LINEAR AND MULTILINEAR FRACTIONAL OPERATORS: WEIGHTED INEQUALITIES, SHARP BOUNDS, AND OTHER PROPERTIES. Kabe Moen

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1 LINEAR AND MULTILINEAR FRACTIONAL OPERATORS: WEIGHTED INEUALITIES, SHARP BOUNDS, AND OTHER PROPERTIES By Kabe Moen Submitted to the Department of Mathematics and the Faculty of the Graduate School of the University of Kansas in partial fulfillment of the requirements for the degree of Doctor of Philosophy Rodolfo H. Torres, Chairperson Carlos Pérez, Co-Chairperson Committee members Estela Gavosto William Paschke James Miller Date defended: April 4, 2009

2 The Dissertation Committee for Kabe Moen certifies that this is the approved version of the following dissertation: LINEAR AND MULTILINEAR FRACTIONAL OPERATORS: WEIGHTED INEUALITIES, SHARP BOUNDS, AND OTHER PROPERTIES Committee: Rodolfo H. Torres, Chairperson Carlos Pérez, Co-Chairperson Estela Gavosto William Paschke James Miller Date approved: April 4,

3 To Andrea and Mom 3

4 Contents Abstract 6 Acknowledgments 7 Introduction 9 Preliminaries 3. L p spaces Operators on L p spaces The main operators Maximal operators Fractional integral operators Calderón-Zygmund operators Weights Sharp weighted bounds for fractional operators Sharp bounds for the fractional maximal operator Extrapolation Sharp bounds for fractional integral operators Examples Sharpness of the strong bounds

5 2.4.2 Sharpness of the weak bounds Sobolev inequalities Weighted inequalities for general maximal operators Maximal operators with respect to a general basis One-weight inequalities Two-weight inequalities Sharp bounds Reverse Hölder class Weighted inequalities for multilinear fractional operators Multilinear fractional operators Banach function spaces Multilinear weights Weights for multilinear fractional operators One-weight theory Multilinear Sobolev inequalities Multilinear BMO Operators on mixed Lebesgue spaces Preleminaries Calderón-Zygmund operators on Lx p Ly q An off-diagonal extrapolation theorem for Lx p Ly q spaces Conclusions 38 Bibliography 43 5

6 Abstract In this work we consider various fractional operators, including the classical fractional integral operators, related fractional maximal functions, multilinear fractional integral operators, and multisublinear fractional maximal functions. We characterize the weighted inequalities for the multilinear fractional operators, and examine more general two-weight inequalities giving sufficient conditions for their boundedness. For the classical fractional integral operator we obtain sharp bounds on the operator norm between weighted Lebesgue spaces in terms of the constant associated to the weight. We also introduce a more general fractional maximal operators, characterize their boundedness on weighted Lebsegue spaces, and obtain sharp bounds on the operator norms in terms of the weighted constants. Finally, we examine singular integral operators and fractional integral operators acting on mixed Lebesgue spaces with weights. We provide endpoint estimates for singular integrals and an off-diagonal extrapolation theorem. 6

7 Acknowledgments I am eternally grateful to my advisors Professor Carlos Pérez and Professor Rodolfo Torres. Professor Torres has taught me so much. His supervision, support, and advice have made me not only a better mathematician but a better person. Under the guidance of Professor Pérez I have also grown. Through him I have gained a deep passion for mathematics and the fruitful interactions with him have greatly improved the quality of my research. I am thankful to Professor Estela Gavosto all of her advice and guidance during my time at the University of Kansas. I thank Professor William Paschke for his wonderful real analysis classes and for sparking my interest in the area of mathematical analysis. I would like to thank the Chair of the Department of Mathematics, Professor Jack Porter for his support and leadership during these past years. I also thank my family and friends for all they have given me while I have been in graduate school. Finally, I would like to thank two important women in my life. My mother Regina has given me love and support through out my whole life. My wife Andrea has provided me with the strength, love, friendship and companionship that has helped my achieve this great accomplishment. This work is dedicated to them and I could not have done this without them. 7

8 This work has been completed with the support of NSF Grants DMS and DMS

9 Introduction Given an operator that is bounded on L p (R n ), a natural problem arises when the measure is changed from Lebesgue measure to a general measure µ. More specifically, what conditions must one assume so that the same operator is bounded on L p (µ)? One approach to this problem is to consider measures that are absolutely continuous, i.e. dµ = w dx for some non-negative function or weight w. This is the essence of studying weighted inequalities, a subject that can probably be traced back to the beginning of integration. Weighted inequalities are not mere generalizations, but also have far reaching applications. For instance, a weighted theory plays a big part in the study of boundary value problems for Laplace s equation on Lipschitz domains. Other applications include vector-valued operators and extrapolation of operators. Weighted inequalities for fractional operators have applications to potential theory and uantum Mechanics. Multilinear operators also appear naturally in fundamental problems and applications of harmonic analysis. Bilinear operators can be used as a tool to analyze nonlinearities where products of functions take place. The main content of this dissertation is concerned with weighted inequalities for various linear and multilinear fractional operators. The groundbreaking work of Muckenhoupt [37] introduced the A p class and used it to characterize the weighted inequalities for the Hardy-Littlewood maximal operator. This spurred the study of weighted estimates for other operators. Hunt, Muckenhoupt, 9

10 and Wheeden [26] showed that weighted inequalities for the Hilbert transform are also characterized by the A p class and Coifman and Fefferman [7] then extended the weighted theory to Calderón-Zygmund operators. Then, Muckenhoupt and Wheeden [38] showed the A p,q class characterizes the weighted inequalities for fractional operators. In Chapter One we start with some basic facts about Lebesgue spaces that will be used in the following chapters. We introduce the main operators that pertain to this work. These operators include: maximal operators, operators that control the average of a function; fractional integral operators, operators that smooth a function; and singular integral operators, operators that are central to Calderón-Zygmund theory. We will be working with these operators or generalizations of these operators throughout the rest of this work. We conclude this chapter by introducing the corresponding class of weights for these operators and pointing out some of the basic properties of these weights. Buckley [4] was the first person to consider the problem of finding sharp bounds on the operator norm in terms of the A p constant. He found the sharp weighted bound for the Hardy-Littlewood maximal operator. Petermichl [42],[43], motivated by applications to partial differential equations (see Astala, Iwaniec, and Saksman []), found the sharp weighted bound on the operator norms of the Hilbert and Riesz transforms in terms of the A p constant. Inspired by these results, Chapter Two is devoted to finding sharp bounds on the operator norms of fractional operators acting on weighted Lebesgue spaces. One of the main tools is a sharp off-diagonal extrapolation theorem of Harboure, Macias, and Segovia [24]. We use techniques of Sawyer and Wheeden [49] to obtain a sharp bound for a pair of exponents (p 0,q 0 ) and then extend this bound to a range of exponents (p,q). We also present some improved Sobolev estimates. Some of the techniques in Chapter Two lead to a more general theory of maximal functions. In Chapter Three we consider maximal functions with respect to a general basis. We obtain one- and two-weight characterizations for the boundedness of the 0

