A TWO WEIGHT LOCAL T B THEOREM FOR FRACTIONAL SINGULAR
INTEGRALS AND REFINED CONSTANTS FOR THE AVERAGING HARDY

OPERATOR

By

Michail Paparizos

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics — Doctor of Philosophy

2021

ABSTRACT

A TWO WEIGHT LOCAL T B THEOREM FOR FRACTIONAL SINGULAR INTEGRALS

AND REFINED CONSTANTS FOR THE AVERAGING HARDY OPERATOR

By

Michail Paparizos

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

sional fractional Calderón-Zygmund operators. Our proof follows the proof for the corre-

sponding one-dimensional T b theorem in [54], but facing a number of new difficulties, most

of which arise from the failure of Hytönen’s one-dimensional two weight A2 inequality in
higher dimensions. We provide a counterexample in two dimensions that shows why the

analogue of Hytönen’s one-dimensional result does not extend to higher dimensions. Thus,

in order to obtain a local Tb theorem in higher dimensions, we use new arguments to control
the difficult nearby form.

We also provide refined constants for strong (p, p) inequality of the averaging Hardy

operator with respect to a probability measure as well as when two measures that satisfy a

special weak type inequality are involved. We obtain these results as corollaries of a more

general theorem for operators with the property

µ{x ∈ X : |T f (x)| > λ} ≤ c
λ

on a probability space (X, µ).

(cid:90)

{|T f|>λ}

|f (x)|dµ(x)

ACKNOWLEDGMENTS

Being in graduate school is not an easy task. I think it would very difficult if had to be

alone in this five-year journey. I want to thank everyone who has supported me in this long

trip.

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

for the fruitful discussions and edifying suggestions all these years. I also thank them for

their patience and their support. I am very grateful to professor Gabor Francsics for his time

and discussions about Radon and Lax-Phillips transforms. I also want to thank professors

Alexander Volberg, Dapeng Zhan and Jeff Schenker.

I want to express my gratitude to Tsveta Sendova and Andy Krause for their contribution

to become a better teacher. Their advice, trust and support are really appreciated and the

time they spent to prepare me for job interview was highly valuable.

I also thank all my friends who played an important role in my life at Michigan State.

First, I want to express my gratitude to Christos, without whom the PhD would be even

harder, to my roommate Yiorgos with whom I have spent many hours cooking and talking

about math, life, sports, movies, literature and politics, to Yiannis for the infinite discussions

about everything, to Ana-Maria for the nice moments we had and her support, to Dimitris,

Andriana, Ilias and Eleni, Christina, Alexandros (x2), Christiana, and Neofytos for the time

we spent together. Finally, I thank Arman, Gora, Dan and Tim.

Last, I thank a lot my parents, who support me in every step I make and without them

nothing would have happened.

iii

TABLE OF CONTENTS

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

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 T b theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Weighted spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 T b theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

1
1
3
6

Chapter 2 Hytönen’s off-testing constant in higher dimensions is unbounded 9
11
11
12
16
19
20
21
21
23

2.1 The One-Dimensional Construction . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Testing Constant is Unbounded . . . . . . . . . . . . . . . . . .
2.1.2 The ¨A2 Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 The Energy Constants ¨E and ¨E∗ . . . . . . . . . . . . . . . . . . . . .
2.2 The Two Dimensional Construction . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The A2 conditions.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Off-Testing Constant . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 The Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 The Riesz transform lemma . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.4.1 Punctured Aα
2

conditions

3.0.1

3.1.1

3.1 The local T b theorem and proof preliminaries

Chapter 3 A two weight local T b theorem for n-dimensional Fractional
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
Standard fractional singular integrals . . . . . . . . . . . . . . . . . .
3.1.1.1 Defining the norm inequality . . . . . . . . . . . . . . . . .
3.1.2 Weakly accretive functions . . . . . . . . . . . . . . . . . . . . . . . .
b-testing conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3
3.1.4 Poisson integrals and the Muckenhoupt conditions
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
3.1.5 Energy Conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.6 The two weight local T b Theorem . . . . . . . . . . . . . . . . . . . .
3.1.7 Reduction to real bounded accretive families . . . . . . . . . . . . . .
3.1.8 Reverse Hölder control of children . . . . . . . . . . . . . . . . . . . .
3.1.8.1 Control of averages over children . . . . . . . . . . . . . . .
3.1.8.2 Control of averages in coronas . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
3.1.9.1 The Calderón-Zygmund corona decomposition . . . . . . . .
3.1.9.2 The accretive/testing corona decomposition . . . . . . . . .
3.1.9.3 The energy corona decompositions
. . . . . . . . . . . . . .
3.1.10 Iterated coronas and general stopping data . . . . . . . . . . . . . . .
3.1.11 Reduction to good functions . . . . . . . . . . . . . . . . . . . . . . .
3.1.11.1 Parameterizations of dyadic grids . . . . . . . . . . . . . . .

3.1.9 Three corona decompositions

28
28
31
31
31
32
33
33
35
36
38
40
50
50
57
60
60
62
67
69
74
75

iv

3.1.14 Organization of the proof

3.2 Form splittings

3.1.12 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.13 Monotonicity Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.13.1 The smaller Poisson integral . . . . . . . . . . . . . . . . . .
3.1.13.2 The Energy Lemma . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78
83
89
91
96
97
3.2.1 The Hytönen-Martikainen decomposition and weak goodness . . . . . 103
3.2.1.1 Good cubes with ‘body’
. . . . . . . . . . . . . . . . . . . . 104
3.2.1.2 Grid probability . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2.1.3 Weak goodness . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.3 Disjoint form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.3.1 Long range form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.3.2
Short range form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.4 Nearby form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.4.1 The case of δ-separated cubes. . . . . . . . . . . . . . . . . . . . . . . 134
. . . . . . . . . . . . . . . . . . . . . . . . 136
3.4.2 The case of δ-close cubes.
3.4.2.1 Return to the original testing functions . . . . . . . . . . . . 153
3.4.2.2 A finite iteration and a final random surgery.
. . . . . . . . 157
3.5 Main below form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
. . . . . . . . . . . . . 172
. . . . . . . . . . . . . . . . . . . . . . 176
Intertwining Proposition . . . . . . . . . . . . . . . . . . . . . . . . . 177
. . . . . . . . . . . . . . . 191
3.5.4.1 The paraproduct form . . . . . . . . . . . . . . . . . . . . . 195
3.5.4.2 The neighbour form . . . . . . . . . . . . . . . . . . . . . . 197
3.6 The stopping form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
3.6.1 The bound for the second sublinear inequality . . . . . . . . . . . . . 209
3.6.2 The bound for the first sublinear inequality . . . . . . . . . . . . . . 213
3.6.3
(cid:91)Straddling, Substraddling, Corona-Straddling Lemmas . . . . . . . . 220
3.6.4 The bottom/up stopping time argument of M. Lacey . . . . . . . . . 236
3.6.5 The indented corona construction . . . . . . . . . . . . . . . . . . . . 243
. . . . . . . . . . . . . . . . . . . . . . 245
3.6.5.1 Flat shifted coronas
Size estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

3.5.1 The canonical splitting and local below forms
3.5.2 Diagonal and far below forms
3.5.3
3.5.4 Paraproduct, neighbour and broken forms

3.6.6

3.7 Finishing the proof

Chapter 4 Refined constants for the averaging Hardy operator . . . . . . 266
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
4.1
4.2 Proof Of Theorem 4.1.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
4.4 Two Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
4.4.1 A three-weight norm inequality . . . . . . . . . . . . . . . . . . . . . 287

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

v

LIST OF FIGURES

Figure 1.1.1:History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.1.2:Theory development

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

3

8

Figure 2.2.1:The two measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Figure 3.4.1:Nearby form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

vi

Chapter 1

Introduction

1.1 T b theory

Boundedness properties of Calderón-Zygmund singular integrals arise in the most critical

cases of the study of virtually all partial differential equations, from Schrödinger operators

in quantum mechanics to Navier-Stokes equations in fluid flow, as well as in the investigation

of a number of topics in geometry and analysis. In particular, the study of boundedness of

these operators from one weighted space L2 (Rn; σ) to another L2 (Rn; ω), not only extends

the scope of application in many cases, but reveals the important properties of the kernels

associated with the individual operators under consideration, often hidden without such

investigation into two weight norm inequalities. The purpose of this monograph is to prove

a general characterization regarding boundedness of Calderón-Zygmund singular integrals

from L2 (Rn; σ) to L2 (Rn; ω), for locally finite positive Borel measures σ and ω, subject to
some natural buffer conditions. This result, a so-called local two weight T b theorem in Rn,

includes much, if not most, of the known theory on two weight L2-boundedness of singular

integrals. We now digress to a brief history of that part of this theory that is relevant to our

purpose here.

Given a Calderón-Zygmund kernel K (x, y) in Euclidean space Rn, a classical problem

for some time was to identify optimal cancellation conditions on K so that there would exist

1

an associated singular integral operator T f (x) ∼ (cid:82) K (x, y) f (y) dy bounded on L2 (Rn).

After a long history, involving contributions by many authors1, this effort culminated in the

decisive T 1 theorem of David and Journé [10], in which boundedness of an operator T on
L2 (Rn) associated to K, was characterized by

T 1, T∗1 ∈ BM O,

together with a weak boundedness property for some η > 0,

(cid:114)
(cid:107)ϕ(cid:107)∞ |Q| + (cid:107)ϕ(cid:107)Lipη |Q|1+

η
n

(cid:114)
(cid:107)ψ(cid:107)∞ |Q| + (cid:107)ψ(cid:107)Lipη |Q|1+

η
n ,

(cid:12)(cid:12)(cid:12)(cid:12) (cid:46)

T ϕ (x) ψ (x) dx

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

Q

(cid:90)

Q

for all ϕ, ψ ∈ Lipη with suppϕ, suppψ ⊂ Q, and all cubes Q ⊂ Rn;

equivalently by two testing conditions taken uniformly over indicators of cubes,

(cid:12)(cid:12)T 1Q (x)(cid:12)(cid:12)2 dx (cid:46) |Q| and

(cid:12)(cid:12)T∗1Q (x)(cid:12)(cid:12)2 dx (cid:46) |Q| ,

all cubes Q ⊂ Rn.

(cid:90)

Q

The optimal cancellation conditions, which in the words of Stein were ‘a rather direct con-

sequence of’ the T 1 theorem, were given in [55, Theorem 4, page 306], involving integrals of

the kernel over shells:

(cid:90)

|x−x0|<N

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

(cid:90)

(cid:90)
dx ≤ AKα
for all 0 < ε < N and x0 ∈ Rn,

Kα (x, y) dy

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

ε<|x−y|<N

dy,

|x0−y|<N

together with a dual inequality.

1see e.g. [55, page 53] for references to the earlier work in this direction

2

Figure 1.1.1: History

We now come to a point of departure for two separate threads of further research on

cancellation conditions. The first thread treats extensions of these testing conditions to the
boundedness of Calderón-Zygmund operators on more general weighted spaces L2 (w) →
L2 (w), and even from one weighted space to another, L2 (σ) → L2 (ω). The second thread
Q∈D more amenable to
the boundedness of the operator at hand, subject of course to some sort of nondegeneracy

replaces the family of testing functions(cid:8)1Q

(cid:9)

(cid:9)

Q∈D with families(cid:8)bQ

conditions. Finally the two threads recombine in the theorem of this paper. See diagram

above.

1.1.1 Weighted spaces

An obvious next step was to replace Lebesgue measure with a fixed A2 weight w,

(cid:19)(cid:18) 1

(cid:90)

|Q|

w (x)

Q

(cid:19)

1

dx

(cid:46) 1 ,

(cid:90)

(cid:18) 1

|Q|

sup

cubes Q⊂Rn

w (x) dx

Q

3

and ask when T is bounded on L2 (w), i.e. satisfies the one weight norm inequality. For

elliptic Calderón-Zygmund operators T , this question is reduced to the David Journé theorem

using two results from decades ago, namely the 1956 Stein-Weiss interpolation with change

of measures theorem [56], and the 1974 Coifman and Fefferman extension [7] of the one

weight Hilbert transform inequality of Hunt, Muckenhoupt and Wheeden [20], to a large

class of general Calderón-Zygmund operators T 2. A motivating example, for the case of the

conjugate function H on the unit circle, arose in the Helson-Szegö theorem that characterized

the boundedness of H on L2 (w) by the existence of bounded functions u and v on the circle
with (cid:107)v(cid:107)∞ < π
and w = eu+Hv. The equivalence with the A2 condition on w follows from
the results just mentioned, and the question of a direct argument linking the Helson-Szegö

2

condition to the A2 condition has remained a tantalizing puzzle for decades since. See [55,
pages 222-227] for this and other applications of one weight theory, such as to the Dirichlet

problem for elliptic divergence form operators with bounded measurable coefficients.

However, for a pair of different measures (σ, ω), the question is wide open in general,

and we now focus our discussion on the main problem considered in this monograph, that

of characterizing boundedness of a general Calderón-Zygmund operator T from one L2 (σ)

space to another L2 (ω) space, subject to natural buffer conditions on the weight pair (σ, ω).

First we note that for the primordial singular integral, namely the Hilbert transform H in

dimension one, the two weight inequality was completely solved by establishing the NTV

conjecture (of Nazarov-Treil-Volberg) in the two part paper [29];[26], see also [21] for the

general case permitting common point masses, where it was shown that H is bounded from

2Indeed, if T is bounded on L2 (w), then by duality it is also bounded on L2(cid:16) 1

(cid:17)

, and the Stein-Weiss
interpolation theorem with change of measure shows that T is bounded on unweighted L2 (Rn). Conversely,
if T is bounded on unweighted L2 (Rn), the proof in [7] shows that T is bounded on L2 (w) using w ∈ A2.

w

4

L2 (σ) to L2 (ω) if and only if the testing and one-tailed Muckenhoupt conditions hold, i.e.

(cid:90)
(cid:32)(cid:90)

I

|H (1I σ)|2 dω (cid:46)

dσ and

(cid:90)

I

(cid:90)
(cid:33)(cid:18) 1

I

|I|

R

|I|2 + |x − cI|2 dσ (x)

|I|

dω

I

(cid:90)

|H (1I ω)|2 dσ (cid:46)
(cid:90)

(cid:19)

dω,

I

(cid:46) 1, and its dual,

uniformly over all intervals I ⊂ Rn. For α-fractional Riesz transforms in higher dimensions
n ≥ 2, it is known (except when α = n − 1) that the two weight norm inequality with
doubling measures is equivalent to the fractional one-tailed Muckenhoupt and T 1 cube testing

conditions, see [30, Theorem 1.4] and [51, Theorem 2.11]. Here a positive measure µ is

doubling if

(cid:90)

(cid:90)

dµ (cid:46)

dµ,

all cubes Q ⊂ Rn.

2Q

Q

However, these results rely on certain ‘positivity’ properties of the gradient of the kernel
y−x > 0 for x (cid:54)= y), something that
is not available for general elliptic, or even strongly elliptic, fractional Calderón-Zygmund

(which for the Hilbert transform kernel

is simply d
dx

1
y−x

1

operators.

Then in [Saw] this T 1 theorem was extended to arbitrary smooth Calderón-Zygmund
operators and A2 measure pairs (σ, ω) with doubling comparable measures, where a pair of
doubling measures σ and ω are comparable in the sense of Coifman and Fefferman [7], if

the measures are mutually absolutely continuous, uniformly at all scales - i.e. there exist

0 < β, γ < 1 such that

|E|σ
|Q|σ

< β =⇒ |E|ω
|Q|ω

< γ for all Borel subsets E of a cube Q.

5

1.1.2 T b theorems

The 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 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 < ∞, which could be
chosen in a way that the verification of the b-testing conditions is easy, while verifying the

1-testing conditions could be more difficult.

Then, M. Christ [6] obtained a local T b theorem for homogeneous spaces, in which the

testing functions are bQ1Q , where the accretive functions bQ can be chosen to differ for
each cube Q. Many authors, including G. David [8]; Nazarov, Treil and Volberg [38], [37];

Auscher, Hofmann, Muscalu, Tao and Thiele [3], Hytönen and Martikainen [24], and more

recently Lacey and Martikainen [27], 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 and

were thus constrained to a single weight - a setting in which both the Muckenhoupt and

energy conditions follow from the upper doubling condition. Good references for that are

Mattila, Melnikov and Verdera [34], G. David [8], [9], X. Tolsa [57], and also Volberg [58].

Applications of the local T b theorem included boundedness of layer potentials, see e.g.

[1]

and references there; and the Kato problem, see [19], [18] and [2].

More recently, E. Sawyer, C.Y. Shen and I. Uriarte-Tuero [54] obtained a general two

weight T b theorem for the Hilbert transform on the real line. In this dissertation, we extend

[54] to higher dimensions.

6

The main two weight local T b theorem:

Theorem 1.1.1 (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)

I

|T α

σ bI|2 dω ≤(cid:16)

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

(cid:17)2 |I|σ and

(cid:12)(cid:12)T α,∗

ω b∗

Tb
T α

J

J

(cid:90)

(cid:17)2 |J|ω ,

Tb∗,∗

T α

(1.1.1)

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∗

functions themselves, namely Lp for some p > 2.

The families of test functions {bI}I∈P and(cid:8)b∗

(cid:9)

J∈P in the T b theorem above are nonde-
generate and slightly better than L2 functions, but otherwise remain at the disposal of the

J

reader. It is this flexibility in choosing families of test functions that distinguishes this char-

acterization as compared to the corresponding T 1 theorem. The T b theorem here generalizes

many of the one-weight T b theorems, since in the upper doubling case, the Muckenhoupt A2
condition and the energy condition easily follow from the upper doubling condition. Recall

that in the one-weight case with doubling and upper doubling measures µ, there has been a

long and sustained effort to relax the integrability conditions of the testing functions: see e.g.

