On Sharp Extrapolation Theorems

ON SHARP EXTRAPOLATION THEOREMS

Dariusz Panek

M.Sc., Mathematics, Jagiellonian University in Krak´ow,Poland, 1995 M.Sc., Applied Mathematics, University of New Mexico, USA, 2004

DISSERTATION

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy Mathematics

The University of New Mexico

Albuquerque, New Mexico

December, 2008 °c 2008, Dariusz Panek

iii Acknowledgments

I would like ﬁrst to express my gratitude for M. Cristina Pereyra, my advisor, for her unconditional and compact support; her unbounded patience and constant inspiration. Also, I would like to thank my committee members Dr. Pedro Embid, Dr Dimiter Vassilev, Dr Jens Lorens, and Dr Wilfredo Urbina for their time and positive feedback.

iv ON SHARP EXTRAPOLATION THEOREMS

Dariusz Panek

ABSTRACT OF DISSERTATION

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy Mathematics

The University of New Mexico

Albuquerque, New Mexico

December, 2008 ON SHARP EXTRAPOLATION THEOREMS

Dariusz Panek

M.Sc., Mathematics, Jagiellonian University in Krak´ow,Poland, 1995 M.Sc., Applied Mathematics, University of New Mexico, USA, 2004

Ph.D., Mathematics, University of New Mexico, 2008

Abstract

Extrapolation is one of the most signiﬁcant and powerful properties of the weighted theory. It basically states that an estimate on a weighted Lpo space for a single expo- p nent po ≥ 1 and all weights in the Muckenhoupt class Apo implies a corresponding L estimate for all p, 1 < p < ∞, and all weights in Ap. Sharp Extrapolation Theorems track down the dependence on the Ap characteristic of the weight.

In this dissertation we generalize the Sharp Extrapolation Theorem to the case where the underlying measure is dσ = uodx, and uo is an A∞ weight. We also use it to extend Lerner’s extrapolation techniques. Such Theorems can then be used to extrapolate some known initial weighted estimates in L2(wdσ). In addition, for some −1 operators this approach allows us to specify the weights w = uo and to use known weighted results in Lp(wdσ) to obtain some estimates on the unweighted space.

This work was inspired by the paper [Per1] where the L2 weighted estimates for

vi the dyadic square function were considered to obtain the sharp estimates for the so-called Haar Multiplier in L2.

vii Contents

Introduction x

1 Preliminaries 1

1.1 Weights and doubling measures ...... 1

1.2 Distribution Function and Real Interpolation ...... 2

1.3 Hardy-Littlewood Maximal Function ...... 5

1.4 A1(dσ) weights ...... 6

1.5 Ap(dσ) weights ...... 9

1.6 Ap weights and classical results ...... 10

1.7 A∞(dσ) weights ...... 13

1.8 RHp(dσ) weights ...... 14

1.9 Rubio de Francia Extrapolation Theorem ...... 17

1.10 Calder´on-Zygmund Decomposition and Besicovitch Covering Lemma 19

2 Buckley type estimates for Maximal Function 21

viii Contents

2.1 Weak Estimates ...... 22

2.1.1 Strong Doubling ...... 24

2.2 RHp(dσ) and Gehring’s Estimates ...... 27

2.3 Reverse H¨olderInequality for the Ap(dσ)...... 32

2.3.1 C. P´erez’new proof ...... 36

2.3.2 Coiﬀman-Feﬀerman-Buckley Theorem dσ: a precise version . . 39

2.4 Strong Estimates and Interpolation ...... 40

2.4.1 Gehring’s Estimates and Interpolation ...... 40

2.4.2 Coifman-Feﬀerman-Buckley dσ and Interpolation ...... 43

2.5 A. Lerner’s approach ...... 45

2.6 Strong estimates for Mσ revisited ...... 47

3 Sharp Extrapolation Theorems dσ. 48

3.1 Weight Lemmas ...... 49

3.2 Sharp Extrapolation Theorem dσ...... 55

3.3 Lerner’s type extrapolation dσ ...... 58

4 Applications 63

References 66

ix Introduction

Extrapolation is one of the most signiﬁcant and powerful properties of the weighted theory. It basically states that an estimate on a weighted Lpo space for a single exponent po and all weights in the Muckenhoupt class Apo implies a corresponding p L estimate for all p, 1 < p < ∞, and all weights in Ap. This is contained in the celebrated theorem due to Rubio de Francia [Ru]. Precise statements of this and other classical theorems can be found in the Preliminaries Chapter 1.

In 1989, Buckley [Buc1] obtained the following sharp-estimates for the Hardy-

Littlewood Maximal Function, namely for w ∈ Ap

p0 p kMfk p ≤ C [w] kfk p . L (w) Ap L (w)

p0 p This result is sharp in the sense that [w] cannot be replaced by φ([w]Ap ), for any p0 positive non-decreasing function φ(t), t ≥ 1, growing slower than t p .

In [DrGrPerPet] Buckley’s result was used to track down the dependence of the estimates on [w]Ap in the Rubio de Francia Extrapolation Theorem. More precisely, if for a given 1 < r < ∞ the norm of a sub-linear operator on Lr(w) is bounded by p a function of the characteristic constant [w]Ar , then for p > r it is bounded on L (v) p by the same increasing function of the [v]Ap , and for p < r it is bounded on L (v) r p0 r−1 by the same increasing function of the r0 p = p−1 power of the [v]Ap . We will refer to this result as the Sharp Extrapolation Theorem.

x Introduction

For some operators the bounds, extrapolated from an initial sharp-estimate, are sharp but not always. For example, if T is the maximal function and we use Buckley’s initial estimates r0 r kMfk r ≤ C [w] kfk r , L (w) Ar L (w)

p0 r0 max{ p , r } then the sharp extrapolation will give kMfk p ≤ C [w] kfk p , which is L (w) Ap L (w) r0 r an optimal bound p < r. However, for p > r, it implies kMfk p ≤ C [w] kfk p , L (w) Ap L (w) p0 1 1 r0 and p = p−1 < r−1 = r , so sharp extrapolation just preserves the initial estimates and is not optimal for p > r despite the fact that the initial Lr(w) estimate was sharp.

If T is any of the Hilbert transform, the Riesz transforms, the Beurling transform, the martingale transform, the dyadic square function or the dyadic paraproduct, then sharp extrapolation guarantees that for any 1 < p < ∞ there exists a positive constant Cp such that for all weights w ∈ Ap we have

αp kT k p p ≤ C [w] , L (w)→L (w) p Ap

p0 where αp = max{1, p }. The exponent is sharp for the Hilbert transform, the Riesz transforms, the Beurling and the martingale transforms for all 1 < p < ∞. For the dyadic square function the exponent is sharp for 1 < p ≤ 2. These estimates are obtained by extrapolating from known linear estimates in L2(w), namely,

2 2 kT kL (w)→L (w) ≤ C[w]A2 .

These linear estimates were proved using the technique of Bellman functions, [Pet,Witt,HTV,PV], [Pet1], [Wit1], [HTrVo], [PetVo]. My academic sister, Oleksan- dra Beznosova [Be1], just proved a linear estimate for the dyadic paraproduct in L2(w), therefore some extrapolated bounds will hold. We do not know yet if any of the extrapolated exponents are sharp for the dyadic paraproduct.

C. P´erez[P2] conjectures a similar result for Calder´on-Zygmund operators.

xi Introduction

Conjecture 1 (The A2 conjecture). Let 1 < p < ∞ and let T be a Calder´on- Zygmund singular integral operator. There is a constant c = c(n, T ) such that for any Ap-weight w p0 max{1, p } kT k p p ≤ c p [w] . (0.1) L (w)→L (w) Ap

Clearly, as the name suggests, it suﬃces to prove this conjecture for p = 2 and apply the Sharp Extrapolation Theorem to get (0.1). It is not known whether the Bellman function techniques could be also extended to this class of operators.

For p > 2 these bounds were improved for S, the square function (both continuous and dyadic), by A. Lerner [Le1] who proved a two weight estimate kSkL2(u)→L2(v) and then extrapolated obtaining a better than linear bound

p 1 max{1, 2 } p−1 kSk p p ≤ C[w] . L (w)→L (w) Ap

In particular, this holds for the dyadic square function deﬁned as

Ã ! 1 X |m f − m f|2 2 Sdf(x) = Il Ir χ , |I| I(x) I where mI f is the average of f over the interval I ∈ D, and D denotes the dyadic intervals. It is not known yet whether Lerner’s estimates are optimal. In fact, Lerner himself conjectures they are not. Finding the optimal power for the square function and p > 2 is a very interesting open problem.

In this dissertation we are interested in obtaining Sharp Extrapolation Theorems dσ which could be used to extrapolate some known estimates on the weighted space 2 2 L (wdσ) to L (wdσ). We deﬁne dσ = uodx, for some uo ∈ A∞, and we consider the class of weights Ap(dσ). We also need the Buckley-type sharp-estimates for the

Maximal Function Mσ, associated with measure σ. First we prove the weighted weak-type inequalities and then use interpolation (as in [Buc2]) to obtain the strong sharp-estimates. Since this argument is true for any sublinear operator satisfying the

xii Introduction same initial weak-estimates as the maximal function does, we state the theorem in a more general form, and then give the corresponding result for the maximal operator

Mσ.

When trying to implement this outline some technical diﬃculties were encoun- tered. More precisely, we are seeking strong-estimates in Lp(wdσ) for the maximal function Mσ, when w ∈ Ap(dσ). It is clear that w ∈ Ap(dσ) implies w ∈ Ap+ε(dσ) for any positive ε, and [w]Ap+ε(dσ) ≤ [w]Ap(dσ). It is a more delicate issue to show that there is an ε > 0 such that w ∈ Ap−ε(dσ). (This is a self-improving integrability condition of weights that we will call the Coifman-Feﬀerman property). Moreover, we want ε as large as possible with uniform control over [w]Ap−ε(dσ) in terms of [w]Ap(dσ). Once such an ε has been established, we have sharp weak-estimates on Lp±ε(wdσ) that we can interpolate, keeping track of the constant [w]Ap(dσ) to obtain strong Lp(wdσ) estimates.

Buckley claims ([Buc1], [Buc2]) that in the case dσ = dx one can choose ε ∼ 1−p0 [w] to get a uniform control [w]Ap−ε ≤ C[w]Ap . He references the classical [CoFe], however we could not find the justification there. When talking to Carlos Pérezhis first reaction was the proof is not there([CoFe]); and he sent us a preprint with his own argument which we also extended to the dσ case. We had to work on a proof valid also in the general dσ case. What seemed to work was an argument using another self-improvement property (proved in [CoFe]) that says that w ∈ Ap(dσ), satisfies the

Reverse Höldercondition. More precisely, if w ∈ Ap(dσ), then w ∈ RH1+γ(dσ). The game now is to introduce [w]Ap into the picture, and carefully track the dependence of γ on [w]Ap(dσ). This knowledge could then be used to track down ε in the Coifman- Fefferman property. However, the proof of the Reverse HölderInequality we used, yields the expected order of γ, but it did not seem to guarantee a uniform control over [w]Ap−ε(dσ).

Instead we took advantage of the added ﬂexibility of having the dσ measure.

xiii Introduction

−1 There is what we call a tautology w ∈ Ap(uodx) ⇐⇒ w ∈ RHp0 (wuodx). We can use now self-improving properties of the RHq(dµ) weights. This time v ∈ RHq(dµ) implies immediately v ∈ RHq−ε(dµ) for any 0 < ε < q−1. What is now more delicate is to go in the other direction. This is the celebrated Gehring Lemma saying there is an ε > 0 such that v ∈ RHq+ε(dµ). Again, we need ε to be as large as possible with uniform control on [w]RHq+ε(dµ) in terms of [w]RHq(dµ).

We carefully analyzed Gehring’s proof [Ge] in the dµ = dx case, modulo some reﬁnements made by Iwaniec [Iw], and generalized it to the dµ case. With the dµ version of Gehring lemma at hand, we can now obtain strong-estimates for Mσ in p −1 L (dσ) and w ∈ Ap(dσ). By the tautology w ∈ RHp0 (dµ), where dµ = wdσ, with −1 0 v = w , q = p , we get v ∈ RHp0±ε(dµ) for an appropriate ε with a uniform control

0 0 over [v]RHp0±ε(wdσ) in terms of [v]RHp0 (dµ). It also means that w ∈ A(p ±ε) (dσ), and

po p1 0 0 0 0 weak-estimates hold in L (wdσ),L (wdσ) for po = (p + ε) < p < p1 = (p − ε) .

This approach leads to the right power of [w]Ap(dσ) (as in [Buc2]) but with an extra constant D(µ).

We return to the Reverse H¨olderProperty and use a new proof ([P1]) which allows us to have a uniform control over [w]Ap−ε(dσ) in terms of [w]Ap(dσ), and relate

ε and [w]Ap(dσ). This time the constant D(σ) appears and we obtain the same power of [w]Ap(dσ). This also helps us justify and understand Buckley’s classical result for the dσ = dx case.

At the end of Chapter 2 we also include the estimates for Mσ inspired by A. Lerner’s very recent result (see [Le2]). It allows us to obtain the optimal power of

[w]Ap(dσ) without using interpolation. Instead it is based on the basic properties of maximal functions and Besicovitch Theorem for the centered maximal function. The doubling constant D(σ) also appears but with a diﬀerent exponent.

Both in the proofs of the weak-estimates and the strong-ones we encounter dif-

xiv Introduction ferent doubling constants, D(σ) and D(µ) respectively. Their role, especially the one of D(µ), is not fully understood so we decided to keep them both to get some insight. They may be only the artifact of the method used in the proofs. They come directly from the theorem which says that for a doubling measure σ the Maximal

Function Mσ is of weak-type (1, 1) with weak-norm at most D(σ), or as result of the Calder´on-Zygmung decomposition used in the proof. To the best of our knowledge the question whether or not the weak-norm can be replaced with a constant independent of the measure is still open (see [StStr], [GrM-S], [GrK]). Even if it has to be there, it is not sure how this may aﬀect the power of the characteristic constant

[w]Ap(dσ). Moreover, in some of our applications when dµ = dx, D(µ) is known and even easier to handle than D(σ).

In Chapter 3 we use the estimates for Mσ to build the Sharp Extrapolation The- orem dσ. We follow closely the idea in [DrGrPerPet] tracking down the dependence on [w]Ap(dσ). We also state it in an operator-free form (with T f replaced with an arbitrary Lp integrable function g) which is more convenient in some applications and more general (see p.48 and [CrMPe1]). For example, the theorem in this form can d 2 be applied directly to some known inverse estimates for Sσ on the weighted L (wdσ) obtained in [Per4], as it is not important on which side the operator is located.

Moreover, the ﬁrst lemma used in the proof of Sharp Extrapolation dσ will also help us to generalize the above mentioned Lerner’s extrapolation technique to the dσ case, for any po > 1 and not just po = 2, and with diﬀerent initial two weight estimates.

In [Per1] this extrapolation dσ (not sharp) was used to obtain Lp estimates for

Haar multipliers by comparing to square function Su. In that case the trick is to think of dσ = udx, and dµ = dx = wdσ so that w = u−1 and Lp(wdσ) estimates for p Sσ become L estimates for Su. This example was an inspiration to start thinking about this problem.

xv Introduction

In the Preliminaries Chapter 1 we list the basic concepts and tools which will be used, some of them already adapted to the dσ case. In Chapter 2 we show weak- estimates for Mσ, involving both D(σ) and D(µ) constants. We use them to derive strong sharp-estimates using interpolation. First we interpolate on RHq(dµ) side, and then we obtain similar estimates repeating the procedure on the Ap(dσ) side. At the end we show a diﬀerent way of obtaining such estimates without using interpolation.

In Chapter 3 we use the estimates for Mσ to build the Sharp Extrapolation Theorem and then use it to extend Lerner’s extrapolation technique to the dσ case and for any po > 1. At the end we show some examples and possible applications of our results.

xvi Chapter 1

Preliminaries

In this chapter we recall the reader basic definitions and well known results. In particular we recall the definitions of weights, doubling measures, distribution functions, Hardy-Littlewood maximal functions. We also state the basic tools such as: interpolation, how to recover Lp-norms from distribution functions, how to construct A1(dσ) weights from locally integrable functions and the maximal function (Coifman-Rochberg dσ, Lemma 1.4.1), Calderón-Zygmund decomposition and Besi- covitch Covering Lemma. We list basic properties of weights, interplay between them and connections with maximal function. We also recall the classical Rubio de Francia Theorem and its weak-variant.