11 general maximal functions, extending the work of Jawerth [27]. We also discover a new testing condition for fractional maximal functions. As a consequence of our techniques we obtain sharp two-weight bounds on the operator norm of the Hardy-Littlewood maximal operator in terms of Sawyer s testing condition [47]. We also find a new sharp bound on a weighted maximal operator in terms of the Reverse Hölder constant. The results contained in Chapter Four are multilinear versions of the fractional integral operator and fractional maximal function. Spurred by the work of Lerner, Ombrosi, Pérez, Torres, and Trujillo-Gonzalez [33] we develop a weighted theory for the multilinear fractional operators. We find new two-weight conditions for the multilinear operators that provide interesting contrast to the linear results. As a consequence of the two-weight theory we obtain the one-weight theory. Finally we end the chapter with some applications of the boundedness of the multilinear fractional integral operator including Sobolev inequalities for products of functions. Chapter Five deals with operators acting on mixed Lebesgue spaces. We present some results for Calderón-Zygmund operators that are more general than those given by Stefanov and Torres, [50] and Kurtz [29]. The weak-type mixed norm endpoint for general Calderón-Zygmund operators is a new result. We also introduce a mixed A p,q class of weights and provide an off-diagonal extrapolation theorem for mixed Lebesgue spaces with product weights. This dissertation contains results from the articles [35], [36], and [30] as well as some additional material. We present the work in a more comprehensive manner and sometimes, if possible, provide a different proof of a result than that contained in the articles. The material in Chapter Two is from our collaboration [30]. We have presented these results at the Analysis Seminar at Kansas State University, March The material in Chapter Three contains some results from the article [36]. These results were presented at the Eighth Prairie Analysis Seminar, Lawrence, November Finally,

12 Chapter Four contains results from the article [35] and these results have been presented at the Eighth International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Spain), June 2008, and the AMS-MAA Joint Mathematics Meetings, Washington, DC, January The research contained in this work has been completed with the partial support of NSF Grants DMS and DMS as well as Summer Graduate Scholarships provided by the Department of Mathematics. 2

13 Chapter Preliminaries In this chapter we introduce the basic theory required for the later chapters. We will mainly be working on, although not limited to, Lebesgue spaces. They are examples of classic Banach spaces and are fundamental to analysis. Many of the basic definitions, propositions, and theorems will be presented without reference, although the interested reader may find more information in the books by Folland [4], Grafakos [8], or Rudin [46]. In this chapter we also introduce various classical operators including, maximal operators, fractional operators, and singular integral operators.. L p spaces Definition... Let 0 < p < and (X,M, µ) be a measure space with associated σ-algebra M, and measure µ. Then L p (X, µ) consists of all complex-valued measurable functions, f, on X such that /p f L p (X,µ) = f dµ) p <. X 3

14 Furthermore L (X, µ) will denote the set of measurable functions, f, that are essentially bounded, that is f L (X,µ) = inf{m 0 : µ({x : f (x) > M}) = 0} <. For p, L p (X,µ) defines a complete norm on L p (X, µ) making it a Banach space. For < p <, p will denote the dual or conjugate exponent defined by the equation p + p =, and we use the convention = and =. For < p < the dual space of L p (X, µ) is isometrically identified with L p (X, µ) under the pairing f,g = X f g dµ. Furthermore, f L p (X,µ) = sup X f g dµ (.) where the supremum is taken over all g L p (X, µ) with norm one. Another useful fact is the layer-cake principle f p L p (X,µ) = p λ p µ({x X : f (x) > λ}) dλ. 0 Definition..2. When 0 < p <, L p, (X, µ) will denote the weak-l p (X, µ) space, consisting of all measurable functions, f, that satisfy f L p, (X,µ) = sup λ µ({x X : f (x) > λ}) /p <. λ>0 4

15 Note that L p, (X,µ) is in general not a norm, however for < p it is equivalent to a norm that makes L p, (X, µ) into a Banach space. Also notice that L p (X, µ) is a proper subset of L p, (X, µ). When X = R n and the measure is Lebesgue measure, dµ = dx, we will write L p, L p, for the respective spaces. When µ is absolutely continuous with respect to Lebesgue measure, i.e. dµ = wdx, for some measurable function w, then we write L p (w) and L p, (w). A non-negative locally integrable function will be called a weight. Given a measurable set E R n, E denotes the Lebesgue measure of E, w(e) = E w dx is the weighted measure of E. Most of the time we will be working on L p (w) where w is a weight... Operators on L p spaces Let (X, µ) and (Y,ν) be two measure spaces, and suppose that T is an operator defined on the space of all µ-measurable functions, taking values in the set of all ν-measurable functions. We say that T is linear if T (λ f + g) = λt f + T g for all f,g and λ C. The operator T is sublinear if for all f,g, and λ C T ( f + g) T f + T g T (λ f ) = λ T f. Given two Lebesgue spaces L p (X, µ) and L q (Y,ν) we say a linear or sublinear operator T is bounded from L p (X, µ) to L q (Y,ν), if there exists a constant C = C p,q such that for 5

16 all functions f L p (X, µ) we have T f L q (Y,ν) C f L p (X,µ). (.2) Occasionally we will write T : L p (X, µ) L q (Y,ν) to indicate T is bounded from L p (X, µ) to L q (Y,ν). The operator norm of T, denoted T L p (X,µ) L q (Y,ν) or simply T when the ambient spaces are clear, is given by T L p (X,µ) L q (Y,ν) = sup T f L q (X,µ) where the supremum is taken over all f L p (X, µ) of norm one. In light of (.) we may also compute an operator norm via, T L p (X,µ) L q (Y,ν) = sup Y gt f dν, with the supremum taken over all f L p (X, µ) and g L q (Y,ν) of norm one. An operator T is bounded from L p (X, µ) to L q, (Y,ν) if T f L q, (Y,ν) C f L p (X,µ). In this case we will say T is weak (p,q) and write T : L p (X, µ) L q, (Y,ν). We will also consider the operator norm of weak (p,q) operators defined by T L p (X,µ) L q, (Y,ν) = sup T f L q, (X,µ) where the supremum is taken over all f L p (X, µ) of norm one. 6

