INEQUALITIES ASSOCIATED TO RIESZ POTENTIALS AND NON-DOUBLING MEASURES WITH APPLICATIONS MUKTA BAHADUR BHANDARI

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1 INEQUALITIES ASSOCIATED TO RIESZ POTENTIALS AND NON-DOUBLING MEASURES WITH APPLICATIONS by MUKTA BAHADUR BHANDARI M.A., Tribhuvan University, Nepal, 992 M.S., Kansas State University, USA, 2005 AN ABSTRACT OF A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Mathematics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 200

2 Abstract The main focus of this work is to study the classical Calderón-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling. We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis-the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in (R n, dµ), where µ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.

3 INEQUALITIES ASSOCIATED TO RIESZ POTENTIALS AND NON-DOUBLING MEASURES WITH APPLICATIONS by MUKTA BAHADUR BHANDARI M.A., Tribhuvan University, Nepal, 992 M.S., Kansas State University, USA, 2005 A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Mathematics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 200 Approved by: Major Professor Dr. Charles N. Moore

4 Copyright Mukta Bahadur Bhandari 200

5 Abstract The main focus of this work is to study the classical Calderón-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling. We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis-the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in (R n, dµ), where µ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.

6 Table of Contents Table of Contents Acknowledgements Dedication Preface vi vii viii x Maximal Function in Non-doubling Metric Measure Space.. Introduction Some Useful Definitions Covering and Interpolation Theorems Non-doubling Measures Estimates for Riesz Potentials by Maximal Function 9 2. Introduction Generalization of the Hedberg Inequalities Modified Centered Maximal Function Good Lambda Inequality Introduction The A p Weights Future Motivation 67 Bibliography 75 A Notations 76 vi

7 Acknowledgments I would like to express my deepest gratitude to my advisor Professor Charles N. Moore for his invaluable time, effort, advice, and continuous support which helped me accomplish this dissertation. Dr. Moore is a major source of encouragement and motivation. I am deeply indebted to him for everything he has done for me. I would like to thank my committee members Professor Pietro Poggi-Corradini and Professor Marianne Korten. They have always been helpful and supportive throughout my student career at K-State. I would also like to thank Dr. Bimal K. Paul, a Professor in the Department of Geography, who kindly accepted my request to be a committee member of my Ph.D defense. My thanks goes to Dr. Hikaru Peterson for supporting me as a outside chairperson of my committee. My special thanks goes to Professor David Yetter for his continuous support as a Graduate Chair and Dr. Louis Pigno, the Head of the Department of Mathematics, for providing me the opportunity to come to K-State to pursue my Ph.D. studies. I would like to thank my family members Bina Bhandari (wife), Dristi Bhandari (daughter) and Prashish Bhandari (son) for their enormous family support during my student career at K-state. vii

8 Dedication I dedicate this work to my parents Kamal Bhandari (Father) and Dila Bhandari (Mother). viii

9 Preface The area of this work falls in Harmonic Analysis. The space of homogeneous type has been one of the most important tools of Harmonic Analysis for more than last three decades. These spaces were formally introduced in [5] by R. Coifmann and G. Weiss. The space of homogeneous type is a metric space equipped with a measure µ satisfying the so-called doubling condition, which means that there exists a constant C = C(µ), such that, for every ball B(x, r) of center x and radius r µ (B(x, 2r)) Cµ (B(x, r)). () It was believed that space of homogeneous type was the base for the optimal level of generality to study harmonic analysis. The reason for its success is that E. M. Stein chose space of homogeneous type to develop the basic fundamental theory of harmonic analysis ([43], [42]). In recent years it has been ascertained that central results of classical Calderón-Zygmund Theory hold true in a very general situation in which the underlying measure is not necessarily doubling but only satisfies a mild condition, known as the growth condition. It came as surprise to many when F. Nazarov, S. Treil and A. Volberg announced that The doubling condition is superfluous for most of the classical theory of harmonic analysis ([50]). They meant that a rather complete theory of Calderón-Zygmund operators could be developed if, in some sense, condition () is replaced by the following condition: A Borel measure µ on a measure metric space (X, d) is said to satisfy the growth condition if µ (B(x, r)) Cr N (2) where the constant C is independent of x and r. This allows, in particular, non-doubling measures. Sometimes we shall refer to condition (2) by saying that the measure µ is N- dimensional. Some other prominent mathematician working on this area are X. Tolsa ([47]), ix

