand the square and integrate to obtain that the term Back

(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(|M| + |y − cM|)n+1−α

(cid:13)(cid:13)(cid:13)♠2

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:0)|M(cid:48)| +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:1)n+1−α dσ (y) .

F(cid:48),M(cid:48)x

L2(ω)

(cid:88)

M∈W(F )

F∈F
M⊂I

(cid:90)

R

(cid:88)
F(cid:48)(cid:17)
M(cid:48)∈W(cid:16)

F(cid:48)∈F:
M(cid:48)⊂I

By symmetry we may assume that (cid:96)(cid:0)M(cid:48)(cid:1) ≤ (cid:96) (M ). We fix a nonnegative integer s, and

359

consider those cubes M and M(cid:48) with (cid:96)(cid:0)M(cid:48)(cid:1) = 2−s(cid:96) (M ). For fixed s we will control the

expression

(cid:90)

Rn

(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(|M| + |y − cM|)n+1−α

(cid:13)(cid:13)(cid:13)♠2

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:0)|M(cid:48)| +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:1)n+1−α dσ (y)

F(cid:48),M(cid:48)x

L2(ω)

(.0.43)

Us ≡(cid:88)

(cid:88)
M(cid:48)∈W(cid:16)
F(cid:48)(cid:17)
M(cid:48)(cid:17)
(cid:16)
M,M(cid:48)⊂I, (cid:96)

M∈W(F )

F,F(cid:48)∈F

=2−s(cid:96)(M )

by proving that

(cid:26)

(cid:18)

Eα

2 +

(cid:113)

Aα,energy

2

(cid:19)(cid:113)

Aα,punct

2

(cid:27)(cid:90)
(cid:98)I

t2dµ,

Us (cid:46) 2−δs

Aα
2 +

where δ =

1
2

.

(.0.44)

(cid:16)(cid:98)I
(cid:17). We now

With this accomplished, we can sum in s ≥ 0 to control the term Back
decompose Us = T proximal

into three pieces.

+ T dif f erence

+ T intersection

s

s

s

Our first decomposition is to write

Us = T proximal

s

+ V remote

s

,

(.0.45)

s

where in the ‘proximal’ term T proximal

we restrict the summation over pairs of cubes M, M(cid:48)
we re-
strict the summation over pairs of cubes M, M(cid:48) to those satisfying the opposite inequality

to those satisfying d(cid:0)cM , cM(cid:48)(cid:1) < 2sδ(cid:96) (M ); while in the ‘remote’ term V remote
d(cid:0)cM , cM(cid:48)(cid:1) ≥ 2sδ(cid:96) (M ). Then we further decompose

s

V remote
s

= T dif f erence

s

+ T intersection

s

,

360

where in the ‘difference’ term T dif f erence

s

we restrict integration in y to the difference R \

B(cid:0)M, M(cid:48)(cid:1) of R and

B(cid:0)M, M(cid:48)(cid:1) ≡ B

(cid:18)

d(cid:0)cM , cM(cid:48)(cid:1)(cid:19)

,

cM ,

1
2

2 d(cid:0)cM , cM(cid:48)(cid:1); while in the ‘intersection’ term T intersection
we restrict integration in y to the intersection Rn∩B(cid:0)M, M(cid:48)(cid:1) of Rn with the ball B(cid:0)M, M(cid:48)(cid:1);

the ball centered at cM with radius 1

s

(.0.46)

(.0.47)

i.e.

T intersection
s

≡ (cid:88)

F,F(cid:48)∈F

(cid:88)
M∈W(F ), M(cid:48)∈W(cid:16)
F(cid:48)(cid:17)
M(cid:48)(cid:17)
(cid:16)
(cid:16)
(cid:17)≥2s(1+δ)(cid:96)
(cid:16)
M(cid:48)(cid:17)
M,M(cid:48)⊂I, (cid:96)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
cM ,cM(cid:48)

=2−s(cid:96)(M )

F,M x

L2(ω)

d

(cid:90)

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:0)|M(cid:48)| +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:1)n+1−α dσ (y) .
We will exploit the restriction of integration to B(cid:0)M, M(cid:48)(cid:1), together with the condition

(|M| + |y − cM|)n+1−α

(cid:13)(cid:13)(cid:13)♠2

B(M,M(cid:48))

F(cid:48),M(cid:48)x

L2(ω)

d(cid:0)cM , cM(cid:48)(cid:1) ≥ 2s(1+δ)(cid:96)(cid:0)M(cid:48)(cid:1) = 2sδ(cid:96) (M ) ,

which will then give an estimate for the term T intersection

s

using an argument dual to that

used for the other terms T proximal

s

and T dif f erence

s

, to which we now turn.

361

The proximal and difference terms

We have

≡ (cid:88)

F,F(cid:48)∈F

T proximal
s

(cid:88)
M∈W(F ), M(cid:48)∈W(cid:16)
F(cid:48)(cid:17)
(cid:16)
M(cid:48)(cid:17)
(cid:16)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
=2−s(cid:96)(M ) and d
M,M(cid:48)⊂I, (cid:96)
(cid:0)|M(cid:48)| +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:1)n+1−α dσ (y)
(cid:88)
(cid:107)Qω,b∗

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:88)

(cid:17)
(cid:13)(cid:13)(cid:13)♠2

ω = Mproximal

F,M z(cid:107)♠2

F(cid:48),M(cid:48)x

<2sδ(cid:96)(M )

F,M x

t2dµ,

(|M| + |y − cM|)n+1−α

L2(ω)

L2(ω)

cM ,c

M(cid:48)

s

(cid:90)
(cid:98)I

F∈F

M∈W(F )

M⊂I

(cid:90)

Rn

×

≤ Mproximal

s

(.0.48)

where

Mproximal

s

≡ sup
F∈F

(M ) ≡ (cid:88)

F(cid:48)∈F

Aproximal

s

Aproximal

s

(M ) ;

sup
M⊂I

M∈W(F )

(cid:88)
M(cid:48)∈W(cid:16)
M(cid:48)(cid:17)
(cid:16)
(cid:16)
(cid:17)
M(cid:48)⊂I, (cid:96)
=2−s(cid:96)(M ) and
<2sδ(cid:96)(M )

F(cid:48)(cid:17)

d

cM ,c

M(cid:48)

(cid:90)

Rn

SF(cid:48)
(M(cid:48),M) (y) dσ (y) ;

(cid:13)(cid:13)(cid:13)♠2

(cid:13)(cid:13)(cid:13)Qω
(cid:0)|M(cid:48)| +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:1)n+1−α ,

F(cid:48),M(cid:48)x

L2(ω)