17 One important tool in the theory of L p spaces is interpolation. The classical Marcienkiewicz interpolation theorem allows one to obtain strong boundedness of operators from two weak endpoints. We state the off-diagonal version here as it will be useful latter. The proof can be found in [8, p. 62]. Theorem..3. Let 0 < p 0 p and 0 < q 0 q and suppose T is a sublinear operators defined on the space L p 0(X, µ) + L p (X, µ) taking values in the space of ν- measurable functions on Y. If T : L p 0 (X, µ) L q 0, (Y,ν) T : L p (X, µ) L q, (Y,ν), then T : L p (X, µ) L q (Y,ν) for p = θ + θ, p 0 p q = θ + θ q 0 q where θ (0,) and p q..2 The main operators In this section we introduce the main operators we will be working with. They include maximal operators, fractional integral operators and Calderón-Zygmund operators. 7

18 .2. Maximal operators We will often use to denote a cube (either open, closed, or neither) in R n with sides parallel to the axes and B(x,r) will denote a ball in R n centered at x with radius r. The side length of a cube will be l() and given a positive constant c, c will denote the concentric cube with that has side length cl(). The set D is the set of all dyadic cubes, i.e. cubes of the form 2 k (m + [0,) n ) with k Z and m Z n. In this section, and in the rest of this work we will use the notation A B to mean there exists positive constants c and C such that cb A CB. Definition.2.. Let f be a locally integrable function on R n then the Hardy-Littlewood maximal operator with respect to the measure µ is defined by M µ f (x) = sup f dµ, (.3) x µ() where the supremum is over all cubes,, with sides parallel to the axes that contain x. We use the convention that M µ f (x) = 0 if µ() = 0 for all cubes,, that contain x. When µ is Lebesgue measure we drop the subscript and write M to denote the Hardy-Littlewood maximal function. Notice in this case we have M f (x) = sup f (y) dy. x Notice that Lebesgue measure satisfies 3 = 3 n. It is this property that allows one to obtain the L p boundedness of the Hardy-Littlewood maximal operator. General measures that satisfy this are called doubling measures. 8

19 Definition.2.2. We say a measure µ is doubling if there exists a positive constant C such that µ(3) Cµ() (.4) for all cubes. The smallest constant C that satisfies (.4) will be called the doubling constant of µ and denoted D µ. The maximal operator M µ is always bounded on L (µ) and if the measure µ is doubling then M µ is also weak (,). Using Theorem..3 one has the following result. A proof for the case of the Hardy-Littlewood maximal operator can be found in [8] or [0]. Theorem.2.3. Suppose µ is a doubling measure with doubling constant D µ, M µ : L (µ) L, (µ) and M µ : L p (µ) L p (µ) for < p. Furthermore, we have the following relationship between the operator norm of M µ and the doubling constant D µ, M µ L (µ) L, (µ) cd µ and M µ L p (µ) L p (µ) cd /p µ. The proof of Theorem.2.3 is based on the following covering lemma due to Vitali. A proof Lemma.2.4 in the case of balls instead of cubes can be found in [46]. One can see this is where the doubling condition (.4) come into play. Lemma.2.4. Let {,..., n } be a finite collection of cubes in R n. Then there exists a subset S {,...,n} such that the collection { i } i S is pairwise disjoint, n j= j i S 3 i 9

20 We now discuss some variants of the Hardy-Littlewood maximal function. First, the dyadic version of M, M d µ f (x) = sup x D f dµ, µ() where the supremum is over all dyadic cubes that contain x. Using the fact that any two dyadic cubes are either disjoint or one is contained in the other we may write {x R n : M d µ f (x) > λ} = j j, where j are maximal disjoint dyadic cubes that satisfy f dµ > λ. µ( j ) j It follows that µ({x : M d µ f (x) > λ}) λ {M d µ f >λ} f dµ f L (µ). λ Thus M d µ is weak (,) with constant one and it follows that for < p, M µ : L p (µ) L p (µ) with norm that depends only on p. We also examine the centered maximal function, Mµ c f (x) = sup f dµ x µ( x ) x where the supremum is over all cubes centered at x. Notice that when µ is Lebesgue measure M c M. We state another covering lemma due to Besicovitch. 20

21 Lemma.2.5. Suppose E is a bounded subset of R n, for each x E, x is a cube centered at x, and E x E x. Then there exists a subset of { x } x E, { j } such that E j j and χ j C n. j It follows that µ({x : M c µ f (x) > λ}) C n λ R n f dµ. Hence M c µ : L p (µ) L p (µ) for < p with operator norm depending only on the dimension n, and p and not µ. We examine one more variant of the maximal operators. This time a family of maximal functions, M µ,α f (x) = sup x µ() α/n f dµ, 0 α < n. Given 0 < α < n we refer to M µ,α as the fractional maximal operator. The case α = 0 corresponds to the operator from (.3). By Hölder s inequality α/n µ() α/n f dµ f dµ) n/α, which implies M α,µ : L n/α (µ) L (µ). If µ is doubling, then a similar argument to that given in the proof of Theorem.2.3 shows that M α,µ is weak (,(n/α) ). Thus, Theorem..3 implies M α,µ : L p (µ) L q (µ) where < p n/α and q is defined by the equation q = p α n. (.5) 2

22 If M d α,µ denotes the dyadic fractional maximal operator and M c α,µ denotes the centered maximal operator then similar arguments to the case α = 0 show M d α,µ : L p (µ) L q (µ) M c α,µ : L p (µ) L q (µ) with operator norms independent of µ and p < q satisfing (.5)..2.2 Fractional integral operators We now introduce fractional integral operators. These are important operators in analysis pertaining to the smoothness of functions and Sobolev embedding theorems. More information can be found in the books by Grafakos [8] and Stein [5]. In order to define these operators we need to define some function spaces and distribution spaces. Let S = S (R n ) be the space of Schwartz rapidly decreasing functions and its dual space S = S (R n ) the space of all tempered distributions. Definition.2.6. Let f S and define the Fourier transform of f by F f (ξ ) = ˆf (ξ ) = f (x)e 2πix ξ dx, R n and inverse Fourier transform F f (x) = ˇf (x) = ˆf (ξ )e 2πiξ x dξ. R n If f S then we have the inversion property f = ( ˆf )ˇ. Given a tempered distribution u S we may define the Fourier transform and inverse Fourier transform by û, f = u, ˆf and ǔ, f = u, ˇf respectively. 22

