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6/01 cJCIRCIDataDue.p65-p.15

The Theory of Function Spaces with Matrix Weights
By

Svetlana Roudenko

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2002

ABSTRACT

The Theory of Function Spaces with Matrix Weights
By

Svetlana Roudenko

Nazarov, Treil and Volberg defined matrix AP weights and extended the theory of
weighted norm inequalities on LP to the case of vector-valued functions. We develop
some aspects of Littlewood-Paley function space theory in the matrix weight setting.
In particular, we introduce matrix-weighted homogeneous Besov spaces B§q(W) and
matrix-weighted sequence Besov spaces bgq(W), as well as 539({AQ}), where the AQ ’s
are reducing operators for W. Under any of three different conditions on the weight
W, W6 Prove the norm equivalences llflisgum % Higclclltgum % ||{§Q}Q||th({AQ}),
where {5Q}Q is the vector—valued sequence of gp-transform coefficients of f. In the
process, we note and use an alternate, more explicit characterization of the matrix
Ap class. Phrthermore, we introduce a weighted version of almost diagonality and
prove that an almost diagonal matrix is bounded on b§q(W) if W is doubling. We
also obtain the boundedness of almost diagonal operators on ngO/V) under any of

the three conditions on W. This leads to the boundedness of convolution and non-

convolution type Calderon-Zygmund operators (CZOS) on 839(W), in particular,
the Hilbert transform. We apply these results to wavelets to show that the above
norm equivalence holds if the go-transform coefficients are replaced by the wavelet
coefficients. Next we determine the duals of the homogeneous matrix-weighted Besov
spaces B:"(W) and b:q(W). If W is a matrix AP weight, then the dual of B§q(W)
can be identified with BQQQKW'pi/p) and, similarly, [bgq(W)]* z bgaql(W“pI/p).
Moreover, for certain W which may not be in AP class, the duals of ng (W) and
b;q(W) are determined and expressed in terms of the Besov spaces B;°‘q’({A51}) and
5;,aq'({A51}), which we define in terms of reducing operators {AQ}Q associated with
W. We also develop the basic theory of these reducing operator Besov spaces. Fi-
nally, we construct inhomogeneous matrix-weighted Besov spaces B:q(W) and show

that results corresponding to those above are true also for the inhomogeneous case.

ACKNOWLEDGMENTS

I would like to express my deep gratitude to my advisor Prof. Michael Frazier
for introducing the subject matter to me, for his encouragment and belief in me, for
the uncountable number of hours spent sharing his knowledge and discussing various
ideas, and for many useful comments and suggestions while examining my work. I
would also like to express my thanks to Prof. Alexander Volberg for helpful suggestions
and for sharing with me his enthusiasm for and appreciation of mathematics; to Prof.
Peter Yuditskii for fruitful discussions, his availability to help me and his respect; and
to Prof. Fedor Nazarov for conveying his sense of mathematical insight. I thank the
other members of my thesis committee: Prof. William Sledd and Prof. Joel Shapiro.
I would also like to thank Prof. Clifford Weil for introducing me to graduate analysis
and teaching me how to write it well and rigorously.

I would like to say a special thank you to D. Selahi Durusoy for motivation, help
and late night fruit snacks; to Dr. Mark McCormick for his valuable advice at the
beginning of my professional career; to Leo Larsson for helpful discussions as well as
suggestions in technical writing. I want to thank Rebecca Grill and Amy Himes for
their tremendous support throughout my graduate studies. I would also like to thank

Dr. Burak Ozbagci for his support and professional advice. Last but not least, I want

iv

to thank Dr. HsingChi Wang and Dr. David Gebhard for their moral support, time

and help throughout various stages of my doctorate.

 

TABLE OF CONTENTS

1 Introduction 1
1.1 History and Motivation ............................ 1
1.2 Overview of the results ............................ 2
2 Notation and Definitions 17
3 Matrix Weights 19
3.1 The AP metric and reducing Operators ................... 19
3.2 Matrix Ap condition ............................. 21
3.3 Two properties of operators ......................... 22
3.4 The class 8,, ................................. 23
3.5 d—doubling measures ............................. 26
3.6 6-layers of the 8,, class ............................ 27
3.7 Doubling measures .............................. 29
3.8 8,, implies the doubling property ...................... 32
3.9 An alternative characterization of the matrix A? class .......... 33
4 Boundedness of the cp-transform and Its Inverse on Matrix-Weighted
Besov Spaces 36
4.1 Boundedness of the inverse (p-transform .................. 36
4.2 Decompositions of an exponential type function .............. 43
4.3 Boundedness of the cp-transform ...................... 45
4.4 Connection with reducing operators ..................... 55

5 Calderén-Zygmund Operators on Matrix-Weighted Besov Spaces 58

5.1 Almost diagonal operators .......................... 58
5.2 Calderon—Zygmund operators ........................ 69
6 Application to Wavelets 81
7 Duality 84
7.1 General facts on duality ........................... 84
7.2 Duality of sequence Besov spaces ...................... 85
7.3 Equivalence of sequence and discrete averaging Besov spaces ....... 90
7.4 Properties of averaging LP spaces ...................... 95

vi

7.5 Convolution estimates ............................ 99

7.6 Duality of continuous Besov spaces ..................... 104
7.7 Application of Duality ............................ 109
8 Inhomogeneous Besov Spaces 111
8.1 Norm equivalence ............................... 111
8.2 Almost diagonality and Calderon-Zygmund Operators ........... 116
8.3 Duality .................................... 117
9 Weighted ’I‘riebel—Lizorkin Spaces 121
9.1 Motivation . . .. ....... . ......................... 121
9.2 Equivalence of f:q(w) and f:q({wQ}) ................... 122
10 Open Questions 126
A Density and convergence 129
BIBLIOGRAPHY 134

vii

CHAPTER 1

Introduction

1.1 History and Motivation

Littlewood—Paley theory gives a unified perspective to the theory of function spaces.
Well-known spaces such as Lebesgue, Hardy, Sobolev, Lipschitz spaces, etc. are
special cases of either Besov spaces 3:4 (homogeneous), 83‘" (inhomogeneous) or
Triebel-Lizorkin spaces Fl?" (homogeneous), F5"? (inhomogeneous) (e.g., see [T]).
These spaces are closely related to their discrete analogues: the sequence Besov spaces
63", 3" and sequence Triebel-Lizorkin spaces f5”, fg‘q ([FJ 1], [FJW]). Among other
things, Littlewood-Paley theory provides alternate methods for studying singular in-
tegrals. The Hilbert transform, the classical example of a singular integral operator,
led to the extensive modern theory of Calderén—Zygmund operators, mostly studied
on the Lebesgue LP spaces.

Motivated by the fundamental result of M. Riesz in the 19208 that the Hilbert

transform preserves D” for 1 < p < oo, Hunt, Muckenhoupt and Wheeden showed

that the famous AP condition on a weight w is the necessary and sufficient con-
dition for the Hilbert transform to be bounded on If’(w) (1973, [HMW]). More
recent developments deal with matrix-weighted spaces where scalar methods simply
could not be applied. In 1996 Treil and Volberg obtained the analogue of the Hunt-
Muckenhoupt-Wheeden condition for the matrix case when p = 2 ([TV1]). Soon
afterwards, Nazarov and Treil introduced in [NT] a new “Bellman function” method
to extend the theory to 1 < p < 00. In 1997 Volberg presented a different solution to
the matrix weighted LP boundedness of the Hilbert transform via techniques related
to classical Littlewood-Paley theory ([V]).

The purpose of this dissertation is to extend some aspects of Littlewood-Paley
function space theory, previously obtained with no weights and partially for scalar

weights, to the matrix weight setting.

1.2 Overview of the results

We define a new generalized function space: the vector-valued homogeneous Besov
space B§Q(W) with matrix weight W. Let M be the cone of nonnegative definite
operators on a Hilbert space H of dimension m (consider ’H = C" or Rm ), i.e., for
M E M we have (Mx,x);¢ Z 0 for all x E ’H. By definition, a matrix weight W is
an a.e. invertible map W : R" —> M. For a measurable g = (g1, ...,gm)T : R" —-> H,
let ||§||Lp(w) = (A2" HWI/P(t)g'(t)||’;{ dt)l/p. If the previous norm is finite, then
§ 6 LP(W). We say that a function (,0 E SUB") belongs to the class A of admissible

kernels if supp (,5 Q {5 6 IR": %S |§| g 2} and |<p(€)| 2 c > 0 if ES |§| S g. Set

90,,(x) = 2""tp(2”x) for V E Z.

Definition 1.1 (Matrix-weighted Besov space ng(W)) For a 6 IR, 1 S p <
oo, 0 < q S 00, (p E A and W a matrix weight, the Besov space 3:9(W) is the
collection of all vector-valued distributions 1;: (f1, ..., fm)T with f, E S'/’P(R"),1 _<_

i S m (the space of tempered distributions modulo polynomials ) such that

 

 

llf HEW) = “(Talley . f Hamil), = ||{HW”P- (a, * f )Hip}, ,

< oo,
3
where cpu * f: (4,9,, * f1, ...,(pu * fm)T.

The case p = 00 is not of interest to us, since B;"(W) = BS: because of the
fact that L°°(W) = L°°. Since (,0 is directly involved in the definition of B:q(W),
there seems to be a dependence on the choice of 4p: ng(W) = B§q(W, (p). Under
appropriate conditions on W, Theorem 1.8 below shows that this is not the case.
The space B§q(W) is complete, as is discussed at the end of Section 4.4.

We also introduce the corresponding weighted sequence (discrete) Besov space

b:q(W):

Definition 1.2 (Matrix-weighted sequence Besov space b§q(W)) For 01 6

IR, 1 _<_ p < oo, 0 < q S 00 and W a matrix weight, the space b;q(W) consists of

T
all vector-valued sequences .32 {§Q}Q, where §Q = (58), ...,s(m)) , enumerated by

the dyadic cubes Q contained in IR" , such that

.. m -12
llisQlQllbng): 2 2 WI 2SQXQ

1(Q):2-u L”(W) u (q

Z lQI—% (llWl/p(t)§Qlln) XQ(t) < oo,

l(Q)=2—V Lp(dt) u [a
q

where IQI is the Lebesgue measure of Q and l(Q) is the side length of Q.

For I/ E Z and k E Z", let Quk be the dyadic cube {(x1,...,x,,) E R" : k,- g
2"x,- < k,- +1, i = 1,...,n} and xQ = 2“”k is the lower left corner of Quk. Sct
waft) = IQI‘1/2so(2”x — k) = nor/we — 170) for Q = at. For each f with

f,- E S'(1R") we define the cp-transform 5,), as the map taking 1? to the vector-valued

sequence Sid") = {(f, <PQ>}Q = {((fme),---,(fm,<PQ>)T}Q for Q dyadic. We
call §Q(f) 2: <15, 90(2) the (ti-transform coefi'icients of f.

The next question is motivated by the following results:
(i) Frazier and Jawerth ( [FJ 1], 1985) showed that, in the unweighted scalar case,

llfHng % |l{SQ(f)}QHigq,

where {sQ( f )}Q are the sip-transform coefficients. A similar equivalence holds if
{sQ(f)}Q are the wavelet coefficients {( f, wQ)}Q of f with wQ being smooth,

say, Meyer’s wavelets (see [M2]).

(ii) Nazarov, Treil and Volberg ([NT], 1996, [V], 1997) obtained

W, e ll{<f7he>}

where {hQ}Q is the Haar system and f§2(W) is the coefficient (sequence Triebel-

 

 

f

 

 

 

 

T W A 1.1
f32(W) 1 E P) ( )

Lizorkin) space for D” (W) A particular case of (1.1), when m : 1 and w is a

scalar weight, is

llfllsguw) = ||f ||L2(w) % llf<fathlllj32(w) = ll{<fthl}llbg2(w)a

4

where the first equality and the second equivalence hold if w 6 A2.

For our purposes we will use a condition on W that is equivalent to the matrix

AP condition of [NT] (for the proof, refer to Section 3.2):

Lemma 1.3 Let W be a matrix weight, 1 < p < 00, and let p’ be the conjugate of

p (1/p+1/p’ =1). Then

1/ —1/ P’ dt p/p’ d5” n
“W p(x)W ”(t)” — — S cm, for every ball B Q R
B B IBI IBI

(1.2)
if and only if W 6 AP.

In (1.2), llWl/p(x)W‘1/P(t)|| refers to the matrix (operator) norm.

The advantage of condition (1.2) is that it allows us to understand the AP condition
in terms of matrices, avoiding metrics p, p" and their averagings as well as reducing
operators (for definitions and details refer to Section 3.2).

Our first result is the norm equivalence between the continuous matrix-weighted
Besov space B§q(W) and the discrete matrix-weighted Besov space b;q(W) under

the AP condition:

Theorem 1.4 LetaER,0<qSoo,1<p<oo andWEAp. Then

{re (0h

In some cases, the AP requirement on W can be relaxed. Recall that a scalar

-o

(1.3)

 

 

 

 

 

 

 

'0“: N . '
B? (W) l bgq(W)

measure a is called doubling if there exists c > 0 such that for any 6 > 0 and any

z E R",
#(B2a(3)) S C#(Ba(z))e (1.4)
where 86(2) = {x E R": [z — x] < 6}.

Definition 1.5 (Doubling matrix) A matrix weight W is called a doubling matrix
(of order p, 1 S p < 00), if there exists a constant c 2 CW, such that for any y E H,

any6>0 andanyzER",

/ llWl/P(t)ylli.dt s c / llWl/"(t)yll€idt, (1.5)
326(2 36(3)

i.e., the scalar measure wy(t) = ||W1/”(t) y“; is uniformly doubling and not identi-
cally zero (a.e.). If c = 25 is the smallest constant for which (1.5) holds, then B is

called the doubling exponent of W.

It is known that if W 6 AP, then wy is a scalar Ap weight for any y E ’H and the
Ap constant is independent of y (for example, see [V]). This, in turn, implies that
wy is a scalar doubling measure (e.g., see [St2]) and the doubling constant is also
independent of y. Using decomposition techniques, we prove the equivalence (1.3)
under the doubling assumption on W with the restriction that p is large, and with

no restriction on p in the case when W is a diagonal matrix:

Theorem 1.6 Let a E R, 0 < q S 00, 1 S p < 00, and let W be a doubling matrix

of order p with doubling exponent 6. Suppose p > 6. Then the norm equivalence

(1.3) holds. If W is diagonal, then (1.3) holds for all 1 S p < oo.

The case of a scalar weight is a particular case of the diagonal matrix weight case,
and thus, the equivalence (1.3) holds just under the doubling condition. This fact is
essentially known (see [FJ2] for the case of F5”); it is proved here for purposes of

comparison and generalization to the diagonal matrix case.

Remark 1.7 One of the directions of the norm equivalence uses only the doubling
property of W with no restrictions ( see Corollary 4 .6), but the other direction requires
the stated assumptions on W (see Theorem 4.15). Furthermore, the first direction is

obtained from a more general norm estimate involving families of “smooth molecules”

(see Theorem 4.2).

Summarizing Theorems 1.4 and 1.6, the norm equivalence (1.3) holds under any

of the following conditions:
(A1) WEAP with1<p<oo,

(A2) W is a doubling matrix of order p, 1 S p < 00, with p > B, where B is the

doubling exponent of W,
(A3) W is a diagonal doubling matrix of order p with 1 S p < 00.

Now we will state the independence of the space B§q(W, go) from «,0:

Theorem 1.8 Let f6 B§q(W,<p(1)), 99(1) 6 A, a 6 IR, 0 < q S 00, 1S p < 00,

and suppose any of (AU-(A3) hold. Then for any rpm 6 A,

~
~

B$q(W.sp(”)

B$q(W,90(2))

 

 

 

 

 

 

 

 

If we use the language of reducing operators (see [V] or Section 3.2), we extend the
norm equivalence (1.3) to a different sequence space, namely b:q({AQ}). For each
dyadic cube Q, consider a reducing operator AQ corresponding to the LP average

over Q of the norm “WI/P - Hy, i.e.,

1 1/P
HAQUHH e (— / IIWl/PUWII’iidt)
IQI Q
for all vector-valued sequences fl. Note that the assumption that W is a.e. invertible

guarantees that each AQ is invertible. Define the sequence space b:q({AQ}) for

a 6 IR, 1 S p < oo, 0 < q S 00 as the space containing all vector-valued sequences

{Sb }Q With

”resonates”: 2'“ Z IQWHAQeQIIme <00.

z(Q)=2-v mm, V ,,

Theorem 1.9 Let a 6 IR, 0 < q S 00, 1 S p < 00. Suppose W satisfies any of

(AU—(A3). Then

-o

(1.6)

 

 

 

 

I
~

 

{812 (0},

In Chapter 5 we study operators on B§q(W) by considering corresponding opera-

 

 

Bi'q‘W) bzqqun '
tors on b$q(W). In [FJW] it was shown that almost diagonal operators are bounded
on bgq and, thus, on Bg". In Section 5.1 we define a class of almost diagonal matrices
adng) for the weighted case and show the boundedness of these matrices on bgq(W)

if W is a doubling matrix weight:

Theorem 1.10 Leta 6 IR, 0 < q S 00, 1 S p < 00, and let W be a doubling matrix
of order p with doubling exponent B . Consider A E adzqw). Then A : b3“? (W) ———>

b$q(W) is bounded.

We say that a continuous linear operator T : S —> 8’ is almost diagonal, T E
ADSqw) , if for some pair of mutually admissible kernels (90, w) (see (2.1), Section
2) the matrix ((Twp,<pQ)QP)Q,pdyadic 6 adng) (see Section 5.1). Combining the
boundedness of an almost diagonal matrix with the norm equivalence, we obtain the

boundedness of an almost diagonal operator on B;q(W) under any of (A1)-(A3):

Corollary 1.11 Let T E AD:"(,B), a 6 IR, 0 < q < oo,1Sp< 00. Then T is a

bounded operator on B§q(W) if W satisfies any of (AU-(A3).

In Section 5.2 we consider classical convolution and generalized non-convolution
Calderén—Zygmund operators (CZOs). The following criterion is used: if an operator
T maps “smooth atoms” into “smooth molecules” (see Sections 4.1 and 5.2 for defini-
tions), then T is almost diagonal (Lemma 5.13) and, therefore, bounded on B§q(W).
To show this property for a C20, the definition of a “smooth molecule” is modified
in order to compensate for the growth of the weight W (note the dependence of the
decay rate of the molecule on the doubling exponent fl ), and, thus, more smoothness
of a C20 kernel is required (see Theorems 5.25 and 5.19). In particular, for example,
we obtain the boundedness of the Hilbert transform (when the underlying dimension
is n = 1) and the Riesz transforms (n 2 2) on 1.3;"? (W) under any of the conditions
(A1)-(A3).

In Chapter 6 we apply the previous results to Meyer’s wavelets and Daubechies’
DN wavelets with N sufficiently large, to show that, instead of the cp-transform

coefficients, one can use the wavelet coefficients for the norm equivalence:

Theorem 1.12 Let a 6 IR, 0 < q S 00, 1 S p < 00 and let W satisfy any of

{A1 )-(A3), then

 

 

 

 

{re (0}.

where [52, (f) }Q are the wavelet coefficients of f.

,

63"(W)

N
~

 

 

 

83"(W)

The next goal (Chapter 7) is to determine the duals of the Besov function spaces
ng(W) and the corresponding sequence spaces b:q(W) for a 6 IR, 0 < q < 00 and
1 < p < 00.

To understand what properties of W are needed to identify dual spaces, we will
heavily use the technique of reducing operators (for definitions refer to Section 3.2
or [V]). In fact, instead of dealing with matrix weights, we consider a sequence of
matrices enumerated by dyadic cubes and establish properties of Besov spaces with
such sequences of matrix weights. Then, given a matrix W, its reducing operators
constitute such a sequence.

Denote by D the collection of dyadic cubes in IR" and for each Q 6 ’1) let AQ be
a positive—definite (thus, self-adjoint) operator on H. Also denote by R81; (reducing
sequences) the collection of all sequences {AQ}QED of positive—definite operators on
H.

In Chapter 7 as a main tool and a useful object by itself, we define the space

B:q({Aq}) with a sequence of discrete weights {AQ}Q:

Definition 1.13 (Averaging matrix-weighted Besov space B§q({AQ})) For

ozElR,1SpSoo,0<qSoo,{AQ}QE’RSDandwEA,theBesovspace

10

B:q({AQ}) is the collection of all vector-valued distributions 1?: (f1, ...,fm)T with

E S’/’P(IR"), 1 S i S m such that

: 2ua Z HAQ ((,0,, * f )llHXQ < 00.

l"(=Q)2” LP

 

 

 

 

33q({/‘Q})
V (<2

This space is well-defined (i.e., independent of (p E A), see Corollary 7.10, if {AQ}Q

is a doubling matrix sequence defined as follows.

Definition 1.14 (Doubling sequence) We say {AQ}Q E RS1; is a (dyadic) dou-
bling sequence (of order p, 1 S p < 00), if there exists ,6 2 n and c 2 1 such that

for all P,Q dyadic

1,. C|_P|m max 15g)" disuse) ‘3
”AQA “ < IQI (1I1(P)l )(“rnexarpmonl ' (1'7)

Observe that if (1.7) holds for some p, then it holds for 1 S q < p, since the right-hand

 

side is Z 1.

Our main result of this chapter identifies the dual space of ng(W). For W E AP
the result can be expressed in terms of matrix weights. However, even for W E AP but
satisfying (A2) or (A3), we are able to characterize [B§q(W)] * in terms of reducing

operators. Set i+§=1if1<p<ooandp’=ooifp=1; +;11—,=1if1<q<oo

l
q
and q’ = 00 if 0 < q S 1. It is important to emphasize our convention for the duality
pairing. In what follows, we say that a function space Y is a dual of a function space
X , X * x Y, in the sense that each y E Y defines an element ly of X " via the pairing
ly(x)= ==(fRn )dt and every element of X " is of the kind 11, for some
y e Y with “1,,” e Hyny. (For example, [LP(W)]* s LP’(W-P’/P), 1 < p < 00, with

the pairing(f szn (f(t) g(t)> dt; refer to Section 7. 2 for more details.)

11

Theorem 1.15 Let a E IR, 1 S p < oo, 0 < q < 00 and let {AQ}Q be reducing

operators of a matrix weight W.

If W E Ap,1 < p < 00, then [B;q(W)]* a: BQOQXW‘W”). (1.8)

If W satisfies any of {AU-(AS), then [339(W)]* z BQQQI({AE,1}).

(1.9)

(For the proof refer to Section 7.4.)

Next we identify the dual space of the sequence (discrete) Besov space bf,” (W)
Recall that the connection between b$q(W) and ng(W) is that f E ng(W) if
and only if the appropriate wavelet coefficient sequence of f belongs to b$q(W).

Analogously to ng({AQ}) we introduce b;q({AQ}).

Definition 1.16 (Averaging matrix-weighted discrete Besov space
bg‘q({AQ}).) For a E IR, 1 S p S 00, 0 < q S 00 and {AQ}Q E RS1), the

space b;q({AQ}) consists of all vector-valued sequences {§Q}QED such that

_. _l —.
Il{se}eui;«({.,,,= 2'“: Z IQI err/lemmas)
lle=2-" LP(dt) u ,q
= ] {AQSQ}Q b3“ < 00.

 

 

 

If {AQ}Q is a sequence of reducing operators for a matrix weight W, then the

norm equivalence
ism) MSW/1d) (1.10)

holds for any matrix weight W, a E IR, 1 S p < 00 and 0 < q S 00 by Lemma 4.18.

12

Theorem 1.17 Leta E IR, 1 S p < oo, 0 < q < 00 and let {AQ}Q be reducing

operators of a matrix weight W. Then

[13mm] 2 5;,” ({A51}). (1.11)
Moreover, if W E Ap, 1 < p < 00, then

[b;q(W)] z 1);,” (W'P/P). (1.12)

The chapter 7 is organized as follows. In Section 7.2 we discuss the discrete Besov
space b;q(W). We use a “one at a time reduction” approach meaning we reduce the

space bgq(W) in the following order:
I3;:"(Wl —+ 53q({AQ}) —+ l.fiqflRm) —> 53"(R1),

where the last two spaces are unweighted vector-valued and scalar-valued discrete
Besov spaces, and then identify the duals in the opposite order. A similar approach
is used for ng(W).

The fact that each AQ is constant on each dyadic cube Q allows us establish

[bzq<{Ae})]* e arm/12,1» (1.13)

for any {AQ}Q E R813, 0 E IR, 0 < q < 00, 1S p < 00. If AQ’s are generated by a
matrix weight W, then combining (1.10) and (1.13), we get (1.11) of Theorem 1.17.

