matrix ap weights via s-functions 0. introduction the

JOURNAL OF THEAMERICAN MATHEMATICAL SOCIETYVolume 10, Number 2, April 1997, Pages 445–466S 0894-0347(97)00233-6

MATRIX Ap WEIGHTS VIA S-FUNCTIONS

A. VOLBERG

0. Introduction

The statement of the problem. In this paper we study weighted norm inequal-ities with matrix valued weights. Namely, let W be a d × d matrix weight, i.e. aL1-function whose values are selfadjoint nonnegative d × d matrices. We supposethat the weight W is defined on the unit circle T = {z ∈ C : |z| = 1} or the realline R.

Let Lp = Lp(Cd) be the space of all measurable vector functions on T whose Cd-norm is summable to the power of p. Let Hp = Hp(Cd) be the corresponding Hardyspace of analytic functions, and let P+ be projection in Lp onto Hp annihilatingantianalytic functions in Lp which vanish at the origin. Let H denote the Hilbert

transform, H = −iP+ + i(I − P+). Clearly, P+, P−def= I − P+, and H act as usual

operators P+, P−, and H coordinatewise if we fix for all t ∈ T the same coordinatebasis in Cd.

We are interested in the conditions on W under which the following weightednorm inequality for operator H holds (say for all f ∈ Lp ∩ L∞):∫

T(W (t)2/pHf(t), Hf(t))p/2dm(t) ≤ C

∫T(W (t)2/pf(t), f(t))p/2dm(t).

Clearly this inequality is equivalent to the same inequality for P+ (with anotherconstant). Here m denotes Lebesgue measure on T, and W (t)2/p stands for theoperator power of nonnegative selfadjoint operator W (t). If we define a weightedspace Lp(W ) as the space of all measurable Cd-valued functions on T satisfying

‖f‖pLp(W )

def=

∫T(W (t)2/pf(t), f(t))p/2dt <∞

(of course we should factorize it over the subspace of functions of norm 0), then thelast inequality means that H (or, equivalently, P+) is a bounded operator in Lp(W ).The notation is chosen in such a way that Lp(W ) becomes a familiar weighted Lp

space in the scalar situation d = 1. We consider the range (1,∞) of p.In the scalar-valued setting d = 1 the answer is given by the famous Hunt–

Muckenhoupt–Wheeden theorem which says that the Muckenhoupt condition (Ap)

supI

(1

|I|∫I

W

)1/p

·(

1

|I|∫I

W−q/p)1/q

<∞

Received by the editors May 24, 1996 and, in revised form, December 2, 1996.1991 Mathematics Subject Classification. Primary 42B20, 42A50, 47B35.Key words and phrases. Ap weights, matrix-functions, area integrals, singular integrals, Car-

leson measures, Triebel-Lizorkin spaces.This work was partially supported by National Science Foundation Grant DMS9622936.

c©1997 American Mathematical Society

445

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

446 A. VOLBERG

(q = p/(p− 1); supremum is taken over all intervals) is necessary and sufficient inthis case.

The main purpose of this paper is to give a matrix analog of (Ap) and to provethat it is necessary and sufficient for the matrix weighted inequality to hold.

In [TV1] the matrix (A2) condition appeared:

supI

∥∥∥∥∥(

1

|I|∫I

W

)1/2

·(

1

|I|∫I

W−1

)1/2∥∥∥∥∥ <∞.

Here the power 1/2 stands for the operator power of nonnegative operators—averages of W in our case. Later S. Treil came up with the conjecture of whatthe matrix (Ap) should look like.

The matrix (Ap) condition does not have such a simple form if p is not 2. Itcannot be obtained by the combination of averagings and raising to operator powers.It requires the analysis of the geometric meaning of the Muckenhoupt condition(Ap), even though there is no geometry in C1. We explain the matrix (Ap) in thenext section. Now let us remark that very few scalar methods are available in thevector situation.

The difficulties. As an illustration of what kind of difficulties one can encounterwith the vector case, let us present two very simple examples. It is trivial in thescalar case that if we have an integral operator in Lp(W ) with positive (scalar)kernel, and we know that an operator with a bigger kernel is bounded, then theoriginal operator is bounded too. This statement (even for scalar kernels) doesnot hold for weighted Lp spaces with matrix weights. Certainly if W (t) can bediagonalized by the same basis for each t ∈ T, we do not have any difficulties. Butthis is not the case we are interested in. Another difficulty comes from the factthat while any nonzero nonnegative operator is invertible in C1, this is quite falsein Cd. For example, suppose we meet the expression (we do meet such expressionswhile working with matrix weights) (A(I+B)−1x, (I+B)−1x), where A and B arenonnegative operators, B ≤ I, and x is a vector in Cd. If A,B, x are numbers, theestimate from below 1

4A|x|2 would follow. But no estimate δ||x||2, δ > 0, exists foroperators.

The main idea. Nazarov and Treil in [NT] overcome all the difficulties by usingthe Bellman function approach (see below). But first a simple idea which wassuccessfully used in [TV1] comes into play. Roughly speaking the idea is as follows.The first step is to choose a good basis in Lp(W ). It should be good in the sensethat:

1) the coefficient space with respect to this basis should be relatively simple;2) the matrix of the operator H (or P+) in this coefficient space should also be

relatively simple (desirably as close to a diagonal matrix as possible).Let hI stand for a Haar function. It turns out that matrix (Ap) is equivalent

to the fact that {hICd}I∈D forms an unconditional basis in Lp(W ). In the scalarsituation this is well known (see [FJW]), and the coefficient space is the weightedTribel-Lizorkin space f02

p (see [FJW] and below for the notation).

The Bellman function. The main tool in [NT] is the use of the Bellman function.Here is what it is. The desired estimates involve the averages of functions and


MATRIX Ap WEIGHTS VIA S-FUNCTIONS 447

weights over dyadic intervals. Consider these averages as “variables”. Generally,the estimates one encounters will have the form∑

I⊂JaI ≤ CFJ ,

where aI is a certain (depending on the problem) function of our “variables” (read“averages”) corresponding to I. Now try to build a pretty concave function B (letus call it the Bellman function of the problem) of our “variables” in such away that:

1) for all I ∈ D, B(I) ≤ CFI ,2) B(I)− 1/2(B(I−) +B(I+)) ≥ |aI |.

Here B(I) stands for a value of function B on “variables”(i.e. averages) correspond-ing to I. The second requirement above is the strong concavity condition. Afterfunction B is built, the desired estimate is obtained just by repeatedly applyingthe concavity. Actually it happens very often that the existense of B is equivalentto the singular integral estimate under consideration.

The function B is closely related to the Burkholder function from [B1], [B2],[B3], [Str], Section 6.3.15, constructed for obtaining the best constants in scalarmartingale and singular integral estimates.