All the measures discussed in this dissertation are positive Borel measures. We will simply refer to them as measures.

1.1 Weights and doubling measures

A weight is a locally integrable function on Rn that takes values in (0, ∞) almost everywhere. Given a weight w and a measurable set E, the w-measure of the set E

1 Chapter 1. Preliminaries is denoted by Z w(E) = w(x) dx. R E For a measure σ, σ(E) = E dσ. In particular, |E| stands for the Lebesgue measure of E.

Deﬁnition 1.1.1. We say a positive Borel measure σ is a doubling measure, σ ∈ D, if there exists a constant C, independent of Q, such that σ(3Q) ≤ Cσ(Q), where 3Q denotes the cube concentric to Q with side length three times as long. The smallest such constant is called the doubling constant of σ and denoted by D(σ). We say w is a doubling weight if wdx is a doubling measure. Its doubling constant will be denoted by D(w).

1 X For any positive quantities X,Y ”X ∼ Y ” will mean C ≤ Y ≤ C, for some positive constant C depending only on n and p. For any p > 0 and p 6= 1, p0 denotes 0 p the dual exponent i.e. p = p−1 .

Remark 1.1.1. Sometimes the doubling condition is deﬁned as above but with 2Q instead of 3Q. However, 3Q has the advantage that if we consider the dyadic parent ∼ ∼ of Q, say Q, so that Q is one of the 2n children of Q in Rn obtained by subdividing ∼ ∼ each side of Q into 2n equal pieces, then Q ⊂ 3Q, but never in 2Q.

1.2 Distribution Function and Real Interpolation

The distribution function df (λ) provides information about the size of f but not about the behavior of f near any given point.

Deﬁnition 1.2.1. Let (X, σ) be a measurable space and let f : X → C be a measurable function. We call the function df : (0, ∞) → [0, ∞), given by

df (λ) = σ ({x ∈ X : |f(x)| > λ}) ,

2 Chapter 1. Preliminaries the distribution function of f (associated with σ).

Proposition 1.2.1. Let φ : [0, ∞) → [0, ∞) be diﬀerentiable, increasing and such that φ(0) = 0. Then

Z Z ∞ 0 φ (|f(x)|) dσ = φ (λ) df (λ) dλ (1.1) X 0 In particular, if φ(t) = tp and f ∈ Lp(dσ), then will get

Z ∞ p p−1 kfkLp(dσ) = p λ df (λ) dλ. (1.2) 0

If f is a ﬁnite function and df (λ) < ∞ for all λ > 0, then Z Z ∞ p p |f(x)| dσ = − λ ddf (λ). (1.3) X 0

Proofs can be found in [Sa] p.163. Below we prove a variation of (1.3) which will be used in Section 2.1. More precisely,

Z Z ∞ gr dσ = − sr−1dh(s), (1.4) X 0 where Z h(s) = g dσ and E(s) = {x ∈ X : g(x) > s}, E(s) and the function g is positive and locally integrable, as in Lemma 2.2.2.

Proof. If h(t) is finite for all positive t and g(t) also finite, then the integral at the right hand side of (1.4) is well defined. The integral at the left can be expressed by the Lebesgue sums as follows. Taking an ε-subdivision, 0 < ε < 2ε... < mε < ... and letting Xj = {x ∈ X :(j − 1)ε ≤ g(x) < jε}, with the measure dµo := gdσ, we have

µo(Xj) = h((j − 1)ε) − h(jε) and Z Z Z X∞ r r−1 r−1 r−1 g dσ = g gdσ = g dµo = lim (jε) µo(Xj) ε→0 X X X j=1 X∞ Z ∞ = − lim (jε)r−1[h(jε) − h((j − 1)ε)] = − sr−1dh(s), ε→0 j=1 0

3 Chapter 1. Preliminaries where the last integral is a Riemann-Stieltjes Integral (see [R]). In particular, if X = E(t), then h(s) = h(t) for all 0 < s ≤ t, which implies that dh(s) = 0 for all s ≤ t. Thus, Z Z ∞ gr dσ = − sr−1dh(s). (1.5) E(t) t We will use this equation in the proof of Lemma 2.2.2.

Deﬁnition 1.2.2. We say an operator T is of weak-type (p, p), with respect to the measure σ, if µ ¶p Ckfk p d (λ) = σ ({x ∈ X : |T f| > λ}) ≤ L (dσ) (1.6) T f λ

The smallest such C is called weak-type Lp(dσ) norm of T. The following theorem is a useful tool which allows to deduce Lp boundedness from weak inequalities.

Theorem 1.2.1 (Marcinkiewicz interpolation theorem with respect to sigma). Sup- pose 1 ≤ p0 < p1 < ∞ and that T is a sublinear operator of weak-type (p0, p0) and

(p1, p1), with respect to the measure σ, with norms M0 and M1, then T is actually of strong type (p, p) for all p0 < p < p1 and for any 0 < t < 1,

1−t t kT fkLpt (dσ) ≤ KtM0 M1kfkLpt (dσ), (1.7) ³ ´ p where 1 = 1−t + t and Kpt = 2 t p1 + p0 . pt p0 p1 t pt p1−pt pt−p0

The proof can be found for example in [Sa] or [Gr].

We will need this theorem only in two very particular cases. Firstly, when p0 := p − ε and p1 := p + ε, for some ε > 0 such that max{Mp−ε,Mp+ε} ≤ c Mp, for ε 1 0 0 some absolute constant c ≥ 1 and t = 2p + 2 . Secondly, when p0 := (p + ε) and 0 0 1 ε (p−1) p1 := (p − ε) , in which case t = 2 − 2 p . In both cases pt = p.

Then, the inequality (1.7) becomes

Kp Mp kT fkLp(dσ) ≤ 1 kfkLp(dσ), (1.8) (ε) p

4 Chapter 1. Preliminaries

p+1 p+1 p p 2 where Kp = 2 c in the ﬁrst case, and in the second case Kp0 = 2 c (and (p−1) p blows up as p approaches 1).

1.3 Hardy-Littlewood Maximal Function

The study of averages of functions is better understood and simpliﬁed by the introduction of the maximal function.

Deﬁnition 1.3.1. Given a locally measurable function f, and a measure σ ∈ D, the

(uncentered) Hardy-Littlewood maximal function (associated with σ), Mσf, is deﬁned by Z 1 Mσf(x) = sup |f(y)| dσ, Q3x σ(Q) Q for all cubes Q in Rn with sides parallel to the axes.

The function Z c 1 Mσf(x) = sup |f(y)| dσ, Qx σ(Qx) Qx where Qx denotes the cubes centered at x, is called the centered Hardy-Littlewood maximal function.

Remark 1.3.1. Let us observe that

c c Mσf(x) ≤ Mσf(x) ≤ D(σ) Mσf(x). (1.9)

The ﬁrst inequality is obvious. If we take x ∈ Q, and Qx denotes a cube congruent to Q and centered at x, then Q ⊂ 3 Qx and Z Z Z 1 1 σ(3Qx) 1 c |f| dσ ≤ |f| dσ ≤ |f| dσ ≤ D(σ)Mσf(x), σ(Q) Q σ(Q) 3Qx σ(Q) σ(3Qx) 3Qx from which the second inequality follows. In particular,

c kMσfkLp(dµ) ≤ D(σ)kMσfkLp(dµ). (1.10)

5 Chapter 1. Preliminaries

p Theorem 1.3.1. The maximal operator Mσ is bounded on L (dσ), 1 < p ≤ ∞, and is weak (1, 1), with respect to σ, and a weak-norm at most D(σ).

The proof can be found in [Gr] or [Du].

Deﬁnition 1.3.2. A dyadic interval in R is an interval of the form

[m2−k, (m + 1)2−k), where m, k are integers. A dyadic cube in Rn is a product of dyadic intervals of the same length.

Any two dyadic cubes are either disjoint, or one contains the other.

Theorem 1.3.2. The dyadic maximal operator, with respect to Borel measure σ, and deﬁned as Z d 1 Mσ f(x) = sup |f(y)| dσ, Q3x σ(Q) Q Q dyadic cube is weak (1, 1), with respect to σ, and a weak-norm at most 1. (See [Gr], p. 555).

1.4 A1(dσ) weights

In this section we introduce the class A1(dσ) and prove a very useful result which 1 allows to construct such weights from Lloc(dσ) functions.

Deﬁnition 1.4.1. A weight w is said to be an A1(dσ) weight, if

n Mσ(w)(x) ≤ C w(x), for σ-a.e. x ∈ R and some constant C. (1.11)

The inﬁmum of all such C’s is called [w]A1(dσ).

The following characterization ﬁrst appeared in [CoRo] and below we follow the idea from [GaRu] and adapt it to the dσ case.

6 Chapter 1. Preliminaries

1 Lemma 1.4.1 (Coifman-Rochberg for dσ). Let f ∈ Lloc(dσ) be such that Mσf < ∞ γ σ-a.e. If 0 ≤ γ < 1, then (Mσf) is an A1(dσ) whose A1(dσ) constant depends only on γ and D(σ).

∼ Proof. Let Q be a ﬁxed cube and Q := 3Q. We write f = f1 + f2, where f1 = f χ∼ Q ∼ is the restriction of f to Q. Since Mσ is sublinear and 0 ≤ γ < 1, we have

γ γ γ (Mσf) ≤ (Mσf1) + (Mσf2) .

It suﬃces to show that for all x, and Q 3 x, there exists C > 0 such that

Z 1 γ γ (Mσfi) dσ ≤ C (Mσf) (x), σ(Q) Q for i = 1, 2.

We carry out the estimates for Mσf1 and Mσf2 separately.

kf1kL1(dσ) We use Proposition 1.2.1, then split the integral at R = σ(Q) to get

Z Z ∞ 1 γ 1 γ−1 (Mσf1) dσ = γ t σ({x ∈ Q : Mσf1 > t}) dσ σ(Q) Q σ(Q) 0 Z R 1 γ−1 = γ t σ({x ∈ Q : Mσf1 > t}) dσ σ(Q) 0 Z ∞ 1 γ−1 + γ t σ({x ∈ Q : Mσf1 > t}) dσ σ(Q) R

In the ﬁrst integral, the distribution function is clearly less than σ(Q). In the second, using the weak-type estimate stated in Theorem 1.3.1, the distribution −1 function is less than D(σ) t kf1kL1(dσ).

7 Chapter 1. Preliminaries

Thus

Z µ Z ∞ ¶ 1 γ 1 γ γ−2 (M f ) dσ ≤ σ(Q) R + D(σ) kf k 1 γ t dt σ(Q) σ 1 σ(Q) 1 L (dσ) Q µ ¶ µ R ¶ D(σ)γ kf k 1 D(σ) γ = Rγ 1 + 1 L (dσ) = Rγ 1 + (1 − γ) R σ(Q) (1 − γ) µ ¶ ÃR !γ D(σ) γ |f| dσ = 1 + 3Q (1 − γ) σ(Q) µ ¶ Ã R !γ D(σ) γ |f| dσ ≤ 1 + D(σ) 3Q = C (M f(x))γ . (1 − γ) σ(3Q) σ,γ σ ³ ´ D(σ) γ γ where Cσ,γ = 1 + (1−γ) D(σ) , and for any x ∈ Q.

Now we can estimate the second term. By construction of f2, for any x, y ∈ Q we have

Mσf2(y) ≤ D(σ)Mσf2(x). (1.12) ∼ In fact, if Q0 is a cube containing y and intersecting the complement of Q, then x ∈ Q ⊂ 3 Q0, and Z Z 1 D(σ) 0 f2 dσ ≤ 0 f2 dσ ≤ D(σ) Mσf2(x). (1.13) σ(Q ) Q0 σ(3Q ) 3Q0 Now, take the supremum over all cubes Q0 containing y, on the left hand side, and ∼ R 0 1 we get (1.12). (Note that if y ∈ Q ⊂ Q, then σ(Q0) Q0 f2 dσ=0). Thus, raising the inequality (1.12) to the power γ, and taking σ-average over Q with respect to y, for every x ∈ Q we have Z 1 γ γ γ γ γ (Mσf2(y)) dσ ≤ D(σ) (Mσf2(x)) ≤ D(σ) (Mσf(x)) . σ(Q) Q Putting together both estimates we conclude that for all x ∈ Q Z µ ¶ 1 γ D(σ) γ γ γ (Mσf) dσ ≤ 2 + D(σ) (Mσf(x)) . σ(Q) Q (1 − γ)

γ This proves that (Mσf) ∈ A1(dσ).

8 Chapter 1. Preliminaries

Remark 1.4.1. The converse statement is also true, see for example [Du] or [Gr] p. 690.

1.5 Ap(dσ) weights

In this section we deﬁne the class of Ap(dσ) weights and state their main properties.

n Deﬁnition 1.5.1. If σ is a positive measure on R , we say w is an Ap(dσ) weight, 1 < p < ∞, if

µ Z ¶ µ Z ¶p−1 1 1 − 1 sup w(x) dσ w(x) p−1 dσ < ∞. (1.14) Q∈Rn σ(Q) Q σ(Q) Q

The expression in (1.14) is called the Ap(dσ) characteristic constant of w and is denoted by [w]Ap(dσ).

The main properties of these classes of weights include the following:

(a) The classes Ap(dσ) are increasing as p increases; precisely, for 1 < p < q < ∞

we have Ap(dσ) ⊂ Aq(dσ) and [w]Aq(dσ) ≤ [w]Ap(dσ), (This is just a simple consequence of H¨older’sinequality.)

1−p0 (b) w ∈ Ap(dσ) if and only if w ∈ Ap0 (dσ) and

1 [w1−p0 ] = [w] p−1 , (1.15) Ap0 (dσ) Ap(dσ)

This type of statement is what we will call a tautology, and it follows directly from the deﬁnitions.

(c) If w ∈ Ap(dσ), then w ∈ Ap−ε(dσ) for some ε > 0. (This is a deeper result due to Coifman-Feﬀerman which can be found in [CoFe] when dσ = dx, and

9 Chapter 1. Preliminaries

in the general case in [Buc2], [OrPe]). We will prove and analyze closely its dσ

version, in particular the relation between ε, [w]Ap(dσ) and D(σ) in Subsection 2.4.2.

α Remark 1.5.1. The function w(x) = |x| belongs to Ap, p > 1 if and only if

−n < α < n(p − 1); w is an A1 weight if and only if −n < α ≤ 0. However, w is doubling in the larger range −n < α < ∞.

1.6 Ap weights and classical results

In the case dσ = dx, the reference to the measure is usually suppressed, and Ap, M will be used. Below we list the classical results about the Ap weights.

The most important theorem in the theory of the Ap weights, due to Mucken- p houpt, states that the Ap class fully characterizes the boundedness of M on L (w).

The Ap class can be also used to characterize many other classical operators. Fur- thermore, it is of interest to determine how the operator norms are bounded in terms of the [w]Ap constant.

Theorem 1.6.1 (Muckenhoupt). Let 1 ≤ p < ∞. The inequality Z C p w (x : Mf(x) > λ) ≤ p |f(x)| w(x) dx λ Q holds if and only if w ∈ Ap.

Furthermore,

p Theorem 1.6.2. If 1 < p < ∞, then M is bounded on L (w) if and only if w ∈ Ap.

S. Buckley [Buc1] obtained the following sharp result for M in terms of [w]Ap .

10 Chapter 1. Preliminaries

Theorem 1.6.3. If 1 < p < ∞, w ∈ Ap, then

p0 p kMfk p ≤ C[w] kfk p . (1.16) L (w) Ap L (w)

p0 The power [w] p is best possible. Ap

We will revisit these Theorems and their proofs in Chapter 2 when trying to obtain the dσ version. The following factorization theorem provides an interesting representation of the Ap weights.

Theorem 1.6.4 (Jones). Suppose w is an Ap weight, for some 1 < p < ∞. Then there are A1 weights w1, w2 such that

1−p w = w1 w2

The proof can be found in [Jo], and most textbooks [Du], [Gr], [GaRu]. The Ap condition can be also formulated for pairs of weights as follows.