SF(cid:48)
(M(cid:48),M) (x) ≡

1

(|M| + |y − cM|)n+1−α

362

and similarly

T dif f erence
s

≡ (cid:88)

F,F(cid:48)∈F

(.0.49)

(cid:88)
M∈W(F ), M(cid:48)∈W(cid:16)
F(cid:48)(cid:17)
(cid:17)≥2sδ(cid:96)(M )
M(cid:48)(cid:17)
(cid:16)
(cid:16)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
=2−s(cid:96)(M ) and d
M,M(cid:48)⊂I, (cid:96)
(cid:0)|M(cid:48)| +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:1)n+1−α dσ (y)
(cid:90)
(cid:88)
(cid:98)I

(|M| + |y − cM|)n+1−α
(cid:88)
F,M z(cid:107)♠2

ω = Mdif f erence

(cid:107)Qω,b∗

F(cid:48),M(cid:48)x

F,M x

t2dµ;

L2(ω)

L2(ω)

cM ,c

M(cid:48)

s

F∈F

M∈W(F )

M⊂I

(cid:90)

×

Rn\B(M,M(cid:48))
≤ Mdif f erence

s

where

Mdif f erence

s

≡ sup
F∈F

(M ) ≡ (cid:88)

F(cid:48)∈F

Adif f erence

s

Adif f erence

s

(M ) ;

(cid:90)

sup
M⊂I

M∈W(F )

Rn\B(M,M(cid:48))

SF(cid:48)
(M(cid:48),M) (y) dσ (y) .

(cid:88)
M(cid:48)∈W(cid:16)
F(cid:48)(cid:17)
M(cid:48)(cid:17)
(cid:16)
(cid:16)
(cid:17)≥2sδ(cid:96)(M )
M(cid:48)⊂I, (cid:96)
cM ,cM(cid:48)
to Rn \ B(cid:0)M, M(cid:48)(cid:1) will be used to establish

=2−s(cid:96)(M ) and

d

The restriction of integration in Adif f erence
(.0.51) below.

s

Notation .0.13. Since the cubes F, M, F(cid:48), M(cid:48) that arise in all of the sums here satisfy

M ∈ W (F ) , M(cid:48) ∈ W(cid:0)F(cid:48)(cid:1) and (cid:96)(cid:0)M(cid:48)(cid:1) = 2−s(cid:96) (M ) and M, M(cid:48) ⊂ I,

we will often employ the notation

∗(cid:80) to remind the reader that, as applicable, these four

conditions are in force even when they are not explictly mentioned.

363

Now fix M as in Mproximal

s

respectively Mdif f erence

, and decompose the sum over M(cid:48)

in Aproximal

s

(M ) respectively Adif f erence

(cid:90)

Rn

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

s

(cid:88)

F(cid:48)∈F

s
(M ) by

F(cid:48)(cid:17)

(cid:88)
M(cid:48)∈W(cid:16)
M(cid:48)(cid:17)
(cid:16)
(cid:17)
(cid:16)
=2−s(cid:96)(M ) and
M(cid:48)⊂I, (cid:96)
cM ,cM(cid:48)
<2sδ(cid:96)(M )
SF(cid:48)
(M(cid:48),M) (y) dσ (y)

(cid:90)

d

Rn

Aproximal

s

(M ) =

(cid:88)

F(cid:48)∈F

(cid:16)

d

∗(cid:88)
(cid:17)
cM(cid:48)∈2M

cM ,cM(cid:48)

<2sδ(cid:96)(M )

(cid:88)

∞(cid:88)

+

F(cid:48)∈F

(cid:96)=1

d

(cid:90)

Rn

∗(cid:88)
(cid:17)

(cid:16)
cM(cid:48)∈2(cid:96)+1M\2(cid:96)M
cM ,cM(cid:48)
<2sδ(cid:96)(M )

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

=

≡

∞(cid:88)

(cid:96)=0

Aproximal,(cid:96)

s

(M ) ,

364

respectively

Adif f erence

s

(M ) =

(cid:88)

F(cid:48)∈F

(cid:88)

F(cid:48)∈F

d

(cid:16)
cM ,cM(cid:48)

∗(cid:88)
(cid:17)≥2sδ(cid:96)(M )
cM(cid:48)∈2M
∞(cid:88)
(cid:88)

+

F(cid:48)∈F

(cid:96)=1

∞(cid:88)

(cid:96)=0

Adif f erence,(cid:96)

s

(M ) .

=

≡

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

Rn\B(M,M(cid:48))

(cid:90)

(cid:88)
M(cid:48)∈W(cid:16)
F(cid:48)(cid:17)
M(cid:48)(cid:17)
(cid:16)
(cid:17)≥2sδ(cid:96)(M )
(cid:16)
M(cid:48)⊂I, (cid:96)
cM ,cM(cid:48)
SF(cid:48)
(M(cid:48),M) (y) dσ (y)

=2−s(cid:96)(M ) and

Rn\B(M,M(cid:48))

(cid:90)

d

(cid:90)

Rn\B(M,M(cid:48))

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

∗(cid:88)
(cid:16)
(cid:17)≥2sδ(cid:96)(M )
cM(cid:48)∈2(cid:96)+1M\2(cid:96)M
cM ,cM(cid:48)

d

Let m = 2 so that

2−m ≤ 1
3

.

(.0.50)

Now decompose the integrals over Rn in Aproximal,(cid:96)

Aproximal,0

s

(M ) =

(M ) by

(cid:90)

Rn\4M

(cid:90)

s

∗(cid:88)
(cid:17)
cM(cid:48)∈2M
∗(cid:88)
(cid:17)
cM(cid:48)∈2M

(cid:88)
(cid:88)

F(cid:48)∈F

(cid:16)

d

+

cM ,cM(cid:48)
(cid:16)
≡ Aproximal,0

F(cid:48)∈F

d

s,f ar

<2sδ(cid:96)(M )

cM ,cM(cid:48)

<2sδ(cid:96)(M )

(M ) + Aproximal,0

s,near

(M ) ,

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

4M

365

and for (cid:96) ≥ 1

Aproximal,(cid:96)

s

(M ) =

(cid:88)

F(cid:48)∈F

2(cid:96)+2M\2(cid:96)−mM

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

(cid:90)

Rn\2(cid:96)+2M

(cid:90)

(cid:16)
cM(cid:48)∈2(cid:96)+1M\2(cid:96)M
cM ,cM(cid:48)
<2sδ(cid:96)(M )

(cid:16)
cM(cid:48)∈2(cid:96)+1M\2(cid:96)M
cM ,cM(cid:48)
<2sδ(cid:96)(M )

d

(cid:90)

d

(cid:88)

F(cid:48)∈F

(cid:88)

+

+

∗(cid:88)
(cid:17)
∗(cid:88)
(cid:17)
∗(cid:88)
(cid:17)