The Bellman functions have been used for decades in Control Theory. They canbe interpreted as the supremum of price over different plans of distributing theresources. By nature such functions are convex. In our dyadic estimates we haveaverages (or integrals) over J and we wish to show that whatever the distributionof them over the dyadic sons, grandsons, et cetera is, the price will not exceed CFIprovided that the price for distributing from any dyadic I to its dyadic “sons” I−and I+ is aI .

Carleson measures and A∞. In the present paper another approach to provingmatrix weighted inequalities is suggested. The observation is that even thoughmany scalar methods are not available now, there is at least one which still managesto survive. We mean the good λ-inequality, which should be more rigorously calledthe relative distributional inequality (see [St], pp. 206–209). Let us also remind thereader about the role of A∞ weights. A positive function cannot be much biggerthan its average over I on a large portion of I. If we change the word “bigger” to“smaller”, we get the class of (A∞) functions (see various definitions of (A∞) on pp.196–197, and 218 of [St]). In the classical case (see [St], pp. 200–205) the class (A∞)comes into play as follows. The singular integral operator is “factorized” throughthe maximal operatorM in the sense that the weighted estimate ||Hf ||W ≤ C||f ||Wsplits into estimates ||Hf ||W ≤ C||Mf ||W and ||Mf ||W ≤ C||f ||W .

The proof of the first estimate uses the good λ- inequality, which, roughly speak-ing, amounts to the fact that |H | ≤ CM up to a “telescopically graded” set. Moreprecisely, W ({|Hf | > 2λ} \ {Mf < δλ}) ≤ 1

2W ({|Hf | > λ}) for any λ > 0 if δ > 0is small. This property follows from the (A∞) property of W .

To some extent, but in a more mysterious and strange way, the same two ideas(splitting the problem into two inequalities, where one of them amounts to the(A∞) property of weight and uses the good λ-inequality) work in this paper.

But we will really factorize our operator,H = E∗U ·D·EW , where U = W−q/p, q =p/(p − 1), and where E is an embedding operator. It will embed to a (definedbelow) Triebel-Lizorkin space f02

p (Cd). D is an almost diagonal operator in the

sequence space f02p (Cd). Those readers who preferred to read the main body of


448 A. VOLBERG

this article before reading the introduction may recognize the geometric meaningof U : U1/q defines the metric which is conjugate to the metric defined by W 1/p.In this factorization, the (A∞) property of the weight W will be used to prove theboundedness of the first factor EW (and not of the second “factor” as it happenedin the classical case treated on pp. 201–207 of [St]). Actually the property of thematrix weight W corresponding to (A∞) will be called (Ap,∞). For matrix weightsthere is a whole spectrum of (Ap,∞) classes graded by p ∈ (1,∞)—they all coincidewith the usual (A∞) in the scalar situation.

So, as we will see, the boundedness of H follows from W ∈ (Ap,∞), U ∈ (Aq,∞).Together, these two properties of W are equivalent to the matrix (Ap) conditionimposed on W .

In proving this equivalence we come across a Carleson measure characterizationof (scalar) (A∞) weights obtained in [FKP], [Bu1], and [Bu2]. For example, thematrix (A2,∞) condition is equivalent to the requirement that the following measurein the disc is a Carleson measure:

µdef=∑I∈D

||〈W 〉−1/2I (〈W 〉I− − 〈W 〉I+)〈W 〉−1/2

I || · |I| · δcI .

Here cI denotes the center of the Carleson box built on the arc I: QIdef= {z ∈

D : 1 − |I| ≤ |z| < 1, z/|z| ∈ I}. This idea from [FKP], [Bu1], [Bu2] that (A∞)is equivalent to a certain Carleson measure condition for the measures built bythe averages of W , turns out to be extremely useful in [TV1], [TV2], [TV3] forproving that EW and EU are the correct embeddings. This is not surprising becauseCarleson measure is what embeddings are about.

Even though we do not know such a neat characterization of (Ap,∞) if p is not 2,we managed to prove that (Ap,∞) is equivalent to the embedding we need. This isdone in Section 3, after explaining the matrix (Ap) condition in Section 1, and thematrix (Ap,∞) condition in Section 2. Then we prove the boundedness of the Hilberttransform H in Sections 4 and 5. To do this we will need the second ingredientfrom the classical theory: we use a certain variant of the good λ-inequality, the ideaof which one can trace back to the works of Burkholder and Gundy [BG] and Axler,Chang, and Sarason [AChS].

The latter work is especially important because of its operator theory motivation.

Operator theory motivation. There is a part of the theory of singular integralswhich treats the Hardy spaces in Rn. The passage from R1 (or T) to Rn makes thetheory immensely richer. At the same time the theory of vector-valued Hardy spaceson T was developing for the needs of the spectral theory of operators (see [Nik]),because the dilation theory of linear contractions (see [Nik]) reduces questions aboutbounded operators in Hilbert space to functon theoretic questions in a vector Hardyspace. Even a finite-dimensional case is known to be much richer than the scalarcase (see [Nik] again). So the increase in dimension in this direction also enriches thetheory. The connection with singular integrals becomes manifest if one considers

Hankel and Toeplitz operators, given by formulae HF fdef= P−(Ff) = (I−P+)(Ff)

and TF fdef= P+(Ff), where F is a d × d matrix function and f ∈ Hp(Cd). One

such problem which is very difficult already in the scalar case d = 1 was consideredin [AChS], and then a similar problem was considered in [S1], [S2]. It is closelyrelated to a two-weight estimate for the Hilbert transform which is still open (seee.g. [TVZ], and the literature cited there).



There is another classical problem for Toeplitz operators which leads to aweighted estimate (with one weight) for the Hilbert transform. In fact, the invert-ibility of the Toeplitz operator TF on Hp is equivalent (see [Si]) to the factorizationF = G∗1 ·G2, where G1, G2 are d× d outer matrix functions such that the followingestimate holds:∫

T(V (t)P+f(t), P+f(t))dm(t) ≤ C

∫T(W (t)f(t), f(t))dm(t),(0.1)

where we denote W = G1G∗1, V = (G−1

2 )∗G−12 .

On first glance the matrix weights W = G1G∗1, V = (G−1

2 )∗G−12 seem to be

different, but the invertibility of TF implies easily F−1 ∈ L∞d×d, which means thatthe matrix weights V,W are equivalent in the sense that there exists a constant Csuch that for all e ∈ Cd and for almost all t ∈ T

1

C(V (t)e, e) ≤ (W (t)e, e) ≤ C(V (t)e, e) .

This is how one can come to the matrix weighted norm inequality considered inthis paper.

As far as we know, the first results about the matrix weight inequality wereobtained by Steven Blum [Bl1], [Bl2], who noticed that if the matrix weight W isassumed to be appropriately “smooth”, then it can be diagonalized by a “smooth”unitary matrix function; and the operator of multiplication on this unitary matrixfunction commutes with P+ up to a compact term (because of “smoothness”). Thisapproach leads to the pointwise diagonalization of the estimate under considerationand so to the corresponding scalar problem.

In the present work the matrix function W is a priori arbitrary. Rather thandoing pointwise diagonalization (which is not available now) we prefer to come toglobal (almost) diagonalization of our operator in the weighted space Lp(W ).