S. Hofmann [16] and Alfonseca, Auscher, Axelsson, Hofmann and Kim [1]. Subsequently,

Hytönen- Martikainen [24] assumed T b in Ls (µ) for some s > 2, and the one weight the-

orem with testing functions b in L2 (µ) was attained by Lacey-Martikainen [27], but their

argument strongly uses methods not immediately available in the two weight setting.

7

Figure 1.1.2: Theory development

The previous diagram details the relevant history of two weight theory. 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 [3] and references there. As is evident from the diagram, Theorem 1.1.1 (and

its precursor for n = 1) is the first local T b theorem for two weights.

The next two chapters are also part of the dissertation of Christos Grigoriadis as they

constitute joint work with him [13], [14].

8

Chapter 2

Hytönen’s off-testing constant in higher

dimensions is unbounded

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

in [54] 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

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

dition if

2 )2 ≡ sup
(E α
Q= ˙∪Qr

1

σ(Q)

(cid:0)E α,∗

2

(cid:1)2 ≡ sup

Q= ˙∪Qr

1

ω(Q)

∞(cid:88)

r=1

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

n

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

Qr

r=1

|Qr| 1

n

(cid:17) < ∞

Qr

1Qr ω

(cid:17) < ∞

1Qr σ

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

9

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)

(cid:90)

(cid:19)2

(cid:19)2

Definition 2.0.2. The off-testing constants Toff,α and Rj,off,α in R2 by

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 2.0.3. 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

Theorem 2.0.4. 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

We begin with the proof of Theorem 2.0.3. The proof of Theorem 2.0.4 will be very

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

Lemma 2.3.1 being useful.

Proof of Theorem 2.0.3. First we build two measures in R, generalizing the work done in

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

10

2.1 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 | =

k=1

2

j }2k

j=1

¨ω(Ik

j ) = 2−k,

k ≥ 0, 1 ≤ j ≤ 2k.

We denote the removed open middle bth of Ik
j

by Gk
j

and by ¨zk
j

its center. Following closely

[28], we define

(cid:88)

k,j

sk
j δ

¨zk
j

¨σ =

where the sequence of positive numbers sk
j

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

= 1, i.e.

(cid:32)

(cid:33)k

2
s2
0

sk
j =

, k ≥ 0, 1 ≤ j ≤ 2k.

2.1.1 The Testing Constant is Unbounded

. Consider the following operator

¨T f (x) =

(cid:90)

R

f (y)

|x − y|2−α dy

11

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:

(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)

j ·(cid:16)s0

2

sk

(cid:17)2k

∞(cid:88)

2k(cid:88)

k=1

j=1

=

= ∞

1
2k

(2.1.1)

2.1.2 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:33)(cid:96)

(cid:32)

(cid:33)k

(cid:32)

4
s2
0

≈

2
s2
0

= s(cid:96)
r

(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

12

and from this, it immediately follows,

¨σ(I(cid:96)
j )¨ω(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).

(2.1.2)

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:18)
(cid:19)m
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:32)
(cid:33)(cid:96)
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
r| + b
|I(cid:96)
(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)

13

(2.1.3)

(2.1.4)

(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

above calculations depend only on α. From (2.1.3), (2.1.4) and (2.1.2), 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

(2.1.5)

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
(2.1.4).

j | when x /∈ Gk
j
, and for the second inequality we used (2.1.3) 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

m > k, since for m = k we have |I| ≈ |Ik
the proof of (2.1.5). Now let Im
t

touches the boundary of Gk
j

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)

If |xI − ¨zk

j | > dist(xI , ∂Gk

j ), we can assume b

t

14

¨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.

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 (2.1.5). 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) < ∞.

15

(2.1.6)

2.1.3 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)

16

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).

(2.1.7)

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

17

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 (2.1.7). This proves

(cid:88)

r≥1

¨ω(Ir)¨P2(Ir, 1I ¨σ) ≤ C ¨σ(I)

(2.1.8)

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).

(2.1.9)

as if we let {Ir : r ≥ 1} be any partition of I, (2.1.9) 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 (2.1.9). 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

18

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

Finally, ¨σ(I) ≈

(cid:18)

2
s2
0

(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 (2.1.9).
, ¨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
s2
0

|I|2

(cid:46)

+

d¨σ(x)d¨σ(x(cid:48))

2

2.2 The Two Dimensional Construction

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

ω(E) =

¨ωn(E)

∞(cid:88)

n=0

∞(cid:88)

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 2.2.1.

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

n=0

γn

σ

ω

σ

ω

kn

σ
ω

σ
ω

Figure 2.2.1: The two measures

19

2.2.1 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 (2.1.6) 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

20

2.2.2 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, (2.1.1) 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.

2.2.3 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.

21

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 (2.1.8), 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 (2.1.8), 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

22

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 (2.1.9) instead of (2.1.8) as in the first case above.

This completes the proof of the Theorem 2.0.3.

2.3 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 2.0.3. 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 [28] 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 2.0.3. First we need to
define the operator

(cid:90)

¨Rf (x) =

(x − y)f (y)
|x − y|3−α dy

R

Lemma 2.3.1. For k ≥ 1, 1 ≤ j ≤ 2k, write Gk

j = (ak

j , bk

j ). Then there exists 0 < c < 1

23

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

2

2

0

24

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

(2.3.1)

(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 (2.3.1) 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

25

obtain

(cid:90)

Ik+1
1

(cid:18)

ak
1 +c

with C1 > 2A and we conclude our lemma.

Proof of Theorem 2.0.4. 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 2.0.3, 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 2.3.1 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

(2.3.2)

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)

26

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 (2.3.2). Letting

27

Chapter 3

A two weight local T b theorem for

n-dimensional Fractional Integrals

3.0.1

Introduction

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

but we do not have this tool as we have shown in the previous chapter. 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 this chapter. 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

28

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: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)

bI 1I\(1+δ)J

T α
σ

bI 1(I\J)∩(1+δ)J

(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

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.

29

3. Pointwise Lower Bound Property (PLBP). In [54] 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,

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 Section 3.6 (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.

30

3.1 The local T b theorem and proof preliminaries

3.1.1 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|

(3.1.1)

(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).

3.1.1.1 Defining the norm inequality

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

For this we introduce a family(cid:110)

ηα
δ,R

σ

the truncated kernels Kα

(cid:107)T α

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

(3.1.2)

31

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)

Kα,

ηα
δ,R

0<δ<R<∞

(3.1.3)

(cid:17) as an α-

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

pendence on the truncations.

Definition 3.1.1. We say that an α-fractional singular integral operator T α satisfies the

norm inequality (3.1.2) 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 (3.1.7) below, the

norm inequality (3.1.2) is essentially independent of the choice of truncations used, and this

is explained in some detail in [52]. Thus, as in [52], we are free to use the tangent line

truncations described there throughout the proofs of our results.

3.1.2 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 3.1.2. 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)

32

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 < ∞.

(3.1.4)

3.1.3 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,

(3.1.5)

for all cubes Q.

3.1.4 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
given by the following expressions:

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

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

n−α

|Q| 1

n

|Q| 1

(cid:18)

dµ (x) ,

33

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 3.1.3. 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,∗

Pα(cid:16)
Pα(cid:16)

Q, 1Qcσ

Q, 1Qcω

(cid:17) |Q|ω
(cid:17) |Q|σ

|Q|1− α

n

|Q|1− α

n

< ∞,

< ∞.

2 ≡ sup
Aα
Q∈P
≡ sup
Q∈P

2

Aα,∗

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

of the products Pα(cid:16)

Q, 1Qcσ

(cid:17) |Q|ω

|Q|1− α

n

. Recall, the definition of the classical Muckenhoupt

34

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

3.1.4.1 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.

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} ,

(3.1.6)

35

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 (3.1.2) 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

.

(3.1.7)

3.1.5 Energy Conditions

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

the energy conditions. Let

I ≡ 1
mµ
|I|µ

(cid:90)

(cid:90)

(cid:28) 1

|I|µ

xdµ(x) =

x1dµ(x), ...,

1
|I|µ

(cid:90)

(cid:29)

xndµ(x)

36

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

Definition 3.1.4. 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

2 )2 ≡ sup
(E α
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

(3.1.8)

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

(3.1.9)

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 [53] and [54]. It is convenient to define

as well as

2 ≡ E α
Eα

2

2 + E α,∗
(cid:113)

T α,∗ +

NT Vα ≡ Tb

T α + Tb∗

2 + Eα
Aα
2 .

(3.1.10)

37

3.1.6 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 [48], and

in part on the proof of a one weight T b theorem in Hytönen and Martikainen [24]. Recall

from [53] 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 .

(3.1.11)

The Hilbert transform kernel K (x, y) = 1
y−x

satisfies (3.1.11) with α = 0, n = 1.

In

dimension n = 1 the Muckenhoupt conditions are necessary for norm boundedness of elliptic

operators by results in [28], [22] and [51], and the energy conditions are necessary for norm

boundedness of gradient elliptic operators by results in [53]. 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 3.1.5. 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 f = T α (f σ) for any smooth truncation of T α

σ , so that T α

σ is apriori
bounded from L2 (σ) to L2 (ω). Assume the Muckenhoupt and energy conditions hold, i.e.
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

2 < ∞. Finally, let p > 2 and let b = (cid:8)bQ
2 ,E α,∗
,E α

(cid:111)
Q∈P be
Q∈P be a p-weakly
σ is bounded from

, Aα,∗,punct

(cid:110)

2 , Aα,punct

2

(cid:9)

b∗

Q

2

38

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 3.1.6. 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,

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:88)

k=0

∞(cid:88)

k=0

n

|Q| 1
2k |Q| 1

(cid:12)(cid:12)(cid:12)2kQ
(cid:12)(cid:12)(cid:12)µ
(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:16)

(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

39

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 3.1.5, 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 [27, 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 [27] 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 3.1.5, 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.

3.1.7 Reduction to real bounded accretive families

We begin by noting that if bQ satisfies (3.1.4) with µ = σ, and satisfies a given b-testing
condition for a weight pair (σ, ω), then RebQ satisfies

(cid:33) 1
p ≤ Cb (p)

(cid:32)

(cid:90)

1
|Q|µ

Q

(cid:12)(cid:12)RebQ

(cid:12)(cid:12)p dµ

40

and the given b-testing condition for (σ, ω) with RebQ in place of bQ.

Thus we may assume throughout the proof of Theorem 3.1.5 that our p-weakly µ-accretive

Next we show that the assumption of testing conditions for a fractional integral T α and

(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∗

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 (3.1.4) 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

(3.1.12)

Proposition 3.1.7. 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 is apriori
bounded from L2 (σ) to L2 (ω). Finally, define the sequence of positive extended real numbers

σ f = T α (f σ) for any smooth truncation of T α

σ , so that T α



(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

41

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 (3.1.12), and the constants
implied by (cid:46) depend on α and the constant CCZ in (3.1.1).
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
σ from L2 (σ) to L2 (ω), uniformly in smooth truncations
of T α

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 +

where Cb (p) , Cb∗ (p) are the accretivity constants in (3.1.4), and the constants implied
by (cid:46) depend on p, α, and the constant CCZ in (3.1.1).

(p1, p0) = (6,∞), and let b =(cid:8)bQ

Proof of Proposition 3.1.7. 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

and let b∗ =

(cid:110)

(cid:111)

(cid:9)

b∗

Q

Q∈P

42

(to be chosen differently at various points in the argument below) and define

(cid:18) p

(cid:19) 1

p−2

λ = λ (ε) =

p − 2

Cb (p)p 1
ε

(3.1.13)

and a new collection of test functions,

(cid:111)(cid:33)
(cid:12)(cid:12)(cid:12)>λ

,

Q ∈ P,

(3.1.14)

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

λ(cid:12)(cid:12)bQ
(cid:12)(cid:12)1(cid:110)(cid:12)(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

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 (3.1.15)

(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

43

by (3.1.13). 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: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

(cid:90)

Q

bQ

(cid:90)

(cid:33)
(cid:32)
1 − λ(cid:12)(cid:12)bQ
(cid:12)(cid:12)
(cid:12)(cid:12)bQ
(cid:12)(cid:12)2 1(cid:110)(cid:12)(cid:12)(cid:12)bQ

(3.1.16)

(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.

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)

(3.1.17)

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

44

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 (3.1.15),

(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 (3.1.17) and (3.1.18), 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))

(3.1.18)

(cid:111)(3.1.19)

Now we choose

with Γ =(cid:0)2Cimplied

−
(Cb (p) + Cb∗ (p))

p
p−2
2− 1
p−2

1

ε =

1
Γ

(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

45

(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 (3.1.19) 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
(cid:9)
suppose that b = (cid:8)bQ
(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
(cid:17)m
1−(cid:16) 2
{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

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

,

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

1−(cid:16) 2

2

3

.

is equivalent to pm =

46

(3.1.20)

(cid:17)m < 4
p−1

6

,

1−(cid:16) 2

2

3

Now let 0 < ε < 1 (to be fixed later), define λ = λ (ε) as in (3.1.13), and define(cid:98)bQ as in

(3.1.14). Recall from (3.1.15) and (3.1.16) 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

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
(cid:12)(cid:12)1(cid:110)(cid:12)(cid:12)(cid:12)bQ
λ(cid:12)(cid:12)bQ
(cid:12)(cid:12)(cid:12)>λ
 dσ +
(cid:12)(cid:12) > λ(cid:9)(cid:12)(cid:12)µ
λq(cid:12)(cid:12)(cid:8)(cid:12)(cid:12)bQ
 qtq−1dt + Cb (p)p λq−p
 1
(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)bQ
(cid:12)(cid:12)(cid:12)≤λ
(cid:111) dσ
(cid:21)
(cid:20) 1
(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

47

which shows that C(cid:98)b

(q) satisfies the estimate

C(cid:98)b

(q) ≤ 2Cb (p)

(cid:46) Cb (p)

p
q

(cid:19) 1

p−2

1− p

q

Cb (p)p 1
ε
− 1− p
p−2 ,

3
2 ε

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:19)
(cid:18)

p − 2

p−2

ε

q

− 1
p−2

ε

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

as the function x (cid:55)→ x
x−2

is decreasing when x > 2. Moreover, from (3.1.18) 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

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

|Q|σ

1 ≤

(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

T α + 2

(cid:13)(cid:13)(cid:13)Lq(ω)

Q

√

≤ Cb∗ (p)

3
2 ε

− 1− p
p−2 ,

q

Q ∈ P ,

εNT α

48

We now use these estimates, together with the fact that (Sm−1) holds, to obtain

NT α(cid:46)(cid:16)

(q)+C(cid:98)b∗ (q)
C(cid:98)b
(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))
before, by choosing

2 3n √

3

− 1− p
p−2 NT α into the left hand side as

q

εε

 3

2 3n
1− p
p−2 − 1
q

2



ε =

1
Γ

(Cb (p) + Cb∗ (p))

with Γ sufficiently large, depending only on the implied constant, since (3.1.20) 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

.

(3.1.21)

, 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 (3.1.21) 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,∗ +

49

which completes the proof of Proposition 3.1.7.

Thus we may assume for the proof of Theorem 3.1.5 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 ≤

(3.1.22)

Q ∈ P .

3.1.8 Reverse Hölder control of children

Here we begin to further reduce the proof of Theorem 3.1.5 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,

(3.1.23)

3.1.8.1 Control of averages over children

Lemma 3.1.8. 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) .

50

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:16)

≤ 2

bQ
T α (Q)

(cid:12)(cid:12)T α
, there exists a real-valued function(cid:101)bQ supported in Q such
(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:17)
(cid:35)(cid:113)|Q|σ ,

(cid:13)(cid:13)L∞(σ) ≤ Cb.

Qi ∈ C (Q) .

(cid:101)bQdσ

1
4 NT α (Q)

Cb ,

3
4
b δ

Qi

Then for every 0 < δ <

1

2n+1C3
b

Q

that

(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)L∞(σ)
(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQ

≤ 16Cb

(3). 0 <

dω ≤

(cid:34)

(2).

Q

δ

bQ
T
T α (Q) + 2C

Proof. Let 0 < δ < 1 and fix Q ∈ P. By assumption we have

Let Qi be the children of Q. We now define(cid:101)bQ. First we note that the inequality

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

51

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

bQdσ

Q

(cid:13)(cid:13)(cid:13)L∞(σ)

bQ

(3.1.24)

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

(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

|Qi|σ

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

cannot hold for all Qi, since otherwise we obtain the contradiction

If (3.1.24) holds for none of the Qi, then we simply define(cid:101)bQ = bQ, and trivially all the
(cid:12)(cid:12)bQ
(cid:12)(cid:12) dσ
then we define(cid:101)bQ differently according to how large the L1 (σ)-average

conclusions of the Lemma 3.1.8 hold. If (3.1.24) 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 (3.1.24) holds and G the set of

indices for which (3.1.24) 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

52

Now let us check the conclusions of the Lemma 3.1.8. For (1) we have

bQdσ

(cid:88)
(cid:101)bQdσ +
(cid:101)bQdσ +(cid:112)CbδCb

1
|Q|σ

i∈B−

1
|Q|σ

(cid:90)

Qi

ni

(cid:112)Cbδdσ − 1
(cid:88)

|Qi|σ ≤ 1
|Q|σ

|Q|σ

i∈B−

(cid:90)
(cid:88)
(cid:101)bQdσ + C

Qi

pi

i∈B+

(cid:90)

Q

(cid:112)Cbδdσ

√

δ

3
2
b

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

Q

Q

Q

1 ≤

≤

≤

1
|Q|σ
1
|Q|σ
1
|Q|σ

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
(c) For i ∈ B− ∪ B+,(cid:13)(cid:13)(cid:13)1Qi(cid:101)bQi
(cid:13)(cid:13)(cid:13)L∞ ≤ 2(1 +(cid:112)Cb)Cb.