In order to connect bgaq'flAC—QID with bgaq’flAgD % bgaq'(W“P'/P) the matrix
AP condition is needed, though only for one direction of the embedding; the other
direction is automatic. Thus, the following chain of the embeddings holds for bgq(W):

any W

[igqrwiy e [6:q<{AQ}>]’ e arm/1.31» fibrin/iii)

13

anyw ._aq. _,/
N bp, (W P P). (1.14)

This completes the proof of Theorem 1.17.

In Section 7.3 we prove the norm equivalence between 334({AQ}) and bgq({AQ})
for any doubling sequence {AQ}Q. Note that if AQ ’s are generated by a matrix weight
W, then all that is required from the weight is the doubling condition. Compare this
with (A1)-(A3) conditions for the norm equivalence between the original matrix-

weighted spaces.

Theorem 1.18 Let a E IR, 0 < q S 00, 1 S p < 00 and {AQ}Q be a doubling

sequence (of order p). Then

 

 

 

 

 

 

 

3:.“qu bitumen.

In Section 7.4 we establish the correspondence between the continuous Besov

spaces B:Q(W) and B§q({AQ}).

Lemma 1.19 Let a E IR, 0 < q S 00 and 1 S p < 00. If W satisfies any of

(A1 )-(A3) and {AQ}Q is a sequence of reducing operators generated by W, then
BTW) % B§q({AQ})-

For one direction of the above equivalence it suffices to have W doubling.
In Section 7.6 it is shown that if {AQ}Q is a doubling sequence of order p, 1 S

p < 00, then

[334({AQ})]* :2: arm/15,1}; (1.15)

14

Using the above duality and equivalence, we get the following chain:

[BMW 33

AP

[Brawn]: eBr‘I’uAsi) e B;“"’<{At}>

4 . I y
(es) Bf" (W‘P /P), (1.16)

where the equivalences (1) and (4) hold if W and W'pI/p, respectively, satisfy any
of (A1)—(A3). The third equivalence holds under the AP condition, however, the AP
condition is needed only for one direction of the embedding. This proves Theorem
1.15.

So far we have dealt only with homogeneous spaces. However, for a number of
applications it is necessary to consider the inhomogeneous distribution spaces (e.g.,
localized Hardy spaces H]:C = F32,0 < p < oo, in particular, H12“: 2 332, see [Go]).
In Chapter 8 we “transfer” the theory developed up until now to the inhomogeneous
Besov spaces. The main difference is that instead of considering all dyadic cubes,
we consider only the ones with side length l(Q) S 1, and the properties of func-
tions corresponding to l(Q) = 1 are slightly changed. Modifying the definitions of
the (p—transform and smooth molecules, we show that all the statements from the
homogeneous case are essentially the same for the inhomogeneous spaces.

In Chapter 9 we study another class of function spaces - scalar weighted Triebel-
Lizorkin spaces. As a starting point of this part we establish the norm equivalence
between the scalar weighted Triebel-Lizorkin space F:q(w) and the averaging scalar
weighted sequence Triebel-Lizorkin space f;q({wQ}) (see definitions below) if w E

A00 (see Chapter 3).

15

Definition 1.20 (scalar-weighted Triebel-Lizorkin space F:q(w)) For a E
IR, 0 < p < oo, 0 < q S 00, (0 E A and w a scalar weight, the Rebel-Lizorkin space
F:q(w) is the collection of all distributions f E S'/'P(IR") such that

1/q
llfllpgu...) = (Eerie. . f0") < ee,

”52 LP(w)

where the l9 -norm is replaced by the supremum on 1/ if q = 00.
This space is well-defined if w is a doubling measure (see [El 2])

Definition 1.21 (scalar weighted sequence Triebel-Lizorkin space f:q(w))
For a E IR, 0 < p < oo, 0 < q S 00 and w a scalar weight, the ’D‘iebel-Lizorkin
space f:q(w) is the collection of all sequences {3Q}QED such that
q 1/q
IIISQlQIIqu(-w) = (Z (IQ—753mm) ) < 00.
er qu)

where the lq-norm is again replaced by the supremum on 1/ if q = 00.

Definition 1.22 (averaging scalar weighted sequence Triebel-Lizorkin
space f:q({wQ}).) For a E IR, 0 < p < oo, 0 < q S 00 and {11%)}er a sequence
of non-negative numbers, the Triebel-Lizorkin space f:q({wQ}) is the collection of all
sequences {359}er such that

1/q
1 _2_1 1 q
lliSQlQIIququD = llwa/pSQlQIIfgq = (2: (WI " 2wag/pt‘5QXQ) ) < 00,

QED LP

where the lq -norm is again replaced by the supremum on u if q 2 00.

Appendix contains several proofs on convergence and density.

16

CHAPTER 2

Notation and Definitions

Let z E IR". Recall that B(z,6) = {x E IR” : |z ——x| < 6} E 85(2). If the center 2: of
the ball is not essential, we will write 3,; for simplicity. In further notation, < V >3
means the average of V over the set B: I B|1/Qv(t)t')dt Denote W (t )= W(2"’t)
for 1/ E Z.

For each admissible (o E A, there exists if) E A (see e.g. [FJW, p.54]) such that

Zm 2%) :1, if re 0. (2.1)

1162

A pair ((p, 1b) with (a, w E A and the property (2.1) will be referred to as a pair of
mutually admissible kernels.

Similarly to (059, define wQ(x) = [QI_1/21/J(2V£E — k) for Q = Quk. The inverse
(o-transform T.) is the map taking a sequence 8 = {sQ}Q to Tws = 2Q sQi/JQ. In the
vector case, T¢§= 2Q €621,120, where 522%: (s g)wQ,. .. ,st )wQ)T . The go-transform

decomposition (see [FJ2] for more details) states that for all f E S’/’P,
f = Z<f199Q>¢Q =3 Z Sol/Jo- (2.2)
Q Q

17

 

In other words, T2!» o 5,, is the identity on S'/’P. Observe that if (5(x) = (,0(——23)
(note that 95 e A). then So = (f. 8063) = loll/2w. . new).
In order to establish the connection between matrix weighted Besov spaces and

averaging Besov spaces in Chapter 7, we introduce an auxiliary LID-space:

Definition 2.1 (Averaging space LP({AQ},1/)) For 1/ E Z, 1 S p S 00 and
{AQ}Q E R89, the space Lp({AQ}, V) consists of all vector-valued locally integrable

functions f such that

 

 

 

 

 

Hf IILP({Aq},1/) = Z XQ(t)AQ (t) < oo.
t<Q>=2-v ”Qt,
NOte that l|f| ‘ : l{2”a (pl/*f }
33q({AQ}) LP({AQ},u) V ,q

 

 

 

To make notation short, define Q, = {Q E D : l(Q) = 2_"}.

18

CHAPTER 3

Matrix Weights

3.1 The Ap metric and reducing operators

Let t E IR". Consider the family of norms p, : ’H —+ IR+. Then the dual norm p" is
given by

. _Su l(cvty)|
pt (:13) — infill Pt(y) .

 

Following [V] (or [NT], [TV1]), we introduce the norms pug through the averagings

of the metrics pt over a ball B

pierce) = (,7;I [inertial/I).

Similarly, for the dual norm
1 , W
" x = —— * x p dt .
pm ) (,3, [Bi/it )1 )
Definition 3.1 (AP - metric) The metric p is an Ap-metric, 1 < p < 00, if

p;,,B S C (pp,3)* for every ball B Q IR". (3.1)

19

The condition (3.1) is equivalent to
pp,3 S C (p;,,3)* for every ball B (_2 IR",

which means that p" is an Apt-metric.
If p is a norm on ’H, then there exists a positive operator A, which is called a

reducing operator of p, such that
p(x) z ||Ax|| for all x E ’H.

For details we refer the reader to [V]. Let A B be a reducing operator for pp,3, and
Ag for p13,, 8. Then, in the language of the reducing operators, the condition (3.1) for

the A,D class is

 

 

“Ag/13H S C < 00 for every ball B Q IR". (3.2)
PROOF. Since p;1’3(x) % HAExII and (pp,B)*(x) = sup l($,y)l , (3.3) implies
y¢0 pp,B(y)
-1
[IA’gCIIH S C sup l(x,y)| = c sup M, where z = A3 y.
yeéo “AB 31” zeeo HZH

Since A131 is self-adjoint,
l(A’1 x, Z)| _
“/1;an S 6 8:}; fi— = c HAB1 xll.

With u = A131 :13, we obtain

llAfiABUH 3 out”, or “At/ten s c.

20

Note that the opposite inequality ||(AQAQ)-1|| S 0 holds always as a simple

consequence of Holder’s inequality: for any x, y E ”H we have

d 1/P Id l/p'
K1310] S (LIIWl/pftflllp'lél) (fQIIWII/“Uyllp I—Qil) % “14093” “1439”,

which implies HAQ x|| 2 c||(AZ§)—1x|l for any x E ’H and, thus, the above statement

follows.

3.2 Matrix AP condition

The particular case of norms p,, we will be interested from now on, is
pt(:r)=||W1/”(t)$ll, n e u. t e R".

Then the dual metric p: is given by

 

p203) = sun “5‘” y”

= |W_1/p t)x||.
#0 WW) I (

Definition 3.2 (Matrix AP weight) For 1 < p < 00, we say that a matrix weight

W is an AP matrix weight if there exists C < 00 such that for every ball B Q IR”
pi... s 0 (tier, (3.3)
where both averaging metrics are generated by W, i.e.,

1 ,, . ”9
new = (,—,,-I [B ”W P<t>nnpdt)

and

1 , , W
pita) = (,3, / HW‘ ”(t)dvll” dt) .

21

Remark 3.3 pr = 2, the condition A2 simplifies significantly:
H < W >2” < w-1 >1,” n g o for every ball B g R". (3.4)

PROOF.

2 _ 12 2 dt _ dt
[piste] — / IIW/ (t):v|| 131‘ f(t/(anneal

2 (< W >3 $.22) = n < W >1,” x||2.

This means that a reducing operator A B can be chosen explicitly as < W >2”.

Similarly, p;,’B(x) = [I < W‘1 >2” x|| and, thus, A# %< W—1 >13”. Therefore,

(3.4) follows from (3.2). I

Remark 3.4 If w is a scalar weight, the condition A,D is the celebrated Muckenhoupt

AP condition:

l/p l/p’
(/ w(t) dt) (/ w‘pl/p(t) dt) S c for every ball B Q IR”. (3.5)
B B

Denote wx(t) = [IWl/P(t)x||p and w;(t) = llW‘l/p(t)x||”'. Similarly, w(t) =
”WI/”(t)“p and w*(t) = [IW’l/p(t)||p'. Sometimes it is more convenient to work

with these families Of scalar-valued measures.

3.3 TWO properties of operators

Observe the following two useful facts. First, if P and Q are two selfadjoint operators

in a normed space, then

IIPQII = ”(PQYII = llQ‘P‘Il = IIQPII- (3-6)

22

Thus, the operators can be commuted as long as we deal with norms.

Second, we need the following lemma:

Lemma 3.5 (NORM LEMMA) If {e1,.. . ,em} is any orthonormal basis in a Hilbert

space H, then for any linear operator V : H —> ’H and r > 0,

m
WW ,5 Z ”Vet-Hit-
i=1

.m)

PROOF. wnh n.- = (nest. we get IIVII' -——- sup HVZn-elli.

llxllgl i=1

m m
s c. sup Berni/ell; s nZHVetns. s crmllVll’V I
li—l ‘

IIIIIS ;:1

3.4 The class 3,,

Definition 3.6 For 1 < p < 00 the class 8,, is the collection of all matrix weights
W so that for a given fixed 0 < r5 < 1 there exists a constant c 2 Cam," such that for

any 2: E IR” and any 1/ E Z the following inequality holds
. dt ”/1” dx
Wg/P(x)W;1/P(t) P —) — g c, m (3.7)
[36(2) (-/B,;(::) H H [Ba] I36] ’19,
where Wu(t) = W(2“’t).

This condition seems to be dependent on the choice of 6 , though it is not the fact.

PROOF. By changing variables we write (3.7) as

“21’
, dt P d.
/ / le/ptew-I/pmn" —”— g c,,,,,.
82_y6(2‘”z) 82-“,(2-112) le-val IB2—V6l

(3.8)

 

23

Let c > 0. Then there exists V() E Z such that 2“("°+1)6 S e < 2“”06. The following

three simple Observations will show that (3.7) is independent Of 6 .
1. B2—(u0+1)6(2) Q 86(2) Q B2—u06(Z),

2. IB2—"Odl Z 2n|Bz—(u0+l)6|,

1/ dt / dt / dt
3. —— ...————g ...— 32" .
2,. IB2-<vo+1)al B. lBel B the-toil

2"”06

 

Hence, the definition of the 8,, class is equivalent to the following one:

Definition 3.7 The class 13p, 1 < p < 00, is the collection of all matrix weights W
so that there exists a constant c =2 cw, such that for any 2 E IR" and any 6 > 0 the

inequality (1.2) holds, i.e.,

, dt ”/1" dx
wl/P(x)W-1/P(t) ” ) —— g ,. (3.9)
fan) (flue) H H IBeI Wei C!"

Remark 3.8 It is also convenient to write condition {3.9) in terms of metrics p and

 

*

p:

I

, Ni)
10? (31)]? dt div
sup —‘—— — — S c ,n, (3.10)
/B.(z) (L42) W60 lpx(y) '36] '36] p

t P/P’
10431)]? dt dd:
sup Scfi. 3.11
h.(.,(h.(.).eoln.(y) 113.1 lBel P ( ’

PROOF. We will show only (3.10), since (3.11) uses the same argument. The

07‘

 

 

 

left-hand side of (3.10) is equal to

/ (/ sup IIW‘1/P(t)yllp' tit)” dtt
are) B.(z)y¢o|lW‘l/P($)yllp' chl lBel'

24

 

Let u = W‘l/P(x) y, then the last expression is

f U [lW‘l/P(t)W1/P(x)ullp' dt )P/P' dis
sup , ——
B¢(z) B.(z) u#0 “qu I86] chI

I dt P/P’ d1:
: w-l/p t)W1/p($ p ) 1
v/B¢(z) (»/B¢(z) H ( )H IBCI IBCI

which is (3.9), by (3.6). I

 

 

Remark 3.9 Similarly, (3.7) can be written in terms of metrics p and p" .'

 

( ) pr P/p’
10* -. y dt d
f / sup —£3—t—)—— — _x_ S cam, for any u E Z,
36(2) 85(z)y?50 p(g-.,,(y) I36] I36]
(3.12)

07‘

I

r P/P
p(2‘Vx) (10] p dt d3?
sup —-———- — — S ca, ,n for any u E Z.
[135(2) (fawn) y¢0 [PO-vuly) I36] I36] p

(3.13)

PROOF. We will show only (3.12), since (3.13) uses the same argument. The

condition (3.12) is equal to

/ / 311p “Wu—l/pUh/IIPI dt p/p d5”
ah...) ah.) who IIWJI/p(x)yllp' lBtI IBtI'

Let u = WJI/ p (x) y, then the last expression is

_ , /'
l / supIIWV”(OWE/”(riallp dt ”” ch:
85(2) 85(2) "#0 HUMP, IBJI lB5l

, dt ”/1" dx
= IIWJ‘/”(t)Wt/p(x)llp —) —,
/B,(z) (fem) I36] '36]

which is (3.7), by (3.6). I

 

 

25

With the help of the Norm Lemma, we observe

dt
__ l/Pt P l/pt p—
<w>B-/IIW((t)ll,—-B,~/1r:1§gglllw (t)e.-ll,—BI

dt _ p p
~125m/Blpt(€tp)l I—B—I - 121133; Ipp.B(€t)l ~ ”PnBII -

The last equivalence can also be viewed in terms of reducing operators

<w >3: /||W1/”( ”“p|d_13| eliggx IIABe.IIP~IIABH”

Similarly, the dual metrics

/ IIW th-l/P t)|lp' gd—ll~ guise e’aIIP'eupihlIP ~IIAi. H”

3.5 6-doubling measures

First, recall that a scalar measure 11 is called doubling, if there exists c > 0 such that

for any (5 > 0 and any 2 E IR" the inequality (1.4) holds, i.e.,

#(B2a(2)) S cu(Ba(3))-
If the above inequality holds only for a specific 6 > 0, then we say u is 6-doubling.

Definition 3.10 Fix y E ”H. Then wy(t) = [IWI/p(t)y||p is a scalar valued 6-
doubling measure (of order p), 1 S p < 00, if there exists (I > 0 and a constant

c 2 C54,”, such that for any 2: E IR”
w.(B..(z)) s cwy(Bt(z))- (3.14)

Remark 3.11 Note that if wy(t) is 6-doubling of order p for any y E ”H, then

w(t) = ”WI/”(t)“? is also a scalar-valued 6-doubling measure of order p.

26

PROOF. Since (3.14) is true for any e,- - an orthonormal basis vector of ’H, we

have

IWl/p(t t)e,-|pdtSc IIWI/p(t)e,||pdt.
2/86 I M 2

£1,185

By the Norm Lemma, this inequality is equivalent to

/ IIW‘/P(t)ll”dtsc “WI/”(t)”Pdt.
326

3.5
I

The reverse of the previous remark is not always true.

3.6 6-layers of the 8,, class

Lemma 3.12 Fix (5 > 0. Suppose that the condition (3.7) or, equivalently, (3.12)
is true for 1/ = —1. Then w;(t) = ||W"1/”(t)y]|p' is a 6—doubling measure for any

yEH.

PROOF. By HOlder’s inequality
_ lBal / dt / My) dt
letl B“ ) Ithl 8“ )Pt(y)|325|
, dt W 1 dx W
S x t p“ y p ) (f )
(L2, BA H t( H I326] 19,, [10201)]? |B2ts|
_ [w;(Ba)]l/p’ (/ 1 dx )1”
I326] 325 [Pills/ll” IB2al

.. I/p' 1p' m 1p
: ”51:65] ( 3.)”:(“110'1310/ (l, IP;(13/)I” lien) / '

 

 

 

 

 

 

 

27

Raising to the pt” power both sides of the previous chain and specifying 2 as a center

of both balls Ba and 325, we get
,, I 1, , NP
2-... = (IBtI )" < [w,(Bt(z)) [W ./ f [p, (31)]? dt dx
I326] ‘- w;(ng(z)) 326(2) 325(z) 102(9) ' I326] I326]
* ' * I P/P'
= [w,(B.(z)> [W , / / [wit/1]" _di _gtn_
w;(Bg,5(z)) 85(2/2) 13,,(z/2) Piny) I36] I36]

where» [”1”
w;(326(2)) ,

 

 

S Cd,p,n l:

by the 8,, condition in terms of metrics (3.12) with V = —1.

Simplifying the last chain, we get

than») s (2"P’ - cm) than», (3.15)

t

y is a 6—doubling measure. I

i.e., w

Remark 3.13 Repeating the same argument, it can be shown that wy is also a 6-

doubling measure.

PROOF. The proof is similar to the previous one, thougn the splitting Of the initial

equality is slightly tricky. So, by HOlder’s inequality

'BI>”=< -‘”—)"=( — r 1”
2 (IBniI f3,,(z)XB"(”(t)le| lie..(.)XB:‘Z’(”n(t/)Ian
, dx 1 dt W
5 (l...(.,XBt<z>“”pry” its) ((3....) that (32.1)
= wy(B5(z))] 1 dt W
I I326] (L35(z)lpt(y)lp’ IBM)

: [W] (femalanHP '72:?) (426(2) [p,(:,)]pt ligature

28

 

 

 

 

 

 

t p/p’
= [wy(B¢S(Z))]/ / [p2e(y)]p _dt_ _di < C6 [wy(Bs(z))[
“Ii/(326(2)) 35(2/2) 36(2/2) PM?!) I36] I36] _ ’p'n nyB26(3)) ’
by the 8,, condition in terms of metrics (3.13) with V = —1.

Simplifying, we get
wy(326(zll _<_ (271p ' C6,p.n) lug/(86(3)), (3-16)

i.e., wy is a d-doubling measure. I

Generalizing the previous lemmas, we get

Corollary 3.14 Fix 6 > 0. Then w;,y(t) := ||WJ1/p(t)y|lp' and wu,y(t) 2:
IIWJ/p(t) yI|P are 5-doubling measures for any y E H, if the condition (3.7) or, equiv-

alently, (3.12) holds for V — 1.

PROOF. Let V(t) = Wu(t), then V_1(t) = Wu._1(t), and so (3.7) holds for V
with V = —1. Applying previous lemma (3.12) to u;(t) := IlV‘l/p(t)y||p', we get
u; is 6-doubling, or, u;(t) = [IV‘I/P(t)y||”' = ||WJ1/p(t)y||”' = w;,y(t) is 6-doubling.
Analogous proof applies to why. I

So each “layer” of the 8,, condition implies 6-doubling property of the scalar-

valued measures generated by the matrix weight W. Anticipating further results,

one can predict that the whole 8,, class will imply a standard doubling property.

3.7 Doubling measures

Let W be a doubling matrix of order p, i.e., (1.5) holds for any y E H, 6 > 0 and
z E IR". For p = 2 this simplifies to

29

W(t) dt 5 c W(t) dt (3.17)
826 36

for a given 6 , where the inequality is understood in the sense of selfadjoint operators.

Remark 3.15 Note that ||W1/p(t)]|p is independent of p. If wy(t) = [IWI/P(t)y||’,’,
is doubling of order p for any y E H, then w(t) = ”WI/”(t)“? is also a scalar-valued

doubling measure.

PROOF. Fix t E IR”. Then there exist a unitary matrix U and a diagonal matrix
A such that W(t) = UAU“, and so Wl/P(t) = UAl/p U“. Moreover, since the
norm of a positive diagonal matrix is the largest eigenvalue, say A0, “W” ”(t)” = A3,”
and, hence, “WI/”(t)”? = A0, regardless of what p is.

Now, since (1.5) is true with y = e,- - any orthonormal basis vector Of H, by the

Norm Lemma we get the second assertion:

Wl/P(t) pdt~ / Wl/p(t) t),e pdt
/826II t)|| 2 II II

B26

«:8 ||W1/P(t) (t),e|]”dt~c ||W1/P(t)||”dt.

3.5
I
The doubling property of w(t) = ||W1/P(t)|]” is not very helpful if one wants to
understand the nature of W; it only tells us how large the weight is, not how it is
distributed in different directions. Therefore, we use the definition of doubling matrix

n (1.5), which involves different directions of y E H.

30

Remark 3.16 In the scalar case, (1.5) gives the standard doubling measure:

/ w(t)lylpdt st: / w(t)lyl’”dt.
B26

3.5
and if y 75 0, then w(ng) S cw(B,5). In particular, there is no dependence on p in

the scalar situation.

Similar definitions for doubling weights (of order p’) can be analogously given for

the “dual” measure w;(t) = llW‘l/p(t)y||’”.

Remark 3.17 The doubling property (1.4) is equivalent to

as; C _|_F_| 6/"
#(Els (IE!) ’ “'18)

where F is a ball (or a cube) and E Q F is a sub-ball {or a sub-cube) (not any subset

of F; any subset would be equivalent to the A00 condition, see the end of Section 3.9,

also [St2]).

PROOF. Since E Q F, there exists j E N such that 23E a“ F, i.e., l(F) a: 2jl(E).

 

Since )1 is doubling, by (1.4) we have fig; S of z 382%. Noticing that [g =
l(F)

[LIE/Ill", we get (3.18). I

In further estimates, it is more convenient to use (3.18) instead of (1.4).

Observe that the doubling exponent Of the Lebesgue measure in IR” is )8 = n;
moreover, if u is any nonzero doubling measure in IR", then 8(p) Z n.

It is a trivial fact that if W is a doubling matrix weight (of order p), then a
reducing operator sequence {AQ}Q, generated by W, is a doubling sequence (Of

order p). (Recall the definition 1.14.)

31

3.8 8,, implies the doubling property

Corollary 3.18 Let W E 8,,. Then w;’y(t) = IIWJl/p(t)y]|”' and wu,y(t) =

[IWul/p(t)y||” are doubling measures for any y E H and any V E Z.

PROOF. First, by the Lemma (3.14) w;’y(t) and wu,y(t) are 6-doubling for any V.
Second, if W E 8,,, then WV satisfies (3.7) for all V E Z and a given 0 < 6 < 1. But
we know that the 8,, class is independent of the choice Of 6 , which means w‘ (t) and

U,y

wy,y(t) are 6-doubling measures for any 5. Therefore, the corollary follows trivially.