Notation. The symbol f02p denotes the sequences s = {sI}I∈D enumerated by the

set of dyadic intervals D such that

‖s‖pf02p

=

∫ (∑I3x

|sI |2 1

|I|

)p/2

dx <∞.

Similarly one can introduce f02p (Cd) by just replacing numbers sI by Cd vectors sI

and | · | by ‖ · ‖. One can consider f02p (w), which is a weighted Triebel-Lizorkin

space. This amounts to changing the integration dx to w(x)dx. As to the vectorweighted norm one just replaces ‖sI‖2 by ‖W 1/p(x)sI‖2 to obtain the definition off02p (Cd,W ), where W (x) is now a positive selfadjoint operator (matrix).

1. Matrix Ap condition

Let ‖ ‖ denote the standard Hilbert norm in Cd, and let ( , ) be the scalarproduct in Cd. If ρ denotes a norm in Cd, we can notice that ρ(x) �

d‖Ax‖, where

A is a positive selfadjoint operator and constants depend only on the dimension d.In fact, let us consider the ellipsoid of the largest volume contained in the ρ-unitball {x : ρ(x) ≤ 1}, which is a convex subset of Cd. Clearly, {x : ‖Ax‖ ≤ 1} ⊂ {x :ρ(x) ≤ 1} ⊂ Cd{x : ‖Ax‖ ≤ 1}, where A is a positive matrix mapping the ellipsoidonto the standard Hilbert ball.


450 A. VOLBERG

If we have a family ρt of norms and consider Lp(ρ) = {f : (∫ρt(f)pdt)1/p <∞},

then we can also introduce an equivalent norm ‖f‖Lp(ρ) � (∫ ‖A(t)f‖pdt)1/p, with

A(t) as above constructed for ρt.In weighted matrix inequalities one deals usually with Lp(W ) spaces, where

W (t) is matrix function with positive (a.e.) W (t) : Cd → Cd, and ‖f‖Lp(W )def=

(∫ ‖W (t)1/pf‖p)1/p.

As we see one can consider Lp(ρ) as well and reduce it to Lp(W ) by settingA(t) = W 1/p(t).

Sometimes it is more convenient to use the language of ρ and sometimes that ofW .

Recall that the dual metric ρ∗ is given by

ρ∗(x) = supy 6=0

|(x, y)|ρ(y)

.

We can identify (Lp(ρ))∗ with Lq(ρ∗),1

p+

1

q= 1, and (Lp(W ))∗ with Lq(U), where

the W 1/p-metric should be dual to the U1/q-metric, that is,

W 1/p(t) = U−1/q(t), for a.e. t.

To introduce the analog of Ap for matrix weights we use the language of metrics ρand their averagings. Consider the norms (I is any interval)

ρp,I(x)def=

(1

I

∫I

ρpt (x)dt

)1/p

.

The unit ball again can be “reduced” to an ellipsoid, that is, there exists a positiveAI such that

ρp,I(x) �d‖AIx‖.

Let us reduce the dual metric as well:

ρ∗q,I(x) �d‖BIx‖.

Proposition 1.1. 1) |(x, y)| ≤ ρp,I(x)ρ∗q,I(y).

2) (ρp,I)∗ ≤ ρ∗q,I .

Proof. 1) Obvious:

|(x, y)| = 1

I

∫I

|(x, y)| ≤ 1

I

∫I

ρt(x)ρ∗t (y) ≤ ρp,I(x)ρ

∗q,I(y).

2) This is a paraphrase of the first assertion.

Definition. If we have an opposite inequality

ρ∗q,I ≤ C(ρp,I)∗, ∀ I,

then ρ is called an Ap-metric. If ρt(x) = ‖W 1/p(t)x‖ and ρ is an Ap-metric, thenW is called an Ap (matrix) weight.

Proposition 1.2. The following assertions are equivalent:

1) ρ is an Ap-metric;2) ρp,I ≤ C(ρ∗q,I)

∗;



3) if ρp,I(x) � ‖AIx‖, ρ∗q,I(x) � ‖BIx‖, then B2I � A−2

I in the sense of selfad-joint operators.

Proof. Obvious.

Remark. For p = q = 2 the operators AI and BI can be calculated if we use thelanguage of matrix weight W ,

AI = 〈W 〉1/2I , BI = 〈W−1〉1/2I .

〈W 〉−1I ≤ 〈W−1〉I .

The A2-condition obtained in [TV1], [TV2] had a form

〈W−1〉I ≤ C〈W 〉−1I ,

where inequality is understood in the sense of selfadjoint operators.

2. Matrix A∞ condition

Let us introduce the Ap,∞ condition on metric weight which serves as the analogof the A∞ condition for scalar weights. Its definition goes exactly along the linesof the definition of Ap, except ρ∗q,I is replaced by

ρ∗o,I(x)def= e

1I

∫I

log ρ∗t (x)dt.

Proposition 2.1. (ρp,I)∗ ≤ ρ∗o,I .

Proof. The inequality can be rewritten as

(ρ∗o,I)∗ ≤ ρp,I .

Then it follows from two elementary observations:

ρo,I ≤ ρp,I ,

which is just Jensen’s inequality, and

(ρ∗o,I)∗ ≤ ρo,I .(2.1)

This last inequality follows from

log |(x, y)| − 1

I

∫I

log ρ∗t (y)dt ≤1

I

∫I

log ρt(x)dt,

and this is the integration of the logarithm of |(x, y)| ≤ ρ∗t (y)ρt(x).

Definition. If we have an opposite inequality

ρ∗o,I ≤ C(ρp,I)∗, ∀ I(2.2)

which is equivalent to

ρp,I ≤ C(ρ∗o,I)∗, ∀ I,(2.3)

then ρ is called an Ap,∞-metric. If ρt(x) = ‖W 1/p(t)x‖ and ρ is an Ap,∞-metric,then W is called an Ap,∞ matrix weight.

Remark. By Jensen’s inequality ρ∗o,I ≤ ρ∗q,I and thus Ap ⇒ Ap,∞.

Let Bρ denote the unit ball in the metric ρ.

Proposition 2.2. The following assertions are equivalent:

1) ρ ∈ Ap,∞;


452 A. VOLBERG

2) 1I

∫Ilog volBρtdt ≤ C + log volBρp,I for all I.

In particular, W ∈ Ap,∞ if and only if

detAI ≤ Ce1I

∫I

log detW (t)1/pdt, ∀ I,where AI is a positive definite operator reducing ρp,I in the previous sense, namely

ρp,I(x) � ‖AIx‖.Proof. 1) ⇒ 2). It is convenient to use the language of matrices. So let ρt(x) �‖A(t)x‖ and ρp,I(x) � ‖AIx‖. Let {ei}di=1 be an orthonormal basis of eigenvectorsof AI . We apply 1) to each of these vectors:

1

I

∫I

log ‖A(t)−1ei‖ ≤ C + log ‖A−1I ei‖, i = 1, . . . , d.