This completes the proof for (1).

53

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)(cid:101)bQ − bQ
(cid:12)(cid:12)(cid:12)2

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−

+

+

+

(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.

54

(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

1
|Q|σ

≤

≤ (cid:16)

Cbδ + Cb

(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)bQ
(cid:12)(cid:12) dσ
(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

(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

(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 (3.1.24) does not hold and as(cid:101)bQ = bQ there, immediately we

55

obtain

(b) for i ∈ G0 ∪ G+,

(c) for i ∈ B−,

Qi

δ

Qi

≤

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

(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)1Qi(cid:101)bQ
(cid:13)(cid:13)(cid:13)L∞(σ)
(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) dσ
(cid:12)(cid:12)bQ
(cid:112)Cbδ
(cid:82)

1|Qi|σ

1|Qi|σ

nidσ

Qi

≤

≤

Qi

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

(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

1|Qi|σ

Qi

≤ 4Cb
Cbδ

=

,

4
δ

, we have 1 +(cid:112)Cbδ < 2.

as, by taking 0 < δ < 1
4C3
b

(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 3.1.8, 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
(cid:12)(cid:12) ≥ c1 > 0 as well. This completes
Finally, if(cid:12)(cid:12)bQ

(cid:12)(cid:12)(cid:12) ≥(cid:12)(cid:12)bQ

the proof of Lemma 3.1.8.

56

3.1.8.2 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 3.1.9. 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) .

i=1 ⊂ DQ be a collection of pairwise disjoint dyadic subcubes of Q. Suppose that

Let {Qi}∞
bQ is a real-valued function supported in Q such that

(cid:90)
Q(cid:48) bQdσ ≤(cid:13)(cid:13)(cid:13)1Q(cid:48)bQ
(cid:12)(cid:12)2 dω ≤ T

bQ
T α (Q)

(cid:13)(cid:13)(cid:13)L∞(σ)
(cid:113)|Q|σ .

1 ≤ 1
|Q(cid:48)|σ

(cid:115)(cid:90)

(cid:12)(cid:12)T α

Q

σ bQ

≤ Cb ,

Q(cid:48) ∈ CQ ,

57

(cid:34)

Q

0 <

(cid:16)

3
2
b δ

≤ 2

dω ≤

2T

(cid:115)(cid:90)

1
4 NT α (Q)

Q(cid:48) ∈ CQ ,

bQ
T α (Q) + 4C

(cid:17)
(cid:35)(cid:113)|Q|σ ,

Cb ,

Then for every 0 < δ < 1
4C3
b

, there exists a real-valued function(cid:101)bQ supported in Q such that
(cid:90)
(cid:13)(cid:13)(cid:13)L∞(σ)
Q(cid:48)(cid:101)bQdσ ≤(cid:13)(cid:13)(cid:13)1Q(cid:48)(cid:101)bQ
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.

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

1 ≤ i < ∞.

Cb ≤ 2

≤ 16Cb

|Qi|σ

Cb

1

|Q(cid:48)|σ

≤ 2

(cid:16)

(cid:17)

(cid:16)

δ

Qi

(cid:17)

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)
and (cid:3)σ,b

Q(cid:48) associated with these functions as defined in (3.1.38), 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

|h| dσ +

1

|Q(cid:48)|σ

I

(cid:19)

(cid:90)
Q(cid:48) |h| dσ

1I .

(3.1.25)

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.

58

Proof. The proof of Lemma 3.1.9 is similar to that of the Lemma 3.1.8. 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)bQ
(cid:12)(cid:12) dσ
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 3.1.8 can be
applied verbatim. We emphasise only that when estimating the testing condition, we need

(cid:90)

Q

∞(cid:88)

the bound

dσ ≤ C (Cb) δ

(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ
(cid:12)(cid:12)(cid:12)2
Remark 3.1.10. The estimate(cid:82)
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

4(cid:80)∞
i=1 |Qi|σ in the last line of

(cid:12)(cid:12)(cid:12)(cid:101)bQ − bQ

|Qi|σ ≤ C (Cb) δ

dσ ≤ C (Cb) δ

4 |Q|σ .

(cid:12)(cid:12)(cid:12)2

i=1

1
4

Q

1

1

59

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.

3.1.9 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 3.6 by adapting the bottom/up stopping

time and recursion of M. Lacey in [26]. 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

f ∈ L2 (σ) and g ∈ L2 (ω) .

(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

3.1.9.1 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

60

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

(cid:90)
(cid:90)

F

F

|φ| dµ;

|φ| dµ

for F (cid:36) F(cid:48) ⊂ πF F.

1

|πF F|µ
|πF F|µ

1

1
|F|µ
1
|F(cid:48)|µ

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 3.1.11. 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

and then define the Calderón-Zygmund stopping cubes of S0 to be the collection

∞(cid:91)

F = {S0} ∪

Sm

where S0 = S (S0) and Sm+1 = (cid:83)

m=0

S (S) for m ≥ 0.

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) .

61

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.

3.1.9.2 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,

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.

62

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:17)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)T α,∗
dσ ≤ (cid:16)
(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 3.1.12. 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

(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

(a).

(b).

(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

where S0 = S (S0) and Sm+1 = (cid:83)

F = {S0} ∪

∞(cid:91)

m=0

Sm

S (S) for m ≥ 0.

S∈Sm

For ε > 0 chosen small enough depending on p > 2, the b-accretive stopping cubes satisfy

63

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 S ∈ CS0

(cid:111)

,

is essentially in [24], but we include a proof for completeness.

Lemma 3.1.13. 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 .

(3.1.26)

(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

(3.1.27)

Proof. Inequality (3.1.27) is immediate from the definition of F in the definition 3.1.12. We
now address the Carleson condition (3.1.26). 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)

I bS0

dσ

we have

(cid:82)

bQdσ

S

S∈S(Q)

|S|σ ≤ γ |Q|σ ,

(cid:90)

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

bQdσ

E(Q)

(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)

(cid:12)(cid:12)(cid:12)(cid:12) < γ

64

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 ,

65

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 ,

66

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 3.1.13.

3.1.9.3 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. [38], [28], [48] and [49]). 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 3.1.14. 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

(3.1.28)

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

67

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.

(3.1.29)

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

(3.1.30)

is (the square of) the α-stopping energy of the weight pair (σ, ω) with respect to the corona

68

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,

(3.1.31)

where Aα
2

and the energy constant Eα
2

are controlled by the assumptions in Theorem 3.1.5.

3.1.10

Iterated coronas and general stopping data

We will use a construction that permits iteration of the above three corona decompositions

by combining Definitions 3.1.11, 3.1.12 and 3.1.14 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

3.1.5, and so we will use Lemma 3.1.9 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 3.1.15. 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)

69

(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 3.1.9 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

(3.1.32)

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 (3.1.32)
applies, while if I ∈ CQ then the accretivity in the second line of (3.1.32) applies.

70

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 3.1.16. Let C0 ≥ 4 and 1 < Γ < ∞. Let Sshadow (Q) ≡ {Qi}∞
i=1 be as in
Definition 3.1.15. 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 3.1.9 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 3.1.17. Let C0 ≥ 4, 0 < γ < 1 and 1 < Γ < ∞. Suppose that b =(cid:8)bQ

(cid:9)

Q∈P is

71

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 3.1.16.

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 3.1.18. 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)

72

(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.

(3.1.33)

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|.

73

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)

.

(3.1.34)

(cid:82)

Proposition 3.1.19. Let f ∈ L2 (σ), let F be as in Definition 3.1.17, 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 3.1.11 , such that the triple (A0,F, αF ) constitutes stopping data for the
function f.

Proof. This is an easy exercise using (3.1.26) and (3.1.29), and is left for the reader.

3.1.11 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 3.1.20. 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 (3.2.16).

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 ,

74

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 [49] and [48] relied heavily on

the use of a single grid, and this must now be modified to accomodate two independent grids.

3.1.11.1 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 [52]. We momentarily fix a large positive

75

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

76



2−i+(cid:96)βi

(3.1.35)

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

77

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 3.1.21. 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.

3.1.12 Formulas

We need the following formulas defined on Appendix A of [54].

(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 ,

(3.1.36)

f dµ,

Q ∈ P ,

f dµ,

Q ∈ P .

(3.1.37)

Q bQdµ

Q

78

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) (3.1.38)
(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

(3.1.39)

(cid:33)

|f| dµ

(cid:90)

Q

1Q(cid:48),

Qf

(cid:46) (cid:107)f(cid:107)2

L2(µ)

.

(3.1.40)

(cid:88)

Q(cid:48)∈C(Q)

Q(cid:48) f − Fµ,bQ
Fµ,bQ
Q f,

(3.1.41)

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

(3.1.42)

(3.1.43)

(3.1.44)

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,

79

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

(3.1.45)

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

(3.1.46)

(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)

(3.1.47)

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

(3.1.48)

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

80

where(cid:98)Fσ

K

is defined in (3.1.37) 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 ,

(3.1.49)

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 .

(3.1.50)

These modified differences and the identities (3.1.47) and (3.1.48) play a useful role in the

analysis of the nearby and paraproduct forms.

Lemma 3.1.22. 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-

81

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.

(3.1.51)

.

(3.1.52)

(cid:17)∗

:

Ψµ,bB

,

(3.1.53)

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

82

the grid, while the Carleson averaging operators (cid:53)µ
choice of broken children of Q.

Q

depend on the grid only through the

3.1.13 Monotonicity Lemma

As in virtually all proofs of a two weight T 1 theorem (see e.g. [26], [29] , [49] and/or [48]),

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 ([29], [26])

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 [30], and also [48]) 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 3.1.23 (Monotonicity Lemma). Suppose that I and J are cubes in Rn such that

83

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)

,

(3.1.54)

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 (3.1.39), 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) ,

84

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

85

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)

86

having used the reverse Hölder control of children (3.1.23) 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 (3.1.54).

− (cid:3)ω,π,b∗

= (cid:3)ω,b∗

J

J

J,brok

The right hand side of (3.1.54) in the Monotonicity Lemma will be typically estimated

87

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 [54].

K

i=1

Lemma 3.1.24. 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) ,

(3.1.55)

|x − mK|2 dω (x) .

Proof. The first inequality in (3.1.55) 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

(3.1.55) 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)

88

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

3.1.13.1 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 3.2.5 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) ,

89

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

(3.1.56)

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 (3.5.10), 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)

90

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 [49] and/or [48].

3.1.13.2 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)

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

(cid:88)

J∈H

(cid:88)

J∈H

+

+

91

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.

92

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 3.1.25 (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)

(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

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

93

(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 3.1.23, 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 = ∅,

,

(3.1.57)

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

94

(3.1.58)

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

95

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)

3.1.14 Organization of the proof

We adapt the proof of the main theorem in [51], 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 (3.1.3) 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

96

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 [27] relied strongly on a property peculiar

to the one weight setting - namely the fact already pointed out in Remark 3.1.6 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 Appendice A of [14] that follows 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. [51] - but requires a different decomposition of the stopping cubes into ‘Whitney

cubes’ in order to accomodate the weaker notion of goodness used here.

3.2 Form splittings

Notation 3.2.1. 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

97

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)

,

(3.2.1)

where the first inequality is the ‘weak upper half Riesz’ inequality from Appendix A of
[54] for the pseudoprojection Ψµ,b∗
GD
k−bad
inequality in (3.2.10) 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,

(3.2.2)

I(cid:48) ∈ C (I) , I ∈ D,

98

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 (3.2.2) 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 [54] 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 3.1.19 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 3.2.2. 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 3.1.18,

99

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 =

100

(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∗
 (3.2.3)

(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
(3.2.3) 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

101

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 (3.1.51) 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 3.4.3 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

102

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 (3.2.3). We are left to deal with the first

sum in (3.2.3).

3.2.1 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)

103

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,

(3.2.4)

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.

3.2.1.1 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 3.6 below. Let P denote the collection of all cubes in Rn. The content of the

104

next four definitions is inspired by, or sometimes identical with, that already appearing in

the work of Nazarov, Treil and Volberg in [36] and [38].

Definition 3.2.3. 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 3.2.4. 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 3.2.5. 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−


(3.2.5)

and we say it is 
−bad in K if (3.2.5) fails.

Definition 3.2.6. 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).

3.2.1.2 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

105

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 3.2.5 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 3.2.7. 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 [54] 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.

(3.2.6)

D0
Ω (D0 : J is k-good in D0) ≥ 1 − Cεk2−εk.

P

(3.2.7)

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.

(3.2.8)

106

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.

(3.2.9)

Then we obtain from (3.2.9), 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 (3.1.39) 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)

(3.2.10)

for some large positive constant C.

From such inequalities summed for k ≥ r, it can be concluded as in [38] 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

107

(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) . (3.2.11)

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 (3.2.11). 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 (3.2.4) 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 3.7) where the need for using the ‘body’ of a cube will become apparent in dealing

108

with the stopping form, and also in the treatment of the functional energy in Appendix B

of [54].

3.2.1.3 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 3.2.8. 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)

.

109

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)

(3.2.12)

with absolute value signs inside the sum.

Remark 3.2.9. 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

110

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 3.3 and

the neighbour form in Section 3.5 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 B of [54], as well as permitting

control of the stopping form in Section 3.6, 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 3.4 using the energy condition assump-

tion along with the original testing functions borig

Q discarded during the construction of

the testing/accretive corona.

k−good = GD

These remarks will become clear in this and the following sections. Recall that we earlier
defined in Definition 3.2.6, 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 3.2.10. 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

(k,ε)−good

.

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,

111

i.e.

G \ GD

k−good =

(cid:91)

(cid:96)≥k

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 (3.2.16) 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.

(3.2.13)

#

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)

=

(3.2.14)

,

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 (3.2.14) 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

112

as in (3.2.12) 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)

113

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 (3.1.39) 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 (3.1.53) 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 (3.1.51) together with (3.2.13), 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.

114

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 (3.2.10) 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)

(3.2.15)

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

,

(3.2.16)

and then the final term above, namely Cgood2
at the end of the proof in Subsection 3.7. Note that (3.2.16) 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 3.5.

. It is this type of weak

n+1−α

1

2 ≤

115

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

(3.2.17)

in the same way we analyzed the below term B(cid:98)r (f, g) in [48]; 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 [48] respectively.
Θ1 (f, g) in essentially the same way that the disjoint form B∩ (f, g) was treated in [48] and
in earlier papers of many authors beginning with Nazarov, Treil and Volberg (see e.g. [58]),

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

116

disjoint cubes [22].

3.3 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.

3.3.1 Long range form

Lemma 3.3.1. 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

117

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 3.3.1, 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

118

(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.

(3.3.1)

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 (3.3.1), 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)

119

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)

(3.3.2)

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

120

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

121

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

122

(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 (3.3.2). We can now sum in (cid:96) to get (3.3.1) 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

3.3.2 Short range form

The form Θshort

1

(f, g) is handled by the following lemma.

Lemma 3.3.2. 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

123

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 3.1.25 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.

124

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)

125

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

(3.3.3)

in the calculations above, and this completes the proof of Lemma 3.3.2.

3.4 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 3.4.1. 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

126

∞-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

(3.4.1)

(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 (3.4.1).

127

Figure 3.4.1: Nearby form

128

Before we proceed any further let us mention that we will repeatedly use the inequality

Lemma 3.4.2. 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

(3.4.2)

(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

(3.4.3)

(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 (3.4.3) 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

129

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

(3.4.4)

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

130

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 (3.1.39),

(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 (3.4.2). 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

131

(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 (3.4.2) 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 3.4.3. 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∗

(3.4.5)

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

132

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

133

(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 (3.4.1) 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

(3.4.6)

(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 2). 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:

3.4.1 The case of δ-separated cubes.

In this subsection we are estimating I in (3.4.6) by using Lemma 3.4.3.

Definition 3.4.4. We say that the cubes J and I are δ-separated, where δ > 0, if J ∩ (1 +

134

δ)I = ∅.

For the first sum in (3.4.6) we have, following the proof of Lemma 3.4.3, 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

I f

g

J

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

135

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=∅

(3.4.7)

(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.

3.4.2 The case of δ-close cubes.

Now we turn to the second sum in (3.4.6) which we will bound by using random surgery and

expectation.

Definition 3.4.5. 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

(3.4.8)

ω

136

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 (3.4.8), 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)

137

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

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

+

L2(σ)

L2(σ)

J

Next by Lemma 3.4.3,

(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

138

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).

139

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 (3.4.8) 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 (3.4.8) 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

(3.4.9)

(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 (3.4.9), 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)

.(3.4.10)

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) .

(3.4.11)

140

Suppose now that I ∈ CA for A ∈ A, and that J ∈ CB for B ∈ B. Then the inner

product in the third line of (3.4.10) 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 (3.4.11) 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 (3.4.11). To see this, write(cid:10)T α
(cid:68)
+(cid:10)T α

(cid:69)

(cid:68)

(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

141

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 3.4.2, 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 3.4.3, 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

(3.4.12)

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.

142

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

(3.4.13)

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 3.4.3 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

143

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 (3.4.11) 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 3.4.3 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)

(3.4.14)

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

144

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)

(3.4.15)

(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 (3.4.19) 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)

145

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 (3.4.2) 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)|

146

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

(3.4.16)

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 3.4.3,

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)

147

(cid:88)(cid:88)

(cid:113)

Ks ∼
Sep

Ks(cid:48)

Aα
2

(cid:112)|Ks|σ

(cid:113)|Ks(cid:48)|ω (3.4.17)

which plugged into (3.4.10) appropriately, we get the bound B(cid:112)Aα

2

To deal with the adjacent cubes term in (3.4.16), 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 3.4.3 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

(3.4.18)

while summing

∼
I over

T = {I ∈ D, J ∈ N (I), I(cid:48) ∈ Cnat(I), J(cid:48) ∈ Cnat(J)}

148

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)

(3.4.19)

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.

