Weighted estimates of Calderón-Zygmund operators on vector ...

Weighted estimates of Calderón-Zygmund

operators on vector-valued function spaces

by

Amalia Culiuc

B. A., Mount Holyoke College; South Hadley, MA 2011

Sc. M., Brown University; Providence, RI, 2013

A dissertation submitted in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

in the Department of Mathematics at Brown University

Providence, Rhode Island

May 2016

© Copyright 2016 by Amalia Culiuc

This dissertation by Amalia Culiuc is accepted in its present form by

the Department of Mathematics as satisfying the dissertation requirement

for the degree of Doctor of Philosophy.

DateSergei Treil, Ph. D., Advisor

Recommended to the Graduate Council

DateJill Pipher, Ph. D., Reader

DateFrancesco Di Plinio, Ph. D., Reader

Approved by the Graduate Council

DatePeter Weber

Dean of the Graduate School

iii

Vita

Amalia Culiuc was born in Bucharest, Romania and attended Mount Holyoke Col-

lege, where she received a Bachelor’s degree in Mathematics and Economics, magna

cum laude, in 2011. She attended graduate school at Brown University, where she

received a Master of Science degree in mathematics in May 2013.

iv

Acknowledgements

I thank my advisor, Sergei Treil, for all his advice, help, and support, as well as for

suggesting these problems and being there for me at every step along the way. Many

thanks also to my committee members, Jill Pipher and Francesco Di Plinio for all

their feedback and their support, which extends far beyond this thesis.

To my academic brother and collaborator, Brett Wick, for his invaluable help, to

Kelly Bickel, for her patience, encouragement, and an excellent collaboration, and to

Fedja Nazarov, for some of the ideas that went into this thesis.

To Michael Lacey for just about everything. I couldn’t ask for a better mentor

during the next stage of my career.

To my academic sister, Hyun Kwon, for a conversation that changed my academic

path.

To my harmonic analysis family at Brown: Yumeng Ou, Tess Anderson, Jingguo

Lai, Armen Vagharshakyan, and, of course, Francesco Di Plinio. Thank you for

being my friends, my role models, and sources of inspiration. In particular, thank

you, Francesco and Yumeng, for being my closest supporters during this past year.

I will miss you more than words can say.

To my friends in the math department at Brown: Jackie Anderson, Kenny Ascher,

Shamil Asgarli, Alex Barron, Dori Bejleri, Paul Carter, Matt Cole, Sam Connolly,

v

Elizabeth Crites, Diana Davis, Brian Friedin, Victoria Gras Andreu, Alicia Harper,

Vivian Healey, Wade Hindes, Younghun Hong, Tom Hulse, Peihong Jiang, Semin

Kim, Seoyoung Kim, Chan Kuan, Mehmet Kiral, Nhat Le, Jonah Leshin, Li-Mei Lim,

David Lowry-Duda, Numann Malik, Igor Minevich, Sam Molcho, Dinakar Muthiah,

Peter McGrath, Edward Newkirk, Isaac Solomon, Ian Sprung, Minh-Hoang Tran,

Martin Ulirsch, Alex Walker, Dale Winter, Laura Walton, Ashley Weber, Elliot

Wells, Miles Wheeler, Wei Pin Wong, Sunny Xiao, and Ren Yi. Extra thanks to my

office mates, Laura and Dori, for making the office feel like a second home to me.

To Audrey Aguiar, Lori Nascimento, Doreen Pappas, Carol Oliveira, and Larry

Larrivee, the best staff a department could have.

To my students, who, just like me, often doubt themselves: Eren Alkan, Yokabed

Ashenafi, Katie Barry, Nik Baya, Eli Berkowitz, Chantel Brown, Ryan Burke, Kiara

Butrosoglu, Emma Byrne, Sally Cai, Valentina Cano, Chien Teng Chia, Crystal

Chen, Shirin Chen, Juan Colin, Matt Cooper, Emma Currier, Neville Dadina, Bran-

don Dale, Victor Dang, Joshua Daniel, Petros Dawit, Rachaell Diaz, Alex Djorno, Al-

bert Dong, Irene Du, Ercole Durini di Monza, Gloria Essien, Marimar Fletcher, Grant

Fong, Andrew Friedman, Meghan Friedmann, Johanna Garfinkel, Maddie Gaw, Leah

Goldman, Leonard Gleyzer, Aaron Gokaslan, Sam Greenberg, Evan Gross, Jack

Haworth, Phebe Hinman, Nicola Ho, Isiah Iniguez, Lucy Jia, Bailey Jones, Min

Jeong Kang, Emily Kasbohm, Nikki Kaufman, Anand Lalwani, Kaiwen Li, Rebecca

Li, Francesca Lim, Amy Lipman, Susan Liu, Jacinta Lomba, Marco Lorenzo Luy,

Ryan Ma, Molly Magid, Megs Malpani, Amy Miao, Mili Mitra, Jasper Miura, Ri-

cardo Mullings, Dan Murphy, Mia Murphy, Sakura Nakada, Kenta Nakagawa, Zach

Neronha, Valerie Nguon, Kemi Odusanya, Angel Ortiz, Clare Peabody, Shaughn

Pender, Brian Pfaff, Marina Renton, Zach Ricca, Sachin Sastri, Sam SaVaun, Isabel

vi

Scherl, Jon Schlafer, Ned Schweikert, Penelope Shao, Drew Solomon, Zach Spector,

Ellen Sukharevsky, Yashil Sukurdeep, Hans Sun, Heather Sweeney, Brittani Tay-

lor, Valeria Tiourina, Charlotte Tisch, Brian Tung, Carolina Velasco, Fifi Walker,

Joanna Walsh, Ben Winston, Jordan White, Zach Woessner, Mingyi Wu, Jonathan

Yakubov, Amanda Yan, Yuval Yossefy, Mario Zaharioudakis, Wennie Zhang, Favi

Zuniga. You are all capable of so much more than you think and I am so fortunate

to have met all of you.

To my undergraduate institution, Mount Holyoke College, particularly to my

undergraduate advisor, Jessica Sidman, and my role models, Margaret Robinson,

Giuliana Davidoff, and Harriet Pollatsek.

To the person who inspired my first love for math: my high school teacher,

Iolanda Podeanu.

To my parents, who always had more faith in me than I did.

vii

Contents

1 Introduction 1

2 Preliminaries 8

2.1 Atomic filtered spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Calderón-Zygmund operators . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Scalar weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Matrix weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 The matrix A2 class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Scalar weighted estimates 15

3.1 Two weight bounds for M qa . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Proof of the two weight estimate . . . . . . . . . . . . . . . . . . . . . 20

4 The Carleson Embedding Theorem 25

4.1 The Carleson embedding theorem . . . . . . . . . . . . . . . . . . . . 26

4.2 Trivial reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Invertibility of W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 The Bellman functions . . . . . . . . . . . . . . . . . . . . . . . . . . 30

viii

4.5 From Bellman functions to the estimate . . . . . . . . . . . . . . . . . 31

4.6 Final step: estimating ∑QRQ(0) . . . . . . . . . . . . . . . . . . . . 32

4.7 Verifying the properties of Bs . . . . . . . . . . . . . . . . . . . . . . 36

5 Matrix weighted two weight estimates for well-localized operators 42

5.1 Expectations and martingale differences . . . . . . . . . . . . . . . . 44

5.2 Generalized band operators . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4 Weighted martingale differences . . . . . . . . . . . . . . . . . . . . . 51

5.5 Density of simple functions . . . . . . . . . . . . . . . . . . . . . . . . 53

5.6 Well-Localized operators . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.7 From band operators to well-localized operators . . . . . . . . . . . . 57

5.8 Estimates of well-localized operators . . . . . . . . . . . . . . . . . . 60

5.9 Applications to the estimates of Haar shifts . . . . . . . . . . . . . . 63

5.10 The A2 theorem and linear dependence on complexity . . . . . . . . . 66

5.11 Weighted paraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.12 Estimates of the paraproducts . . . . . . . . . . . . . . . . . . . . . . 71

5.13 Estimates of well-localized operators . . . . . . . . . . . . . . . . . . 74

5.14 Estimate of the main part . . . . . . . . . . . . . . . . . . . . . . . . 75

5.15 Estimates of parts involving constant functions . . . . . . . . . . . . . 78

5.16 Estimates of the Haar shifts . . . . . . . . . . . . . . . . . . . . . . . 79

5.17 Comparison of different truncations . . . . . . . . . . . . . . . . . . . 81

5.18 Proof of Lemma 5.9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Bibliography 85

ix

CHAPTER 1

Introduction

The theory of weighted estimates has been an area of active research in harmonic

analysis since the 1960s, when the seminal work of Helson and Szegö [HS60] pro-

vided necessary and sufficient conditions for the boundedness of the Hilbert trans-

form in weighted Lebesgue spaces. In the 1970s, Hunt, Muckenhoupt, and Wheeden

[HMW73] formulated new conditions, giving a completely different description of

the weights for which the Hilbert transform is bounded in the weighted Lp space.

Presently known as the Muckenhoupt Ap conditions, these statements have held, and

continue to hold, a prominent place in the weighted theory literature.

In 1974, Coifman and Fefferman [CF74] showed that the necessity and sufficiency

of the Ap conditions extends to the boundedness of a wider family of singular integral

operators, which includes, but is not limited to the Hilbert transform. Namely, it

was shown that a Calderón-Zygmund operator is bounded on the weighted Lp(w)

space if and only if the weight w belongs to the so-called Ap class, or, equivalently,

if a quantity known as the Ap characteristic of the weight (denoted [w]Ap) is finite.

1

2

A natural question that arises from these estimates regards a more precise charac-

terization of Lp boundedness. If w is an Ap weight, one may ask in what way does the

norm of a Calderón-Zygmund operator on Lp(w) depend on the Ap characteristic of

w. Such norm estimates have applications in the study of elliptic partial differential

equations and quasiconformal mappings (see for instance[FKP91], [FG13])

Of particular importance is the case when the weight w is in A2. Rubio de Fran-

cia’s groundbreaking work from 1984 [RdF84] introduced the extrapolation theorem,

which essentially reduces problems in Lp to the weighted L2 case. As summarized

informally by Rubio de Francia’s colleague Antonio Córdoba, “there are no Banach

function spaces, just weighted L2". As a result, any estimate of the dependence of

the L2(w) norm of an operator on [w]A2 can be translated into a statement about Lp

norms.

In the context of extrapolation, a significant amount of research during the 1990s

and 2000s went into the so-called A2 conjecture, which claims that not only is any

Calderón-Zygmund operator T bounded on the weighted space L2(w) if and only if

w is an A2 weight, but the sharp norm dependence is in fact linear, i.e.

‖T‖L2(w)→L2(w) . [w]A2 .

Throughout the 1990s and 2000s, a series of important contributions were made,

starting with S. Buckley’s work from 1993 [Buc93], which proved that the weighted

norm of the Hardy-Littlewood maximal operator depends linearly on the A2 char-

acteristic and that the estimate is sharp. Throughout the following two decades,

sharp linear dependence was established for various other operators. In 2000, J.

Wittwer proved the statement for Haar multipliers [Wit00], and then, in 2002, S.

Petermichl and A. Volberg proved it for the Ahlfors-Beurling transform [PV02]. In

3

2007 [Pet07] and 2008 respectively [Pet08], Petermichl showed that the linear bound

also holds for the Hilbert and Riesz transforms. Further advances were made by O.

Beznosova ([Bez08] for the dyadic paraproduct), M. Lacey, S. Petermichl, and M.

Reguera ([LPR10] for the Haar shifts), D. Cruz-Uribe, J. Martell, and C. Pérez (a

simplified proof for the Haar shifts [CUMP10], [CUMP12]), and, finally, C. Pérez, S.

Treil, and A. Volberg (a proof for general Calderón- Zygmund operators, but with

the bound [w]A2 log(1 + [w]A2) instead of the optimal [w]A2 [PTV10]).

The conjecture was finally proved in full generality in 2010 by Hytönen [Hyt12b]

(see also [Hyt12a]) through an argument that reduces the study of Calderón-Zygmund

operators to the uniform boundedness of simpler objects called dyadic shifts, which

exhibit the desired linear dependence. Various different and substantially simpler ap-

proaches have been provided since. In 2012, A. Lerner [Ler13] showed that Calderón–

Zygmund norms could be estimated by a special class of operators called sparse op-

erators, and, as such, the A2 Theorem is a simple consequence of the “local mean

oscillation decomposition" introduced in [Ler12]. Yet another approach was given

by M. Lacey in [Lac15], who gave a straightforward way of establishing pointwise

control of a Calderón–Zygmund operator by a sparse operator, further simplifying

Lerner’s argument. Developments in the area of pointwise estimates for multilinear

singular integrals by sparse form are also presented in [CDPO16].

While the A2 conjecture has been settled in the scalar case, the question remains

open in the setting of vector-valued function spaces with matrix weights. Other than

its intrinsic interest, such a setting is important for its applications to geometric

function theory, Toeplitz operators, multivariate prediction theory, and even the

study of finitely generated shift invariant subspaces of unweighted Lp(Rd) spaces (see

[IM01] [NT96], [Nie10], [Vol97]). In the matrix weighted setting, the A2 conjecture

4

claims that if W is a weight in the appropriately defined matrix A2 class and T is a

Caldéron-Zygmund operator, then T is bounded on the weighted space L2(W), and

the constant appearing in the L2(W) norm estimate for T depends linearly on the

A2 characteristic.

In spite of a great amount of recent work, progress in the area of matrix weighted

inequalities has been slow. The setup imposes a variety of difficulties, from issues

of commutativity and preserving homogeneity, to challenges regarding the very def-

initions of objects that appear in the scalar setting. While the boundeness of T on

L2(W) was settled almost a decade ago through the work of Nazarov and Treil, and

respectively Volberg ([NT96], [Vol97]), the explicit dependence on the A2 character-

istic is far from being fully understood. The problem of proving linear bounds in

[W]A2 for singular integral operators is currently the object of ongoing research.

Some recent estimates have been obtained for specific operators by adapting

arguments from the scalar setting. For the Hilbert transform H, it was shown by

Bickel, Petermichl, and Wick [BPW14] that

‖H‖L2(W)→L2(W) . [W]32A2 log[W]A2 .

For the Riesz transform R and the Ahlfors-Beurling transform B, the author,

together with B. Wick [CW15], proved similar bounds:

‖R‖L2(W)→L2(W ) . [W]32A2 log[W]A2

‖B‖L2(W)→L2(W) . [W]32A2 log[W]A2

Clearly, these estimates are suboptimal and, in fact, they can be further improved,

5

as we will show in the final chapter of this thesis. Their proof relies on the fact that

the operators above can all be represented as averages of dyadic shifts. For the Hilbert

transform, this was shown by S. Petermichl in [Pet00]. For the Riesz transform, it

is a result of S. Petermichl, S. Treil, and A. Volberg [PTV02], and for the Ahlfors-

Beurling transform it is a result of O. Dragičević and A. Volberg [DV03]. Providing

uniform bounds for the dyadic shifts, which in this situation are significantly simpler

objects, yields the results for H, R, and B. However, linear dependence does not

follow as it would in the scalar case.

Another family of operators which have been studied in the matrix setting are

the so-called sparse operators. An operator S : L2(R,Cd)→ L2(R,Cd) is said to be

sparse if

Sf =∑Q∈S〈f〉Q 1Q(x),

where S is a collection of dyadic cubes satisfying

∑R∈ChS(Q)

|R| ≤ 12 |Q|,

for all Q ∈ S and ChS(Q) stands for the children of Q in S, i.e. the maximal

dyadic subcubes of Q in S. In [BW14], Bickel and Wick showed that if S is a sparse

operator, then

‖S‖L2(W )→L2(W ) . [W ]32A2 .