Use the following elementary observation: for any positive definite d × d matrix

Q and any orthonormal basis {ei}di=1 in Cd, detQ ≤ ∏di=1(Qei, ei) ≤

∏di=1 ‖Qei‖.

This inequality is just the consequence of arithmetic mean-geometric mean inequal-ity. Summing up the inequalities above we get

1

I

∫I

log detA(t)−1dt ≤ C + log detA−1I ,

which means

1

I

∫I

log volBρt ≤ C + log volBρp,I .

2) ⇒ 1). Let g be a unit vector. Choose g1(t) ≡ g, g2(t), . . . , gd(t) to be unit

vectors such that for C(t)def= A−1

I A(t) we have detC(t)−1 ≥ Cd∏d

i=1 ‖C(t)−1gi(t)‖.We know that

1

I

∫I

‖C(t)gi(t)‖p ≤ Cd,p

d∑j=1

1

I

∫I

|(C∗(t)ej , gj(t))|p

≤ Cd,p

d∑j=1

‖A(t)A−1I ej‖p � Cd,p

d∑j=1

ρpp,I(A−1I ej)

< C ′d,p <∞.

In particular,

d∑i=1

1

I

∫I

log+ ‖C(t)gi(t)‖dt ≤ 1

pC ′d,p.

We are given1

I

∫I log detC(t)−1dt ≤ C which implies

1

I

∫I

log+ ‖C(t)−1g‖dt ≤ C +

d∑i=1

1

I

∫I

log− ‖C(t)−1gi(t)‖

≤ C +d∑i=1

1

I

∫I

log+ ‖C(t)gi(t)‖dt ≤ C +1

pC ′d,p = C′′ <∞.



This implies

1

I

∫I

log ‖A(t)−1g‖dt ≤ C′′ + log ‖A−1I g‖.

The matrix A2,∞ condition can be reformulated (see [TV1]) in terms of the

Carleson property of a certain measure built using ‖W−1/2I (WI1 − WI2)W

−1/2I ‖

exactly as it is done for scalar A∞ in Fefferman, Kenig and Pipher’s work [FKP].The same should probably be true for Ap,∞ if some AI -built measure is used.

3. Embedding theorem

In this section we deal with ρ ∈ Ap,∞. We consider the “reducing” selfadjointoperators AI : (

1

I

∫I

ρt(x)pdt

)1/p

� ‖AIx‖.(3.1)

Then the Ap,∞ condition can be written as (ρp,I)∗ ≥ cρ∗o,I , that is,

∀ J, 1

J

∫J

log ρ∗t (x)dt ≤ C + log ‖A−1J x‖,(3.2)

with the inverse inequality being always valid as well with C = 0 (see Proposition2.1).

Lemma 3.1. Let ρ ∈ Ap,∞, and given a dyadic I let Fk denote the family of dyadicJ, J ⊂ I, such that

‖A−1J AI‖ ≥ ek.

Then ∣∣∣∣∣ ⋃J∈Fk

J

∣∣∣∣∣ ≤ c

k|I|.(3.3)

Proof. Let {ei}di=1 be a standard basis in Cd. Denote by F ik the set of dyadic

intervals with

‖A−1J AIei‖ ≥ ek.

First use (3.2) to conclude that

1

I

∫I

log ρ∗t (AIei) ≤ C.(3.4)

Then use the inverse to (3.2) to conclude that for J ∈ F ik

1

J

∫J

log+ ρ∗t (AI , ei) ≥ log ‖A−1J AIei‖ ≥ k.(3.5)


454 A. VOLBERG

If we could replace log by log+ in (3.4) we would be done by standard argument.To do this replacement let us notice that

1

I

∫I

log+ ρ∗t (AIei) =1

I

∫I

log ρ∗t (AIei) +1

I

∫I

log+ 1

ρ∗t (AIei)

≤ C +1

pI

∫I

log+ 1

(ρ∗t (AIei))p≤ C +

1

pI

∫I

log+(ρt(A−1I ei))

p

≤ C +1

pI

∫I

ρpt (A−1I ei) ≤ C +

1

p(C‖ei‖)p ≤ C′.

In this chain we used an elementary inequality

ρt(A−1I ei)ρ

∗t (AIei) ≥ |(ei, ei)| = 1.

The lemma is proved.

Remark. It is needless to say that for the scalar situation all Ap,∞ just coincidewith A∞. Lemma 3.1 reflects a characteristic property of A∞ functions: the setwhere the function is much smaller than its average is small.

Let hJ denote a standard Haar function normalized in L2(dt).Before stating the theorem we need to introduce a new type of weighted Triebel-

Lizorkin spaces. Given a family of linear operators {CJ}J∈D we consider the spacef02p (Cd, {CJ}) to consist of sequences {s}J∈D such that {CJsJ} ∈ f02

p .The following is the embedding theorem which gives the name to the section.

As always1

p+

1

q= 1.

Theorem 3.2. Let ρ ∈ Ap,∞ and let {AJ} be its reducing operators as in (3.1).Let sJ =

∫J g hJ . Then the following inequality holds:

‖{sJ}‖f02q ({A−1

J }) ≤ C‖g‖Lq(ρ∗).

Remarks. 1) Even in the simplest case p = q = 2 and the scalar weight w ∈ A∞this result seems to be rather new (although we did not check all the literature). Itamounts to the inequality

w ∈ A∞ ⇒∑J∈D

〈w〉−1J |(g, hJ)|2 ≤ C‖g‖2

L2( 1w ),

which was mentioned in [TV1], [TVZ]. Notice that when w ∈ A2 the left part isequivalent to∑

〈w−1〉J |(g, hJ)|2 =

∫ ∑J3x

|(g, hJ)|2 1

|J |w−1(x)dx

= ‖{(g, hJ)}‖2f022 (w−1) � ‖g‖2

L2(w−1).

2) There are interesting general relationships between

f02p (Cd, ρ)

def=

{sI} :

∫ (∑I3t

ρ2t (sI)

1

|I|

)p/2

dt <∞



and

f02p (Cd, {CI}) =

{sI} :

∫ (∑I3t

‖CIsI‖2 1

|I|

)p/2

dt <∞ .

Quite naturally one should choose CI reducing ρt:(1

I

∫I

ρpt (x)dt

)1/p

� ‖CIx‖,(3.6)

and then

f02p (Cd, {CI}) =

{sI} :

∫ (∑I3t

(1

I

∫I

ρps(sI)ds

)2/p1

|I|

)p/2

dt <∞ .

It is clear that for p = 2 these spaces coincide. It will be proved elsewhere that forp ≥ 2, ρ ∈ Ap,∞ they also coincide. Clearly, many questions appear: what aboutp < 2?, what if we change exponent p to another one in (3.6)?, etc.

Notice that if W ∈ Ap these spaces coincide.

3) The space f02q (Cd, {A−1

J }) is not of the type considered in 3). The difference

is that now CJ = A−1J are not generated as reducing operators as it was in (3.6).