Deﬁnition 1.6.1. A pair of weights (u, v) is said to belong to the class Ap, 1 < p < ∞, if µ Z ¶ µ Z ¶p−1 1 1 − 1 sup u(x) dx v(x) p−1 dx < ∞. (1.17) Q∈Rn |Q| Q |Q| Q

The expression in (1.17) is called the Ap characteristic constant of (u, v) and is denoted by [u, v]Ap . We will consider pairs of the Ap weights when proving Lerner’s dσ Extrapolation Theorem in Section 3.3.

Deﬁnition 1.6.2. A pair of weights (u, v) is said to be an A1 weight, if

M(u)(x) ≤ Cv(x) a.e. x ∈ Rn, for some constant C. (1.18)

When u = v these are exactly the A1 and Ap conditions deﬁned in (1.3) and (1.4).

Necessary and suﬃcient conditions on u and v, such that the weak-type inequality (1.19) holds, are similar to the one-weighted case, presented in Theorem 1.6.1.

11 Chapter 1. Preliminaries

Theorem 1.6.5 (Muckenhoupt). Given p, 1 ≤ p < ∞, the weak-type inequality

Z C p u ({x : Mf(x) > λ}) ≤ p |f(x)| v(x) dx (1.19) λ Q holds if and only if (u, v) ∈ Ap.

This was proved in [Mu]. However, for the strong (p, p) boundedness the Ap condition is necessary but not suﬃcient. The characterization of strong inequalities with two weights is due to E. Sawyer and is called Sp condition. The pair (u, v) ∈ Sp if and only (1.20) holds.

Theorem 1.6.6 (Sawyer). M is bounded from Lp(v) to Lp(u), 1 < p < ∞, if and only if Z Z ³ ´p 1−p0 1−p0 M(χQv )(x) u(x) dx ≤ C v(x) dx. (1.20) Q Q for all cube Q, and C independent of Q, χQ the characteristic function of the cube Q.

For the proof we refer the reader to [Saw1]. For equal weights, Ap and Sp conditions are equivalent [HuKN].

Remark 1.6.1. A few years before the Ap theory, in 1960, H. Helson and G. Szeg¨o (see [HeSz]) gave necessary and suﬃcient condition for the boundedness of the Hilbert transform in L2(µ).

Theorem 1.6.7. The Hilbert Transform H is bounded in L2(µ) if and only if µ is absolutely continuous with respect to Lebesgue measure, dµ = w(x)dx, and w is of the form π log w = u + H v, with u, v ∈ L∞ and kvk < . ∞ 2

2 This condition works only for L and must be equivalent to A2 but there is not direct proof of this equivalence (see [Du2] or [Sa]). The Lp versions and other

12 Chapter 1. Preliminaries important generalizations were proved by Cotlar and Sadosky in a sequence of papers (see [Sa]).

1.7 A∞(dσ) weights

Deﬁnition 1.7.1. We say w is an A∞(dσ) weight if, for all cubes Q, and all E ⊂ Q, we have µ ¶ µ(E) σ(E) ε ≤ C (1.21) µ(Q) σ(Q) for some C, ε > 0, where dµ = w dσ.

Lemma 1.7.1. The following is equivalent to w ∈ A∞(dσ) For all cubes Q, there exists a constant C independent of Q such that, Z µ Z ¶ 1 1 w dσ ≤ C exp log w dσ . σ(Q) Q σ(Q) Q

The proof can be found in [Buc2]. Thus, the A∞(dσ) characteristic constant can be deﬁned as follows. ½µ Z ¶ µ Z ¶¾ 1 1 −1 [w]A∞(dσ) := sup w dσ exp log w dσ < ∞. (1.22) Q∈Rn σ(Q) Q σ(Q) Q

Theorem 1.7.1. (Jensen’s Inequality) Let σ be a positive measure on a sigma-algebra M in a set Ω, so that σ(Ω) = 1. If f is a real function in L1(σ), if a < f(x) < b for all x ∈ Ω, and if φ is convex on (a, b), then µZ ¶ Z φ f dσ ≤ (φ ◦ f) dσ. (1.23) Ω Ω The cases a = −∞ and b = ∞ are not excluded.

The proof can be found for example in [R].

13 Chapter 1. Preliminaries

Remark 1.7.1. If we take φ(x) = ex, and f = log g, then (1.23) becomes ½Z ¾ Z exp log g dσ ≤ g dσ (1.24) Ω Ω Corollary 1.7.1. For all 1 ≤ p < ∞ we have

[w]A∞(dσ) ≤ [w]Ap(dσ). (1.25)

This follows from the deﬁnitions and Jensen’s inequality (1.24), applied to

−1 g = w p−1 .

1.8 RHp(dσ) weights

The Reverse H¨olderproperty is a fundamental feature of the Ap(dσ) weights. It is of independent interest and plays an important role in the theory weights.

Deﬁnition 1.8.1. If σ ∈ D and 1 < p < ∞, we say w is a RHp(dσ) weight or w ∈ RHp(dσ) if µ Z ¶ 1 Z 1 p 1 wp dσ ≤ C w dσ (1.26) σ(Q) Q σ(Q) Q for all cubes Q. The smallest such C is referred to as RHp(dσ)-characteristic of w and is denoted by [w]RHp(dσ).

Some properties of these classes include the following:

(a) The classes RHp(dσ) are decreasing as p increases; precisely, for 1 < p < q < ∞

we have RHp(dσ) ⊃ RHq(dσ). (This is just a consequence of H¨olderinequality.)

(b) If w ∈ RHp(dσ), then w ∈ RHp+ε(dσ) for some ε > 0. This is a much deeper result and it is usually referred to as the Gehring Lemma.

14 Chapter 1. Preliminaries

It was ﬁrst proved in [Ge], and for general measures in [St], [Buc2], [Mil] or [Iw]. We will revisit the proof of Gehrings Lemma dσ in Section 2.2, tracking

very carefully the relation between ε, [w]RHp(dσ) and D(σ).

Remark 1.8.1. In the study of partial diﬀerential equations or quasi-conformal mappings the inequality (1.26) is usually replaced with a weaker and a more natural condition (sometimes called the weak-RHp(dσ), see [UrNeu])

µ Z ¶ 1 Z 1 p C wp dσ ≤ w dσ. (1.27) σ(Q) Q σ(2Q) 2Q

More details can be found in [UrNeu], [Mil] or [Iw].

α n Remark 1.8.2. The function w(x) = |x| belongs to RHp if and only if α > − p .

Deﬁnition 1.8.2. If σ1 and σ2 are positive doubling measures, we say σ1 is compa-

σ1(E) σ2(E) rable to σ2 if there exist α, β ∈ (0, 1) such that < β whenever < α for σ1(Q) σ2(Q) every E ⊂ Q, and every Q.

Proposition 1.8.1. The following conditions are equivalent.

a) There exist δ > 0, C > 0 such that for every measurable set A contained in a cube Q µ ¶ σ (A) σ (A) δ 2 ≤ C 1 , σ2(Q) σ1(Q)

b) σ2 is comparable to σ1,

c) σ1 is comparable to σ2,

d) dσ2 = w dσ1 with w ∈ RHp(dσ1) for some p > 1.

This was proved in [CoFe].

Corollary 1.8.1. The comparability of measures is an equivalence relation.

15 Chapter 1. Preliminaries

Corollary 1.8.2. If σ1 and σ2 are comparable, then σ1(A) = 0 ⇐⇒ σ2(A) = 0.

Corollary 1.8.3. If uo ∈ A∞ and dσ := uodx, and if dµ := wdσ, where w ∈ Ap(dσ), then the measures: dµ, dσ and dx are comparable.

In particular, when talking about a.e. statements, this will allow us to omit the underlying measure: σ, µ.

Lemma 1.8.1. Let σ be an arbitrary but ﬁxed doubling measure on Rn.

If w ∈ A∞(dσ), then w ∈ D. Furthermore, [ [ A∞(dσ) = Ap(dσ) = RHq(dσ) 1≤p<∞ 1≤q<∞

When dσ = dx this a result of Coifman and Feﬀerman [CoFe]. The general case can be found in [Buc2]. Thus, w ∈ Ap(dσ) ⇐⇒ w ∈ RHq(dσ) for some p and q. There is no possible relationship between p and q as the example of a power function w(x) = |x|α demonstrates. In Section 2.3 we will prove and analyze closely the dσ version of what we call Reverse H¨olderInequality (or the RH1+γ(dσ) property) for the Ap(dσ) weights. In particular, the relation between γ,[w]Ap(dσ) and D(σ).

However, for positive functions u, v on Rn and 1 < p < ∞ the following tautology 1 1 holds (see [Gr]), where p + p0 = 1,

1 [uv−1] = [vu−1] p (1.28) RHp0 (vdx) Ap(udx) that is uv−1 satisﬁes a reverse H¨oldercondition of order p0 with respect to vdx −1 if and only if vu is in Ap(udx).

16 Chapter 1. Preliminaries

Proof.

³ ´ 1 ³ ´ 1 R 0 R 0 1 −1 p0 p 1 p0 1−p0 p v(Q) Q(uv ) vdx v(Q) Q u v dx [uv−1] (vdx) = sup R = sup RHp0 1 −1 u(Q) Q3x (uv )vdx Q3x v(Q) Q v(Q) µ ¶ 1 Z 0 v(Q) 1 0 0 p = sup up v1−p dx Q3x u(Q) v(Q) Q 1 µZ ¶ p−1 p [v(Q)] p 0 = sup (vu−1)1−p udx Q3x u(Q) Q " # 1 µ Z ¶p−1 p v(Q) 1 0 = sup (vu−1)1−p udx Q3x u(Q) u(Q) Q " # 1 µ Z ¶ µ Z ¶p−1 p 1 1 0 = sup (vu−1) udx (vu−1)1−p udx Q3x u(Q) Q u(Q) Q 1 = [vu−1] p Ap(udx)

In particular,

−1 w ∈ RHp0 (dx) ⇐⇒ w ∈ Ap(wdx), (1.29)

−1 w ∈ Ap(dx) ⇐⇒ w ∈ RHp0 (wdx). (1.30)

1.9 Rubio de Francia Extrapolation Theorem

In this section we present the classical Rubio de Francia and its weak-version.

Theorem 1.9.1 (Rubio de Francia extrapolation). Given an operator T , suppose that for some po, 1 ≤ po < ∞, and every w ∈ Apo , there exists a constant C depending on [w] such that Ap0 Z Z |T f(x)|po w(x) dx ≤ C |f(x)|po w(x) dx. (1.31) Rn Rn

17 Chapter 1. Preliminaries

Then for every p, 1 < p < ∞, and every w ∈ Ap there exists a constant depending on [w]Ap such that Z Z |T f(x)|pw(x) dx ≤ C |f(x)|pw(x) dx. (1.32) Rn Rn

The original proof was rather complex and nonconstructive. Garc´ıa-Cuerva gave a more direct proof of this result, involving only weighted norm inequalities (see

[GaRu]). For po > 1 a simpler proof is due to Duoandikoetxea (see [Du]). In a very recent monograph on extrapolation (see [CrMPe1]) the authors presented the history of this theorem and provided a new and more direct proof. In addition, they included a number of their results on the generalizations of this theorem in various directions.

Both Garc´ıa-Cuerva (see [Cu]) and Rubio de Francia (see [Ru] or [CrMPe1]) were able to adapt their proofs for the strong-type inequalities to get the weak-type versions.

Theorem 1.9.2. Given an operator T , suppose that for some po, 1 ≤ po < ∞, and

po p all w ∈ Apo , T is of L (w) weak-type (po, po), then T is of L (w) weak-type (p, p) for all 1 < p < ∞, and all w ∈ Ap.

n Proof. If Eλ := {x ∈ R : |T f(x)| > λ}, then we can write the weak-type (po, po) inequality as

p p kλχEλ kL o (w) ≤ CkfkL o (w).

Now, we can apply the Extrapolation Theorem to get

p p kλχEλ kL (w) ≤ CkfkL (w),

for all p and w ∈ Ap, which is equivalent to T being of weak-type (p, p), for any 1 < p < ∞.

18 Chapter 1. Preliminaries

As J. Duoandikoetxea points out (see [Du2]) it is possible to start with weak p (1, 1) inequalities with respect to all A1 weights to deduce strong L (w) inequalities with w ∈ Ap. However, it is not possible to obtain weak (1, 1) inequalities in Theorem p 1.9.1. There exist operators bounded on L (w) for all w ∈ Ap, and 1 < p < ∞, which are not weak (1, 1) (even unweighted). In [CrMPe] Rubio de Francia Ap Theorem was generalized to A∞ weights in the context of Muckenhoupt bases, which allows to prove weak-endpoint inequalities starting from strong-type inequalities.

1.10 Calder´on-Zygmund Decomposition and Besi- covitch Covering Lemma

Calder´on-Zygmund Decomposition was invented by Calder´onand Zygmund and proved to be extremely useful in real variable analysis of singular integrals.

Theorem 1.10.1. Given σ ∈ D, and a function f which is σ-integrable and nonnegative, and given a positive number λ, there exists a sequence {Qj} of disjoint dyadic cubes such that

S (i) f(x) ≤ λ for σ-a.e. x∈ / j Qj; ³ ´ S D(σ) kfkL1(dσ) (ii) σ j Qj ≤ λ ; R (iii) λ < 1 f dσ ≤ D(σ)λ. σ(Qj ) Qj

The proof can be found in [Cu] or [Gr].

The Besicovitch covering lemma has the advantage of being applicable even if the underlying measure is non-doubling.

Lemma 1.10.1 (Besicovitch covering lemma). Suppose A ⊂ Rn is bounded and that for each x ∈ A, Qx is a cube centered at x. Then we can choose, from among

19 Chapter 1. Preliminaries

{Qx : x ∈ A}, a possibly ﬁnite sequence {Qi} and an associated sequence of integers

{mi} such that

S (a) A ⊂ i Qi;

(b) 1 ≤ mi ≤ Nn, where Nn depends only on n;

The proof can be found in [Gu]. The sub-points (b) and (c) say that the sequence of cubes can be distributed into a ﬁnite number of disjoint families. More precisely, given j,

1 ≤ j ≤ Nn we can deﬁne Fj := {k ∈ N : mk = j}, and then write

[ [Nn [ Qi = Qk,

i j=1 k∈Fj

where the family {Qk}k∈Fj consists of pairwise disjoint cubes.

As a consequence of Besicovitch lemma we get a very useful theorem.

c Theorem 1.10.2. The centered Hardy-Littlewood maximal function Mσ, deﬁned in Section 1.3, is of weak type (1, 1) with respect to σ, with the weak norm at most C(n), where the constant C(n) depends only n.

The proof can be found for example in [Gu] p. 37.

Remark 1.10.1. The theory of Ap weights has an analogue when the underlying measure σ is nondoubling but satisﬁes σ(δQ) = 0 for all cubes Q in Rn with sides parallel to the axes, where δQ is the boundary of Q, see [OrPe].

20 Chapter 2

Buckley type estimates for Maximal Function

In this chapter we obtain sharp-estimates for the uncentered Maximal Function associated with measure σ. We are interested in the Buckley-type estimates, where the dependence on the Ap(dσ) (or RHq(dµ), where dµ = wdσ) characteristic constant is established, with possibly the best power. As in the classical case (see [Buc2]), we start first with the weak-type estimates and then use the self-improvement properties of the Ap(dσ) (or RHq(dµ)) weights and the Interpolation Theorem 1.2.1 to get the strong-type estimates. The Interpolation Theorem, as clearly seen in (1.8), requires a careful study of ε as a function of Ap(dσ) (or RHq(dµ)) characteristic constant. Since the Buckley’s Theorem, especially the fact used in his theorem, stated without a proof, that the ε appearing in the Coifman-Fefferman-Buckley Theorem should be 0 of order [w]1−p was not that clear to us, we first took advantage of the tautology Ap (1.28), and obtained the Gehring type estimates. Then, we used them to interpolate on the RHq(dµ) side instead. We obtained the same power as in Theorem 1.6.3 but the constant D(µ) appeared as a result of the Calderón-Zygmung decomposition used in the proof.

21 Chapter 2. Buckley type estimates for Maximal Function

In the second part of this chapter, as we try to avoid the constant D(µ), we go back to Theorem 2.3.1, to obtain the Coiffman-Fefferman-Buckley Theorem dσ. We arrive at the same conclusions but with a different doubling constant i.e. D(σ) this time. Finally, we present yet another proof, based on A. Lerner’s very recent result (see [Le1]). It allows us to obtain the strong-estimates without using interpolation. Instead, it relies only on the properties of the Maximal Function and the Theorem 1.10.2. It also yields a doubling constant D(σ) but with a different exponent.