The importance of estimating sparse operators is clear by analogy with the scalar

setup. Recent proofs of the scalar A2 conjecture (such as those provided by Lerner

and Lacey) have taken the approach of controlling Calderón-Zygmund operators by

sparse operators. Therefore, if one can prove linear bounds for sparse operators,

one will potentially have all the required ingredients for a proof of the matrix A2

6

conjecture.

In the final chapter of this thesis, which overlaps with the text of [BCTW16], we

will prove the bound [W ]32A2 for another important class: the well-localized operators.

A discussion of the difficulties that prevent us from obtaining linear bounds, and thus

proving the matrix A2 conjecture will also be included. We will, however, extend the

setup to include not only dyadic lattices, but also general atomic filtrations. This

chapter generalizes the result presented in [NTV08] to the context of matrix weighted

spaces.

The rest of the thesis is organized as follows. In Chapter 2 we introduce the no-

tion of atomic filtered spaces and give examples of atomic filtrations, including, but

not limited to, the dyadic case. We also define the objects to be studied throughout:

scalar A2 weights, matrix weighted spaces, and the corresponding matrix A2 class.

The following three chapters constitute two distinct parts. The first, comprised of

Chapter 3, whose content overlaps with [Cul15], discusses a scalar problem: the two

weight boundedness of a class of operators given by `q norms in the space of se-

quences indexed by atoms. The second part, comprised of the final two chapters,

provides new results and various improvements to the known arguments in the matrix

weighted literature. We begin by introducing an essential tool for matrix weighted

estimates, which was up until recently an open problem: the matrix Carleson em-

bedding theorem. This theorem was obtained in [CT15] by the author and S. Treil,

following a suggestion of F. Nazarov. Unlike previous embedding results, the one

presented in this thesis (and in [CT15]) gives a complete analogue of the scalar case,

with no additional assumptions on the weight or the structure of the space. We

employ this tool in the last chapter to study two weight bounds for well-localized on

vector-valued function spaces. Our results parallel those in [NTV08], but provide an

7

improvement of the arguments in [NTV08] even when restricted to the scalar setting.

CHAPTER 2

Preliminaries

In this chapter we introduce some of the definitions and notation to be used through-

out the thesis. Further definitions, as well as any potential changes in notation will

be provided as they arise, usually at the beginning of each chapter. However, the

general atomic filtration setting is to be assumed in all that follows, whether or not

it is explicitly mentioned.

2.1 Atomic filtered spaces

Let (X ,F , σ) be a sigma-finite measure space with a filtration {Fn}, that is, a

sequence of increasing sigma-algebras Fn ⊂ F . Here F is taken to be the smallest

sigma-algebra containing ⋃Fn.We make the assumption that Fn is atomic, meaning

that there exists a countable collection Dn of disjoint sets Q of finite measure (which

we will call atoms or cubes) with the property that every set in Fn can be written

as a union of atoms (cubes) Q ∈ Dn.

8

9

Denote by D the collection of all atoms, D =⋃n∈ZDn. A set Q could belong

to multiple generations Dn, so atoms Q ∈ Dn should formally be represented as

pairs (Q, n). However, to simplify notation, we will suppress the dependence on n

and write Q instead of (Q, n); if the “time” n is needed, it will be represented by

rkQ, i.e. if Q stands for the atom (Q, n) then we will say n = rkQ. The inclusion

R ⊂ Q for atoms should be understood as set inclusion, together with the inequality

rkR ≥ rkQ. In particular, for any r ∈ Z, ChrQ will stand for the children of order

r of Q, the collection of atoms R ⊂ Q with rkR = r + rkQ. For r = 1, we write

ChQ and avoid the superscript.

For a measurable set E, and the underlying measure σ on the space X , we will

often use the notation |E| to represent σ(E) and dx for dσ(x).

Example 2.1.1. A standard example of an atomic filtration is the standard dyadic

filtration in Rd with the Lebesgue measure.

For n ∈ Z, let

Dn := {2−n((0, 1]d + k

): k ∈ Zd}

be the collection of dyadic cubes of size 2−n. Then each Fn is the σ-algebra generated

by Dn, and F is the Borel σ-algebra.

Note that in this example we do not have atoms of different ranks coinciding as

sets.

The standard dyadic filtration also leads to more interesting examples, such as

the one below:

Example 2.1.2. Consider a measurable set X ⊂ Rd, again endowed with the Lebesgue

measure. For each n ∈ Z, define the collection of atoms Dn as the collection of all

non-empty intersections Q∩X , where Q runs over all dyadic cubes of size 2−n from

10

the previous example. If, for example, X = Q0 = (0, 1]d, then Q0 ∈ Dn for all n ≤ 0,

so we have cubes of different ranks, coinciding as sets.

By taking more complicated sets X , we can have more complicated structures of

atoms and their ranks. Furthermore, we can expand upon these examples by letting

the underlying measure σ be any arbitrary Radon measure.

2.2 Calderón-Zygmund operators

Let σ be a measure on Rd. A singular integral operator is an operator T on the

function space Lp(σ) given formally by the expression

Tf(x) =ˆK(x, y)f(y)dσ(y),

where the kernel K(x, y) exhibits a singularity near x = y (i.e. K(x, ·) and K(y, ·)

are not locally integrable as functions of x and y respectively). A classical example

of such an operator is the Hilbert transform on the real line:

Hf(x) =ˆR

1x− y

f(y)dy.

In particular, a singular integral operator is said to belong to the Calderón-

Zygmund class if it is bounded on L2 and its kernel K satisfies the following growth

and cancellation conditions:

• |K(x, y)| ≤ C|x− y|−d, x 6= y

• |K(x, y)−K(x′, y)|, |K(y, x)−K(y, x′)| ≤ C |x−x′|δ|x−y|d+δ , for |x− x′| ≤ |x−y|

2 ,

for some constants C > 0 and δ > 0.

11

One can easily see that the Hilbert transform is a Calderón-Zygmund operator.

Other examples of such operators are the Riesz transforms

Rf(x) =ˆRn

xj − yj|x− y|d+1f(y)dσ(y), 1 ≤ j ≤ d

and the Ahlfors-Beurling transform

Bf(z) =ˆC

f(w)(z − w)2 dσ(w).

2.3 Scalar weights

A (scalar) weight w is a nonnegative, locally integrable function. For a weight w,

one can define the space Lp(w) as the normed function space with norm given by

‖f‖Lp(w) :=(ˆ|f(x)|pw(x)dx

)1/p

<∞.

One can also define the Ap characteristic of w, denoted [w]Ap :

[w]Ap := supQ

( Q

w(x)dx)(

Q

w−p′p (x)dx

) pp′

where the supremum is taken over all atoms Q and p′ is the Hölder conjugate expo-

nent to p, 1p

+ 1p′

= 1. We will say that w belongs to the Muckenhoupt Ap class if the

quantity [w]Ap is finite. In particular, w is said to be an A2 weight if

[w]A2 := supQ

( Q

w(x)dx)(

Q

w−1(x)dx)<∞.

12

2.4 Matrix weights

We can extend the weighted theory to the setup of vector-valued function spaces.

Let F0 be the collection of sets E ∩ F where E ∈ F and F is a finite union of

atoms. A d× d matrix-valued measure W on X is a countably additive function on

F0 with values in the set of non-negative operators on Fd, where F is either C or

R. Equivalently, W = (wj,k)dj,k=1 is a d × d matrix whose entries wj,k are (possibly

signed or even complex-valued) measures, finite on atoms, and such that for any

E ∈ F0 the matrix (wj,k(E))dj,k=1 is positive semidefinite. Note that the measure W

is always finite on atoms.

Given such a measure W and measurable functions f = (f1, f2, . . . , fd)T and

g = (g1, g2, . . . , gd)T with values in Fd, we can define the integrals

ˆX

⟨dWf, g

⟩Fd

:=d∑

j,k=1

ˆXfkgjdwj,k

ˆX

dWf

where the second integral is a vector whose jth coordinate is given by

d∑k=1

ˆXfkdwj,k.

Remark 2.4.1. Readers not comfortable with matrix-valued measures can always,

without loss of generality, restrict themselves to working with absolutely continu-

ous measures and matrix-valued functions. Namely, it is an easy corollary of the

non-negativity of the matrix measure W that all the measures wj,k are absolutely

13

continuous with respect to the measure

w := tr W :=d∑

k=1wk,k.

Therefore, we can write dW = Wdw, where W is a w-a.e. positive semi-definite

matrix-valued function and

ˆX

dWf =ˆXWfdw,

ˆX

⟨dWf, g

⟩Fd

=ˆX

⟨Wf, g

⟩Fd

dw.

For a measure W, the weighted space L2(W) is defined as the set of all measurable

Fd-valued functions (where F is R or C) such that

‖f‖2L2(W)

:=ˆX

⟨dWf, f

⟩Fd<∞.

As usual, we will consider the quotient space over the set of functions of norm 0.

2.5 The matrix A2 class

As in the scalar case, if W is a matrix measure, one can define a version of the A2

characteristic:

[W]A2 = |Q|−2 supQ

∥∥∥W(Q) 12 W− 1

2 (Q)∥∥∥2<∞,

where the supremum is taken over all atoms Q

Remark 2.5.1. Although the matrix A2 condition looks like a natural extension of its

scalar counterpart, the matrix Ap condition has a more complicated form for p 6= 2

and requires the introduction of so-called Ap metrics. See [Vol97] for a more in depth

14

discussion.

The matrix A2 conjecture claims that if T is Calderón-Zygmund and W is a

weight in the A2 class, then

‖T‖L2(W)→L2(W) . [W]A2 .

CHAPTER 3

Scalar weighted estimates

The content of this chapter overlaps with that of [Cul15] by the author.

Let (X ,F , σ) be a σ-finite measure space with an atomic filtration Fn as described

in the previous chapter and let D be the collection of atoms. Since a classical example

of such a filtration is that given by a dyadic lattice on Rd, we may often refer to D

as a lattice and to its elements as cubes. In spite of this language, we will not be

making any further assumptions on the underlying structure of the space, including

for example any assumptions about the homogeneity of X with the measure σ. Let

µ and ν be measures, finite on all Q ∈ D.

For a sequence of functions a = {aQ}Q∈D , aQ : X → [0,∞) indexed by the

sequence of cubes, define the operator M qa , given by

M qafµ(x) =

∑Q∈Dx∈Q

∣∣∣∣∣(ˆ

Q

fdµ

)aQ

(x)1Q

(x)∣∣∣∣∣q

1/q

for 1 < q <∞

15

16

and

M∞a fµ(x) = sup

{∣∣∣∣∣(ˆ

Q

fdµ

)aQ

(x)1Q

(x)∣∣∣∣∣ : Q ∈ D such that x ∈ Q

}.

In this chapter we will show that under a so-called Sawyer type testing condition,

the operator f 7→ M qafµ is bounded Lp(µ) → Lp(ν) for p ≤ q. Testing conditions

of this type were named after E. Sawyer, who introduced them in [Saw82] for the

purpose of studying the two weight estimates for the classical maximal operator M .

The testing condition presented in [Saw82] essentially amounts to a uniform estimate

on characteristic functions of dyadic cubes. Later, in [Saw88], Sawyer proved that

for operators such as fractional integrals, Poisson kernels, and other nonnegative

kernels, the two weight estimate still holds if one assumes the testing condition not

only on the operator itself, but also on its formal adjoint. For the positive martingale

operators such results were obtained in [NTV99] (p = 2) and later in [LS09] (general

p) (see also [Tre12] for an easier argument).

3.1 Two weight bounds for M qa

The main result presented this chapter is a simple proof of the theorem below. The

novelty of this theorem, compared to Sawyer’s result in [Saw82] is that it allows aQ

to be nonnegative functions and also considers the case q < ∞ (as opposed to just

q = ∞). Furthermore, our argument is not restricted to the dyadic case, and will

also hold in a nonhomogeneous setting.

17

For Q ∈ D, define the truncated operator M q

a,Qby

M q

a,Qfµ(x) :=

∑R∈D,R⊂Q

x∈R

∣∣∣∣∣(ˆ

R

fdµ

)aR

(x)1R

(x)∣∣∣∣∣q

1/q

,

with the obvious modification for q =∞.

We have the following theorem:

Theorem 3.1.1. Let 1 < p ≤ q ≤ ∞. The operator M qa satisfies

‖M qafµ‖pLp(ν)

≤ A‖f‖Lp(µ) ∀f ∈ Lp(µ)(3.1)

if and only if the following testing condition holds for the truncation M q

a,Q:

(3.2) ‖M qa,Q(1

Qµ)‖Lp(ν) ≤ Bµ(Q)1/p, for any Q ∈ D.

Moreover, for the best constants A and B, we have B ≤ A ≤ C(p)B,

C(p) =((1 + 1/p)p+1 p

)1/pp′,

where p′ is the Hölder conjugate of p, 1/p+ 1/p′ = 1.

3.2 Observations

Before proving the theorem, we make a few observations. First, it is easy to see that

for p = q =∞, the result is trivial with A = B. Note that limp→∞

C(p) = 1.

A second observation is that the classical dyadic (martingale) maximal operator

18

M is a particular case of operator M qa where q = ∞, a

Q≡ σ(Q)−1

1Q, and D is

a dyadic lattice in Rn. Therefore, one can view the M qa as a generalization of the

classical martingale maximal function.

In [Saw82], E. Sawyer considered slightly more general maximal operators M =

Mα which are a particular case of our M qa with q =∞ and a

Q≡ σ(I)−α1

Q, 0 < α ≤

1. He characterized the measures µ, ν and σ for which the inequality

‖Mfσ‖Lp(ν) ≤ A‖f‖

Lp(µ)∀f ∈ Lp(µ)(3.3)

holds.

Note that without loss of generality one can assume that µ is absolutely continu-

ous with respect to σ, dµ = wdσ (adding a singular part to µ does not change (3.3)).

So, making the standard change of weight f 7→ wp′/pf and denoting µ := w−p

′/pσ we

transform the above estimate (3.3) to (3.1). Then in this notation the necessary and

sufficient condition obtained by E. Sawyer is exactly the testing condition (3.2).

For the classical dyadic maximal operator M = Mα, the truncation M∞a,Q1Q

µ

defined above is equivalent to 1QM1

Qµ, so up to a change of measure, our setup is

identical to [Saw82].

Finally, we remark that the reduction to the two measure setup, eliminating

the underlying measure σ, is now considered standard for weighted estimates. In

the three measures setup for the classical maximal operator as in [Saw82] all the

information about σ is captured by the (constant in this case) functions aQ.

To obtain Sawyer’s estimate for the non-martingale maximal function, one can use

the two weighted estimate for the dyadic case and proceed by an averaging argument.

This reasoning is fairly standard and will not be discussed in this chapter.

Our proof simplifies the argument in [Saw82] and gives a stronger result: in

19

particular, the coefficients aQdo not need to be constant. It also has the additional

benefit of placing the Hardy-Littlewood maximal function in the context of a wide

range of similar operators.

The proof we present relies on the stopping time construction presented in [Tre12]

and the Martingale Carleson Embedding Theorem stated below.

Denote Q

fdµ = 1µ(Q)

ˆQ

fdµ.

Theorem 3.2.1. (Martingale Carleson Embedding Theorem) Let µ be a measure on

X and let {wQ}Q∈D , wQ ≥ 0 be a sequence satisfying the following condition:

∑Q⊂R,R∈D

wQ≤ Aµ(R), for any cube R ∈ D and some constant A.

Then for any measurable function f ≥ 0 and for any p ∈ (1,∞),

∑Q∈D

( Q

fdµ

)pwQ≤ (p′)pA‖f‖pLp(µ).

The Carleson Embedding Theorem with the constant (p′)p can be proved as a

straightforward consequence of the one weight Lp boundedness of the classical Hardy-

Littlewood maximal function (see [Tre12]). Other arguments that include the sharp

constant have been given by Nazarov, Treil, and Volberg [NTV01] for p = 2 and Lai

[Lai15] for p 6= 2 using Bellman function techinques. The exact Bellman function for

p > 1 was originally computed by Melas in [Mel05], but the sharp constant was not

explicitly stated.