F(cid:48)∈F

(cid:16)
cM(cid:48)∈2(cid:96)+1M\2(cid:96)M
cM ,cM(cid:48)
<2sδ(cid:96)(M )
≡ Aproximal,(cid:96)
(M ) + Aproximal,(cid:96)

d

s,near

s,f ar

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

2(cid:96)−mM

(M ) + Aproximal,(cid:96)

s,close

(M )

Similarly we decompose the integrals over the difference

B∗ ≡ Rn \ B(cid:0)M, M(cid:48)(cid:1)

in Adif f erence,(cid:96)

s

(M ) by

Adif f erence,0

s

(M ) =

(cid:90)

∗(cid:88)
(cid:17)≥2sδ(cid:96)(M )
cM(cid:48)∈2M
∗(cid:88)
(cid:17)≥2sδ(cid:96)(M )
cM(cid:48)∈2M

(cid:88)
(cid:88)

d

F(cid:48)∈F

(cid:16)
cM ,cM(cid:48)
(cid:16)
≡ Adif f erence,0

F(cid:48)∈F

+

d

s,f ar

cM ,cM(cid:48)

(cid:90)

(M ) + Adif f erence,0

s,near

(M ) ,

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

B∗\4M

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

B∗∩4M

366

and

Adif f erence,(cid:96)

s

(M ) =

(cid:88)

F(cid:48)∈F

+

F(cid:48)∈F

(cid:88)

∗(cid:88)
(cid:17)≥2sδ(cid:96)(M )
(cid:16)
cM(cid:48)∈2(cid:96)+1M\2(cid:96)M
∗(cid:88)
cM ,cM(cid:48)
(cid:16)
(cid:17)≥2sδ(cid:96)(M )
cM(cid:48)∈2(cid:96)+1M\2(cid:96)M
cM ,cM(cid:48)
≡ Adif f erence,(cid:96)

(cid:88)

F(cid:48)∈F

+

d

d

s,f ar

(cid:90)

B∗\2(cid:96)+2M

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

(cid:17) SF(cid:48)
(M(cid:48),M) (y) dσ (y)

d

∗(cid:88)
(cid:17)≥2sδ(cid:96)(M )
(cid:16)
cM(cid:48)∈2(cid:96)+1M\2(cid:96)M
(cid:90)
cM ,cM(cid:48)
B∗∩(cid:16)
(cid:90)

2(cid:96)+2M\2(cid:96)−mM

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

B∗∩2(cid:96)−mM

(M ) + Adif f erence,(cid:96)

s,near

(M ) + Adif f erence,(cid:96)

s,close

(M ) ,

(cid:96) ≥ 1.

We now note the important point that the close terms

(M ) and Adif f erence,(cid:96)

s,close

(M ) both vanish for (cid:96) > δs because of the decomposition

Aproximal,(cid:96)
s,close
(.0.45):

Aproximal,(cid:96)

s,close

(M ) = Adif f erence,(cid:96)

s,close

(M ) = 0,

(cid:96) ≥ 1 + δs.

(.0.51)

Indeed, if cM(cid:48) ∈ 2(cid:96)+1M \ 2(cid:96)M, then we have

1
2

2(cid:96)(cid:96) (M ) ≤ d(cid:0)cM , cM(cid:48)(cid:1) .
(M ) satisfy d(cid:0)cM , cM(cid:48)(cid:1) < 2δs(cid:96) (M ), which by (.0.52) is

(.0.52)

Now the summands in Aproximal,(cid:96)
impossible if (cid:96) ≥ 1 + δs - indeed, if (cid:96) ≥ 1 + δs, we get the contradiction

s,close

2δs(cid:96) (M ) =

1
2

21+δs(cid:96) (M ) ≤ 1
2

2(cid:96)(cid:96) (M ) ≤ d(cid:0)cM , cM(cid:48)(cid:1) < 2δs(cid:96) (M ) .

367

s,close

s,close

(M ) = 0. Thus we are left to consider the term

It now follows that Aproximal,(cid:96)
Adif f erence,(cid:96)
also restricted in Adif f erence,(cid:96)

(M ), where the integration is taken over the set Rn \ B(cid:0)M, M(cid:48)(cid:1). But we are
in B(cid:0)M, M(cid:48)(cid:1) by (.0.52). Indeed, the smallest ball centered at cM that contains 2(cid:96)−mM
2 d(cid:0)cM , cM(cid:48)(cid:1),
the radius of B(cid:0)M, M(cid:48)(cid:1). Thus the range of integration in the term Adif f erence,(cid:96)

22(cid:96)−m(cid:96) (M ), which by (.0.50) and (.0.52) is at most 1

(M ) to integrating over the cube 2(cid:96)−mM, which is contained

42(cid:96)(cid:96) (M ) ≤ 1

has radius 1

(M ) is the

s,close

s,close

empty set, and so Adif f erence,(cid:96)

s,close

(M ) = 0 as well as Aproximal,(cid:96)

s,close

(M ) = 0. This proves (.0.51).

s

s

s,close

s,close

and T dif f erence

(M ) and Adif f erence,(cid:96)

From now on we treat T proximal

in the same way since the terms
(M ) both vanish for (cid:96) ≥ 1 + δs. Thus we will sup-
Aproximal,(cid:96)
press the superscripts proximal and dif f erence in the f ar, near and close decomposi-
tion of Aproximal,(cid:96)

d(cid:0)cM , cM(cid:48)(cid:1) < 2sδ(cid:96) (M ) and d(cid:0)cM , cM(cid:48)(cid:1) ≥ 2sδ(cid:96) (M ) in the proximal and difference terms

(M ), and we will also suppress the conditions

(M ) and Adif f erence,(cid:96)

s

s

since they no longer play a role. Using the pairwise disjointedness of the shifted coronas
CG,shif t

, we have

F

(cid:13)(cid:13)(cid:13)Qω,b∗

F(cid:48),A

(cid:13)(cid:13)(cid:13)♠2

x

L2(ω)

(cid:88)

F(cid:48)∈F

(cid:46) |A|2 |A|ω ,

for any cube A.