23 Definition.2.7. Let 0 < α < n, we define the fractional integral operator or Riesz potential by I α f (x) = R n f (y) dy. x y n α We notice that the function α n is locally integrable for 0 < α < n, so I α is well defined by an absolutely convergent integral if, say, f S. We may also define I α on the Fourier transform side by (I α f )ˆ(ξ ) = c n ξ α ˆf (ξ ) where c n is an appropriate dimensional constant. In this sense I α acts as the α-th order anti-derivative. The operator, I α is also intimately related to M α. First, it is a pointwise bigger operator, i.e., M α f ci α f almost everywhere for all non-negative f. As we shall see later, the reverse inequality also holds in L p norm. Furthermore, I α has the same boundedness properties as M α. We state the following theorem, a proof can be found in [8]. Theorem.2.8. Suppose 0 < α < n, and p < q < satisfy (.5). Then I α : L L q, and I α : L p L q when p >. One of the most important applications of the boundedness of I α is the Sobolev Embedding Theorem. For < p < define the Sobolev space W s,p, to be the space of 23

24 all tempered distributions, u, with the property that (( + ξ 2 ) s 2 û(ξ ))ˇ is in L p. We may norm this space by f W s,p = (( + 2 ) s 2 ˆf ) ˇ L p, and this norm makes W s,p into a Banach space, see [8], or [5] for details. A non-trivial fact, at least when p 2, is that if s = k for some non-negative integer k, then f W k,p s f L p f L p + s f L p s k s =k with the convention that (0,...,0) f = f. In this case W k,p corresponds to the space of functions whose derivatives up to order k are in L p. A proof of this can be found in [8]. We now state the Sobolev embedding Theorem. A proof, which depends heavily on the boundedness of I α (Theorem.2.7) can be found also in [8]. Theorem Let 0 < α < n and < p < n/α. Then the Sobolev space W α,p continuously embeds in L q where q satisfies (.5). 2. If < p < and 0 < α = n/p, then W α,p continuously embeds in L q for any q > p. 3. If < p < and n/p < α, then every element of W α,p can be modified on a set of measure zero so that it is uniformly continuous. 24

25 .2.3 Calderón-Zygmund operators In this section we examine an important class of operators in analysis, Calderón- Zygmund operators. We say that a function K defined away from the diagonal of R n R n is standard kernel if it satisfies the size condition K(x,y) C x y n (.6) and regularity conditions K(x,y + h) K(x,y) + K(x + h,y) K(x,y) C h δ x y n+δ (.7) for some 0 < δ, whenever x y 2 h. Definition.2.0. An operator T is a Calderón-Zygmund operator if T is bounded on L q for some < q < and is associated with a standard kernel K, in the sense that T f (x) = K(x,y) f (y) dy, Rn whenever f L q has compact support and x is not in the support of f. Some of the main examples of Calderón-Zygmund operators are ones that are given as convolution, K(x,y) = k(x y), where k is locally integrable away from zero and satisfies the corresponding estimates (.6) and (.7). Example.2.. Let f S (R), and define the Hilbert transform as H f (x) = π lim ε 0 + y >ε f (x y) y dy = π p.v. f (y) dy. (.8) x y 25

26 The higher dimensional versions of H are the Riesz transforms given by R j f (x) = c n p.v. x j y j f (y) dy (.9) x y n+ for j n and f S (R n ). Notice that (H f )ˆ(ξ ) = isgn(ξ ) ˆf (ξ ) (.0) and if c n is chosen correctly (R j f )ˆ(ξ ) = i ξ j ξ ˆf (ξ ). (.) It follows from the Plancherel theorem that H : L 2 (R) L 2 (R) and R j : L 2 (R n ) L 2 (R n ) for j n. It can be easily checked that the kernels of the Riesz and Hilbert transform satisfy (.6) and (.7), and hence they are Calderón-Zygmund operators. As the following theorem states, these operators are also bounded on all L p spaces for < p <. A proof can be found in [8] and [0]. Theorem.2.2. Suppose that T is a Calderón-Zygmund operator then T : L L, 26

27 and T : L p L p for < p <..3 Weights Muckenhoupt [37] introduced the A p class of weights and used it to characterize the boundedness of the Hardy-Littlewood maximal operator, on L p (w). In this section we introduce the Muckenhoupt or A p weights and present some of the fundamental results concerning weighted inequalities for the operators introduced in the previous sections. A comprehensive guide to much of the material presented in this section can be found in the books by Duoandikoetxea [0] or Grafakos [9]. A weight w belongs to the class A p, < p <, if ) p [w] Ap = sup w(y) dy w(y) dy) p <, (.2) where the supremum is over all cubes,. We refer to [w] Ap as the A p constant of w. For p =, we say w A if w(y) dy C essinf w, where the smallest constant C will be denoted [w] A. This is equivalent to saying Mw(x) [w] A w(x) for almost every x R n. We notice a few properties about the class A p : 27

28 [w] Ap, by Hölder s inequality. A p A q, for p < q. w A p if and only if w p A p, with [w] Ap = [w p ] /(p ) A p. If u,v A, then uv p A p. If w A p then w is a doubling measure with D µ [w] Ap. Example.3.. The function w(x) = x a is in A p if and only if n < a < n(p ). Also, w(x) = log x for x < /e and otherwise is in A. We notice that if p in (.2) then we get ) w dy C exp logw dy. (.3) We will say say that w A if inequality (.3) holds. However, one may also define A in a number of other ways, including A = p>a p, which we will usually use as the definition of A. The fact that these two definitions are equivalent can be found in [6]. The fundament result concerning A p weights is due to Muckenhoupt [37]. Theorem.3.2. Let < p <. Then M : L p (w) L p (w) if and only if w A p. 28

29 For p <, it is not difficult to show that w A p if and only M : L p (w) L p, (w). The key step in Muckenhoupt s approach to showing the strong inequality holds is showing that A p = A q. (.4) q<p From here it follows that if w A p then w A q for some q < p, hence M : L q (w) L q, (w). Since M : L (w) L (w), we obtain Theorem.3.2. The equality in (.4) follows from what is known as Reverse Hölder condition. Theorem.3.3. Let w A p, p <. The there exists a constant C and r >, depending on p and the A p constant of w, such that for any cube, ) /r w r dy C w dy (.5) When a weight w satisfies (.5) we say w belongs to the class RH r and write w RH r. We give a short proof of Theorem.3.2 by Lerner [3] that avoids Theorem.3.3 and yields sharp constants. We give this proof because some of these techniques will be used later. Proof of Theorem.3.2. Let < p < and w A p with σ = w p. Notice the A p condition for w can now be written as sup w()σ() p p <. We also notice that M M c, the centered Hardy-Littlewood maximal function. So it suffices to prove it for the centered maximal operator. Let x R n and be any cube 29