20

3.3 Proof of the two weight estimate

We aim to prove Theorem 3.1.1. Again, we note that while we may refer to the

elements Q of D as cubes, they may not be dyadic cubes, and D is not necessarily

assumed to be the dyadic lattice on Rd.

Proof. First notice that the necessity of the testing condition and the estimate B ≤ A

are trivial: if M qa is bounded on Lp(µ) functions, it is, in particular, bounded on

caracteristic functions. Thus, by testingM qa on the functions 1Q, we obtain condition

(3.2).

To prove sufficiency, we begin by constructing a collection of stopping cubes

G ⊂ D, following the definitions and notation in [Tre12]. For any cube Q ∈ D, define

D(Q) to be the collection of subcubes of Q in D. For a fixed r > 1, let G∗(Q) be the

set of stopping cubes of Q, that is,

G∗(Q) ={R ∈ D(Q), R maximal, 1

µ(R)

ˆR

fdµ ≥ r1

µ(Q)

ˆQ

fdµ

},

where maximality is considered with respect to the partial ordering given by inclu-

sion.

Denote by E(Q) the collection of descendants of Q that are not stopping cubes

or descendants of the stopping cubes:

E(Q) = D(Q) \⋃

P∈G∗(Q)D(P ).

21

Note that, by definition, for any R ∈ E(Q),

(3.4) R

fdµ < r

Q

fdµ.

Also note that

(3.5)∑

R∈G∗(Q)µ(R) = µ

⋃R∈G∗(Q)

R

≤ µ(Q)r

.

To construct the collection G of stopping cubes, let N be a fixed large positive

integer and define the first generation G1 as

G1 = D−N .

Then, to obtain the subsequent generations, apply the inductive formula

Gn+1 =⋃

Q∈GnG∗(Q).

Define the collection of stopping cubes G to be the union

G =∞⋃n=1Gn.

Equation (3.5) implies that

(3.6)∑

R∈G,R⊂Qµ(R) ≤ r

r − 1µ(Q), ∀Q ∈ D.

To prove the theorem it is sufficient to prove uniform bounds inN for the operator

22

M q,Na ,

M q,Na f :=

∑n≥−N

∑Q∈Dn

(( Q

fdµ

)µ(Q)a

Q(x)1

Q(x))q1/q

(with the obvious change for q =∞) and then let N →∞.

Given the construction of stopping moments, it is easy to see that

∞⋃n=−N

Dn =⋃Q∈GE(Q).

and that the sets E(Q) are disjoint.

In the proof below we use notation for 1 < q < ∞. The proof for q = ∞ is

absolutely the same (up to obvious changes in the notation).

Denoting

FQ

(x) = ∑R∈E(Q)

(( R

fdµ

)µ(R)a

R(x)1

R(x))q1/q

,

we can writeM q,Na f =

∑Q∈G

F q

Q

1/q

, so the proof amounts to bounding

∥∥∥∥∥∥∥∑Q∈G

F q

Q

1/q∥∥∥∥∥∥∥Lp(ν)

.

Since ‖x‖`q ≤ ‖x‖`p for q ≥ p, we can estimate

∥∥∥∥∥∥∥∑Q∈G

F q

Q

1/q∥∥∥∥∥∥∥Lp(ν)

≤

∥∥∥∥∥∥∥∑Q∈G

F p

Q

1/p∥∥∥∥∥∥∥Lp(ν)

(3.7)

=ˆ ∑

Q∈GF p

Qdν

1/p

=∑Q∈G

ˆF p

Qdν

1/p

=∑Q∈G‖F

Q‖pLp(ν)

1/p

.

23

By definition,

∥∥∥FQ

∥∥∥pLp(ν)

=

∥∥∥∥∥∥∥ ∑R∈E(Q)

(( R

fdµ

)µ(R)a

R1R

)q1/q∥∥∥∥∥∥∥p

Lp(ν)

(3.8)

≤

∥∥∥∥∥∥∥ ∑R∈E(Q)

((r

Q

fdµ

)µ(R)a

R1R

)q1/q∥∥∥∥∥∥∥p

Lp(ν)

by (3.4)

= rp(

Q

fdµ

)p ∥∥∥∥∥∥∥ ∑R∈E(Q)

((ˆR

1Qdµ

)aR1R

)q1/q∥∥∥∥∥∥∥p

Lp(ν)

≤ rpBp

( Q

fdµ

)pµ(Q) by (3.2) .

Therefore, from inequalities (3.7) and (3.8) we obtain

∥∥∥∥∥∥∥∑Q∈G

F q

Q

1/q∥∥∥∥∥∥∥Lp(ν)

≤

∑Q∈G‖F

Q‖pLp(ν)

1/p

≤ rB

∑Q∈G

( Q

dµ

)pµ(Q)

1/p

.

The final step is to apply Theorem 3.2.1, taking

wQ

=

µ(Q) : Q ∈ G

0 : Q /∈ G

Equation (3.6) shows that the sequence wQsatisfies the Carleson measure condi-

tion. Hence ∑Q∈G

( Q

fdµ

)pµ(Q) ≤ r

r − 1(p′)p‖f‖pLp(µ).

Consequently,

24

∥∥∥M q,Na fµ

∥∥∥pLp(ν)

≤ rp+1

r − 1(p′)pBp‖f‖pLp(µ).

In particular, since no assumption was made on r other than r > 1, one can consider

the minimal value of the constant on the right hand side, which is attained when

r = p+1p. Then

∥∥∥M q,Na fµ

∥∥∥pLp(ν)

≤(

1 + 1p

)p+1

p(p′)pBp‖f‖pLp(µ).

Observe that the right hand side above does not depend on the choice of N .

Taking the limit as N approaches ∞ completes the proof for M qa .

CHAPTER 4

The Carleson Embedding Theorem

The content of this chapter overlaps with that of [CT15], by the author and her

advisor, S. Treil.

As evident from the previous chapter, the Carleson Embedding Theorem is an

invaluable tool in the proof of weighted estimates in scalar-valued function spaces.

As one may expect, a similar embedding theorem is essential for proving analogous

matrix weighted bounds. In what follows we introduce this result, obtained in [CT15]

by the author and S. Treil. It is important to mention that while earlier versions of

the matrix weighted Carleson Embedding Theorem were known through the work of

F. Nazarov, S. Treil, and A. Volberg in [TV97] and, more recently, Isralowitz, Kwon,

and Pott in [IKP14], and Bickel and Wick in [BW15], all these results required strong

additional assumptions, such as the weight belonging to the A2 class. It quickly

becomes clear that the [W ]A2 dependence is a particular concern for the problem of

proving sharp bounds.

25

26

The weighted embedding theorem presented below does not assume any prop-

erties for the matrix weight except local boundedness, and produces an embedding

constant that depends polynomially on the dimension of the space. As in the scalar

case, our embedding theorem states that the Carleson measure condition, which is

just a simple testing condition, implies the embedding.

For matrix weights, the Carleson measure condition (condition (ii) in Theorem

4.1.1 or condition (iii) in Theorem 4.1.2) is an inequality between positive semidefi-

nite matrices. For scalar weights in the domain, the right hand side of the inequality

is a multiple of the identity matrix I: in this situation, sacrificing constants, one can

replace matrices by their norms, and the matrix embedding theorem trivially follows

from the scalar one. Of course, the constants obtained by such trivial reduction are

far from optimal: constants of optimal order were obtained using more complicated

reasoning in [NPTV02]. For our setup, both sides of the Carleson measure condition

are general positive semidefinite matrices, so the simple strategy of replacing matri-

ces by norms or traces will not work. A more complicated idea, in the spirit of the

argument in [NPTV02], is used to obtain the result. We will introduce a family of

Bellman functions depending on a nonnegative parameter s. The convexity proper-

ties of these functions, together with an observation on their behavior as functions

of s, will give the desired estimates.

4.1 The Carleson embedding theorem

The main result of this chapter is the following theorem:

Theorem 4.1.1. Let W be a d × d matrix-valued measure and let AQ

be positive

semidefinite d× d matrices. The following statements are equivalent:

27

(i)∑Q∈D

∥∥∥∥∥A1/2Q

ˆQ

W(dx)f(x)∥∥∥∥∥

2

≤ A‖f‖2L2(W)

for all f ∈ L2(W);

(ii)∑Q∈DQ⊂Q0

W(Q)AQ

W(Q) ≤ BW(Q0) for all Q0 ∈ D.

Moreover, for the best constants A and B, we have B ≤ A ≤ CB, where C = C(d)

is a constant depending only on the dimension d.

Note that the underlying measure σ is absent from the statement of the theorem:

we do not need σ in the setup, we only need the filtration Fn. Alternatively, we can

pick σ to make the setup more convenient. For example, if we define

σ := tr W :=d∑

k=1wk,k,

then the measures wj,k are absolutely continuous with respect to σ. Thus, we can

always assume that our matrix-valued measure W is an absolutely continuous mea-

sure Wdσ, where W is a matrix weight, i.e. a locally integrable (meaning integrable

on all atoms Q) matrix-valued function with values in the set of positive semidefinite

matrices.

As before, for a measurable function f , we will denote by 〈f〉Qits average on Q,

〈f〉Q

:= σ(Q)−1ˆQ

fdσ,

and if σ(Q) = 0, we will say that 〈f〉Q

= 0. The same definition is used for both

vector and matrix-valued functions.

The theorem below is the restatement of Theorem 4.1.1 in this setup, obtained

by setting AQ

= |Q|−1AQ. More precisely, Theorem 4.1.1 is just the equivalence

(ii)⇐⇒ (iii) in Theorem 4.1.2. The equivalence (i)⇐⇒ (ii) will be explained below.

28

Theorem 4.1.2. Let W be a d × d matrix-valued weight and let AQ, Q ∈ D be a

sequence of positive semidefinite d× d matrices. Then the following are equivalent:

(i)∑Q∈D

∥∥∥∥A1/2Q〈W 1/2f〉

Q

∥∥∥∥2|Q| ≤ A‖f‖2

L2 .

(ii)∑Q∈D

∥∥∥∥A1/2Q〈Wf〉

Q

∥∥∥∥2|Q| ≤ A‖f‖2

L2(W ).

(iii) 1|Q0|

∑Q∈DQ⊂Q0

〈W 〉QAQ〈W 〉

Q|Q| ≤ B〈W 〉

Q0for all Q0 ∈ D.

Moreover, B ≤ A ≤ CB, where C = C(d) = e · d3(d+ 1)2.

4.2 Trivial reductions

The equivalence of (i) and (ii) is trivial. In (i), perform the change of variables

f := W 1/2f to obtain (ii) and similarly, in (ii) set f := W−1/2f to obtain (i). Note

that here we do not need to assume that the weight W is invertible a.e.: we just

interpret W−1/2 as the Moore–Penrose inverse of W 1/2.

The implication (i) =⇒ (iii) and the estimate A ≥ B become obvious if one sets

f = W 1/21Qe, e ∈ Fd in (i) (recall that F stands for either R or C). Equivalently,

to show that (ii) =⇒ (iii) it suffices to apply (ii) to the test functions f = 1Qe.

Therefore, it remains to prove that (iii) =⇒ (i), or equivalently, that (iii) =⇒ (ii).

4.3 Invertibility of W

Let us notice that without loss of generality we can assume that the weight W is

invertible a.e., and even more, that the weight W−1 is uniformly bounded. To show

29

this, define, for α > 0, the weight Wα by Wα(s) := W (s) + αI, and let

AαQ

:= 〈Wα〉−1Q〈W 〉

QAQ〈W 〉

Q〈Wα〉−1

Q.

If (iii) is satisfied, then trivially

1|Q0|

∑Q∈DQ⊂Q0

〈Wα〉QAα

Q〈Wα〉Q |Q| ≤ B〈W 〉

Q0≤ B〈Wα〉Q0

.

If Theorem 4.1.2 holds for invertible weights W , we get that for all f ∈ L2(W ) ∩ L2

∑Q∈D

∥∥∥∥(AαQ)1/2〈Wαf〉Q∥∥∥∥2|Q| ≤ A‖f‖2

L2(Wα).

Noticing that

‖f‖L2(Wα)

→ ‖f‖L2(W )

〈Wαf〉Q → 〈Wf〉Q

AαQ→ A

Q

as α → 0+ we immediately get (ii) for all f ∈ L2(W ) ∩ L2. Note that in this case

taking the limit inside the sum is justified, because an infinite sum of non-negative

numbers is the supremum of all finite subsums, and finite sums commute with limits.

Since the estimate (ii) holds on a dense set, extending the embedding operator by

continuity, we trivially obtain that (ii) holds for all f ∈ L2(W ).

30

4.4 The Bellman functions

By homogeneity, we can assume without loss of generality that B = 1. As discussed

above, we only need to prove the implication (iii) =⇒ (i).

Following a suggestion by F. Nazarov, we will do so by a “Bellman function with

a parameter” argument similar to one presented in [NPTV02]. Denote

FQ

= ‖f‖2L2(Q)

:= 〈|f |2〉Q

(4.1)

MQ

= 1|Q|

∑R⊂Q〈W 〉

RAR〈W 〉

R(4.2)

xQ

= 〈W 1/2f〉Q.(4.3)

For any real number s, 0 ≤ s <∞, define the family of Bellman functions

Bs(Q) = Bs(FQ , xQ ,MQ) =

⟨(〈W 〉

Q+ sM

Q

)−1xQ, x

Q

⟩Fd.(4.4)

Notice that FQ

is not explicitly involved in the definition of Bs(Q). However, we

retain it as a variable because it will be used in the estimates.

The functions Bs(Q) satisfy the following properties:

(i) The range property: 0 ≤ Bs(Q) ≤ FQ;

(ii) The key inequality:

Bs(Q) + sRQ

(s) ≤∑

Q′∈Ch(Q)

|Q′||Q|Bs(Q′)(4.5)

31

where

RQ

(s) = ‖A1/2Q〈W 〉

Q(〈W 〉

Q+ sM

Q)−1x

Q‖2.

The inequality Bs(Q) ≥ 0 is trivial, and the inequality Bs(Q) ≤ FQfollows immedi-

ately through an application of the Cauchy-Schwarz inequality. The details of this

computation are presented in the proof of Lemma 4.7.1 below. The key inequality

(4.5) is a consequence of Lemma 4.7.3, which we also prove below, in the final section

of this chapter.

4.5 From Bellman functions to the estimate

Let us assume for now that the properties of Bs(Q) hold true. Then we can use them

to prove our main theorem. Rewrite (4.5) as

|Q|Bs(Q) + |Q|sRQ

(s) ≤∑

Q′∈Ch(Q)|Q′|Bs(Q′).

Then, applying this estimate to each Bs(Q′), and then to each descendant of each

Q′, we get, going m generations down,

|Q|Bs(Q) +∑

Q′∈D:Q′⊂QrkQ′<rkQ+m

sRQ′

(s)|Q′| ≤∑

Q′∈D:Q⊂QrkQ′=rkQ+m

|Q′|Bs(Q′) ≤ ‖f1Q‖2L2 .

In the last inequality we used the fact that

Bs(Q) ≤ FQ

= 〈‖f( · )‖2〉Q

= |Q|−1‖f1Q‖2L2 .

32

Letting m→∞ and ignoring the non-negative term sBs(Q) in the left hand side,

we get that

s∑

Q′∈D:Q′⊂QRQ′

(s)|Q′| ≤ ‖f1Q‖2L2 .

Furthermore, summing the above inequality over all Q ∈ Dn, we obtain

s∑

Q′∈D: rkQ′≥nRQ′

(s)|Q′| ≤ ‖f‖2L2 .

Then, letting n→ −∞ and replacing Q′ by Q, we arrive to the estimate

(4.6) s∑Q∈DRQ

(s) ≤ ‖f‖2L2 .