They are inverses to reducing operators.Notice that if W ∈ Ap, then inverses to reducing operators are also reducing for

the dual weight U = W−q/p and in this particular case we are going to prove in thenext section that

Lp(W ) ≈ f02p (Cd, {AJ}),(3.7)

where AJ are reducing operators introduced in (3.1).

Proof of Theorem 3.2. Fix g ∈ Lq(ρ∗) and a test function f ∈ Lp. We need to showthat ∑

J∈D

∣∣∣∣(∫ g hJ , A−1J

∫f hJ

)∣∣∣∣ ≤ C‖f‖Lp‖g‖Lq(ρ∗).(3.8)

The sum on the left can be rewritten as∫ (∑J3t

∣∣∣∣(∫ g hJ , A−1J

∫f hJ

)∣∣∣∣ 1

|J |

)dt

def=

∫S(t)dt.

Let us consider a nonnegative function h(t) and introduce

Sh(t)(t) =∑J3t

|J|≤h(t)

∣∣∣∣(∫ g hJ , A−1J

∫f hJ

)∣∣∣∣ 1

|J | .

Sh(t)(t) ≤ S(t) but ∫Sh(t)(t)dt ≥ a

∫S(t)dt

if the function h(t) has the following property:

∀ J ∈ D |{t ∈ J : h(t) ≥ |J |}| ≥ a|J |.(3.9)

We are going to choose now a function h satisfying (3.9) and such that

Sh(t)(t) ≤ B (M‖f‖p∗)1/p∗ (t) (M (ρ∗(g))q∗)1/q∗ (t)(3.10)


456 A. VOLBERG

for certain B < ∞, 1 < p∗ < p, 1 < q∗ < q. M denotes the maximal function.After (3.10) we are done:∫

S(t)dt ≤ a−1

∫Sh(t)(t)dt ≤ a−1

∫(M‖f‖p∗)1/p∗ (t) · (Mρ∗(g)q∗)1/q∗ (t)

≤ a−1

(∫(M‖f‖p∗)p/p∗

)1/p(∫(Mρ∗(g)q∗)q/q∗

)1/q

≤ C(a, p/p∗, q/q∗)(∫

‖f‖p)1/p(∫

ρ∗(g)q)1/q

= C(a, p/p∗, q/q∗)‖f‖Lp · ‖g‖Lq(ρ∗),which is what we need.

To choose h(t) satisfying (3.9) and (3.10) we follow the algorithm below.Fix a dyadic interval I. Fix a large k and delete from I all the dyadic intervals

for which

‖AIA−1J ‖ ≥ ek.

By Lemma 3.1 the set EI which rests has the property that

|EI | ≥ 1

2|I|.

Let us introduce

t ∈ I, SA−1I (g)(t) =

∑J3tJ⊆I

‖A−1I

∫g hJ‖2 1

|J |

1/2

;

t ∈ I, SAIA−1J (f)(t) =

∑J3t,J⊆I

‖AIA−1J

∫f hJ‖2 1

|J |

1/2

.

We are interested in proving that there are relatively large subsets of I on whichthese S-functions are smaller than maximal functions from (3.10).

To find such sets it is sufficient to estimate any averages of these S-functions.Fix α > 1 and very close to 1. Then

1

I

∫I

(SA

−1I (g)(t)

)αdt ≤ C(α)

1

I

∫‖A−1

I g · χI‖αdt

= C(α)1

I

∫I

‖A−1I g‖αdt.

The first inequality follows from the classical Lα estimate of the S-function by theLα norm of the function when α > 1 (see e.g. [St]). To continue the estimate weuse that we are in the space of finite dimension to write ({ei} is a standard basisof Cd):

‖A−1I g(t)‖ ≤ Cd

d∑i=1

|(A−1I g(t), ei)| ≤ Cd

d∑i=1

ρ∗t (g)ρt(A−1I ei).



Thus

‖SA−1I (g)‖Lα( dtI ) ≤ C(α, d)

d∑i=1

(1

I

∫I

ρ(p+ε)αt (A−1

I ei)

) 1(p+ε)α

·(

1

I

∫I

ρ∗t (g)(q−δ)α

) 1(q−δ)α

.

(3.11)

Here (p + ε)−1 + (q − δ)−1 = 1, and ε is chosen to be very small to guaranteeq − δ > 1. The choice of ε and α is also dictated by the observation that thepositive function t→ ρpt (x) is an A∞-function if ρ ∈ Ap,∞.

In fact, inequalities (2.1) and (2.3) imply that

1

I

∫I

ρpt (x)dt ≤ Ce1I

∫I

log ρpt (x)dt,(3.12)

which is equivalent to A∞ (see [GCRF] and [St], p. 218). As ρpt (x) is (uniformlyin x) in A∞ it satisfies (uniformly in x) a reverse Holder inequality. In particular,there exist a p > p such that

∀ I, ∀ x,(

1

I

∫I

ρpt (x)dt

)1/p

≤ C

(1

I

∫I

ρpt (x)dt

)1/p

.(3.13)

Now we choose α close to 1 and ε close to 0 to guarantee that

(p+ ε)α ≤ p.

In particular, (3.13) implies(1

I

∫I

ρ(p+ε)αt (A−1

I ei)dt

) 1(p+ε)α

≤ C

(1

I

∫I

ρpt (A−1I ei)

)1/p

≤ C′‖AIA−1I ei‖ = C′,

(3.14)

where we use the definition (3.1) of AI .Now we can continue (3.11) using (3.14):

‖SA−1I (g)‖Lα(I, dtI ) ≤ C(α, d) · d · C′ · inf

t∈I[M(ρ∗(g)q∗)(t)]

1q∗ ,(3.15)

where q∗ = (q − δ) · α ∈ (1, q) if δ = δ(ε) and α are chosen properly.This last inequality ensures that for any τ , τ ∈ (0, 1), we can find the subset

E(τ, I) ⊂ I, |E(τ, I)| ≥ (1 − τα)|I| such that

t ∈ E(τ, I) ⇒ SA−1I (g)(t) ≤ C ′′(α, d)

τinft∈I

[M(ρ∗(g)q∗)]1/q∗ .(3.16)

This is the main estimate of our first S-function.Now we will give a similar estimate for the second S-function SAIA

−1J (f)(t).

Again choose α > 1 and close to 1 to estimate1

|EI |∫EI

(SAIA

−1J (f)(t)

)αdt. For

t ∈ EI all J 3 t, J ⊂ I have the property

‖AIA−1J ‖ ≤ ek


458 A. VOLBERG

and so

t ∈ EI , SAIA−1J (f)(t) ≤ SI(f)(t)

def=

∑J3tJ⊂I

∥∥∥∥∫ f hJ

∥∥∥∥21

|J |

1/2

.

Thus

1

|EI |∫EI

(SAIA

−1J (f)(t)

)αdt ≤ 2

I

∫I

(SI(f)(t)

)αdt ≤ C(α)

1

I

∫I

‖f‖αdt.