Note that if cM(cid:48) ∈ 2M, then M(cid:48) ⊂ 3M. Then with

M ≡ (cid:91)

Ws

F(cid:48)∈F

(cid:8)M(cid:48) ∈ W(cid:0)F(cid:48)(cid:1) : M(cid:48) ⊂ 3M and (cid:96)(cid:0)M(cid:48)(cid:1) = 2−s(cid:96) (M )(cid:9) ,

(.0.53)

368

we have

A0

∗(cid:88)

(cid:90)
(cid:88)

cM(cid:48)∈2M

Rn\4M

F(cid:48)∈F: A∈W(F(cid:48))

F(cid:48)∈F

s,f ar (M ) ≤ (cid:88)
(cid:46) (cid:88)
(cid:46) (cid:88)
 (cid:88)

A∈Ws
M

A∈Ws
M

=

(cid:90)

A∈Ws
M

SF(cid:48)
(M(cid:48),M) (y) dσ (y)
(cid:90)

(cid:13)(cid:13)(cid:13)Qω,b∗

F(cid:48),M(cid:48)x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(|M| + |y − cM|)2(n+1−α)

dσ (y)

Rn\4M
|A|2 |A|ω

Rn\4M

(|M| + |y − cM|)2(n+1−α)

dσ (y)

|A|2 |A|ω

Rn\4M

1

(|M| + |y − cM|)2(n+1−α)

dσ (y) .

Now we use the standard pigeonholing of side length of A to conclude that

(cid:88)

A∈Ws
M

(cid:88)

|A|2 |A|ω =

≤

≤

|A|2 |A|ω
M : (cid:96)(A)=2−k(cid:96)(M )
(cid:88)

A∈Ws
2−2k |M|2

A∈Ws
2−2k |M|2 |3M|ω

M : (cid:96)(A)=2−k(cid:96)(M )
(cid:46) 2−2s |M|2 |3M|ω ,

|A|ω

(.0.54)

(cid:90)

(cid:90)

k=s

∞(cid:88)
∞(cid:88)
∞(cid:88)

k=s

k=s

so that combining the previous two displays we have

1

s,f ar (M ) (cid:46) 2−2s |M|2 |3M|ω
(cid:90)
A0
(cid:90)
|4M|1−αPα(cid:16)

≤ 2−2s |4M|ω
≈ 2−2s |4M|ω
|4M|1−α
(cid:46) 2−2s |4M|ω

Rn\4M

Rn\4M

1

dσ (y)

(cid:32)
(|M| + |y − cM|)2(1−α)

(|M| + |y − cM|)2(n+1−α)
(cid:33)1−α
(cid:17) (cid:46) 2−2sAα

(|M| + |y − cM|)2

|M|

2 .

4M, 1Rn\4M σ

Rn\4M

dσ (y)

dσ (y)

369

To estimate the near term A0
write

s,near (M ), we initially keep the energy (cid:13)(cid:13)(cid:13)Qω,b∗
∗(cid:88)

(cid:90)

F(cid:48),M(cid:48)z

(cid:13)(cid:13)(cid:13)2

L2(ω)

and

cM(cid:48)∈2M

4M

1

(cid:18)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)Qω,b∗

|M|n+1−α
∗(cid:88)
∗(cid:88)

F(cid:48),M(cid:48)x

F(cid:48),M(cid:48)x

|M(cid:48)| 1

L2(ω)

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

F(cid:48),M(cid:48)x

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
n +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:19)n+1−α dσ (y)
(cid:90)
(cid:13)(cid:13)(cid:13)♠2
(cid:18)
Pα(cid:0)M(cid:48), 14M σ(cid:1)
(cid:13)(cid:13)(cid:13)♠2

|M(cid:48)| 1

L2(ω)

4M

.

L2(ω)

|M(cid:48)| 1

n

dσ (y)

n +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:19)n+1−α

A0

s,near (M ) ≤ (cid:88)
(cid:90)

∗(cid:88)

F(cid:48)∈F

M(cid:48)∈2M

c

4M

1

|M|n+1−α

cM(cid:48)∈2M

1

|M|n+1−α

cM(cid:48)∈2M

F(cid:48)∈F

≈ (cid:88)
(cid:88)
(cid:88)

=

F(cid:48)∈F

=

F(cid:48)∈F

In order to estimate the final sum above, we must invoke the ‘prepare to puncture’ argument

above, as we will want to derive geometric decay from(cid:13)(cid:13)(cid:13)Qω,b∗
‘nonenergy’ term(cid:12)(cid:12)M(cid:48)(cid:12)(cid:12)2(cid:12)(cid:12)M(cid:48)(cid:12)(cid:12)ω
an augmented cube(cid:102)M ∈AD satisfying (cid:83)
(cid:101)ω = ω − ω ({p}) δp where p is an atomic point in (cid:102)M for which

(cid:13)(cid:13)(cid:13)♠2
M(cid:48) ⊂ 4M ⊂ (cid:102)M and (cid:96)

cM(cid:48)∈2M

M(cid:48) x

L2(ω)

by dominating it by the

(cid:16)(cid:102)M
(cid:17) ≤ C(cid:96) (M ). Define

, as well as using the Muckenhoupt energy constant. Choose

ω ({p}) =

q∈P(σ,ω): q∈(cid:102)M

sup

ω ({q}) .

(If ω has no atomic point in common with σ in (cid:102)M, set (cid:101)ω = ω). Then we have (cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:16)(cid:102)M , P(σ,ω)

ω

=

(cid:17) and(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)1− α

n

(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)1− α

n

(cid:17)

(cid:16)(cid:102)M , P(σ,ω)
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)1− α

n

(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)1− α

n

ω

=

≤ Aα,punct

2

.

370

From (.0.39) and (.0.19) we also have

(cid:13)(cid:13)(cid:13)Qω,b∗

F(cid:48),A

(cid:13)(cid:13)(cid:13)♠2

x

L2(ω)

(cid:88)

F(cid:48)∈F

(cid:46) (cid:96) (A)2 |A|(cid:101)ω ,

for any cube A.

Now by Cauchy-Schwarz and the augmented local estimate (.0.35) in Lemma .0.12 with

M = (cid:102)M applied to the second line below, and with Ws

M

as in (.0.53), and noting (.0.54),

the last sum in (.0.55) is dominated by

(.0.55)

2

2

L2(ω)

L2(ω)

F(cid:48)∈F

F(cid:48)∈F

c

F(cid:48),M(cid:48)x

F(cid:48),M(cid:48)x

c(M(cid:48))∈2M

∗(cid:88)
∗(cid:88)

(cid:88)
(cid:88)
 (cid:88)
(cid:113)|4M|(cid:101)ω

 1
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qω,b∗
2 1
Pα(cid:0)M(cid:48), 14M σ(cid:1)
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qω,b∗
 1
M(cid:48)∈2M
(cid:114)(cid:12)(cid:12)(cid:12)(cid:102)M
2(cid:113)(cid:0)Eα
(cid:12)(cid:12)(cid:12)σ
(cid:1)2 + Aα,energy
|A|2 |A|(cid:101)ω
(cid:114)(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:113)(cid:0)Eα
(cid:12)(cid:12)(cid:12)σ
(cid:1)2 + Aα,energy
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)
(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)n+1−α
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)n+1−α
(cid:113)
(cid:1)2 + Aα,energy