Note that

RQ

(0) = ‖A1/2QxQ‖ = ‖A1/2

Q〈W 1/2f〉

Q‖,

so to prove (i) we need to estimate ∑QRQ(0). However, at this point in the proof,

we only have the estimate of s∑QRQ(s).

In the scalar case, the proof would be complete: since MQ≤ 〈W 〉

Q, we have

RQ

(0) ≤ 4RQ

(1), which gives us (i) with constant 4B. Due to non-commutativity,

such an estimate fails in the matrix case, so an extra step is needed.

4.6 Final step: estimating ∑QRQ

(0)

The final piece of the proof of Theorem 4.1.2 is the following lemma:

Lemma 4.6.1. For ε > 0

RQ

(0) ≤ C(ε, d)1ε

ˆ ε

0sR

Q(s)ds.

33

Moreover, for ε = 2/d we can have C(d) = e · d3(d+ 1)2.

Remark 4.6.2. Applying Lemma to (4.6), we get

∑Q∈DRQ

(0) ≤ e · d3(d+ 1)2‖f‖2L2 ,

which proves Theorem 4.1.2.

We conclude this section by providing a proof of the lemma above.

Proof. Observe that it follows from the cofactor inversion formula that the entries of

the matrix (〈W 〉Q

+ sMQ

)−1 are of the form pj,k(s)Q(s) , where

Q(s) = QQ

(s) = det(〈W 〉Q

+ sMQ

)

is a polynomial of degree at most d, and pj,k(s) are polynomials of degree at most

d− 1.

Therefore RQis a rational function in s,

RQ

(s) =PQ

(s)|Q

Q(s)|2 ,

where PQ

(s) is a polynomial of degree at most 2(d− 1) and PQ

(s) ≥ 0. We can then

write PQ

(s) = |PQ

(s)|2, where PQhas degree at most d− 1. Therefore,

RQ

(s) =∣∣∣∣∣∣PQ

(s)QQ

(s)

∣∣∣∣∣∣2

.

By hypothesis, MQ≤ 〈W 〉

Q, so the operator 〈W 〉

Q+ sM

Qis invertible for all s

such that Re(s) > −1. Thus the zeroes of QQ

(s) are all in the half plane Re(s) ≤ −1.

34

Let λ1, λ2, ..., λd be the roots of the polynomial QQ

(s) counting multiplicity. We

have ∣∣∣∣∣∣QQ

(s)QQ

(0)

∣∣∣∣∣∣ =d∏

k=1

∣∣∣∣∣s− λkλk

∣∣∣∣∣ .For a fixed s and Reλk ≥ −1 the term |s−λk|/|λk| attains its maximum at λk = −1.

Therefore, on the interval [0, ε],

(4.7)∣∣∣∣∣∣QQ

(s)QQ

(0)

∣∣∣∣∣∣ ≤ (1 + ε)d .

From the estimate above,

(4.8)ˆ ε

0s

∣∣∣∣∣∣PQ

(s)QQ

(0)

∣∣∣∣∣∣2

ds ≤ (1 + ε)2dˆ ε

0sR

Q(s)ds.

It will suffice then to find a constant C1 = C1(ε, d) such that for any polynomial

p of degree at most d− 1

(4.9) |p(0)|2 ≤ C1

ˆ ε

0s |p(s)|2 ds

ε.

Note that if we are not interested in determining the exact constant C(d), the argu-

ment is complete: we can just consider the space of polynomials of degree at most d

endowed with the norm

‖p‖ := ε−1ˆ e

0s|p(s)|2ds

and the linear functional p 7→ p(0). Since any linear functional on a finite-dimensional

normed space is bounded, we will immediately get (4.9).

If we want to estimate the constant C(d), some additional work is needed. First,

making the change of variables x = 2s/ε we can see that (4.9) is equivalent (with

35

the same constant C1) to

|p(0)|2 ≤ C1ε

4

ˆ 2

0x |p(x)|2 dx

or, equivalently, to the estimate

|p(1)|2 ≤ C1ε

4

ˆ 1

−1(1− x) |p(x)|2 dx(4.10)

for all polynomials p, deg p ≤ d− 1.

Consider the Jacobi polynomials P (1,0)n , which are orthogonal polynomials with

respect to the weight w,

w(x) = (1− x) = (1− x)1(1 + x)0.

Denote by J (1,0)n the normalized Jacobi polynomials,

J (1,0)n := ‖P (1,0)

n ‖−1L2(w)

P (1,0)n .

Since P (1,0)n (1) = n+ 1 and

∥∥∥P (1,0)n

∥∥∥2

L2(w)= 2

(n+1) , we have that

J (1,0)n (1)2 = (n+ 1)3

2 .(4.11)

Writing P =d−1∑n=0

cnJ(1,0)n we obtain

ˆ 1

−1(x− 1) (P (x))2 dx = ‖P‖2

L2(w)=

d−1∑n=0|cn|2

36

and by (4.11)

P (1) =d−1∑n=0

cn(n+ 1)3/2√

2.

From Cauchy–Schwarz,

|P (1)|2 ≤(d−1∑n=0|cn|2

)(d−1∑n=0

(n+ 1)3

2

)= 1

8d2(d+ 1)2‖P‖2

L2(w).

Comparing this inequality with (4.10), we can see that (4.10) and consequently (4.9)

hold with

C1 = C1(ε, d) = ε−1d2(d+ 1)2/2.

From (4.9) and (4.8),

RQ

(0) ≤ C(ε, d)1ε

ˆ ε

0sR

Q(s)ds,

with C(ε, d) = ε−1d2(d+ 1)2(1 + ε)2d/2.

By letting ε = 1/(2d), we have indeed that

C(d) = d3(d+ 1)2(

1 + 12d

)2d≤ e · d3(d+ 1)2.

4.7 Verifying the properties of Bs

It remains to show that the functions in the family Bs satisfy the Bellman function

properties. The range property (i) is proved in the following lemma:

Lemma 4.7.1. For Bs defined above in (4.4), Bs(Q) ≤ FQ.

37

Proof. Let e ∈ Fd be fixed. Since W is self-adjoint, an application of the Cauchy-

Schwarz inequality gives

∣∣∣∣∣ Q

〈W 1/2f, e〉∣∣∣∣∣ ≤

( Q

〈f, f〉)1/2 (

Q

〈W 1/2e,W 1/2e〉)1/2

.

Therefore, recalling the notation (4.1), (4.3), we get that for any vector e,

(4.12)

∣∣∣〈xQ, e〉∣∣∣2

〈〈W 〉Qe, e〉

≤ FQ.

Using Lemma 4.7.2 below we can write

〈(〈W 〉Q

+ sMQ

)−1x, x〉 = supe6=0

|〈xQ, e〉|2

〈(〈W 〉Q

+ sMQ

)e, e〉

≤ supe6=0

∣∣∣〈xQ, e〉∣∣∣2

〈〈W 〉Qe, e〉

≤ FQ.

which means exactly that Bs(Q) ≤ FQ.

Lemma 4.7.2. Let A ≥ 0 be an invertible operator in a Hilbert space H. Then for

any vector x ∈ H

〈A−1x, x〉 = supe∈H: e 6=0

|〈x, e〉|2

〈Ae, e〉

38

Proof. By definition,

〈A−1x, x〉 = ‖A−1/2x‖2

= supa∈H: ‖a‖6=0

|〈A−1/2x, a〉|2

‖a‖2

= supa∈H: ‖a‖6=0

|〈x,A−1/2a〉|2

‖a‖2 .

Performing the change of variables a = A1/2e, we conclude

〈A−1x, x〉 = supe∈H: ‖e‖6=0

|〈x, e〉|2

〈Ae, e〉.

Having verified the range property, we now turn to the main estimate (4.5). This

inequality is the consequence of the following lemma:

Lemma 4.7.3. Let H be a Hilbert space. For x ∈ H and for U being a bounded

invertible positive operator in H define

φ(U, x) := 〈U−1x, x〉H .

Then the function φ is convex, and, moreover, if

x0 =∑k

θkxk, ∆U := U0 −∑k

θkUk

where 0 ≤ θk ≤ 1, ∑k θk = 1, then

∑k

θkφ(Uk, xk)− φ(U0, x0) ≥ 〈U−10 ∆UU−1

0 x0, x0〉H(4.13)

39

Remark 4.7.4. To see that this lemma implies (4.5), fix s > 0. Denoting

U s

Q= 〈W 〉

Q+ sM

Q, x

Q= 〈W 1/2f〉

Q,

we observe that

Bs(Q) = φ(U s

Q, x

Q).

Let Qk, k ≥ 1 be the children of Q, and let θk = |Qk|/|Q|. Notice that

〈W 〉Q

=∑k

θk〈W 〉Qk

MQ

=∑k

θkMQk+ s〈W 〉

QAQ〈W 〉

Q,

so

U s

Q−∑k

θkUQk =: ∆U s = s〈W 〉QAQ〈W 〉

Q.

Therefore, applying Lemma 4.7.3 with

U0 = U s

Q

x0 = xQ

Uk = U s

Qk

xk = xQk

∆U = ∆U s,

we get (4.13), which translates exactly to the estimate (4.5).

We end this chapter with the proof of the lemma stated above.

Proof of Lemma 4.7.3. The function φ and the right hand side of (4.13) are invariant

40

under the change of variables

x 7→ U−1/20 x,(4.14)

U 7→ U−1/20 UU

−1/20 ,

so it is sufficient to prove (4.13) only for U0 = I.

In this case, define function Φ(τ), 0 ≤ τ ≤ 1 as

Φ(τ) =∑

θk

⟨(I + τ∆Uk)−1 (x0 + τ∆xk), (x0 + τ∆xk)

⟩H− 〈x0, x0〉H ,

where ∆xk = xk − x0 and ∆Uk = Uk − U0 = Uk − I.

Using the power series expansion of (I + τ∆U)−1 we get

Φ(τ) =τ(2∑

θk〈∆xk, x0〉H −∑

θk〈∆Ukx0, x0〉)

+ τ 2(∑

θk〈∆U2kx0, x0〉+

∑θk〈∆xk,∆xk〉 − 2

∑θk〈∆Ukx0,∆xk〉H

)+ o(τ 2)

Notice that ∑θk∆xk =

∑θk(xk − x0) = 0

and also ∑θk∆Uk = −∆U.

Hence

Φ(τ) =τ〈∆Ux0, x0〉+ τ 2∑ θk(‖∆Ukx0‖2 + ‖∆xk‖2 − 2〈∆Ukx0,∆xk〉

)(4.15)

+ o(τ 3).

41

Using the above formula for

x0 = x1 + x2

2U0 = U1 + U2

2 ,

we get that that the second differential of φ at U = I is non-negative. Note that the

function φ is clearly analytic, so all the differentials are well defined.

The change of variables (4.14) implies that the second differential of φ is non-

negative everywhere. In particular, we obtain that Φ′′(τ) ≥ 0, for all τ , so Φ is

convex.

Returning to the general choice of arguments U , x, we can see from (4.15) that

Φ′(0) = 〈∆Ux0, x0〉H .

Since Φ is convex and Φ(0) = 0,

Φ(1) ≥ Φ′(0) = 〈∆Ux0, x0〉H ,

which concludes the proof

CHAPTER 5

Matrix weighted two weight estimates for

well-localized operators

The content of this chapter overlaps with [BCTW16] by the author and K. Bickel,

S. Treil, and B. Wick.

In what follows we will give necessary and sufficient conditions for the two weight

L2 estimates of the so-called well-localized operators with matrix-valued weights. The

main examples of such operators are the Haar shifts, and their different generaliza-

tions, considered in the weighted spaces. More specifically, in this chapter we will

study two weight estimates of the form

‖T (Wf)‖L2(V)

≤ C‖f‖L2(W)

(5.1)

with matrix-valued measures V and W. Here T (Wf) is defined for the integral

42

43

operator T given formally by

T (Wf)(x) =ˆX

dW(y)K(x, y)f(y) =ˆXK(x, y)W (y)f(y)dw(y),(5.2)

where the kernel K(x, y) = k(x, y)⊗Id, for a scalar-valued kernel k(x, y) Recall that,

as before, W is the density of W with respect to the scalar measure w.

The main result presented here is that the Sawyer type testing conditions are

necessary and sufficient for the boundedness of the operators T . We will show that

it is sufficient to verify the estimates of the operator and its formal adjoint only on

characteristic functions of atoms.

The proof will follow the lines of [NTV08]. The main part of the operator is

estimated by bounding the corresponding paraproduct, and the bound on the para-

product follows from the Carleson embedding theorem presented in the previous

chapter. It is important to mention that the matrix weighted version of this the-

orem, with matrix-valued weights both in the domain and in the target space, is

essential for our argument. As such, this result is only possible in the context of the

theorem proved in Chapter 4.

The context of matrix-valued weights requires several technical steps that would

not be present in the scalar setting. However, even restricting to the scalar case,

the arguments remain interesting and new. To begin with, unlike previous works,

this chapter studies a very general filtration, and not necessarily the standard dyadic

setup. Furthermore, and perhaps more importantly, by carefully estimating the

“easy” parts and slightly changing the definition of well-localized operators, we are

able to get (even in the case of scalar measures) better estimates and stronger results

than ones in [NTV08].

One of our main results, Theorem 5.9.2 is specifically adapted to estimating the

44

so-called Haar shifts. Earlier, similar results were obtained (only in the scalar case)

with significant extra work from the results of [NTV08], or just by modifying the

proofs from [NTV08].

We will use the symbol TW for the operator f 7→ T (Wf) and, similarly, for the

scalar measure w, we will use Tw to denote the operator f 7→ T (fw), where

Twf(y) :=ˆXK(x, y)f(y)dw(y).(5.3)

The above operator, Tw, is defined for the scalar-valued functions as well as for the

functions with values in Fd; we will use the same notation for both cases, although,

formally, in the latter case we should write Tw ⊗ Id.

If dW = Wdw and dV = V dv, we can rewrite estimate (5.1) as

‖V 1/2TwW1/2f‖

L2(v)≤ C‖f‖

L2(w).(5.4)

A particular interesting case is that when the measures V and W are absolutely

continuous with respect to the underlying measure σ, so in (5.4) we have v = w = σ.

5.1 Expectations and martingale differences

Before stating the results, we will introduce some more notation and terminology.

Throughout this chapter, a function f will be called locally integrable if it is integrable

on every atom Q ∈ D.

For an atom Q and a locally integrable function f , we will denote by 〈f〉Q

its

45

average (with respect to the underlying measure σ),

〈f〉Q

:= σ(Q)−1ˆQ

fdσ,

adopting the convention that 〈f〉Q

= 0 if σ(Q) = 0.

Define the averaging operator, or expectation EQby

EQf := 〈f〉

Q1Q,(5.5)

and the martingale difference operator ∆Qby

∆Q

:=∑

P∈Ch(Q)EP− E

Q.(5.6)

Note that EQ

and ∆Q

are orthogonal projections in L2(σ), and that the subspaces

generated by the ∆Qare orthogonal to each other.

We will think of EQ

and ∆Q

as of operators in Lebesgue spaces (that is, EQf

and ∆Qf are defined only a.e.), so if for atoms Q1 ⊂ Q2 we have σ(Q2 \ Q1) = 0,

then ∆Q2

= 0.

5.2 Generalized band operators

To the collection D, one can associate a tree structure where each Q is connected to

the elements of the collection ChQ (the children of Q). Given this tree, let dtree(Q,R)

denote the “tree distance” between atoms Q and R, namely, the number of edges of

the shortest path connecting Q and R. If Q and R share no common ancestor, then

take dtree(Q,R) =∞.

The operators of interest possess a band structure related to this tree distance,

46

as defined below. These operators are called generalized band operators because they

generalize the band operators studied in [NTV08] and [BW14].