149

Thus now we are left only with the first term of (3.4.16), 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 = ∅ (3.4.20)

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 ,

(3.4.21)

and Kin the 2n grandchildren of K that do not intersect the boundary of K while Kout the

150

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} .

(3.4.22)

Note that the first two terms on the right hand side of (3.4.22) 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 (3.4.22) 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)

(3.4.23)

151

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

(3.4.24)

(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)

152

(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 (3.4.24) 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

.

3.4.2.1 Return to the original testing functions

From the discussion above, we recall the identity (3.4.24) and the estimate (3.4.25). 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

(3.4.25)

153

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

(3.4.26)

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

−

154

(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

155

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∗

156

The second term II is bounded using Lemma 3.4.3. 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.

3.4.2.2 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

+

+

(3.4.27)

(cid:113)|Kin|ω

157

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 (3.4.22) 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

158

surgery again. Using Lemma 3.4.3,

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)

ω

out

K(cid:96)

bA1

, 1Km
out
b∗
out|ω + NT α

, 1Km
out

B

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

(3.4.28)

(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

159

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 (3.4.22),

|{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) ,(3.4.29)

160

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 (3.4.13). 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)

(3.4.30)

(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

161

|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

(3.4.31)

M∈Mν(Ki)

(cid:113)|I(cid:48)|σ

(cid:113)|J(cid:48)|ω,

162

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 (3.4.2),

(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 (3.1.28),

163

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 (3.4.27), 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

164

(3.4.13),

(3.4.32)

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 (3.4.2) 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

165

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) .

(3.4.33)

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 (3.4.33) 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

166

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

(3.4.34)

(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

(3.4.35)



+ |Min|σ

Indeed, in this last inequality (3.4.35), 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

167

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
(3.4.35) 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 (3.4.34) 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 (3.1.40) and (3.1.51), 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

168

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.

√

(3.4.36)

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

√

(3.4.37)

i=1

j=1

(cid:96)=1

Note here that upon choosing δ small enough there is no repetition in the different terms

169

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

(3.4.38)

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

(3.4.7) in the separated case and after an initial application of random surgery, we reduced

the proof of Proposition 3.4.1 to establishing inequality (3.4.11). Then using the bounds in

(3.4.12), (3.4.14), (3.4.15), (3.4.16), (3.4.17), (3.4.18) we reduced P (I, J) to getting a bound
for {K, K} in the notation used in (3.4.21). Then using the estimates in (3.4.30), (3.4.31),
(3.4.32) and (3.4.34) together with (3.4.29), (3.4.36), (3.4.37) and (3.4.38) establishes prob-
abilistic control of the sum of all the inner products {K, K} taken over appropriate cubes
K, yielding (3.4.11) as required if we choose ε, λ, η0 and δ sufficiently small. And combining

170

all the above bounds we proved proposition 3.4.1, 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.

3.5 Main below form

Now we turn to controlling the main below form (3.2.17),

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 [48]. 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 (3.2.16).

171

Definition 3.5.1. 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

(3.5.1)

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

(3.5.2)

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).

3.5.1 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

172

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) .

(3.5.3)

(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)

173

(3.5.4)

(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 [48], which in turn follows the treatment originating in [38], 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 [26].

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

174

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 (??) in Appendix B of [54] for more detail.

Definition 3.5.2. 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) ,

(3.5.5)

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 [48]. 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

175

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

(3.5.6)

, 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

3.5.2 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 [48] 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

(3.5.7)

(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) .

176

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 (3.4.1) in

, with J

B

Proposition 3.4.1 , 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 3.5.7 can now be applied to this latter form to show that it
2 +E α
is bounded by NT Vα +Fα. Then Proposition ?? 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) .

(3.5.8)

3.5.3

Intertwining Proposition

First we adapt the relevant definitions and theorems from [48].

177

Definition 3.5.3. 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 3.5.4. 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 3.5.1 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

.

(3.5.9)

(cid:88)

H∈H

(cid:88)

H∈H: H⊂K

Recall from Definition 3.5.2 that Fα = Fα (D,G) = Fb∗
i.e.

α (D,G) is the best constant in (3.5.5),

(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 3.5.5. If in (3.5.5), 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

178

condition in Definition ??, but with Pω,b∗

CG,shif t
is larger than Pω,b∗

F

;M

doprojection Pweakgood,ω

J

in place of Pweakgood,ω

J

. However, the pseu-

CG,shif t

, and so we just miss obtaining the deep

;J

F

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 3.5.6. 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)

(3.5.10)

(3.5.11)

(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

179

The proof of this lemma is similar to those of Lemmas 3.3.1 and 3.3.2 in Section 3.3

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 3.5.7 (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.

180

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

(3.5.12)

(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

181

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 3.4.3 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 (3.5.12), 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∞

182

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

183

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| ,

(3.5.13)

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 (3.5.11)
(cid:122) ⊂ F ⊂ Ik, which

implies J ⊂ J

F

Eσ
F f
Eσ
F bF

184

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) ,

(3.5.14)

where we have defined

Φ ≡ (cid:88)

F∈F

αF (F ) 1F\∪CF (F ) .

185

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

ε

F

ε

containment will be used in the analysis of the term IIB below.

186

Notation 3.5.8. 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.

F∈F

(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 (??) in Appendix B of [54]. 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 (3.1.54) in the Monotonicity
Lemma 3.1.23 using (3.5.14) 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

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

J(cid:48)

g

L2(ω)

+

F∈F

F∈F

F ;M

L2(ω)

♠

x

L2(ω)

L2(ω)

F

F

F∈F

F∈F

L2(ω)

1J(cid:48) ω

L2(ω)

F

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 B of [54] .

187

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 (3.5.1) 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 (3.1.58) 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

(3.5.15)

(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

188

(3.1.8), 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

(3.5.16)

2
(cid:19)
2

.

where in the last inequality we used the Cauchy-Schwarz inequality. Now we again apply

189

Cauchy-Schwarz and (3.5.16) 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 (3.5.16) and the strong energy condition (3.1.8),

190

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 3.5.7.

The task of controlling functional energy is taken up in Appendix B of [54] below.

3.5.4 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
(3.5.4) 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

(3.5.17)

,

ω

(cid:69)

IJ(cid:98)(cid:3)σ,(cid:91),b

I

Eσ

f

191

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

192

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 (3.1.47) 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

193

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 (3.5.4) and (3.5.2) 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 )

194

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

(3.5.18)

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 3.5.9. 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 (3.2.15) for

the bad form Θbad(cid:92)

2

defined in (3.2.12).

3.5.4.1 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 (3.1.48), 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,

=

=

=

195

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)

196

(cid:33)(cid:41)

A(cid:98)Fσ,b

A f

f − Eσ

(3.5.19)

(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)

3.5.4.2 The neighbour form

neighbour (f, g) is easily controlled by the Aα

Next, the neighbour form BA
pivotal estimate in Energy Lemma 3.1.25 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 (3.5.25) 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

(3.5.20)

(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)

,

(3.5.21)

197

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 [58].

Lemma 3.5.10. 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 ).

(3.5.22)

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) ,

198

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 (3.5.22).

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 3.1.25 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)

199

Here we are using (3.5.22) in the third line, which applies since J ⊂ I0, and we have used
(3.5.21) 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

;(3.5.23)

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

200

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 (3.4.2) 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

(3.5.24)

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-

201

(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 (3.5.23) 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

(3.5.25)

202

is complete since Eσ

Now if we sum in A ∈ A the inequalities (3.5.19), (3.5.25) and (3.5.18) 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.

3.6 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 3.1.23, 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

203

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 [26], [49],

[51] and [52]. 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

,

(3.6.1)

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)

204

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 3.6.1. 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,

205

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

(3.6.2)

Definition 3.6.2. 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:17)

(cid:16)

(cid:16)

T α
σ

J

Eσ

πI f

g

.

ω

(I,J)∈P

Proposition 3.6.3. 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)

(3.6.3)

With the above proposition in hand, we can complete the proof of (3.6.1) by summing

206

(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 3.6.3, 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-

207

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:

(3.6.4)

(3.6.5)

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 3.1.23 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) ,

(3.6.6)

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

208

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 (3.6.4) and (3.6.5) 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)

,

(3.6.7)

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

(3.6.8)

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 (3.6.8) that

does not require recursion.

3.6.1 The bound for the second sublinear inequality

Now we turn to proving (3.6.8), 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

209

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 [28], 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

210

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 (3.6.8) follows upon summing in s ≥ 0. Now using both

(3.6.9)

(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)

,

211

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

(3.6.9) 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)

212

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 3.1.14, as (I, J) ∈ P , we have that I is not a stopping cube in A,
(cid:17)
and hence that (3.1.28) 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)

3.6.2 The bound for the first sublinear inequality
Now we turn to proving the more difficult inequality (3.6.7). 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)

,

(3.6.10)

where f ∈ L2 (σ) satisfies Eσ
{πI : I ∈ Π1P}. We refer to NA,P
Inequality (3.6.7) 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 (3.6.7). 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 3.6.4. Suppose that {Qm}∞

m=0 is a set of admissible collections for A that are

213

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

214

Now we turn to proving inequality (3.6.7) 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 (3.1.30),

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 [26], 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

(3.6.11)

(cid:17)

B, 1I\Aσ
|B| 1

n

I\A

I\A

where the implied constants depend on n, α.

Fix A ∈ A. Following [26] we will use a ‘decoupled’ modification of the stopping energy

215

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 3.6.5. 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 3.6.6. 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 [26], 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 3.6.7. 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

216

.

(3.6.12)

The following key fact is essential.

Key Fact #1:

If K ⊂ A and K /∈ CA, then ΠK

2 P = ∅ .

(3.6.13)

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 (3.6.13) 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

(3.6.14)

.

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)

(3.6.15)

(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

217

(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 3.6.8. 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

.

218

(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 (3.6.15),

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 [26]. 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 3.6.9. If PA is as in (3.6.2) and P ⊂ PA, then

Sα,A
augsize (P) ≤ Xα (CA) (cid:46) E α
2 +

219

(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 (3.1.31).

There is an important special circumstance, introduced by M. Lacey in [26], 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

(3.6.41), where it is explained how this applies to the proof of (3.6.40). 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 (3.6.7) to the three special cases addressed by

3.6.3 (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

220

collection PA - see (3.6.2). 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)

.

(3.6.16)

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 3.6.10. 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

221

(cid:122)
J

(cid:17)
(cid:17)

and we have

if A(cid:48) = J (cid:91)

if A(cid:48) = J

(cid:122)
J

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 (3.6.4), 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 (3.1.8).

Definition 3.6.11. We say that an admissible collection of pairs P is reduced if it contains
no pairs from PA

cor, i.e.

P ∩ PA

cor = ∅.

222

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 3.6.12. 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 3.6.13. 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

3.6.15 below. For S ∈ S, let QS ≡(cid:110)

(I, J) ∈ Q : J (cid:91) ⊂ S ⊂ I

(cid:111).

Proposition 3.6.14. 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

223

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 (3.6.4)

. 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)

224

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 (3.6.4) 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 (3.6.15) shows that 3J (cid:91) ⊂ J
J (cid:91) ⊂ K = K [J] for some K ∈ W (S) and so by (3.6.4) 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

225

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 (3.1.41), 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 3.6.14.

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 3.1.22 (where

I

I∈CA

A

is a family of

projections, and the second line of Lemma 3.1.22 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

,

226

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

(3.6.17)

(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 3.6.14 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

(3.6.18)

(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)

227

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

(3.6.19)

Eσ
I

Returning to (3.6.18), we have from (3.6.17) and (3.6.19) 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

(3.6.20)

(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 [26].

Lemma 3.6.15. 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

228

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) ,

(3.6.21)

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)

(3.6.22)

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 (3.6.20),

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 .

229

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 [26]. 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

230

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) (3.6.23)

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)

,

(3.6.24)

which below we will see reduces matters to proving inequality (3.6.21) 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.

231

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 3.6.14 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 (3.6.15) 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 (3.6.14)
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 (3.6.4), and so the sum over S ∈ SH of the first term on the right side of the conclusion

QH

(J)

A

1

A

J

232

of Proposition (3.6.14) 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 3.6.8, 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

233

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
(3.6.24); 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

(3.1.43), 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 (3.1.44) - which relies on the reverse

H∈HS

234

Hölder property for children in Lemma 3.1.9 - 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)

(3.6.25)

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 3.6.15.

In a similar fashion we can obtain the following Substraddling Lemma.

Definition 3.6.16. 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 3.6.17. Let L be a D-cube contained in A, and suppose that Q is an admissible

235

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 (3.6.15), 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 3.6.15 to obtain

the required bound (cid:98)NA,Q

stop,(cid:52)ω ≤ CSα,A

augsize (Q).

3.6.4 The bottom/up stopping time argument of M. Lacey

Before introducing Lacey’s stopping times, we note that the Corona-straddling Lemma 3.6.10
allows us to remove the ‘corona straddling’ collection PA

cor of pairs of cubes in (3.6.16) from

236

the collection PA in (3.6.2 ) 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.

(3.6.26)

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

(3.6.27)

(3.6.28)

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 (3.6.15). Note also that the atomic
contained in one of J
measure ω(cid:91)P differs from the measure µ in (??) in Appendix B of [54] - 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

J

+

237

measure ω(cid:91)P. We can get away with this here, as opposed to in Appendix B of [54], 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:16)
(cL, (cid:96) (L)) ∈ Rn+1
iff J (cid:91) ⊂ K iff
of P in Definition 3.6.8 as

J(cid:91), (cid:96)

(cid:16)

c

(cid:110)

2

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)) .

(3.6.29)

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 3.6.18. The functional ω(cid:91)P (T (K)) is increasing in K, while the functional

238

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 (3.6.7) 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 (3.6.13), that we can restrict the collection to Πbelow

P ∩ C(cid:48)

1

1

A

is used for the augmented size functional.

239

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 (3.6.37). 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 ,

(3.6.30)

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 [26]. However, in our
P are no longer ‘good’ in any sense, and we must

situation the cubes I belonging to Πbelow

1

240

include an additional top/down stopping criterion in the next subsection to accommodate

this lack of ‘goodness’. The argument in [26] 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.

(3.6.31)

As in [49], [51] and [52], we follow Lacey [26] 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

(3.6.32)



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 (3.6.32) for A(cid:48) ∈ LM +1 in this case, but we do have that (3.6.32) 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

241

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 (3.6.7).

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.

242

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} ,

(3.6.33)

...

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 (3.6.36) in Proposition 3.6.19 below.

3.6.5 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

243

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}

244

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 3.2.1 above.

3.6.5.1 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)

245

,

,

with one exception: if L ∈ H0 we set P(cid:91)H−small
≡ ∅ since in this case
L fails to satisfy (3.6.32) 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)

(3.6.34)

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 (3.6.35) 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 (3.6.36) 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

246

3.6.6 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 3.6.19. Suppose ρ in (3.6.31) 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

(3.6.35)

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

(3.6.36)

t ≥ 1.

Using this proposition on size estimates, we can finish the proof of (3.6.7), and hence the

247

proof of (3.6.1).

Proof. Recall that (cid:98)NA,P
(cid:110)P(cid:91)L−small

(cid:111)

L,0

L∈L

Corollary 3.6.20. The sublinear stopping form inequality (3.6.7) holds.

stop,(cid:52)ω is the best constant in the inequality (3.6.10). Since

is a mutually orthogonal family of A-admissible pairs, the Orthogonality

Lemma 3.6.4 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 (3.6.36) 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 (3.6.2) 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)

.

248

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 (3.6.35) in the last line. If we choose ρ > 1 so that

C(cid:112)ρ − 1 <

1
2

,

(3.6.37)

then we obtain L ≤ 2C 1√

ρ−1

. Together with Lemma 3.6.9, 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 (3.6.7).

(cid:16)E α

(cid:113)

(cid:17)

2 +

Aα
2

Thus, in view of Conclusion 3.6.4, it remains only to prove Proposition 3.6.19 using the

Orthogonality and Straddling and Substraddling Lemmas above, and we now turn to this

task.

Proof of Proposition 3.6.19. We split the proof into three parts.

Proof of (3.6.35): To prove the inequality (3.6.35), suppose first that L /∈ LM +1. In
the case that L ∈ L0 is an initial generation cube, then from (3.6.30) and the fact that every

249

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

1

L,0

A

L

L,0

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 ,

250

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 ( 3.6.32) 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)) .

(3.6.38)

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)

251

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 (3.6.28), 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 (3.6.38), 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))

252

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 (3.6.35).

To prove the other inequality (3.6.36) in Proposition 3.6.19, 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 (3.6.36): 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 (3.6.36) 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,

(3.6.39)

which recovers the key geometric gain obtained by Lacey in [26], except that here we are

253

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 3.6.15 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 3.6.4 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 3.6.4.

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 3.6.15 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

254

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)) ,

(3.6.40)

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 (3.6.29), 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

(3.6.41)

(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 (3.6.40), 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

255

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 [26], 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 (3.6.32) 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

(3.6.42)

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

256

for L0 ∈ Lm (that appears in (3.6.32)) 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 (3.6.32)), 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 (3.6.42), 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 (3.6.33). 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

257



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 (3.6.42) 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)

258

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 (3.6.40), and hence that of (3.6.41). Finally, an application of

the Orthogonality Lemma 3.6.4 proves (3.6.39).

Proof of the first line in (3.6.36): At last we turn to proving the first line in (3.6.36).