Lemma 3.19 Let x E H and W E A,,. Then v,,(t) :2 ||W1/P(x)W'1/P(t)||”' =
||W‘1/p(t)W1/p(x)||p' is a doubling measure, i.e., there exists a constant c such that

for any 6 > 0

/ IIWI/pIIElVV—VWUIIPIWSC ||W1/P(x)W"/P(t)||"'dt. (3-19)
326

Ba
PROOF. Applying the Norm Lemma to the Operator norm in the left-hand side,

we obtain

w(t) % Z IIW"1/p(t)W1/p($)etll”' = Z ||W_1/"(t)yt($)llp' = Z w;.(x)(t)1
i=1 i=1 i=1

where y,(x) = Wl/p(x)e,-. Then

m

02(326) x Z]

w;i(3)(t) dt S 20L w;.(x)(t) dt S CUZ(BJ)1
i=1

6

since w; is doubling (W‘pI/p E Apt). I

32

3.9 An alternative characterization of the matrix

A,, class

or WHAT IS THE 8,, CONDITION INDEED?

Now we are ready to reveal what the class 8,, really is, or, in other words, we give
a proof of the equivalence of condition (3.9), or (1.2), to the A,, condition.

PROOF OF LEMMA 1.3. By property (3.6) and the Norm Lemma

, dt ‘W dx

Wl/PW 1-/pt I" _) _

f (l H V” I8! lBl
, dt 1W dx

: w-l/p t Wl/p 1" _) _
l(l” () “3)” IBI IBI

z/B(A:llW‘l/P(t)W1/p($)ei”p' IiBtT)P/p ldgl
TEL (LI [p2(W1/P($ ) 6.)]p' ray/”Eda; 224/5) p,,13((W‘/p(~’ve) )IP [1%

Now, in terms of the reducing operators, the last expression is equivalent to

Z/ [[A:(W1/p(x)e) e, p [Liza /BHA#W1/p($
i=1 3

eff/B ”wt/mug...) 1,3,2” ~V:[pp.(t>nA 8.)]?
i=1

i=1
x Z HAB(A§ e.)
i=1

p
Therefore, (1.2) is equivalent to HA2;é ABH S c, i.e., the A,, condition. I

)de

IEI

 

 

 

 

 

 

 

:2 p
I zllABAf—S .

 

Thus, the 8,, class is nothing else but the matrix AP class. Therefore, we will

not use the notation 8,, anymore, though it was useful to understand what layers

33

 

this class consists of (as well as A,,) and that each layer implies a certain doubling

property.

Remark 3.20 Rephrasing Corollary 3.18, we obtain that A,, implies doubling.
Moreover, if W E A,,, then by (3.16) W is the doubling matrix weight of order p

log on")
P

and the doubling exponent 8 S np + log2 c,,,, = p (n + , where cm, is the
constant in (1.2).

Also W E A,, implies that the “dual” weight W‘PVP is a doubling matrix of order

1082 can

p’ with the doubling exponent 8’ S p’ (n +
P

) by using {3.15), where again

c,,,,, is the constant from (1.2).

Corollary 3.21 (SYMMETRY OF MATRIX A,, CONDITION) The following state-

ments are equivalent:
(2') W e A,,,-
(22') w-p’/p E A,,;

1/ _1/ p’ dt P/p’ d3: 11.
(iii) ”W p(x)W ”(t)” — —— S c for every ball B Q IR ,'
B B IBI lBl

. 1/ _1/ p div pI/p (it n
(iv) ”W p(x)W ”(t)“ — — S c for every ball B Q IR .
B B IBI lBl

PROOF. Recall that p E A,, if and only if p" E Apt. In terms of matrix weights,
W E A,, if and only if W’pI/P E A,,: (note that p;(x) = ||(W’p’/p)1/p’(t)x||). By
Lemma 1.3, the third statement is equivalent to W E A,,, whereas the fourth is

equivalent to WWI/P E A,,t. I

34

Observe that the scalar classes A,, are increasing in p, i.e., A,, Q A,, if p S q.
This Observation brings us to the definition Of the scalar A00 class.
Definition 3.22 (Scalar A00 class) Let w 2 0. Then A00 2 U A,,.
1Sp<oo

Equivalently, w E Aoo if there exists 6 > 0 such that given a cube ( or a ball) F

and any subset E Q F,

 

We.)

(See [St2/ for equivalence and other details.)

This property of scalar weights will be used in Chapter 9.

35

CHAPTER 4

Boundedness of the ge-transform
and Its Inverse on

Matrix-Weighted Besov Spaces

4.1 Boundedness of the inverse go-transform

Consider B§q(W) with parameters a E R, 0 < q S 00, 1 S p < 00 fixed. For
0 < 6 S 1, M > O and N E Z define (as in [FJ2]) mQ to be a smooth (6,M,N)-

molecule for Q E D if:

(M1 xlm (:1: dx=0, for 7 SN,
Q

ICE _ le ) — max(‘M,AI1—o)

l(Q)

,

(M2) lmq(~’r)| s IQI"‘/2 (1+

lx—inl

l(Q)

_1_L‘Ll_é
(M4) lDlmQW) - Dlmdyll S IQI 2 " "III? - ylé

(M3) IDimdxMsIQI-1/2-'i'/"(1+ ) if m _<_ [a],

36

 

l1“ Z—wol)—M.
x sup 1+ 1f7|=a.
IzISIz-yl ( l(Q) I [ ]

It is understood that (M1) is void if N < O; and (M3), (M4) are void if a < 0.
Also, [0] stands for the greatest integer S a; 7 is a multi-index 7 = (71,...,’7n)
with 7,- E N U {0}, 1 S i S n, and the standard notation is used.

We say {mQ}Q is a family of smooth molecules for BEWW) if each mQ is a

((5, M, N )-molecule with
(M.i) a— [a] < 6 S 1,

(M.ii) M > J, where J = g + 1‘— (ifpz 1, then n/p’ 2:0 and J=fl),

(M.iii) N = max([J — n — a], —1).

Remark 4.1 Note that, in contrast to the case in [17.12], there is a dependence of
the family of smooth molecules for B§q(W) on the weight W (more precisely, on the

doubling exponent fl).

Theorem 4.2 Let a 6 IR, 1 S p < oo, 0 < q S 00, and let W be a doubling matrix

weight of order p. Suppose {mQ}Q is a family of smooth molecules for B:q(W).

Then

S C llnglQllbgqm/y (4-1)
33"(W)

 

 

25‘}; mo
Q

The proof uses the following estimates for Q dyadic with I (Q) = 2"”, ,u E Z, and

 

 

90V, uEZ,with<,oE.A:
ifu>1x,then forsome o>J—a
Irv * moon 3 c IQI‘W 2““‘"’° (1 + 2‘19: — lob—M; (4.2)

37

if uSV,then for some r>a

lsou * mo($)l s c Ion-”2 WW (1 + 2m: - man—M. (4.3)

The proofs are entirely elementary, but quite tedious (see [FJ2, Appendix 8]).
Note that in the statement of Lemma B.1 in [FJ2], it should say j S It. For (4.2),
for N79 —1,app1yLemmaB.1withj=1/,kzu,L=N,R=M,S=M—a,
g=2“""/2cpu, hzmQ with l(Q)=2"‘, x1=xQ, J—n—oz—[J—n—oz] <6S 1.
Letting o = N + n + 0 > J — a, we obtain (4.2). For N = —1, apply Lemma B2
in [FJZ] with o = n > J — a to get (4.2). Now for (4.3), for a > 0, apply Lemma
B.1withk=u,j=u, L=[a], R=1W,6=9, S=[a]+n+5,x1 =0,
g(x) = mq(x+xQ), h = 2“"""/2 90V, and observe that pu*mQ(x) = 2""/2g* h(x —xQ)
to get (4.3) with r = 6 + [a] > a. For a < 0, Lemma B2 in [FJ2] gives (4.3) with

r=0>a.

Lemma 4.3 (SQUEEZE LEMMA) Fix a dyadic cube Q and let w : IR" ——> IR+ be a

scalar doubling measure with the doubling exponent fl . If L > H , then for r 2 I (Q),

/. w(x) (1+ I‘D—13‘il)-L dx 3 c5 [Rib—dig [Q w(x) dx. (4.4)

PROOF. Decompose IR" into the annuli Rm:

:- U{x: 2m"1rS|x—xQ|<2mr}U{x: lx—xQ|<r} =: U73".-
m=0

m=1

Then the left-hand side of (4.4) is bounded by

in? 1+2m 1) Lw(7?,m)+w(720). (4.5)

38

 

Using the doubling property of w, we get

|B($Q,2"‘7‘)|
lRol

 

fi/n
w(Rm) S w(B(xQ,2mr)) S c( ) w(Ro) = c2m‘3w(7€0).

Thus, (4.5) is bounded by

e Z 2mB_mL w(Ro) S c5 w(RO),

1n=0

since L > 5. Note that B(xQ,l(Q)) _C_ 3Q and so w(B(xQ,l(Q))) S Cfi w(Q). If

r > l(Q), then

”30'
SEQ)“

 

was s c (11% Q,)I)B/nw(B<xQ.z<Q))) s CH [,(iQflBMQ),

which is (4.4). I

Lemma 4.4 (SUMMATION LEMMA) Let u, 1/ E Z and y 6 JR". Then for M > n,
2: (.___.) W. (....
l(Q)=2“‘

PROOF. If u 2 u, i.e., 2‘” 2 2‘“, there are 2(p‘V)" dyadic cubes of size 2‘” in a
dyadic cube of size 2“". Fix l E Z" such that y E Q”. Then the left-hand side of
(4.6) is

Z (1 + 2"ly - ~’L‘(.2..,.|)"M

kEZ"

= Z Z (1+2”|y—$ka|)"M

1'62" k! kath/(l-H)

g Z(1+|i|)'M x 20H)" g 0,, WW)",
iEZ"

again since M > n. I

39

PROOF OF THEOREM 4.2. By definition,

21% mo
Q

 

 

 

 

 

 

 

[339047) I

Q LP V If;
= Z Z (W1/p§0l(¢u*mQ)
#62 1(0):?“ L. V ,3.

By Minkowski’s (or the triangle) inequality, the last expression is bounded by

Z Z (WI/p5Q)(%*mQ)

[JEZ l(Q):2—l‘

LP V If;
p 1/P
s 2 f 2 “WI/”(arlé‘oll|<¢u*mo)(x)l da:
#62 R" 1(0):?" 1
z; {mags/P} . (4.7)
#>V #9! y ,0,

Using estimates (4.2) and (4.3) with 61 = —(u — V)0', 02 = —(1/ — ,u)r and r1 = 2"”,

r2 = 2‘”, we bound each J,, i = 1,2:

P

—M
J" 5 0/ Z IIVl/1/"’(~’L‘)§'c2IIHIQI"1/2 29" (1+ w) dx-
R" t(Q)=2-u ‘

If p > 1, split M = M1 +M2, where [V11 > fl/p and N12 > n/p' (this is possible since
M > J). If p = 1, M = M1 > 5 (and n/p’ = 0 in further calculations). Then by

the discrete Holder inequality with wQ(x) = IlWl/P(x)§Q||’;i, we get

J,‘ S Cp/
Rn

—/2 9' '33—le _Mlp
Z wQ(I)IQ| P 2m 1+--—————

l(Q)=-2“‘ T‘
p/p’
~M2P'
(I: — (I:
x Z (1 + L—fl) dx.
1(0):?“ r”

40

By the Summation Lemma 4.4 (with 1/ = u in (4.6)), we have

stcpmrw Z Ion-W] wQ(x>(1+2~Ix—xQI>-M1de,

z<Q>=2-» 1““

since M2 > n/p’. Applying the Squeeze Lemma 4.3 with r = 2““ = l(Q) and

L = Mlp (and so L > B), we get

J2 S Cp.n,e WWW" Z lQl—pflwdQl-
z(Q)=2-#

By the Summation Lemma 4.4 (with u > 1/ in (4.6)), we have

J1 S Cp,n2(V-“)(a—n/p’)p Z lQl'm/ wQ($)(1+ 2"I113 — $Qll—M‘p d9?)
l(Q)=2’“ R”

again since M2 > n/p’ . Applying the Squeeze Lemma 4.3 again with r = 2’” >

2'“ = l(Q) and L = Mlp, we get

J1 S Cp,n,B 2(V'#)(0—"/P —fi/p)p Z lQl_p/2wQ(Q)-
1(0):?“

p

Observe that the last sum is equal to “Emma-u lQl_1/2§QXQH (W). Combining the
Lp

estimates for J1 and J2 (recall that J = 1% + g), we have

2m (2 Jll/p + 23/10) S Cp,nfi:2(V—#)a (2(V-#)(a-J)X{V_#<O}

p>u pSu pEZ

+2‘(""‘”X{u—u20}) X 2'” Z lQl-lflgQXQ - (4'8)
t(Q)=2-u LP(W)
Denote
az- = 2'“ (2i(U—J)X{i<0} + 247mm)
and

b. = 2*“ Z lQl"l/28'bxo
l(Q):2-—u Lp(W)

41

Then the right side of (4.8) is nothing else but c E aw,‘ by = c (a*b)(z/). Substituting
#62
this into (4.7), we get

ESQ mQ
Q

 

 

 

 

5 {Z Z 4.1/P} g cm, Ila * b||,.,. (4.9)
3°(W)

V 0:
lg

Observe that

Ila * bllw S llallzlllbllza for q .>_ 1 (4-10)
and

HG * bllzq S Hallqulbllza for q < 1 (4-11)

(to get the last inequality, apply the q-triangle inequality followed by ”a * bllp S

“annual“ ). For any 0 < q < 00, ”any, = Erma-”q + 22-47-000. Both sums
i<0 £20

converge, since 7 > a and o + a > J by (4.2) and (4.3). Hence, ||a|lzq S 0,, for any

q > O. (In fact, here we only need 0 < q S 1.) Combining all the estimates together

into (4.9), we obtain

Scllbllza=c 2.... Z lQl’l/25QXQ
B$q(w) l(Ql=2_" LP(W)

 

 

 

Zgomo
Q

 

ulq

= C llnglll63Q(W),

where c = cmmfi. I

Remark 4.5 Since 1b 6 A, observe the following properties of wQ :

1. 0 ¢ supp ibQ for any dyadic Q, and, therefore, fxle(x)dx = O for any
multi-index ’7;

42

 

1 111 l$—le —L—l7|
2. IDTle S c,,,L|Q|“§_ .. (1+ _l(—Q)_) for each L E N U {0} and 7 as

before.

Hence, {wQ}Q is a family of smooth molecules for B§q(W), and for f: 2Q §Q ibQ,

we obtain the boundedness of the inverse cp-transform Ty, :

Corollary 4.6 Let W be a doubling matrix of order p, and consider the sequence

§={§Q}Q€b:q(W). ThenforalllSp<oo, O<qSoo andaER,

ZgQ wQ
Q

S C ll{§Q}Qllth(wy (4-12)
83"(W)

llngllB$q(W) =

 

 

 

 

In particular, given f6 B§Q(W), consider 5': Swf. Then by (2.2)

ZgQ ¢Q
Q

83"(W)

 

 

s c ”sexism, = c ||5.f

 

f

 

 

 

63"(W) '

 

 

 

 

83W)
4.2 Decompositions of an exponential type func-
tion

Definition 4.7 For V E Z, let E, = {f: f,- E 8’ and supp f,- C_: {5 E R" : |§| S
2V+1},i = 1, ...,m}. Then we say that E, consists of vector functions of exponential

type 2"+1 .

Consider the following lemma on the decomposition of an exponential type function

(for the proof the reader is referred to [FJW, p.55]):

43

Lemma 4.8 Suppose g E S’(R"),h E 8(R") and suppg,supph g {|§| < 2V7r} for
some V E Z. Then
(9 * h)(x) = Z 2‘”"g(2_"k) h(x — 2%). (4.13)
keZ"
Note: if h({) = 1 on suppg, then 9 * h = g.
Let I‘ = {768:‘7zlon {€ER":|€|S2} andsupp&§{§ER":|€|<7r}}.
Define 7,,(x) :2 2""7(2"x) for V E Z. Since ”7,, = fi(2"§), suppfiu C {5 6 IR" :

|£| < 2V7r}.

Lemma 4.9 For V E Z let 9' 6 E, and fix :1: E Quk where k E Z". Then for any
31 E R" and '7 E 1"
@131) = Z 2‘”"§(2’”l + a?) My - (TV1 + $))- (4-14)
162"
PROOF. Denote g’x(y) = g(y + x). Trivially, g(y) = g”’(y — x). Note that

A
—o

(§$)’(§) = 6359(6), and so supp (gr) = supp 5'. Therefore, by (4.13) applied to gt:

so) = my — as) = Z 2—""g“=‘(2"”l) My — a: — 2W),

(62"

which is (4.14). I

Lemma 4.10 Let g E 8’ with suppg Q {Iél S 3} and 7 E P. Then for any

x,sElR" andjENU{O}

0,9(2: + s) = Z g(x + mono. — k). (4.15)

kEZ"

44

PROOF. Let gx(s) 2: g(x + 3). Observe that Slippy} E “El S 3}, since 9}“) =

eix£g(€) and suppg Q {Igl S 3}. Applying the decomposition from Lemma (4.8)

(withV=0) g(s :Wg’) (s—k k) to gx(s),we get
keZ"
= 2 9.007(8 — k) = Z g(s: + we — k).
keg" keZ"

Note the following two implications:
1° 92: =gx*7 :’ ngx :gx’ijV
2. (DJ-gm)“: g} - (Dnl‘ =¢ SUPp(ng..-)‘§ suppg‘x Osuppwn)“; {Iél S 3}

Therefore,

Dj9(-73+3)= ng$(8 2M: 017(193— :(ZQCC +19) )Dj7(3_k)

kEZ" kEZ"

Remark 4.11 Let f E 8’. Recall the dilations of (,0: <p,,(x) = 2""(p(2"x). Since
supp (,5 Q {E E IR": % S |§| S 2}, supp 95,, Q {E E R": 12"‘1 S [6| S 2"“} as well
as supp (ch * f)“Q {E 6 IR" : 2‘”1 S |§| S 2"“}. Observe that (90,, >1: f) E 5’ and

o (ch =1: f) E E,,. Thus, all previous lemmas apply to 4,9,, * f.

4.3 Boundedness of the cp-transform

Before we talk about the boundedness of the g()-transform, we develop two “maximal

operator” type inequalities:

45

 

Lemma 4.12 Let 1<p< 00, W6 A,, and ge E0. Then

2] )IW‘/P() 4vllde<cpnII4H:.m (4.16)

keZn QOk

Remark 4.13 Note that in terms of reducing operators, (4.16) is equivalent to

1/p
(P - (Z: “14001.9( klllp) S Cm II§IIL4(W)- (4-17)

11:62"

 

 

IRAQ.» W )lkezn

PROOF. Let '7 E P. Then for g’ E E0, we have (7 = 7 * g’, and the left-hand side

of (4.16) is
10

dx

 

 

 

 

52/4... W”P(4 ()f 4(4))(4—y)dy

<CM4§1/Qm. (/n Ill/VIE? Egg/(Sill iv)? dx,

for some M > n + 5p/ p’ , where B is the doubling exponent of W, since ’7 E S . Since

 

0m and m,- S ,-< m,+1,i = 1,...,n, on each 0m, the last sum is
3/

mEZ"
meQ, ”WWW )9 (yllldy p
623;,L0,( (1+lk- m|)M >44.

 

bounded by
mEZ"

Writing M = M / p + M / p’ and using the discrete Holder inequality (note that M >

n), we bound the last expression by

(f4... )IWP/P(4)4(y)n 4.4)”
c E: / (1+ We _ ml)“ dx. (4.18)

kEZn Q0}: mEZ”

 

Observe that

p

(AW llW1/”(:v)4‘(y)ll 44)? s (l... l)Wl/P(4)W-‘/P<y)n (WP/4444(4)!) 44)

p/p’
s (onm llW‘/”(4)W‘”p(y)ll”'dy) (wi llWl/p(y)4‘(y)ll”dy) ,

46

again by Holder’s inequality. By Lemma 3.19, v,(y) = llW’l/P(x)W‘1/P(y)||’" is a

doubling measure with the doubling exponent B:

v4(Q0m) S 714(B(m)|k — m| + fill S C(1+ lk - mllfideul-

Thus, (4.18) is bounded by

0 Z (1 + )4 — m))PP/P’-M U0“ U4... I)W‘/P(4)W~”P(4)))P’44)p/p' 44]

kJflEZ"

(4.19)

x (WNW/Pu) )4 (4 )))P44

By Lemma 1.3, the expression in the square brackets of (4.19) is bounded by a
constant independent of k. Since M > fip/p’ + n, the sum on k converges and,

therefore, (4.19) is estimated above by

42 f ))W“P(4) )4())Pd4=c ... ))W‘/P(4 )4 (4 )))P44=c))4)):.,w

m€Zn Q0711

Lemma 4.14 Let W be a doubling matrix of order p, 1 S p < 00, with doubling
exponent S such that p > B, and let 4 E E0. Then (4.16) holds. Furthermore, if W

is a diagonal matrix, then (4.16) holds for any 1 S p < oo.

PROOF. First, assume (4),- E S with supp(g )9 Q {|€| < 7r}, i = 1,...,m.
We want to show that for such 9', the sum on the left-hand side of (4.16) is finite.

Choosing r > B + n, we have

Z]... ))WVP(4) (4))(Pd4: ZW/Q )IWP/P(4)))Pd4.

kEZ" k6 Z“

47

Since w(x) = ||W1/P(x)||” is a scalar doubling measure, w(QOk) S c (1 + |k|)5w(Q00).