Definition 5.2.1. A bounded operator T : L2(σ) → L2(σ) is said to be a generalized

band operator with radius r if T can be written as

T =2∑

j,k=1

∑Q,R∈D

P j

RT jkR,Q

P k

QT jkR,Q

: P k

QL2(σ)→ P j

RL2(σ),(5.7)

where T j,kR,Q

= 0 if dtree(R,Q) > r and for any Q,

P 1Q

= ∆Q

P 2Q

= EQ.

Remark 5.2.2. We usually assume that the sum in (5.7) has only finitely many

nonzero terms, but in a more general situation convergence will be considered in

the weak operator topology with respect to some ordering of the pairs R,Q.

Remark 5.2.3. Each block T j,kR,Q

:= P j

RT jkR,Q

P k

Qis an operator bounded in L2(σ), and

can be represented as an integral operator with kernel Kj,k

R,Q. The kernel Kj,k

R,Qcan

be computed in the following way: for y ∈ Q, let Qy ∈ ChQ be the unique child of

Q containing y. Defining

Kj,k

R,Q(x, y) :=

σ(Q(y))−1

(T j,kR,Q

1Qy

)(x), y ∈ Q

0 y /∈ Q,(5.8)

one can easily see that

(T j,kR,Q

f)

(x) :=ˆXKj,k

R,Q(x, y)f(y)dσ(y)

47

for all functions f ∈ L2(σ) such that 1Qf is supported on a finite union of cubes

S ∈ ChQ. Note also that the kernel Kj,k

R,Qis supported on R×Q and is constant on

sets R′ ×Q′, for R′ ∈ ChR and Q′ ∈ ChQ.

Because the operators P kQ are orthogonal projections in L2(σ), T being a gener-

alized band operator of radius r implies that its adjoint T ∗ is also a generalized band

operator of radius r.

5.3 Examples

In this section, we give several examples of generalized band operators of various

radii.

Example 5.3.1. For a numerical sequence a = {aQ}Q∈D , define the “dyadic” oper-

ator T : L2(σ)→ L2(σ) by

Tf = Taf :=∑Q∈D

aQEQf.

Trivially, T is a generalized band operator of radius 0 as long as Ta is bounded on

L2(σ). To see this, set j = k = 2 in the definition of generalized band operators

(so expectations will appear on both sides), and let TQQ = aQ for each Q ∈ D, and

TR,Q

= 0 for each R ∈ D with R 6= Q.

Remark. For a sequence |a| := {|aQ|}Q∈D one can easily see that the pointwise

estimate

|Taf | ≤ T|a||f |

holds for all x ∈ X . Therefore, in the scalar case, the two weight estimates for Ta

48

follow from the two weight estimates for T|a|. Consequently, in the scalar case, the

operators with all aQ≥ 0 (the so-called positive dyadic operators) play a special role

in weighted estimates.

In the case of matrix-valued measures, it is not clear that the weighted estimates

of T|a| imply any corresponding estimates for Ta. In fact, we suspect that this is not

true, so we will not reserve any special place in our setup for the positive dyadic

operators.

Example 5.3.2. For r ∈ Z+ and a locally integrable function b, define a paraproduct

Π = Πrb of order r on L2(σ) as

Πf =∑Q∈D

EQf

∑R∈Chr Q

∆Rb.(5.9)

Clearly, if it is bounded on L2(σ), the above paraproduct is a generalized band operator

of radius r.

Remark. Since Πf is defined by the sum of an orthogonal series, the convergence of

the sum defining Π in the weak operator topology implies its unconditional conver-

gence in the strong operator topology.

Example 5.3.3. A Haar shift of complexity (m,n) is an operator T : L2(σ)→ L2(σ)

defined by

Tf =∑Q∈D

∑R∈Chn(Q),S∈Chm(Q)

∆RTR,S

∆S,(5.10)

TR,S

: ∆SL2(σ)→ ∆

RL2(σ),

where for each block TR,S

= ∆RTR,S

∆S, the canonical kernel K

R,Sof T

R,S(defined

49

by (5.8) and supported on R× S) satisfies the estimate

‖KR,S‖∞ ≤ σ(Q)−1.(5.11)

If T is a Haar shift of complexity (m,n), then trivially its adjoint T ∗ is also a Haar

shift of complexity (n,m). Any Haar shift of complexity (m,n) ( and in fact, any

bounded operator given by (5.10)) is a generalized band operator of radius r = m+n.

Remark. An operator defined by (5.10) is bounded if and only if all blocks TQ

TQ

:=∑

R∈Chn(Q),S∈Chm(Q)

∆RTR,S

∆S

are uniformly bounded: in this case the series in (5.10) converges unconditionally

(independently of the ordering) in the strong operator topology.

Notice that the normalization condition (5.11) implies that ‖TQ‖ ≤ 1. Indeed,

(5.11) implies that the block TQ

can be represented as an integral operator with

kernel KQ(supported on Q×Q) satisfying ‖K

Q‖∞ ≤ |Q|−1, so ‖K

Q‖L2(Q×Q)

≤ 1.

The concept of Haar shift can be generalized.

Definition 5.3.4. A generalized Haar shift of complexity (m,n) is an operator T ,

T : L2(σ)→ L2(σ)

Tf =2∑

j,k=1

∑Q∈D

∑R∈Chn(Q),S∈Chm(Q)

P j

RT j,kR,S

P k

S,(5.12)

T j,kR,S

: P k

SL2(σ)→ P j

RL2(σ),

50

where the kernel Kj,k

R,Sof the operator T

R,S:= P j

RT j,kR,S

P k

Ssatisfies

‖Kj,k

R,S‖∞ ≤ |Q|−1.(5.13)

It is convenient to introduce an alternative representation of a (generalized) Haar

shift by grouping terms T j,kR,Q

. Denoting

TQ

=2∑

j,k=1

∑R∈Chn(Q),S∈Chm(Q)

P j

RT j,kR,S

P k

S

(or taking the inner sum in (5.10) for regular Haar shifts), we can represent a gen-

eralized Haar shift as ∑Q∈D TQ . Note that the kernel KQ

of the integral operator

TQ

is supported on Q × Q and constant on R × S, where R, S ∈ Chr+1Q and

r = max{m,n}. Since the sets R × S with R ∈ ChnQ and S ∈ ChmQ are disjoint,

the kernel KQ satisfies the estimate

‖KQ‖∞ ≤ |Q|−1(5.14)

for Haar shifts, and the estimate ‖KQ‖∞ ≤ 4|Q|−1 for the generalized Haar shifts

(the constant 4 appears here because for each pair R ∈ ChnQ, S ∈ ChmQ there are

four possible operators T j,kR,S

). This discussion motivates the following general object

of study:

Definition 5.3.5. A generalized big Haar shift of complexity r is a bounded operator

T : L2(σ)→ L2(σ) defined by

T :=∑Q∈D

TQ,(5.15)

51

where each block TQ

is an integral operator with kernel KQ, such that K

Qis sup-

ported on Q×Q, constant on R×S with R, S ∈ Chr+1Q, and satisfies the estimate

(5.14). If, in addition, each block TQand its adjoint T ∗

Qannihilate constants 1

Q, we

will call the operator a big Haar shift of complexity r, removing the word generalized.

Finally, if an operator T admits the above representation but does not satisfy the

estimates (5.14), we will say that the operator T has the structure of a (generalized)

big Haar shift of order r.

Remark 5.3.6. It is easy to see that a (generalized) band operator of radius r has the

structure of a (generalized) big Haar shift of order r. Moreover, if the kernels Kj,k

R,Q

of the blocks T j,kR,Q

admit the estimate ‖Kj,k

R,Q‖∞ ≤ 1/4 (or the estimate ‖K

R,Q‖∞ ≤ 1

for kernels of the blocks TR,Q

for the case of a band operator), then the operator T

is a (generalized) big Haar shift. To see that we can just define

TQ

:=∑

R∈Chr Q

∑S∈D(Q)

rkS≤rkR

2∑j,k=1

T j,kR,S

+∑

S∈Chr Q

∑R∈D(Q)

rkR<rkS

2∑j,k=1

T j,kR,S

.(5.16)

5.4 Weighted martingale differences

For the matrix measure W (or V) discussed in a previous chapter, one can define

the W-weighted expectation EWQ

and the martingale difference ∆WQ

as

EWQf := 〈f〉W

Q1Q

(5.17)

〈f〉WQ

:= W(Q)−1(ˆ

Q

dWf

)

52

and

∆WQ

=∑

R∈ChQEWR− EW

Q.(5.18)

respectively, for all atoms Q ∈ D.

We will not assume here that the matrix W(Q) is invertible. Instead, given that´Q

dWf ∈ Ran W(Q), the operators will be well defined if we interpret W(Q)−1 as

the Moore–Penrose inverse. As before, if W(Q) = 0, we set EWQf = 0.

It is easy to see that EWQ

is the orthogonal projection in L2(W) onto the subspace

of constants {1Qe : e ∈ Fd}. It can also be easily shown that

EWQ

∆WQ

= ∆WQEWQ

= 0,

that ∆WQ

is an orthogonal projection, and that the subspaces generated by ∆WQ

and

∆WR

are orthogonal whenever Q 6= R.

For any Q ∈ D, let HQdenote the space of (non-weighted) Haar functions, H

Q:=

∆Q

(L2(σ)). HQis the subspace of L2(σ) spanned by functions h

Qsupported on Q,

constant on each element of ChQ, and orthogonal to constant vectors. Similarly,

for the weighted case, let HWQ

:= ∆WQ

(L2(W)) be the analogous subspace of L2(W)

spanned by functions hWQ.

Remark 5.4.1. The vector-valued Haar functions hQ

and hWQ

should not be inter-

preted as scalar-valued Haar functions times constant vectors in Fd. Throughout

this chapter, if any reference to the scalar Haar functions is needed, it will be clearly

indicated and reflected in the notation.

53

5.5 Density of simple functions

Recall that the operator TW , with TWf := T (Wf) was given by the integral rep-

resentation (5.2), provided that the integral was well defined. However, to verify

boundedness, we only need to know the bilinear form of the operator on a dense set.

We will proceed to show that (finite) linear combinations of functions 1Qe, Q ∈ D,

e ∈ Fd are dense in L2(W) in the following lemma:

Lemma 5.5.1. Let F be the smallest σ-algebra containing an increasing sequence

of atomic σ-algebras Fn, with sets of atoms Dn. Let L denote the space of linear

combinations of functions 1Qe with Q ∈ D = ∪nDn and e ∈ Fd. If W is a d × d

matrix valued measure defined on F , then L is dense in L2(W).

Proof. We claim that if f ∈ L2(W) satisfies 〈f, e〉L2(W)

= 0 for all e ∈ Fd and Q ∈ D,

then f = 0. To see this, fix e ∈ Fd and suppose that for any Q ∈ D,

ˆX

⟨dWf,1

Qe⟩Fd

= 0.

Define a scalar measure w by w := tr W = ∑dj=1 wj,j. As mentioned earlier, W

is absolutely continuous with respect to w and so there is a measurable, positive

semidefinite function W (x) such that dW = W (x)dw.

Then, by assumption, we have

ˆQ

〈W (x)f(x), e〉Fd dw = 0, ∀ Q ∈ D,

which implies the function 〈W (x)f(x), e〉Fd = 0,w-a.e. To see this easily, assume

that Fd = Rd (similar arguments work for Cd). As 〈W (x)f(x), e〉Fd is measurable,

the pre images Q1 := 〈W (x)f(x), e〉−1Fd ((0,∞)) and Q2 := 〈W (x)f(x), e〉−1

Fd ((−∞, 0])

54

are both in F . Thus, we can write Q1, Q2 as countable unions of disjoint atoms:

Q1 = ∪jQji and Q2 = ∪kQk

2. Then, we can compute

ˆQ1

〈W (x)f(x), e〉Fddw =∑j

ˆQj1

〈W (x)f(x), e〉Fddw = 0.

As 〈W (x)f(x), e〉Fd ≥ 0 on Q1, this implies that 〈W (x)f(x), e〉Fd1Q1 = 0, w-a.e.

Similar results hold on Q2 and so 〈W (x)f(x), e〉Fd = 0 w-a.e. As this works for each

e ∈ Fd, we obtain W (x)f(x) = 0, w-a.e. If W (x) is invertible a.e., then W 1/2 is

invertible and we can conclude that

W 1/2(x)f(x) = W−1/2(x)W (x)f(x) = 0

w-a.e. This immediately implies that

‖f‖2L2(W) =

ˆX〈dWf, f〉 =

ˆX

⟨W 1/2f,W 1/2f

⟩dw = 0,

so f is the zero element in the space L2(W).

By Lemma 5.5.1, to verify the boundedness of TW , it is enough to be able to

compute

〈TW

1Qe,1

Rv〉L2(V )

=¨ ⟨

dW(y)K(x, y)1Q

(x)e, dV(x)1R

(y)v⟩Fd

=¨ ⟨

W (y)K(x, y)e, V (x)1R

(y)v⟩Fd

dw(y)dv(x).

for all Q,R ∈ D and all e, v ∈ Fd. Thus, we say that an operator TW acts formally

55

from L2(W) to L2(V) if the bilinear form

〈TW1Qe,1

Rv〉L2(V)

(5.19)

is well defined for all Q,R ∈ D and all e, v ∈ Fd. Then the formal adjoint T ∗Vis given

by

〈TW1Qe,1

Rv〉L2(V)

= 〈1Qe, T ∗

V1Rv〉L2(W)

.

We also assume a very weak continuity property, namely that

〈TW1Qe,1

Rv〉L2(V)

=∑

S∈ChQ〈TW1

Se,1

Rv〉L2(V)

(5.20)

=∑

S∈ChR〈TW1

Qe,1

Sv〉L2(V)

;

this property is non-trivial only if Q or R have infinitely many children, which is

a possibility in our atomic filtration (and not necessarily dyadic) setup.

Consider the set L of all finite linear combinations of functions 1QeQ, Q ∈ D,

eQ∈ Fd. If the bilinear form (5.19) is defined, then

〈TWf, g〉L2(V)

= 〈f, T ∗Vg〉L2(W)

f, g ∈ L(5.21)

is well defined for all f, g ∈ L. Since ∆WQf ∈ L for f ∈ L, the expression

〈TW∆WQf,∆V

Rg〉L2(V)

is also well defined for all f, g ∈ L. Thus the expression

∆VRT∆W

Qis well defined, in the sense that its bilinear form is well defined for all

f, g ∈ L

56

5.6 Well-Localized operators

To state and prove the main results, it is convenient to introduce the formalism

of well-localized operators between weighted spaces, rather than work directly with

generalized band operators.

Definition 5.6.1. An operator TW acting formally from L2(W) to L2(V) is said to

be localized if for all e, v ∈ Fd,

〈TW1Qe,1

Rv〉L2(V)

= 0

whenever Q,R ∈ D share no common ancestors.

Definition 5.6.2. An operator TW acting formally from L2(W) to L2(V) is called

r-lower triangular if for all R,Q ∈ D and e ∈ Fd,

∆V

RTW1

Qe = 0

if either of the following conditions hold:

(i) R 6⊂ Q and rkR ≥ r + rkQ

(ii) R 6⊂ Q(r+1) and rkR ≥ rkQ− 1.

Here Q(r+1) is the ancestor of Q of order r + 1 , i.e. Q(r+1) is the unique atom in D

which contains Q and has the property that rkQ(r+1) = rkQ− (r + 1).

Definition 5.6.3. An operator TW acting formally from L2(W) to L2(V) is said to

be well-localized with radius r if it is localized and if both TW and its formal adjoint

T ∗V

are r-lower triangular.