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

(3.6.43)

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

259

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

260

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 3.6.15. 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 3.6.4
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)

(3.6.44)

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 3.6.15 to the reduced admissible collection Qout,in

k,i
out,in

k,i
out,in

k,i
out,in

L

L

L

261

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) (3.6.45)

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

}.

(3.6.46)

In either case in (3.6.46), 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 (3.6.26) of Conclusion
collection P A
3.6.4 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 (3.6.46) 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

262

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 3.6.17 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)

.

(3.6.47)

Combining the bounds (3.6.44), (3.6.45) and (3.6.47), we obtain (3.6.43).

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 3.6.4 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 3.6.19 is now complete.

This finishes the proofs of the inequalities (3.6.7) and (3.6.1).

263

3.7 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 3.1.5.

The task of controlling functional energy is taken up in Appendix B of [54], after first

264

establishing 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

265

Chapter 4

Refined constants for the averaging

Hardy operator

4.1 Introduction

Let µ be a non-atomic measure on (0,∞). We define the µ-averaging Hardy operator as

Aµf (x) =

1

µ(0, x)

(0,x)

f (t)dµ(t),

x ∈ (0,∞)

(4.1.1)

for any non-negative function f. If L is the Lebesgue measure, (4.1.1) becomes

(cid:90)

(cid:90) x

0

f (t)dt

ALf (x) =

1
x

and the classical Hardy inequality holds:

(cid:107)ALf(cid:107)Lp ≤ p
p − 1

(cid:107)f(cid:107)Lp .

for all non-negative f ∈ Lp(cid:0)(0,∞)(cid:1) and the constant

(4.1.2)

p
p−1

is sharp. This result is due to

Hardy [15] in the course of attempts to simplify the proof of Hilbert’s double series theorem.

This inequality has been studied a lot and a complete discussion is included in [25] and [42].

266

More recently, Nikolidakis [40] improved inequality (4.1.2) by proving a sharp integral

inequality valid for non-negative functions defined [0, 1] with given L1 norm:

Theorem A. Let f : [0, 1] → R+ be in Lp([0, 1]), p > 1 with (cid:82) 1

0 f dt = φ. Then for any

1 ≤ q ≤ p,

(cid:90) 1

(cid:18) 1

(cid:90) x

0

x

0

(cid:19)p

f dt

dx <

(cid:18) p

p − 1

(cid:19)q(cid:90) 1

(cid:18) 1

(cid:90) x

0

x

0

(cid:19)p−q

f dt

f (x)qdx − q

p − 1

φp

(4.1.3)

Moreover, inequality (4.1.3) is sharp in the sense that, the constant ( p

p−1)q cannot be de-

creased, while the constant

q
p−1 cannot be increased for any fixed φ.

Meanwhile, Melas [31] calculated the Bellman function

(cid:26)(cid:90)

X

Dp(φ, Φ) := sup

(MT f )pdµ : f ∈ Lp(X, µ),

(cid:90)

X

(cid:27)

f pdµ = Φ

(cid:90)

X

f dµ = φ,

where (X, µ) is a non-atomic probability space, 0 < φp ≤ Φ and MT a tree like maximal
operator, and showed that

(cid:18) φp

(cid:19)

Φ

Φ,

Dp(φ, Φ) = ψ−1

p

where ψp(z) = pzp−1 − (p − 1)zp. Melas [32] also showed that
(cid:90) 1

(cid:26)(cid:90) 1

(cid:90) x

(cid:19)p

(cid:18) 1

Dp(φ, Φ) =

f dt

dx :

f dx = φ,

0

x

0

0

sup

f :(0,1]→R+
decreasing
continuous

(cid:27)

f pdx = Φ

(cid:90) 1

0

via a symmetrization principle of dyadic maximal operator with respect to the averaging

Hardy operator. Finally, Nikolidakis [41] characterized the extremal sequences of functions
for the latter expression of Dp related to the averaging Hardy operator.

267

In this note, we calculate

(cid:40)(cid:90)

Bp

(cid:0)µ, φ, Φ(cid:1) := sup

(cid:90)
where(cid:0)(0,∞), µ(cid:1) is a non-atomic probability space.

|Aµf (x)|pdµ :

(0,∞)

f≥0

(cid:41)

f pdµ = Φ

(cid:90)

(0,∞)

f dµ = φ,

(0,∞)

Definition 4.1.1. Let 1 < p < ∞. A pair of two positive numbers (φ, Φ) is called p-
admissible if φp ≤ Φ.

Let (X, µ) be a probability space and (φ, Φ) a p-admissible pair. We may write

(cid:91)

Lp(X, µ) =

Fφ,Φ
p

(X, µ)

(φ,Φ)

p−admissible

where

(cid:26)

f ∈ Lp(X, µ) :

(cid:90)

(cid:90)

(cid:27)

|f|pdµ = Φ

.

|f|dµ = φ and

Fφ,Φ
p = Fφ,Φ

p

(X, µ) =

In these smaller classes of functions we have refined bounds:

Theorem 4.1.2. Let µ be a non-atomic probability Radon measure on (0,∞). For any
non-negative f ∈ Fφ,Φ

p

(cid:0)(0,∞), µ(cid:1),
(cid:13)(cid:13)Aµf(cid:13)(cid:13)Lp((0,∞),µ) ≤ ψ−1

p

(cid:18) φp

(cid:19)

Φ

(cid:107)f(cid:107)Lp((0,∞),µ)

where ψp(z) = pzp−1 − (p − 1)zp. Moreover, the inequality is sharp.

268

Corollary 4.1.3. ([32]) For non-negative f ∈ Fφ,Φ

(cid:107)ALf(cid:107)Lp((0,1)) ≤ ψ−1

p

(cid:0)(0, 1),L(cid:1), we have the sharp inequality:
(cid:19)
(cid:18) φp

p

(cid:107)f(cid:107)Lp((0,1)) .

Φ

On the other hand, it is known that the dyadic maximal function Md satisfies the following

sharp special weak type (1,1) inequality

L{x ∈ Rn : |Mdf (x)| > λ} ≤ 1
λ

(cid:90)

{|Mdf|>λ}

|f (x)|dx

for every f ∈ L1(Rn) and every λ > 0, from which is easy to get the inequality

(cid:107)Mdf(cid:107)Lp(Rn) ≤ p
p − 1

(cid:107)f(cid:107)Lp(Rn) ,

for every p > 1 and f ∈ Lp(Rn). The constant
result [31] refines this inequality when restricted to functions on [0, 1]n.

p
p−1

is the best possible [4, 5, 59]. Melas’

Being inspired from that, let T be an operator defined on a space (X, µ) that satisfies

the special weak type inequality

µ{x ∈ X : |T f (x)| > λ} ≤ [µ]
λ

(cid:90)

{|T f|>λ}

|f (x)|dµ(x)

(4.1.4)

for any λ > 0 and f ∈ L1(X, µ). By [µ] denote the best possible constant in (4.1.4). Then,
we easily conclude,

for every 1 < p < ∞ and every f ∈ Lp(X, µ) provided that (cid:82) |T f|pdµ < ∞. Inequality

(cid:107)T f(cid:107)Lp(X,µ) ≤ [µ]p
p − 1

(cid:107)f(cid:107)Lp(X,µ)

(4.1.5)

(4.1.5) can be refined as the following theorem shows. To state it we need to define a

269

function on (0,∞):

A (cid:55)→ kp,f,T (A) =

[µ]pAp−1(cid:82){|T f|>A} |f|dµ − (p − 1)Ap

(cid:82) |f|pdµ

.

Theorem 4.1.4. Let (X, µ) be a non-atomic probability space and T be an operator satisfying
(4.1.4). Then for f ∈ Fφ,Φ

(X, µ),

p

(cid:18)

(cid:19)

(cid:107)T f(cid:107)Lp(X,µ) ≤ ˜ψ−1

p

max
A>0

kp,f,T (A)

(cid:107)f(cid:107)Lp(X,µ) ,

(4.1.6)

provided that(cid:82) |T f|pdµ < ∞. Here ˜ψp(z) = [µ]pzp−1 − (p − 1)zp defined on(cid:2)[µ], [µ]p

(cid:3). In

p−1

the special case that |T f (x)| > [µ]φ for all x ∈ X, then

p

(cid:107)T f(cid:107)Lp(X,µ) ≤ ˜ψ−1
(cid:18)[µ]pφp

(cid:19)

≤ ˜ψ−1

p

[µ] ≤ ˜ψ−1

p

Φ

(cid:19)

(cid:18)[µ]pφp
(cid:18)

Φ

(cid:107)f(cid:107)Lp(X,µ) .
(cid:19)

max
A>0

kp,f,T (A)

≤ [µ]p
p − 1

.

Moreover,

Theorem 4.1.2 and Corollary 4.1.3 easily follow now from Theorem 4.1.4.

We also have a result for the two-weight setting, which is an application of Theorem 4.1.4

but it is not anywhere near as developed as the one-weight case. In particular,

Theorem 4.1.5. Suppose two non-atomic Radon measures ω, µ satisfy the special weak type

inequality

ω{x ∈ (0,∞) : Aµf (x) > λ} ≤ K
λ

(cid:90)
{x∈(0,∞): Aµf (x)>λ} f (t)dµ(t),

with µ being a probability measure.

If L = ω(0,∞), then for every non-negative f ∈

270

Fφ,Φ
p

(cid:0)(0,∞), µ(cid:1),
(cid:90)

(0,∞)

|Aµf (x)|pdω ≤

(cid:20)

(cid:18)

K

ψ−1

p

(cid:19)(cid:21)p − (K − L)

(cid:18) φp

Φ

φp
Φ

(cid:19)(cid:90)

(0,∞)

f pdµ.

4.2 Proof Of Theorem 4.1.4

The idea of the proof has been used in [33], [12] and [41].

Proof. Let 0 <(cid:82) |f|pdµ < ∞. For A to be determined later, using (4.1.4) we have,

(cid:90)

|T f|pdµ =

(cid:90) ∞

0

≤ Ap +

≤ Ap + [µ]

= Ap + [µ]p

= Ap +

= Ap +

A

A

(cid:90)

pλp−2

(cid:90) ∞
pλp−1µ{|T f| > λ}dλ
(cid:90) ∞
pλp−1µ{|T f| > λ}dλ
(cid:90) |T f|
(cid:90)
(cid:19)
(cid:18)(cid:12)(cid:12)T f(cid:12)(cid:12)p−1 − Ap−1
|f|(cid:12)(cid:12)T f(cid:12)(cid:12)p−1dµ − [µ]p

{|T f|>λ}
|f|

λp−2dλdµ

|f|dµdλ

{|T f|>A}

{|T f|>A}

(cid:90)
(cid:90)

[µ]p
p − 1
[µ]p
p − 1

p − 1

{|T f|>A}

|f|

A

dµ

Ap−1

(cid:90)

{|T f|>A}

|f|dµ

[µ]p
p − 1

(cid:18)(cid:90)

(cid:19)1
p(cid:18)(cid:90)

|T f|pdµ ≤ Ap +

Set EA = {|T f| > A}. Using Hölder’s inequality with exponents p and p
p−1
(cid:19)1− 1
(cid:90)
Dividing both sides by(cid:82) |f|pdµ and rearranging we obtain,
(cid:18)(cid:82) |T f|pdµ
(cid:82) |f|pdµ

(cid:82)
(cid:82) |f|pdµ

(cid:82) |f|pdµ

p − [µ]p
p − 1

(cid:19)1− 1

≤ [µ]p
p − 1

− Ap

|T f|pdµ

[µ]p
p − 1

|f|pdµ

Ap−1

Ap−1

|f|dµ

p −

EA

(cid:82) |T f|pdµ
(cid:82) |f|pdµ

, we obtain

|f|dµ

(cid:90)

EA

271

or equivalently,

[µ]pAp−1(cid:82)

|f|dµ − (p − 1)Ap

(cid:82) |f|pdµ

EA

(cid:18)(cid:82) |T f|pdµ
(cid:82) |f|pdµ

≤ [µ]p

(cid:19)1− 1

p − (p − 1)

(cid:82) |T f|pdµ
(cid:82) |f|pdµ

(4.2.1)

Consider for any p > 1 the function

˜ψp(z) = [µ]pzp−1 − (p − 1)zp, z > 0.

Notice that ˜ψ(cid:48)

p(z) = p(p − 1)zp−2([µ] − z). Thus, ψp(z) ≤ [µ]p for all z > 0. Set

[µ]pAp−1(cid:82)

|f|dµ − (p − 1)Ap

(cid:82) |f|pdµ

EA

.

kp,f,T (A) =

Rewriting inequality (4.2.1), we have, for all A > 0,

(cid:33)
(cid:105) since otherwise we have nothing
to prove. Here is the place where (cid:82) |T f|pdµ has to be finite, because the proof of (4.1.5)

(cid:32)(cid:107)T f(cid:107)Lp(µ)
∈ (cid:104)

By (4.1.5), we may assume that

(cid:107)T f(cid:107)Lp(µ)
(cid:107)f(cid:107)Lp(µ)

kp,f,T (A) ≤ ˜ψp

(cid:107)f(cid:107)Lp(µ)

[µ], [µ]p
p−1

(4.2.2)

.

requires it. Note that

(cid:32)

kp,f,T (A) = ˜ψp

|f|dµ

A(cid:82)
(cid:104)

EA

EA

(cid:17)p

(cid:17)p

|f|dµ

(cid:16)(cid:82)

(cid:33)(cid:16)(cid:82)
(cid:82) |f|pdµ
(cid:105) → [0, [µ]p] is strictly decreasing and onto. Since the

(cid:82) |f|pdµ

≤ [µ]p

≤ [µ]p

|f|dµ

EA

and that the restriction ˜ψp :

[µ], [µ]p
p−1

272

inverse of ˜ψp, ˜ψ−1

p

: [0, [µ]p] →(cid:104)

[µ], [µ]p
p−1

(cid:105), is also strictly decreasing,
(cid:0)kp,f,T (A0)(cid:1)(cid:107)f(cid:107)Lp(µ)

p

(cid:107)T f(cid:107)Lp(µ) ≤ ˜ψ−1

where A0 is chosen so that 0 < kp,f,T (A0) ≤ [µ]p and kp,f,T (A0) is maximum.

It is easy to see that

[µ]pAp−1(cid:82) |f|dµ − (p − 1)Ap
(cid:105) with

kp,f,T (A) ≤ Lp,f (A) :=

(cid:82) |f|pdµ
and the function A (cid:55)→ Lp,f (A) is increasing on(cid:104)
0, [µ](cid:82) |f|dµ
(cid:0)[µ](cid:82) |f|dµ(cid:1)p
(cid:82) |f|pdµ

and Lp,f (0) = 0.

|f|dµ

Lp,f

[µ]

(cid:18)

,

(cid:90)

(cid:19)

=

In the special case that EA1
X}, thus kp,f,T (A0) = Lp,f (A0).
Lp,f (A) ≤ [µ]p, inequality (4.2.2) implies

= X, for some A1 > 0, let A0 = sup{A : |T f (x)| > A, for all x ∈

If A0 < [µ](cid:82) |f|dµ, since ψ−1 is decreasing and 0 <

(cid:107)T f(cid:107)Lp(µ) ≤ ˜ψ−1

p

(cid:0)Lp,f (A0)(cid:1)(cid:107)f(cid:107)Lp(µ) ,

while if A0 ≥ [µ](cid:82) |f|dµ, we have {|T f| > [µ](cid:82) |f|dµ} = X and inequality (4.2.2) implies

(cid:107)T f(cid:107)Lp(µ) ≤ ˜ψ−1

p

(cid:18)[µ]p((cid:82) |f|dµ)p
(cid:82) |f|pdµ

(cid:19)

(cid:107)f(cid:107)Lp(µ) .

273

Remark 4.2.1. Inequality (4.1.4) together with Hölder’s inequality imply that

µ{x ∈ X : |T f (x)| > λ} ≤ [µ]
λq

{|T f|>λ}

|f (x)|qdµ(x)

for 1 ≤ q < p. Using this in the proof of Theorem 4.1.4, one can show that for 1 ≤ q < p,

we have

(cid:18)(cid:90)

(cid:19)1/p ≤ ˜ψ−1

p

|T f|pdµ

(cid:19)1/p

|f|pdµ

(cid:90)

(cid:0)kp,q,f,T (A0)(cid:1)(cid:18)(cid:90)
(cid:33)

≤ ˜ψ−1

p

for some A0 > 0 and a function kp,q,f,T that depends on k, p, q, f and T . Moreover,

(cid:32)

q[µ]p/q(cid:0)(cid:82) |f|qdµ(cid:1)p/q

(cid:82) |f|pdµ

[µ] ≤ ˜ψ−1

p

(cid:0)kp,q,f,T (A0)(cid:1) ≤ [µ]p

p − 1

.

Remark 4.2.2. If we assume the special weak type (r, q) inequality

(cid:32)(cid:90)

(cid:33)q/r

µ{x ∈ X : |T f (x)| > λ} ≤ [µ]
λq

|f (x)|rdµ(x)

{|T f|>λ}

for 1 ≤ r ≤ q < p, then again we have

(cid:18)(cid:90)

(cid:19)1/p ≤ ˜ψ−1

p

(cid:0)kp,q,f,T (A0)(cid:1)(cid:18)(cid:90)

(cid:19)1/p

|f|pdµ

|T f|pdµ

for some A0 > 0 and a function kp,q,f,T that depends on k, p, q, f and T . Moreover,

(cid:32)

q[µ]p/q(cid:0)(cid:82) |f|qdµ(cid:1)p/q

(cid:82) |f|pdµ

(cid:33)

(cid:0)kp,q,f,T (A0)(cid:1) ≤ [µ]p

p − 1

.