Hence,

1p p CWIQOO)
Z/MIHW/w(14)”d4SZ—(l+lkl),_,Sww(Qoo)<

kEZ" kEZ"

since r — [5’ > n.
Now we will prove (4.16) for g’ with (g),- E S and supp (4),“ Q {IE} S 3}, and then
generalize it to (g‘), E 5’. Let 0 < (5 < 1. Then 86(19) Q 3Q0k. Using the doubling

property of wk(x) = ||W1/P(4:)g'(k)||P, we “squeeze” each Q0), into 8,504):

)5/"
wk(QOk) S w4(3Q04) S C [llgfaQfilfill] w4(36(k)) S 09 5nfiw4(Ba(kll-

Hence, the left-hand side of (4.16) is bounded by

455 P 2 f4 Ilwl/P(4) (4)“de (4.20)

keZn 845(k)

To estimate the integral, we will use the trivial identity 4(4) 2 fix) + [4(4) — g’(x)]

for x 6 35(19). Apply the decomposition from Lemma 4.8 with 7 E F:

=Zg(m) 7(k— m)andg(x =Zg(m) m.)

mEZ" mEZ“

Using the Mean Value Theorem for [7(k -— m) — 7(x — m)] and the properties of 7 E 8

(note that Ix — kl < 6), we have

 

WP/P(4 m))IP
Wl/P P < wl/p p 6p II
II (W(lll 64” (WM ).M||+Cp H2424 (1+|k— ngW’
(4.21)
for some M > S + n. Integrating (4.21) over B,5(l:), we get
/ )IWP/P(4 )4 (4 )IIP44:c. IIW‘/P(4 )4 (4 )IIP44
315(k) 345(k)

48

 

 

fgm “WI/‘1 1‘)9 (771 )||”dl‘
1" 6( 4.22
+65 2 (1+|k- m|)M ( )
mEZ"
Apply the doubling property of wm(x )— - ||W1/’D ( ) g(m )ll” again:
(6+Ik—ml)" W"
4434(4)) s 'wm(B(m,lk-m| +6)) 5 6,, ] 4484(4))

 

= 65 g(1+lk- (”Wu MBA ))

Substituting this estimate into (4.22) and summing over k E Z", we have

EL IIW‘/”() (k)|lpd4<cpZ/m IIW‘/’”(:v) )(4IIPd4
1:62" 6

kEZ"

+44P 52/”, HM”) )4( )IIPd4(Z(1+I4—mI)"-M)

mEZ" kEZ"

where the last sum converges since M > fl + n. If p > 5, by choosing 0 < 6 < 1/2
such that 1 — cdp‘fi > 0, we subtract the last term from both sides (note that it is
finite because of our estimates above for g;- E S ), substitute it into (4.20) and get the

estimate of the left—hand side of (4.16) (note that ZkEZ" [Bo-(k) S flR" ...:)

Z] IIW‘/P(4) (4)IIP44<(1‘:5—;—3’)4 Z/B IIW‘/P(4) (4)IIP44

kEZ" QOk )kEZ" 60°)

3 4.4,,fgIIW1/P(4)4(4)IIP44 -—— 64,4,pll9llip(w)- (4.23)

Now let (9‘),- E S’,z' = 1,...,m. Since {7' E E0, it follows that (g),- E C°°, and g
and all its derivatives are slowly increasing. Pick a scalar-valued '7 E S such that
7(0) = 1 and supp”) g B(0,1). Then for O < e < 1, the function g"(x) := {(4)7(44)
has its components in 8. Observe that (g()A = (g)" * [7(ex)]A, with [1(a)]A (6) =

(1/6) ”7(5/6), and, therefore,

-o

SUPP (9")A C; supp(g)A + Supp(1/€)i(-/€) Q {fir |€| < 3}-

49

We can apply the result (4.23) to g“:

23/ IIWWIa) (k)ll”d:v<cl|g ‘II’ZW

kEZ"

or

Z/' “WWW WIek)IIPda:<c nllW””(a:)§(a:)||”|)(e:v)l"dx.

keZn Q0}:

Taking lim inf as e —> O of both sides and using Fatou’s Lemma on the left-hand side
(with a discrete measure for the sum) and the Dominated Convergence Theorem on

the right-hand side, we obtain

ZlimgafIaIaW / IlWl/P(r)§(k)llpdrr

k€Zn Q0}:

<c / IIWl/PIa) )IaIIanIaIaaMPda

Since 7(a) —-> 7(0), we obtain (4.16) for all g E E0.

c——>O

To get the second assertion of the Lemma, we consider the scalar case with w a

scalar doubling measure. Then (4.22) becomes

wIBaIk))IgIk)IP 5 cp / w(w)lg(x)l”dx (4.24)

860‘)

 

+cp6Pw(B<s(k)) Z (1 +|£|]l(€n:)l;l)M

or

p 6—1 211:1: 1:93: C” |g(m)|”

mEZ"

 

We want to estimate the last sum on m. Fix l E Z". Dividing everything by
(1+ Ik — l|)M and summing on k E Z", we get
lg(k)|” faaa) (NW
2 s a 2
(1+ lk - ll)“ (1+ lku-(ZIVIVMBAM)

kEZ" kEZ"

 

 

50

 

 

+C(SI’Z(1+|k1—|)M 2( (1+||km )ImIW

keZ" mEZ"

Note that in the last term

 

 

Z 1 < C ’,
kezn (1+lk_ll) (1+Ik—m|)M(1+|1_m|)M

since M > 72. Therefore,

19(kllp f36(k)w(x)|g(:1:)|”dx p [g(me
2 (1+ lk — lllM S Cp 2 (1+ Ik — l|)Mw(B(5(k)) + C6 2 (1+ ll“ ml)M'

kEZ” kEZ" InEZ"

 

 

 

Choose 0 < 6 <1/2 such that 1— c6” > 0. Then

l9( )1 Cp [136071) w(x)|g(:c)|1’dx
2 (1+ ll— m|)M S l— 66? mg” (1 + ll _ mI)M’w(B¢5(m))'

1nEZ"

 

 

Substituting this into (4.24) and summing on k E Z" (again using Zkezn [136(k) _<_

fRn ...), we obtain

 

Z w(Ba(k))lg(k)l"
keZ"
p f( 6 m )(IPda:
s c. IIgIIW) + WE MBA“) Z (1 f kaw m|)Mw(Ba(m ))'

Use the doubling property of w to shift 85(k) to Ba(m). Since (5 is fixed, w(Ba(k)) 3

C5," (1 + Ik — m|)5w(B5(m)), and thus, the last term is dominated by

u)(:1:)|g(:r)|p d2: x (Z (1 + lk -— m|)5_M) , (4.25)

kEZ"

where the sum on k converges, since M > B + n. Thus, (4.25) is estimated by

Cpanvfi HgHIZP(w) ' Hence,

Z/W k)_|pda: < cpna 2 III IBII Ik))IaIk)I” S Can)? 'lg'liww)

ICEZ" QOk kezn

51

 

Now if W is a diagonal matrix, then

and thus, applying the scalar case, we get

2] IIWVPI) I)IIPda~ZZ/ 10:1(13 a)Ig.II-a )Ide

kez" Q0), 1': 1 keZ"

m
_<_ 2 C “9:"le (w,,)~ NCpmflm llglliqw
i=1

Theorem 4.15 Let a E R, O < q S 00, 1 g p < 00, and let W satisfy any of

{AU-(A3). Then
ll{§Q}Qllb;,‘q(W) S Cllf—‘HBgfiwy (426)
where E'Q = Swf = (f, IpQ> for a given f.

PROOF. By definition,

IIIaPQ)QII.-,ga(m = Z )0)“ IIWl/P - 5all... m
IIQ)=2-P U,

V If;
=: HUI/Mtg. (4.27)

Fix V E Z. Then Q : Quk : [[[EJ’T

i=1
lQll/2(¢u * f)(2’"k) and

Jr: 2 IQI-P/Pf IIW‘/PIa)aQIIPda
Q

1(62):?”

ll

Wl/p(t * _. 2"”k pdt.
‘é/QMH( am f)( )II

52

—o

Let f;(:::) = (2‘V33). Then ((5,, * f)(2"’k) = (95 * fi)(k). We substitute this in the
last integral and note that the change of variables y = 2"t (with Wu(t) z: VV(2"’t))

will yield

J: = 2-... 2 / IIWJ/PIa)IIa .. fi)Ik)IIPdt. (4.28)

kezn
Observe that (Ifi * fl),- 6 S’, 2' = 1,...,m, and 95 a): f: E E0, since supp 93 Q {g E

R" : S |§| g 2}. Using either Lemma 4.12 or Lemma 4.14 with 5: 922* f; and WV

1
2
instead of W (both the A,, condition and the doubling condition are invariant with

respect to dilation), we obtain
J5 S 62”" ”WE/”UNIX? * f:)(t)||pdt-
IR”
Changing variables, we get

J?) S c R llWl/p(t)(95u * f)(t)ll" dt = C ”(,5, * f)||’£p(w)-

Combining the estimates of JV for all V into (4.27), we get
2 ll{§Q}Qllbg‘1(w): ”{Jv}ulll§‘

l({<f?Pa>}Q ,,.,W,
{New ...,}

where c = 6(1), )3, n).
To finish the proof of the theorem, we have to establish the equivalence between

 

 

g c = c (4.29)

 

 

 

 

 

 

 

 

 

 

BgP(w,§3) ’

’3‘

B;q(W, Ip) and B§q(W, 95). As we mentioned in Section 2, 95 E A, and so the pair

(€3,213) satisfies (2.1), since 95 2 IE and 1]}: Z/J. By (2.2), f: 2 <f-', @Q> l/SQ. Since
Q

53

{i/SQ}Q is a family of smooth molecules for B;q(W) (see Remark 4.5), by Theorem

4.2 we have

 

 

 

 

 

 

 

 

 

 

IIf’II ~ g c ||{(f‘, aaQ>} . (4.30)
B” (WM) Q 53"(W)
Applying (4.29) to the right-hand side of the last inequality, we bound it by
c _. z = c f , . (4.31)
ng(W,Ip) BS"(W,,p)

Finally, combining (4.29) with (4.30) and (4.31), we obtain

ll{<f7aa>}.

 

 

2 IIIa‘Q}QII,-,gq(w, s a HIP

 

 

. 'aq '
b;9<w) 8p (st)

Remark 4.16 The fact that go and If) were interchanged in the last step of the previ-
ous theorem can be generalized into Theorem 1.8 about the independence of the space

B§q(W) from the choice of go:

PROOF OF THEOREM 1.8. Let {99(1),1,D(1)} and {90(2), /(2)} be two different sets

of mutually admissible kernels. Decompose f in the second system:

" " 2 2 #2 2
a=2<aag>)av=:aaag>.
Q Q

Observe that 108 ) is a molecule for Q and, therefore, by Theorem 4.2,

_. g 2 ..
llfllsgq(w,,pI1)) S Cllfsh)}Qllbg‘1(W) S Cllf HB;"(W¢”)’

where the last inequality holds by Theorem 4.15. Interchanging gem with 90(2), we get

the norm equivalence between 339(W, 90(1)) and B§q(W, 90(2)). In other words, the

54

space B§q(W) is independent of the choice of go under any of the three assumptions

onW. I

Remark 4.17 Combining boundedness of the Ip-transform (Theorem 4.15) and that
of the inverse Ip-transform (Corollary 4.6), we get the norm equivalence claimed in

Theorems 1.4 and 1.6.

4.4 Connection with reducing operators

Now we connect the weighted sequence Besov space with its reducing operator equiv-
alent. Recall that for each matrix weight W, we can find a sequence of reducing

operators {AQ}Q such that for all a E H,

1 1 p 1/P
ppaIa) = (,3, / IIW /PIa) alumna) z IIAQa'a'IIaa. (4.32)

Lemma 4.18 Let a 6 IR, 0 < q S 00, 1 g p < 00, and let {AQ}Q be reducing

operators for W. Then

||{§Q}QHI'.3PIW) % llfgoblltgnmqn (4-33)

PROOF. Using (4.32), we get the equivalence

_. _l , _.
llfSQlQllbg‘PM): Z IQI PllWl/p-SQIIHXQ

IIQ)=2-P L,

a
qu

Z lQlJ'l [Pp,Q(§Q)lp IQI

l(Q)=2"’"

—
lI-l

a
qu

55

3

Z IQI-PIIAaa‘QIIt [moat

1(0):?”

22

a
qu

_l —. -'
= E IQI P “AQSQHHXQ = ||{PQ}Q”I'>$PIIAQ})'
l(Q)=2—” LP V 1°
(1

Finally, combining Theorems 1.4 and 1.6 with (4.33), we get Theorem 1.9.

Corollary 4.19 The space B:q(W) is complete when a 6 IR, 0 < q S 00, 1 S p <

00 and W satisfies any of (AU-{A3}.

PROOF. If {fl} is Cauchy in B§q(W), then {{E'Q (fl)} } is Cauchy

nEN Q nEN
in b:q({AQ}) by Theorem 4.15 and Lemma 4.18 (or just Theorem 1.9). This implies

that
p

 

 

 

 

l(Q)=2_u LP
W‘” l“ (”l“ (”)1 ,
2 Z )le Q .. Q fm ., ...—42.0)

—) 0 for each Q. Since

ll’H rum—>00

for each l/ E Z. Hence, ”AQ [so (fl) — 522 0%)]

the AQ ’s are invertible, {5Q (fl)} N is a vector-valued Cauchy sequence in ’H for

 

 

 

 

 

 

 

 

 

ne
each Q. Therefore, we can define s0 = lim §Q(f;). Set f = 2Q s’Q wQ. Observe
that
a; — a) = H: Ia. (f2) — a.) a.
Q B°q(W)
IaIa2)—~a)
Q bSPIIAQ»
gcliminf {“Q(:,)—§Q(f:n)} —) O,
m—mo Q b;q({Aq}) n—mo

 

 

 

 

by Corollary 4.6 and Lemma 4.18, the discrete version of Fatou’s Lemma and the fact

—9 _.

that { {sq (fl)} } is Cauchy in b§q({AQ}). Furthermore, f: (f— fn) + fn E

nEN

B§q(W). Thus, B:q(W) is complete. I

Recall (Chapter 3) the A,, condition in terms of reducing operators: HAQAgH S c
for any cube Q 6 IR”; in other words, HAlel S c||(A:§)—ly[| holds for any y E
H. Also, the inverse inequality ||(AQA$)'1|I S c (or, equivalently, ”(Agrl y|| S
c IIAQ y|| for any y E ’H) holds automatically. This implies the following imbeddings

of the sequence Besov spaces:

Corollary 4.20 For a 6 IR, 1 < p < oo, 0 < q S 00, and W a matrix weight with

corresponding reducing operators AQ and Ag,
1. bzPIIAQ})§6:PI{IA§)*1}) always,

2. bzPIIIA3)—1}) g bra/10)) if We A,,.

57

CHAPTER 5

Calderén—Zygmund Operators on

Matrix-Weighted Besov Spaces

5.1 Almost diagonal operators

Consider b:q(W) with parameters a,p,q fixed (a 6 IR, 1 S p < oo, 0 < q S 00) and
W a doubling matrix of order p with doubling exponent ,8 . Also, if p = 1, then the

convention is that 1/ p’ = 0.

Definition 5.1 A matrix A = (an)Q,pEp is almost diagonal, A E adzqw), if there

exist M > J = 5,- +5 and c> 0 such that for all Q,P,

'“QP' 5 C mi“ (ll%lm’ l%l l I” maLIlIb)IIIIIIP)))-M’ (5'1)

withal>a+§ andag>J—(a+%).

 

Remark 5.2 This definition difiers from the definition of almost diagonality in
[FJW], since both 02 and .M depend on the doubling exponent B.

58

To simplify notation for the matrix A above, we will only write (an) without

specifying indices Q, P.

Example 5.3 (AN ALMOST DIAGONAL MATRIX) Let to E A. If {mQ}Q is a family

of smooth molecules for B§q(W), then
(an) E adzqw), (5-2)

where aoa = Imam), by I42) and (4.3), (mare) = IQIP/PIIa. . mp)IaQ) a)

l(Q) = 2‘”-

Now we show that almost diagonal matrices are bounded on bgq(W), i.e., Theorem

1.10. First we need the following approximation lemma:

Lemma 5.4 Let P, Q be dyadic cubes and t E Q. Then

It — JIPI

maX(l(Q)a l(F))

:1: —:1:
1+ IQ Pl z 1+

maxIIIQ).IIP)) (n) (5'3)

 

 

PROOF. First suppose that l(Q) Z l(P). If P g 3Q, then 0 S (13;: — le S

2\/ii_l(Q) = cl(Q) and so

Iaa — apI IaQ — apI
lgl+————S1+c <::> 1+——z1.
l(Q) l(Q)
Also 0 S pr — t| S 2\/hl(Q) = cl(Q) and thus
V—$M V—xfl
lSl+———S1+C <=> 1+—-—-~1,
l(Q) l(Q)

and (5.3) follows.

59

If PflBQ = 0, then |atp — t| 2 l(Q) and |a:p — :rQI _>_ l(Q). Since IxQ - t) S GHQ),

by the triangle inequality we get both
Iap—aQI s Iap—tI+It-aQI s Iaa—aI+cIIQ) s lap—aI+cIap—aI 5 (1+0) Iap—tI
and

IJIP — tl S IJIP — $Q| + ICUQ — t| S ICEP — SEQI + cl(Q) S (1 + 6) IIL‘P — SEQI-

Therefore, |xp — ccQ| z Imp — t|.
Now assume that l(Q) < l(P). Choose P dyadic with l(P) = l(P) and Q g P.

If Pfl3P = (ll, then Imp — t) Z cl(Q) and Imp — 1‘le cl(Q). Hence,
|:I:p—:1:Q| S lxp—tl+|t—$Q| S |a:p—t|+cl(Q) S Imp—t|+c|a:p—t| S (1+c)|1:p—t|,
and

|$P — tl S |$P — le + live? ‘15) S W - $Q| +Cl(Q) S (1 +C)|$P ‘30],

and we again get Imp — xQ| a: |a:p — t|.
If P g 313, then 0 g IxQ — xp| g C1l(P) = c1l(P) and o g |t — xp| g c21(P) =

c21(P); thus,

_ t—
].Sl'l-MSI'I‘C] and 1S1+l——a‘7!;|31+62
l(P)
which means
IIIIQ —a:p| lt-CPPI
1 —— z 1 z 1 ——
+ W”) 1’ l(P)

6O

PROOF OF THEOREM 1.10. Let A = (an) with A E ad:q(,8). We want to show

that

{ZQQPgP} S Cams ll{§Q}Qllb$q(W)' (54)
Q 534

P (W)

|l{;a5p§p}

By definition,

Q 53"(W)
p 1/P
s E: IQWQ / (ZIanIIIWP/PIt)apII) at
l(Q)=2-v Q P
V If;
1/P
= 2W 2””2 Z JQ . (5.5)
l(Q)=2“’
V 1‘1
Substituting the estimate (5.1) for an in Jq, we get
p
JQ s cam / Z2-PPP Z IIWP/PIt)a*aIII1+2PIaQ—apI)‘M at
Q :20 ((P):2-(v+1)
)0
. _. V . -M
+cp,M / 22m 2 I|w1/P(t).apn(1+2< +P>|xQ—xp|) dt.
Q j<0 l(P)=2-(v+j)

Pick 6 > 0 sufficiently small such that (i) a1— 6 > a +n/2, (ii) a2 — e > J—a — n/2
and (iii) M > fl / p + (n + e)/p' . Apply the discrete Holder inequality twice, first

with 0:,- = e + (01,- — e) for the sum on j (note that 01,02 > O) and second with

 

M = n; + (IV! -— "7“?) for the sum on P (if p’ = 00, then the LP'-norm is replaced

by the supremum):

p/p’
JQ S Cp,M / (Z 27"”) [Z 2'j(°2“)”
Q

320 1'20

x Z IIWP/PIt)a'pIII1+2PIaQ—apl)"” at
l(P):2-Iu+a‘)

61

p/P’
+Cp,M/ (2: 23w) [2: WWI—Q”
Q

 

j<0 j<0
,.
X Z: IIWP/PIa)a'pII (1+ PPPPPIaQ — apI)”” dt
l(P):2"IV+J) _
‘ P/p’
S Cp,M,e Z 2—j(a2—c)p 2 (1+ 2V|£CQ — (CPD—The
j>0 l(P)=2—(v+j) _
x [[0 ||W1/p(t(t).§'p||p (1+ 2"le — atpl) M'EPPPP dt
(:(P) 2- (II-H)
p/IP’
+Cp,M,£ Z 21Iai—clp Z (1 + 2(V+j)l$Q _ $130.."—C
j<0 l(P)-_-2—(v+j)
(M
x Z [5 ||W1/p(t(t)8p||p (1+ 2 P+PP IxQ — $10))“ 73*)? dt.
up): 2- Ma)

Use the Summation Lemma 4.4 to estimate the square brackets and denote wp(t) =

||W1/P(t)§p||”. By Lemma 5.4, pg; can be replaced by any t E Q, and so we get

JQ S Cp,M Z: 2-j(02—e)p+jnP/p’

 

J20
5: / PPPI WWI
X ’LUp(t dt
[(P): 2- (PH-j) l(Q)
-(M-m}5)p
' -6 t-x P
W2 P 2 twat) I '71:“)

J'<0 l(P) =-2 (v+2')

Summing on Q and applying the Squeeze Lemma 4.3 (recall M > E / p + (n + e) / p’ ),

we get

:2 JQ < cmeZ— JI 02- -)cp+jnp/p'
lIQ)=

j>0

X Z Z leIP )I-1+2"|t apI)‘ MPP‘PPdt

l(P): 2 (”P”) 1(Q) 2 "

+cmeTI‘“ P)? E Z /wp(t)( )(1+2"+P|t— xpl)’(M‘"TP+’£"’ dt

j<0 l(P): 2- <P+PPI(Q)= 2 "

62

S 6pm 2 (g-J'Iaz—c)p+jnp/p’+jfix{j20} + 21(a1—e)px{j<0}) Z W(p).
jEZ ((p)=2—(v+j)

Observe that 2""9/22 [PI ”/22 an/z for l(P) = 2‘0“”), and

p

Z |P|“”/2wp(P)= Z lPl‘l/25pxP

l(P)=2'("+1) l(P)=2““’+” LP(W)

Then, using 1 < p < 00 to take the power l/p inside the sum on j, we get
1/?

21/0: 2(In/2 2 Jo S 6 Z [Q—ja2-J’n/2 (2-j(az—e)p+jnp/p’+jBX{jZO}
1(0):?" 1'62

‘ a ——e 1/ u ' a — -‘
+2“ 1 )pX{j<0}) P] X 2( +1) 2: lpl 1/25PXP
((P)=2—(v+2’) LP(W)

z: c Za_,~ x by“ = C(a * b)(u).
jez
Use (4.10) and (4.11) to estimate the norm of the convolution Ila * szq. Then for
q S 1,

HaHqu— Z2.“ a+n/2+( 02— e) J)q _+_ :2j(a+n/2— (01—6))q < an
j<0 j>0

since 01 — 6 > a + 71/2 and 02 — 6 > J — (oz + n/2). Using the ||a||p estimate for

q 2 1 and the Ilallzq estimate for q < 1, and substituting into (5.5), we obtain

{zaqup} S C “bllzq : C 2’“! Z IPI—l/zngp
P Q baq

p (WI) l(P)-42"“ LP(W) I‘qu

= C ||I§P}P|l5g°(wp
where c = cmmfi. I

Now we will show that the class of almost diagonal matrices is closed under com-

position. For 6 > 0,6 > O, J = 57 + g and P,Q E D, denote

“’Q”(6’ 6) z [gig] min (BI—g] ’ [%l W) I1 + mafIfiéfliIlPDyj—J'

63

Mln

 

Theorem 5.5 Let A, B E ad;q(,8). Then A o B E adg‘qw).

We need the following lemma, which is a modification of [F J2, Theorem D2]

adjusted to the weighted ad condition:

Lemma 5.6 Let 6,71,72 > O, 71 76 72, and 26 < 71 + 72. Then there exists a

constant c = Cn.5,’71.’72. J such that

Z wQR(6171) 1031905172) S C wQP(61min(71172))' (56)
R

PROOF. Without loss of generality, we may assume that oz : -n/ 2, since the

]a+n/2

terms [l(R)]°‘+"/2 cancel in the sum of (5.6), leaving [591

,(P) for the right-hand side

of the inequality. Denote '7 = min('71,72). With l(Q) = 2‘q,l(P) = 2’P,l(R) = 2",

first assume I (P) S I (Q) Then the sum in (5.6) can be split into the following terms:

Z + Z + Z =I+II+III.

lIR)<1(P)SIIQ) l(P)SIIR)SIIQ) l(P)S1IQ)<1IR)

Then

I = ; [%]71/2+J (1+ [$3(’Q;I3Rl)_J—6 [%]12/2(1+ l$:(;2;lipl)—J_6

00 ”n+7;
= [IIQH—(Vl/HJ)[KPH—W2 Z T“ 1’ +J)9P.Q.J+6,rI$P)

r=p+1
—(71/2+J) -72/2 n IxQ — $1" -.,—6 00 —r(1%2. J_n)
some» [l(P)] + (1+—1(Q) ) Z 2 + ,

r=p+l

by [F J2, Lemma D.1]. Since J > n, the geometric progression sum is bounded by
c2-P<3%u+J-PP> = c[l(P)](n‘:lz+J“"). Thus,

[—l(——>[—J(—)

64

substituting 71 with 7, since l(P) _<_ l(Q).

Similarly, using [FJ2, Lemma D.1], we have

2H (111—WW H <11—-—'w:,::~)—’-6

R

WWW ”
[mm/2+1

—-J—6 ‘72 2+ _
«(Him—LP» W H/ J[1(P)]”2—u

71‘72)

S 2T1 2 9P,Q,J+6,r(113p)

l(C2) [l(Q )J’W'”
71/2+J :1: —:cp —J—6
-[—l <1+-—-————-' '> 1

and '71 can be replaced by 7, since l(P) g l(Q).

The estimate of I I I is also similar:

q—l
, 7 +7
111 s [l(Q)l”‘/2[1(P)l”/2+J Z 2W +’)gp,Q,J+a,r(xp)

q—l 7+7 '37 —$ I —J—6
s [l(Q)l‘“/2[1(P)l”/2+J Z 2“?” (1+ "32727) '

r=—oo

Observe that

 

< ”T—W as film: W

:l’E—ii 3W ”2sz ”)1”-

~J-6 9—1
[[1 < l(Q )7—1/2 J- 6 [p 72/2+J 1 M 2r(11;—"1+J)2r(-J—5)
cl(Q )] [l( )J + ,(Q) 2 ,

Then

r=-oo

(11:2... _5)

Where the last sum converges since L213- > (5 and is bounded by 02" Sim-

plifying, we get

fl 72/2+J( IxQ'—$P|)—J—6
IIISC[Q] 1+___I(Q) .

65

Combining I, II and III, we get the right-hand side estimate of (5.6), if l(P) S l(Q).

The case I (P) > I (Q) follows by exact repetition of the steps above. I

PROOF OF THEOREM 5.5. Since A = (an),B = (pr) E ad:q(,8), for each
i = A,B there exist 0 < 6,1 < min(al—(a+n/2),ag—J+a+n/2) and O < 6 < Ill—J

such that lanl 3 chp(6,eA) and lePI g chp(6, 63). Without loss of generality,

€A+63

2 . Then

 

we may assume 6A < 63 and 6 <

l(AB)QPl S lZaQR bRPl S C ZwQR(6)€A)wRP(6?€B) S chP(6)€A)a
R R

by Lemma 5.6, which means that A o B 6 adng). I

Definition 5.7 Let T be a continuous linear operator from S to 8’ . We say that
T is an almost diagonal operator for B§q(W), and write T E AD:"(,B) , if for some

pair of mutually admissible kernels (9041)), the matrix (an) 6 adng) , where an =

(71wa (pQ) '

Remark 5.8 The definition of T E Angw) is independent of the choice of the pair

(«a 1101

PROOF. Define 80 = {f E S : 0 $4 supp f}. Observe that 11) E A implies

N
¢7¢u,¢q E 50 f01‘ V E Z and Q dyadic. Moreover, if g E 80, then gN := Z 95,, *
Vz-N
1,0,, * g converges to g as N ——> oo in the S-topology (for proof refer to Appendix,

Lemma A.1). Since T is continuous from S into 8’, we have T9 = 271065" *
uEZ

1,0,, * g). Fhrthermore, for g E 80 and fixed 1/ E Z, we have 2 (g,¢Quk)1/2ka
IkISM
converges to «6,, * it” * g as M —> oo in the S-topology (again refer to Appendix,

66

Lemma A2). Hence, T9 = Z Z (9,904)”) TibQuk = Z<g,goQ) Ti/JQ. Now,

VEZ kEZ" Q
suppose ((Tzlzp, WQlopl 6 adng) for some fixed pair (90, w) of mutually admissible

kernels. Take any other such pair (95,16). Then 2,6,: = Z <ibp,goL> VII. and @Q =

L

2 (950,103) 903, which gives
R

<T¢3P795Q> = Z<1LPMPL> (leLaSORl (9504401

R,L

Since both Hugh; and {ch}L constitute families of smooth molecules for B§q(W),

by (5.2) the matrices (<ibp,goL>LP), ((ng,ibR)QR) 6 adng). By Theorem 5.5,
- ~ aq

(<T¢P,¢Q>Qp) 6 ad. (a. I

A straightforward consequence of Theorem 1.10 is the following statement:

Corollary 5.9 Let T E AD:"(B), a 6 IR, 13 p < 00,0 < q < 00. Then T

extends to a bounded operator on B§q(W) if W satisfies any of (AU-(A3).