30 centered at x. Then f dy = ( w(3)σ() p ) /(p ) p p w() /(p ) f dy σ(3) ( ) ) 3 np [w] /(p ) p /(p ) A p f σ σ dy w() σ(3) 3 np [w] /(p ) A p M w() σ( c f σ ) p/p w w dy ) p /p Taking the supremum over all centered at x we have the following pointwise inequality, M c f (x) 3 np [w] /(p ) A p M c w{m c σ( f σ ) p/p w )(x) p /p. From the comments after Lemma.2.5 we have that M c w : L p (w) L p (w) and M c σ : L p (σ) L p (σ) with operators norms independent of w and σ respectively. Thus we have M f L p (w) C M c f L p (w) C[w] /(p ) A p Mw{M c σ( c f σ ) p/p w ) p /p L p (w) = C[w] /(p ) A p M c w{m c σ( f σ ) p/p w ) p /p L p (w) C M c w p /p [w] /(p ) A p M c σ( f σ ) p/p w p /p L p (w) = C Mw c p /p [w] /(p ) A p Mσ( c f σ ) L p (σ) C Mw c p /p Mσ [w] c /(p ) A p f σ L p (σ) = C Mw c p /p Mσ [w] c /(p ) A p f L p (w). This completes the proof of the Theorem. 30

31 Notice that from Lerner s proof of Theorem.3.2 we have the following relationship between the operator norm of M and the A p constant of w M L p (w) L p (w) cpp [w] /(p ) A p. (.6) We shall see later that (.6) is sharp. For Calderón-Zygmund operators the A p class of weights is also the natural class of weights. Hunt, Muckenhoupt, and Wheeden [26] showed that A p also characterizes the class of weights for which the Hilbert transform is bounded on L p (w). Then Coifman and Fefferman [7] extend the A p theory to general Calderón-Zygmund operators. For fractional operators which map off-diagonally, weighted inequalities are simplified by treating the weight as a multiplier rather than a measure. More specifically, the weighted inequalities we will be concerned with are ) /q ) /p R n(wt α f ) q dx C R n( f w)p dx where p and q satisfy (.5) and T α is either the fractional integral operator I α or fractional maximal function M α. The weights for these operators is the A p,q class of weights, w, that satisfy ) ) q/p [w] Ap,q = sup w q dx w p dx <. This class is defined for any < p q <, which is the case when p and q satisfy (.5). For p =, A,q is the class of weights w such that w q A and [w] A,q = [w q ] A. We make a few observations: [w] Ap,q since p q, by Hölder s inequality. 3

32 w A p,q if and only if w q A +q/p with [w] Ap,q = [w q ] A+q/p, w A p,q if and only if w A q,p, with [w] A p,q = [w ] q/p A q,p. w A p,q if and only if w A +/p RH q. We provide a quick proof of the last point as it does not seem to be in the literature. Suppose w A p,q and set r = + /p, then for any cube ) r w dx w dx) r = [w] /q A p,q <. )( w dx ) /q w q dx ) /p w p dx ) /p w p dx Moreover, /q w dx) q = = [w] /q A p,q ( ) /q ( w q dx ) /q ( w q dx ) /q ( w q dx ) w dx. ) r w /r w /r dx )( w dx ) /p w p dx ) r w r dx ) w dx This shows w A p,q implies w A r RH q. On the other hand, if w A r RH q then ) /q ) /p w q dx w p dx ) ) /p C w dx w p dx C[w] A+/p <. 32

33 Muckenhoupt and Wheeden [38] characterized the weighted inequalities for the operators I α and M α in the following theorems below. Theorem.3.4. Suppose 0 < α < n, then I α : L (w) L n/(n α), (w n/(n α) ) if and only if w A,n/(n α). If < p < n/α and q is defined by /q = /p α/n then I α : L p (w p ) L q (w q ) if and only if w A p,q. Theorem.3.5. Suppose 0 α < n, then M α : L (w) L n/(n α), (w n/(n α) ) if and only if w A,n/(n α). If < p < n/α and q is defined by /q = /p α/n then M α : L p (w p ) L q (w q ) if and only if w A p,q. 33

34 Chapter 2 Sharp weighted bounds for fractional operators In this chapter we find sharp weighted bounds for the operators M α and I α. We use techniques similar to those developed in [3] for M α. These techniques, in turn lead to a more general theory which will be presented in the next chapter. Our main motivation for finding the sharp bound on the operator norm of I α is the following result of Petermichl [42], [43]. If T is either the Hilbert transform (.8), or the Riesz transform in R n (.9), then T L p (w) L p (w) c[w] max{,/(p )} A p. The problem of finding sharp bounds on the weighted operator norm of singular integral operators is also of interest because of applications to partial differential equations. More specifically it has applications to the regularity Beltrami equations in the plane see Astala, Iwaniec, and Saksman [] and Petermichl and Volberg [44]. For I α different techniques are used to find sharp bounds for operator norms. We use a dyadic decomposition to view the operator as a discrete operator. This decomposition lets us obtain a sharp bound for a fixed p 0 and q 0. We then use a sharp off-diagonal extrapolation theorem to obtain our results. We also present a weak extrapolation 34

35 theorem and as an application we obtain sharp weak inequalities for I α. This leads to an improved Sobolev estimate. Finally, we provide some examples to show that the bounds are indeed sharp. For the most part, the content of this chapter overlaps with of the work [30], as originally started jointly with Pérez and Torres (see also Chapter 6). We include all of the Theorems and proofs with some of them similar to what appears in [30]. Our aim, however, is to provide a more comprehensive account of the work. We have included more detail and expanded in many places. Moreover, we have included different approaches or proofs when possible. 2. Sharp bounds for the fractional maximal operator Theorem 2... Suppose 0 α < n, < p < n/α and q is defined by the relationship /q = /p α/n. If w A p,q, then p q wm α f L q c[w] ( α n ) A p,q w f L p (2.) and the exponent p q ( α n ) is sharp. Proof. First notice that M α M c α where M c α is the centered version. Let x R n, a cube centered at x, u = w q, σ = w p and r = + q/p. Noticing that p /q( α/n) = r /q, we proceed as in [3] to obtain ( ) α/n f dy 3 nr /q [w] p /q( α/n) p /q( α/n) A p,q u() c[w] p /q( α/n) A p,q M c u() α,σ( f /σ) q/r dy σ(3) α/n ) r /q. f σ σ dy 35