≤ ˜ψ−1

p

[µ] ≤ ˜ψ−1

p

274

4.3 Applications

Lemma 4.3.1. For any Radon measure µ on (0,∞), 1 < p < ∞ and f ≥ 0, we have

(cid:90)

(0,∞)

|Aµf (x)|pdµ(x) < ∞.

Proof. First of all note that inequality (4.1.4) is satisfied. Indeed, the set Eλ = {x ∈ (0,∞) :
Aµf (x) > λ} is open for any λ > 0, because of the regularity of µ. This implies that Eλ can

be written as Eλ =(cid:83) Ij, where Ij are maximal pairwise disjoint open intervals. It follows

that

(cid:88)

j

µ(Ij) =

1
λ

µ(Eλ) =

(cid:90)

(cid:88)

j

Ij

Let N > 0 and fN = min(f, N ). Then, by (4.1.4)

(cid:90)

(cid:90)

f dµ =

1
λ

f dµ =

1
λ

∪Ij

f dµ

Eλ

(cid:90)

(0,∞)

(cid:90) ∞

0
p
p − 1

p
p − 1

(cid:90)
{AµfN >λ} fN dµdλ
fN (AµfN )p−1dλdµ
p(cid:32)(cid:90)
(cid:33) 1

pλp−2
(cid:90)
(cid:32)(cid:90)

(0,∞)

(fN )pdµ

(0,∞)

(0,∞)

(AµfN )pdµ =

=

≤

(cid:33)1− 1

p

(AµfN )pdµ

With the left-hand side being positive and finite, this inequality gives

(cid:90)

(cid:19)p(cid:90)

(cid:18) p

p − 1

(AµfN )pdµ ≤

(0,∞)

(fN )pdµ

(0,∞)

Letting N → ∞, the conclusion follows by the monotone convergence theorem.

Now, recall the distribution function of f with respect to µ is the function µf (λ) :

275

[0,∞) → (0,∞] defined by

µf (λ) = µ{x ∈ X : |f (x)| > λ}

and the decreasing rearrangement of f is the function f∗ : [0,∞) → (0,∞] defined by

f∗(t) = inf{λ ≥ 0 : µf (λ) ≤ t}

The functions f and f∗ are equimeasurable, that is,

µ{x ∈ X : |f (x)| > λ} = L{t > 0 : f∗(t) > λ}

for any λ > 0.

For a probability space (X, µ) define the quantities

B(cid:0)µ, Fφ,Φ

p

(cid:26)(cid:90)

X

(cid:1) = sup
(cid:26)(cid:90)

X

and

˜B(cid:0)µ, Fφ,Φ

p

(cid:1) = sup

|Aµf (x)|pdµ : 0 ≤ f ∈ Fφ,Φ

p

Lemma 4.3.2. For any non-negative decreasing f ∈ Fφ,Φ

p

|Aµf (x)|pdµ : 0 ≤ f ∈ Fφ,Φ

.

p

(cid:0)X, µ(cid:1)(cid:27)
(cid:0)X, µ(cid:1), decreasing
(cid:0)(0, 1),L(cid:1),

(cid:27)

.

(cid:18) φp

(cid:19)

Φ

(cid:107)f(cid:107)Lp(0,1) ,

(cid:107)ALf(cid:107)Lp((0,1)) ≤ ψ−1

p

276

where ψp(z) = pzp−1 − (p − 1)zp. Moreover, if Ep(f ) = 1 − φp/Φ,

(cid:104)

ψ−1

p

(cid:0)1 − Ep(f )(cid:1)(cid:105)p

(cid:18) p

<

p − 1

(cid:19)p − p

p − 1

(cid:0)1 − Ep(f )(cid:1).

(4.3.1)

Proof. It is easy to see that

(cid:90)
{ALf >λ} f (x)dµ(x).
Let f be a decreasing function. Then, ALf (x) ≥(cid:82) 1

L{x ∈ (0, 1) : ALf (x) > λ} =

1
λ

0 f (t)dt for all x ∈ (0, 1) and Theorem

4.1.4 and Lemma 4.3.1 imply

(cid:107)ALf(cid:107)Lp((0,1)) ≤ ψ−1

p

(cid:0)1 − Ep(f )(cid:1)(cid:107)f(cid:107)Lp((0,1))

(cid:16) φp

Φ

(cid:17), it is easy to

α . For α = ψ−1

p

Now, consider the decreasing function fα(x) = φ
see that

(cid:90) 1

0

fαdx = φ and

α x−1+ 1
(cid:90) 1

f p
αdx =

0

φp

ψp(α)

= Φ

An easy calculation shows that

(cid:90) 1

(cid:18) 1

(cid:90) x

0

x

0

(cid:19)p

fαdt

dx =

(cid:18)
(cid:104)

ψ−1

p

ψ−1

p

(cid:18)φp
(cid:19)(cid:19)p(cid:90) 1
(cid:17)(cid:105)p
(cid:16) φp

Φ

0

Φ.

f p
α dx.

Thus we have shown that ˜B(L|(0,1), Fφ,Φ
To obtain (4.3.1), consider the function

p

) =

(cid:104)
ψ−1

p

g(y) =

(1 − y)

Φ

(cid:19)p

(cid:105)p −

(cid:18) p

p − 1

+

p
p − 1

(1 − y),

277

for 0 ≤ y < 1. Then

g(cid:48)(y) =

(cid:16)

−ψ−1

p

(1 − y)
p (1 − y)

1 − ψ−1

(p − 1)

(cid:17) − p

p − 1

> 0

which implies that g is strictly increasing on (0, 1). Since lim
at 0, we proved (4.3.1) for 0 ≤ Ep(f ) < 1.

y→1− g(y) = 0 and g is continuous

Proof of Corollary 4.1.3. Due to Lemma 4.3.2 and the inequality

(cid:90) t

(cid:90) t

f (x)dx ≤

f∗(x)dx,

t ∈ (0, 1),

0

0

we obtain

(cid:90) 1

0

(ALf∗)pdt =
≥

=

(cid:90) ∞
(cid:90) ∞
(cid:90) 1

0

0

0

(cid:26)
(cid:26)

pλp−1L
pλp−1L

t ∈ (0, 1) :

t ∈ (0, 1) :

(cid:27)
(cid:27)
f∗dx > λ

dλ

f dx > λ

dλ

(cid:90) t
(cid:90) t

0

0

1
t
1
t

(ALf )p dt

which implies that B(L|(0,1), Fφ,Φ
lated the sharp constant of Corollary 4.1.3.

) = ˜B(L|(0,1), Fφ,Φ

p

p

(cid:104)

ψ−1

p

(cid:16) φp

(cid:17)(cid:105)p

Φ

) =

Φ, and we have calcu-

To the best of our knowledge, the proof of Corollary 4.1.3 as a consequence of Theorem

4.1.4 is the simplest.

If we restrict Theorem A to Fφ,Φ

p

better bound. Indeed, set

(cid:0)(0, 1),L(cid:1) and let q = p, Corollary 4.1.3 provides a

φp
Φ

= 1 − Ep(f )

278

where 0 ≤ Ep(f ) < 1 (by Hölder’s inequality). Then, inequality (4.1.3) can be rewritten as

(cid:90) 1

(cid:18) 1

(cid:90) x

0

x

0

(cid:19)p

f dt

dx <

(cid:20)(cid:18) p

p − 1

(cid:19)p − p

p − 1

(cid:21)(cid:90) 1

0

(1 − Ep(f ))

f p dt.

and inequality (4.3.1) provides an improvement to (4.1.3).

Lemma 4.3.3. Let µ be a non-atomic probability Radon measure on (0,∞). Then

(cid:18) φp

(cid:19)

Φ

(cid:13)(cid:13)Aµf(cid:13)(cid:13)Lp((0,∞),µ) ≤ ψ−1
(cid:0)(0,∞), µ(cid:1).

p

(cid:107)f(cid:107)Lp((0,∞),µ)

for decreasing f ∈ Fφ,Φ

p

Proof. Let f be a decreasing function. Then Aµf (x) ≥(cid:82)
(cid:19)

Theorem 4.1.4 and Lemma 4.3.1 give

(cid:13)(cid:13)Aµf(cid:13)(cid:13)Lp((0,∞),µ) ≤ ψ−1

(cid:18) φp

p

(cid:107)f(cid:107)Lp((0,∞),µ) ,

Φ

(0,∞) f (t)dµ(t) for all x ∈ (0,∞).

which implies that

˜B(cid:0)µ, Fφ,Φ

p

(0,∞)(cid:1) ≤

(cid:20)

ψ−1

p

(cid:18) φp

(cid:19)(cid:21)p

Φ

Φ = B(L|(0,1), Fφ,Φ

p

).

Proof of Theorem 4.1.2. Now consider a decreasing function f ∈ Fφ,Φ
t ∈ (0, 1), let

p

((0, 1),L). For every

St := {x ∈ (0,∞) : µ(0, x) = t}.

For λ ≥ 0 and for all t ∈ (0, 1) and x ∈ St, define a non-negative function g on (0,∞) with

279

the property

g (λ) := µ{y ∈ (0, x) : g(y) > λ} = L{y ∈ (0, t) : f (y) > λ}
µx

Then, for every x ∈ St,
(cid:90) ∞

(cid:90)

gdµ =

(0,x)

0

(cid:90) ∞

0

µx
g (λ)dλ =

L{y ∈ (0, t) : f (y) > λ}dλ =

(cid:90) t

0

f (y)dy.

and

(cid:90)

1

µ(0, x)

(0,x)

g(u)dµ(u) =

(cid:90) t

0

1
t

f (u)du, for f ≥ 0,

Notice that for any λ > 0,

(cid:26)

L

t ∈ (0, 1) :

(cid:27)

f (u)du > λ

(cid:90) t

0

1
t

= L(0, tλ) = tλ

for some tλ ∈ (0, 1). From the discussion above we obtain that for all x ∈ Stλ

(cid:40)
x ∈ Stλ

µ

(cid:90)

(cid:41)

1

:

µ(0, x)

(0,x)

gdµ > λ

= µ(0, sup Stλ

) = tλ

,

(cid:41)

This implies that

(cid:90)

(0,∞)

|Aµg|pdµ =

≥

=

=

(cid:40)

y ∈ (0,∞) :

(cid:90)

1

µ(0, y)

(0,y)

gdµ > λ

dλ

(cid:27)

f dx > λ

dλ

(cid:90) t

0

1
t

0

(cid:90) ∞
(cid:90) ∞
(cid:90) ∞
(cid:12)(cid:12)(cid:12)(cid:12)1
(cid:90) 1

t

0

0

0

pλp−1µ

(cid:26)
pλp−1tλdλ
pλp−1L
(cid:90) t

y ∈ (0, 1) :

(cid:12)(cid:12)(cid:12)(cid:12)p

280

f (x)dx

dt

0

Additionally, g ∈ Fφ,Φ

shows that B(cid:0)µ, Fφ,Φ

p

p

(µ) as(cid:82)
(0,∞) gdµ =(cid:82) 1
(0,∞)(cid:1) ≥ B(L|(0,1), Fφ,Φ

p

).

0 f dy = φ and(cid:82)

(0,∞) gpdµ =(cid:82) 1

0 f pdy = Φ. This

For every x ∈ (0,∞), there exists t ∈ (0, 1) such that µ(0, x) = t. Notice that there could
exist y (cid:54)= x, such that µ(0, y) = t (which means that µ(x, y) = 0 in the case that x < y).
Let f ∈ Fφ,Φ

(cid:0)(0,∞), µ(cid:1). Then from the well-known inequality

p

(cid:90)

(cid:90) t

(4.3.2)

(cid:40)

µ(0, x)

0

1

(0,x)

(cid:90)

f (t)dµ(t) ≤ 1
t

f∗(u)du, for f ≥ 0
(cid:41)
(0,x) f dµ > λ} and tλ = L(cid:110)
(cid:82)
(0,∞)(cid:1) ≤ B(L|(0,1), Fφ,Φ

t ∈ (0, 1) : 1

f dµ > λ

µ(0, x)

x ∈ (0,∞) :

(0,x)

1

p

).

(0,∞)(cid:1) ≤ B(cid:0)µ, Fφ,Φ

p

≤ µ(0, xλ) ≤ tλ
(cid:82) t
0 f∗(u)du > λ

(cid:111). This
(0,∞)(cid:1). Now take a function

t

(0,x) dµ, Then the pushforward measure

G#µ(E) := µ(cid:0)G−1(E)(cid:1)

we get

µ

1

µ(0,x)

where xλ = sup{x :

implies that B(cid:0)µ, Fφ,Φ
On the other hand, trivially ˜B(cid:0)µ, Fφ,Φ
(cid:0)(0,∞), µ(cid:1) and let G(x) =(cid:82)

h ∈ Fφ,Φ

p

p

p

for any E ⊂ (0, 1) is equal to Lebesgue measure of E.
Indeed, let (c, d) ⊂ (0, 1). Since
µ(0,∞) = 1, there exist α, β ∈ (0,∞) such that µ(0, α) = c and µ(0, β) = d. In other words,

G(α) = c and G(β) = d. Then G−1(cid:0)(c, d)(cid:1) = (α, β) and

(cid:16)

G−1(cid:0)(c, d)(cid:1)(cid:17)

µ

= µ(α, β) = µ(0, β) − µ(0, α) = d − c = L(c, d).

Since (c, d) is an arbitrary interval, the pushforward measure G#µ is the Lebesgue measure

281

on (0, 1).

Notice that G is increasing and onto (0, 1), but it may not be invertible. However, G has

an inverse, G−1, when restricted on supp µ. Since G(cid:0)(0,∞)\ supp µ(cid:1) is an at most countable

set, say {x1, x2, . . .}, the function h◦ G−1 is defined on (0, 1)\{x1, x2, . . .} and by changing
variables

(cid:90)

(cid:90)

hdµ =

supp µ

supp µ

(h ◦ G−1) ◦ Gdµ =

(cid:90)
(0,1)\{x1,x2,...} h ◦ G−1dx

Let (h ◦ G−1)∗ be the decreasing rearrangement of h ◦ G−1. Notice that

µ{h > λ} = |{h ◦ G−1 > λ}| = |{(h ◦ G−1)∗ > λ}| = µ{(h ◦ G−1)∗ ◦ G > λ}|,

thus, (h ◦ G−1)∗ ◦ G is decreasing and equimeasurable to h with respect to µ. Notice also
that for every x ∈ St,

(cid:90)

hdµ =

(0,x)

(cid:90)
(0,t)\{x1,x2,...} h ◦ G−1dx ≤

(cid:90)
(0,t)\{x1,x2,...}(h ◦ G−1)∗dx
(cid:90)

(h ◦ G−1)∗ ◦ Gdµ

Therefore, we obtain B(µ, Fφ,Φ

These imply B(µ, Fφ,Φ

p

p

) ≤ ˜B(µ, Fφ,Φ
) = B(L|(0,1), Fφ,Φ

p

p

=

(0,x)

(cid:104)

).

) =

ψ−1

p

(cid:16) φp

(cid:17)(cid:105)p

Φ

Φ.

Remark 4.3.4.

(i) Let us point out that the supremum with respect to any probability

measure is attained and is equal to

supB(µ, Fφ,Φ

p

) = B(L|(0,1), Fφ,Φ

p

)

282

where the supremum is taken over all probability measures µ.

(ii) If µ(0,∞) = L < ∞, then the measure σ = µ/L is a probability measure. By Theorem

4.1.2,

(cid:107)Aσf(cid:107)Lp((0,∞),σ) ≤ ψ−1

p

(cid:32)((cid:82)
(cid:82)

(cid:33)

(0,∞) f dσ)p
(0,∞) f pdσ

(cid:107)f(cid:107)Lp((0,∞),σ) ,

which implies that for f ∈ Fφ,Φ

((0,∞), µ),

p

(cid:13)(cid:13)Aµf(cid:13)(cid:13)Lp((0,∞),µ) ≤ ψ−1

p

(cid:18) φp

Lp−1Φ

(cid:19)

(cid:107)f(cid:107)Lp((0,∞),µ) .

(iii) If µ(0,∞) = ∞ for a σ-finite measure µ, we consider the measure

1(0,N )

N µ, and letting

N → ∞, we get

Here we point out that

(cid:13)(cid:13)Aµf(cid:13)(cid:13)Lp((0,∞),µ) ≤ p
(cid:13)(cid:13)Aµf(cid:13)(cid:13)Lp((0,∞),µ)

(cid:107)f(cid:107)Lp((0,∞),µ) .

p − 1

sup

f∈Lp((0,∞),µ)

(cid:107)f(cid:107)Lp((0,∞),µ)

≤ p
p − 1

=

sup

f∈Lp(0,∞)

(cid:107)ALf(cid:107)Lp(0,∞)
(cid:107)f(cid:107)Lp(0,∞)

Corollary 4.3.5 ([31]). Let (X, µ) be a non-atomic probability space and f ∈ Fφ,Φ

(X, µ).

p

Then

(cid:107)MT f(cid:107)Lp(X,µ) ≤ ψ−1

p

(cid:107)f(cid:107)Lp(X,µ)

where MT is the dyadic-like maximal function defined by

(cid:18) φp

(cid:19)

Φ

(cid:90)

MT φ(x) = sup
x∈I∈T

1

µ(I)

I

|φ|dµ

for every φ ∈ L1(X, µ) where T is a family of measurable subsets of X such that

283

(a) X ∈ T and for every I ∈ T we have µ(I) > 0.

(b) For every I ∈ T there corresponds an at most countable subset C(I) ⊂ T containing at
least two elements such that the elements of C(I) are pairwise disjoint subsets of I and
I = ∪C(I).

(c) T =(cid:83)

m≥0 T(m) where T(0) = {X} and T(m+1) =(cid:83)

C(I).