These estimates will be later used in Chapter 3 to obtain Sharp Extrapolation Theorems.

2.1 Weak Estimates

In this section we generalize Muckenhoupt-Buckley type weak-estimates to the case dσ, where dσ = uodx for some uo ∈ A∞.

p Lemma 2.1.1. If f ∈ L (dµ), where dµ = wdσ and w ∈ Ap(dσ), and if R 1 f dσ ≥ λ > 0 for each of the disjoint cubes Qk, then σ(Qk) Qk

µ ¶p X kfk p µ(Q ) ≤ C [w] L (dµ) (2.1) k n Ap(dσ) λ k

Proof. Without loss of generality, we can assume f ≥ 0, and normalized in Lp(dµ), i.e.

¡R ¢ 1 p p kfkLp(dµ) = f w dσ = 1.

− 1 Since w ∈ Ap(dσ) ⇐⇒ w p−1 ∈ Ap0 (dσ), and

p0−1 υ(Q )µ(Q ) 0 k k ≤ [w]p −1 (2.2) p0 Ap(dσ) σ(Qk)

22 Chapter 2. Buckley type estimates for Maximal Function

R p0 − p0 1 for any such Qk, where υ(Qk) := w p dσ. (Note that = ). Qk p p−1

Now, using ﬁrst the hypothesis, then H¨olderinequality plus the fact that kfkLp(dµ) =

1, and ﬁnally using (2.2) and the fact that Qk’s are disjoint we get : X X Z µ(Qk) µ(Qk) ≤ f dσ λ σ(Qk) k k Qk Z X µ(Qk) 1 1 p − p = χQk (fw )w dσ λ σ(Qk) k 1  ¯ ¯ 0  µZ ¶ 1 Z p p0 p ¯X ¯ 0 ¯ µ(Qk) ¯ − p p  p  ≤ f w dσ ¯ χQk ¯ w dσ ¯ λ σ(Qk) ¯ k Ã ! 1 1 Ã ! 1 X p0 p0 [w] p X p0 υ(Qk)µ(Qk) Ap(dσ) ≤ p0 p0 ≤ µ(Qk) λ σ(Qk) λ k k

Thus, X [w] µ(Q ) ≤ Ap(dσ) k λp k

Now we can use Lemma 2.1.1 to prove the following lemma.

c Lemma 2.1.2. The centered maximal function Mσ is weak (p, p), with respect to measure µ, where dµ = wdσ, and w ∈ Ap(dσ),

µ ¶p kfk p µ({x ∈ Rn : M cf > λ}) ≤ C [w] L (dµ) . σ n Ap(dσ) λ

Proof. Without loss of generality, we can assume f ≥ 0 and kfkLp(dµ)=1. Suppose c now Mσf(x) > λ, then there must exist some maximal Qx centered at x such that R 1 f dσ > λ. (By letting Qe be a large enough dilation of a cube Q, we can σ(Qx) Qx make µ(Qe) very large because dµ is comparable to dx). By the Besicovitch covering

23 Chapter 2. Buckley type estimates for Maximal Function

Lemma 1.10.1 [Nn [ c Ar = {x : |x| < r, Mσf > λ} ⊂ Qk

j=1 k∈Fj where, each family {Qk}k∈Fj is formed by pairwise disjoint cubes. Thus,

[Nn [ XNn [ [ X µ(Ar) ≤ µ( Qk) ≤ µ( Qi) ≤ Nn max µ( Qk) = Nn µ(Qk), 1≤j≤Nn j=1 k∈Fj j=1 k∈Fj k∈Fj k∈F0 where F0 is a collection whose union has maximal µ-measure. By letting r → ∞ and using the Lemma 2.1.1 we obtain the desired inequality: Ã ! [ n c Cn [w]Ap(dσ) µ({x ∈ R : M f > λ} ≤ N µ Q ≤ kfk p . (2.3) σ n k λp L (dµ) k

By (1.9) we get a similar result for the uncentered maximal function.

Corollary 2.1.1.

p n Cn D(σ) [w]Ap(dσ) µ({x ∈ R : M f > λ} ≤ kfk p . (2.4) σ λp L (dµ)

2.1.1 Strong Doubling

In this section we present a lemma comparing the measures µ and σ, which will be used a few times in the next sections.

Lemma 2.1.3 (Strong Doubling). Let w ∈ Ap(dσ) for some 1 ≤ p < ∞ and let 0 < α < 1. Then there exists 0 < β < 1 such that whenever S is a measurable subset of a cube Q that satisﬁes σ(S) ≤ ασ(Q), we have µ(S) ≤ βµ(Q).

Proof. Applying H¨older’sinequality with exponents p and p0 we obtain

24 Chapter 2. Buckley type estimates for Maximal Function

µ Z ¶p µ Z ¶p 1 1 1 − 1 f(x) dσ = f(x)w(x) p w(x) p dσ σ(Q) Q σ(Q) Q

µ Z ¶ µ Z ¶ p 0 p0 1 p 1 − p ≤ f(x) w(x) dσ w(x) p dσ σ(Q) Q σ(Q) Q µ Z ¶ µ Z ¶ µ Z ¶p−1 1 p 1 1 − 1 = f(x) dµ w(x) dσ w(x) p−1 dσ µ(Q) σ(Q) σ(Q) µQ Z ¶ Q Q 1 p ≤ [w]Ap(dσ) f(x) dµ µ(Q) Q Thus, µ Z ¶p µ Z ¶ 1 1 p f(x) dσ ≤ [w]Ap(dσ) f(x) dµ (2.5) σ(Q) Q µ(Q) Q

If we set f = χA in (2.5), we obtain µ ¶ σ(A) p µ(A) ≤ [w] (2.6) σ(Q) Ap(dσ) µ(Q)

If we write S = Q\A we get µ ¶ µ ¶ σ(S) p µ(S) 1 − ≤ [w] 1 − . (2.7) σ(Q) Ap(dσ) µ(Q)

Given 0 < α < 1, set

(1 − α)p β = 1 − (2.8) [w]Ap(dσ) σ(S) and use (2.7) to obtain the required conclusion. More precisely, if σ(Q) ≤ α, then µ ¶ µ(S) (1 − α)p ≤ [w] 1 − . Ap(dσ) µ(Q)

From here we deduce µ(S) (1 − α)p ≤ 1 − = β. µ(Q) [w]Ap(dσ)

25 Chapter 2. Buckley type estimates for Maximal Function

Corollary 2.1.2. Using (2.5) with the function f = χQ and averaging over 3 Q in the place of Q we get µ ¶ µ(3Q) σ(3Q) p ≤ [w] , (2.9) µ(Q) Ap(dσ) σ(Q) p which implies µ ∈ D and D(µ) ≤ [w]Ap(dσ) [D(σ)] .

As a consequence of (2.5) we also obtain an alternative and a much simpler proof of the weak-estimates for Mσ but with D(µ) constant.

Corollary 2.1.3. If dµ = wdσ, w ∈ Ap(dσ), then Z n D(µ)[w]Ap(dσ) p µ({x ∈ R : Mσf > λ}) ≤ p |f(x)| dµ. (2.10) λ Rn

Proof. By (2.5) we get

1 1 M f(x) ≤ [w] p [M (f p)(x)] p for a.e. x. (2.11) σ Ap(dσ) µ

By Theorem 1.3.1 Mµ is of weak-type (1, 1), with respect to measure µ, thus we obtain µ ¶ 1 1 µ({x ∈ Rn : M f(x) > λ}) ≤ µ {x ∈ Rn :[w] p [M (f p)] p > λ} σ Ap(dσ) µ µ p ¶ n p λ = µ {x ∈ R : Mµ(f ) > } [w]A (dσ) Z p D(µ)[w]Ap(dσ) p ≤ p |f(x)| dµ. (2.12) λ Rn

This implies that Mσf is of weak-type (p, p) with respect to µ, with the weak-type 1 1 norm at most D(µ) p [w] p . Ap(dσ)

Remark 2.1.1. We can actually avoid D(µ) constant, in (2.12), by using Theorem 1.10.2. The inequality (2.5) also implies the same inequality as (2.11) for the centered maximal functions. Thus we have

1 £ ¤ 1 M cf(x) ≤ [w] p M c(f p)(x) p for a.e. x. (2.13) σ Ap(dσ) µ

26 Chapter 2. Buckley type estimates for Maximal Function

Thus,

µ({x ∈ Rn : M f(x) > λ}) ≤ µ({x ∈ Rn : D(σ) M cf(x) > λ}) σ µ σ ¶ 1 £ ¤ 1 ≤ µ {x ∈ Rn : D(σ )[w] p M c(f p) p > λ} Ap(dσ) µ µ p ¶ n c p λ = µ {x ∈ R : Mµ(f ) > p } D(σ) [w]Ap(dσ) p Z C(n) D(σ) [w]Ap(dσ) p ≤ p |f(x)| dµ, λ Rn where C(n) is the dimensional constant as in Theorem 1.10.2.

The weak-norm here is exactly the same as in (2.4).

What we have obtained both for the centered and uncentered maximal function are the following weak-estimates.

If dµ = wdσ, w ∈ Ap(dσ), then

 1 p C(σ, µ, p)[w] p Z µ({x ∈ Rn : T f > λ}) ≤  Ap(dσ) |f(x)|p dµ , (2.14) λ Rn

c where T = Mσ or T = Mσ, and the constant C(σ, µ, p) depends only on p, D(σ) or D(µ). More precisely, for

1 1 c p p for T = Mσ : C1(σ, µ, p) = min{C(n) ,D(µ) }; (2.15) 1 1 for T = Mσ : C2(σ, µ, p) = min{C(n) p D(σ),D(µ) p }. (2.16)

2.2 RHp(dσ) and Gehring’s Estimates

In this subsection we are going to adapt Gehring’s Lemma to the dσ case, keeping track of D(σ) and the size of ε. For that we need some preliminary lemmas.

27 Chapter 2. Buckley type estimates for Maximal Function

Lemma 2.2.1. Suppose that q ∈ (0, ∞) and a ∈ (1, ∞),

h : [1, ∞) → [0, ∞) is non-increasing, right-continuous with

lim h(t) = 0 t→∞ and that Z ∞ − sq dh(s) ≤ atqh(t). (2.17) t for t ∈ [1, ∞).

Then Z µ Z ¶ ∞ q ∞ − tp dh(t) ≤ − tq dh(t) , (2.18) 1 aq − (a − 1)p 1 £ q ¢ for p ∈ q, q + a−1 . This inequality is sharp.

This lemma can be found in [Ge] p.266, and will be used to prove the next lemma.

Lemma 2.2.2. Suppose that b, q ∈ (1, ∞), Q is an n-cube in Rn, g : Q −→ [0, ∞] is Lq(dσ) integrable in Q and

Z µ Z ¶q 1 q 1 0 g dσ ≤ b 0 g dσ (2.19) σ(Q ) Q0 σ(Q ) Q0 for each parallel n-cube Q0 ⊂ Q.

Then g is Lp(dσ)-integrable in Q with

Z µ Z ¶ p 1 c 1 q gp dσ ≤ gq dσ (2.20) σ(Q) Q q + c − p σ(Q) Q for p ∈ [q, q + c), where c is a positive constant which depends only on q, b, D(σ) and n.

Proof. Without loss of generality we can assume Z gq dσ = σ(Q). (2.21) Q

28 Chapter 2. Buckley type estimates for Maximal Function

This is because if go satisﬁes (2.19), then g (x) g(x) := Ro , (2.22) 1 q σ(Q) Q g dσ will satisfy both (2.19) and (2.21). If we have shown the Lemma for such a g, then substituting (2.22) into (2.20) will also imply (2.20) for go.

Therefore all we need to show is Z 1 c gp dσ ≤ , (2.23) σ(Q) Q q + c − p for a g that satisﬁes (2.21).

Deﬁne E(t) = {x ∈ Q : g(x) > t}. We can begin by showing Z Z gq dσ ≤ atq−1 g dσ (2.24) E(t) E(t) for t ∈ [1, ∞), where a is a constant which depends only on q, b, D(σ) and n.

³ ´q q q Next for t ∈ [1, ∞) choose s > t such that s := b q−1 t , and apply the Calder´on-Zygmund decomposition, introduced in Section 1.9, at the level sq, to obtain a disjoint sequence of parallel n-cubes Qj ⊂ Q such that Z 1 sq < gq dσ ≤ D(σ)sq (2.25) σ(Qj) Qj S for all j, and such that g ≤ s a.e. in Q\ j Qj. This implies that the set E(s) is S contained, except for a subset of measure zero, in j Qj. The cubes Qj are disjoint, therefore with (2.25) we have Z X Z X q q q g dσ ≤ g dσ ≤ s D(σ) σ(Qj). (2.26) E(s) j Qj j

The inequalities (2.25) and (2.19) imply that µ ¶ Ã Z !q q q 1 b t ≤ b g dσ , q − 1 σ(Qj) Qj

29 Chapter 2. Buckley type estimates for Maximal Function therefore, Z Z q tσ(Qj) ≤ g dσ ≤ g dσ + tσ(Qj). q − 1 Qj Qj ∩E(t) The second inequality holds because if x∈ / E(t), then g(x) ≤ t, and Z

g dσ ≤ t σ(Qj). Qj \E(t)

We now get, Z q − 1 σ(Qj) ≤ g dσ t Qj ∩E(t) for each j. Combining this inequality with (2.26) yields Z Z Z X q − 1 q − 1 gq dσ ≤ D(σ) sq g dσ ≤ D(σ) sq g dσ, (2.27) t t E(s) j Qj ∩E(t) E(t) If x ∈ E(t)\E(s), then g(x) ≤ s, therefore Z Z gq dσ ≤ sq−1 g dσ. (2.28) E(t)\E(s) E(t)

By (2.28) and (2.27) we obtain Z Z Z · ¸ Z q − 1 gq dσ = gq dσ + gq dσ ≤ D(σ) sq + sq−1 g dσ t E(t) ·E(t)\E(s) E(s) ¸ Z E(t) ³s´q ³s´q−1 = D(σ) (q − 1) + tq−1 g dσ t t E(t) This proves (2.24) with

³s´q ³s´q−1 a = D(σ) (q − 1) + t t µ ¶q µ ¶q−1 q q q−1 = D(σ) b (q − 1) + b q q − 1 q − 1 µ ¶ q q−1 ≤ b q [D(σ) + 1] ≤ 6D(σ) b q. q − 1

³ ´q−1 1 The last inequality holds because D(σ) + 1 < 2D(σ) and 1 + q−1 ≤ e < 3 for any q > 1.

30 Chapter 2. Buckley type estimates for Maximal Function

R Now, for t ∈ [1, ∞), set h(t) = E(t) g dσ. Then h : [1, ∞) → (0, ∞) is non- increasing, right-continuous and limt→∞ h(t) = 0.

Then, using (1.5) we get the following,

For any r, t ∈ [1, ∞)

Z Z Z ∞ gr dσ = gr−1 gdσ = − sr−1dh(s). E(t) E(t) t Thus inequality (2.24), can be rewritten in the form,

Z ∞ − sq−1dh(s) ≤ atq−1h(t) t which shows that the function h satisﬁes the remaining hypothesis (2.17).

Then we can apply Lemma 2.2.1 with q and p , replaced by q − 1 and p − 1, to obtain Z µ Z ¶ ∞ q − 1 ∞ − tp−1dh(t) ≤ − tq−1dh(t) . 1 a(q − 1) − (a − 1)(p − 1) 1 This is equivalent to Z Z q − 1 gp dσ ≤ gq dσ E(1) a(q − 1) − (a − 1)(p − 1) E(1) for p ∈ [q, q + c), where q − 1 q − 1 c = > . (2.29) a − 1 6D(σ) b q

p q a(q−1) (a−1+1)(q−1) Since g (x) ≤ g (x) in Q\E(1), and a−1 = a−1 = q + c − 1, therefore Z Z Z gp dσ ≤ gp dσ + gp dσ Q E(1) Z Q\E(1) Z c ≤ gq dσ + gq dσ q + c − p ZE(1) Q\E(1) c c ≤ gq dσ = . q + c − p Q q + c − p for p ∈ [q, q + c). This together with (2.23) yields (2.20).