36 Taking the supremum over all cubes centered at x we have the pointwise estimate M c α f (x) c[w] p /q( α/n) A p,q M c u{m c α,σ( f /σ) q/r u }(x) r /q. Using the fact that M u : L r (u) L r (u) and M σ : L p (σ) L q (σ) with operator norms independent of u and σ respectively, we get wm α f L q c M c α f L q (u) c[w] p /q( α/n) A p,q M c u{m c α,σ( f /σ) q/r u } r /q L r (u) c[w] p /q( α/n) A p,q f w L p, which is the desired estimate. We show that the bound (2.) is sharp in Section Extrapolation The celebrated extrapolation theorem of Rubio de Francia [45] is one of the most important theorems in the modern harmonic analysis. It allows one to obtain boundedness of an operator on a wide class of function spaces from a starting point. To obtain sharp bounds for singular integral operators on L p (w) (Petermichl [42], [43]) one only needs to obtain the bound for p = 2. The general case p 2 then follows by the sharp version of the Rubio de Francia extrapolation theorem given by Dragi cević, Grafakos, Pereyra, and Petermichl [9]. In this section we present an off-diagonal extrapolation theorem with sharp constants. The original off-diagonal extrapolation is due to Harboure, Macias, and Segovia [24]. We begin with a lemma about L p space for 0 < p <. 36

37 Lemma Let f 0, g > 0 be measurable functions, 0 < s < and s = s/(s ), then As a consequence, X f g dµ f L s (µ) g L s (µ). f L s (µ) = inf f g dµ X where the infimum is over all g with g L s =, and the infimum is attained. Proof. Since 0 < s <, /s > so we may use Hölder s inequality with /s and /( s), X /s /s f dµ) s = ( f g) s g dµ) s X X ) /s f g dµ g s dµ. X Equality is attained by taking g = f s / f s L s (µ). The following result is due to Harboure, Macías, and Segovia [24], we repeat the proof to show the dependence on the constants. Theorem Suppose that T is an operator defined on an appropriate class of functions such as S, or w Lloc p L p (w p ). Suppose further that p 0 and q 0 are exponents with p 0 q 0 <, and wt f L q 0 (R n ) c[w] γ A p0,q 0 w f L p 0 (R n ) for all w A p0,q 0 and some γ > 0. Then, γ max{, q 0 p wt f L q (R n ) c[w] p q 0 } A p,q w f L p (R n ) 37

38 holds for all p and q satisfying < p q < and p q = p 0 q 0, and all w A p,q. To prove Theorem we need the following lemma whose proof can be found in [9]. Lemma Suppose that r > r 0, v A r, and g is a non-negative function in L (r/r 0) (v). Then, there exists a function G such that. G g, 2. G L (r/r 0 ) (v) 2 g L (r/r 0 ) (v), 3. Gv A r0 with [Gv] Ar0 c[v] Ar. Proof of Theorem First suppose w A p,q and p 0 < p, which implies q > q 0. Then, ) /q T f q w q = R n = R n( T f q 0 ) q/q 0 w q ) T f q 0 gw q q0 R n ) q 0 q q0 for some non-negative g L (q/q 0) (w q ) with g L (q/q 0 ) =. Now, let r = + (w q q/p ) and r 0 = + q 0 /p 0. Since p > p 0 we have r > r 0. Furthermore, by the relationship p q = p 0 q 0, we have q/q 0 = r/r 0. Hence by Lemma and using that w q A r, there exists G with G g, G L ( r/r 0 ) (w q ) 2, Gwq A r0, and [Gw q ] Ar0 c[w q ] Ar = c[w] Ap,q. Also, 38

39 since Gw q A r0 then (Gw q ) /q 0 A p0,q 0 since, ( [(Gw q ) /q 0 ] Ap0,q 0 = sup = sup ( = [Gw q ] Ar0. (G /q 0 w q/q 0 ) q 0 ) Gw q (Gw q ) p 0 /q 0 ) ) q0 (G /q 0 w q/q /p 0 ) p 0 0 ) q0 /p 0 Then, we can proceed with ) /q T f q w q = R n = T f q 0 gw q R n T f q 0 Gw q R n ) q0 ) q0 R n T f q 0 (G /q 0 w q/q 0 ) q 0 ) q0 c[g /q 0 w q/q 0 ] γ A f p 0 p0 (G /q 0 w q/q 0 ) p 0,q 0 R n ) = c[gw q ] γ A f p 0 r0 w p 0 G p 0/q 0 w q/(p/p 0) p0 R n ) p0 c[w] γ A p,q R n f p w p ) /p R n G(r/r 0) w q ) (p p0 )/pp 0 c[w] γ A p,q R n f p w p ) /p, where we have used the relationship p q = p 0 q 0. 39

40 For the case < p < p 0, and hence q < q 0, notice that we can write ) /p ) /p f p w p = R n R n( f wp p 0 ) p/p 0 w p. Since p/p 0 <, by Lemma 2.2. exists a function g 0 satisfying R n gp/(p p 0) w p = such that ) /p ) /p0 f p w p = f R n R n wp p 0 gw p. Let h = g p 0 /p 0, r = + p /q and r 0 = + p 0 /q 0, so that r > r 0. Notice that p q = p 0 q 0 implies r/r 0 = p /p 0, which in turn yields p 0 p 0 ( r r 0 ) = p p 0 p. (2.2) Hence, h(r/r 0) w p = R n gp/(p p0) w p =. R n Observe that w p A r, so by Lemma we obtain a function H such that H h, H L (r/r 0 ) (w p ) 2, and Hw p A r0 with [Hw p ] Ar0 c[w p ] Ar = c[w] p /q A p,q. Now, for Hw p A r0 we claim that (Hw p ) /p 0 Ap0,q 0 with [(Hw p ) /p 0 ]Ap0,q 0 = [Hw p ] q 0/p 0 A r0. Indeed, ) ) [(Hw p ) /p 0] Ap0,q 0 = sup (H /p 0w p /p 0) q 0 q0 /p (H /p 0w p /p 0) p

41 ) ) = sup (Hw p ) q 0/p q0 /p 0 Hw p 0 = [Hw p ] q 0/p 0 A r0. Finally expressing g in terms for h and using (2.2), working backwards we have ) /p f p w p = R n = ) /p0 f p 0 h p 0/p 0w p (p 0 ) R n ) /p0 f p 0 H p 0/p 0w p (p 0 ) R n [(Hw p ) /p γ 0 ] A p0,q 0 [(Hw p ) /p 0] γ f p 0 (H /p 0w p /p 0) p 0 A R n p0,q 0 c [(Hw p ) /p 0] γ T f q 0 (H /p 0w p /p 0) q 0 A R n p0,q 0 ) c /q [(Hw p ) /p 0] γ T f q w q A R n p0,q 0 ) c /q [(Hw p ) /p 0] γ T f q w q. A R n p0,q 0 ) /p0 ) /q0 R n H(r/r 0) w p ) q q0 /qq 0 In the second to last inequality we have used Lemma Thus we have shown, ) /q ) /p T f q w q c[(hw p ) /p 0] γ R n A f p p0 w p.,q 0 R n From here we have T c[(hw p ) /p 0] γ γ q 0 A p0 = c[hw p p ] 0,q 0 γ q 0 p 0 A r0 c[w p ] A = c[w] +p /q γ q 0 p p q 0 A p,q. This proves the theorem. 4