A∈Ws
M

Aα,punct

.

|M(cid:48)| 1

n

2

2

2

2

2

2

(Eα)2 + Aα,energy

2

1

|M|n+1−α

×

1

(cid:46)

|M|n+1−α
(cid:46) 2−s |M|
|M|n+1−α
(cid:46) 2−s(cid:113)
(cid:113)(cid:0)Eα

(cid:46) 2−s

2

Similarly, for (cid:96) ≥ 1, we can estimate the far term A(cid:96)
A0
s,f ar (M ) but applied to 2(cid:96)M in place of M. For this need the following variant of Ws

s,f ar (M ) by the argument used for
in

M

371

(cid:110)
M(cid:48) ∈ W(cid:0)F(cid:48)(cid:1) : M(cid:48) ⊂ 3

(cid:16)

2(cid:96)M

(cid:17) and (cid:96)(cid:0)M(cid:48)(cid:1) = 2−s−(cid:96)(cid:96)

(cid:16)

2(cid:96)M

(cid:17)(cid:111)

.

(.0.56)

(.0.53) given by

M ≡ (cid:91)

Ws,(cid:96)

F(cid:48)∈F

Then we have

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

(cid:13)(cid:13)(cid:13)Qω,b∗

F(cid:48),A

(cid:13)(cid:13)(cid:13)♠2

x

L2(ω)

Rn\4

(|M| + |y − cM|)2(n+1−α)

dσ (y)

M(cid:48)∈(cid:16)

∗(cid:88)
(cid:88)

2(cid:96)+1M

(cid:17)\(cid:16)

(cid:17)

2(cid:96)M

(cid:90)

(cid:90)

F(cid:48)∈F: A∈W(F(cid:48))
(cid:17)

(cid:16)

Rn\4

2(cid:96)M

(cid:90)

Rn\2(cid:96)+2M

(cid:16)
(cid:17)
2(cid:96)M
|A|2 |A|ω

A(cid:96)

c

F(cid:48)∈F

s,f ar (M ) ≤ (cid:88)
(cid:46) (cid:88)
(cid:46) (cid:88)
 (cid:88)

A∈Ws,(cid:96)
M

A∈Ws,(cid:96)
M

=

(cid:90)

|A|2 |A|ω

A∈Ws,(cid:96)
M

(|M| + |y − cM|)2(n+1−α)

dσ (y)

(cid:16)

(cid:17)

2(cid:96)M

1

(|M| + |y − cM|)2(n+1−α)

dσ (y) ,

Rn\4

where, just as for the sum over A ∈ Ws,0

, we have

(cid:88)

A∈Ws,(cid:96)
M

|A|2 |A|ω =

≤

≤

M

A∈Ws,(cid:96)

(cid:16)
M : (cid:96)(A)=2−k−(cid:96)(cid:96)

(cid:88)
(cid:12)(cid:12)(cid:12)2
2−2k−2(cid:96)(cid:12)(cid:12)(cid:12)2(cid:96)M
(cid:12)(cid:12)(cid:12)2(cid:12)(cid:12)(cid:12)3
2−2k−2(cid:96)(cid:12)(cid:12)(cid:12)2(cid:96)M

(cid:16)

A∈Ws,(cid:96)

k=s

∞(cid:88)
∞(cid:88)
∞(cid:88)

k=s

k=s

2(cid:96)M

(cid:17)|A|2 |A|ω
(cid:88)
(cid:17)|A|ω
(cid:12)(cid:12)(cid:12)2(cid:12)(cid:12)(cid:12)3
(cid:46) 2−2s−2(cid:96)(cid:12)(cid:12)(cid:12)2(cid:96)M

(cid:16)
M : (cid:96)(A)=2−k−(cid:96)(cid:96)
2(cid:96)M

(cid:17)(cid:12)(cid:12)(cid:12)ω

2(cid:96)M

(.0.57)

(cid:16)

2(cid:96)M

(cid:17)(cid:12)(cid:12)(cid:12)ω

.

372

(cid:12)(cid:12)(cid:12)2(cid:96)M

(cid:12)(cid:12)(cid:12)2

dσ (y)

2(cid:96)M

A(cid:96)

(cid:83)

1

2(cid:96)M

Rn\2(cid:96)+2M

Rn\2(cid:96)+2M

2(cid:96)M

≈ 2−2s2−2(cid:96)

(cid:46) 2−2s2−2(cid:96)

dσ (y)

2 .

Now using

tinue with

2(cid:96)+2M, 1Rn\2(cid:96)+2M

σ

(cid:12)(cid:12)(cid:12)(cid:17)2(1−α)

for y /∈ 2(cid:96)+2M, we can con-

(cid:12)(cid:12)(cid:12)2(cid:96)+2M
(cid:12)(cid:12)(cid:12)ω
(cid:90)
(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)1−α
(cid:12)(cid:12)(cid:12)2(cid:96)+2M
(cid:12)(cid:12)(cid:12)ω

(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)1−αPα(cid:16)

(cid:12)(cid:12)(cid:12)y−c
(cid:12)(cid:12)(cid:12)+
(cid:16)(cid:12)(cid:12)(cid:12)2(cid:96)M
(|M|+|y−cM|)2(n+1−α) ≤
(cid:90)
(cid:12)(cid:12)(cid:12)ω
s,f ar (M ) (cid:46) 2−2s2−2(cid:96)(cid:12)(cid:12)(cid:12)2(cid:96)+2M


(cid:12)(cid:12)(cid:12)(cid:17)2(1−α)
(cid:12)(cid:12)(cid:12)y − c
(cid:16)(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12) +
1−α
(cid:12)(cid:12)(cid:12)2(cid:96)M
(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:17)2
(cid:12)(cid:12)(cid:12)y − c
(cid:16)(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12) +
(cid:17) (cid:46) 2−2s2−2(cid:96)Aα
(cid:16)(cid:102)M
(cid:17) ≤ C2(cid:96)(cid:96) (M ) and
ture’ argument. Choose an augmented cube (cid:102)M ∈ AD such that (cid:96)
M(cid:48) ⊂ 2(cid:96)+2M ⊂ (cid:102)M. Define (cid:101)ω = ω − ω ({p}) δp where p is an atomic point
in (cid:102)M for which ω ({p}) =
(cid:12)(cid:12)(cid:12)(cid:101)ω
(If ω has no atomic point in common with σ in (cid:102)M set (cid:101)ω = ω.) Then we have (cid:12)(cid:12)(cid:12)(cid:102)M
(cid:17), and just as in the argument above following (.0.55), we have from (.0.39)
(cid:16)(cid:102)M , P(σ,ω)
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)1− α
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)1− α

s,near (M ) we must again invoke the ‘prepare to punc-

(cid:46) (cid:96)(cid:0)M(cid:48)(cid:1)2(cid:12)(cid:12)M(cid:48)(cid:12)(cid:12)(cid:101)ω .