31 Chapter 2. Buckley type estimates for Maximal Function

Corollary 2.2.1. If w ∈ RHq(dσ), then w ∈ RHq+ε(dσ) with

1 [w] ≤ 2[w] , for some ε ∼ . RHq+ε(dσ) RHq(dσ) D(σ)[w]q RHq(dσ)

If we take g = w, and choose b = [w]q , then we can use Lemma 2.2.2. RHq(dσ)

Thus, for any Q, by (2.20) and (2.19) we have

µ Z ¶ 1 µ ¶ 1 µ Z ¶ 1 1 p c p 1 q wp dσ ≤ wq dσ σ(Q) Q q + c − p σ(Q) Q µ ¶ 1 µZ ¶ c p ≤ [w]RHq(dσ) w dσ , (2.30) q + c − p Q

c which implies w ∈ RHq+ε(dσ), with ε = p − q < c. Furthermore, if we choose ε = 2 , ³ ´ 1 1 c p then by (2.29) we obtain ε ∼ D(σ)[w]q and q+c−p ≤ 2, for any p > 1, which RHq(dσ) ends the proof.

2.3 Reverse H¨olderInequality for the Ap(dσ)

In this section we return to the properties of the Ap(dσ) class. We analyze and adapt the proofs of the Reverse HölderInequality for the Ap(dσ) weights, to determine the connection between the order of γ in Theorem 2.3.1 and the order of ε in Theorem 2.3.2. This will also help us understand better the classical Buckley’s Theorem 1.6.3, in particular, his result stated without a proof in [Buc1], that the optimal ε appearing 0 in the Coiffman-Fefferman Theorem should be ε ∼ [w]1−p . This way we can avoid the Ap constant D(µ), but D(σ) appears instead as a consequence of the Calderón-Zygmund decomposition used in the proof.

Theorem 2.3.1 (Reverse H¨olderInequality). Let w ∈ Ap(dσ) for some 1 ≤ p < ∞. Then there exist constants C and γ > 0 that depend only on the dimension n, on p,

32 Chapter 2. Buckley type estimates for Maximal Function

D(σ) and on [w]Ap(dσ) such that for every cube Q we have

µ Z ¶ 1 Z 1 1+γ C w(t)1+γ dσ ≤ w(t) dσ (2.31) σ(Q) Q σ(Q) Q

Proof. Let us fix a cube Q and set Z 1 αo = w(t) dσ. σ(Q) Q We also fix an 0 < α < 1. We define an increasing sequence of scalars

αo < α1 < α2 < ... < αk < ... for k ≥ 0 by setting

−1 −1 k αk+1 = (D(σ)α )αk or αk = (D(σ)α ) αo, and for each s ≥ 1 we divide the cube Q into a mesh of 2ns subcubes of side length equal to 2−sl(Q). Among all these cubes, we select those with the property that the average of w over them is strictly greater than αk and we isolate all maximal cubes with this property. In this way we obtain a collection {Qk,j}j so that the following are satisﬁed:

R 1 1. αk < w(t) dσ ≤ D(σ)αk σ(Qk,j ) Qk,j

S 2. On Q\Uk we have w ≤ αk σ-a.e., where Uk = j Qk,j; for each k, {Qk,j}j are disjoint

3. Each Qk+1,j is contained in some Qk,l

Property (i) is satisﬁed since the unique dyadic parent of Qk,j was not chosen in the selection procedure, while (ii) follows from the Lebesgue diﬀerentiation theorem (for

33 Chapter 2. Buckley type estimates for Maximal Function

doubling measures) using the fact that for almost all x∈ / Uk there exists a sequence of non selected cubes of decreasing lengths whose intersections is {x}. Property (iii) is satisﬁed since each Qk,j is the maximal subcube of Q with the property that the average of w over it is bigger than αk. And since the average of w over Qk+1,j is also bigger than αk, it follows that Qk+1,j must be contained in some maximal cube that has this property.

We will now compute the portion of Qk,l that is covered by cubes of the form Qk+1,j for some j. We have

Z 1 D(σ)αk ≥ w(t) dσ σ(Qk,l) Qk,l∩Uk+1 Z 1 X 1 = σ(Qk+1,j) w(t) dσ σ(Qk,l) σ(Qk+1,j) Qk+1,j j : Qk+1,j ⊂Qk,l

σ(Qk,l ∩ Uk+1) σ(Qk,l ∩ Uk+1) −1 > αk+1 = D(σ)α αk. σ(Qk,l) σ(Qk,l)

It follows that σ(Qk,l ∩ Uk+1) ≤ ασ(Qk,l). Thus, applying Lemma 2.1.3, we obtain

µ(Q ∩ U ) (1 − α)p k,l k+1 < β = 1 − µ(Qk,l) [w]Ap(dσ) from which, summing over all l, we obtain

µ(Uk+1) ≤ βµ(Uk).

k The latter gives µ(Uk) ≤ β µ(Uo). We also have σ(Uk+1) ≤ ασ(Uk); hence σ(Uk) → 0 0 as k → ∞. Therefore, the intersection of the Uk s is a null set. We can therefore write [∞ Q = (Q\Uo) ∪ ( Uk\Uk+1) k=0 modulo a set of Lebesgue measure 0. Let us ﬁnd a γ > 0 so that the Reverse H¨older

34 Chapter 2. Buckley type estimates for Maximal Function

Inequality (2.3.1) holds. We have w(x) ≤ αk for almost all x in Q\Uk and therefore Z Z X∞ Z w(t)γ+1 dσ = w(t)γw(t) dσ + w(t)γw(t) dσ Q Q\Uo k=0 Uk\Uk+1 X∞ γ γ ≤ αo µ(Q\Uo) + αk+1µ(Uk) k=0 X∞ γ ¡ −1 k+1 ¢γ k ≤ αo µ(Q\Uo) + (D(σ)α ) αo β µ(Uo) Ã k=0 ! X∞ γ −1 γ −1 γk k ≤ αo 1 + (D(σ)α ) (D(σ)α ) β (µ(Q\Uo) + µ(Uo)) µ Z ¶ kµ=0 ¶ Z 1 γ (D(σ)α−1)γ = w(t) dσ 1 + −1 γ w(t) dσ σ(Q) Q 1 − (D(σ)α ) β Q provided γ > 0 is chosen small enough so that (D(σ)α−1)γ β < 1. Keeping track of the constants, as in [Gr], we conclude the proof with ¡ ¢ ¡ ¢ − log β log [w] − log [w] − (1 − α)p γ < = Ap(dσ) Ap(dσ) log D(σ) − log α log D(σ) − log α and   1 µ γ ¶ 1 γ γ+1 (D(σ)α−1) γ+1 (D(σ)α−1) C = 1 + = 1 + ³ ´ −1 γ p 1 − (D(σ)α ) β 1 − (D(σ)α−1)γ 1 − (1−α) [w]Ap(dσ)

Note that since,

x2 x3 − log(1 − x) = x + + + ... 2 3 if |x| < 1, and µ ¶ (1 − α)p 1 γ < − log 1 − D(σ) , [w]A (dσ) p log α 1 therefore if we set α = D(σ) < 1, then 1 γ ∼ . (2.32) [w]Ap(dσ) log(D(σ))

35 Chapter 2. Buckley type estimates for Maximal Function

The parameter α can be chosen to optimize the constant γ. However, we would also like to have a uniform bound on C in terms of [w]Ap(dσ). This will be much easier to see in the next Section.

2.3.1 C. P´erez’new proof

In this subsection we present a simpler proof of the Theorem 2.3.1 adapted to the dσ case. It was inspired by Carlos P´erez’recent paper, (see [P1]) who was interested in obtaining similar and even more precise estimates for the Ap weights. The order of γ is the same as in (2.32) but the constant C here is much better. We need it to be uniformly bounded, and actually we can choose C = 2 (as observed in [Buc1] or [Wit1] p. 6). This fact will be crucial in Theorem 2.3.2, and then in the Interpolation Theorem 1.8.

Proof. Let Z 1 µ(Q) wQ := w dσ = . σ(Q) Q σ(Q) By Proposition 1.2.1, with φ(x) = xγ, the underlying measure dµ = wdσ,

|f(x)| = w(x) and X = Q, we get Z Z 1 γ ∞ w(x)γw(x) dσ = λγ−1 µ({x ∈ Q : w(x) > λ}) dλ σ(Q) σ(Q) Q Z0 γ wQ = λγ−1 µ({x ∈ Q : w(x) > λ}) dλ σ(Q) 0 Z γ ∞ + λγ−1 µ({x ∈ Q : w(x) > λ}) dλ σ(Q) wQ = I + II. (2.33)

Clearly, Z γ wQ I = λγ−1 µ({x ∈ Q : w(x) > λ}) dλ σ(Q) Z0 Z wQ wQ γ γ−1 γ−1 γ+1 ≤ λ µ(Q) dλ = wQ γ λ dλ = (wQ) . σ(Q) 0 0

36 Chapter 2. Buckley type estimates for Maximal Function

Now for any Q deﬁne 1 EQ = {x ∈ Q : w(x) ≤ p−1 wQ}. 2 [w]Ap(dσ) Then we claim σ(Q) σ(E ) ≤ . (2.34) Q 2

By (2.6) for EQ ⊂ Q we have µ ¶ σ(E ) p µ(E ) Q ≤ [w] Q σ(Q) Ap(dσ) µ(Q)

Since by deﬁnition of EQ Z Z wQ wQ µ(EQ) = w dσ ≤ p−1 dσ ≤ p−1 σ(EQ), EQ EQ 2 [w]Ap(dσ) 2 [w]Ap(dσ) therefore µ ¶p σ(EQ) µ(EQ) wQ σ(EQ) σ(EQ) ≤ [w]Ap(dσ) ≤ [w]Ap(dσ) p−1 = p−1 , σ(Q) µ(Q) µ(Q) 2 [w]Ap(dσ) 2 σ(Q) from which the claim follows.

Notice that (2.34) implies σ(Q) σ(Q\E ) > , (2.35) Q 2

1 where Q\EQ = {x ∈ Q : w(x) > p−1 wQ}. 2 [w]Ap(dσ)

The second claim is the following:

For every λ > wQ we have λ µ({x ∈ Q : w(x) > λ}) ≤ 2D(σ) λ σ({x ∈ Q : w(x) > p−1 }). (2.36) 2 [w]Ap(dσ) By Calder´on-Zygmund decomposition of w at the level λ, we obtain a family of

µ(Qi) disjoint cubes {Qi} contained in Q and satisfying λ < wQ = ≤ D(σ)λ for i σ(Qi) each i. Hence, except for a set of measure zero, we get [ d {x ∈ Q : w(x) > λ} ⊂ {x ∈ Q : Mσ,Qw(x) > λ} = Qi, i

37 Chapter 2. Buckley type estimates for Maximal Function

d where Mσ,Q is the dyadic maximal function, associated with measure σ and restricted to Q. Hence this together with (2.35) applied to each Qi gives X X µ({x ∈ Q : w(x) > λ}) ≤ µ(Qi) ≤ D(σ) λ σ(Qi) i i X w ≤ 2D(σ)λ σ({x ∈ Q : w(x) > Qi }) i 2p−1 [w] i Ap(dσ) λ ≤ 2D(σ)λ σ({x ∈ Q : w(x) > p−1 }), 2 [w]Ap(dσ) since wQi > λ. This proves the second claim.

Now we estimate the second integral in (2.33) denoted by II. By (2.36) and integration by substitution we get Z γ ∞ dλ II = λγ µ({x ∈ Q : w(x) > λ}) , σ(Q) wQ λ Z ∞ 2D(σ) γ γ+1 λ dλ ≤ λ σ({x ∈ Q : w(x) > p−1 }) , σ(Q) wQ 2 [w]Ap(dσ) λ Z ∞ ¡ p−1 ¢1+γ 2D(σ) γ γ+1 dλ = 2 [w]Ap(dσ) λ σ({x ∈ Q : w(x) > λ}) , σ(Q) wQ λ 2p−1 [w] Z Ap(dσ) ¡ ¢ 2D(σ) γ ∞ ≤ 2p−1 [w] 1+γ λγ σ({x ∈ Q : w(x) > λ}) dλ, Ap(dσ) σ(Q) 0 Z ¡ p−1 ¢1+γ 2D(σ) γ 1 1+γ = 2 [w]Ap(dσ) w(x) dσ. 1 + γ σ(Q) Q If we choose 1 γ = 2p+1 < 1, (2.37) 2 D(σ)[w]Ap(dσ) 1 [w] then [w]γ ≤ [w] Ap(dσ) ≤ 2 because the function Ap(dσ) Ap(dσ)

1 f(t) = t t ≤ 2, for t ≥ 1.

Therefore,

¡ p−1 ¢1+γ γ p−1 (p−1)γ γ 2 [w]A (dσ) 2 D(σ) ≤ 2 [w]A (dσ)2 [w] 2 D(σ) γ p 1 + γ p Ap(dσ)

2p 1 1 ≤ 2 D(σ)[w]Ap(dσ) 2p+1 ≤ . 2 D(σ)[w]Ap(dσ) 2

38 Chapter 2. Buckley type estimates for Maximal Function

Combing the estimates for I, II, with γ as in (2.37) we obtain Z µ Z ¶ 1 γ+1 γ+1 1 1 γ+1 w(x) dσ ≤ (wQ) + w(x) dσ , σ(Q) Q 2 σ(Q) Q which is equivalent to

µ Z ¶ 1 µ Z ¶ 1 γ+1 1 w(x)γ+1 dσ ≤ 2 w dσ , (2.38) σ(Q) Q σ(Q) Q which ends the proof.

2.3.2 Coiﬀman-Feﬀerman-Buckley Theorem dσ: a precise version

Now, we can prove a precise version of the Coiﬀman-Feﬀerman-Buckley Theorem dσ.

Theorem 2.3.2 (Coiﬀman-Feﬀerman-Buckley dσ). Suppose that w ∈ Ap(dσ), then w ∈ Ap−ε(dσ) for some ε > 0. Moreover, ε can be chosen so that

p−1 1−p0 1 [w]A (dσ) ≤ 2 [w]A (dσ) with ε ∼ [w] . p−ε p Ap(dσ) D(σ)

− 1 Proof. If w ∈ Ap(dσ), then ν = w p−1 ∈ Ap0 (dσ), therefore by Theorem 2.3.1

ν ∈ RH1+γ(dσ) for some γ > 0. More precisely, by (2.38) γ can be chosen so that Z µ Z ¶ ³ ´1+γ 1+γ 1 − 1 2 − 1 w p−1 dσ ≤ w p−1 dσ . σ(Q) Q σ(Q) Q We can now conclude that

µ Z ¶ µ Z ¶ p−1 µ Z ¶ 1+γ 1 1 − 1+γ p−1 1 w dσ w p−1 dσ ≤ 2 w dσ σ(Q) Q σ(Q) Q σ(Q) Q µ Z ¶p−1 1 − 1 × w p−1 dσ σ(Q) Q p−1 ≤ 2 [w]Ap(dσ). (2.39)

39 Chapter 2. Buckley type estimates for Maximal Function

γ p−1 If we choose ε = (p − 1) 1+γ , then γ+1 = p − ε − 1, and by (2.39) w ∈ Ap−ε(dσ). p−1 Moreover, [w]Ap−ε(dσ) ≤ 2 [w]Ap(dσ).

Once again, since γ is small enough, we have 1 = 1 − γ + γ2 − γ3 + ... 1 + γ Therefore, γ ε ∼ ∼ γ. (2.40) 1 + γ Taking into account (2.40), (2.37) and (1.15), we obtain

1 1 0 1 ε ∼ = [w]1−p (2.41) Ap(dσ) [ν]Ap0 (dσ) D(σ) D(σ)

2.4 Strong Estimates and Interpolation

In this section we are going to use the Interpolation Theorem 1.2.1 and self-improving properties of the Ap(dσ) (or RHq(dµ)) weights to obtain strong and sharp-estimates for the operator Mσ. In view of the tautology (1.28), we can do it both on the Ap(dσ) side, or on the RHp0 (dµ) side. Each time we get essentially the same power but with diﬀerent doubling constants, D(σ) and D(µ) respectively. In fact, the same argument can be applied to any sublinear operator T that satisﬁes the weak-inequalities (2.14).

2.4.1 Gehring’s Estimates and Interpolation

In this subsection we use Interpolation and the Gehring’s type estimates obtained in Section 2.2 to get the strong-estimates starting from the weak-ones. We can actually state the theorem for any sublinear operator T , and at the end we show the c corresponding estimates for the operators Mσ and Mσ.