42 Using an idea of Grafakos and Martell [20] we may extend our extrapolation theorem to the weak case. Corollary Suppose that for some p 0 q 0 <, an operator T satisfies the weak-type (p 0,q 0 ) inequality T f L q 0, (w q 0) c[w] γ A p0,q 0 w f L p 0 (R n ) for every w A p0,q 0 and some γ > 0. Then T also satisfies the weak-type (p,q) inequality, γ max{, q 0 p T f L q, (w q ) c[w] p q } 0 A p,q w f L p (R n ) for all < p q < that satisfy p q = p 0 q 0 and all w A p,q. Proof. Note that Theorem does not require T to be linear. We can simply apply the result to the operator T λ f = λ χ { T f >λ}. Fix λ > 0, then wt λ f L q 0 = λw q 0 ({x : T f (x) > λ}) /q 0 T f L q 0, (w q 0) c[w] γ A p0,q 0 w f L p 0, 42

43 with constant independent of λ. Hence by Theorem if w A p,q, T λ maps L q (w q ) L p (w p ) for all /p /q = /p 0 /q 0 and with bound γ max{, q 0 p wt λ f L q c[w] p q } 0 A p,q f w L p. with c independent of λ. Hence, γ max{, q 0 p T f L q, (w q ) = sup wt λ f L q c[w] p q } 0 A p,q f w L p. λ>0 2.3 Sharp bounds for fractional integral operators We now present our main result of this chapter, the sharp bounds for the operator norm of I α. We need the following packing condition lemma see [49]. Lemma Suppose ε > 0, c > 0 and f is a locally integrable function. Let 0 be a cube and G be the dyadic grid associated to 0, then there exists a constant C ε > 0 such that ε f (y) dy C ε 0 ε f (y) dy G c c 0 0 Proof. Let a = l( 0 ), then ε f (y) dy = G c 0 = k=0 G, 0 l()=2 k a a εn 2 knε k=0 ε c G, 0 l()=2 k a f (y) dy c f (y) dy. 43

44 Notice that at most c n of the cubes c overlap if G with side length c2 k a. It follows that f (y) dy C f (y) dy. G, c c 0 0 l()=2 k a Proceeding we have, k=0 2 knε G, c 0 l()=2 k a f (y) dy Ca εn c 0 f (y) dy = C ε 0 ε c 0 f (y) dy. 2 knε k=0 Theorem Let < p < n/α and q be defined by the equation /q = /p α/n, and let w A p,q. Then, wi α f L q (R n ) c[w] η(p /q) A p,q w f L p (R n ), (2.3) where η(x) = min{max( α/n,x),max(,( α/n)x)}. The relationship I α c[w] η(p /q) A p,q is sharp for p /q in the range (0, α/n] [n/(n α), ) (see Figure 2.). 44

45 α n ( α n ) p Figure 2.: The graph of the function η. q Proof. We use Theorem with base exponents q 0 /p 0 = α/n. This along with the fact that /p 0 /q 0 = α/n yields p 0 = 2 α/n α/n (α/n) 2 + and q 0 = 2 α/n α/n. We will show the linear estimate wi α f L q 0 c[w] Ap0,q 0 w f L p 0. (2.4) Notice that (2.4) is equivalent to I α ( f σ) L q 0 (u) c[w] Ap0,q 0 f L p 0 (σ), (2.5) where u = w q 0 and σ = w p 0. Moreover, by duality, showing (2.5) is equivalent to proving ) /p0 ) /q I α( f σ)gu dx c[w] f p 0 0 Ap0 R n,q 0 σ dx 0u R n R n gq dx, (2.6) 45

46 for all f and g non-negative bounded functions with compact support. We first discretize the operator I α as follows. Given a non-negative function f, f (y) I α f (x) = k Z 2 k < x y 2 k x y c k n α dy χ (x) l() n α D l()=2 k c χ (x) α/n D 3 f dy x y l() f (y) dy where the last inequality holds because if x, then B(x,l()) 3. One immediately gets then R n I α( f σ)gu dx c D α/n 3 f σ dx gudx. We may pass the sum to smaller set of dyadic cubes that are better suited for our calculations. We combine ideas from the work of Sawyer and Wheeden in [49], together with some techniques from Pérez [40]. Fix a > 2 n. Since g is bounded with compact support, for each k Z, one can construct a collection { k, j } j of pairwise disjoint maximal dyadic cubes (maximal with respect to inclusion) with the property that a k < gudx. k, j k, j By maximality, the above also gives gudx 2 n a k. k, j k, j 46

47 For a fixed k the family { k, j } j is disjoint in j. If we define for each k the collection C k = { D : a k < gudx a k+ }, then each dyadic cube belongs to only one C k or gu vanishes on it. Moreover, each C k has to be contained in one of the maximal cubes k, j0 and verifies for all k, j gudx a k+ a gudx. k, j k, j Lemma 2.3. shows that for any dyadic cube 0, α/n f σ dx c α 0 α/n f σ dx. D, Thus one easily deduces as in [49] that D α/n 3 k, j f σ dx gudx ac α α/n f σ dx gudx. k, j k, j 3 k, j k, j Notice also that, [w] Ap0,q 0 = sup u() ( ) σ() α/n <, so we can estimate I k, j α( f σ)gudx c α/n f σ dx gudx R n k, j k, j 3 k, j k, j = c k, j σ(5 k, j ) α/n f σ dx gu dx 3 k, j u(3 k, j ) k, j u(3 ( ) k, j) σ(5k, j ) α/n k, j k, j k, j c[w] Ap0,q 0 k, j σ(5 k, j ) α/n f σ dx gu dx k, j, 3 k, j u(3 k, j ) k, j 47

48 (2.7) where we have set up things to use, in a moment, certain centered maximal functions. Before we do so, we need one last property about the Calderón-Zygmund cubes k, j. We need to pass to a disjoint collection of sets E k, j each of which retains a substantial portion of the mass of the corresponding cube k, j. Define the sets E k, j = k, j {x R n : a k < M d (gu) a k+ }, where M d is the dyadic maximal function. The family {E k, j } k, j is pairwise disjoint for all j and k. Moreover, set Ω k = {x : M d (gu)(x) > a k } so Ω k = j k, j where a k < gu dx 2 n a k. k, j k, j Then, k, j Ω k+ = k, j k+,i i = k+,i k+,i k, j k+,i k, j a k+ = k, j a k+ k, j a k+ k, j k+,i k, j k+,i gu dx k, j k, j gu dx k+,i gu dx 48