Choosing 1 < α < p and denoting it by p∗ we get

‖SAIA−1J (f)‖Lα(EI ,

dt|EI | )

≤ C(α) inft∈I

(M‖f‖p∗)1/p∗(t).

As before it implies for any τ ∈ (0, 1) the existence of e(τ, I) ⊂ EI , |e(τ, I)| ≥(1− τα)|EI | ≥ 1

2(1− τα)|I| such that

t ∈ e(τ, I) ⇒ SAIA−1J (f)(t) ≤ C(α)

τinft∈I

(M‖f‖p∗)1/p∗ .(3.17)

Now choose τ so small that

|e(τ, I) ∩ E(τ, I)| ≥ 1

4|I|.(3.18)

We are ready to choose the height function h(t) satisfying (3.9) and (3.10). Put

h(t)def= sup

{h : Sh(t)

def=∑J3t|J|≤h

1

|J |∣∣∣∣(∫ g hJ , A

−1J

∫f hJ

)∣∣∣∣≤ B(M‖f‖p∗)1/p∗(t)(Mρ∗(g)q∗(t)

},

(3.19)

where p∗, q∗ appeared in (3.17) and (3.16) and where B =C′′(α, d)C(α)

τ2, with

constants taken from (3.16) and (3.17).Then (3.10) is fulfilled automatically. To check (3.9) let us fix any I ∈ D and let

us consider for t ∈ I

SI(t)def=∑J3tJ⊆I

1

|J |∣∣∣∣(∫ g hJ , A

−1J

∫f hJ

)∣∣∣∣ .Notice that

SI(t) =∑J3tJ⊆I

∣∣∣∣(A−1I

∫g hJ , AIA

−1J

∫f hJ

)∣∣∣∣ ≤ SA−1I (g)(t)SAIA

−1J (f)(t).

Thus (3.16) and (3.17) applied for t ∈ e(τ, I) ∩ E(τ, I) show that on at least aquarter of I (see (3.18))

SI(t) ≤ B(M‖f‖p∗)1/p∗(t)(Mρ∗(g)q∗)1/q∗(t).

Thus on a quarter of I the function h just defined in (3.19) is greater than |I|.Because I was arbitrary, (3.9) holds and Theorem 3.2 is finished.



4. T± theorem

Theorem 3.2 means that the operator∑A−1J hJ ⊗ hJ given by

g →∑J∈D

A−1J

(∫g hJ

)· hJ

is bounded as Lq(ρ∗) → f02q . If we use the language of matrix functions, meaning

that ρt(x) = ‖W 1/p(t)x‖, ρ∗t (x) = ‖U1/q(t)x‖, where U1/q = W−1/p, we can makea change of variable g(t) = W 1/p(t)G(t) now having G ∈ Lq with no weight. Theoperator now can be written in the form

∑A−1J hJ ⊗W 1/phJ . If AJ is given by

(3.1), the sum above is proved to be a bounded operator from Lq to f02q . The same

proof certainly shows that operators∑±A−1

J hJ ⊗W 1/phJ are uniformly bounded.All these claims are true under the condition that ρ (or W ) belongs to Ap,∞.Now if ρ (or W ) belongs to Ap one can say more. We summarize it in two

theorems.

Theorem 4.1. Let ρ ∈ Ap. Then if AJ are from (3.1) and sJ =∫f hJ we have

‖{sJ}‖f02p ({AJ}) � ‖f‖Lp(ρ).(4.1)

Proof. It follows immediately from the definition that ρ ∈ Ap ⇒ ρ ∈ Ap,∞ andρ ∈ Ap ⇒ ρ∗ ∈ Aq ⇒ ρ∗ ∈ Aq,∞.

Let us also introduce the reducing operators for ρ∗:(1

J

∫J

ρ∗t (x)qdt

)1/q

� ‖CJx‖.

Let us apply Theorem 3.2 to ρ∗ ∈ Aq,∞ to obtain∥∥∥∥{∫ f hJ

}∥∥∥∥f02p ({C−1

J })≤ C‖f‖Lp(ρ).

Notice that ρ ∈ Ap means that C−2J � A2

J and so we get∥∥∥∥{∫ f hJ

}∥∥∥∥f02p ({AJ})

≤ C‖f‖Lp(ρ).

To obtain the inverse inequality we use duality and again Theorem 3.2 for ρ ∈ Ap,∞to obtain

‖f‖Lp(ρ) = sup‖g‖Lq(ρ∗)≤1

|〈f, g〉| = sup

∣∣∣∣∑(∫f hJ ,

∫g hJ

)∣∣∣∣= sup

∣∣∣∣∑(AJ

∫f hJ , A

−1J

∫g hJ

)∣∣∣∣≤

∥∥∥∥{∫ f hJ

}∥∥∥∥f02p ({AJ})

·∥∥∥∥{∫ g hJ

}∥∥∥∥f02q ({A−1

J })

≤ C

∥∥∥∥{∫ f hJ

}∥∥∥∥f02p ({AJ})

.

The last inequality is Theorem 3.2. We are done.


460 A. VOLBERG

Remark. It seems to us that this theorem gives new information about Lp(w), w ∈Ap, even in the scalar case. The reader should notice that we organized a canon-ical isomorphism of Lp(w) and f02

p . This is not the classical isomorphism f →{(f, hJ)}J∈D between Lp(w) and f02

p (w).

Corollary 4.2. Operators

T±def=∑

±hJ ⊗ hJ

defined by the formula

T±ϕ =∑J∈D

±hJ ·∫ϕhJ

are uniformly bounded in Lp(ρ) if and only if ρ ∈ Ap.

Proof. The “if” part follows immediately from Theorem 4.1. In fact, we just usethe equivalent norm ‖·‖f02

p ({AJ}) in Lp(ρ) which does not change if ϕ is replaced by

T±ϕ because∫T±ϕ · hJ = ± ∫ ϕhJ . The “only if” part follows from the uniform

boundedness of rank one operators hJ ⊗ hJ . It follows exactly as the first part ofTheorem 5.1 below. So we send the reader to this coming proof.

The next result is an analog of the classical isomorphism Lp(w) ≈ f02p (w).

Theorem 4.3. Let ρ ∈ Ap. Let sJ =∫f hJ . Then

‖{sJ}‖f02p (ρ) � ‖f‖Lp(ρ).(4.2)

Proof. This follows easily from the corollary. In fact, we know that

‖f‖pLp(ρ) �∫ρpt

(∑J∈D

±hJ ·∫f hJ

)dt.

We integrate over the signs and use the Khintchin theorem which is valid here. Weobtain

‖f‖pLp(ρ) �∫ (∑

J3tρ2t (

∫f hJ)

1

|J |

)p/2

dt

= ‖{sJ}‖pf02p (ρ).