To estimate the near term A(cid:96)

(cid:13)(cid:13)(cid:13)Qω,b∗

F(cid:48),M(cid:48)x

(cid:13)(cid:13)(cid:13)♠2

q∈P(σ,ω): q∈(cid:102)M

sup

and (cid:88)

F(cid:48)∈F

cM(cid:48)∈2(cid:96)+1M\2(cid:96)M

≤ Aα,punct

2

n

n

ω

and (.0.19) that both

=

L2(ω)

ω ({q}) .

373

Thus using that m = 2 in the definition of A(cid:96)

s,near (M ), we see that

∗(cid:88)

(cid:90)

s,near (M ) ≤ (cid:88)

A(cid:96)

c

(cid:90)

F(cid:48)∈F

≈(cid:88)
∗(cid:88)
(cid:88)
(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)n+1−α

(cid:46)

1

F(cid:48)∈F

cM(cid:48)∈2(cid:96)+1M\2(cid:96)M

∗(cid:88)

F(cid:48)∈F

M(cid:48)∈2(cid:96)+1M\2(cid:96)M

2(cid:96)+2M\2(cid:96)−mM

2(cid:96)+2M\2(cid:96)−mM

(cid:18)

1

(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)n+1−α
(cid:90)
(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(cid:13)(cid:13)(cid:13)Qω,b∗

F(cid:48),M(cid:48)x

|M(cid:48)| 1

2(cid:96)+2M

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

L2(ω)

dσ (y)

|M(cid:48)| 1

F(cid:48),M(cid:48)x

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
n +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:19)n+1−α dσ (y)
n +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:19)n+1−α
(cid:18)
(cid:17)
Pα(cid:16)
 1

M(cid:48), 1
|M(cid:48)| 1

2(cid:96)+2M

σ

n

2

L2(ω)

(cid:17)

2 1

2

.

2(cid:96)+2M

σ

M(cid:48), 1
|M(cid:48)| 1

n

cM(cid:48)∈2(cid:96)+1M\2(cid:96)M

is dominated by

F(cid:48)∈F

(cid:88)
(cid:88)
∗(cid:88)

F(cid:48)∈F

1

(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)n+1−α
(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)n+1−α
(cid:88)

1

×

F(cid:48)∈F

cM(cid:48)∈2(cid:96)+1M\2(cid:96)M

∗(cid:88)
∗(cid:88)
(cid:13)(cid:13)(cid:13)Qω,b∗

F(cid:48),M(cid:48)x

L2(ω)

F(cid:48),M(cid:48)x

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
Pα(cid:16)
(cid:13)(cid:13)(cid:13)♠2

F(cid:48),M(cid:48)x

L2(ω)

cM(cid:48)∈2(cid:96)+1M\2(cid:96)M

cM(cid:48)∈2(cid:96)+1M\2(cid:96)M

≤

This can now be estimated as for the term A0

s,near (M ), along with the augmented local

374

estimate (.0.35) in Lemma .0.12 with M = (cid:102)M applied to the final line above, to get
(cid:114)(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)σ

s,near (M ) (cid:46) 2−s2−(cid:96)
A(cid:96)

2

(cid:12)(cid:12)(cid:12)2(cid:96)M
(cid:12)(cid:12)(cid:12)
(cid:114)(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)n+1−α
(cid:113)(cid:0)Eα
(cid:1)2 + Aα,energy
(cid:113)(cid:0)Eα
(cid:1)2 + Aα,energy

2

2

2

2

2

(cid:113)(cid:0)Eα
(cid:1)2 + Aα,energy
(cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116)
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)1− α
(cid:12)(cid:12)(cid:12)(cid:102)M
(cid:12)(cid:12)(cid:12)1− α
(cid:113)

n

n

Aα,punct

.

2

(cid:46) 2−s2−(cid:96)

(cid:46) 2−s2−(cid:96)

Each of the estimates for A(cid:96)

s,f ar (M ) and A(cid:96)

s,near (M ) is summable in both s and (cid:96).

Now we turn to the terms A(cid:96)

s,close (M ), and recall from (.0.51) that A(cid:96)

s,close (M ) = 0 if

(cid:96) ≥ 1 + δs. So we now suppose that (cid:96) ≤ δs. We have, with m = 2 as in (.0.50),

∗(cid:88)

(cid:90)

cM(cid:48)∈2(cid:96)+1M\2(cid:96)M

SF(cid:48)
(M(cid:48),M) (y) dσ (y)

2(cid:96)−mM

(cid:13)(cid:13)(cid:13)♠2

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)n+1−α dσ (y)

F(cid:48),M(cid:48)x

L2(ω)

A(cid:96)

F(cid:48)∈F

s,close (M ) ≤ (cid:88)
∗(cid:88)
∗(cid:88)

c

M(cid:48)∈2(cid:96)+1M\2(cid:96)M

≈ (cid:88)
(cid:88)

F(cid:48)∈F

=

F(cid:48)∈F

cM(cid:48)∈2(cid:96)+1M\2(cid:96)M

(cid:90)

(cid:90)

×

(cid:18)

|M| 1

n + |y − cM|

1

(cid:19)n+1−α
(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12)n+1−α
(cid:19)n+1−α dσ (y) .

1



1

(cid:13)(cid:13)(cid:13)♠2

F(cid:48),M(cid:48)x

L2(ω)

2(cid:96)−mM

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:18)

2(cid:96)−mM

|M| 1

n + |y − cM|

The argument used to prove (.0.57) gives the analogous inequality with a hole 2(cid:96)−1M,

(cid:88)

∗(cid:88)

F(cid:48)∈F

M(cid:48)∈2(cid:96)+1M\2(cid:96)M

c

(cid:13)(cid:13)(cid:13)Qω,b∗

F(cid:48),M(cid:48)x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(cid:46) 2−2s(cid:12)(cid:12)(cid:12)2(cid:96)M

n(cid:12)(cid:12)(cid:12)2(cid:96)+2M \ 2(cid:96)−1M
(cid:12)(cid:12)(cid:12) 2

(cid:12)(cid:12)(cid:12)ω

.