40 Chapter 2. Buckley type estimates for Maximal Function

Theorem 2.4.1. If w ∈ Ap(dσ), dµ = wdσ, and T is a sublinear operator of weak- type (p, p) i.e.

 1 p C(σ, µ, p)[w] p Z µ({x ∈ Rn : T f > λ}) ≤  Ap(dσ) |f(x)|p dµ , (2.42) λ Rn then p0 1 p kT fk p ≤ K 0 C(σ, µ, p)(D(µ)) p [w] kfk p , L (dµ) p Ap(dσ) L (dµ) where the constant Kp0 comes from the Interpolation Theorem 1.8 and C(σ, µ, p) is, 1 as in (2.16), that is a product of functions which either do not depend on p or are p powers of such functions.

−1 Proof. Substituting w := vuo into (2.3), since dσ = uo dx, we obtain

µ ¶p kfk p µ{x ∈ Rn : T f > λ} < Cp(σ, µ, p)[w] L (dµ) Ap(dσ) λ which is, by the tautology (1.28), equivalent to

µ ¶p kfk p µ({x ∈ Rn : T f > λ}) < Cp(σ, µ, p)[w−1]p L (dµ) RHp0 (wdσ) λ

−1 Suppose now p > 1. Then, by the Corollary, 2.2.1 w ∈ RHp0+ε(wdσ) with

1 −1 comparable norm, where ε ∼ 0 , trivially w ∈ RHp0−ε(wdσ) and [w−1]p D(µ) RHp0 (wdσ) −1 −1 [w ]RHp0−ε(wdσ) ≤ [w ]RHp0 (wdσ).

We can now interpolate between p0 and p1, where 0 0 p+ε(p−1) 0 0 p−ε(p−1) 1 p0 := (p + ε) = 1+ε(p−1) < p, and p1 := (p − ε) = 1−ε(p−1) > p, if ε < p−1 . 2ε(p−1)2 1 Also, p1 − p0 = 1−ε2(p−1)2 . Since ε < p−1 we can expand the last expression in a geometric series

2 3 4 5 6 p1 − p0 = 2ε(p − 1) + 2ε (p − 1) + 2ε (p − 1) + ...

41 Chapter 2. Buckley type estimates for Maximal Function

1 1−t t Thus p1 − p0 ∼ ε. We choose t so that = + . p p0 p1 By the Marcinkiewicz Interpolation Theorem 1.2.1 we obtain the strong-type result of the form −1 0 Kp C(σ, µ, p)[uov ]RHp0 (vdx)kfkLp(dµ) kT fkLp(dµ) ≤ Ã ! 1 p 1 0 −1 p D(v)[uov ] RHp0 (vdx)

p0 1 1+ −1 p 0 p kT fkLp(dµ) ≤ Kp C(σ, µ, p)(D(v)) [uov ] kfkLp(dµ) RHp0 (vdx)

1 −1 p0 0 p kT fkLp(dµ) ≤ Kp C(σ, µ, p)(D(µ)) [uov ] kfkLp(dµ) RHp0 (vdx)

This can be rewritten, using tautology (1.28), in the following form

p0 1 p kT fk ≤ K 0 C(σ, µ, p)(D(µ)) p [w] kfk , (2.43) Lp(dµ) p Ap(dσ) Lp(dµ) which is exactly what we set out to prove.

In particular, Theorem 2.4.1 and the weak estimates (2.14) imply the following estimates for maximal operators.

Corollary 2.4.1. If w ∈ Ap(dσ), and dµ = wdσ, then

p0 1 c p kM k ≤ K 0 C (σ, µ, p)(D(µ)) p [w] kfk , (2.44) σ Lp(dµ) p 1 Ap(dσ) Lp(dµ) and

p0 1 p kM k ≤ K 0 C (σ, µ, p)(D(µ)) p [w] kfk , (2.45) σ Lp(dµ) p 2 Ap(dσ) Lp(dµ) where as in (2.15) and (2.16),

1 1 C1(σ, µ, p) = min{C(n) p ,D(µ) p },

1 1 C2(σ, µ, p) = min{C(n) p D(σ),D(µ) p }.

42 Chapter 2. Buckley type estimates for Maximal Function

Corollary 2.4.2. In particular, if we specify dσ = wdx and dµ = w−1dσ = dx, then p0 0 n −1 p p D(µ) = 3 , [w ] = [w] and we get the following estimates for Mσ = Mw, Ap(wdx) RHp0

p0 kMwfkLp ≤ Cn,p [w] kfkL . (2.46) RHp0 p

This generalizes results in [Per3] to p 6= 2. Similar results have been found in [P] and [Mo]. Furthermore, the power p0 is sharp (see [Mo]).

2.4.2 Coifman-Feﬀerman-Buckley dσ and Interpolation

This subsection is analogous to Subsection 2.4.3. Here again we use Interpolation and the Coifman-Feﬀerman-Buckley Theorem dσ, obtained in Section 2.3, to get the strong-estimates starting from the weak-ones. We ﬁrst state the theorem for any sublinear operator T , and at the end we show the corresponding estimates for the c operators Mσ and Mσ

Theorem 2.4.2. If w ∈ Ap(dσ), dµ = wdσ, and T is a sublinear operator of weak- type (p, p) i.e.

 1 p C(σ, µ, p)[w] p Z µ({x ∈ Rn : T f > λ}) ≤  Ap(dσ) |f(x)|p dµ , (2.47) λ Rn then p0 1 p kT fk p ≤ K C(σ, µ, p)(D(σ)) p [w] kfk p , (2.48) L (dµ) p Ap(dσ) L (dµ) where the constant Kp comes from the Interpolation Theorem 1.8 and C(σ, µ, p) is, 1 as in (2.16), a product of functions which either do not depend on p or are p powers of such functions..

Proof. We can interpolate again between the corresponding weak-estimates for p0 := p − ε and p1 := p + ε to obtain another version of Buckley’s type estimate. Clearly, if

43 Chapter 2. Buckley type estimates for Maximal Function

w ∈ Ap(dσ), then w ∈ Ap+ε(dσ) and [w]Ap+ε(dσ) ≤ [w]Ap(dσ). Moreover, by Theorem 2.3.2 ε can be chosen so that

0 [w] ≤ 2p−1[w] , with ε ∼ [w]1−p 1 . Ap−ε(dσ) Ap(dσ) Ap(dσ) D(σ)

Thus, by the Marcinkiewicz Interpolation Theorem 1.2.1 we obtain the strong -type result of the form

1 K C(σ, µ, p)[w] p kfk p Ap(dσ) Lp(dµ) kT fkLp(dµ) ≤ ³ ´ 1 0 [w]1−p 1 p Ap(dσ) D(σ)

p0 1 kT fk ≤ K C(σ, µ, p)D(σ) p [w] p kfk (2.49) Lp(dµ) p Ap(dσ) Lp(dµ)

In particular, Theorem 2.4.1 and the weak-estimates (2.14) imply the following estimates for maximal operators. This is an analogue of Corollary 2.4.1.

Corollary 2.4.3. If w ∈ Ap(dσ), and dµ = wdσ, then

p0 1 c p kM fk ≤ K C (σ, µ, p)(D(σ)) p [w] kfk , (2.50) σ Lp(dµ) p 1 Ap(dσ) Lp(dµ) and

p0 1 kM fk ≤ K C (σ, µ, p)(D(σ)) p [w] p kfk , (2.51) σ Lp(dµ) p 2 Ap(dσ) Lp(dµ) where as in (2.15) and (2.16),

1 1 C1(σ, µ, p) = min{C(n) p ,D(µ) p },

1 1 C2(σ, µ, p) = min{C(n) p D(σ),D(µ) p }.

44 Chapter 2. Buckley type estimates for Maximal Function

2.5 A. Lerner’s approach

In this section we adapt a very recent result, due to A. Lerner to the case dσ, (see

[Le1]). We obtain yet another proof of the boundedness of Mσ, and sharp estimates, but without using interpolation. More precisely we will prove.

Theorem 2.5.1. If w ∈ Ap(dσ), dµ = wdσ, then

0 2p−1 p p kM fk p ≤ C(n)(D(σ)) p−1 [w] kfk p . (2.52) σ L (dµ) Ap(dσ) L (dµ)

− 1 Proof. Let ν := w p−1 and deﬁne

µ ¶ Z µ(Q) υ(3Q) p−1 Ap(Q) := , where υ(Q) := ν dσ. (2.53) σ(Q) σ(Q) Q

Then, using the doubling of σ and the deﬁnition of [w]Ap(dσ) we get

µ ¶ 1 µ ¶ 1 µ(Q) p−1 υ(3Q) [A (Q)] p−1 = p σ(Q) σ(Q) µ ¶ 1 µ ¶ µ(Q) p−1 υ(3Q) ≤ D(σ) D(σ) σ(3Q) σ(3Q) µ ¶ 1 µ ¶ p µ(Q) p−1 υ(3Q) = [D(σ)] p−1 σ(3Q) σ(3Q) Ã ! 1 µ ¶p−1 p−1 p µ(3Q) υ(3Q) ≤ [D(σ)] p−1 σ(3Q) σ(3Q)

p 1 ≤ [D(σ)] p−1 [w] p−1 . (2.54) Ap(dσ)

45 Chapter 2. Buckley type estimates for Maximal Function

Now, using (2.54) yields

Ã ! 1 Z µ Z ¶p−1 p−1 1 1 σ(Q) 1 |f| dσ = [Ap(Q)] p−1 |f| dσ σ(Q) Q µ(Q) υ(3Q) Q Ã ! 1 µ Z ¶p−1 p−1 p 1 σ(Q) 1 ≤ [D(σ)] p−1 [w] p−1 |f| dσ Ap(dσ) µ(Q) υ(3Q) Q Ã ! 1 µ Z ¶p−1 p−1 p 1 p−1 σ(Q) 1 −1 = [D(σ)] p−1 [w] (|f|ν )ν dσ Ap(dσ) µ(Q) υ(3Q) Q µ Z ¶ 1 p 1 p−1 p−1 1 c −1 p−1 ≤ [D(σ)] p−1 [w] M (fν ) dσ , (2.55) Ap(dσ) υ µ(Q) Q µ Z ¶ 1 p 1 p−1 p−1 1 c −1 p−1 −1 = [D(σ)] p−1 [w] (M (fν ) w )wdσ , Ap(dσ) υ µ(Q) Q µ Z ¶ 1 p 1 p−1 p−1 1 c −1 p−1 −1 = [D(σ)] p−1 [w] (M (fν ) w ) dµ (2.56) Ap(dσ) υ µ(Q) Q c where Mυ denotes the weighted centered maximal function associated with measure dυ = ν dσ. Note that the inequality (2.55) comes from the following geometrical observation: for any x ∈ Q, if Qx denotes the cube congruent to Q and centered at x, then

Q ⊂ 2Qx ⊂ 3Q. Hence, using this and the deﬁnition of the maximal function, for any x ∈ Q Z Z c −1 1 −1 1 Mυ(fν )(x) ≥ (|f|ν )ν dσ ≥ |f| dσ. υ(2Qx) Qx υ(3Q) Q Let us recall (1.9) which says

c c Mσf(x) ≤ Mσf(x) ≤ D(σ) Mσf(x). (2.57)

Thus, if we take the supremum in (2.56) over all cubes Q centered at x we get

p 1 ¡ ¢ 1 c p−1 c c −1 p−1 −1 M f(x) ≤ [D(σ)] p−1 [w] M (M (fν ) )w )(x) p−1 . (2.58) σ Ap(dσ) µ υ

Finally, using (2.58) and (2.57) yields

0 2p−1 p ¡ ¢ 1 p c c −1 p−1 −1 M f(x) ≤ [D(σ)] p−1 [w] M (M (fν ) )w )(x) p−1 . (2.59) σ Ap(dσ) µ υ

46 Chapter 2. Buckley type estimates for Maximal Function

c c By Theorem 1.10.2 both kMµkLp0 (dµ) and kMυkLp(νdσ) are ﬁnite with constant uniformly in w. Therefore,

0 2p−1 p p kM fk p ≤ C(n)(D(σ)) p−1 [w] kfk p . (2.60) σ L (dµ) Ap(dσ) L (dµ)

2.6 Strong estimates for Mσ revisited

In this Section we summarize the estimates obtained for uncentered maximal function

Mσ, associated with measure σ. Taking into account the estimates (2.45), (2.51) and (2.52), we conclude with

p0 p kM f(x)k p ≤ K [w] kfk p , (2.61) σ L (dµ) σ,µ,p Ap(dσ) L (dµ) where the constant Kσ,µ,p may depend on the doubling constants D(σ) or D(µ), and the constants Kp and Kp0 come from the Interpolation Theorem 1.8. More precisely,

1 1 2p−1 Kσ,µ,p = min{Kp0 C2(σ, µ, p)(D(µ)) p ; Kp C2(σ, µ, p)(D(σ)) p ; C(n)[D(σ)] p−1 }, (2.62) p+1 1 1 p+1 p p p p 2 where C2(σ, µ, p) = min{C(n) D(σ),D(µ) },Kp = 2 and Kp0 = 2 . (p−1) p

47 Chapter 3

Sharp Extrapolation Theorems dσ.

In this chapter we use the estimates obtained for the Maximal Function Mσ in

Chapter 2 to build Sharp Extrapolation Theorems dσ, where dσ = uodx, for some uo ∈ A∞. We begin with the classical Rubio de Francia algorithm and follow closely [DrGrPerPet] to obtain a similar theorem, tracking down the dependence on the

[w]Ap(dσ) characteristic constant. Furthermore, we use the fact observed in [CrMPe1] that the proofs of extrapolations theorems depend not on the properties of the operators, but rather on duality, the structure of the Ap weights, and norm inequalities for the Hardy-Littlewood maximal operator. Therefore, we can eliminate the super- ﬂuous operators and replace inequality (1.9) with Z Z g(x)rw(x) dx ≤ C f(x)rw(x) dx, (3.1) and concentrate on F, a family of ordered pairs (g, f) of non-negative measurable functions such that the left-hand side of (3.1) is ﬁnite. As a consequence of adopting this approach we can apply this theorem directly to some known inverse estimates, presented in Chapter 4. Furthermore, vector-valued inequalities and many other results follow from extrapolation.

In order to prove these theorems we will need some technical lemmas. The lem-

48 Chapter 3. Sharp Extrapolation Theorems dσ. mas and their proofs follows closely the corresponding ones in the case dσ = dx in [DrGrPerPet].

3.1 Weight Lemmas

Careful examination of the extrapolation theorem justiﬁes the need for lemmas discussed in this Section. We will use the strong-estimates, obtained in Chapter 2, for q Mσ in L (νodη), for dη = νodσ and νo ∈ Aq(dσ). More precisely, in two cases when q = p and dη = dµ; or when q = p0 and dη = w1−p0 dσ. Let us recall the constant

Kσ,η,q obtained there is of the form

1 1 2q−1 Kσ,η,q = min{Kq0 C2(σ, η, q)(D(η)) q ; Kq C2(σ, η, q)(D(σ)) q ; C(n)[D(σ)] q−1 }, (3.2)

q+1 1 1 q+1 q q q q 2 where C2(σ, η, q) = min{C(n) D(σ),D(η) },Kq = 2 and Kq0 = 2 . (q−1) q

s Lemma 3.1.1. Take p, s > 1, w ∈ Ap(dσ) and u ∈ L (dµ), dµ = wdσ. Let

³ ´ p0 s s −1 0 Sσ(u) := w Mσ(|u| p w)

s (a) Then Sσ is bounded in L (dµ). Moreover,

0 p0 1 p 0 s s kS k s s ≤ C(p ) s K 0 [w] , σ L (dµ)→L (dµ) σ,η,p Ap(dσ)

1−p0 where dη = w dσ and the constant Kσ,η,p0 as in (3.2).

p s (b) Let p, s be such that r := s0 ∈ [1, ∞). Take a nonnegative function u ∈ L (dµ).