Remark and question. In particular, if ρ ∈ Ap and AJ are reducing operatorsfor ρ considered in (3.1), we see from (4.1) and (4.2) the following equivalence ofnorms in sequence spaces:

‖{sJ}‖f02p (ρ) � ‖{sJ}‖f02

p ({AJ}).(4.3)

But it has already been mentioned in the remarks to Theorem 3.2 that this equiv-alence does not require ρ ∈ Ap. For p = 2 it holds always. For p ≥ 2, ρ ∈ Ap,∞would be sufficient for (4.3). This can be proved along the lines of Theorem 3.2.

We would like to ask for the criterion for (4.3) to hold.



5. Hilbert transform

Theorem 4.1 and Corollary 4.2 allow one to show that any Calderon Zygmundoperator with T 1 = T ∗1 = 0 is bounded in Lp(ρ) if ρ ∈ Ap. But still the reductionof any T to T± is a very technical and difficult task. In this section we show thatat least for the Hilbert transform one can give a relatively easy and straightforwardproof very similar to the one of Corollary 4.2.

Theorem 5.1. Let ρ ∈ Ap. Then

‖P+f‖Lp(ρ) ≤ C‖f‖Lp(ρ).(5.1)

Conversely if (5.1) holds, then ρ ∈ Ap.

Before proving Theorem 5.1 let us prove two simple lemmas. Consider the Pois-

son kernel PI =|I|

x2 + |I|2 and define kI,q = Pq/2I · |I|q/2−1 (1 < q < ∞). Then∫

kI,q � 1 and here is the first lemma.

Lemma 5.2. If u is a scalar Aq weight, then∫ukI,q ≤ C

1

|I|∫Iu.

Proof. We can write kI,q �∑∞

k=1 εk1

|2kI|χ2kI , where εk � (2−k)q/p. Now if u ∈ Aq

let w = u−p/q ∈ Ap and we can write the following chain of inequalities:

(wI)q/p

(∫kI,qu

)≤

∞∑k=1

(wI)q/pu2kIεk

≤∞∑k=1

(wI)q/p

(w2kI)q/pεk =

∞∑k=1

(w(I)

w(2kI)

)q/p2k·

qp εk

≤ C

∞∑k=1

(w(I)

w(2kI)

)q/p≤ C

∞∑k=1

(1 + ε)−k ≤ C′.

The last inequality follows from the doubling property of Ap weights. We get∫kI,qu ≤ C(wI)

−q/p ≤ C′uI by using the Ap condition again.

Lemma 5.3. If W is a matrix Ap weight, then t→ ‖W 1/px‖p is a scalar Ap weightfor any vector x.

Proof. It is convenient to use the language of ρt(x) = ‖W 1/p(t)x‖. Fix x and choose

y(x) in such a way that (ρp,I)(x) = supy|(x, y)|

(ρp,I)∗(y)≤ 2

|(x, y(x)|(ρp,I)∗(y(x))

. Now

(1

|I|∫I

ρpt (x)

)1/p(1

I

∫I

ρ−qt (x)

)1/q

≤ ρp,I(x)

(1

I

∫I

(ρ∗t (y(x))|(x, y(x))|

)q)1/q

≤ ρp,I(x)ρ∗q,I(y(x))

|(x, y(x))| ≤ Cρp,I(x)(ρp,I)

∗(y(x))|(x, y(x))| ≤ 2C.

Proof of Theorem 5.1. (5.1) ⇒ ρ ∈ Ap.


462 A. VOLBERG

Operator P+ has the kernel1

1− ζz. We are given that it is a bounded operator.

Fix an a in the unit disk and consider the operator with the kernel

1

1− ζz

[1−

(ζ − a

1− aζ

)z − a

1− az

].

Obviously it is a bounded operator in Lp(ρ) too, actually it is given by the formula

P+ − baP+ba, where ba(ζ) =ζ − a

1− az. Notice that pointwise multiplication by ba

does not change the norm in Lp(ρ). Now let us make the simplification in the kernel

to see that it equals1− |a|2

(1− ζa)(1− az)= ka(z)⊗ ka(ζ), where ka(ζ) =

(1 − |a|2)1/21− aζ

is a normalized Szego kernel in L2. Notice that

|ka(ζ)|2 = Pa(ζ),

where Pa(ζ) denotes the Poisson kernel.Now let us use the fact (which we just derived from the boundedness of P+)

that ka ⊗ ka are uniformly bounded. Let ϕ(t) ∈ Lq, ψ ∈ Lp, and∫ ‖ϕ‖qdt ≤

1,∫ ‖ψ‖pdt ≤ 1. Then∣∣∣∣(∫ ka(t)W

1/p(t)ϕ(t)dt,

∫ka(s)U

1/q(s)ψ(s)ds

)∣∣∣∣ ≤ C

uniformly in a. Denoting k(p)a = (1− |a|2)1/2−1/pka and k

(q)a = (1− |a|2)1/2−1/qka,

we have ∫|k(p)a (t)|p

∥∥∥∥W 1/p(t)

∫k(q)a (s)U1/q(s)ψ(s)ds

∥∥∥∥p ≤ C.

Denoting z =∫k

(q)a (s)U1/q(s)ψ(s)ds we conclude that(∫

|k(p)a |p‖W 1/p(t)z‖p

)1/p

≤ C‖ψ‖Lp .

This is an averaging because∫ |k(p)

a |p � 1. Denote it by ρp,a:

ρp,a(z) ≤ C‖ψ‖Lp .We want to show that

(ρ∗q,a)∗(z) ≥ C inf

{‖ψ‖Lp : z =

∫k(q)a (s)U1/q(s)ψ(s)

}.

But

(ρ∗q,a)∗(z) = max

y|(y, z)|/

(∫‖U1/q(s)y‖q|k(q)

a (s)|q)1/q

= maxy|〈k(q)

a (s)U1/q(s)y, ψ(s)〉|/‖k(q)a (s)U1/q(s)y‖Lq

= inf

{‖ψ‖Lp : z =

∫k(q)a (s)U1/q(s)ψ(s)ds

}.

Thus

ρp,a(z) ≤ C(ρ∗q,a)∗(z).



But |k(p)a |p ≥ C(p)

χIa|Ia| , where Ia is the interval of length 1 − |a| centered at a/|a|.

Thus

ρp,I ≤ C(ρ∗q,I)∗.

Now let us prove that ρ ∈ Ap ⇒ (5.1).