I∈T(m)

(d) limm→∞ supI∈T(m)

µ(I) = 0.

The operator MT satisfies (4.1.4) with [µ] = 1 and the result follows from Theorem 4.1.4.
The sharpness of the constant has been proven by Melas [31], by calculating a Bellman

function.

4.4 Two Weights

Now we turn our attention to inequalities of two measures. We will need the following

lemmas whose proofs are provided in [23].

Lemma 4.4.1. For any t ∈ R, any measure w on [t,∞) and α ∈ (0, 1), we have

w[x,∞)−αdw(x) ≤ w[t,∞)1−α
1 − α

.

[t,∞)

Lemma 4.4.2. For any t ∈ R, any measure σ on (0, t] and α ∈ (0, 1), we have

(cid:90)

(cid:90)

(0,t]

σ(0, x]−αdσ(x) ≤ σ(0, t]1−α
1 − α

.

284

Definition 4.4.3. Let µ be a measure. We define the µ-Hardy operator as

Hµf (x) ≡ H(µf ) =

f (t)dµ(t),

f ≥ 0.

(cid:90)

(0,x]

Theorem 4.4.4. The two-measure (˜σ, ω) Hardy inequality, for 1 ≤ p < ∞,

(cid:32)(cid:90)

(0,∞)

(cid:33)1/p

(cid:33)1/p

(cid:32)(cid:90)

(0,∞)

|H(˜σg)|pdω

≤ Np(˜σ, ω)

|g|pd˜σ

g ≥ 0

,

(4.4.1)

holds if and only if

(cid:16)

ω[r,∞)1/p ˜σ(0, r]1/p(cid:48)(cid:17)

< ∞

Gp(˜σ, ω) ≡ sup

r>0

Moreover, Gp(˜σ, ω) ≤ Np(˜σ, ω) ≤ p1/p(p(cid:48))1/p(cid:48)
N1(˜σ, ω).

Gp(˜σ, ω), for 1 < p < ∞ while G1(˜σ, ω) =

The proof is essentially due to [35], while the proof for p = 2 is written in [23]. We write

Proof. For 1 < p < ∞ and h(t) =

(0,x] d˜σ)d˜σ(x)

pp(cid:48) we have,

(cid:17) 1

(cid:33)p

it here for general p for completeness.

(0,t]((cid:82)

(cid:16)(cid:82)
(cid:32)(cid:90)
(cid:32)(cid:90)
(cid:32)(cid:90)

(0,∞)

(0,x]

(0,∞)

(0,x]

(0,∞)

(0,∞)

(0,x]

g(t)ph(t)p

(cid:90)

(cid:90)

(0,∞)

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

|H(g˜σ)|pdω =

=

≤

=

dω(x)

g(t)d˜σ(t)

(cid:33)p
g(t)h(t)h(t)−1d˜σ(t)
(cid:33)(cid:32)(cid:90)

g(t)ph(t)pd˜σ(t)

dω(x)

(cid:32)(cid:90)

h(t)−p(cid:48)
(cid:33)p−1

(0,x]

h(t)−p(cid:48)

d˜σ

d˜σ(t)

dω(x)

(cid:33)p/p(cid:48)
 d˜σ(t)

dω(x)

[t,∞)

(0,x]

285

By Lemma 4.4.2 and definition of Gp,

(cid:90)

|H(g˜σ)|pdω ≤

(0,∞)

g(t)ph(t)p

(0,∞)

(cid:90)
(cid:90)

≤

(0,∞)
= Gp−1

p

(p(cid:48))p−1

g(t)ph(t)p

(cid:90)

(cid:90)
(cid:34)(cid:90)

[t,∞)

[t,∞)

d˜σ

p(cid:48)
(cid:41)1/p(cid:48)p−1
(cid:40)(cid:90)
p(cid:48)Gp · ω[x,∞)−1/p(cid:17)p−1
(cid:16)
(cid:34)(cid:90)

ω[x,∞)−1/p(cid:48)

(0,x]

g(t)ph(t)p

(0,∞)

[t,∞)

 d˜σ(t)
(cid:35)
(cid:35)

d˜σ(t)

dω(x)

dω(x)

dω(x)

d˜σ(t)

By Lemma 4.4.1 and definition of Gp,

(cid:90)

(0,∞)

|H(g˜σ)|pdω ≤ Gp−1

p

(cid:90)

(0,∞)

(p(cid:48))p−1p
(cid:90)
(cid:90)
(cid:90)

p(p(cid:48))p−1p
p(p(cid:48))p−1p
p(p(cid:48))p−1p

≤ Gp

= Gp

= Gp

(0,∞)

(0,∞)

(0,∞)

g(t)ph(t)pω[t,∞)−1/pd˜σ(t)
(cid:33)−1/p(cid:48)

(cid:32)(cid:90)

g(t)ph(t)p

d˜σ

d˜σ(t)

(0,t]

g(t)ph(t)ph(t)−pd˜σ(t)

g(t)pd˜σ(t)

Thus, Np(˜σ, ω) ≤ p1/p(p(cid:48))1/p(cid:48)

Gp(˜σ, ω). For p = 1, by changing the order of integration,

(cid:90)

(cid:90)

H(g˜σ)dω ≤ sup

ω[r,∞)

r>0

(0,∞)

g(t)d˜σ(t)

(0,∞)

So, N1 ≤ G1.

Conversely, for 1 ≤ p < ∞, letting g(t) = 1(0,r](t) and since for x ≥ r, H(g˜σ)(x) ≥

286

H(g˜σ)(r) =(cid:82)
(cid:32)(cid:90)

(0,r] d˜σ,

(cid:33)p−1(cid:90)

d˜σ

(0,r]

[r,∞)

(cid:90)

dω

(cid:32)(cid:90)

d˜σ =

(0,r]

(0,r]

(cid:33)p(cid:90)
(cid:90)

d˜σ

(0,∞)

≤ N p

p (˜σ, ω)

which implies that(cid:16)(cid:82)

(cid:17)p−1(cid:82)

(0,r] d˜σ

(cid:90)

dω ≤

|H(g˜σ)|pdω

(0,∞)

[r,∞)
|g|pd˜σ = N p

p (˜σ, ω)

(cid:90)

|g|pd˜σ

(0,r]

[r,∞) dω ≤ N p

p (˜σ, ω). Taking supremum over all r > 0,

Gp(˜σ, ω) ≤ Np(˜σ, ω).

4.4.1 A three-weight norm inequality

Now consider the inequality

(cid:32)(cid:90)

(cid:33)1/p

≤ Kp(µ, σ, ω)

(cid:32)(cid:90)

(0,∞)

(cid:33)1/p

|f (x)|pdσ(x)

,

(4.4.2)

|Hµf (x))|pdω(x)

(0,∞)

for f ≥ 0 and the three measures µ, σ, ω.

It is easy to check that (4.4.2) holds only if dµ(t) = m(t)dσ(t) and so it implies that

(cid:90)

(cid:90)

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

(0,∞)

(0,x]

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)p

(cid:90)

(0,∞)

f (t)m(t)dσ(t)

dω(x) ≤ Kp

p (µ, σ, ω)

|f (x)|pdσ(x).

Setting f (t)m(t)dσ(t) = g(t)d˜σ and f (x)pdσ = g(x)pd˜σ, which imply that

d˜σ =

f (t)m(t)

g(t)

dσ(t) =

f (t)p
g(t)p dσ(t) and m(t) =

f (t)p−1
g(t)p−1 ,

287

we have that (4.4.2) is equivalent to the two measure (˜σ, ω) Hardy inequality (4.4.1). More-

Corollary 4.4.5. ([35]) If ω and σ are Borel measures and 1 ≤ p < ∞, then

Gp(˜σ, ω).

ω[r,∞)1/p

(cid:32)(cid:90)

(cid:33)1/p(cid:48).

mp(cid:48)

dσ

(0,r]

f ≥ 0

,

over,

where Gp(˜σ, ω) = sup
r>0

(cid:32)(cid:90)

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

0

(0,∞)

if and only if

= sup
r>0

Gp(˜σ, ω) ≤ Kp(µ, σ, ω) ≤ p1/p(p(cid:48))1/p(cid:48)
ω[r,∞)1/p˜σ(0, r]1/p(cid:48)(cid:17)
(cid:16)
(cid:33)1/p
(cid:12)(cid:12)(cid:12)(cid:12)p
ω[r,∞)1/p

(cid:32)(cid:90)

(cid:32)(cid:90)

f (t))dt

dω(x)

mp(cid:48)

dσ

(0,∞)

≤ C

B = sup
r>0

(0,r]

|f (x)|pdσ(x)

(cid:33)1/p
(cid:33)1/p(cid:48) < ∞,

where dx = m(x)dσ(x). Moreover, B ≤ C ≤ p1/p(p(cid:48))1/p(cid:48)
p = 1.

B, for 1 < p < ∞ while B = C for

Corollary 4.4.6. The inequality

(cid:32)(cid:90)

(cid:33)1/p

(cid:32)(cid:90)

(0,∞)

(cid:33)1/p

|f (x)|pdσ(x)

≤ Mp(µ, σ, ω)

f ≥ 0 (4.4.3)

,

|Aµf (x))|pdω(x)

(0,∞)

holds if and only if

Gp ≡ sup

r>0

(cid:34)(cid:90)

[r,∞)

(cid:0)(cid:82)
(0,x] m(t)dσ(t)(cid:1)p

dω(x)

(cid:35)1/p(cid:32)(cid:90)

(cid:33)1/p(cid:48)

mp(cid:48)

dσ

< ∞

(0,r]

where dµ(t) = m(t)dσ(t). Moreover, Gp ≤ Mp(µ, σ, ω) ≤ p1/p(p(cid:48))1/p(cid:48)

Gp.

Proof. The inequality (4.4.3) is equivalent to the two-measure (τ, ν) Hardy inequality (4.4.1)
for dτ = mp(cid:48)

(0,x] mdσ(cid:1)p. The result then follows from Theorem 4.4.4.

dσ and dν = dω/(cid:0)(cid:82)

288

[r,∞)

(cid:90)
where g(x) =(cid:82) x
(cid:18)(cid:90) ∞

In the special case that ω = σ = µ with dµ(x) = m(x)dx for m ∈ L1

loc(µ),

(cid:90)

µ(0, x]−pdµ(x) =

g(cid:48)(x)
g(x)p dx =

g(r)1−p
p − 1

µ(0, r]1−p

p − 1

=

[r,∞)

0 m(t)dt, we have that Gp =

(cid:19)1/p ≤ p

1

(p−1)1/p

(cid:18)(cid:90) ∞

and the (4.4.3) becomes

(cid:19)1/p

|f (x)|pdµ(x)

f ≥ 0

,

p − 1

0

|Aµf (x))|pdµ(x)

0

Thus, the inequality of Corollary 4.1.2 is a refinement of this.

Theorem 4.4.7. Let ω, µ be two Radon measures on (0,∞) and define

s := inf{x ∈ (0,∞) : µ charges the interval (0, x)}.

The following are equivalent:

(i) For λ > 0 and f ≥ 0, the special weak type (1,1)

ω{x ∈ (0,∞) : Aµf (x) > λ} ≤ K
λ

(cid:90)
{x∈(0,∞): Aµf (x)>λ} f (t)dµ(t)

(ii) For any collection of open intervals {(aj, bj)}j∈N in (s,∞),

(iii) The restriction ω(cid:12)(cid:12)(s,∞) of ω at (s,∞) is absolutely continuous with respect to µ and

the density

dω|(s,∞)

dµ

∈ L∞ with

(cid:88)

j

ω(aj, bj) ≤ K

µ(aj, bj).

j

(cid:88)
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L∞(µ)

≤ K.

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)dω|(s,∞)

dµ

289

(4.4.4)

(4.4.5)

Remark: The definition of s is not needed to show the equivalence of the conditions (ii)

and (iii) but it is important to circumvent the cases where the measures are not absolutely

continuous and satisfy (4.4.4) trivially. For example, dω = 1(0,1)dx and dµ = 1(2,∞)dx.

Proof. Without loss of generality, assume that s = 0, that is, µ charges every open interval

j µ(aj, bj) < 
. The hypothesis implies that (cid:80)

µ(E) < 
. Since E is open, it can be written as a union of open intervals, i.e. E =(cid:83)
Thus, (cid:80)
ω(E) < K
. Since 
 is arbitrary, ω(A) = 0. Thus, ω (cid:28) µ and ω(A) =(cid:82)

of the form (0, δ). Let us show first that (ii) implies (iii). Let A be a Borel set such
that µ(A) = 0. As µ is regular, for any 
 > 0, there exists an open set E ⊃ A such that
j(aj, bj).
j ω(aj, bj) < K
, or equivalently,
A m(t)dµ(t) for some
non negative function m and any Borel set A. If we assume that there exists a Borel set A
such that µ(A) > 0 and m(t) > K, for all t ∈ A, then using Borel regularity, we cover A by

j µ(aj, bj) = µ(A) + 
 and we obtain

open intervals (aj, bj) such that(cid:80)
(cid:90)

(cid:88)

ω(aj, bj) ≥ ω(A) =

j

m(t)dµ(t) > Kµ(A) = K

A

µ(aj, bj) − 


(cid:88)

j

 .

As 
 is arbitrary, we have a contradiction.

m such that for any measurable set A, ω(A) =(cid:82)

Now we show that (iii) implies (ii). By the hypothesis, there exists a non negative function
A m(t)dµ(t) with (cid:107)m(cid:107)L∞(µ) ≤ K. Consider

any collection of open intervals {(aj, bj)}j∈N. Then

(cid:88)

j

(cid:90)

(cid:88)

ω(aj, bj) =

j

(aj ,bj )

m(t)dµ(t) ≤ K

(cid:88)

j

µ(aj, bj)

To prove that (ii) implies (i), notice that the set Eλ = {x ∈ (0,∞) : Aµf (x) > λ} is
open for any λ > 0. Thus, Eλ = ∪Ij where Ij are maximal pairwise disjoint open intervals

290

and

ω{Aµf > λ} =

(cid:88)

j

ω(Ij) ≤ K

(cid:88)

j

µ(Ij) =

=

K
λ

K
λ

(cid:90)

(cid:88)
(cid:90)
{Aµf >λ} f (t)dµ(t).

f (t)dµ(t)

Ij

j

Finally, we prove that (i) implies (ii). It is enough to show (4.4.4) implies (4.4.5) for an

interval (a, b). Because then, for any N ∈ N,

N(cid:88)

N(cid:88)

ω(aj, bj) ≤ K

µ(aj, bj) ≤ K

j=1

j=1

and letting N → ∞, we obtain (4.4.5).

∞(cid:88)

j=1

µ(aj, bj)

Fix λ > 0. Given an interval (a, b) with µ(a, b) (cid:54)= 0, we find a function f and an interval
J ⊃ (a, b) such that J = {Aµf > λ} and µ(J) = µ(a, b) + δ, for δ ≥ 0. As a first case,
suppose that µ charges the subintervals (a, a + c) and (b − c, b) for some 0 < c < b − a.
Consider the function

f = λ1(0,a) + (λ + η)1(a,t) + (λ − 2η)1(t,k) + λ1(k,b) ,

where a < t < k < b, 0 < 2η < λ and(cid:82)
(cid:90) b

(cid:90) a

(cid:90) a

(a,t) dµ = 2(cid:82)
(cid:90) b

f dµ = λ

dµ,

f dµ = λ

(t,k) dµ. Then, f satisfies the conditions

(cid:90) x

(cid:90) x

dµ, and

f dµ > λ

dµ

0

0

0

0

0

0

for all x ∈ (a, b) with the reverse inequality holding otherwise. Thus, {Aµf > λ} = (a, b)

291

and

ω(a, b) = ω{Aµf > λ} ≤ K
λ

(cid:90)
{Aµf >λ} f dµ =

K
λ

(cid:90)

(a,b)

f dµ = Kµ(a, b).

On the other hand, suppose that µ(a, a+c) = 0 while µ(b−c, b) (cid:54)= 0, for some 0 < c < b−a.
By definition of p, there is a point a1 < a such that µ(a1, a) = 
. Then, following the previous
case, we construct a function f such that {Aµf > λ} = (a1, b) and

ω(a, b) ≤ ω(a1, b) ≤ Kµ(a1, b) = K(cid:0)µ(a, b) + 
(cid:1)

As 
 is arbitrary, we obtain (4.4.5).

In the case that µ(a, a + c) (cid:54)= 0 while µ(b − c, b) = 0, for some 0 < c < b − a, either
there is f such that {Aµf > λ} = (a,∞), when µ(b,∞) = 0, or there is f such that
{Aµf > λ} = (a, b1), where b1 > b with µ(b, b1) = 
. Thus, ω(a, b) ≤ Kµ(a, b).

Finally, if µ(a, b) = 0, as above, we find an open interval J ⊃ (a, b) and a function f such

that {Aµf > λ} = J and µ(J) = 
. This implies that ω(a, b) = 0.

For measures ω, µ that satisfy (4.4.4), define

(cid:26)(cid:90)

Bp(µ, ω, Fφ,Φ

p

) = sup

|Aµf (x)|pdω : 0 ≤ f ∈ Fφ,Φ

p

(cid:0)(0,∞), µ(cid:1)(cid:27)

and

(cid:26)(cid:90)

˜Bp(µ, ω, Fφ,Φ

p

) = sup

|Aµf (x)|pdω : 0 ≤ f ∈ Fφ,Φ

p

(cid:27)
(cid:0)(0,∞), µ(cid:1) decreasing

Theorem 4.4.8. Suppose two non-atomic Radon measures ω, µ on (0,∞) satisfy the special

292

weak type inequality (4.4.4), where µ is a probability measure and L = ω(0,∞). Then for
every non-negative f ∈ Fφ,Φ

p

ψ−1

p

(cid:19)(cid:90)

(cid:19)(cid:21)p − (K − L)

(cid:18) φp
(cid:0)(0,∞), µ(cid:1) such that the equality in (4.4.6) is attained

(4.4.6)

f pdµ.