375

Thus we get that A(cid:96)

(cid:46) 2−2s(cid:12)(cid:12)(cid:12)2(cid:96)M
(cid:46) 2−2s(cid:12)(cid:12)(cid:12)2(cid:96)M

(cid:90)

s,close (M ) is bounded by

1
n (n+1−α)

(cid:12)(cid:12)(cid:12)ω
n(cid:12)(cid:12)(cid:12)2(cid:96)+2M\2(cid:96)−1M
(cid:12)(cid:12)(cid:12) 2
(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12) 1
(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)2(cid:96)−mM
(cid:12)(cid:12)(cid:12)ω
(cid:12)(cid:12)(cid:12)2(cid:96)+2M \ 2(cid:96)−1M
(cid:12)(cid:12)(cid:12) 2
(cid:12)(cid:12)2(cid:96)M(cid:12)(cid:12) 1
(cid:12)(cid:12)(cid:12)ω
(cid:12)(cid:12)(cid:12)2(cid:96)+2M \ 2(cid:96)−1M
(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)2(cid:96)−mM
(cid:12)(cid:12)2(cid:96)−mM(cid:12)(cid:12)1− α
(cid:12)(cid:12)2(cid:96)+2M(cid:12)(cid:12)1− α

n (n+1−α)

n (n+1−α)

|M| 1

n

n

n

(cid:46) 2−2s2(n+1−α)(cid:96)

(cid:46) 2−2s2(n+1−α)(cid:96)Aα
2 ,

(cid:18)

2(cid:96)−mM

(cid:19)n+1−α

dσ (y)

|M| 1

n +|y − cM|

provided that m = 2 > 1. Note that we can use the offset Muckenhoupt constant Aα
here
2
since 2(cid:96)+2M \ 2(cid:96)−1M and 2(cid:96)−mM are disjoint. If (cid:96) ≤ s, then we have the relatively crude
estimate A(cid:96)
without decay in (cid:96). But we are assuming (cid:96) ≤ δs here, and so
we obtain a suitable estimate for A(cid:96)
. Indeed,
we then have

s,close (M ) provided we choose 0 < δ ≤

s,close (M ) (cid:46) 2−sAα

n+1−α

2

1

δs(cid:88)

l=1

2−2s2(n+1−α)(cid:96)Aα

2 = 2−2s

 δs(cid:88)

l=1

 Aα

2

2(n+1−α)(cid:96)

(cid:46) 2−2s2(n+1−α)δsAα

2 ≤ 2−sAα
2 ,

provided δ ≤ 1
prove

n+1−α

, and in particular we may take δ = 1
2

. Altogether, the above estimates

T proximal
s

+ T dif f erence

s

(cid:46) 2−s

which is summable in s.

(cid:18)

Aα
2 +

(cid:113)(cid:0)Eα

2

(cid:1)2 + Aα,energy

2

(cid:113)

Aα,punct

2

(cid:19)(cid:90)
(cid:98)I

t2dµ,

376

The intersection term

Now we return to the term T intersection

s

(cid:90)

(cid:88)

(cid:88)
F(cid:48)(cid:17)
M(cid:48)∈W(cid:16)
F,F(cid:48)∈F
M(cid:48)(cid:17)
(cid:16)
(cid:17)≥2s(1+δ)(cid:96)
(cid:16)
(cid:16)
M,M(cid:48)⊂I, (cid:96)
cM ,cM(cid:48)

M∈W(F )

d

=2−s(cid:96)(M )

M(cid:48)(cid:17)

≡

(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

(cid:13)(cid:13)(cid:13)♠2

B(M,M(cid:48))

(|M| + |y − cM|)n+1−α

L2(ω)

(cid:13)(cid:13)(cid:13)♠2

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:0)|M(cid:48)| +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:1)n+1−α dσ (y) .

F(cid:48),M(cid:48)x

L2(ω)

It will suffice to show that T intersection

satisfies the estimate,

T intersection
s

(cid:46) 2−sδ

= 2−sδ

(cid:113)(cid:0)Eα
(cid:113)(cid:0)Eα

2

2

s

(cid:113)
(cid:1)2+Aα,energy
(cid:113)
(cid:1)2 + Aα,energy

2

2

Aα,punct

2

(cid:88)
M(cid:48)∈M(ρ,ε)−deep

M(cid:48)⊂I

(cid:88)
F(cid:48)∈F(cid:48)
(cid:90)
(cid:98)I

Aα,punct

2

t2µ .

(cid:107)Qω,b∗
(cid:16)
F(cid:48)(cid:17)
F(cid:48),M(cid:48)x(cid:107)♠2

L2(ω)

377

(cid:16)

L2(ω)

(cid:18)

F,M x

cM , 1

|M| 1

F(cid:48),M(cid:48)x

(cid:13)(cid:13)(cid:13)♠2

L2(ω)
n + |y − cM|

2 d(cid:0)cM , cM(cid:48)(cid:1)(cid:17), we can write (suppressing some notation for
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:18)
(cid:19)n+1−α
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)♠2
(cid:90)
(cid:12)(cid:12)cM − cM(cid:48)(cid:12)(cid:12)n+1−α
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:90)
(cid:13)(cid:13)(cid:13)♠2
(cid:88)
(cid:88)
(cid:12)(cid:12)cM − cM(cid:48)(cid:12)(cid:12)n+1−α
(cid:13)(cid:13)(cid:13)♠2
(cid:0)M(cid:48)(cid:1) ,

(cid:13)(cid:13)(cid:13)Qω,b∗
n +(cid:12)(cid:12)y − cM(cid:48)(cid:12)(cid:12)(cid:19)n+1−α dσ (y)
(cid:19)n+1−α
(cid:18)
(cid:19)n+1−α

dσ (y)
n + |y − cM|
(cid:18)

n +|y − cM|

B(M,M(cid:48))

B(M,M(cid:48))

F(cid:48),M(cid:48)x

|M(cid:48)| 1

|M| 1

|M| 1

F,M x

dσ (y)

L2(ω)

L2(ω)

L2(ω)

L2(ω)

F

M

Ss

L2(ω)

(cid:18)

(cid:19)n+1−α ≈ Pα

dσ(y)

|M| 1

n +|y−cM|

(cid:18)

M,1

B(M,M(cid:48))σ
|M| 1
n

(cid:19)

, it remains to show that

Recalling B(cid:0)M, M(cid:48)(cid:1) = B

clarity) T intersection

s

as

B(M,M(cid:48))

F,M x

M,M(cid:48)

(cid:88)

(cid:90)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:88)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:88)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:88)

F(cid:48),M(cid:48)x

F(cid:48),M(cid:48)x

M,M(cid:48)

M(cid:48)

M(cid:48)

=

F,F(cid:48)

F,F(cid:48)

(cid:88)
≈(cid:88)
≤(cid:88)
≡(cid:88)
and since(cid:82)

F(cid:48)

F(cid:48)

B(M,M(cid:48))
for each fixed M(cid:48),

(cid:0)M(cid:48)(cid:1) ≈ (cid:88)

Ss

F

∗(cid:88)
(cid:17)≥2s(1+δ)(cid:96)
(cid:16)
(cid:113)
(cid:1)2 + Aα,energy

M(cid:48)

2

cM ,c

(cid:16)
(cid:113)(cid:0)Eα

2

M : d

Aα
2 .