Let ρ be given by W 1/p and ρ∗ by U1/q = W−1/p. We have to prove that

f →W 1/pP+W−1/pf

is a bounded operator as Lp → Lp. Fix g ∈ Lq. We need to show∫|(∇P+W

−1/pf(z),∇P+W1/pg(z))|δ(z)dA(z) ≤ C‖f‖p‖g‖q,

where δ(z) means the distance to the boundary. In other words, introducing a Stolzcone Γt, and

S(t)def=

∫Γt

|(∇P+W−1/pf(z),∇P+W

1/pg(z))|dA(z),

one needs to prove that ∫S(t)dt ≤ C‖f‖p‖g‖q.(5.2)

We follow closely the lines of the proof of Theorem 3.2. Let us consider a nonneg-ative function h(t) and

Sh(t)(t) =

∫Γt,h(t)

|(∇P+W−1/pf(z),∇P+W

1/pg(z))|dA(z),

where

Γt,h(t) = Γt ∩ {z : δ(z) ≤ h(t)}.As in Section 3 ∫

Sh(t)dt �∫S(t)dt(5.3)

if the function h(t) has the following property:

∀ J |{t ∈ J : h(t) ≥ |J |}| ≥ a|J |.(5.4)

Let us choose h to be maximal such that

Sh(t)(t) ≤ B(M‖f‖p∗)1/p∗(t)(M‖g‖q∗)1/q∗(t),(5.5)

where B, p∗ ∈ (1, p), q∗ ∈ (1, q) will be chosen later.If this h satisfies (5.4), then (5.3) and (5.2) imply what we need.To choose B, p∗, q∗ and to prove that h satisfies (5.4) we follow the algorithm

below.Fix an interval I. Consider f1 = f ·χ2I , g1 = g ·χ2I and f2 = f − f1, g2 = g− g1.

Let us introduce AI as in (3.1):

t ∈ I, SAI (fi)(t) =

(∫Γt,|I|

‖∇P+AIW−1/pfi‖2dA(z)

)1/2

, i = 1, 2;

t ∈ I, SA−1I (gi)(t) =

(∫Γt,|I|

‖∇P+A−1I W 1/pgi‖2dA(z)

)1/2

, i = 1, 2.


464 A. VOLBERG

Then exactly as in Section 3 we choose α > 1, α close to 1, and

‖SAI (f1)‖Lα(I, dtI ) ≤ C(α, d)

(1

I

∫2I

‖f‖(p−δ)α) 1

(p−δ)α

·d∑

i=1

(1

I

∫I

‖W−1/p(t)AIei‖(q+ε)αdt

) 1(q+ε)α

.

We can now use the fact that the function t → ‖U1/q(t)x‖ = ‖W−1/p(t)x‖ isuniformly in A∞ to conclude that we have a uniform inverse Holder inequality withq > q. Now choosing ε and α to satisfy (q + ε)α < q we conclude that the last

integral is bounded by

(1

I

∫I ‖W−1/pAIei‖qdt

)1/q

. But W ∈ Ap, so A−1I reduces

averages of the metric ρ∗ given by W−1/p = U1/q. In other words,(1

I

∫I

‖W−1/px‖q)1/q

� ‖A−1I x‖.(5.6)

Thus finally

‖SAI (f1)‖Lα(I, dt|I| )≤ C′(α, d) inf

t∈I(M‖f‖p∗)1/p∗(t),

where p∗ = (p− δ(ε))α is ensured to be in (1, p). Similarly

‖SA−1I (g1)‖Lα(I, dt|I| )

≤ C′(α, d) inft∈I

(M‖g‖q∗)1/q∗(t).

Now let us estimate SAI (f2)(t), SA−1I (g2)(t), for t ∈ I.

Let CI be a center of the square QI built on I. It is easy to see that for t ∈ Iwe have

(∫Γt,|I|

‖(∇P+AIW−1/p(t)f2(t))(z)‖2

)1/2

≤ a

∫PCI (t)‖AIW

−1/p(t)f2(t)‖dt,

(5.7)

where PCI denotes the Poisson kernel. Let δ be small, p−δ > 1, and let (p−δ)−1 +

(q+ ε)−1 = 1. We can continue the estimate as follows (k(r)a

def= (1− |a|2) 1

2− 1r · ka):∫

PCI‖AIW−1/pf2‖ ≤

d∑i=1

∫‖W−1/pAIei‖‖f2‖|kCI |2(t)

≤ C(p, q)

d∑i=1

∫‖W−1/pAIei‖‖f2‖k(q+ε)

CIk

(p−δ)CI

≤ C(p, q)

d∑i=1

(∫‖W−1/pAIei‖q+ε|k(q+ε)

CI|q+ε

)1/(q+ε)

·(∫

‖f2‖p−δ|k(p−δ)CI

|p−δ)1/(p−δ)

.

(5.8)

Notice that |k(q+ε)CI

|q+ε ≤ C|k(q)CI|q def

= CkI,q.



As t→ ‖W−1/p(t)x‖q is uniformly Aq weight (see Lemma 5.3) we can conclude

that there exists a small ε such that t→ ‖W−1/p(t)x‖q+ε is uniformly Aq weight bya classical inverse Holder inequality. Lemma 5.2 claims that if u is a scalar weight

in Aq, then∫u kI,q ≤ C

1

I

∫I u. Combining this knowledge we conclude that

d∑i=1

(∫‖W−1/pAIei‖q+ε|k(q+ε)

CI|q+ε

)1/(q+ε)

≤ C

d∑i=1

(1

I

∫I

‖W−1/pAIei‖q+ε)1/(q+ε)

≤ C

d∑i=1

(1

I

∫I

‖W−1/pAIei‖q)1/q

.

The last inequality is another application of inverse Holder inequality.Finally, using (5.6) we estimate the last sum by a constant.Combining this with (5.7) and (5.8) we get (p∗ = p− δ)

t ∈ I ⇒ SAI (f2)(t) ≤ C(p, q, d) (M‖f‖p∗)1/p∗ (t).

Similarly

t ∈ I ⇒ SA−1I (g2)(t) ≤ C(p, q, d)(M‖g‖q∗)1/q∗ (t).

Combining with our previous estimates for SAI (f1), SA−1I (g1) we get

‖SAI (f)‖Lα(I, dt|I| )≤ C(p, q, α, d)(M‖f‖p∗)1/p∗(t),

‖SA−1I (g)‖Lα(I, dt|I| )

≤ C(p, q, α, d)(M‖g‖q∗)1/q∗(t).In particular, on a quarter of I

SI(t)def=

∫Γt,|I|

|(∇P+W−1/pf(z),∇P+W

1/pg(z))|dA(z)

=

∫Γt,|I|

|(AI∇P+W−1/pf,A−1

I ∇P+W1/pg)|dA(z)

=

∫Γt,|I|

|(∇P+AIW−1/pf,∇P+A

−1I W 1/pg)|dA(z)

≤ SAI (f)(t)SA−1I (g)(t) ≤ C2(p, q, α, d)

·(M‖f‖p∗)1/p∗(t)(M‖g‖q∗)1/q∗(t).For such t, h(t) ≥ |I|, if we choose in the definition of h to have B = C2(p, q, α, d)

and p∗, q∗ as above. Thus (5.4) is satisfied and the Theorem is proved.

Acknowledgements

The author is grateful to the participants of the Analysis Seminar at MichiganState University, and especially to Sheldon Axler, Michael Frazier, Fedja Nazarov,Serguei Treil and Dechao Zheng for fruitful discussions and constant enthusiasm.The author is also grateful to Robert Fefferman and Peter Jones for valuable dis-cussions.


466 A. VOLBERG

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Department of Mathematics, Michigan State University, East Lansing, Michigan48824

E-mail address: [email protected]


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