10 J. Verdera ([54]), C. Peréz ([39]) J. Mateu, P. Matila, A. Nicolau, J. Orobitg ([38]), Y. Sawano, H. Tanaka ([40] and José García-Cuerva, A. Eduardo Gatto ([23]). The list of results that one can establish without resorting to the doubling condition is quite amazing and encouraging for further research in this area. For example, the T(b)- Theorem ([7] and [49]), the Calderón-Zygmund decomposition and the derivative of weak L and L p bounds, < p <, from L 2 bounds ([50], [46], and [48]), Cotlar s inequality for the maximal singular integral ([50], and [45]) and many others ([38], [39], and [24]). This is the motivating factor of this work. The first chapter deals with some basics of our work. We discuss maximal functions with their variations and introduce non-doubling measures with examples. Chapter 2 deals with the Riesz potentials and its estimates by maximal functions. In the sequel, we generalize the well known Hedberg inequality associated to non-doubling measures. Finally, we give an application of this deriving an exponential inequality. In Chapter 3, we review the well known good-lambda inequalities and its variations. We generalize this associated to a non-doubling measure with applications. In Chapter 4, we conclude our work with motivation to future work. x

11 Chapter Maximal Function in Non-doubling Metric Measure Space.. Introduction. In this chapter, we will review the history of the Hardy-Littlewood maximal operator. It was first introduced in 930 by G. H. Hardy and J. E. Littlewood [27] in R in order to apply this as a tool in the theory of Complex analysis. Then N. Weiner [55] in 939 introduced this operator in higher dimensions R n (n > ). The purpose was to apply this in Ergodic theory. Since then the operator has been widely studied and used. One of its applications is Lebesgue s differentiation theorem which can be deduced from the boundedness of the maximal operator. The generalization of the differentiation theorem to averages over a variety of families of sets leads to the definition of several variants of the Hardy-Littlewood maximal operator ([20]). Important cases are averages over balls or cubes (the usual Hardy- Littlewood maximal function), averages over rectangles with sides parallel to the coordinate axes, and averages over arbitrary rectangles. In 97, R. Coifman and G. Weiss [5] introduced the maximal operator on a quasi-metric measure space satisfying the doubling condition which we call homogeneous space. It was in 998, F. Nazarov, S. Treil and A. Volberg [50] introduced modified Hardy-Littlewood maximal operators on quasi-metric spaces possessing a Radon measure that do not necessarily satisfy a doubling condition which we

12 call nonhomogeneous spaces. In recent years it has been ascertained that the central results of classical Calderón- Zygmund Theory hold true in very general situations in which the standard doubling condition on the underlying measure is not satisfied. This came as great surprise to the authors who felt that homogeneous spaces were not only a convenient setting for developing Calderón-Zygmund Theory, but that they were essentially the right context. The T(b)-theorem ([7]), the Calderón-Zygmund decomposition and the derivation of L and L p bounds, < p <, from L 2 bounds ([50], [46], [48] ), Cotlar s inequality for the maximal singular integral ([50], [45]) and many others ([38], [39], [24]) are among those theorems which hold without resorting to the doubling condition..2 Some Useful Definitions Definition. Let (X, µ) and (Y, ν) be measure spaces, and let T be an operator from L p (X, µ) into the space of measurable functions from Y to C. We say that T is weak (p,q), q <, if there exists a constant C such that for every λ > 0, ( ) q C f p ν ({y Y : T f(y) > λ}), λ and we say that it is weak (p, ) if it is bounded operator from L p (X, µ) to L (Y, ν). We say that T is strong (p,q) if it is bounded from L p (X, µ) to L q (Y, ν). Remark 2. If T is strong (p,q) then it is weak (p,q). Proof. Let E λ = {y Y : T f(y) > λ}. Then ν(e λ ) = dν T f(x) E λ λ E λ q dν T f q q λ q ( ) q C f p. λ 2