If r > 1, then the pair (uw, Sσ(u)w) ∈ Ar(dσ) Furthermore,

µ Z ¶ µ Z ¶r−1 1 p 1 1 − 1− 0 sup uw dσ (S (u)w) r−1 dσ ≤ [w] s . (3.3) σ Ap(dσ) Q σ(Q) Q σ(Q) Q

49 Chapter 3. Sharp Extrapolation Theorems dσ.

If r = 1, the A1(dσ) condition on the pair (uw, Sσ(u)w) also holds and trans- lates into

Mσ(uw) ≤ Sσ(u)w. (3.4)

Proof. (a) Estimating directly the norm we obtain

µZ ¶ 1 µZ ¶ 1 s s s 0 s 0 0 −1 0 p 0 p 1−p kSσ(u)kLs(dµ) = [w Mσ(|u| p w)] w dσ = [Mσ(|u| p w)] w dσ

0 s p p0 s ≤ kM (|u| w)k 0 0 σ Lp (w1−p dσ) 0 p s s p0 ≤ kM k 0 0 0 0 k|u| wk p0 1−p0 σ Lp (w1−p dσ)→Lp (w1−p dσ) L (w dσ) p0 s ≤ kM k 0 0 0 0 kuk s . σ Lp (w1−p dσ)→Lp (w1−p dσ) L (dµ)

Now, it only remains to insert the estimate from Section 2.6 and to recall (1.15) to obtain

0 0 1 (p ) 0 p0 1−p0 p0 kMσk p0 1−p0 p0 1−p0 ≤ C(p )K 0 [w ] L (w dσ)→L (w dσ) σ,η,p Ap0 (dσ) p µ 0 ¶ 1 p p0 0 p0 p = C(p )K 0 [w] σ,η,p Ap(dσ)

1 0 p0 = C(p )Kσ,η,p0 [w]Ap(dσ).

0 p0 1 p 0 s s Thus, kS k s s ≤ C(p ) s K 0 [w] , as claimed. σ L (dµ)→L (dµ) σ,η,p Ap(dσ)

0 (ii) If s = p , we have r = 1, then Sσ(u)w = Mσ(uw). Automatically the two weight A1(dσ) condition, Mσ(uw) ≤ Sσ(u)w, holds by (3.4). If s > p0 > 1, then p > s0 > 1 and r > 1.

Note that

0 s µ ¶ µ 0 ¶ p − s p − s−1 (p − 1)s − p p − 1 p p (r − 1) = 0 = s = = s − = (p − 1) 1 − , s s−1 s s p − 1 s and in particular µ ¶ 1 1 p0 − = − 1 − . (3.5) p − 1 r − 1 s

50 Chapter 3. Sharp Extrapolation Theorems dσ.

By deﬁnition of the maximal function, if x ∈ Q,

³ s ´ s s σ p0 p0 σ p0 mQ u w ≤ Mσ(u w)(x) = sup mQ(u w), (3.6) Q3x where mσ (f) denotes the mean, with respect to measure dσ, of the function f over Q ³ ´ −1 σ r−1 cube Q. Consequently, we can estimate mQ (Sσ(u)w) ³ ´ ³ ´ −1 s p0 −1 σ r−1 σ −1 p0 s r−1 mQ (Sσ(u)w) = mQ [(w Mσ(u w)) w] ³ ´ s p0 −1 −1 p0 σ p0 s ( r−1 ) ( r−1 )(1− s ) = mQ [Mσ(u w)] w ³ 0 ´ s p −1 ( −1 ) σ p0 s ( r−1 ) p−1 = mQ [Mσ(u w)] w (3.7)

But if we raise (3.6) to a negative power we get

³ ´ 1 p0 ³ ´ 1 p0 s (− r−1 ) s s (− r−1 ) s p0 σ p0 Mσ(u w) ≤ mQ(u w) . (3.8)

Hence, by (3.7), (3.8) and (3.5)

h ³ ´i h i p0 h i −1 r−1 s − s −1 r−1 σ r−1 σ p0 σ p−1 mQ (Sσ(u)w) ≤ mQ(u w) mQ(w )

h i p0 h i p0 s − s −1 (p−1)(1− s ) σ p0 σ p−1 ≤ mQ(u w) mQ(w ) (3.9)

s 0 s Using H¨older’sinequality with exponents q = p0 > 1 and q = s−p0 , we get the estimate ³ ´ p0 p0 σ σ s 1− s mQ(u w) = mQ u w w 0 ³ ´ p 0 s s p σ p0 1− s ≤ mQ(u w) (mQw) (3.10)

Combining (3.9) and (3.10) we get

0 · ¸1− p ³ ´r−1 ³ ´p−1 s p0 σ σ −1 σ σ −1 1− m (u w) m (S (u)w) r−1 ≤ m (w) m w p−1 ≤ [w] s Q Q σ Q Q Ap(dσ)

Taking supremum on the left-hand-side, over all cubes Q with sides parallel to the axes, we obtain the desired inequality (3.3).

51 Chapter 3. Sharp Extrapolation Theorems dσ.

Lemma 3.1.2. Let p, s, r and w be as in the Lemma 3.1.1(b). Then for each u ≥ 0, u ∈ Ls(dµ), there exists v ∈ Ls(dµ) such that

(a) u(x) ≤ v(x) a.e. and kvkLs(dµ) ≤ 2kukLs(dµ);

p0 1 0 s s (b) vw ∈ Ar(dσ), and [vw]Ar(dσ) ≤ 2C(p ) Kσ,η,p0 [w]Ap(dσ) 1−p0 where dη = w dσ and the constant Kσ,η,p0 as in (3.2).

Proof. Deﬁne v via the following convergent Neumann series:

X∞ Snu S (u) v = σ = u + σ + ..., 2nkS kn 2kS k n=0 σ σ where kSσk := kSσkLs(dµ)→Ls(dµ).

Since

X∞ n X∞ kSσ ukLs(dµ) kSσk kukLs(dµ) kvk s ≤ ≤ = 2kuk s , L (dµ) 2nkS kn 2nkS kn L (dµ) n=0 σ n=0 σ thus (a) is clearly satisﬁed.

(b) It follows from the deﬁnition of v and the sublinearity of Sσ that Ã ! X∞ Snu X∞ Sn+1u 2 kS k S v = S σ ≤ σ σ σ σ 2nkS kn 2nkS kn 2 kS k n=0 σ n=0 σ σ X∞ Sn+1u ≤ 2 kS k σ ≤ 2kS kv, σ 2n+1 kS kn+1 σ n=0 σ thus Sσ(v) ≤ 2kSσkv. If we take a negative power, we will get

−1 −1 v ≤ 2kSσk (Sσv) . (3.11)

Suppose r > 1. By the previous lemma, since v ∈ Ls(dµ) and p > s0 > 1, the pair 0 1− p (vw, S (v)w) ∈ A (dσ) with the A (dσ)-constant bounded by [w] s . σ r r Ap(dσ)

52 Chapter 3. Sharp Extrapolation Theorems dσ.

Also, since 0 p0 1 p 0 s s kS k s ≤ C(p ) s K 0 [w] , σ L (dµ) σ,η,p Ap(dσ)

using (3.11) and (3.3) we can estimate [vw]Ar(dσ): ³ ´ ³ ´ 1 r−1 1 r−1 σ σ − r−1 σ σ − r−1 mQ(vw) mQ (vw) ≤ mQ(vw) mQ (Sσ(v)w) 2kSσk 0 0 p p0 1 p 1− s 0 s s ≤ [w] 2C(p ) s K 0 [w] Ap(dσ) σ,η,p Ap(dσ) p0 1 0 s s = 2C(p ) Kσ,η,p0 [w]Ap(dσ).

Taking supremum on the left hand side, over all cubes Q with sides parallel to the axes, we obtain the desired estimate for [vw]Ar(dσ), r > 1. 1 0 0 p0 When r = 1, then s = p and kSσk ≤ C(p )Kσ,η,p0 [w]Ap(dσ), futhermore

1 0 p0 Mσ(vw) ≤ Sσ(v)w ≤ 2kSσkvw ≤ 2C(p )Kσ,η,p0 [w]Ap(dσ) vw 1 0 p0 We conclude that [vw]A1(dσ) ≤ 2C(p )Kσ,η,p0 [w]Ap(dσ).

Lemma 3.1.3. Fix r satisfying 1 ≤ r < ∞, dµ = wdσ.

p (a) Let 1 ≤ r < p < ∞ and r := s0 . Let w ∈ Ap(dσ), then for every u ≥ 0, u ∈ Ls(dµ), there exists v ≥ 0, v ∈ Ls(dµ), such that u(x) ≤ v(x) a.e. and

kvkLs(dµ) ≤ 2kukLs(dµ). p−r p−r 0 p−1 p Moreover, vw ∈ Ar(dσ) and [vw]Ar(dσ) ≤ 2C(p ) Kσ,η,p0 [w]Ap(dσ),

1−p0 where dη = w dσ and the constant Kσ,η,p0 as in (3.2).

p (b) Let 1 0. Let w ∈ Ap(dσ), then for every u ≥ 0, u ∈ Ls(dµ), there exists v ≥ 0, v ∈ Ls(dµ) such that, u(x) ≤ v(x), a.e. and

r−1 kvkLs(dµ) ≤ 2 kukLs(dµ). −1 Furthermore, v w ∈ Ar(dσ) and

µ ¶ r−1 r−p p−1 −1 r−1 r−p p [v w]Ar(dσ) ≤ 2 C(p) Kσ,µ,p[w]Ap(dσ) ,

53 Chapter 3. Sharp Extrapolation Theorems dσ.

where dµ = wdσ, the constant Kσ,µ,p as in Section 2.6, and C(p) denotes the constants in (2.49).

Proof. (a) Clearly r ≥ 1 implies s0 ≤ p, and we can now use Lemma 3.1.2 after p0 p−r observing that s = p−1 . (b) Take p, r and s as in the formulation of the lemma. (Notice that everything that is being said still holds if 0 < s < 1). Now the dual exponents satisfy the opposite 0 0 ∗ p0 0 inequality, r 1, then 0 p p p 0 ∗ r0 p p−1 p−1 p(r−1) s = 0 = 0 0 = p r = = = s(r − 1). p −1 p −r − p(r−1)−r(p−1) r−p r0 p−1 r−1 (p−1)(r−1)

0 0 1−p0 We apply the previous case with p , r and w ∈ Ap0 (dσ) instead of p, r and 0 s s p s∗ 1−p0 w ∈ Ap(dσ), respectively. If u ≥ 0, u ∈ L (dµ), then u0 = u s∗ w s∗ ∈ L (w dσ) s∗ 1−p0 and by (a) there exists v0 ∈ L (w dσ) such that

u0 ≤ v0 a.e., kv0kLs∗(w1−p0 dσ) ≤ 2ku0kLs∗(w1−p0 dσ), (3.12) and

0 0 p0−r0 0 0 p −r 0 1−p 1−p 0 p0 1−p 0 p −1 v0w ∈ Ar (dσ), [v0w ]Ar0 (dσ) ≤ 2C(p) Kσ,µ,p [w ]Ap0 (dσ) (3.13) r−p r−p 1 p(p−1) p−1 = 2C(p) p−1 Kσ,µ,p [w] . Ap(dσ)

s p0 s∗ p0 ∗ s − Deﬁne v so that v0 = v s w s∗ , that is, v = v0 w s . Then clearly r−1 u(x) ≤ v(x) a.e., and 3.12 implies kvkLs(dµ) ≤ 2 kukLs(dµ).

s∗ p0 p r−p p−1−p+r 0 Since s = r − 1 and 1 + s = 1 + p−1 p = p−1 = (1 − r)(1 − p ), then by −s∗ 0 p 0 −1 s 1+ 1−p 1−r (1.15), v w = v0 w s = (v0w ) ∈ Ar(dσ).

Thus, (3.13) implies

−1 1−p0 1−r [v w]Ar(dσ) = [(v0w ) ]Ar(dσ) µ ¶ r−1 1 r−p p−1 1−p0 r0−1 r−1 r−p p = [v0w ] ≤ 2 C(p) Kσ,µ,p[w]Ap(dσ) . Ar0 (dσ)

54 Chapter 3. Sharp Extrapolation Theorems dσ.

3.2 Sharp Extrapolation Theorem dσ.

Now we are ready to present and prove the Sharp Extrapolation Theorem dσ, in an operator-free form, as mentioned at the at the beginning of this chapter.

Theorem 3.2.1. Given a family F, suppose there is 1 ≤ r < ∞ and functions r (f, g) ∈ F such that g ∈ L (udσ) for all weights u ∈ Ar(dσ) and

kgkLr(udσ) ≤ CkfkLr(udσ), for any (f, g) ∈ F, (3.14)

where the constant C depends only on [u]Ar(dσ), and possibly on D(σ) but not on (g, f).

Then for all 1 < p < ∞, (f, g) ∈ F

kgkLp(wdσ) ≤ Kσ,w,p kfkLp(wdσ), (3.15)

with Kσ,w,p depending possibly only on [w]Ap(dσ), p and D(σ).

More precisely, suppose for each B > 1 and (f, g) ∈ F there is a constant

Nr(B) > 0 such that

kgkLr(udσ) ≤ Nr(B)kfkLr(udσ), (3.16)

for all u ∈ Ar(dσ), with [u]Ar(dσ) ≤ B. Then for any 1 1 there is

Np(B) > 0 such that for all weights w ∈ Ap(dσ) with [w]Ap(dσ) ≤ B

kgkLp(wdσ) ≤ Np(B)kfkLp(wdσ). (3.17)

Also, if we assume that Nr(B) is the smallest constant satisfying (3.16), then for any 1 1 one can choose Np(B) in (3.17) so that

 µ ¶ p−r p−r  1 p−1  r 0 p−1 2 Nr 2C(p ) Kσ,η,p0 B if p > r, µ ¶ Np(B) ≤ r−p  r−1 r−1  r−1 p−r p p−1 2 r Nr 2 (C(p) Kσ,µ,pB) if p < r.

55 Chapter 3. Sharp Extrapolation Theorems dσ.

Here C(p) is the constant appearing in (2.49).

Theorem 3.2.1 is a consequence of Lemma 3.1.3.

1 r Proof. Case 1: Assume 1 ≤ r < p, w ∈ Ap(dσ), dµ = wdσ, and s = 1 − p , i.e. 0 p s = r . Then

µZ ¶ r Z p r p r r kgkLp(dµ) = |g(x)| w(x) dσ = k|g| kLs0 (dµ) = sup |g(x)| u(x)w(x) dσ, u≥0 kukLs(dµ)=1 where the last equality holds by duality. For each such u, by Lemma 3.1.3a, there is s v ∈ L (dµ) such that u ≤ v, a.e. , and kvkLs(dµ) ≤ 2kukLs(dµ) = 2.

p−r p−r 0 p−1 p Furthermore, vw ∈ Ar(dσ) and [vw]Ar(dσ) ≤ 2C(p ) Kσ,η,p0 [w]Ap(dσ). Then, Z Z r r r |g(x)| u(x)w(x) dσ ≤ |g(x)| v(x)w(x) dσ ≤ kgkLr(vwdσ). (3.18)

Now, by the hypothesis,

r r r kgkLr(vwdσ) ≤ Nr ([vw]Ar(dσ))kfkLr(vwdσ). (3.19)

Moreover, by H¨older’sinequality we obtain Z r r 1− r kfkLr(vwdσ) = |f(x)| v(x)w(x) p w(x) p dσ

µZ ¶ r µZ ¶ 1 p s p s r ≤ |f(x)| dµ v(x) dµ ≤ 2kfkLp(dµ) (3.20)

Since, by deﬁnition, Nr is an increasing function and

p−r p−r 0 p−1 p [vw]Ar(dσ) ≤ 2C(p ) Kσ,η,p0 [w]Ap(dσ), thus combining (3.18), (3.19) and (3.20) we obtain Z µ ¶ p−r p−r r r 0 p−1 p r |g(x)| u(x)w(x) dσ ≤ 2Nr 2C(p ) Kσ,η,p0 [w]Ap(dσ) kfkLp(dµ).

56 Chapter 3. Sharp Extrapolation Theorems dσ.

Taking the supremum over all admissible u we obtain the desired inequality, ³ ´ 1 0 p−r 1−p0 p−r p r p−1 p p kgkL (dµ) ≤ 2 Nr 2C(p ) K(σ, w dσ) [w]Ap(dσ) kfkL (dµ)

r r In particular, if kgkL (ωdσ) ≤ Cσ [ω]Ar(dσ)kfkL (ωdσ), where Cσ may depend only on D(σ), then for p > r we get

p−r r+1 p−r 0 p p r p−1 p kgkL (wdσ) ≤ C2 C(p ) Kσ,η,p0 [w]Ap(dσ)kfkL (wdσ).