57

Remark 5.6.4. This definition is very similar to the definition of well-localized oper-

ators from [NTV08], with two exceptions. First, the definition in [NTV08] contained

a typographical error: in the language of this chapter, the definition in [NTV08] only

required that rkR ≥ rkQ in condition (ii) of Definition 5.6.2 above. This was not

correct, as pointed out in [BW14] , and the inequality rkR ≥ rkQ is not sufficient

to get the results in [NTV08]. For further details about the necessity of condition

(ii), see the discussion in [BW14].

The other difference between the two statements, which is more significant, is

that in [NTV08], the operator TW was not required to be localized in the sense of

the above definition. By imposing this requirement on our operators, we are able

to obtain better estimates than those in [NTV08]. In particular, unlike the result

in [NTV08], which specifically studied the dyadic case, assumed that each cube

had at most 2N children, and produced estimates depending on this bound, we do

not assume any bounds on the number of children of a cube Q ∈ D. Since all the

examples we are interested in (presented in the previous section) give rise to localized

operators, we do not lose generality by including this requirement into the definition

of well-localized operators. Thus, even for the case of scalar measures, our bounds

are stronger than the ones presented in [NTV08].

5.7 From band operators to well-localized opera-

tors

We will now show that if T has a structure of a generalized big Haar shift of complex-

ity r, as described in definition 5.3.5, then the operator TW , given by TWf = T (Wf),

is a well-localized operator of radius r.

58

We assume that in the representation (5.15) there are only finitely many terms

TQ, and that each block T

Qis represented by an integral operator with a bounded

kernel. Note that the latter assumption is always true if Q and R have finitely many

children; for the generalized big Haar shifts it is just postulated (together with the

uniform estimate for the norm of the kernels).

The above two assumptions imply that the bilinear form (5.21) is well defined

for f, g ∈ L, so TW is well defined as an operator acting formally from L2(W) to

L2(V). In fact, it can even be shown that the bilinear form (5.21) is well defined

for all f ∈ L2(W), g ∈ L2(V), so one can conclude that TW is a bounded operator

L2(W)→ L2(V).

Lemma 5.7.1. Let T have a structure of a generalized big Haar shift of complexity

r, satisfying the assumptions above. Then for matrix-valued measures W and V

the operator TW, TWf = T (Wf), acting formally from L2(W) to L2(V) is a well-

localized operator of radius r.

Proof. The fact that the operator TW is localized, in the sense of Definition 5.6.1, is

obvious. We will show that TW is also r-lower triangular. Then, by symmetry, the

same result will hold for T ∗V.

To prove that TW is r-lower triangular it is sufficient to show that for any e ∈ Fd,

the function T (W1Qe) outside of Q(r+1) is constant on cubes R, rkR ≥ rkQ − 1,

and that outside of Q it is constant on cubes R, rkR ≥ rkQ+ r.

Let us analyze how the non-zero blocks TSact on W1

Qe. First, observe that

TS

(W1Qe) is non-zero outside of Q only if Q ( S. Since rkS ≤ rkQ− 1 for Q ( S,

the condition rkR ≥ rkQ+ r implies that

rkR ≥ rkQ+ r ≥ rkS + 1 + r.(5.22)

59

We know that the kernel of TSis constant on sets S ′ × S ′′ with S ′, S ′′ ∈ Chr+1 S.

Therefore TS

(W1Qe) is constant on cubes R such that

rkR ≥ rkS + r + 1.

So if R ∩Q = ∅ and rkR ≥ rkQ+ r, we can conclude from (5.22) that TS

(W1Qe)

is constant on R. Similarly, if TS

(W1Qe) does not vanish outside of Q(r+1), then

Q(r+1) ( S, so

rkS ≤ rkQ(r+1) − 1 = rkQ− (r + 1)− 1 = rkQ− r − 2.

or equivalently

rkS + r + 1 ≤ rkQ− 1

The condition rkR ≥ rkQ− 1 then implies that

rkR ≥ rkQ− 1 ≥ rkS + r + 1.(5.23)

But, as discussed above, TS

(W1Qe) is constant on cubes R such that rkR ≥ rkS +

r+ 1, so (5.23) implies that outside of Q(r+1) the function TS

(W1Qe) is constant on

cubes R, rkR ≥ rkQ− 1.

Remark 5.7.2. In Lemma 5.7.1 we assumed that the operator T is a sum of finitely

many blocks TQ, and that each block is an integral operator with bounded kernel.

However, if we assume that the matrix-valued measures V and W are absolutely

continuous with respect to the underlying measure σ, i.e. dV = V dσ, dV = Wdσ,

60

it is enough to require that the sum over Q ∈ D converge (in the weak operator

topology and with respect to some ordering) to a bounded operator in L2(σ).

The reasoning is fairly straightforward if V,W ∈ L2loc(σ), and can be extended to

the case V,W ∈ L1loc(σ) (where the index index “loc” means “finite on atoms.”) via

a limiting argument similar to [NTV08]. The paper [NTV08] studied scalar weights.

A corresponding line of reasoning for matrix-valued weights V , W is presented in

[BW14].

5.8 Estimates of well-localized operators

For a cube Q ∈ D let DW,kQ be the collection of functions of the form

∑R∈D(Q)

rkR=rkQ+k

∆WRf =

∑R∈Chk Q

∆WRf, f ∈ L.

We have the following theorem:

Theorem 5.8.1. Let TW be a well-localized operator of radius r acting formally from

L2(W) to L2(V). Then TW extends to a bounded operator from L2(W) to L2(V) if

and only if the following conditions

(i) ‖1QTW1

Qe‖

L2(V)≤ T1‖1Qe‖L2(W)

for all e ∈ Fd;

(ii) ‖1QTWf

Q‖L2(V)

≤ T2‖fQ‖L2(W)for all f ∈ DW,r

Q ;

and their dual counterparts (corresponding conditions for T ∗V

with V and W inter-

changed) hold for all Q ∈ D.

Furthermore,

‖TW‖L2(W)→L2(V)≤(C(d)1/2 + 1/2

)(T1 + T∗1) + (r + 1)1/2(T2 + T∗2).

61

Here T∗1, T∗2 are the constants appearing in the duals to the testing conditions (i)

and (ii) respectively, and C(d) is the dimensional constant from the matrix Carleson

embedding theorem presented in the previous chapter (Theorem 4.1.2). Moreover, for

the best possible bounds Tk, T∗k we have

Tk, T∗k ≤ ‖TW‖L2(W)→L2(V)

k = 1, 2.(5.24)

Remark 5.8.2. In the case when each Q ∈ D has at most N children (N < ∞), the

condition (ii) follows from the testing condition ‖TW1Qe‖

L2(V)≤ T3‖1Qe‖L2(W)

. In

this case, one can estimate T2 ≤ C(r,N)T3 and obtain similar inequalities for the

dual condition. This is exactly the approach that was used in [NTV08].

Condition (i) and its corresponding dual from Theorem 5.8.1 can be slightly re-

laxed. Given a well-localized operator TW and an atom Q ∈ D, define the truncation

TQW

by

TQWf =

∑R∈D(Q)

∆VRTWf,(5.25)

and define a similar truncation for the dual T ∗V.

The theorem above can be rephrased in terms of truncations:

Theorem 5.8.3. Let TW be a well-localized operator of radius r acting formally from

L2(W) to L2(V). Then TW extends to a bounded operator from L2(W) to L2(V) if

and only if the conditions

(i) ‖TQW

1Qe‖

L2(V)≤ T1‖1Qe‖L2(W)

for all e ∈ Fd;

(ii) ‖TQWfQ‖L2(V)

≤ T2‖fQ‖L2(W)for all f ∈ DW,r

Q ,

62

their dual counterparts (corresponding conditions for T ∗Vwith V and W interchanged)

and the following weak type estimate

(iii)∣∣∣∣〈TW1

Qe,1

Qv〉L2(V)

∣∣∣∣ ≤ T3‖1Qe‖L2(W)‖1

Qv‖

L2(V)for all e, v ∈ Fd,

hold for all Q ∈ D.

Moreover,

‖TW‖L2(W)→L2(V)≤ C(d)1/2(T1 + T∗1) + (r + 1)1/2(T2 + T∗2) + T3,

where again C(d) is the constant from the Matrix Carleson Embedding Theorem.

Moreover, for the best possible bounds Tk, T∗k we trivially have

Tk, T∗k ≤ ‖TW‖L2(W)→L2(V)

k = 1, 2, 3.(5.26)

There is no dual condition to (iii) in this theorem, because this condition is self-

dual. Note that Theorem 5.8.1 follows immediately from Theorem 5.8.3, because

trivially for all f ∈ L

‖TQWf‖

L2(V)≤ ‖1

QTWf‖

L2(V).

Note also that condition (i) of Theorem 5.8.1 implies condition (iii) of Theorem 5.8.3,

with the trivial estimate for the corresponding bounds,

T3 ≤T1 + T∗1

2 ;

here T3 is the bound from Theorem 5.8.3, and T1, T∗1 are the bounds from condition

(i) and its dual in Theorem 5.8.1.

Remark 5.8.4. The condition (iii) of Theorem 5.8.3 can be further relaxed. First,

63

we do not need this condition to hold for all cubes Q ∈ D: it is sufficient if this

condition holds for arbitrarily large cubes Q, meaning that for any Q0 ∈ D one

can find Q ∈ D, Q0 ⊂ Q for which (iii) holds. Secondly, if for any Q0 ∈ D we

have that for the increasing sequence of cubes Qn, where Qn+1 is the parent of Qn,

W(Qn) ≥ αnI as αn → +∞ and similarly for V, the condition (iii) can be removed

from Theorem 5.8.3.

5.9 Applications to the estimates of Haar shifts

While conditions (i) from Theorems 5.8.1 and 5.8.3 are fairly standard testing condi-

tions, and the condition (iii) from Theorem 5.8.3 is the standard weak boundedness

condition, the condition (ii) seems unnecessarily complicated.

However, if the measures V, W satisfy the two weight matrix A2 condition

supQ∈D|Q|−2‖V(Q)1/2W(Q)1/2‖2 =: [V,W]

A2<∞,(5.27)

and the operator T is a generalized big Haar shift ( see definition 5.3.5), then the

condition (ii) follows from the testing condition (i) and the A2 condition (5.27).

Let us introduce some notation. For a generalized big Haar shift T = ∑Q∈D TQ

and a cube Q ∈ D, define the localized operator TQ,

TQ :=∑

R∈D(Q)TR

(5.28)

For a matrix measure W, define the weighted version (TQ)W of TQ by

(TQ)Wf = TQ(Wf).

64

Note that (TQ)W is different from TQW

defined above in (5.25). The following lemma

holds:

Lemma 5.9.1. Let T be a a generalized big Haar shift of complexity r with finitely

many terms, and let the matrix measures V and W satisfy the two weight matrix A2

condition (5.27). Assume that for all Q ∈ D

‖(TQ)W1Qe‖

L2(V)≤ T‖1

Qe‖

L2(W)∀e ∈ Fd.(5.29)

Then for all Q ∈ D

‖TQW

1Qe‖

L2(V)≤(d1/2r[V,W]1/2

A2+ T

)‖1

Qe‖

L2(W)∀e ∈ Fd,(5.30)

‖TQWf‖

L2(V)≤(d1/2(2r + 1)[V,W]1/2

A2+ T

)‖f‖

L2(W)∀f ∈ DW,r

Q.(5.31)

Moreover, for all sufficiently large Q ∈ D

‖TW1Qe‖

L2(V)≤ T‖1

Qe‖

L2(W)(5.32)

Lemma 5.9.1 implies that for a generalized big Haar shift T of complexity r with

finitely many terms, the bounds in the testing conditions in Theorem 5.8.3 can be

estimated as

T1 ≤ d1/2r[V,W]1/2A2

+ T,

T2 ≤ d1/2(2r + 1)[V,W]1/2A2

+ T,

T3 ≤ T,

with similar estimates for the dual bounds T∗1,2. Note that T3 ≤ T∗, so T3 ≤

65

(T+T∗)/2. Using these estimates and applying Theorem 5.8.3, we get the following

result.

Theorem 5.9.2. Let T be a generalized big Haar shift of complexity r (with finitely

many terms), and let the matrix measures V and W satisfy the A2 condition (5.27).

Let

(i) ‖(TQ)W1Qe‖

L2(V)≤ T‖1

Qe‖

L2(W)for all Q ∈ D and all vectors e ∈ Fd,

and also let the corresponding condition for T ∗ (with V and W interchanged) hold

with constant T∗.

Then

‖TW‖L2(W)→L2(V)≤(C(d)1/2 + (r + 1)1/2 + 1/2

)(T + T∗)

+ 2d1/2(C(d)1/2r + (2r + 1)(r + 1)1/2

)[V,W]1/2

A2;

here again C(d) is the constant from the Matrix Carleson Embedding Theorem.

The testing condition (i) of Theorem 5.9.2 is necessary, and moreover satisfies

the following estimate:

Proposition 5.9.3. The best possible constants T, T∗ in (i) of Theorem 5.9.2 satisfy

T, T∗ ≤ ‖TW‖L2(W)→L2(V)+ C1(d) · r · [V,W]1/2

A2.(5.33)

We postpone the proof of this proposition and Lemma 5.9.1 till Section 5.16.

66

5.10 The A2 theorem and linear dependence on

complexity

Theorem 5.9.2 can be directly applied to estimating dependence of the norm of Haar

shifts (and consequently of the Calderón–Zygmund operators) in the weighted space

L2(W ) on the A2 characteristic [W ]A2

of the weight W ([W ]A2

is the exactly the

A2 characteristic [V,W]A2

from (5.27) with dW = Wdσ, dV = W−1dσ). In the

scalar case, by the A2 theorem [Hyt12b], the norm depends linearly on [W ]A2, and

this estimate is optimal. In the matrix case, as specified in the introduction, the A2

conjecture remains open. The best known estimate so far is [W ]3/2A2

. Theorem 5.9.2

above reduces the problem to finding the optimal estimate in the testing condition

(i) and its dual.

It should be mentioned here that the best known estimates for Haar shifts in

the weighted space L2(w) with a scalar weight w satisfying the Muckenhoupt A2

condition grow linearly in the complexity r of the shift. It appears that in the

scalar case our theorem gives us the growth rate r3/2 in terms of the complexity r,

because the testing constants in condition (i) and its dual are usually estimated by

C · (r + 1)[w]A2. However, the standard splitting technique allows to get a growth

that is linear in complexity. Namely, one can split the operator T as T = ∑rk=0 Tk

Tk =∑j∈Z

∑Q∈D

rkQ=k+(r+1)j

TQ

;

then each Tk is a generalized big Haar Shift of complexity 0, with respect to the

rarefied filtration given by σ-algebras Fk+(r+1)n, n ∈ Z.

In the scalar case estimates of the testing bounds T and T∗ in terms of [w]A2

do

67

not depend on the filtration, so as a result we get a linear (in r) estimate of the norm

of the Haar shifts.

5.11 Weighted paraproducts

We are now ready to prove our main results from this chapter. The essential part of

the proof is the estimate of the weighted paraproducts, which we will present in this

section.

Let f = 1Se be a characteristic function with S ∈ D and e ∈ Fd. Then, for each

fixed n ∈ Z, f has the orthogonal decomposition

(5.34) f =∑

Q∈D,rkQ≥−n∆WQf +

∑Q∈D,rkQ=−n

EWQf.

To prove this equality, just observe that if m ≥ rkE, then f1Q = EWQf for all cubes

Q ∈ D with rkQ ≥ m. Then for each x ∈ X ,

f(x)−∑Q∈D

m≥rkQ≥−n

(∆WQf)

(x)−∑Q∈D

rkQ=−n

(EWQ f

)(x) = f(x)−

∑Q∈D

rkQ=m+1

(EWQ f

)(x) = 0.

Letting m→∞ gives the desired result. By orthogonality, it follows that

‖f‖2L2(W) =

∑Q∈D,rkQ≥−n

‖∆WQf‖2

L2(W) +∑

Q∈D,rkQ=−n‖EW

Qf‖2

L2(W).