13 If (X, µ) = (Y, ν) and T is the identity operator then the weak (p, p) inequality is the classical Chebyshev inequality. Definition 3. A measure µ in a metric space is called doubling if balls have finite and positive measure and there is a constant C = C(µ) such that µ(2b) Cµ(B) (.) for all balls B. The constant C is independent of the center and radius of the balls in X. We also call a metric space (X, µ) doubling or homogeneous if µ is a doubling measure. Note that if µ is a doubling measure then µ(λb) C(µ, λ)µ(b) for all λ. Example 4. Lebesgue measure in R n is doubling. In general dµ(x) = x a dx, a > n is a doubling measure in R n. Proof. The condition a > n ensures that the measure under consideration is finite in R n. Let ν n denote the volume of the unit ball in R n. We divide the balls B(x 0, R) in R n into two categories as follows: T = {B(x 0, R) : x 0 3R}, and T 2 = {B(x 0, R) : x 0 < 3R}. For balls in T we have x 0 3R. It follows that x 0 + 2R 4( x 0 R), and x 0 2R 4 ( x 0 + R). (.2) For balls in T and a 0, µ (B(x 0, 2R)) = x a dx ( x 0 + 2R) a B(x 0, 2R) = ν n (2R) n ( x 0 + 2R) a, (.3) B(x 0,2R) 3

14 and µ (B(x 0, R)) = x a dx ( x 0 R) a ν n R n. (.4) B(x 0,R) Combining the inequalities (.2),(.3), and (.4) we obtain µ (B(x 0, 2R)) Cµ (B(x 0, R)) (.5) where C = 2 3n 4 a. Similarly, for the balls in T and a < 0, we obtain the following inequalities µ (B(x 0, 2R)) ( x 0 2R) a ν n (2R) n, and µ (B(x 0, R)) ( x 0 + R) a ν n R n. Combining the above two inequalities with (.2) we obtain the same (.5) inequality. Note that B(x 0, 2R) B(0, 5R) for x 0 < 3R. So, for the balls in T 2 we µ (B(x 0, 2R)) = B(x 0,2R) x a dx x 5R x a dx (5R) a ν n (5R) n = C n R n+a. Also note that the function x a is radially increasing for a 0 and radially decreasing for a < 0. So, we have For x B { x a dx B(0,R) x a dx, when a 0, (.6) B(x 0,R) B(3R x 0 x 0,R) x a dx when a < 0. ( ) 3R x 0, R x 0 we must have x 2R. Thus both integrals in the inequality (.6) are at least a multiple of R n+a. This establishes the doubling condition (.5) for the balls in T 2. This completes the proof. Definition 5. Let (X, µ) be a metric measure space. The Hardy-Littlewood Maximal function of a locally integrable function f on X is defined by M(f)(x) = sup r>0 f(y) dµ(y). µ(b(x, r)) B(x,r) 4

15 Theorem 6. let µ be a Lebesgue measure in R n. If f L (R n, µ) and is not identically 0, then Mf / L (R n, µ). This is true in any Euclidean space R n. But it is not true in arbitrary metric spaces. We can see this by the following example: Example 7. Let X = [0, ] and let µ be a Lebesgue measure on X. Then Mf L loc (X, µ) for any f L loc (X, µ). The integrability of Mf in the above remark fails at infinity, and does not exclude local integrability. However, the following example shows that even local integrability can fail. Example 8. Let f(x) = {, x(log x) 2 if 0 < x /2; 0, otherwise The local integrability of M f fails for this function. The following theorem provides a partial converse of the Theorem 6 that characterizes when M f is locally integrable. Theorem 9. ([9]) If f is an integrable function supported on a compact set B, then Mf L (B) if and only if f log + f L (B). In a metric measure space (X, d, µ) with µ (X) < we can find an integrable function such that its maximal function is also integrable. For example, any constant function defined on X has this property. It is remarkable to find such a function when µ (X) =. Consider the following example: Example 0. Let dµ(x) = e x dx, f(x) = χ (,) (x), x R. Clearly, µ (R) = and f L (µ). Let x / (, ). We may take x > and let B = B(x, R) be a ball with center at x and radius R. For x R 0, µ(b) = x+r x R e y dy = x+r x R 5 e y dy = e x+r e x R.