Case 2: p p r−p Assume 1 < p < r and deﬁne s := r−p . For f ∈ L (dµ) deﬁne u = |f| .

s r−p Then u ∈ L (dµ) and kukLs(dµ) = kfkLp(dµ). By Lemma 3.1.3b there exists a function v such that

u ≤ v a.e., (3.21)

r−1 r−1 r−p kvkLs(dµ) ≤ 2 kukLs(dµ) = 2 kfkLp(dµ). (3.22)

−1 Furthermore v w ∈ Ar(dσ), and

µ ¶ r−1 r−p p−1 −1 r−1 r−p p [v w]Ar(dσ) ≤ 2 C(p) Kσ,µ,p[w]Ap(dσ)

r 0 r q0 Now, using H¨older’sinequality, with q = p > 1, q = r−p and q = s, we obtain

µZ ¶ r p p p r p − r r kgkLp(dµ) = |g(x)| v(x) v(x) w(x) dσ

r µZ ¶ Z r ³ ´q0 p q0 ³ ´ p p − p p ≤ v(x) r w(x) dσ |g(x)| v(x) r w(x) dσ Z r −1 = kvkLs(dµ) |g(x)| v(x) w(x) dσ

−1 By construction v(x) w(x) ∈ Ar(dσ), and we can use the hypothesis to get Z Z r −1 r ¡ −1 ¢ r −1 |g(x)| v(x) w(x) dσ ≤ Nr [v w]Ar(dσ) |f(x)| v (x)w(x) dσ (3.23)

57 Chapter 3. Sharp Extrapolation Theorems dσ.

Furthermore, since by (3.21) v−1(x) ≤ |f(x)|p−r a.e. we get Z Z r −1 r p−r r |f(x)| v (x)w(x) dσ ≤ |f(x)| |f(x)| dµ = kfkLp(dµ) (3.24)

Thus combing (3.22), (3.23) and (3.24) yields

Ã µ ¶ r−1 ! r−p p−1 r r−1 r r−1 r−p p r kgkLp(dµ) ≤ 2 Nr 2 C(p) Kσ,µ,p[w]Ap(dσ) kfkLp(dµ).

We conclude that Ã µ ¶ r−1 ! r−p r−1 p−1 r−1 r−p p p r p kgkL (dµ) ≤ 2 Nr 2 C(p) Kσ,µ,p[w]Ap(dσ) kfkL (dµ).

r r In particular, if we know that kgkL (ωdσ) ≤ Cσ[ω]Ar(dσ)kfkL (dµ), for all ω ∈ Ar(dσ) and all (g, f) ∈ F, where Cσ may depend on the doubling constant D(σ), then for 1 < p < r we have

r−1 (r−p)(r−1) r2−1 (r−p)(r−1) p−1 p(p−1) kgk p ≤ C 2 r C(p) p−1 [w] Kσ,µ,p kfk r . L (dµ) σ Ap(dσ) L (dµ)

2 2 Specializing further, when r = 2 and kgkL (dµ) ≤ Cσ [w]A2(dσ)kfkL (dµ), then

α p p kgkL (wdσ) ≤ Cσ(p)[w]Ap(dσ)kfkL (wdσ),

p0 where α = max{1, p } and   p−2 p−2  0 p−1 p √ C(p ) Kσ,η,p0 if p ≥ 2, Cσ(p) = 2 2Cσ × 2−p 2−p  p(p−1) C(p) p−1 Kσ,µ,p if 1 < p ≤ 2.

3.3 Lerner’s type extrapolation dσ

In this section we use Lemma 3.1.1 to generalize Lerner’s Theorem, (see [Le1]). First, we use ordered pairs (f, g) instead of (f, T f). We also have initial two weight

58 Chapter 3. Sharp Extrapolation Theorems dσ.

po po estimates from L (vo dσ) to L (wo dσ) for any 1 < po < ∞, not just po = 2 and dσ = dx.

Theorem 3.3.1. Let p > p0 > 1. Suppose

0 − po po α β kgkLpo (w dσ) ≤ C[vo ] [wo, vo] kfkLpo (v dσ) o A∞(dσ) Apo (dσ) o

for all weights (wo, vo) ∈ Apo (dσ), and some constant C depending possibly on D(σ).

Then for any w ∈ Ap(dσ),

p−po+α+β(po−1) po(p−1) kgk p ≤ C [w] kfk po , L (w dσ) σ Ap(dσ) L (w dσ)

p−po p po where Cσ = CKσ,η,p0 , and Kσ,η,p0 as in (3.2).

0 p Proof. Let s := > 1. Take an arbitrary function u ≥ 0 with kuk s = 1, and po L (wdσ) set n o p−po −1 p−1 p−1 Sσ(u) = w Mσ(u p−po w) .

By Lemma 3.1.1a we have

p−p p−p p−po 0 o 0 p−1 p p−1 kS (u)k s ≤ C(p ) K 0 [w] , (3.25) σ L (wdσ) σ,η,p Ap(dσ) where the constant Kσ,η,p0 as in (3.2). Furthermore, by Lemma 3.1.1b when r := p s0 = po we get po−1 [uw, S (u)w] ≤ [w] p−1 . (3.26) σ Apo (dσ) Ap(dσ)

Now, by duality

µZ ¶s0 Z 0 po po s po po 0 kgkLpo (w dσ) = g wdσ = k|g| kLs (w dσ) = sup |g| uw dσ Rn u≥0 kukLs(wdσ)=1

Applying the assumption with vo := Sσ(u)w and wo := uw we obtain Z p0 − o β p po po α o |g| uw dσ ≤ C[(Sσ(u)w) ] [uw, Sσ(u)w] kfk po , A∞(dσ) Apo (dσ) L (Sσ(u)wdσ) Rn

59 Chapter 3. Sharp Extrapolation Theorems dσ.

By H¨olderinequality Z po po kfk p = |f| S (u)w dσ L o (Sσ(u)wdσ) σ

µZ ¶ po µZ ¶ p−po p p p p po ≤ (|f| ) po w dσ (Sσ(u)) p−po w dσ

= kfkLpo (wdσ) kSσ(u)kLs(wdσ)

0 0 − po − po po po In order to estimate [(Sσ(u)w) ]A∞(dσ) it suﬃces to show (Sσ(u)w) ∈ Ar(dσ) for some r > 1 and use (1.25). ³ ´ 0 Since 1 − p−po (− po ) = − 1 we have p−1 po p−1

0 po µ Z ¶ Z · p−1 ¸− p0 ³ ³ ´´ po 1 − o 1 −1 p−1 po−1 (Sσ(u)w) po dσ = w Mσ u po−1 w dσ σ(Q) Q σ(Q) Q 0 Z p−po po p0 ³ ´ (− ) 1 (1− p−po )(− o ) p−1 p−1 po = w p−1 po Mσ u po−1 dσ σ(Q) Q µ Z ¶ h ³ ´ i p−po (− 1 ) p−1 p−1 po−1 1 − 1 ≤ essinf Mσ u p−po w (x) w p−1 dσ Q σ(Q) Q

Next, by H¨olderinequality, with the exponents q := (p − 1)(r − 1) > 0 and 0 q (p−1)(r−1) q = q−1 = (p−1)(r−1)−1 :

µ Z 0 ¶r−1 1 po 1 (Sσ(u)w) po r−1 dσ σ(Q) Q  1 r−1 " 0 #  Z ³ ´ po r−1  1 −1 p−1 p−po po = w w Mσ(u p−po w) p−1 dσ σ(Q) Q  µ ¶ Z 0 r−1 1 1 p−po p−po po 1 = w (p−1)(r−1) Mσ(u p−1 w) p−1 po r−1 dσ σ(Q) Q · Z ³ ´ ¸ r−1 1 1 (p−1)(r−1) (p−1)(r−1) ≤ w (p−1)(r−1) dσ σ(Q) Q µ Z ¶ (p−1)(r−1)−1 1 p−1 p−po (p−1) × Mσ(u p−po w) (po−1)[(p−1)(r−1)−1] dσ σ(Q) Q

60 Chapter 3. Sharp Extrapolation Theorems dσ.

µZ ¶ 1 µ Z ¶ (p−1)(r−1)−1 p−1 1 p−1 p−po (p−1) ≤ w dσ Mσ(u p−po w) (po−1)[(p−1)(r−1)−1] dσ Q σ(Q) Q

If we choose r > po , then p−po < 1, and hence by Lemma 1.4.1 po−1 (po−1)[(p−1)(r−1)−1]

µZ ¶ (p−1)(r−1)−1 ³ ´ p−po p−1 p−1 (po−1)[(p−1)(r−1)−1] Mσ u p−po w dσ Q h ³ ´ i p−po p−1 (po−1)(p−1) p−po ≤ Cp,po,n essinf Mσ u w (x) Q

Combing three latter estimates yields

µ Z ¶ µ Z ¶r−1 0 p0 1 − po 1 o 1 (S (u)w) po dσ (S (u)w) po r−1 dσ σ(Q) σ σ(Q) σ Q µ Z Q ¶ µ Z ¶ 1 1 1 − 1 p−1 ≤ C w p−1 dσ w dσ ≤ C[w] . Ap(dσ) σ(Q) Q σ(Q) Q

Therefore by (1.25),

0 0 α po po − α − α p−1 [(S (u)w) po ] ≤ [(S (u)w) po ] ≤ C[w] σ A∞(dσ) σ Ar(dσ) Ap(dσ)

Unifying the latter estimate with (3.25) and (3.26) gives Z p−p p−po+α+β(po−1) p−po o po 0 p−1 p p−1 po |g| uw dσ ≤ C(p ) K 0 [w] kfk σ,η,p Ap Lpo (wdσ) Rn

Taking the supremum over all u ≥ 0 with kukLs(w) = 1, we obtain

p−po p−po+α+β(po−1) po p p−1 po kgk p ≤ CK 0 [w] kfk p . L (wdσ) σ,η,p Ap(dσ) L o (wdσ)

The extrapolation theorems require some initial estimates. O. Beznosova [B1] d was able to obtain the following estimates for the dyadic square function Sσ with d d Aq (dσ) condition instead of A∞(dσ).

61 Chapter 3. Sharp Extrapolation Theorems dσ.

d Theorem 3.3.2 (Beznosowa). Let v and w be weights, such that v ∈ Aq (dσ) for d d some q > 1 and the pair (v, w) ∈ A2(dσ), then the dyadic square function Sσ is bounded from L2(v−1) to L2(wdσ). Furthermore, for all f ∈ L2(v−1dσ),

d 2 2q − 1 d q+1 kS k 2 ≤ C [D (dσ)] [v]Ad(dσ) [v, w]Ad(dσ)kfkL2(v−1dσ). (3.27) σ L (wdσ) q q 2

When analyzing the proof of Theorem 3.3.1 we realized that we only needed

(w , v ) ∈ A , and r > po , and it suﬃces to assume that the initial two- o o p0(dσ) po−1 weight estimates are of the form

po po kgkL (dσ) ≤ C [vo]Ar(dσ) [wo, vo]Apo (dσ)kfkL (vo dσ).

In fact, we have a variant of the Theorem 3.3.1 that can be used in this case.

Theorem 3.3.3. Let p > p0 > 1. Suppose

0 − po po α β kgkLpo (w dσ) ≤ C[vo ] [wo, vo] kfkLpo (v dσ) o Ar(dσ) Apo (dσ) o

for all weights (wo, vo) ∈ Apo (dσ), some constant C depending possibly on D(σ); and some positive exponents α, β.

Then for any w ∈ Ap(dσ),

p−po+α+β(po−1) po(p−1) kgk p ≤ C [w] kfk po , L (w dσ) σ Ap(dσ) L (w dσ)

p−po p po where Cσ = CKσ,η,p0 , and the constant Kσ,η,p0 as in (3.2).

Now, we can apply this version of the theorem to Beznosowa estimates (3.27) 1 −1 when po = 2, α = β = 2 with wo = w and vo = v .

d p d Corollary 3.3.1. Sσ is bounded in L (wdσ), for w ∈ Ap(dσ), 1 < p < ∞ and

max{1, p } 1 d 2 p−1 kS k p ≤ C [w] kfk po . σ L (w dσ) σ Ap(dσ) L (w dσ)

Note that for 1 2.

62 Chapter 4

Applications

In this chapter we present some examples and applications which motivated this work. In the paper [Per1] sharp bounds on the L2 norms for the Haar multiplier X w(x) Twf(x) = hf, hI ihI (x) mI I∈D were obtained. First, the bound on the norm of the dyadic square function in the 2 weighted space L (w) was lifted to the case dσ, where dσ = udx for some u ∈ A∞, d and Sσ is deﬁned using averages with respect to dσ instead of dx, similarly A2(dσ), ³ R ´ ³ R ´ 1 1 −1 and [v] d = sup vdσ v dσ . Using A2(dσ) I∈D σ(I) I σ(I) I

d 2 2 kSσkL (vdσ)→L (vdσ) ≤ C[v]A2(dσ), and then taking dσ = wdx and v = w−1 allowed to push the weight into the operator, which is equivalent [Per4] to

∗ −1 2 2 2 2 2 −1 kTwkL = kTwkL ∼ kSwkL = kSwkL (w wdx) ≤ D(w)[w ]A2(wdx) = D(w)[w]RH2 .

d Tw are known [Per3] to be bounded if and only if w ∈ RHp .

p We are interested in obtaining sharp L estimates for Tw of the form:

63 Chapter 4. Applications

αp kT fk p ≤ C[w] kfk p . w L RHp L Sharp extrapolation dσ and Lerner type dσ theorems provide the ﬁrst approximation 2 to αp. Since Sσ is bounded in L (vdσ) for any v ∈ A2(dσ) with a bound that depends p0 linearly on [v]A2(dσ) [Per4], then by extrapolation Sσ is bounded in L (νdσ) for any

ν ∈ Ap0 (dσ). Also,

Corollary 4.0.2. Given σ ∈ D, ν ∈ Ap0 (dσ)

αp0 kSσfk p0 ≤ Cσ [ν] kfk p0 , L (νdσ) Ap0 (dσ) L (νdσ)

0 1 0 where the exponent αp = max{ p0−1 , 1} is sharp only for p ≤ 2 and Cσ

−1 ∗ If we choose ν := w , then we also will obtain that the adjoint operator Tw is p0 −1 bounded in L (dx) for any w ∈ Ap0 (dσ) = Ap0 (wdx) if and only if w ∈ RHp(dx). 1 −1 p0 p Recalling that [w]RHp(dx) = [w ] , we obtain the boundedness in L for Haar Ap0 (wdx) multiplier.

Corollary 4.0.3. 0 p αp0 kT k p ≤ C [w] , w L σ RHp(dx)

where Cσ depends only on the doubling constant of the measure σ.

As we mentioned at the beginning of Chapter 3, the operator-free form of the extrapolation theorems is more convenient and general. The Theorem 3.2.1 can be thus applied directly to extrapolate some known inverse estimates for the dyadic d square function Sσ. They were established in [PetPo], and [Per2] for the measure dσ.

d Given σ ∈ D and v ∈ A2(dσ) the following holds

1 2 d kfk 2 ≤ C [v] kS fk 2 . (4.1) L (vdσ) σ A2(dσ) σ L (vdσ)

Therefore by the Theorem 3.2.1 we get

64 Chapter 4. Applications

Corollary 4.0.4. Let σ ∈ D, w ∈ Ap(dσ), then

αp 2 d kfk p ≤ K [w] kS fk p , (4.2) L (wdσ) σ Ap(dσ) σ L (wdσ)

1 where αp = max{ p−1 , 1} and the constants Cσ,Kσ may depend on the D(σ).

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[P1] C. P´erez.Sharp Reverse H¨olderProperty of the A1 and Ap weights. To appear.

[P2] C. P´erez.The growth of the Ap constant for Calder´on-Zygmund singular integral operators. To appear. [Pet1] S. Petermichl. The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of classical Ap characteristic. Amer. J. Math. 129:1355- 1375, 2007. [Pet2] S. Petermichl. The sharp bound for the Riesz transforms. Proc. AMS, 136:1237-1249, 2008. [PetPo] S. Petermichl, S. Pott. An estimate for weighted Hilbert transform via square functions. Trans. Amer. Math. Soc., 354:281-305, 2002. [PetVo] S. Petermichl, A. Volberg. Heating of the Beurling operator: weakly quasiregular maps on the plane are quasilinear. Duke Math. J., 112:281-305, 2002.

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