PROOF. First, consider f with (f) 6 80. Let (9031)) be a pair of mutually admis-
sible kernels. Denote {Q = 2,, (Twp, goQ) §p(f) and observe that ((Tibp, ‘Pqupl E
adzqm). Using the cp-transform decomposition f = Z}, §p(f )ibp and taking T

inside the sum as in the previous remark, we get

llellBg‘1(W) =

 

 

Z§P(f)T¢P
p

 

 

BMW)

 

 

Z (Z (TI/1mm) 5pm) we

Q P

 

 

339W)

 

 

= II ZtQ¢QIIB;1(W) s c “can
Q

S C ll{§Q}Q|lng(W) 3 CW llegnwy

t'1;."'(W)

67

by Corollary 4.6, Theorem 1.10 and Theorem 4.15.

Note that So is dense in B§q(W) if oz 6 IR, 0 < q < oo, 1 g p < 00 and W
satisfies any of (A1)-(A3) (for the proof, refer to Appendix). Thus, T extends to all
of ng(W). I

Note that if q : 00, then T extends to a bounded operator on the closure of So
in B;°°(W).

Remark 5.10 Let {mQ}Q be a family of smooth molecules for B§q(W). Apply the

<p-transform to 2 sp mp:
P

t2) 3: ¢( §P me) = <2 3'10 mP,90Q> = Z (mPaSOQ> 5P
P

P P

Then ((mp,gpQ)QP) forms an almost diagonal matrix by (5.2), and therefore, by

Theorem 1.10,

|l{fo}olligv(m 5 HS (8210711 P()llif;qw S C |l{§P}Pllig‘1(W),
P
if W is doubling.
Corollary 5.11 Let T, S E AD:"(B). Then T o S E AD:"(,B).

PROOF. Since T,S' 6 Angm), it follows that (tQp) 2: ((Tibp,<pQ)QP) is in
adng), and 50 i3 (SOP) i: ((swPHPQlQpl Thus, for Q,P dyadic we have 51/21: =

Z (Si/JP, 903) $3) and SO

R

(T 0 51.013, <PQ> = Z HSI/JPWR (TI/JRMPQ) = :tQR SRP E adzqtfi),
R R

by Theorem 5.5 (composition of almost diagonal matrices). I

68

5.2 Calderén—Zygmund operators

In this section we show that Calderén-Zygmund operators (CZOs) are bounded on
B1?" (W) for certain parameters a,p,q,[3. First we recall the definition of smooth
atoms and the fact that a CZO maps smooth atoms into smooth molecules. Then we
use a general criterion for boundedness of operators: if an operator T maps smooth
atoms into molecules, then its matrix ((Twp, ‘PQlQPl forms an almost diagonal oper-

ator on b;q(W), and therefore, T is bounded on B§q(W).

Definition 5.12 Let N E N U {0}. A function aQ E D(IR") is a smooth N-atom

for Q 2T
1. supp aQ Q 3Q,
2. /x7aQ(x)dx = 0 for [7| g N, and
3. Imam): s worm-n” for an l7! 2 0.
Let 0<6g 1, M>0, NENU{0,—1}, NOENU{O}.

Lemma 5.13 (BOUNDEDNESS CRITERION) Suppose a continuous linear operator
T : S ——> 8’ maps any smooth N0 -atom into a fixed multiple of a smooth (6, M,N)-
molecule for B;q(W), a E IR, 13 p < oo, 0 < q g 00 with 6, M, N satisfying (M.i),
(M.ii) and (M.iii) (see Section 4.1). Suppose W satisfies any of (AU-(A3). Then

T E ADSQW) and, if q < 00, T extends to a bounded operator on B§q(W).

PROOF. By Corollary 5.9, it suffices to show that ((Twp,wQ)Qp) E adzqw) for
some 90,112 E A satisfying (2.1). Observe that if if) E A, then there exists 9 E S with

69

SUPP 0 9 81(0), frv79(w)drv = 0, if HI S No, and 29(2‘”€)¢(2’”€) = 1 for E at 0

1162

([FJW, Lemma 5.12]). Using wp = Z 6,, * 90,, * wp as in the atomic decomposition
uEZ

theorem ([FJW, Theorem 5.11]), we have
w(x) = thpag’la) (5.7)
Q

with t p = Q 1/2sup (4,9,, * wp)(y for l(Q) = 2‘”, and each a(P) is an No-atom
Q EQ Q
y

defined by

1181(1) = 1221? [Q 6.11: — y) (991/ 1 1mm dy if top 11 o (5.8)

and agp) = 0 if tQp = 0. Using (4.2)-(4.3) (valid because {wp}p is a family of

molecules for ngU/V) ), we get

 

W” * My)! S C 'P'_1/2mi“ (I%l I'I’Igl) (1+ mall/«billed —M’

for some r>a and o> J—a. In fact, go,,*wp=0 if lu—VI > 1 (2"‘=l(P)),

since 90,11) E A, but all we require is the previous estimate. Since y E Q, y can be

replaced by xQ in the last expression by Lemma 5.4, and so

”11' (RD/2m (BRIT [RD (1+ ..:??(aleo111M1

 

which is exactly (5.1). Thus (tQp) E adng). Using (5.7), we obtain

WP, 10.» = (213mg), a) = Zap (Tap, 1,10).
R R

Since T maps any No-atom a)? into a fixed multiple of a smooth (6, M, N)-molecu1e

mR: Tag?) 2 cm}; and c depends neither on R nor on Q, we get

P 4.
<Ta), ),<,oQ> = c (7723,9063) =2 ctQR,

70

and by (5.2), since in}; is asmooth (6, M, N)-molecule for B§q(W), (tQR) E ad:q(fl).

Hence,

(<Twp,soolqp) = (0 ZEQRtRP) E adng)’
R

since the composition of two almost diagonal operators is again almost diagonal by

Theorem 5.11. I

Let T be a continuous linear operator from 8(IR") to S’(IR"), and let K = K (x, y)
be its distributional kernel defined on 1R2" \ A, where A = {(x, y) E IR" XIR" 2 1’5 = y}
(for definitions refer to [FJW], Chapter 8). Then T E CZ 0(6), 0 < e g 1, if K has

the following properties:

(I) IK($13/)IS W,

Ix - x'l‘

(11.) |K($,y) -K(iv',y)| + IKIyfl) -K(3/,1")| S C if2lx—1’E'l S livryl-

I 113 _ yln+c
To show that a C20 maps atoms into molecules we start with the following result

from [FJ W]:

Theorem 5.14 ([FJW], Theorem 8.13) Let 0 < e S 1 and 0 < a < 1. If
T E CZO(e) fl WBP and T1 = 0, then T maps any smooth O-atom aQ into a

fixed multiple of a smooth (6, n + e, —1)-molecule mQ.

Thus, if aQ is a smooth O-atom for Q, then TaQ = ch, where mQ satisfies

_ —(n+e)
1. Imo($)|S IQl‘1/2(1+%§f—l) ,

 

IL‘ _ Z _ 1Q|)—(n+c)
7

2. lmQU?) —mQ(y)| S IQI—% (M) sup (1+ I l(Q)

and c is uniform for all Q. Moreover, an (e, n + e, —1)—molecule is a smooth molecule

for B§q(W) (see Section 4.1) if1§p<oo, 0<q§oo, O<a<eandfi<n+paz
(i) 6=eand0<a<e§1,

(ii) J=3+§<n+a<n+e=M,

(iii) J—n—a=B—;'—'—a<0 => N=max([J—n—a],—1)=—1.

The next theorem follows by combining the two statements mentioned above, and
gives the boundedness of certain Calderon-Zygmund operators on B§q(W) with some

restriction on the weight W:

Theorem5.15 SupposeO<e_<_1, 0<a<e,1§p<oo,0<q<oo, andlet
W satisfies any of (AU-(A3). Assume B < n +pa. If T E 020(6) 0 WBP and

T 1 = 0, then T extends to a bounded operator on BS“? (W)

Remark 5.16 If also T‘“ = 0 in Theorem 5.14, then a = 0 can be included in the
range, since fTa(x) dx = (Ta, 1) = (a,T“1) = 0 and so T maps any smooth O-atom

into a smooth (an + €,0)-molecule (see also [FJW, Cor. 821]).

Corollary5.l7 Let1§p<oo, 0<q<oo, 0<6_<_1andOSa<e. Assume
B < n+pe. IfT E CZO(e)flWBP and T1 = T‘l = 0, then T extends to a bounded

operator on B0“? W , in particular, for a = O.
p

PROOF. Since N = O, the bound on {3 from the previous theorem can be relaxed

tofi<n+pe I

72

Remark 5.18 The condition T*(y7) = O for |7| g N ,N 2 1, produces more van-
ishing moments of a molecule Ta, so it is not difficult to satisfy (M.iii). But (M.ii)
M = n + e > J = n + a? 4:) B < n +pe creates a major restriction on the dou-
bling exponent of W. Note that in this case, we get that T maps any smooth O-atom
into a smooth (an + e, N )—molecule, but this molecule is not a smooth molecule for

839W).

From now on N Z 0, since the case N = —1 is completely covered by Theorem 5.15.
Next we want to show that the restriction on the weight W (to be more precise the
restriction on the doubling exponent B) can be removed in some cases by requiring
more smoothnes than (II,,) on the kernel K.
We say that T E CZO(N+6), N E NU{O}, 0 < e S 1, if T is a continuous linear
operator from 8(IR") to S’(IR"') and K, its distributional kernel defined on 1R2" \A,

has the following properties:

(I) |K($,y)| S

 

3

la? - :1!"

am lDlg)K(rv,y)l s for m s N,

Irv — Wl ’

(UN...) 1032mm) — 03,,K(:c',y>| + IDz,,K(y,x> - 022mm!»

['6

 

lx—x
<c

_ |a:—y|n+l7|+e’ for 2|x—x’lSlx—y| and |7|=N,

where the subscript 2 in DEE) refers to differentiation with respect to the second

argument of K (x, y) .

Note that 020(6) 2 CZO(N + e) for N 2 O.

73

Theorem 5.19 Let 0 S (1 <1, 0 < 6 g 1, N0 E NU {0}. Suppose T E 020(N0 +
6) fl WBP, T1 = 0 and T*(y7) = O for [7| 3 N0. Then T maps any No-atom aQ

into a fixed multiple of a smooth (6, N0 + n + 6, N0) -molecule.

More precisely, we will show that TaQ = ch with c independent of Q and

(i) fxiTaQ(x)dsc=0, for lilSNo,

 

_ “(N0+n+€)
(ii) Images 3 clot-“2 (1+ '5” “’Q') ,

 

 

l(Q)
_l [3; _ y| 6 [Z _ (III _ IQ)| —(No+n+£)
(m) [TaQ(33)—T0Q(l/)|SC|Q| 1| ,(Q) | H.311: |(1+ l(Q) ) .

Before we start the proof, we quote the following estimate, due to Meyer:

Lemma 5.20 ([M1]) Let T : D —> D’ be a continuous linear operator with T E
020(6) flWBP, O < 6 g 1 and T1 = 0. Then T maps D into L°° and there exists

a constant c such that for any fixed 2 E IR" , t > 0, «,0 E ’D with supp (,0 E B,(z)

||T<e||Loo S 6(lele + t H V sells»)-

PROOF OF THEOREM 5.19. For simplicity, we give the proofs of (i), (ii) and
(iii) for Q = Q00. The same methods apply to the general cube because of the
dilation-translation nature of the estimates. Thus, consider the unit atom a = aQ00
with xQOO = O and l(Qoo) = 1. First property (i) immediately follows from the fact
that T*(y7) = O for [7| S No. To get (ii) we consider two cases: |x[ 5 6x/fi and

[x[ > 6\/n. For |x| 3 6\/1_i, use Lemma 5.20 to obtain

lTa(III)| S llTallioo S C(||a||L°° + H \7 alltoo) S c.

74

If [II > 6V'rri. we get

[T0(I)! = |/1\'(I.y)a(y)dy|

' D.“ .K(I.0)
= / I\(.r.y) — Z Iy‘ ’3' y” a(y)dy . (5.9.)
3 00 "

:7‘7 ‘ S No

 

since aQ is an .VO-atom. and thus. has NO vanishing moments I y‘aQ( y) dy = O for

[7| 3 31,. Then (5.9) is bounded by

hi

Note that if y E supp a, then 2 [6(y)| g 2 [y[ g 2-3V’h < |I|. and. using the property

|D(y)1\(x 6(y ))— D I\'(r 0)|

 

 

00 {‘7=\o

(HA-1,) of the kernel K to estimate the difference. we get

2 [Dgy,A'(.r, 9a)) — ny,A'(.r. on g c

i‘7i=1\"o

Thus,

Cn V ' 4.5 C
[TOW )I S ———-O— |y|‘\° [0(ylldy S T
3Q00 [Iin ‘ 0 (

|I|n+. 30-7-6

In order to show (iii), we prove that

 

 

[Ta(x)—Ta(x')| _<_ clx—x'l‘ < .1 . + . , .1 .- ). (5.10)

(1+|I|)n+.\0+6 (1+ |Ilt)n+.VU+-6
In the case [x — x'l _>_ 1, the estimate (5.10) follows trivially from (ii) and the triangle
inequality. For [x — x’[ < 1 and [x[ > 10 [75, we use vanishing moments of 0(1) and

the integral form of the remainder to get

 

[Tam — Ta(1:’)| = | [(1171.11) — Katy» am dy| =

/ [K(I. y)
3Qoo

D7 A'(2:,0) D A'( .l‘ 0)
- Z I” , y -K(I’ y) + Z I” y" a(y)dy
'7.

[7|5N0-1 "7 .<\o- 1

 

 

75

1)“ 1 I_ I
ZID” K($,83/) _D, mesa y |a(y )ldsdy
</Qoo[;1(1_ (No _1)I | |§;No (y) (y) 7|
If [x| 2 10¢}; and y E supp a, then [x—syl Z |x|—s[y| Z 10\/ii—3\/ii_>_ 2|x—x’l
and also [x — sy| > |x| — slyl > |x| — 3f > [x|—— l_2_a:| > L—x'. By (IIN+,) the last

2

integral is bounded by

37'l N I: — ___33_:'|.
00

In case [x — x’| < 1 and |x| 3 10 J17, an exact repetition of the argument on p. 85

 

of [FJW] or part (c) on p.62 of [FTW] shows that
|Ta(x) — Ta(x’)| S c [x — x'[‘

by using the decay property (I) and the Lipschitz condition (Hold) of the kernel K,

which holds for any 020(N0 + 6), N0 2 0. This completes the proof. I

Corollary 5.21 Let 1 S p < oo, 0 < q < 00, and let W satisfy any of (A1)-
(A3). Suppose 0 S a S [if — [fifn], where B is the doubling exponent of W. Let
N: [fig—a] and fi—gfl—[Eg] <5 1. IfTECZ0(N+6)r7WBP, T1 =0 and

T*(y'7) = O for [7| 3 N, then T extends to a bounded operator on B§q(W).

PROOF. By the previous theorem T maps any smooth N -atom into a smooth
(6, N + n + 6, N )-molecule. This molecule is a smooth molecule for BI?" (W) if

(i) a < 6 g 1,

(ii) M=N+n+6>J=n+é—;—’l 4:) [‘tfn—a[=[g;—"[>é§3—6 and

(iii) N = max([J — n — a], —1) 2 [LEE — a], which are all true.

By the boundedness criterion (Lemma 5.13), T is bounded on B§q(W). I

76

Corollary 5.22 Let 1 S p < oo, 0 < q < 00, and let W satisfy any of {AU-(A3).
Suppose O S _B_;g — [5?] < a < 1, where B is the doubling exponent of W. Let
N =[13—1—D'2—a] anda < e 31. IfT e CZO(N+1+6)r7WBP, T1: 0 and

T*(y7) = 0 for [7| S N +1, then T is bounded on B§q(W).

PROOF. By Theorem 5.19, T maps any smooth (N + 1)-atom into a smooth
(6,N + 1 + n + 6, N + 1)-molecule, which is also a smooth (6,N + 1 + n + 6,N)-
molecule. This one, in its turn, is a smooth molecule for B§q(W), since

(i) a < 6 S 1,

(ii) M=N+1+n+6>J=n+L;—"- 4:: [%§—a[+1>@—;—"—6 and

(iii) N = max([J — n — a[,—1)= Bf"- — a].

By the boundedness criterion (Lemma 5.13), T extends to a bounded operator on

830(W). I

Remark 5.23 Note that the condition T*(y7) = 0, [7| S N, can be very restrictive;
for example, the Hilbert transform does not satisfy this condition for [7| > 0. 0n the
other hand, we have considered a general class of 02 Os, not necessarily of convolution

type. Utilizing the convolution structure will let us drop the above condition.

Let N E N U {0}. Let T be a convolution operator, i.e., the kernel K(x,y) =

K (x — y) is defined on IR"\{O} and satisfies

(0.1) |K(a:)| < C

_ |$|n,

 

(C2) [D7K(x) for [7| S N +1,

I < __C_
_ |$|n+|7ll

77

(C.3) / K(x)dx=0, for allO<R1<R2<oo.
R1<|$|<R2

Remark 5.24 We replace (UN) and (HA/+6) of the general 020 kernel with the
slightly stronger smoothness condition (0.2) to make the proof below more concise.
The reader can check that conditions corresponding to (UN ) and (IINH ) in the con-

volution case would sufiice for the statements below.

Now we obtain an analog of Theorem 5.19 saying that T maps smooth atoms into

smooth molecules, and then we show the boundedness of T.

Theorem 5.25 Let 0 S a < 1, 0 < 6 S 1, N E N U {0}. Let T be a convolution
operator with a kernel K satisfying (0. 1)-(0.3). Then T maps any smooth N -atom

aQ into a fixed multiple of a smooth (6, N + 1 + n, N) -molecule.

More precisely, we will show that
(i) /x7TaQ(x)dx = O, for [7| S N, and

[LE _ le —(N+fl+1)
(ii) lDlTaQ($)l s cIQI-W-W" (1+ —-,(—Q—)—) for l7| = 0,1.

By the Mean Value Theorem, (ii) with [7| 2 1 implies the Lipschitz condition (M4)
for [7| 2 0.
PROOF. To obtain (ii) we first consider x E 10\/fiQ. Then

ITaQIxII,IvTaQIchI= Z /3QD”°K(x-y)ao(y)dy

[7o[=00r1

: Z [362 [D70K(x — y)

[70|=0 07‘1

78

 

DTDVOK x — x
‘ Z | Q)(37Q - II)" adv) dy (5.11)
lnSN—hd 7'

since aQ is an N -atom, and thus, has N vanishing moments f yV'aQ(y)dy = O for

[7| S N. Then (5.11) is bounded by

'IxQ — yl”"”°'“ lao(v)l dy,

 

Z Z |Dl+l°K($ —$Q +9(y ‘40))
l
l7o|=00r1 3‘9 |7I=N+1-|7o| 7'

for some 0 S 9 S 1. Since x E 10 WQ and y E 3Q, [y—le S 2\/r_zl(Q) S %|x—xQ|.

Using property (C2) of the kernel K, we get

| S C m C .
[2: - 5'30 + 9(1/ - $Q)|"+'7' [x - $Ql"+""

 

 

[177K017 — 930 + 9(3/ - 1130))

So,

 

Cn,N _
[TGQ(IE)|, l V TaQ(‘T)| S xQ|n+N+1/3Q|‘TQ — le |10|+1|GQ(y)|dy

I1: —
”MN—W IQI-1/2IQI — CIQ|“1/2—I7ol/n| “62) |"”“
I11 — xQIrNH ’ Ix — zQI

 

Sc

1

by the properties of aQ.
If x E 10\/7_iQ and y E 3Q, then [x — y| S 13nl(Q), so by the cancellation

property (C3) of K (using D7(K 1I< aQ) = K * (D7aQ)), we obtain

 

|Tao($)|1l V Tao($)| S E:

[70[:0 or]

/ K(x — y) DraQIyIdyl
3Q

= >:

|‘70|=0 0T1

1 — —: n— 1:
sc / IQI ”2 '70” V Ix—yldy
|

y—xlS13nl(Q) [33 — y|n

 

[w K(x — y) IDVOaQIy) — Dl°ao(:v)l dy|

 

l3nl(Q)
0

This concludes the proof of (ii).

79

Property (i) comes from the fact that T is a convolution operator and aQ has
vanishing moments up to order N . Property (ii) guarantees the absolute convergence

of the integral in (i). I

Corollary 5.26 Convolution operators with kernels satisfying (0. 1)-(0.3) are
bounded on B§q(W) if W satisfies any of (AU-(A3) and 0 S a < 6 S 1, O < q < oo,
1 < p < 00. In particular, the Hilbert transform IHI (n = 1) is bounded on B;q(W)

and the Riesz transforms 72,-, j = 1, ...,n (n 2 2), are bounded on B§q(W).

PROOF. This is an immediate consequence of Theorem 5.25 and Lemma 5.13:
choose N = [@;—" — (1| in Theorem 5.25; then T maps any smooth N -atom into a
smooth (6, N+1+n, N)—molecule, which is either a smooth (6, N+1+n, N)-molecule

for B§q(W), if oz S gig — [gI—Tn] or an (6, N +1 + n, N — 1)-molecule for B§Q(W), if

 

1 > a > %‘- — [fin]. Note that both Hilbert and Riesz transforms are convolution

type operators with kernels satisfying (C.1)-(C.3). I

80

CHAPTER 6

Application to Wavelets

Consider a pair (99,7,b) from A with the mutual property (2.1). Then the family

{90¢}, ’l/JQ} behaves similarly to an orthonormal system because of the property

f = Z (moo) 170 = 2: So It for all f e 8773.
Q Q

However, this system does not constitute an orthonormal basis. This can be achieved

by the Meyer and Lemarié construction of a wavelet basis with the generating function

6 E 5 (see [LM] and [M1[):

Theorem 6.1 There exist real-valued functions 6“) E 5(IR"), i = 1, ...,2" — 1, such
that the collection {68,3} 2 {2”"/26(i)(2”x — k)} is an orthonormal basis for L2(IR”).

The functions 6(1) satisfy

and, hence,

/ x76(x) dx = O for all multi-indices 7.
IR

81

2"—1

Thus, we have f = Z Z (f, 63’) 93 for all f e L2(IR"). This identity extends to
£21 Q

all f E S'/’P(IR").

Theorem 6.2 Leta E IR,0 < q S 00, 1S p < 00, and let W satisfy any of (A1)-

(A3). Let I9“), i = 1, ...,2" — 1, be generating wavelet functions as in Theorem 6.1.

Then

I

 

 

 

 

 

 

 

 

2n—1
1 79‘”>}
BSQUV) g2; {<Cf Q (Q

PROOF. Assume i = 1, ..., 2" — 1. Since {68)}Q1i is a family of smooth molecules

6$°IWI

for ng(W), the inequality

E r 9m> 9(1)
83"(W) ||;<f’ Q Q

follows immediately from Theorem 4.2. Therefore, we need to focus only on the

f'

 

 

 

 

s c 2: II{(fi68))IQII.-.;1(w, (6.1)

 

 

BMW)

opposite direction.

Let (,0 E A be such that Zuez|cp(2"{)[2 = 1 for g 75 0. Let E'Q = (fjcpq)

Applying the boundedness of the go-transform (Theorem 4.15), we obtain

Now for each i and Q, define t3” = (f, 08)). Since 08) E 8, by the cp-transform

f . (6.2)

 

 

 

 

 

 

 

{ngQ )SC

131M 8::qu

decomposition with w = Ip, we have I98) = Z <08), gop> pp, which gives
P

5Q“) = 2 (69.9911) (Mp) = 211231.571.