49 2n a k, j. It follows that E k, j ( 2n a ) k, j. Recalling now that = u n α n n σ = u q0 n α q σ 0, we can use Hölder s inequality to write k, j E k, j = E k, j u q 0 n n α q σ 0 u(e k, j ) /q 0σ(E k, j ) /q 0, (2.8) since q 0 q 0 n n α =. With (2.8) we go back to the string of inequalities to estimate I α ( f σ)gudx. Using the discrete version of Hölder s inequality, we can estimate in (2.7) ( ) /q0 q0 c[w] Ap0,q 0 ( k, j σ(5 k, j ) α/n f σ dx σ(e k, j)) 3 k, j ( ) q /q 0 0 gu dx u(ek, j)) k, j u(3 k, j ) k, j ) /q0 ( c[w] Ap0,q 0 ( (Mα,σ c f ) q 0 σ dx k, j E k, j (Mug) c q 0u dx k, j E k, j ) /q0 ) /q c[w] Ap0,q 0 R n(mc α,σ f ) q 0 0 σ dx ug) q 0u dx R n(mc ) /p0 ) /q c[w] f p 0 0 Ap0,q 0 σ dx 0u R n R n gq dx. ) /q 0 49

50 We have also used the boundedness of M c u and M c α,σ with operator norms independent of the corresponding measure. We obtain then the desired linear estimate wi α f L q 0 c[w] Ap0,q 0 w f L p 0. (2.9) From this estimate we can extrapolate (Theorem 2.2.2) to get, wi α f L q c[w] max{,( α/n)p /q} A p,q w f L p (2.0) for all < p < q < with /p /q = α/n. This proves one of the estimates in Theorem The estimate (2.0) is equivalent to saying that the linear operator T ( f ) = wi α ( f w ) is bounded T : L p (R n ) L q (R n ) with bound [w] max{,( α/n)p /q} A p,q. By duality, the transpose operator given by T t ( f ) = w I α ( f w) is bounded T t : L q (R n ) L p (R n ). Furthermore the bound is the same as above, namely less than [w] max{,( α/n)p /q} A p,q. Hence we have w I α f L p (R n ) c[w]max{,( α/n)p /q} A p,q w f L q (R n ). (2.) Since /p /q = /q /p = α/n we can replace this by wi α f L q (R n ) c[w ] max{,( α/n)q/p } A q,p w f L p (R n ). (2.2) 50

51 Thus we have I α L p (w p ) L q (w q ) c[w] min{max( α n, p q ),max(,( α n ) p q )} A p,q. The sharpness of the bounds obtained for p /q (0, α/n] [n/(n α), ) will be shown with an example in Section 2.4. Remark One may also take a different approach by examining the dyadic fractional operator defined in [49] by Iα d f (x) = χ (x) α/n f dy. D Notice that for a given f 0 if we let {a (x)} D be the function from R n to the space of sequences indexed by D, defined by a (x) = χ (x) α/n f dy. Then I d α f (x) = a (x) l (D) and M d α f (x) = a (x) l (D), where l (D) and l (D) are the spaces of absolutely summable sequences index by D and bounded sequences indexed by D respectively. Minor modifications to the proof (using the boundedness of M d α,σ and M d u instead of M c α,σ and M c u) give I d α L p 0 (w p 0) L q 0(w q 0) c[w] Ap0,q 0. 5

52 From here one could use the shifting lemma in [49] similar to Lemma below to conclude I α L p 0 (w p 0) L q 0(w q 0) c[w] Ap0,q 0. Continuing from here one obtains the results of the Theorem 2.3. Remark Examples indicate that the sharp bound for I α should be )max{, p q } I α L p (w p ) L q (w q ) c[w] ( α n A p,q. (2.3) To prove (2.3) using extrapolation (Theorem 2.2.2) one would need to consider the case p 0 = q 0 and show that the estimate I α L p 0 (w p 0) L q 0(w q 0) c[w] α n A p0,q 0 holds. We do not know if this approach can be modified to work. See again Chapter 6. We also have the following theorem for the sharp bound on the weak operator norm of the fractional integral operator. Theorem Let 0 < α < n, /q 0 = α/n, and w be a weight with u = w q 0 I α f L q 0, (u) C f L ((Mu) α n ). (2.4) Remark Estimate (2.4) is a fractional version of the Muckenhoupt-Wheeden conjecture. The Muckenhoupt-Wheeden conjecture states that given a weight and a Calderón-Zygmund operator T, T f L, (w) C f L (Mw). (2.5) 52

53 This remains a difficult open problem in the theory of weights. Another version of the Muckenhoupt-Wheeden conjecture for I α is the following estimate I α f L, (w) c f L (M α w). However, in general this estimate is false, see Carro, Pérez, F. Soria, and J. Soria [5] for a counter example. Since w A,q implies u = w q A we have M(w q ) /q = M(u) /q [u] /q A u /q = [w] /q A,q w. Combining this with (2.4) we have I α f L q 0, (w q 0) [w] α/n A,q0 f L (w), where q 0 = n/(n α). From here we may apply Theorem to obtain the following Theorem. Theorem Suppose that p < n/α and that q satisfies /q = /p α/n. Then I α f L q, (w q ) c[w] α n A p,q w f L p (R n ) (2.6) and the exponent α n is sharp. Remark Theorem is the fractional version of the linear growth conjecture for Calderón-Zygmund operators: T f L p, (w) [w] Ap f L p (w). 53

54 This conjecture was formulated by Lerner, Ombrosi, and Pérez in [32] and is another unsolved problem in the theory of weights. Proof of Theorem In order to prove (2.4), we note that L q 0, (u) is equivalent to a norm since q 0 >. Hence, we may use Minkowski s integral inequality as follows I α f L q 0, (u) c q R n f (y) y α n L q 0, (u) dy. (2.7) We can finally calculate the inner norm by y α n L q 0, (w q ) = sup λu({x : x y α n > λ}) /q 0 λ>0 = (sup t>0 t n u({x : x y < t}))/q 0 = cmu(y) /q 0. Once again, the sharpness of the exponent α/n will be shown with an example in Section Examples As mentioned in the introduction, power functions such as x a with n < a < n(p ) are important examples of A p weights. It is with these examples that we will show Theorems 2.3.2, 2.3.7, and 2.. are sharp. This technique was first used by Buckley [4], to show (.6) is sharp. We state one lemma that will be used through out this section. 54