(0,∞)

φp
Φ

Φ

(cid:0)(0,∞), µ(cid:1),
(cid:20)
(cid:18)

(cid:90)

|Aµf (x)|pdω ≤

K

(0,∞)

There is a decreasing function g ∈ Fφ,Φ

p

if and only if ω = Kµ.

Proof. First, let f be a decreasing function. Notice that if L = K, by (4.4.5), ω = Kµ. Let
us assume that ω(0,∞) = L < K. For A, such that {Aµf > A} = (0,∞) to be determined
later and using (4.4.4), we have

(cid:90)

|Aµf|pdω =

(cid:90) ∞

0

≤ LAp +

≤ LAp + K

= LAp + Kp

A

A

(cid:90) ∞
pλp−1ω{|Aµf| > λ}dλ
(cid:90)
(cid:90) ∞
pλp−1ω{|Aµf| > λ}dλ
pλp−2
(cid:90) |Aµf|
(cid:90)
{|Aµf|>λ} f dµdλ
(cid:19)
(cid:18)(cid:12)(cid:12)Aµf(cid:12)(cid:12)p−1 − Ap−1
(cid:90)
λp−2dλdµ
(cid:90)
f(cid:12)(cid:12)Aµf(cid:12)(cid:12)p−1dµ − Kp
dµ
Ap−1

Kp
p − 1
Kp
p − 1

p − 1

A

f

f

= LAp +

= LAp +

(cid:90)

f dµ

Using Hölder’s inequality with exponents p and p(cid:48) = p
p−1

, we obtain

(cid:90)

(cid:18)(cid:90)

(cid:19)1
p(cid:18)(cid:90)

(cid:19)1− 1

(cid:90)

Ap−1

f dµ

|Aµf|pdω ≤ LAp +

Kp
p − 1

f pdµ

|Aµf|pdµ

p − Kp
p − 1

293

Dividing both sides by(cid:82) f pdµ and rearranging we obtain,
KpAp−1(cid:82) f dµ − L(p − 1)Ap

(cid:82) |Aµf|pdω
(cid:82) f pdµ

+

(cid:82) f pdµ

(p − 1)

(cid:18)(cid:82) |Aµf|pdµ
(cid:82) f pdµ

(cid:19)1− 1

p

(4.4.7)

≤ Kp

Now, by Corollary 4.1.2, the right hand side is bounded by

(cid:20)
ψ−1
and as A (cid:55)→ KpAp−1(cid:82) f dµ − L(p − 1)Ap

Kp

p

Φ

(cid:19)(cid:21)p/p(cid:48)

(cid:18) φp
is increasing on(cid:104)
(Kp − Lp + L)((cid:82) f dµ)p
(cid:82) f pdµ

(cid:82) f pdµ
(cid:82) |Aµf|pdω
(cid:82) f pdµ

+

gives

(p − 1)

(cid:105), inequality (4.4.7)
(cid:82) f dµ
(cid:19)(cid:21)p/p(cid:48)
(cid:18) φp

0, K
L

(cid:20)

≤ Kp

ψ−1

for A = (cid:82) f dµ (we choose this A, as {Aµf (x) > A} = (0,∞) since f decreasing). This
(cid:33)p
(cid:90)

(cid:32)(cid:90)

implies

Φ

p

|Aµf|pdω ≤ Kp
p − 1

(0,∞)

(cid:20)
ψ−1
(cid:32)
Kp(cid:48)(cid:20)

ψ−1

p

p

(cid:19)(cid:21)p/p(cid:48)(cid:90)
(cid:18) φp
(cid:19)(cid:21)p/p(cid:48)
(cid:18) φp

Φ

Φ

=

f pdµ − Kp − Lp + L
(cid:33)(cid:90)

(0,∞)
− Kp − Lp + L

p − 1

p − 1

φp
Φ

(0,∞)

f dµ

(0,∞)

f pdµ

(4.4.8)

We claim that

p(cid:48)(cid:20)

ψ−1

p

(cid:18) φp

(cid:19)(cid:21)p/p(cid:48)

Φ

(cid:20)

ψ−1

p

(cid:18) φp

(cid:19)(cid:21)p

Φ

− 1
p − 1

φp
Φ

=

Indeed, consider the function

g(y) = p(cid:48)(cid:104)

ψ−1

p

(1 − y)

(cid:105)p/p(cid:48)

(1 − y) −(cid:104)

ψ−1

p

(cid:105)p

(1 − y)

− 1
p − 1

0 ≤ y < 1

,

294

Then, for 0 < y < 1,

−p(cid:2)ψ−1

p (1−y)(cid:3)p−2
(cid:105)p−2(cid:16)

g(cid:48)(y) =

p(p−1)

(cid:104)

=− 1
p−1

+

ψ−1
p (1−y)
1
p−1

= 0

(cid:17) +
p (1−y)

1 − ψ−1

1
p−1

+

(p−1)

(cid:16)

ψ−1
p (1−y)
1 − ψ−1

(cid:17)
p (1−y)

which implies that g is constant and as g is continuous at 0, we obtain that g ≡ 0 on [0, 1).
Thus, inequality (4.4.8) gives

(cid:90)

|Aµf|pdω ≤

(0,∞)

Thus, ˜Bp(µ, ω, Fφ,Φ

p

) ≤(cid:16)

(cid:104)

ψ−1

p

K

p

Φ

K

φp
Φ

ψ−1

(cid:19)(cid:90)

(cid:19)(cid:21)p − (K − L)

(cid:20)
(cid:18)
(cid:18) φp
(cid:16) φp
(cid:17)(cid:105)p − (K − L) φp
(cid:82){Aµg>λ} gdµ. Choose g to be the extremizer of

(0,∞)

f pdµ

(cid:17)

Φ.

Φ

Φ

To obtain equality in (4.4.6), notice that ω = Kµ if and only if there is a decreasing

function g such that ω{Aµg > λ} = K
Theorem 4.1.2.

λ

Finally, if h ∈ Fφ,Φ

p

(cid:0)(0,∞), µ(cid:1) is such that Bp(µ, ω, Fφ,Φ
(0,∞) |Aµg|pdω ≥(cid:82)

(0,x) gdµ. This implies that(cid:82)

p

proof of Theorem 4.1.2, we obtain a decreasing function g ∈ Fφ,Φ

(cid:82)

(0,x) hdµ ≤(cid:82)

) is attained, following the

(cid:0)(0,∞), µ(cid:1) such that

p

(0,∞) |Aµh|pdω which in turn

gives ˜Bp(µ, ω, Fφ,Φ

p

) = Bp(µ, ω, Fφ,Φ

p

).

Conjecture 4.4.9. Suppose two non-atomic Radon measures (ω, µ) satisfy the special weak
type inequality (4.4.4), µ is a probability measure and ω(0,∞) = L ≤ K. For the p-admissible

pair (φ, Φ),

(cid:18)

Kψ−1

p

(cid:18) φp

Φ

(cid:19)p−(K−L)

φp
Φ

(cid:19) 1
p ≤ p1/p(p(cid:48))1/p(cid:48)

sup
r>0

(cid:34)(cid:90)

[r,∞)

p(cid:32)(cid:90)
(cid:35) 1

(cid:33) 1

p(cid:48)

dµ

(0,r]

(cid:0)(cid:82)

dω(x)

(0,x] dµ(cid:1)p

295

If ω = Kµ is absolutely continuous, the conjecture is true.

296

BIBLIOGRAPHY

297

BIBLIOGRAPHY

[1] M. A. Alfonseca, P. Auscher, A. Axelsson, S. Hofmann, and S. Kim, Ana-
lyticity of layer potentials and L2 solvability of boundary value problems for divergence
form ellliptic equations with complex L∞ coefficients, arXiv:0705.0836v1.

[2] Auscher, P., Hofmann, S., Lacey, M., McIntosh, A., and Tchamitchian, P.,
The Solution of the Kato Square Root Problem for Second Order Elliptic Operators on
Rn, Ann. of Math. 156 (2002), 633–654.

[3] P. Auscher, S. Hofmann, C. Muscalu, T. Tao, and C. Thiele, Carleson mea-

sures, trees, extrapolation, and T (b) theorems, Publ. Mat. 46 (2002), no. 2, 257–325.

[4] Burkholder D.L. Martingales and Fourier analysis in Banach spaces, Lecture Notes

in Mathematics, vol 1206, Springer (1986)

[5] Burkholder D.L. Explorations in martingale theory and its applications, Ecole d’Eté

de Probabilités de Saint-Flour XIX — 1989 (pp.1-66)

[6] Christ, M., A T (b) theorem with remarks on analytic capacity and the Cauchy integral,

Colloq. Math. 60/61 (1990), 601–628.

[7] R. R. Coifman and C. L. Fefferman, Weighted norm inequalities for maximal

functions and singular integrals, Studia Math. 51 (1974), 241-250.

[8] David, G., Unrectifiable 1-sets have vanishing analytic capacity, Rev. Mat. Iberoamer-

icana 14 (2) (1998), 369-479.

[9] David, G., Analytic capacity, Calderón-Zygmund operators, and rectifiability, Publ.

Mat. 43 (1) (1999), 3-25.

[10] David, Guy, Journé, Jean-Lin, A boundedness criterion for generalized Calderón-

Zygmund operators, Ann. of Math. (2) 120 (1984), 371–397, MR763911 (85k:42041).

[11] David,G.,Journé,J.-L.,and Semmes,S.,Opérateurs de Calderón-Zygmund, fonctions

para-accrétives et interpolation. Rev. Mat. Iberoamericana 1 (1985), 1–56.

298

[12] Delis A., Nikolidakis E., Sharp and general estimates for the Bellman function of three
integral variables related to the dyadic maximal operator, Colloq. Math. 153 (2018), no.
1, 27-37.

[13] Grigoriadis C., Paparizos M. Counterexample to the Hytöonen’s off-testing condi-

tion in two dimensions, arXiv:2004.06207 (To appear in Coloq. Math.).

[14] Grigoriadis C., Paparizos M., Sawyer E., Shen C-Y, Uriarte-Tuero
fractional singular integrals,

I. A two weight local Tb theorem for n-dimensional
arXiv:2011.05637 .

[15] Hardy, G.H., Note on a theorem of Hilbert, Math. Z. 6 (1920), 314–317.

[16] Hofmann, S., A proof of the local T b theorem for standard Calderón-Zygmund opera-

tors, arXiv:0705.0840v1.

[17] Hofmann, S., Local Tb theorems and applications in PDE. International Congress of

Mathematicians. Vol. II, 1375–1392, Eur. Math. Soc., Zürich, 2006.

[18] Hofmann, S., Lacey, M., and McIntosh, A., The solution of the Kato
problem for divergence form elliptic operators with Gaussian heat ker-
nel bounds. Ann. of Math. 156 (2002), 623–631.

[19] Hofmann, S., and McIntosh, A., The solution of the Kato problem in two dimen-
sions, In Proceedings of the Conference on Harmonic Analysis and PDE (El Escorial,
2000), Publ. Mat. Extra Vol. (2002), 143–160.

[20] R. Hunt, B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for
the conjugate function and the Hilbert transform, Trans. Amer. Math. Soc. 176 (1973),
227-251.

[21] Hytönen, Tuomas, On Petermichl’s dyadic shift and the Hilbert transform, C. R.

Math. Acad. Sci. Paris 346 (2008), MR2464252.

[22] Hytönen T., The two-weight inequality for the Hilbert transform with general mea-

sures. Proceedings of the London Mathematical Society (2013). 10.1112/plms.12136.

[23] Hytönen T., The two weight inequality for the Hilbert transform with general measures,

Proc. Lond. Math. Soc. Vol 117, 483-526, 2018.

299

[24] Hytönen, Tuomas and H. Martikainen, On general

local T b theorems,

arXiv:1011.0642v1.

[25] Kufner A., Maligranda L. and Persson L.-E.,The Hardy inequality: about its

history and some related results, Praha: Vydavatelsk
y servis ,2007, pp. 161

[26] Lacey, Michael T., Two weight inequality for the Hilbert transform: A real variable

characterization, II, Duke Math. J. Volume 163, Number 15 (2014), 2821-2840.

[27] M. T. Lacey and H. Martikainen, Local T b theorem with L2 testing conditions

and general measures: Calderón–Zygmund operators, arXiv:1310.08531v1.

[28] Lacey, Michael T., Sawyer, Eric T., Uriarte-Tuero, Ignacio, A Two Weight
Inequality for the Hilbert transform assuming an energy hypothesis, Journal of Func-
tional Analysis, Volume 263 (2012), Issue 2, 305-363.

[29] M. Lacey, E. Sawyer.,C. Shen, I. Uriarte-Tuero, Two weight inequality for
the Hilbert transform: A real variable characterization I, Duke Math. J, Volume 163,
Number 15 (2014), 2795.

[30] Lacey, Michael T., Wick, Brett D., Two weight inequalities for Riesz transforms:

uniformly full dimension weights, arXiv:1312.6163v1,v2,v3.

[31] Melas A., The Bellman function of dyadic-like maximal operators and related inequal-

ities, Advances in Mathematics 192 (2005), 310-340.

[32] Melas A., Sharp general local estimates for dyadic-like maximaloperators and related

Bellman functions, Advances in Mathematics 220 (2009) 367–426.

[33] Melas A., Nikolidakis E. A sharp integral rearrangement inequality for the dyadic
maximal operator and applications, Appl. Comput. Harmon. Anal. 38 (2015), no. 2,
242-261.

[34] Mattila, P., Melnikov, M., and Verdera, J., The Cauchy integral, analytic

capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1) (1996), 127–136.

[35] Muckenhoupt B. Hardy’s inequality with weights, Studia Mathematica T.XLIV

(1972)

300

[36] F. Nazarov, S. Treil and A. Volberg, The Bellman function and two weight
inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), 909-928, MR{1685781
(2000k:42009)}.

[37] Nazarov, F., Treil, S. and Volberg, A., The T b-theorem on non-homogeneous

spaces, Acta Math. 190 (2003), no. 2, MR 1998349 (2005d:30053).

[38] Nazarov, F., Treil, S., and Volberg, A., Accretive system T b-theorems on non-

homogeneous spaces, Duke Math. J. 113 (2) (2002), 259–312.

[39] F. Nazarov, S. Treil and A. Volberg, Two weight estimate for the Hilbert
transform and corona decomposition for non-doubling measures, preprint (2004)
arxiv:1003.1596

[40] Nikolidakis E., A Hardy inequality and applications to reverse Holder inequalities for

weights on R, J. Math. Soc. Japan, 70 (2018), no.1, 141-152.

[41] Nikolidakis E., Extremal Sequences for the Bellman Function of the Dyadic Maximal
Operator and Applications to the Hardy Operator, Canadian Journal of Mathematics,
69(6), 1364-1384, (2017)

[42] Pachpatte B. G., Mathematical Inequalities, North Holland Mathematical Library,

Vol 67, (2005).

[43] E. Sawyer, A characterization of two weight norm inequalities for fractional and Pois-

son integrals, Trans. A.M.S. 308 (1988), 533-545, MR{930072 (89d:26009)}.

[44] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero,

Ignacio, A two
weight theorem for α-fractional singular integrals with an energy side condition,
arXiv:1302.5093v8.

[45] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, A geometric con-
dition, necessity of energy, and two weight boundedness of fractional Riesz transforms,
arXiv:1310.4484v1.

[46] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, A note on failure
of energy reversal for classical fractional singular integrals, IMRN, Volume 2015, Issue
19, 9888-9920.

[47] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, A two weight
theorem for α-fractional singular integrals with an energy side condition and quasicube
testing, arXiv:1302.5093v10.

301

[48] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, A two weight the-
orem for α-fractional singular integrals with an energy side condition, quasicube testing
and common point masses, arXiv:1505.07816v2,v3.

[49] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, A two weight
theorem for α-fractional singular integrals with an energy side condition, Revista Mat.
Iberoam. 32 (2016), no. 1, 79-174.

[50] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, The two weight
T 1 theorem for fractional Riesz transforms when one measure is supported on a curve,
arXiv:1505.07822v4.

[51] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, A two weight
fractional singular integral theorem with side conditions, energy and k-energy dispersed,
arXiv:1603.04332v2.

[52] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, A good-λ lemma,
two weight T 1 theorems without weak boundedness, and a two weight accretive global T b
theorem.

[53] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, Energy counterex-

amples in two weight Calderón-Zygmund theory, IMRN Vol 2019.

[54] Sawyer, Eric T., Shen, Chun-Yen, Uriarte-Tuero, Ignacio, A two weight local

T b theorem for the Hilbert transform, to appear in Revista Mat. Iberoam 2021.

[55] E. M. Stein, Harmonic Analysis: real-variable methods, orthogonality, and oscillatory

integrals, Princeton University Press, Princeton, N. J., 1993.

[56] E. M. Stein, G. Weiss Interpolation of operators with change of measures, Trans.

Amer. Soc. 87 (1958), 159-182.

[57] Tolsa, X., Painlevé’s problem and the semiadditivity of analytic capacity, Acta Math.

190 (1) (2003), 105–149.

[58] A. Volberg, Calderón-Zygmund capacities and operators on nonhomogeneous spaces,
CBMS Regional Conference Series in Mathematics (2003), MR{2019058 (2005c:42015)}.

[59] Wang G. Sharp Maximal Inequalities for Conditionally Symmetric Martingales and
Brownian Motion, Proceedings of the American Mathematical Society, vol. 112, no. 2,
1991, pp. 579–586

302