P P
Since supp 95p (7 supp 68) yé {(6} only if l(Q) = 2jl(P) with j = 1,2,3,4 (recall
that supp 96;: Q {5 E IR" : 2’“1 S |§| S 2"“} when l(P) = 2"“), we see that

82

 

aglvz<68)1‘PP>= 0 unless 2 S fi—g) S 16, in which case

V

|in — xpl

—M
for each III > 0,
1(Q) )

[an| S CM (1+

as was shown in [FJW], p. 72. Let M > 1% + 3. Then A“) 2: (ago) is an almost

diagonal matrix for each i, and, by Theorem 1.10,

II{?.§"}QII.;1(W, s c III§QIQII.;1(W,- (63)

Combining (6.3) with (6.2) we get the opposite direction of (6.1). I

Corollary 6.3 Let {Ni/Ail}, i = 1,...,2" — 1, be a collection of Daubechies DN
generating wavelet functions for L2(IR") with compact supports linearly dependent on

N (for more details, see [D]). Then for any f with f, E S’/’P(IR"), j = 1,...,m.,

{<6 W82},

2"—1

BSQIW) g,

for sufiiciently large N.

(6.4)

 

 

f

 

 

 

 

 

 

531M)

PROOF. First, observe that there exists a constant e such that for all i = 1, ..., 2" —
(,7 N (i)
are smooth molecules, and so Q is a family of
c
Q

 

 

1, the functions 0
smooth molecules for B§q(W) if we choose N sufficiently large to have the necessary
smoothness and vanishing moments. Second, if cp E A, then (<NilIg),Ipp>QP) E
adng) by (5.2). Applying these two facts in the proof of the previous theorem, we

get (6.4). I

83

CHAPTER 7

Duality

7.1 General facts on duality

An important tool that we need is the duality on lq(X) with X being a Banach space.
By definition lq(X), O < q < 00 is the set of all sequences {fu},,€z with fV E X,
1/q
u E Z such that (Z ||f,,||}) < 00. If 1 S q < 00, then (lq(X))" = lq'(X*) (see
1162'.

[D, Chapter 8]), and if g is a continuous linear functional on lq(X) identified with

{gu}ueZ E lq'(X*), then the duality is represented as

g(f) = (f,g) = Z <f.,g.>x,

VEZ

where (fu,g,,)x = g,,(f,,) is the pairing between X and X *. we will mainly be
concerned with X = LP,1S p < 00, or L”(W), 1< p < 00, and, thus, X“ 2 LP' or
LP'(W_P'/P), respectively, with the pairing (f,g)X = f (f(w),g(:r))H dx.

If 0 < q < 1, and X = L", IS p < 00, then (lq(Lp))* : l°°(L”') (see [T, p.177])

and the pairing is defined as above.

84

7 .2 Duality of sequence Besov spaces

Theorem 7.1 Let W be a matrix weight, a E R, 0 < q < oo, 1< p < 00. Then
(i) hgaq’(W—pI/p) g [53"(VV)]* always
(a) [bgamli grim-1071’) if w e A,,.

We will prove this theorem, which implies (1.12) of Theorem 1.17, in several steps.
The use of reducing operators is essential and helps to understand why certain con-
ditions on the weight W are necessary.

PROOF OF (1) OF THEOREM 7.1. For each {E hgaq'(W‘P'/P) define a functional

l; on h$q(W) by

l(g‘) = (:t“) = 2 (so, “0),, for any 5': {5Q}Q e b;q(W).
Q

i

The calculations below show that this sum converges and It E [h$q(W)] :

S 2 Z l<§QiiQ>Hl = Z/Rn Z l<§QwFQ>Hl |Ql_1XQ(tldt

VeZ QEQu VEZ QEQu

 

 

Z <§Qi€Q>u
Q

Z/Rn Z lQl—l I<W_l/p($lW1/p($)§o,fq)ul xq(a:)dx. (7.1)

V62 QEQV

Using the self-adjointness of W and the Cauchy-Schwarz inequality, we bound (7.1)
by

23/an Z (lQl—i—ixotv) ”WI/”(ftlé'ollu) (lQl%-%XQ(CC)[|W_1/p($)FQIIH) d113,

VEZ QEQV

Applying Holder’s inequality several times, we estimate l,(§) by

2“on

VEZ QEQV

 

a_l _. P
’XQ($)||W1/p($)solln) )

85

 

 

x (Z (IQli'ixQCv) Hw-VMFQHHY')” dx

 

 

 

 

 

 

 

 

QeQu
_g_l a 2-1 ~
S: 2 IQI " 2X0 Q 2 WI" 2XQtQ (7.2)
VEZ QEQu Lp(W) QEQV LP’(W-p'/p)

s “slum, ||5l|,;,aqi(w-,.l,,.,,
for 1 < q < 00. In case of 0 < q S 1, we bound (7.2) by ||§||5g1(w) ||fl|5;’aoo(w_pr/p).
Since 1" is embedded into I1 when 0 < q S 1, we estimate the previous product by
llgllbgflm Hillbgfflw—p’my I

In terms of reducing operators (or using (1.10)) the previous lemma states

brie/13» g [brawn] (7.3)

If we follow along the lines of the proof again but instead of W‘l/ ” (t) W” p (t) in (7.1)

use A51 AQ, then we obtain the following statement.

Lemma 7.2 Let aER, O<q<oo,1Sp< 00 and {AQ}Q ERSD. Then

arm/4.31» g [brawn]? (7.4)

In fact, if we have only proven (7.4), then (7.3) (and equivalently part (i) of Theorem

7.1) could have been obtained as a consequence of (7.4) and Corollary 4.20, i.e.,
brim/12$» g Emma» ; [bra/1a)] . (7.5)

Observe that (7.4) holds for any {AQ}Q E 7281;, not necessarily generated by W.

Now we will study the opposite embeddings. By Lemma 7.3 below, we will get

[62"({AQ})]' g b;""’({A5‘}) (7.6)

86

without any additional assumptions on the sequence {AQ}Q. Note that combining
(7.4) and (7.6), we obtain (1.13). Applying Corollary 4.20 again, (7.6) is continued

as

lb:q({AQ})l C b7aq({AQ 1}) C b {WM/12$ }) ~b‘a" (W W.)

with the second embedding being held under Ap condition. Thus, the embedding (ii)

of Theorem 7.1 holds if W E A,,.
Lemma 7.3 Let {AQ}Q E R813, (1 E IR, 0 < q < oo, 1 S p < 00. Then {7.6) holds.

PROOF. Let I E [530({AQ})]*. We show that there exists I? E hgaq'({A51}) such

that for any 5E h$q({AQ})

z<§)=(*,t">=Z<“Q,Q>.. and lli'll-aq({A1,,<lllll
Q

Let e J )denote a vector-valued sequence enumerated by dyadic cubes such that in
the k“ component (kth row) of this vector the J ‘h entry (corresponding to the dyadic

cube J) is equal to 1 and all other entries are zero:

..k
>= (..., {0}Q, {...o...1,..e,,,,...o...}Q — k‘hrow, {0}Q1---)T-

Now if 3’ has only finitely many non-zero entries, i.e.,

§= Z :30 53k,

innite k: 1

then by linearity

87

By continuity, since finitely non-zero sequences are dense (p, q < 00), we get

=2 2.3% )tgl— — Z <SQ’tQ>H for any 3 E bo‘ p.q({AQ})

QED k=l QED

Now everything is set up to show that F :2 ({tg )}Q, {tg )}Q,...,{tgn)}Q)T

bgaq’({AC—21}). For 5E b;q(lRm), set Q; = A515 and define
l~(33?) ==l({AQSQlQ)=l({8Q}Q)-Z<8QatQ>H=Z<AQ§Q1A515Q>H
Q
= 2 82,50
Q < >.
where trq = A5156). By above,

”WI 5 C ||{§Q}Qllng({AQ}) = C ||{5Q}Qlligqmm),

e, l induces a continuous linear functional l on bgq(Rm). By Lemma 7.4 below

{tQ}Q E bl?” (Rm). Since the inside LP -norm of the b1?” (Rm)-norm of t is

 

 

 

 

 

 

 

 

 

 

 

 

Z IQI 5th Z gran/15150117.)... 2 1621-5?on ,
QEQV Lp’ QEQV Lp’ QGQV Lp’ ({A51})
5 E by“ {A51}) and the lemma is proved. I
Lemma7.4 LetaER,0<q<oo,1Sp<oo. Then
[bgqmm] z 1);,“ (W). (7.7)

PROOF. It suffices to show only the scalar case (m = 1) of (7.7), since §E b:q(Rm)
means that each component 5“ )belongs to bag and by making zero all but one of the

components of an arbitrary 53’ we obtain (7.7).

88

. .a * °-—aq’ . . . . . .. , . .
The embeddlng [bpq] 2 bp, 18 a tr1v1al application of Holder 8 mequality plus
the embedding b1?" —> bf,” for q < 1, so we concentrate only on the opposite embed-
ding.

Suppose I E [bgq]) . Using linearity and continuity, I can be represented by some

 

 

sequence {tQ}Q as [(5 =ZQ thQ for any s— - {8Q} E bag and
”(SH = stb s lllll “slag. (7.8)
Q
Case q_ > 1: For each V E Z let fu(s =2: lQl’a n szXQ(:c ). Define a map
QEQV

I : b3“ -—> l"(L”) by 13( )= {fu(s )luez- Observe that ||I(s)||,q(Lp) = Hsllbgq, in
other words, by the natural construction I is a linear isometry onto the subspace

1(1):") of lq(Lp). Then I induces a continuous linear functional l on I (bgq) g lq(Lp)

~

(continuous in lq(Lp)-norm) by l(I(s)) = [(3). Since l"(LP) is a Banach space, by
the Hahn-Banach Theorem l extends to a continuous linear functional last on all of
lq(LP) by i... IRES"): lwith Hie...“ = “in g lllll. Since [lq(LP)]"' = lq'(LP'), in. is
represented by a sequence g = {gubez E lq'(Lp’) with ||g|| = ||{g,,}V||,qr(L,,:) S ”I”

and

Zth—Q = [(3) =((l~{f,,s) })=Z/( fu(s dx, for any §E bgq,
Q

uEZ
OI‘

Zth‘Q = Z Z |o|-%~%3Q [qu(:13)dr.

Q VEZ QEQU
Taking 3Q = O for all but one cube, we get tQ = |Q|_%+% < g” >Q. Using Hblder’s

inequality, we have

2: <9u>QXQ

”tuba. =
p QEQu

 

 

 

 

S ||{§u}u||m'( (up) _ <-||l||

LP u (‘1'

89

Case 0 < q < 1: Suppose I < p < 00. Fix 1/ E Z and let FV denote a finite
a_l l P, .
collection of cubes from QV. Set TV = ZQEFV (|Ql3 2+E’ItQO . Since the sum 1S
finite, TV < 00. Let sQ = |Q|(%-%+ 31’) p thl” 2tq, if Q E F and tQ 75 0; otherwise
let sQ = 0. Note that ||{sQ}QH53q = 7.3/P. Observe that 2Q thQ = TV and by (7.8)
S Hill ||s||53q= ”III I”). Since TV is finite, we get rfi/p’ S “III and the estimate
holds independently of the collection FV taken. Hence, we can pass to the limit from
FV to QV. Then,
p, l/p’
a l l , .
“tilt-0°0— — sup 2: (IQli‘WItQI) = supTJ/p s ”In or te b,;°°°.
uZE QEQu VEZ
Now assume p = 1. Fix P E D and set 3(1)) = {s .sgp)}Q by 38)) = IQlfi‘isgnt—Q

m}
S
{a Q

= III” for any P E D. Hence,

.0q
51

= 1 and lPl%—%|tp| =

.0“)
b1

if Q = P and 380) = 0 otherwise. Then
:35); (i—P’)

IItII.—aoo=supIPI%—%Itpis11111 or tea.“-
°° PeD

 

 

 

 

{803)}

 

 

               

7 .3 Equivalence of sequence and discrete averag-

ing Besov spaces

In this section we discuss norm equivalence between B§q({AQ}) and b:q({AQ}). We
suppose a E IR, 0 < q S 00 and 1 S p < 00 for all statements in this section. If

q=oo, then set q’= 1.

90

Lemma 7.5 If {AQ} is a doubling sequence of order p, then for 5Q = <fd, <pQ>

<0 Hf ”33% (-{AQ}) (7-9)

                    

 

PROOF. Note that 5Q: |Q|1/2(,5V *f)(2 ”“k) for Q = QVk, where @(x) = w(—:c).

Let ||{§Q}Q”bgq({Aq}) =1 ”{Jj/phnlg, where
J — Z LII/1am * f1I2 "1:11sz (7.10)
kEZ"
Since @V * fE EV, Lemma 4.9 implies
Isa. * fie-"m = 2m * fie-"1+ 12:)ka — I — 2’22), x 6 Q”.
15sz

for some 7 E F. Then

JV<Z/

kEZ" Quk

(Z IIAQ..(¢V * JOB—”l + 117)“ Mk - l - 2%)!) da:

IEZ"

 

SCZ/

kEZ" Quk

p
|le( (%*f)(2 "l+$)ll
u]: f .
(,EEZ..( (1+|k—l—2"x|M d2: orsome M>B+n

Using the discrete Holder inequality and the fact that M > n, we bring the p“ power

inside the sum on l. Furthermore, since {AQ}Q is doubling, (1.7) implies

 

“Ac..."ill” s c(1+ Ill)‘3llAQ.(..1,fiI|”, for any a e H. (7.11)
Thus,
1+ Ill) )fillAQ (9% *f)(2 ”l+$)l|”
JV < V(k—H) d
02/: (1+|k—l—2V:c|)M “3

keZ" W (52"

Changing variable (t = a: + 2"”l) and reindexing the sum on I , we get

J _<_c Z] ZI (1+lk— l)‘3 MIIAQ..I¢.*f>It>IIPdt

kEZn Qul 1E2"

91

S 0 Z / Imam. =1 f)(t)ll”dt = c115. =1 f‘ Hun/1.1a

(the sum on k converges since M — 6 > n). Thus,

 

||{§Q}Q||534((AQ}) S CHf | B§q({AQ},¢)°

Now we need an independence of the space ng({AQ}) on the choice of 4,0 (or (b). We
apply the same strategy as in Theorem 4.15, namely, we use the proof of Corollary
7.10 below, which will imply that the last expression is equivalent to c H f” BS"({Aq}.so)

and, thus, (7.9) is proved. I

Corollary 7.6 If {AQ} is a doubling sequence of order p, then for E'Q = <f, Ipq>

 

 

|l{§Q}Qllb$q({A5‘}) S C Hf l ng({A51})‘ (7-12)
and
||{§Q}Q||5;aqr({AC—21}, s c Ilf “Brawn. (7.13)
PROOF. For (7.12) repeat the previous proof with each AQ replaced by A51 and
instead of the estimate (7.11) use
”Aiken? g c(1 + (1|)5 ||A5:(k+nu||”, for any u e H, (7.14)
- - - _1_. _ l(izifill
Wthh follows from the doubhng property (1.7) and duality ||AQ u H — sup H A 17“ .
#0 Q

For (7.13) use the obvious replacements for a, p, q and AQ. If 1 < p < oo, choose

M > fip’/p + n and replace (7.11) by

“113,11”? 3 c(1+ III)B"/"IIA5:(,+,,1IIIP, for any at e n, (7.15)

92

which is obtained from (7.14) by raising to the power p’ / p. If p = 1 ( p’ = 00), then

replace (7.10) with the L°°-norm:

J— — sup 2 IIAa:.I (woo "k)IIxQ..I:v)

”ER kEZ"

and use (7.14) instead of (7.11) to get

JV < C sup 2: ”AQV,( (Pu * fllt l” XQwUl = C “851/ * f llLoo({Ag‘},u)-

tElRn lEZ"

Lemma 7.7 Suppose {AQ}Q is a doubling sequence of order p. Then

 

 

 

 

 

 

 

IIfIIagnaQy Sc {5am} , . (716)
Q bgqflAQl)
PROOF. Usingf =.§'Q(Z f)wq, we get
Q
Z§Q(f)1bQ
Q BSQMAQD
:2 1/P
_<. Z [Z IIApé‘aIIIIawaImI da:
#62 l(P): 2- v l(Q): 2- u
u If,"
v+1 p l/p
= Z [Z IIApé'aIIIIawame a
p=u-1 IIP1=2- " l(Q): 2- u I

-——= Ins/11.1,,

since IpV * wQ = 0 if [u — VI > 1. Using the convolution estimates (4.2) and (4.3), we

get (for any M > O)

IIawaon s on IQI'1/2(1+2"lw-$Ql)‘M, if u = u—1,u,u+1. (7-17)

93

If 1 < p < oo, choose M = M1+ M2 with M1> fi/p+n/p and A12 > n/p’;
if p = 1, let M = All > fl + n. Then applying the above estimate and Holder’s

inequality, we obtain

v+1

1.3.; Z Z Z IIAPEQIIPIPIIQI‘P/2(1+2”IwP-$Q|)‘M”’-

#=V-11(P)=2‘"1(Q)=2‘"

Shifting Ap to AQ by doubling, we get

u+1
age :3 Z |Q|“”/2llAQ§Q||”|Q| Z caI1+2Vpr—xaI)-MIP+3.
#=V-11(Q)=2“‘ l(P)=2"’

Applying Lemma 4.4 (Summation Lemma) to the sum on P, we have

u+1 u+1

7.52: 2 Z IQI-P/2IIAa5aIIPIQI=c Z Z IQI-Wé'axa
#:11—1 I(Q):2-I‘ I-‘ZV—l l(Q)-12"“ Lp({AQ},#)

Combining the estimates for all JV and reindexing when necessary, we get

 

 

 

 

BanIA})S3c 2"“ Z lQl_1/2§QXQ =C“{§Q}H53"({AQ})'
p Q l(Q)=2“’ ”((2101.22) V1?

Remark 7.8 Theorem 1.18 is obtained by combining Lemmas 7.5 and 7. 7.

Corollary 7.9 If {AQ}Q is doubling (of order p), then

 

 

 

 

 

(7.18)

 

 

ng({Ac—gl}) — bSQ({AC-21})

and

._, _, Sc
8,,QIIAQ»

f

 

 

 

 

 

 

 

 

{§Q(f)}Q (7.19)

6;.” Imam

94

PROOF. For (7.18) use the previous proof with the following shifting of A p to AQ

(similar to (7.14)):
IIAE1§Q II10 S Cum (1 + 2"|a:p — lelfi “/15ng H”, (720)

where l(P) = 2‘” and l(Q) = 2‘” with u = V — 1,1/ or 1/ + 1; for (7.19) use the
above proof with the indices —a, q’, p’; if 1 < p < 00, take M > Bp’/p + n and

apply (7.20) raised to the power p’/p; if p’ = 00, then

(1+1

“sup 2: Z Z llAialé‘Qlll(sou*¢Q)(:v)le(x)-

:1:ElRfl p:u—1((P)=2_"1(Ql=2-”

Using the convolution estimate (7.17) (with M = M1 > B +n) and (7.20) for shifting

A131 to A51, we get

u+1
Ju S 0 Z Z “21515;?”ch 1
[1211—1 l(Q):2-I‘ Loo

which gives (7.19). I

Corollary 7.10 The spaces B§q({AQ}), B§q({A51}) and Bgaqlfl/lél” are inde-

pendent of the choice of the admissible kernel, if {AQ}Q is doubling ( of order p )

PROOF. Repeat the proof of Theorem 1.8 with W replaced by AQ and use Lemmas
7.5 and 7.7 for the space B§q({AQ}); for the space B§q({A51}) apply (7.12) and

(7.18); and for the space Bgaq’flAélD use (7.13) and (7.19). I

7 .4 Properties of averaging LP spaces

In this section we study the connection between LP({AQ}, V) and LP(W) and the
dual of LP({AQ}, V).

95

Lemma 7.11 Let W be a doubling matrix weight of order p, 1 S p < 00. Then for

fEEV,VEZ

Hf IILP(W) S CIIfIILP({AQ},1/)i (7.21)
where {AQ}Q is a sequence of reducing operators generated by W and c is independent

OfV.

—O

PROOF. Using the notation WV(t) = W(2‘Vt) and fV(t) = f(2"’t), we write

IIf‘IIipnn 22/... IIW‘/”(t) )llpdt- 2277/ IIWJ/PIt)f;It.)IIPdt

kEZ" kezn Qm.

Since f; E E0, there exists 7 E P such that f; = fV * 7. Using the decay of 7 and

Holder’s inequality, we get

(711....) < z ,-.,. /

keZ" Q0). mezn

for some M > B + n. Observe that IIAQkaV(y)IIP a: fQo). IIWJ/p(t)f:,(y)IIpdt. Using

 

1/P(t p
Z/ “W iii/()II a.) 0,,

1+ Im)—kI)M

the doubling property of W to shift AQuk to AQm (see (7.11)), we bound the previous

line by

Z Z2 ”(of +17” kl) M ”’IIAQ...f;Iy )llpdy
mEZ" 11:62"
< C E fom llAQumfl/(y )Pll d3},

mEZ"

where the sum on k converges, since M > 3 + n. Changing variables a: = 2‘Vy, we

get the desired inequality (7.21). I

Corollary 7.12 Leta E R, 0 < q S 00 and 1 S p < 00. If W is doubling (of order
p) and {AQ}Q is a sequence of reducing operators generated by W, then
BTU/loll E B§q(W)-

96

PROOF. Since IpV >1: f E EV, the previous lemma implies

LP(W)}V

 

”f IIBEHW) = ”{2m

{2” " }
LP({AQ}.V) V

 

 

 

 

 

¢u*f

(q

 

 

 

 

= CIIf IIBgQ({AQ})-

[q

wu*f

   

 

 

Lemma 7.13 Let 1 < p < 00 and W satisfies any of (AU-(A3). Suppose fE EV,

V E Z. Then

Hf IILP({AQ}.V) S C||f “mm, (722)

where {AQ}Q is a sequence of reducing operators produced by W and c is independent

ofV.

PROOF. Using the definition of reducing operators, we write

HfHLPI WOW) ~12]

kEZ" Quk IQVkI Quk

=Z//IWW Iymma
Q01: Quk

kEZ"

WWMwww

 

—-o

by changing variables :1: = 2””y and denoting f_V(y) = (2“’y). Note that f; E E0.
Applying the decomposition of an exponential type function (Lemma 4.8) to fV =

fV * 7 for 7 E F and Hélder’s inequality (choose M > 5 + n), the last expression is

 

 

bounded by
”WI/”(t mlll”
CZ/W/izlu ZWMMy
kEZ" QOL Quk mEZ" y—
szzanaM/Iwww<wm,
mEZ" 1662" um

97

by applying the doubling property of W (any of (A1)-(A3) imply that W is doubling).

Integrating on y and summing on k (M > fl + n), we obtain

WWW <cZ/ Iwrm m)Wt

mEZ"

=c2-W 2: f... “Wt/”U m)ll”dt

mEZ"

again by changing variables. Now applying Lemma 4.12 and Lemma 4.14 (this
is where (A1)-(A3) come into play), we bound the above by c2 ”" ||f,, “U, (W..) =

c “f ||Lp(W), which gives (7. 22). I

Corollary 7.14 Let a E IR, 0 < q S 00 and 1 < p < 00. If W satisfies any of

(AU—(A3) and {AQ}Q is a sequence of reducing operators generated by W, then
33"(W) g Bg‘IIIAQI).

PROOF. As in the proof of Corollary 7.12, use the fact that 90,, * f E E, and

Lemma 7.13. I

Remark 7.15 Combining Corollaries 7.12 and 7.14, we have Lemma 1.19.

In order to establish the dual of Lp({AQ},1/), 1 < p < 00, we consider the following

idea:

IIf’II‘g.({..Q},.)= 2 / IIAQf‘ )IIidx=/ (ZIIAQI‘I Wham) d2:

QEQu QEQV
p
=/ d1: =: f

’H

—o

Us/PIxI Ix)” ch: = If Ii. U,

 

 

Z AQXQ(I (I)

060

 

 

 

 

98

i.e., LP({AQ},V) = L”(U ), where U( =2: AQXQ() is a matrix weight. Since
QEQu
the dual [LP(UV)]* can be identified with Lp'(Ulf) with Ul/p(:c) = (U;)"1/”'(:c) (e.g.

 

 

 

 

see [NT] or [V]), i..,e U( =2: AQ” XQ(a: ,we obtain
QEQu
pl
Hf'uPz’W — -/ 2: AQ mm: ’22:) da:
Q€Qu H

=2 / HAQ litre MIde=IIfII:; (QM }

QeQu

OI‘

[UM/1Q}, or z LP’<{AQP}, v). (7.23)

If p = 1, then the standard duality argument gives [L1({AQ}, V)]"' % L°°({A61},V)

The details are left to the reader.