(cid:46) 2−δs

(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:12)(cid:12)cM − cM(cid:48)(cid:12)(cid:12)n+1−α

F,M x

L2(ω)

Pα(cid:16)

(cid:17)

M, 1B(M,M(cid:48))σ

|M| 1

n

M(cid:48)(cid:17)

378

We write

(cid:0)M(cid:48)(cid:1) ≈ (cid:88)
(cid:0)M(cid:48)(cid:1) ≡ (cid:88)

k≥s(1+δ)

Ss

Sk
s

F

M : d

s

1

(cid:0)2k |M(cid:48)|(cid:1)n+1−α Sk
∗(cid:88)
(cid:17)≈2k(cid:96)(M(cid:48))
(cid:16)

cM ,c

M(cid:48)

(cid:0)M(cid:48)(cid:1) ;
(cid:13)(cid:13)(cid:13)Qω,b∗

F,M x

Pα(cid:16)

(cid:13)(cid:13)(cid:13)♠2

L2(ω)

(cid:17)

,

M, 1B(M,M(cid:48))σ

|M| 1

n

where by d(cid:0)cM , cM(cid:48)(cid:1) ≈ 2k(cid:96)(cid:0)M(cid:48)(cid:1) we mean 2k(cid:96)(cid:0)M(cid:48)(cid:1) ≤ d(cid:0)cM , cM(cid:48)(cid:1) ≤ 2k+1(cid:96)(cid:0)M(cid:48)(cid:1). More-
over, if d(cid:0)cM , cM(cid:48)(cid:1) ≈ 2k(cid:96)(cid:0)M(cid:48)(cid:1), then from the fact that the radius of B(cid:0)M, M(cid:48)(cid:1) is
2d(cid:0)cM , cM(cid:48)(cid:1), we obtain

1

B(cid:0)M, M(cid:48)(cid:1) ⊂ C02kM(cid:48),

where C0 is a positive constant (C0 = 6 works).

For fixed k ≥ s (1 + δ), we invoke yet again the ‘prepare to puncture’ argument. Choose

an augmented cube (cid:102)M(cid:48) ∈ AD such that C02kM ⊂ (cid:102)M(cid:48) and (cid:96)
(cid:101)ω = ω − ω ({p}) δp where p is an atomic point in (cid:102)M(cid:48) for which

(cid:16)(cid:102)M(cid:48)(cid:17) ≤ C2k(cid:96)(cid:0)M(cid:48)(cid:1). Define

ω ({p}) =

sup

q∈P(σ,ω): q∈(cid:102)M(cid:48)

ω ({q}) .

ω

(If ω has no atomic point in common with σ in (cid:102)M(cid:48), set (cid:101)ω = ω.) Then we have (cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:16)(cid:102)M(cid:48), P(σ,ω)
(cid:17) and so from (.0.39) and (.0.19), for any cube A,
(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:13)(cid:13)(cid:13)♠2
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)1− α

(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)σ
(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)1− α

(cid:46) (cid:96) (A)2 |A|(cid:101)ω

and (cid:88)

≤ Aα,punct

F,A x

F∈F

L2(ω)

n

n

2

=

Now we are ready to apply Cauchy-Schwarz and the augmented local estimate (.0.35) in

379

1
2

2

2

Sk
s

Ss

×

(cid:46)

(cid:17)

1
2

(cid:16)

2

2

|M| 1

n

2

F,M x

L2(ω)

F,M x

L2(ω)

F

M : d

cM ,cM(cid:48)

F

M : d

cM ,cM(cid:48)

M, 1B(M,M(cid:48))σ

(Eα

2 )2 + Aα,energy

2

2

(cid:0)M(cid:48)(cid:1) ≤

to the first line below, to get the following estimate for Sk
s

Lemma .0.12 with M = (cid:102)M(cid:48) to the second line below, and to apply the argument in (.0.57)
(cid:0)M(cid:48)(cid:1) defined in (.0.58) above:
(cid:88)

(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:88)
(cid:17)≈2k(cid:96)(M(cid:48))
(cid:88)
Pα(cid:16)
(cid:13)(cid:13)(cid:13)Qω,b∗
(cid:13)(cid:13)(cid:13)♠2
(cid:88)
(cid:17)≈2k(cid:96)(M(cid:48))
(cid:16)
(cid:19) 1
(cid:18)
22s(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)2(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:105)(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)σ
2(cid:16)(cid:104)
(cid:17) 1
(cid:114)(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)σ
2s(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)(cid:114)(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)(cid:101)ω
(cid:46) (cid:113)(cid:0)Eα
(cid:1)2 + Aα,energy
(cid:46) (cid:113)(cid:0)Eα
(cid:113)
2s(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:102)M(cid:48)(cid:12)(cid:12)(cid:12)1−α
(cid:113)(cid:0)Eα
(cid:113)
2s2k(1−α)(cid:12)(cid:12)M(cid:48)(cid:12)(cid:12)n+1−α
(cid:16)(cid:102)M(cid:48)(cid:17) ≈ 2k(cid:96)(cid:0)M(cid:48)(cid:1) .
(cid:88)
(cid:0)2k |M(cid:48)|(cid:1)n+1−α Sk
(cid:46) (cid:113)(cid:0)Eα
(cid:113)
(cid:1)2 + Aα,energy
(cid:113)(cid:0)Eα
(cid:113)
(cid:1)2 + Aα,energy
(cid:113)(cid:0)Eα
(cid:1)2 + Aα,energy

(cid:0)M(cid:48)(cid:1)
(cid:88)
(cid:88)

2s2k(1−α)

(cid:0)2k |M(cid:48)|(cid:1)n+1−α

(cid:1)2 + Aα,energy
(cid:1)2 + Aα,energy

2

(cid:12)(cid:12)M(cid:48)(cid:12)(cid:12)n+1−α

(cid:0)M(cid:48)(cid:1) =

Altogether then we have

(cid:46) 2−δs

Aα,punct

2

because (cid:96)

k≥(1+δ)s

k≥(1+δ)s

k≥(1+δ)s

Aα,punct

Aα,punct

Aα,punct

2

(cid:113)

2

1

2

2

Aα,punct

,

2

2

2

s

2s−k

2

2

2

2

=

≈

,

380

which is summable in s. This completes the proof of (.0.44), and hence of the estimate for

(cid:16)(cid:98)I
(cid:17) in (.0.40).

Back

The proof of Proposition .0.1 is now complete.

381

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