For an operator TW acting formally from L2(W) to L2(V) define the paraproduct

ΠW = ΠWT

of complexity r as

ΠWf =∑Q∈D

∑R∈Chr(Q)

∆VR

(TWEW

Qf)

=∑Q∈D

∑R∈Chr(Q)

∆VR

(TW〈f〉

WQ

1Q

);(5.35)

68

clearly, the bilinear form of the paraproduct ΠW is well defined for f, g ∈ L, i.e. ΠW

also acts formally from L2(W) to L2(V). Similarly, for the adjoint T ∗V

of TW, define

the paraproduct ΠV = ΠVT ∗

by

ΠVg =∑Q∈D

∑R∈Chr(Q)

∆WR

(T ∗

VEVQf).

Lemma 5.11.1. Let T = TW be a well-localized operator of radius r, acting formally

from L2(W) to L2(V). Then for any cubes Q ⊂ S and for any R ∈ ChrQ and for

all e ∈ Fd,

∆VRTW1

Se = ∆V

RTW1

Qe.

Remark 5.11.2. The above lemma states that in the formula (5.35) for paraproducts,

one can replace 〈f〉WQ

1Qby 〈f〉W

Q1Swith any arbitrary cube S ⊃ Q. So formally we

can write in the right hand side of (5.35) the expression 〈f〉WQ

1 instead of 〈f〉WQ

1Q,

which looks more in line with the definition of the paraproduct in the scalar case.

To make it even more similar to the scalar representation we should use TW(1⊗

IFd

) instead of TW1 (to apply the operator TW to a matrix-valued function one just

needs to apply it to each column), and write the paraproduct ΠW as

ΠWf =∑Q∈D

∑R∈Chr Q

(TW(1⊗ I

Fd))〈f〉W

Q(5.36)

which is an alternative way of writing (5.35). The expression TW(1⊗ IFd

) should be

understood as TW(1S⊗ I

Fd) where S is an arbitrary cube with Q ⊂ S.

Proof of Lemma 5.11.1. Let P be a cube such that P 6= Q and rkP = rkQ. Since

69

TW is r-lower triangular,

∆VRTW1

Pe = 0

for any cube R 6⊂ P , rkR ≥ rkP + r. In particular, the equality holds for any

R ∈ ChrQ.

Since for a cube S ⊃ Q the set S \ Q is a (countable) union of cubes P with

rkP = rkQ, we conclude, using the weak continuity property (5.20), that for any

R ∈ ChrQ

∆VRTW1

S\Qe = 0,

which proves the lemma.

Remark 5.11.3. As one can see, in the above proof we only used the fact that T is

r-lower triangular; more precisely, only a part of the definition was used.

The following lemma states that the paraproducts ΠW and ΠV exhibit the same

behavior as TW and T ∗V

respectively.

Lemma 5.11.4. Let TW be a well-localized operator of radius r (acting formally

from L2(W) to L2(V), and let ΠW = ΠWT

be the paraproduct of complexity r defined

as above. Then for Q,R ∈ D

(i) If rkR ≤ r + rkQ, then

∆VR

ΠW∆WQ

= 0.

(ii) If R 6⊂ Q, then

∆VR

ΠW∆WQ

= 0.

70

(iii) If rkR > r + rkQ, then

∆VR

ΠW∆WQ

= ∆VRTW∆W

Q

and in particular if R 6⊂ Q, both sides of the equality are zero.

Proof. Using summation indices Q′ and R′, we have

ΠW∆WQ

=∑Q′∈D

∑R′∈Chr(Q′)

∆VR′

(TWEW

Q′∆WQ

),

and since ∆VR

is orthogonal to ∆VR′

for all choices of R′ except for R, we have

∆VR

ΠW∆WQ

= ∆VRTWEW

Q′∆WQ

where Q′ = R(r) is the rth order ancestor of R. Notice that EWQ′

∆WQ6= 0 only if

Q′ ( Q, so if rkR ≤ r + rkQ, then rkR(r) ≤ rkQ, which implies EWQ′

∆WQ

= 0, and

consequently

∆VR

ΠW∆WQ

= 0,

proving the first statement. Also, if R 6⊂ Q, then Q′ = R(r) 6⊂ Q. As above, this

implies EWQ′

∆WQ

= 0, and consequently

∆VR

ΠW∆WQ

= 0,

which proves the second statement.

To prove the third statement, assume rkR > r+ rkQ. If R 6⊂ Q, we can use our

71

previous result and the fact that T is well-localized to conclude:

∆VR

ΠW∆WQ

= 0 = ∆VRTW∆W

Q.

It now suffices to consider the case R ⊂ Q. Recall that Q′ = R(r). Since Q∩Q′ 6= ∅,

we can look at ranks to conclude that Q′ ( Q. Choose Q ∈ ChQ with Q′ ⊆ Q.

Then, using the fact that T is r-lower triangular, we have

∆VRTW∆W

Q= ∆V

RTW

∑S∈ChQ

EWS− EW

Q

= ∆VRTW

(EWQ− EW

Q· 1Q

).

Using earlier arguments and Q′ ( Q, we can write ∆VR

ΠW∆WQ

as

∆VRTWEW

Q′∆WQ

= ∆VRTW

(EWQ· 1Q′ − EW

Q· 1Q′

)= ∆V

RTW

(EWQ− EW

Q· 1Q

),

where the last equality follows by Lemma 5.11.1, completing the proof.

5.12 Estimates of the paraproducts

We restate the Carleson embedding theorem, presented in the previous chapter and

in [CT15]. This result will be used to control the norms of the paraproducts:

Theorem 5.12.1 (The matrix weighted Carleson Embedding Theorem). Let W be

a d × d matrix-valued measure and let AQ

be positive semidefinite d × d matrices.

The following statements are equivalent:

(i)∑Q∈D

∥∥∥∥∥A1/2Q

ˆQ

dWf

∥∥∥∥∥2

≤ A‖f‖2L2(W)

(ii)∑

Q∈D(Q0)W(Q)A

QW(Q) ≤ BW(Q0) for all Q0 ∈ D.

72

Moreover, for the best constants A and B we have B ≤ A ≤ C(d)B, where C(d) is

a constant depending only on the dimension d.

Remark. As shown in the previous chapter, C(d) = e · d3(d+ 1)2, where e is the base

of the natural logarithm. We are not sure if this estimate gives optimal asymptotic

on terms of dimension d, but that seems unlikely.

We now bound the paraproducts as follows:

Lemma 5.12.2. Let ΠW be the paraproduct defined earlier and assume that the

well-localized operator TW satisfies the testing condition

∑R∈D(Q)

rkR≥rkQ+r

‖∆VRTW1

Qe‖2

L2(V)≤ T2

1‖1Qe‖2L2(W)

(5.37)

for all Q ∈ D and e ∈ Fd. Then ΠW is bounded from L2(W) to L2(V) and

∥∥∥ΠW∥∥∥L2(W)→L2(V)

≤ C(d)1/2T1,

where C(d) is the constant in Theorem 5.12.1.

Remark 5.12.3. The testing condition (5.37) is clearly weaker than the testing con-

dition (i) from Theorem 5.8.3; the constant T1 from (5.37) is majorized by the

corresponding constant from (i).

Proof of Lemma 5.12.2. Fix f ∈ L2(W) and in the dense set L. Then by orthogo-

nality,

‖ΠWf‖2L2(V) =

∑Q∈D

∑R∈Chr(Q)

‖∆VR

(TWEW

Qf)‖2L2(V).

73

To control this expression, we use Theorem 5.12.1. First, for each Q ∈ D, define the

linear map BQ

: Fd → L2(V) by

BQe =

∑R∈Chr(Q)

∆VRTW(W(Q)−11Qe), ∀e ∈ Fd.

Then defining AQ

:= B∗QBQ : Fd → Fd, we can write

‖ΠWf‖2L2(V) :=

∑Q∈D

∥∥∥∥∥A1/2Q

ˆQ

dWf

∥∥∥∥∥2

,

so we are able to apply Theorem 5.12.1.

To prove condition (ii) in Theorem 5.12.1, fix Q0 ∈ D, e ∈ Fd, and use the

definitions of AQ, B

Qto obtain

∑Q∈D(Q0)

∥∥∥∥A1/2Q

W(Q)e∥∥∥∥2

=∑

Q∈D(Q0)

∥∥∥BQ

W(Q)e∥∥∥2

=∑

Q∈D(Q0)

∑R∈Chr(Q)

‖∆VR

(TW1

Qe)‖2L2(V).

Then using Lemma 5.11.1 and the testing condition (5.37) we get

∑Q∈D(Q0)

∥∥∥∥A1/2Q

W(Q)e∥∥∥∥2

=∑

Q∈D(Q0)

∑R∈Chr(Q)

‖∆VR

(TW1

Q0e)‖2L2(V) by Lemma 5.11.1

≤ T1‖1Q0e‖2

L2(W)by (5.37),

so condition (ii) of Theorem 5.12.1 is verified. Thus

‖ΠWf‖2L2(V) ≤ C(d)T2

1‖f‖2L2(V)

,

which completes the proof.

74

5.13 Estimates of well-localized operators

In this section we will prove Theorem 5.8.3. Theorem 5.8.1 will follow automatically,

since the bounds T1,2 and their duals T∗1,2 from Theorem 5.8.3 are trivially majorized

by the corresponding bounds from Theorem 5.8.1, and the bound T3 from Theorem

5.8.3 is dominated by the minimum of T1 and T∗1 from Theorem 5.8.1. We will also

explain Remark 5.8.4, claiming that the weak estimate (iii) of Theorem 5.8.3 can be

relaxed and sometimes ignored.

To prove Theorem 5.8.3, we estimate the bilinear form of the operator TW . Let

f ∈ L2(W) and g ∈ L2(V), with ‖f‖L2(W)

= ‖g‖L2(V)

= 1, be from the dense set L

of finite linear combinations of characteristic functions of atoms, i.e.

f =N∑j=1

aj1Qj ej and g =M∑k=1

bk1Rkvk,(5.38)

where Qj, Rk ∈ D and ej, vk ∈ Fd. As such functions are dense in L2(W) and L2(V),

to obtain the result, we need to show that

|〈TWf, g〉L2(V)

| ≤ C‖f‖L2(W)

‖g‖L2(V)

.(5.39)

Let us first perform some simplifications. Define an equivalence relation ∼ on D,

by saying that Q ∼ R if Q and R have a common ancestor (i.e. if Q,R ⊂ S for some

S ∈ D).

Since T is a localized operators, 〈T1Q,1

R〉L2(V)

= 0 if Q and R are in different

equivalence classes. Therefore, it is sufficient to prove (5.39) under assumption that

all Qj, Rk in the representation (5.38) are in the same equivalence class. Then,

taking the direct sum over equivalence classes, we get the general case.

75

Let Q0 ∈ D be a common ancestor of all Qj, Rk. Then, by (5.34), we can write

f , g using the orthogonal decompositions:

f =∑

Q∈D(Q0)∆WQf + EW

Q0f =: f1 + f2;(5.40)

g =∑

R∈D(Q0)∆VRg + EV

Q0g =: g1 + g2.(5.41)

We will estimate the four terms 〈TWfj, gk〉L2(V) for 1 ≤ j, k ≤ 2 separately.

5.14 Estimate of the main part

To estimate 〈TWf1, g1〉L2(V)let us first notice that by Lemma 5.12.2, the testing

condition (i) of Theorem 5.8.3 and its dual counterpart imply that the paraproducts

ΠW = ΠWT

and ΠV = ΠVT ∗

are bounded and that

‖ΠW‖L2(W)→L2(V) + ‖ΠV‖L2(V)→L2(W) ≤ C(d)1/2(T1 + T∗1).

Thus, it is sufficient to estimate the operator TW := TW − ΠW − (ΠV)∗. Lemma

5.11.4 implies that

∆VRTW∆W

Q=

∆VRTW∆W

Q, | rkQ− rkR| ≤ r;

0, | rkQ− rkR| > r,

76

so

〈TWf1, g1〉L2(V)=

∑Q,R∈D(Q0)| rkQ−rkR|≤r

〈TW∆WQf,∆V

Rg〉L2(V)

=∑

Q,R∈D(Q0)rkQ≤rkR≤rkQ+r

〈TW∆WQf,∆V

Rg〉L2(V)

+∑

Q,R∈D(Q0)rkR<rkQ≤rkR+r

〈TW∆WQf,∆V

Rg〉L2(V)

.

Let us estimate the first sum. The second sum will be treated similarly, by considering

the dual operator T ∗V. To estimate the first sum, we need to estimate the operator

T+W

:=∑

Q,R∈D(Q0)rkQ≤rkR≤rkQ+r

∆VRTW∆W

Q.

Since T is r-lower triangular, we can see that ∆VRTW∆W

Q= 0 if rkR ≥ rkQ and

R 6⊂ Q(r). So we can rewrite T+W

as

T+W

=∑

S∈D(Q(r)0 )

∑Q∈Chr S

∑R∈D(S)

rkQ≤rkR≤rkQ+r

∆VRTW∆W

Q=:

∑S∈D(Q(r)

0 )

T+,SW

,

where

T+,SW

=∑

Q∈Chr S

∑R∈D(S)

rkQ≤rkR≤rkQ+r

∆VRTW∆W

Q.

The testing condition (ii) of Theorem 5.8.3 implies that

‖T+,SW‖L2(W)→L2(V)

≤ T2.(5.42)

77

Note that if S ∩ S ′ = ∅ or | rkS − rkS ′| > r then

Ran T+,SW⊥ Ran T+,S′

W,

(ker T+,S

W

)⊥⊥(ker T+,S′

W

)⊥.

Therefore for fixed k ∈ Z, the operator T+,kW

defined by

T+,kW

:=∑j∈Z

∑S∈D(Q(r)

0 )rkS=k+(r+1)j

T+,SW

is the direct sum of the corresponding operators T+,SW

, and the estimate (5.42) implies

‖T+,kW‖L2(W)→L2(V)

≤ T2.(5.43)

Since T+W

= ∑rk=0 T

+,kW

, we can easily conclude from (5.43) that

‖T+W‖L2(W)→L2(V)

≤ (r + 1)T2.

However, by being more careful, we can obtain the following better dependence on

r:

‖T+W‖L2(W)→L2(V)

≤ (r + 1)1/2T2.(5.44)

To get this, observe that for for 0 ≤ j < k ≤ r

(ker T+,j

W

)⊥⊥(ker T+,k

W

)⊥.

78

Then, decomposing f1 = ∑rk=0 f

k, where

fk :=∑n∈Z

∑S∈D

rkS=k+(r+1)n

∑Q∈Chr S

∆WQf,

we get that

‖T+Wf1‖L2(V)

=∥∥∥∥∥T+

W

r∑k=0

fk∥∥∥∥∥L2(V)

=∥∥∥∥∥

r∑k=0

T+,kW

fk∥∥∥∥∥L2(V)

≤r∑

k=0‖fk‖

L2(W)≤(

r∑k=0‖fk‖2

L2(W)

)1/2

(r + 1)1/2;

here the last inequality is a consequence of the Cauchy–Schwarz inequality.

5.15 Estimates of parts involving constant func-

tions

Estimates

|〈TWf2, g1〉L2(V)| ≤ T1

|〈TWf1, g2〉L2(V)| ≤ T∗1

follow immediately from the testing condition (i) and its dual. Estimate

|〈TWf2, g2〉L2(V)| ≤ T3

is a direct corollary of the assumption (iii).

Note that in decompositions (5.40) and (5.41), we can replace Q0 by any of its

79

ancestors, so, as pointed out in Remark 5.8.4, it is sufficient that the estimate (iii)

hold only for sufficiently large cubes Q (meaning that for any Q0 ∈ D we can find

Q ∈ D, Q0 ⊂ Q such that (iii) holds for Q).