7 .5 Convolution estimates

The boundedness of the convolution operator with a decaying kernel on averaging

V({AQ}, V) will be helpful in the next section. We establish it here.

Lemma 7.16 Let |<I>(t )|_ for some M > B/p+n and for V E Z define

(1 + WM
<I>V(t) = 2""<I>(2"t). Let {AQ}Q be a doubling matrix sequence of order p, 1 S p < 00.

F221: A,,u,1/ E Z. Then
(2) ”(I)” * f ”LP({AQ},/\) S CO (Cl)A—V(C2)#_Vi|f HL”({AQ},V) :

(W ”(I)” * f “LP({A51},/\) 5 CO (Cay—”(QVWW lle({Ag‘},u):

99

where 01 = 2"/px{,\>u} + 2("‘B’/”X{Agu}, 62 = 2nX{Iu>z/} + 2"’MX{p3v}: C3 =

2(5“")/"X{,\>Q} + 2’"/pX{A§V}, and co is independent of A, ,u and V.

21/".
PROOF. U ' th d f<I> 1 CD — < k , 'h
Sing e ecay o , name y, I ”(as y)| _ c 2(1+ 2"]:1: _ 90M vv ere

 

k2 = 2(“_")"X{#>V} + 2(V‘“)(M‘")X{#SV}, we have

II<I> *fIIIiPQAQfi, Z/IIAQI «P *f)( )IIde

 

QEQA
<qu/(/ IIAQI‘ yIIII<I> Ix— my)de
.2. I (I titlriniIil .) «a

 

.. p
k 2"" A f y

-.:] (Z I 1:.ILP:(II'MP)
keZ" QM mEZ" QV'"

Since {AQ}Q is doubling, we “shift” AQM to AQQm:
IIAQikf(y) H S ck1(1+2"|$—$le)fi/” liAQume) II, for a: E QM, (7-24)

where k1 = 2(’\_”)"/px{,\>u} + 2(V’A)(fi‘")/px{,\gu}- Substituting (7.24) into the convo-
lution estimate, we get

-' P
* 2(1; k22”"||AQ f(y)”
p um

mEZ"

 

Using the discrete Holder inequality on the sum inside and then Jensen’s inequality

to bring pm power inside of the integral (if p > 1), the last line is bounded above by

1 2""IIAQ fix/)II”
[CD um d
Ck," /. (I;(1+|P”P-ll>”"P/P)m (gawk (1+I2”x-m|)P"P/P y dz

«m; X [421mm WW M, [Q IIAQ.mf(y)ll”dyd-r,

mEZ"

 

 

 

100

since M — fl / p > n, the sum on l converges (independently of x). Changing variables
(t = 2"x) and observing that the integral on t converges (independently of m), again

since M — B/p > n, we obtain

II<I>.*I’IIP2.Q..},QScAn~£ 2 [Q IIAQ.-f'<yIIIde.

mEZ"
Put c1 = lei/(AW) and 02 = leg/(“fl”). Then part (i) is proved.
For the second part observe that (1.7) (“shift” AQm to AQM) together with

l(iIUfi'H

”fl-15H = sup .. imply
Q #0 |leu ||

 

—o

“Ac-31.17.71)” S ck3(1+ 2"|x - $Q.m|)B/p||Ac3imf(3/) II, It E Q», (7-25)

where 13;; = 2(’\"’)(5‘")/I’X{A>Q} +2(”‘*)"/”X{ASV}. Note that (7.25) is similar to (7.24),
so previous estimates with each AQ replaced by A51 prove (ii) with c3 = leg/(AW).

Remark 7.17 Recall that ”Ag—2117“ g ell/1317“ for any if E H (since ”(Ag/1&4“ _<_
c). Suppose that W'PI/P is a doubling matrix of order p’, 1 < p’ < 00, with the

doubling exponent fi‘ (instead of the assumption that W is doubling of order p).

Then

”Ag-21,1711)” S c IIASMfly) II S ckI(1+ 2"]:1: - $Q.ml)fi'/p'llA3,mf(3/) II,

(where [cf 2 2(’\_”)"/p'x{,\>,,} + 2("_A)(B"")/pIX{ASV}, i.e., 131 with 5 replaced by 5"
and p by p’) holds instead of (7.24). Choosing M > B*/p’+n in the previous lemma,

we get

(”2) Hép * f ”LP({A51},A) S CU (Cl)A—V(CQ)#—V ”f “LP({AZ§},V) ’ 1 < p < 00'

101

Remark 7.18 A similar convolution estimate can be proved for LP(W) spaces, 1 <

p<oo:

“‘1’ * 1;”me S C HfHLP(W)- (7.26)

Recall that if (I) were to be a Caldero’n-Zygmund singular kernel K, then HK *
flle(w) S cllflle(w) if W E A,, (see [NT], [TVI], [V]). Conversely, if {7.26)

holds for every <1) 6 S , then W E A,, is necessary (see the scalar case below).

Before we show the necessity of the A,, condition, we demonstrate that, for exam-

ple, having just a doubling weight is not sufficient for (7.26).

Example 7.19 Suppose w is a doubling scalar measure such that there exist E C
[0,1] with 0 7Q IE] 2 a < 1 but w(E) = 0. Such a measure exists (cf. [St2] or [FM]).

Let (,0 = X[—1,1]- Choose f = XE- Then ||fI|Lp(w) = 0.

Now (go * f)($)=f[x_1,x+1]XE(t)dt=[[113 — 1,313 + 1] D E]. So if x 6 [0,1], then
(90* f)(x) = IE] 2 a and ifx é [—1,2], then (90* f)(x) = 0.

Thus, ||w*f]|ip(w) = f[-l,0]u[1,2] |(cp*f)(x)|pw(x) d$+j]0,1]\E osz(x) dx > 0, since

the second term is for sure positive.

Proposition 7.20 Let w be a scalar weight and 1 < p < 00. Suppose that for every

(I) E S the inequality

“‘1’ * f Huh») 5 Cd» Hf HLPIw) (7-27)

holds for any f E Lp(w). Then

</)(/)

102

holds for any interval of side length l(I) S 64., where Cd. is determined by Q.

Remark 7.21 Observe that if {7.28) holds for any interval, then w E A,,.

PROOF OF PROPOSITION 7.20. Choose Q E 8 such that Q(x) 2 1 for x E [—2, 2].

Take f 2 0. Then (Q*f)(x) Z/ f(y)dy for x 6 [—2,2].
[x—2,x+2]
Consider 11,12 6 ’D such that l(I,) = 1, i = 1,2, and 12 is right adjacent to
11. Let f = My Then for x E 12, we have (Q * f)(x) 2/ X1,(y)dy =1,
[x—2,x+2]

w(y)dv- A180 llfll’ipm = / w(y)dy- By (7-27) we get

11

and so II<I> . fun...) 2 /

12

/ w(y) dy g (C(p)” / w(y) dy. By symmetry (suppose x E 11), we get
[2 I]

12

/ w(yIczysICQIP / w(y)dy- (7.29)
11

Note that the above two inequalities say that w is at least doubling.

Next, let f(y) 2 w‘PI/P(y) X11(y). Then for x E 12, we have (Q * f)(x) =

p
/ w‘p'/”(v)dv. and so ll‘IP * fH'pr) 2 / w(y)dv (/ w”’”“’(v)dv) - A180
12 [1

11

llfllin.) = / w-P'/P(y> dy. Again by (7.27),

/12 w(y) dy ([11 w’p'/”(v) day S (CQ)P f1. w—P’/P(y) dy,

Substituting (7.29) and simplifying, we get

I

([11 w(y) dy) (A w—p'/p(y) dy) W S (0402”,

which is the scalar A,, condition for intervals of side length 1 (since 11 was arbitrary).

Now, let l(Il) = [(12) = 2‘”, V Z 0. Repeating the same argument as above

w(y)dy _<_ (2"C¢)” / w(y)dy- US-

with f = X11 and using symmetry, we get /
12

11

:0
ing f = w‘l”/")(11 again as before, we obtain /w(y)dy (/ uI‘P'/p(y)dy) g
[2 I]

103

(2" C4,)” / w—p'/p(y) dy, and thus,

h
1 1 ’ P/P'
(- / w(y)dy) (—/ w’“”(y)dy) S (04,)”,
Nil 11 Hi] 11

which gives (7.28) for intervals of side length 5 1 (0.1> = 1 in this case). I

7 .6 Duality of continuous Besov spaces

Now we shift our attention to continuous Besov spaces and our task is to construct

]B§q({AQ})] It and eventually ]B§q(W)] *.

Lemma 7.22 Let {AQ}Q be a doubling matrix sequence of order p, 1 _<_ p < 00. Let
OER and0<q<oo. Then

B;PP’I{AQ}) ; [BsqIIAQD] (7.30)

 

PROOF. Take gong E A with the mutual property (2.1). Let w(x) = w(—x).

Note that 113(5) 2 w(f). Let f6 B§q({AQ}) and g’ E B;°¢({A51}). First consider

80 = {f E 8 : 0 ¢ supp f} a dense subspace of B§q({AQ}) (see Appendix) and take

—9

fwith (f), E 80, i=1,...,m (and g with (g‘), E 8'). Then

§=Z§*(w*¢7u), Z(¢.*z&.>‘(§>=1, by (2.1).

VEZ VEZ

and

Q-

g(f) = 2 (§* (awn) If) = Zen (9% mt!)

V62 V62

104

= Z Z L(AQA51(§* sol/)(1"), (f* W(t))“ dx

VEZ QEQU

3 Z Z f"||A51(§*ru)(x)lluIIAQ(f*wV)(x)lluxo(x)dx,

VEZ QEQV

by the self-adjointness of each AQ and the Cauchy-Schwarz inequality. Using Holder’s

inequality several times, we obtain

 

 

Ib‘II’II :22” Ifw.)

V62

3 H {zWIIIfl ¢’u)llLP({AQ}»V)},,

 

 

'Zw 4* u , -1 7.31
Lp({AQ},l/) ”(g ‘P )“LP ({AQ }W) ( )

if 1<q<oo,

 

 

 

 

 

 

{2“’"|l(§ * Wullle'({A51}'”)} u

(q (q’ ’

and if 0 < q s 1, we bound (7.31) by

]{2“’°‘Il(§* so.IIIQlQ.,-,x},.)},]]

 

 

 

]] {2"°ll(fa* ¢V)]]LP({AQ}IV)}U

I1 [oo

 

 

 

{2"’°ll(§* Wulllxp’({A5‘}.u)},,

 

 

 

3 || {2"“ll(f * AVIIIWQM}, ,, ...-

Combining cases and using the independence of B;q({AQ}) and BQOQIHAEZID on

the choice of the admissible kernel if {AQ}Q is doubling (Corollary 7.10), we get

|§(f )l S ||f||3$P({AQ}) ||§||B;.,,({A51}y

Since 80 is dense in B§q({AQ}), we get the above inequality for any f E 839({AQ}).
Thus, 9' e Bgiaq'HAQll) belongs to [ngIIAbv] and llé‘lloper s ll§||370.'({,,51},-

Lemma 7.23 Let a 6 1R, 1 _<_ p < oo, 0 < q < 00 and {AQ}Q be a doubling

sequence of order p Then

[ngliAQlly g Bria/451D. (7.32)

105

PROOF. Let l E [ng({AQ})]*. We show that there exists 9‘ E BQQQ’HACSID
such that If I = g(f I = If: a) for any I’e BzPIIAQII.

Case 1 S q < 00. Take f6 B§q({AQ}), and for any V E Z denote f; = f* «pp.
Set T by T({f:,},,) = l(f), so T is defined on a subspace of [g(Lp({AQ},I/)) Since I

is bounded, so is T:

|T({fn}u)l = |l(f)l S c llfllggan» = CllifilulllngG/‘own

Extend T, denote the extension T, onto all of lg(Lp({AQ},V)) (note: q 2 1).
Since [lq(X)]* z lq'(X*) (refer to Section 7.2 or [D, Ch.8]), [lg(Lp({AQ},V))]‘
x Q" ([LP({AQ},V)]') z 1,;“(LP'({A51},V)) by (7.23). Thus, there exists a vector-
valued sequence {91,},62 E 1;,“(LP’({A51},V)) such that ||{g’V}V€Z]|,;Q(LPI(”51L”) g

”I“ and for any fe ng({AQ})

III’ I = TIIILII = TIIfLII = {asIIfZlII = 2: {:(g (fl(x).§u(x))H db

VEZ

= 23/" ((f*<.0u)(:v ).9’ da- — Z/(II maIInI), dx.

V62 V62

--0

Define g(x) = Z(§Q * ¢Q)(x). Then l(f) 2 (fig). Moreover, for any w E A (by

VEZ
LVN/15041)}

Corollary 7.10)
{2 ”fit! * 95V * wpllbp’({A51},p)}
p.

 

 

:guwwwn

VEZ

 

 

ll 1.1-'0

IIgIIQ .. (0,, 1%}, {
VEZ

 

 

 

 

 

 

n+1
{ 2: “Six * 85:; * wflllLP'({Aal},u)}

u=p-l

 

p

106

since supp 1b,, Q {6: 2’“1 3 lg] g 2"“} and so pun/1,, = 0 if lu—Vl > 1. Reindexing

the inner sum, we get

1
H glqu-aql({Ac-?l}) S C 2 2—yaq Z i|§#+J * $5" * ¢fl+ji|:p’({A51},#)-
p, #62 j=-l

Since {AQ}Q is doubling and sum on j is finite, we apply Lemma 7.16 (ii) to get

2-Pag' .-. }l] 31.
{ II.IIQQ.,,,,., II II

 

 

.4 ._a, _ <:CI
Hg ”8?, q ({AQ1}) _

-0

Case 0 < q < 1. Take f with (f), E 80. Since (,0 E 80, for V E Z by definition of

convolution and then boundedness of l , we have

|(l .. saVIIf‘ II = IlIf' . IaVII s lllll IIf' . bullagnQAQ}, (7.33)

Note that each component of l * 99,, is a C°°-function and also I] f * 95,,” 33°({AQ}) S
V+1
2"“ Z Hf *QQIILP({AQ},H) g c2"°]|f lle({AQ},Q) by Lemma 7.16 (i). Substituting this
p=V—l
estimate into (7.33), we get |(l * gay)(f)| g c2"°‘||l|] ]]f]]Lp({AQ},u). By duality,

 

sup |(l . aw II

- s S 6 Hill,
fESo llf IILPIIAQ}.uI

2-1/0]” * <Pulle’({Ac—?1},Q) = 2

i.e., the functional l =I< 90,, can be associated with a function g), E HIM/151}, V) such
that 2"”"- ||gV||L,,({A51},U, S c ||l||. Let g‘ = 21/6291, *6”, where 0 is as in the atomic
decomposition theorem [FJW, Lemma 5.12], which implies g’ = Zuez lgbuéu = l - 1

and so 9 = l. Observe that g E B;°°°({A51}):

HQHBQWMAQ‘D : $212) 2”’“Hg * walling/45%) S C III”-

Thus, the functional l E [339({AQ})]‘ can be associated with g E B;°‘°°({A[21})
and l(f) = (f, g). This completes the proof. I

107

Summarizing the results of the previous two sections we get the following embed-

dings of B -spaces:

Corollary 7.24 Let W be a matrix weight and {AQ}Q its reducing operators. Let
aElR, 0<q<oo,1§p<oo. Then

. *(1) . nI2I._a, _ (3)._a,
[BsqIWI] g ]B£q({AQ})] g B. PIIAQPII ; B. PIIAQPII
(é) B—PP' w-P’/P 7 34)
_ p, ( ). ( -
where
(1) holds if W is doubling of order p, 1 < p < oo,
(2) holds if W is doubling of order p, 1 g p < oo,
(3) holds ifW E Ap,1<p< oo,
(4) holds if W‘Pl/p is doubling of order p’, 1 < p < 00.
Also,
.aq . (1‘) .0q . (2‘) ._aq. _, (3') ._a,,. #
[8,. (W)] 2 [8. ({AQ})] 2 8. IIAQ }) 2 B. ({AQII
(4*) ._a . ,
2 3,, PIW—P/PI, (7.35)
where
(1") holds if W satisfies any of (AU-(A3),

(2") holds if W is doubling of order p, 1 S p < oo,

(3‘ ) holds for any matrix weight W,

108

(4*) holds if W‘PI/p satisfies any of (AU-(A3).
In terms of a matrix weight W only, (7.34) and (7.35) are
]B§q(W)]* g B;°"'(W—P'/P) if W e A,,,1 < p < 00
and
[Wm] 2 BgPP’Iw-WP) if W, W‘WP satisfy any of (A1)-(A3).

In particular, if W E A,, (and so W‘PI/P E A,,:), then [334(W)] z
B;°¢(W"p'/P), otherwise, (W still satisfies any of (A1)-(A3), or otherwise there
is a dependance on go) ]B§q(W)]‘ z Tamra/151D, which completes the proof of

Theorem 1.15.

7 .7 Application of Duality

In this section we will study T ,2 and S; and we will briefly show how duality can be
used to prove boundedness of operators T.) and SW Recall that T,), : b:q(W) ——>
B§q(W) by
S = {gale *—* Z 50 1/1Q-
Q
Moreover, T1), is bounded if W is a doubling matrix of order p.

Let 5Q e bgq(W) and g e ngIW). Then

Therefore, T J = 5,], and, similarly, S; = T 9p. So we have

I:

11;; [ngIW)] __, [bngWI]
or, another words,
sQ : qu'Iw-WP) —> 15;,PP'(W-P’/P),

and so S1), is bounded if W E A,,. Reformulating this by changing indices, we get

that under the A,, condition the following operators are bounded:
T; : [BgPP'IW'PVO] ——> [6;P4'(W“P'/P)]

01‘

s,,: ngIW) ——+ 630W).

(This is another proof of Theorem 1.4.)

110

CHAPTER 8

Inhomogeneous Besov Spaces

8. 1 Norm equivalence

In this section we discuss the inhomogeneous spaces. Before we define the vector-
valued inhomogeneous Besov space B§q(W) with matrix weight W, we introduce a
class of functions A“) with properties similar to those of an admissible kernel: we say

QEAU) ifQES(R"),supp QQ KER": |(l 32} and |Q(§)| Zc>0 if |(l g 3.

Definition 8.1 (Inhomogeneous matrix-weighted Besov space B§q(W))
ForaElR,1$p<oo,0<qSoo, W amatrixweight, IpEA andQEAU),

we define the Besov space B:q(W) as the collection of all vector-valued distributions

—0

f = (f1....,fm)T with f.- e S’IRP),1sz‘g m, such that

{2...

Note that now we consider all vector-valued distributions in S’(R") (rather than

 

 

90V*f <00.

 

 

 

 

=]]Q*f

 

 

+
LP(W)

 

 

 

 

 

 

WWW) LP(W)}u31

[q

8’/’P as in the homogeneous case), since Q(0) ;£ 0.

111

The corresponding inhomogeneous weighted sequence Besov space b;q(W) is de-

fined for the vector sequences enumerated by the dyadic cubes Q with I (Q) S 1.

Definition 8.2 (Inhomogeneous weighted sequence Besov space bgq(W))
For a E IR, 1 S p < oo, 0 < q S 00 and W a matrix weight, the space b$q(W)

consists of all vector-valued sequences 5 = {§Q}1(Q)51 such that

" V ..l_.
llsllb$q(W)= 2" Z IQI 23QXQ <oo.

l(Q)=2—u Lp(w) ”>1 (q

Following [FJ2], given Ip E A and Q E A“), we select 1/) E A and ‘11 E A“) such

that

5(a) - @(6) + Ibo-Pa) - Iva-Pa) =1 for an 6. (81)

V21

where Q(x) = Q(—x). Analogously to the g()-transform decomposition (2.2), we have

the identity for f E S’(R")

f= 2(f.‘1>c2)‘1’o+: Z ImaIya, (8.2)

l(Q)=1 ”=1 l(Q)=2""
where QQ(x) = |Q|_1/2Q(2"x — k) for Q = Quk and QC; is defined similarly.

For each f with f,- E S’(R") we define the inhomogeneous go-transform Sh!) :
ng(W) —> ngIW) by setting (Sg’lf'IQ = (f? eQ) if [(62) < 1, and (512,”)? )Q =

(flQQ> if l(Q)=1.

)

The inverse inhomogeneous Ip-transform T1]! is the map taking a sequence

3 = {SQ}I(Q)31 to Tins = Z sQ\IlQ + Z: sQwQ. In the vector case, T]Ӥ =
l(Q)=1 l(Q)<1
.. _. (I) III - - . , n
X sQ‘IJQ + Z SQ’l/JQ. By (8.2), Tw OSQ IS the 1dent1ty on 8 (R ).
l(Q)=1 l(Q)<l

112

Next we show that the relation between B§q(W) and bgq(W) is the same as for

the homogeneous spaces.

Theorem 8.3 Let a E R, 0 < q S 00, 1 g p < 00, and let W satisfy any of

(AU—(AS). Then

I" (8.3)

 

 

 

 

l
N

 

{.., (f) LIQQ

Before we outline the proof we need to adjust the notation of smooth molecules for

 

 

33”“ b.‘I‘PIWI .
the inhomogeneous case. Define a family of smooth molecules {mQ}l(Q)gl for ng(W)

as a collection of functions with the properties:

1. for dyadic Q with l(Q) < 1, each mQ is a smooth (5, AI, N)-molecule with

(M.i)-(M.iii) as for the homogeneous space B§q(W) (see Section 4.1);

2. for dyadic Q with l(Q) = 1, each mQ (sometimes we denote it as MQ to
emphasize the difference) satisfies (M3), (M4) and a modification of (M2) (which

makes it a particular case of (M3) when 7 = 0):

—M
(M?) ImaIaII s IQI-P/P (1+ Bale?) .

Note that MQ does not necessarily have vanishing moments. Now one direction of

the norm equivalence (8.3) comes from the modified version of Theorem 4.2:

Theorem 8.4 Let a E 1R, 1 S p < oo, 0 < q 3 00 and W be a doubling matrix
weight of order p. Suppose {mQ}1(Q)Sl is a family of smooth molecules for B:q(W).

Then

2: 50 m0 3 C llng}l(Q)31|lb$q(W)- (8.4)
l(Q)Sl BSQW,

113

 

 

SKETCH OF THE PROOF. We have

2 ngQ = Z §Q(<1>an)
l(Q)g1 33°(W) l(Q)Sl Lp(w)
+ 2... : §Q(sou*mc2) =I+II-
l(Q)Sl LP(W) ”>1 (q

 

As in Theorem 4.2, which uses the convolution estimates (4.2) and (4.3), we need

similar inequalities for modified molecules (the proofs are routine applications of

Lemmas BI and B2 from [FJ2]):
if l(Q) = 1, then
I<I> . MaIaII s c (1 + In — aaII‘M, (8.5)
if l(Q) = 2"“, u 2 1, then for some a > J — a
I<I> .. maIaII s lel‘iT’” (1 + In - aaII‘M, (8.6)
if V 21 and l(Q) = 1, then for some 7 > a
|%*MQ($)| _<_ 02“”(1+|$-$Q|)-M. (8-7)

if V Z 1 and l(Q) < 1, the estimate of |(I,0,, =1: mQ)(x)| comes from either (4.2)

or (4.3).

To estimate I we use (8.5) and (8.6) (note that (8.5) is a special case of (8.6) for

u = 0) and follow the steps of Theorem 1.10 by using Holder’s inequality twice to

114

bring the pth power inside of the sum, and the Squeeze and the Summation Lemmas

from Section 4.1 (it is essential that o > J — a for convergence purposes) to get

1 S Cl|{§Q}IIQI31Hbgq(w>-

The second term I I is also estimated by llng}l(Q)Slllb$q(W)v which is obtained by
exact repetition of the proof of Theorem 4.2, only restricting the sum over u E Z to
the sum over ,u 2 0. Also note that (8.7) is a particular case of (4.3) when a = 0

and, thus, l(Q) = 1. Therefore, (8.4) is proved. I

In particular, since Q and \II generate families of smooth molecules for B:"(W),

we get

f

 

 

 

 

 

 

 

 

{Po (0 IQ.

which gives one direction of the norm equivalence (8.3). To show the other direction,

S c .
ngU/V) ng(W)
i.e., that the (inhomogeneous) go-transform is bounded, we simply observe that Q* f E
.. _. A 2
E0, which is true since (Q * f) E 8' and supp Q Q {g E 1R" : |{l g 2}. Hence,

Lemmas 4.12 and 4.14 apply to g = Q * f as stated. We have

] {aQ (i) }“mg 2 (<1 . f) (k) ya...

kEZ"
+ : IaI—aQInas

z(Q)=2-v WW)

~
~

bx‘i‘qIW)

 

 

 

 

 

 

 

LP(W)

V21 [q

Using Q * f E E0 and repeating the proof of Theorem 4.15 for both terms (in the

second term we take the [‘7 norm only over V E N), we get the desired estimate:

I {.., (I) IQ. f"

g c
bIi'n"’(W)

33"(W)

 

 

 

 

 

 

 

115

Note that as a consequence we also get independence of B;q(W) from the choices of

Q and 99.