Moreover, if for the increasing sequence of cubes Qn, n ≥ 0, where Qn+1 is the

parent of Qn, we have that W(Qn) ≥ αnI, αn ↗∞ then writing the decomposition

(5.40) with Qn instead of Q0 and letting n→ +∞, we obtain

f =∑Q∈D

∆WQf =: f1.

The analogous condition for V implies a similar representation for g, so the theorem is

reduced to estimating 〈TWf1, g1〉L2(V), which was done using only testing conditions

(i), (ii) and their duals.

5.16 Estimates of the Haar shifts

In this section we will prove Lemma 5.9.1. Theorem 5.9.2 is then a simple corollary of

Theorem 5.8.3. We will need the following lemma, which is well known to specialists;

for the convenience of the reader we present its proof here.

Lemma 5.16.1. Let T be an integral operator with kernel K, Tf(x) =´K(x, y)f(y)dy,

where K is supported on Q × Q (Q ∈ D) and ‖K‖∞ ≤ |Q|−1. Assuming that the

weights V, W satisfy the matrix A2 condition (5.27) we have for the operator TW,

TWf = T (Wf)

‖TW‖L2(W)→L2(V)≤ d1/2[V,W]1/2

A2.

Proof. Take f ∈ L2(W), g ∈ L2(V), ‖f‖L2(W)

= ‖g‖L2(V)

= 1. As discussed above

80

in Chapter 2, we can assume without loss of generality that the measures V and

W are absolutely continuous with respect to scalar measures v and w respectively,

dV = V dv, dW = Wdw. We then can write

∣∣∣∣〈TWf, g〉L2(V)

∣∣∣∣ ≤¨Q×Q

∣∣∣〈V (x)K(x, y)W (y)f(y), g(x)〉Fd

∣∣∣ dv(x)dw(y).

The integral then can be estimated by

|Q|−1¨Q×Q‖V 1/2(x)W 1/2(y)‖ · ‖V 1/2(x)g(x)‖

Fd‖W 1/2(y)f(y)‖

Fddv(x)dw(y)

≤(¨

Q×Q‖V 1/2(x)g(x)‖2

Fd‖W 1/2(y)f(y)‖d

Fdv(x)dw(y)

)1/2

×

×(|Q|−2

¨Q×Q‖V 1/2(x)W 1/2(y)‖2dv(x)dw(y)

)1/2

= ‖f‖L2(W)

‖g‖L2(V)

(|Q|−2

¨Q×Q‖V 1/2(x)W 1/2(y)‖2dv(x)dw(y)

)1/2

.

In the last integral, we can replace the operator norm by the Frobenius (Hilbert–

Schmidt) norm ‖ · ‖S2

(recall that ‖A‖2S2

= tr(A∗A)):

¨Q×Q‖V 1/2(x)W 1/2(y)‖2dv(x)dw(y) ≤

¨Q×Q‖V 1/2(x)W 1/2(y)‖2

S2dv(x)dw(y)

=¨Q×Q

tr(V (x)W (y)

)dv(x)dw(y)

= tr(

V(Q)W(Q))

= ‖V(Q)1/2W(Q)1/2‖2S2

≤ d · ‖V(Q)1/2W(Q)1/2‖2

≤ d · |Q|2[V,W]A2.

81

Combining this with the previous estimate, we get the conclusion of the lemma.

5.17 Comparison of different truncations

In the testing conditions from Theorems 5.8.3 and 5.9.2, we used different trunca-

tions of the operator TW , namely TQW

and (TQ)W respectively. These operators are

generally different, but their difference can be estimated.

To state the estimate, we will require some new notation. Let PVQ

be the or-

thogonal projection in L2(V) onto the subspace of functions supported on Q and

orthogonal to {1Qe : e ∈ Fd}. Then for f ∈ L:

PVQf =

∑R∈D(Q)

∆VRf = 1

Qf − EV

Qf.

In this notation, the operator TQW

defined above can be written as TQW

= PVQTW .

Lemma 5.17.1. For operators TQW

and (TQ)W introduced above, we have for f

supported on Q

∥∥∥∥(TQW − PVQ

(TQ)W

)f∥∥∥∥L2(V)

≤ d1/2r · [V,W]1/2A2‖f‖

L2(W).

Proof. For f ∈ L and supported on Q, we have

(TQ

W− PV

Q(TQ)W

)f =

r∑k=1

PVQTQ(k) (Wf)

(the terms TQ(k) (Wf) with k > r are annihilated by PV

Q).

Each operator TQ(k) is an integral operator with kernel K

Q(k) supported on Q(k)×

Q(k) and satisfying ‖KQ(k)‖∞ ≤ |Q(k)|−1. Therefore applying Lemma 5.16.1 and using

82

the fact that PVQ

is an orthogonal projection (and so a contraction) in L2(V) we get

‖PVQTQ(k) (Wf)‖

L2(V)≤ d1/2[V,W]1/2

A2‖f‖

L2(W).

Summation over k completes the proof.

5.18 Proof of Lemma 5.9.1

We conclude this chapter by proving Lemma 5.9.1. Assume that the testing condition

(5.29) holds. Applying Lemma 5.17.1 with f = 1Qe and noticing that

‖PVQ

(TQ)Wf‖L2(V)

≤ ‖(TQ)Wf‖L2(V)

≤ T‖f‖L2(W)

,

we immediately get (5.30).

To get (5.31), some more work is needed. Write

TQ = T r+1 +r∑

k=0Tk,

where

T r+1 =∑

R∈Chr+1 Q

TR

Tk =∑

R∈Chk Q

TR,

with the obvious agreement that Ch0Q = {Q}.

Following the agreed upon notation, for a scalar integral operator T , we denote

83

by TW the operator given by

TWf := T (Wf),

whenever this expression is well defined.

The operators TR

are R-localized, meaning that TRf = T

R(1

Rf), and T

Rf is

supported on R, and the same holds for TR. The functions f ∈ DW,r

Qare constant

on cubes R ∈ Chr+1Q, so using the testing condition (5.29) and the fact that the

operators TR are R-localized, we get for f ∈ DW,r

Q

‖(T r+1)Wf‖2L2(V)

=∑

R∈Chr+1 Q

‖TR(W1Rf)‖2

L2(V)

x ≤∑

R∈Chr+1 Q

T‖1Rf‖2

L2(W)(5.45)

= T‖f‖2L2(W)

.(5.46)

To estimate the operators Tk, we estimate each block TRby Lemma 5.16.1, and using

the fact that TRis R-localized we get for f ∈ DW,r

Q

‖(Tk)Wf‖2L2(V)

=∑

R∈Chk Q

‖TR

(W1Rf)‖2

L2(V)

≤∑

R∈Chk Q

d · [V,W]A2‖1

Rf‖2

L2(W)

= d · [V,W]A2‖f‖2

L2(W).

Adding these estimates for k = 0, 1, . . . , r and combining them with (5.45) ,we see

84

that for any f ∈ DW,r

Q

‖(TQ)Wf‖L2(V)

≤(d1/2(r + 1)[V,W]1/2

A2+ T

)‖f‖

L2(W).

Since the projection PVQ

is a contraction in L2(V), the same estimate holds for the

norm ‖PVQ

(TQ)Wf‖L2(V)

, so combining it with Lemma 5.17.1, we obtain (5.31).

Finally, to show that (5.32) holds, let us recall that T = ∑R∈R TR , where R ⊂ D

is some finite collection. Then for each Q0 ∈ D we can find a cube Q ⊃ Q0 which

is not contained in any R ∈ R. Then TW1Qe = (TQ)W1

Qe, and (5.32) follows from

(5.29).

Bibliography

[BCTW16] K. Bickel, A. Culiuc, S. Treil, and B. Wick. Two weight estimates with

matrix measures for well localized operations. in preparation, 2016.

[Bez08] O. Beznosova. Linear bound for the dyadic paraproduct on weighted

Lebesgue space L2(w). J. Funct. Anal., 255(4):994–1007, 2008.

[BPW14] K. Bickel, S Petermichl, and B. Wick. Bounds for the Hilbert transform

with matrix A2 weights. preprint arXiv:1402.3886, 2014.

[Buc93] S. Buckley. Estimates for operator norms on weighted spaces and reverse

Jensen inequalities. Trans. Amer. Math. Soc., 340(1):253–272, 1993.

[BW14] K. Bickel and B. Wick. Well-localized operators on matrix weighted L2

spaces. preprint arXiv:1407.3819, 2014.

[BW15] K. Bickel and B. Wick. A study of the matrix Carleson embedding the-

orem with applications to sparse operators. preprint arXiv:1503.06493,

2015.

[CDPO16] A. Culiuc, F. Di Plinio, and Y. Ou. Domination of multilinear singular

integrals by positive sparse forms. preprint arXiv:1603.05317, 2016.

85

86

[CF74] R. R. Coifman and C. Fefferman. Weighted norm inequalities for maxi-

mal functions and singular integrals. Studia Math., 51:241–250, 1974.

[CT15] A. Culiuc and S. Treil. The Carleson embedding theorem with matrix

weights. preprint arXiv:1401.6570, 2015.

[Cul15] A. Culiuc. A note on two weight bounds for the generalized Hardy-

Littlewood maximal operator. preprint arXiv:0911.3437, 2015.

[CUMP10] D. Cruz-Uribe, J. Martell, and C. Pérez. Sharp weighted estimates for

approximating dyadic operators. Electron. Res. Announc. Math. Sci.,

17:12–19, 2010.

[CUMP12] D. Cruz-Uribe, J. Martell, and C. Pérez. Sharp weighted estimates for

classical operators. Adv. Math., 229(1):408–441, 2012.

[CW15] A. Culiuc and B. Wick. The boundedness of the Riesz and Ahlfors-

Beurling transforms in matrix weighted spaces. preprint available upon

request, 2015.

[DV03] O. Dragičević and A. Volberg. Sharp estimate of the Ahlfors-Beurling

operator via averaging martingale transforms. Michigan Math. J.,

51(2):415–435, 2003.

[FG13] F. Farroni and R. Giova. Change of variables for A∞ weights by means of

quasiconformal mappings: sharp results. Ann. Acad. Sci. Fenn. Math.,

38(2):785–796, 2013.

[FKP91] R. A. Fefferman, C. E. Kenig, and J. Pipher. The theory of weights

and the Dirichlet problem for elliptic equations. Ann. of Math. (2),

134(1):65–124, 1991.

87

[HMW73] R. Hunt, B. Muckenhoupt, and R. Wheeden. Weighted norm inequalities

for the conjugate function and Hilbert transform. Trans. Amer. Math.

Soc., 176:227–251, 1973.

[HS60] H. Helson and G. Szegö. A problem in prediction theory. Ann. Mat.

Pura ed App., 51(1):107–138, 1960.

[Hyt12a] T. Hytönen. The A2 theorem: Remarks and complements. preprint

arXiv:1212.3840, 2012.

[Hyt12b] T. Hytönen. The sharp weighted bound for general Calderón-Zygmund

operators. Ann. of Math. (2), 175(3):1473–1506, 2012.

[IKP14] J. Isralowitz, H. Kwon, and S. Pott. A matrix weighted T1 theo-

rem for matrix kernelled Calderon-Zygmund operators. preprint arXiv:

1508.01716, 2014.

[IM01] T. Iwaniec and G. Martin. Geometric function theory and non-linear

analysis. Oxford University Press, Oxford, 2001.

[Lac15] M. Lacey. An elementary proof of the A2 Bound. preprint

arXiv:1501.05818, 2015.

[Lai15] J. Lai. Two weight problems and Bellman functions on filtered spaces.

PhD thesis, Brown University, 2015.

[Ler12] A. Lerner. On an estimate of Calderón-Zygmund operators by dyadic

positive operators. preprint arXiv:1202.1860, 2012.

[Ler13] A. Lerner. A simple proof of the A2 conjecture. Int. Math. Res. Not.,

(14):3159–3170, 2013.

88

[LPR10] M. Lacey, S. Petermichl, and M. Reguera. Sharp A2 inequality for Haar

shift operators. Math. Ann., 348(1):127–141, 2010.

[LS09] M. Lacey and I. Sawyer, E.and Uriarte-Tuero. Two weight inequalities

for discrete positive operators. preprint arXiv:1506.07125, 2009.

[Mel05] A. Melas. The Bellman functions of dyadic-like maximal operators and

related inequalities. Adv. Math., 192(2):310–340, 2005.

[Nie10] M. Nielsen. On stability of finitely generated shift-invariant systems. J.

Fourier Anal. Appl., 16(6):901–920, 2010.

[NPTV02] F. Nazarov, G. Pisier, S. Treil, and A. Volberg. Sharp estimates in vec-

tor Carleson imbedding theorem and for vector paraproducts. J. Reine

Angew. Math., 542:147–171, 2002.

[NT96] F. Nazarov and S. Treil. The hunt for a bellman function: applications to

estimates for singular integral operators and to other classical problems

of harmonic analysis (russian). Algebra i Analiz, 8(5):32–162, 1996.

[NTV99] F. Nazarov, S. Treil, and A. Volberg. The Bellman functions and two-

weight inequalities for Haar multipliers. J. Amer. Math. Soc., 12(4):909–

928, 1999.

[NTV01] F. Nazarov, S. Treil, and A. Volberg. Bellman function in stochastic con-

trol and harmonic analysis. In Systems, approximation, singular integral

operators, and related topics (Bordeaux, 2000), volume 129 of Oper. The-

ory Adv. Appl., pages 393–423. Birkhäuser, Basel, 2001.

89

[NTV08] F. Nazarov, S. Treil, and A. Volberg. Two weight inequalities for indi-

vidual Haar multipliers and other well localized operators. Math. Res.

Lett., 15(3):583–597, 2008.

[Pet00] S. Petermichl. Dyadic shifts and a logarithmic estimate for Hankel opera-

tors with matrix symbol. C. R. Acad. Sci. Paris Sér. I Math., 330(6):455–

460, 2000.

[Pet07] S. Petermichl. The sharp bound for the Hilbert transform on weighted

Lebesgue spaces in terms of the classical Ap characteristic. Amer. J.

Math., 129(5):1355–1375, 2007.

[Pet08] S. Petermichl. The sharp weighted bound for the Riesz transforms. Proc.

Amer. Math. Soc., 136(4):1237–1249, 2008.

[PTV02] S. Petermichl, S. Treil, and A. Volberg. Why the Riesz transforms are

averages of the dyadic shifts? In Proceedings of the 6th International

Conference on Harmonic Analysis and Partial Differential Equations (El

Escorial, 2000), number Vol. Extra, pages 209–228, 2002.

[PTV10] C. Pérez, S. Treil, and A. Volberg. On the A2 conjecture and Corona

decomposition of weights. preprint arXiv:1006.2630, 2010.

[PV02] S. Petermichl and A. Volberg. Heating of the Ahlfors-Beurling operator:

weakly quasiregular maps on the plane are quasiregular. Duke Math. J.,

112(2):281–305, 2002.

[RdF84] J. Rubio de Francia. Factorization theory and Ap weights. Amer. J.

Math., 106(3):533–547, 1984.

90

[Saw82] E. Sawyer. A characterization of a two-weight norm inequality for max-

imal operators. Studia Math., 75(1):1–11, 1982.

[Saw88] E. Sawyer. A characterization of two weight norm inequalities for frac-

tional and Poisson integrals. Trans. Amer. Math. Soc., 308(2):533–545,

1988.

[Tre12] S. Treil. A remark on two weight inequalities for positive dyadic opera-

tors. preprint arXiv:1201.1455, 2012.

[TV97] S. Treil and A. Volberg. Wavelets and the angle between past and future.

J. Funct. Anal., 143(2):269–308, 1997.

[Vol97] A. Volberg. Matrix Ap weights via S-functions. J. Amer. Math. Soc.,

10(2):445–466, 1997.

[Wit00] J. Wittwer. A sharp estimate on the norm of the martingale transform.

Math. Res. Lett., 7(1):1–12, 2000.