8.2 Almost diagonality and Calderén—Zygmund

Operators

Now we will briefly discuss operators on the inhomogeneous spaces. An almost diag-
onal matrix on b;q(W) is the matrix A = (aQP)I(Q)y(p)Sl whose entries satisfy (5.1),
i.e., lan| is bounded by (5.1) only for dyadic Q,P with l(Q),l(P) S 1. Such a
matrix A is a bounded operator on bgq(W) for the following reasons: let s‘ E b3“? (W)
and then define g: {523}er by setting 5Q = sq if l(Q) S 1 and SQ = 0 if l(Q) > 1.
Note that s' is a restriction of E on bgq(W). Also set A = (ClQp)Q,pE’D putting
an = an if l(Q),l(P) _<_ 1 and nQp = 0 otherwise. Then

llAgllb$P(W) = Z aQPgP

1(P)Sl I(Q)S1 bg"(W)

 

 

 

: E aQPEP S C I; .0q 1
. bp (W)
P dyadic Q 53" (W')
by Theorem 1.10. By the construction, llglliguw) = llgllbgnwp and so we get bound-

edness of A on b;q(W).

It is easy to see that the class of almost diagonal matrices on bgq(W) is closed
under composition. The same statements (boundedness and being closed under com-
position) are true for the corresponding almost diagonal operators on B§q(W) by

combining the norm equivalence (8.3) and the above results about almost diagonal

116

matrices on b;q(W). For Calderon-Zygmund operators on inhomogeneous matrix-
weighted Besov spaces, some minor notational changes should be made. The collec-
tion of smooth N -atoms {0Q}QE’D in the homogeneous case ought to be replaced by
the set of atoms {aQ}1(Q)<1 U {AQ}1(Q):1, where the aQ ’s have the same properties as
before and the AQ’s are such that supp AQ g 3Q and IDIAQ(x)| g 1 for '7 E Z1.
This leads to a slight change of the smooth atomic decomposition (see [FJ 2, p. 132]):

f= 2 $000+ 2 30/10-

I(C2)<1 l(Q)=1

With these adjustments, all corresponding statements about CZOs hold with essen-

tially the same formulations for the inhomogeneous spaces.

8.3 Duality

Let RS”) be the collection of all sequences {AQ}I(Q)SI of positive-definite operators

on H. Similar to the homogeneous case, we introduce the averaging space b:q({AQ}).

Definition 8.5 (Inhomogeneous averaging matrix-weighted sequence
Besov space b;q({AQ}).) For a E R, 0 < q 3 oo, 1 S p 3 00 and

{AQ}I(Q)SI E 735”); let
b$q({AQ}) = {5: {{§Q}IIQ)51} 3

_. Q _1_.
IIsIIQaQAQ,,= 2° 2 IQI PSQXQ <oo

l(Q)=2‘” LP({AQ}’”) 120 [Q

117

Let s'E b$q(W). Define E: {£53}er as in the previous section. Applying (1.10),

we get

—0
I

N

N

 

 

 

 

 

 

“gllbgnm Z ]]3 = llglleQM/‘QD’

53°(W) 53"({AQ})

which proves the following proposition.

Proposition 8.6 Let a E R, 1 g p < oo, 0 < q 3 00 and let W be a matrix weight

with reducing operators {AQ}Q. Then
1):“le (ATM/10D,
in the sense of the norm equivalence.

Note that it’s enough to consider reducing operators AQ generated by a matrix weight
W only for dyadic cubes of side length l(Q) g 1, i.e., {AQ}I(Q)§1-

Now we establish the duality.

Theorem 8.7 Let a E R, 1 S p < oo, 0 < q < 00 and let W be a matrix weight

with reducing operators {AQ},(Q)51. Then
[bSPIWIT a 12;,” ({AQ‘D-
Moreover, if W E A,,, 1 < p < 00, then
]b:q(W)]* z bglaq (W—p’ho).
To prove this theorem, one can simply repeat the arguments from Section 7.2

with proper adjustments (for example, consider sums on V taken only over V Z 0).

However, we would like to give a simple proof for the embedding

[bSPIIAQDI‘ g bgPP'IIAQ‘II-

118

PROOF. Let I E ]b;q({AQ})]'. Let P be the projection from b§q({AQ}) to bgq({AQ})
defined by restricting a sequence {5362}er to {.§'Q}I(Q)51. Set lby l(§) = l(Ps') for

each s'E b$q({AQ}). Then lE ]b;q({AQ})]*, since

IIIa'II = IIIP§)I S Hill IIP§IIQ;«({AQ}) 3 Hill IIa‘IIQnQQQ

Then by Lemma 7.3 (or, equivalently, by (7.6)), l is represented by t: E bgaql({AC—21})

such that [(5') (by?) and g ”in 3 Hill. Let i‘: Pf. For g e

”t iib;09'({A51})
b$q({AQ}) define SE b$q({AQ}) as above (i.e., Pg: 5). Then

l(§)=i(§) = (52):; Z SQtQ+ Z SQ5Q= Z SQ§Q=

l(Q)S1 l(Q)>1 l(Q)S1

since SQ = 0 for l(Q) > 1. Moreover, Ilt-‘IIFIQQ (52M 1})_ S Ht llb.’QQI({A61}) S ”III. I
P

Analogously, we introduce the averaging space ng({AQ}).

Definition 8.8 (Averaging matrix-weighted Besov space B§q({AQ})) For

aER,0<qSoo,1SpSoo,IpEA, QEA”) and {AQ}[(Q)S] E728”), let

B:PII={Aa} {f=..If.,..fmIP‘withf.es'IRPI,1313m:

{2” < 00}.
[q

Now the remaining results from Sections 7.3, 7.4 and 7.6 transfer easily to the

=]]¢*f 9011*}?

 

 

 

 

 

 

 

 

 

L"({AQ}.0) ] Lp({AQ}'V)}1/Zl

 

 

 

 

ngflf‘ol)
inhomogeneous Besov spaces by using the properties (discussed earlier in this section)
such as replacing a family {I’m/Lyez with {90V}VEN UQ; observing that Q * f E E0 and

summing over V 2 0 (or I (Q) S 1) in all sums. In particular, we get

119

Theorem 8.9 Leta E IR, 0 < q S 00, 1 S p < 00 and let {AQ}1(Q)SI be a doubling

sequence (of order p). Then for §q(f) = <fy900>

If {PA (I) IQ.

Corollary 8.10 The spaces B:q({AQ}), B§q({A51}) and Bgaq’({A51}) are inde-

 

 

 

 

 

 

Biq‘{"0}) ] b$q({AQ}).

pendent of the choice of the pair of admissible kernels (Ip, Q), if {AQ}1(Q)SI is doubling

(oforderp),1Sp<oo,aElR,0<qSoo.

Lemma 8.11 Let a E IR, 0 < q S 00 and 1 S p < 00. If W satisfies any of

(A 1)-(A3) and {AQ}Z(Q)S1 is a sequence of reducing operators generated by W, then
B§q(W) % B§q({AQ})-

Theorem 8.12 Let a E IR, 0 < q < oo, 1 S p < 00 and let {AQ},(Q)51 be reducing

operators of a matrix weight W. If W E A,,, 1 < p < 00, then
[B;P(W)]* z BgPPIW—P/P).

If W satisfies any of (AU-(A3), then
[BSIIWIIP 2 B;“"({A51})-

Thus, all results obtained for the matrix-weighted homogeneous Besov spaces are

essentially the same for the inhomogeneous case.

120

CHAPTER 9

Weighted Triebel-Lizorkin Spaces

9. 1 Motivation

The study in this chapter was stimulated by the question posed by A. Volberg in [V].

He proved that if W E A,,, 1 < p < 00, then the following equivalence takes place

{so (f) L. {.., 0)}.

where sQ(f) = <f, hQ> with {hQ}Q being the well-known Haar system and {AQ}Q

(9.1)

 

 

 

 

 

 

 

~
I

f32(W) f3’({AQ}) ’
the reducing operators for W. Moreover, he pointed out that the equivalence does not
necessarily require W E A,,. For example, it holds always for p = 2. He conjectured
that for p Z 2, the condition on the metric p generated by W, which is similar to a
scalar A00 condition, p E A,,,co might be sufficient. The criterion for (9.1) was asked.

In the light of our studies of function spaces, we rephrase (and partially answer)

the question of Volberg in familiar terms: what conditions on W are needed for the

121

 

equivalence below to hold?

Our main result of this chapter deals with scalar weights and the matrix case is

(9.2)

 

 

 

 

 

~
~

 

{ngq {EQ}Q

LEW) 'SQ({AQ})
left for future research. We Show in Theorem 9.3 that if a scalar weight w E A00, then
(9.2) holds for a E IR, 0 < q S 00 and O < p < 00. Furthermore, by using the result of
Frazier and Jawerth (see [FJ 2, Proposition 1014]), we connect the reducing operators I

sequence space f:q({wQ}) with the continuous Triebel-Lizorkin space F;q(w), and

 

therefore, obtain the following norm equivalence L

IIfIIF:q(w) R1 II{<fi SOQ>}QIIf:q({wq})'

9.2 Equivalence of f:q(w) and f§q({wQ})

Before we prove the main result, we establish two lemmas for the weighted and
unweighted maximal functions. Denote wQ 2 I327 fQ w(x) dx.

Lemma 9.1 Let EQ = {:L‘ E Q: w(x) S 2wQ}. If w E A00, then Mw(XEQ) Z c XQ-

PROOF. Using the definition of the maximal function, for a: E Q we have

 

_ __1_ _1__ _ w(EQ)
Mama) — 5:16pm”) [lawman 2 W) [Q XEQ<y>wa>dy — w(Q) .
l3
The condition w E A00 implies that M Z c (Ll—Bil) for some fl > 0. Since
w(Q) IQI

EQ = {:13 E Q : w(x) S 2wQ}, the compliment E5 = {as E Q : w(x) > ZwQ}. This

gives us the following chain of inequalities:

w(Q) = wi(:c)da: 2 f. w($)d:1: > Le 2wQ d1: = 2-leEal.

Q Q

122

In other words, IQIwQ = w(Q) > 2wQ|Ef9|, or |Q| > 2|Egl, which implies IEQI >

élQl. Hence, for :1: E Q we have

Mw(Xz-:Q)($) 2 c (-)B = 6' XQ(~’13),

which finishes the proof. I

Lemma 9.2 There exists 0 < 6 < 1 such that if EQ = {x E Q : w(x) 2 6wQ} and

w E A00, then M(XEQ) 2 cXQ.

PROOF. The proof goes similarly to the proof of the first lemma except we will

apply the A00 condition in a slightly different way. If a: E Q, then

_I_EQI
MIXEQ“ :fEEIlII/XEQW ”dy-IQI/XEQW ”dy—— IQI

Considering the compliment of EQ, we have E5 = {27 E Q : w(x) < 6 1052}. Then,

w(EfQ) = /C w(x)d:z: <

Q

6de$S6le dxzwaIQl =6w(Q).
Q

322

 

So w(Ea) S 6 w(Q) implies w(EQ)_>_ (1 —6) w(Q). Since 211 E A()(, and

E
IEQ|<eorl Q|>1

IQI _ IQI —

                            

 

 

— 6. Hence, for :r E Q we have

M(XEQ)($) 2 (1 - 6)XQ($)-

Theorem 9.3 Suppose w E A00 and let a E IR, 0 < q S 00 and 0 < p < 00. Then

 

HfSQ}Q|

fsquwon “ “{SQ}QHf{;“'(w)

123

Moreover, if f E F:q(w), then for sQ(f) = (f,goQ),
||f||pg2qu) *2 ||f8Q(f)}Q||j;,w({wQ})- (9-3)

PROOF. By definition,

l/q
II{3Q(f )lQIIj; Mm): (ZUQl—i—filSQIWXQY’)

Q Lp

We need to show that the last norm is equivalent to

1/q
[{SQ.wg,/2}Q = (ZUQI 2 %I2QIwQ w)

Q
Let EQ = {x E Q: w(x) S 2wQ}. Choose A > 0 such that p/A > 1 and q/A > 1.

 

 

 

.aq
fp LP

l/A
By Lemma 9.1, X a: S c Mw(XA )(x) . Therefore,
Q EQ

p/q l/p
||{8Q}Qllj;2(w)= /[Z(|Q|_2_%ISQ|XQ($ W] w(iBNiB
Q
/A q p/q U”
:(f [2: (IQI-H IsQI(M Mason 22)) )] w(2Id2)
Q
/A p/q l/p
c/[ZX M..<-IQI 2 2I2QIXEQ(2 2)) )° I wanna)

A/q l/A

s c [2 (M..(IQI'%-%mummy/A]

Q LP/A (w)

Since 211 is a doubling measure and the weighted maximal function Mu, satisfies the
vector-valued maximal inequality (see [Stl] or [St2]), the last expression is bounded

above by
1 /A

       

A/q
c... [DIQI-H )0]
Q

LP/A(w)

124

1/p

p/q
= (/ [Ema-2‘2IsQIxEQ(2)wl/2(2>)2] 22)

Q
p/q l/p
S21/p0p.q (/ [ZUQI 2 "ISQle/p XEQ($ 113W] dill)
C?
p/q l/p
go / [20:06?! 2 2IsQIwQ W W] dz =2 {Swa/p}Q f3“

 

 

 

 

In the last inequality we used EQ Q Q.
For the opposite direction, set EQ = {2: E Q : w(x) Z é'wQ} and again choose

A > 0 such that p/A > 1 and q/A > 1. Then by Lemma 9.2,

1/p

p/q
/[Z(IQI 2 "ISQIwQ pHQ( )Iq] da:

C?

p/q l/p
Sc/Z( ZUQI‘HIsaIwQ/PMuiwa] dz)
Q

p/q V"
“(f [E (M( IQI 2 "ISQIwQ/ pHEQ( “WAY/A] d2:
<9
1 WI
gal/2c / ] (IQI—222‘2Isqu1/2<2>XEQ(2)>2] dz)
l/p

p/q
S C (/ [ZUQl‘i‘glsleQMW] WWW) = C I|{SQ}QIII,§‘2(w)'

It is easy to show the second assertion. By [FJ 1, Proposition 10.14]

1/p

IIfIIF:q(w) % II{3Q(f)}QIIf:q(w)

Combining this equivalence with the first result, we get (9.3). The proof is complete.

1125

CHAPTER 10

Open Questions

1. In the unweighted theory of function spaces, special cases of the Besov and
TriebeI-Lizorkin spaces are the Lebesgue spaces: e.g. L302 = L2 and F192 = D”,
1 < p < +00. In the scalar weighted situation it is known that 33201)) = L2(w)
and F£2(w) = Lp(w) if and only if 11) E A,,. In the matrix case it is expected that
the A,, condition is the minimal condition on W needed for this equivalence
to hold. Using vector-valued square function operators might be one of the

approaches to this problem.

2. The crucial step for our theory of Besov spaces is the norm equivalence between
continuous B§q(W) and discrete b3"? (W) spaces. We were able to Show that it
holds for any doubling matrix weights if the order p is greater than the doubling
exponent of W. In the special case of diagonal matrices (equivalently, in the
scalar case) this restriction is removed. A conjecture is that the equivalence

holds for any doubling matrix weight W.

126

3. One major goal is to answer the same “equivalence of norms” questions for
the matrix-weighted 'IriebeI-Lizorkin spaces Flf‘q(W) and f:q(W). The scalar
weighted case is known (see [F J2], [FJW]) and the matrix-weighted case is to
be studied. This will also lead to the question of the boundedness of singular
integral operators on F:q(W). Possible approaches include a variation of the
exponential type estimates used in the Besov space case or Volberg’s factoriza-

tion method mentioned in the introduction.

4. All previous research was done on function spaces with the parameter p being
between 1 and co, quite often not including the end point p = 1, which requires
more careful consideration. Moreover, this raises the question of matrix “A”
weights and their factorization, and whether this can be developed further into
an extrapolation theory of matrix-weighted distribution spaces (similar to Rubio
de Francia results in scalar case). Furthermore, it would be interesting to study
a case when 0 < p < 1. Nothing is known in this area, except for certain scalar

cases .

5. The motivation for the norm equivalence studied above came from the fact
that IILIILP(W) z ||{< f,h1 >}IIIj32(W)2 where {h1}1 is a Haar system and
{< f, h, >}1 constitutes a sequence of the Haar coefficients of f. Recall that
we obtained the norm equivalence when the generators of the expansion (either
the p-transform or wavelet functions) have some degree of smoothness. This

property is lacking for the Haar system. Nevertheless, the Haar system is widely

127

used in applications. This creates the open question (for ceirtain indices 0:, q,
p) of the norm equivalence between continuous and discrete function spaces

with the Haar coefficients.

. A very difficult problem of modern Fourier analysis is to obtain weighted norm
inequalities on the function spaces (at least on LP spaces) with different weights.
Complete answers to the scalar two-weight problem is known only for the Hardy-
Littlewood maximal function (by Muckenhoupt and Wheeden in [MW] and by
Sawyer in [821]). For the Hilbert Transform the necessary condition is given by
Muckenhoupt and Wheeden and the dyadic version is studied by Nazarov, Treil
and Volberg in [NTV]. Furthermore, the necessary and sufficient conditions for
the case p = 2 are obtained by Cotlar and Sadosky in [CS]. Since the Lebesgue
spaces are special cases of Besov and Triebel-Lizorkin function spaces, the same
two—weight questions should be asked in the light of Littlewood-Paley theory.
The hope is to consider at least the scalar case and to obtain the conditions
on the two weights for the boundedness of the cp-transform, almost diagonal
operators, maximal function operators (such as Peetre’s maximal operator), the

Hilbert Transform and possibly other singular integral operators.

128

Appendix A

Density and convergence

Define 80 = {f E S : 0 E supp f}. Observe that 2b E A implies ¢,i,b,,,z/)Q E 50 for

z/ E Z and Q dyadic.

Lemma A.1 Let f E 80. Then fN :2 Z @V *ibu * f = 2 (f, 90ka) ibQuk con-
IVISN IVISN
verges to f in the S -topology as N —+ 00.

PROOF. For 1/ e Z, define f(Q, = Q, *sz * f = Z < f, ,QQWQ. Then f(Q,(g) =
QEQu
$V(€)zr/3V(€)f(€) and <,5.,, dimf E 80 => f(y) E 80. Observe another fact: since f E 80,

there exists N0 E N large such that f = Z f(y). Indeed, O E supp f implies that

112—No
N

f(2T) = 0 if I513] S Z—N" for some No > 0. Thus, for large N, fN = Z f(y) =

112-No

VSN
To prove the lemma, it suffices to show that p.,(fN — f) —+ 0 as N ——> oo. Denote

N
mN(§) = 1— Z (high/3”“). Because of the support of «[9,, and 2b,, and the mutual

u=—No

property (2.1), mN(§) = 0 for TN" < |{l < 2N. Moreover, 0 S mN(€) S 1 for

129

2N < [g] < 2""+1 and mN(§) = 1 for |§| Z 2N“. Using these facts, we obtain

A A

mew-f): sup (1+IEI2)2|D°(m~f)(€)I

IEIZ2", IOIS’Y

5c sup (1+lél2)2 Z ID2‘m~(€)||D22f(€)l-

N
I£IZ2 ’IaIS7 71+7zza

Observe that |Dl2mN(§)| ~1/|§|hll if |§| ~ 2” and that f E 8 implies |Dl2f(§)| S

 

 

 

CL
(1+ |€IIL+I22I for any L > 0. Take L > 27. Then
. . (1 + [5])” CL
f —f Sc sup _<_ ‘—2 0-
"“ N ) "IeeemQIe—QQQ-..(1+I2I)I22I(1+I2I>L+I22I (1+2~)2-22 N222

Thus, fN N——> f in S-topology. I

Lemma A.2 Let f E 80 and fix V E Z. Then 2: (f, «pg/k) W2“. converges to

IkISM
fu = 45,, * 1% * f = Z<f,soQuk)1/quk in S-topology as M —> oo.
keZ"
PROOF. Denote fQ,M = E: (f, (prk) ¢Qek2 Obviously, fMM E 80. Then

kSM

p27(fu,M — fu) : 811p (1+ IxI2)7 IDa(fi/,M — fu)($)I

xER".|aIS‘7

= SUP (1+I$|2)7 DQZ(fI‘PQ>l/’Q($) -

36R",]0]S‘7 k>M

Observe that we can bring DC2 inside of the sum. (A similar argument as below proves

this claim.) Then

mum—ms sup (1+I2I)22Z|¢u*f(2‘”k)lID‘2(2I2(2":v-k))l

$ER",IOIS7 IkI>M
(A.1)
Choose L1 > 27 — [(1], L2 > 27 + n and L3 > max(L1,'7). Using properties of 8

functions, we have

CL1,C¢2VI0|
(1 + [2222: — k[)122+|a|'

 

ID“ (212(2'2 — W = 22"" I<D°Q> (2'22: — I2)I s

130

‘ _EI

 

Applying the convolution estimates (4.2) and (4.3), we bound (4,5,, * f)(k):

~ CL2 2~IVIL3
'V k < _—_.

Substituting the above estimates into (A.1), we obtain

CL1,LQ,Q (1 + [3])27 2VIaI-]V]L3
p7(fu,M _ fu) .<_ SUP (1+ IkDLz (1+ l2u$ _ k])L1+I0I

36R" ’IOIS1IkI>M

(1+ ””27 2VI0I—IVIL3
< c su
_ L1.L2,a (m61R'hlEIS7 (1+ 2u]$])L1+|a|

 

 

 

1
Z <1+I2I>22-22’

|k|>M

by using (1+2”|J:|) S (1+ |2":1:—k|) (1+Ikl). The supremum on :1: and a is bounded

by e., = 2“(7_L3)X{V20} + 2”(I‘3’L‘)X{,,<o}, since L1 > 27 — lal. Thus, we get

1

p7(fu,M — fu) S 61.1.1.2,7 Cu ME)?” (1 + |k|)L2—27 ——> 0 as M —> oo

 

as a tail of a convergent series, since L2-27 > n. Thus, me M—> f” in S-topology.
—>oo

Remark A.3 If T is a continuous linear operator from S into S' and f E 50, then

Tf = Em. 2 w. 2 f) = Z Z (we...) me... = Z (we) Two-

VEZ VEZ kEZ" Q

Lemma A.4 SO is dense in B§q(W) for a E IR, 0 < q < 00, 1S p < 00 and if W

satisfies any of (AU-(A3).

N
PROOF. Let f6 B;Q(W). For N E N denote fN = Z Z <f, <pQ> T/JQ- Then
u=-N QEQu
by Corollary 4.6

‘1

gym): 2 23230.) SC

q

[If— f”

 

 

{2‘2 (2") }QQW

 

 

 

 

232(W)

131

 

 

 

 

2: z: ,Q,-Q,Q,, :22
|u|>N QEQu Lp(W)
as a tail of the convergent series
‘1
ZAV 3: Z 2V0 Z IQI—lflgQXQ : II{§Q}QIIZ;Q(W) (A2)
uEZ uEZ QEQV LP(W)

 

 

 

 

S CIIf IIESRW) < 009
if W satisfies any of (A1)-(A3) by Theorem 4.15.

As in the previous proposition for each V E Z and M E N define sz =

Z <f2cpquk>ibek E80 and recall fl: 2 (fich> ibQ. Note that f” = 2 fl).

kSM QeQu IVISN

 

 

 

 

 

 

Then
2— T. < c {.2 (8)] = c2"°‘ 2m/2s
l f f ,Ml ng(W) — Quk f IkIZM boq(w) Z kaXQuk
l/p
: ua imp/2 l/p -2 p
c2 2 2 [Q ||W (t)eQ,,II dt Mi; 0
IkIZM ”’2

again as a tail of the convergent series 2 2""‘0/2/ ||W1/P(t)§Q,,||Pdt =
IkIEZn Quiz
(Tm/1,1,”)? < 00, since each A,, < 00 (see (A.2)). Thus, each f E B§q(W) is a

limit (in B$q(W)-norm) of 80 functions and so 80 is a dense subset of ng(W).

I
Proposition A.5 SO is dense in 332({AQ}) if {AQ}Q is doubling of order p, 1 S
p<oo andaElR, O<q<oo.

PROOF. Repeat the previous proof with W replaced by {AQ}Q and refer to
Lemma 7.7 instead of Corollary 4.6 and to Lemma 7.5 instead of Theorem 4.15. Both

require {AQ}Q to be only doubling. I

132

BIBLIOGRAPHY

133

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