16 For x R < 0, µ(b) = x+r e y dy = 0 e y dy + x R x R o x+r Next we compute B χ (,)(y)e y dy. For x R 0, e y dye = e R x + e R+x 2. For x R < 0, χ (,) (y)e y dy = B x R χ (,) (y)e y dy = 0 e y dy + B x R 0 e y dy = e e x R. e y dy = e R x + e 2. Let g(r) = χ (,) (y)e y dy. µ(b) B Note that Mf(x) = sup R>0 g(r). We also note that x R x +. So, Mf(x) = max x R x+ g(r). From the above computation, we see that e e x R if x R 0, g(r) = e x+r e x R e R x + e 2 if x R < 0. e R x + e R+x 2 Note that e e x R, e x+r e x R, e R x +e 2, and e R x +e R+x 2 are all increasing functions of R. Because of the inequality x R x +, e e x R e e x (x+) = e e, e x+r e x R e 2x e, e R x + e 2 2e 2, and e R x + e R+x 2 e 2x 2 + e. Thus g(r) = e ex R e 2 e x+r ex R e 2x for x R 0, 6

17 and g(r) = er x + e 2 e R x + e R+x 2 2e(e ) e 2x 2e + for x R < 0. We also note that e 2 2e(e ) and Therefore, for every R [x, x + ] g(r) e 2x 2e + e 2x. 2e e 2x 2e + 2e(e ) e 2x 5. This means that Mf(x) 2e(e ) e 2x 5. By symmetry x/ (,) 2e Mf(x)dµ(x) 2 e 2x 5 ex dx <. For x (, ), let B be a ball with center at x (, ). Then, χ (,) dµ(x) = µ (B (, )). µ(b) B µ(b) Therefore, we conclude that Mf L (µ)..3 Covering and Interpolation Theorems Dyadic Cubes in R n : Definition. (Dyadic Cubes in R n ) The unit cube in R n, open on the right, is defined to be [0, ) n. Let Q 0 be the collection of cubes in R n which are congruent to [0, ) n and whose 7

18 vertices lie in Z n. Let Q k = {2 k Q : Q Q 0 }, k Z. That is, Q k is the family of cubes, open on the right, whose vertices are adjacent points of the lattice ( 2 k Z ) n. The cubes in Q := k Q k are called the dyadic cubes. We get the following properties of the dyadic cubes from this construction: (a) For every x R n there exists a unique cube in Q k which contains it; (b) Any two dyadic cubes are either disjoint or one is wholly contained in the other; (c) A dyadic cube in Q k is contained in a unique cube of each family Q j, j < k, and contains 2 n dyadic cubes of Q k+. The covering lemmas provide the standard approach to prove that the Hardy-Littlewood maximal function in R n is weak type (,). We state here very useful covering lemmas due to Whitney, N. Weiner, A. Besicovitch and A. P. Morse. Note that for any ball B = B(x, r), t > 0 we mean by tb the ball concentric to B with radius tr. That is, tb = B(x, tr). An arbitrary open set in R n can be decomposed as a union of disjoint cubes whose lengths are proportional to their distance from the boundary of the open set. For a given Q in R n, we will denote by l(q) its length and by diam(q) its diameter. Whitney Decomposition: Theorem 2. (Whitney Decomposition) Let Ω be an open nonempty proper open subset of R n. Then there exists a family of dyadic cubes {Q j } j such that (a) Ω = j Q j where Q j s have disjoint interiors. (b) diam(q j ) dist(q j, Ω c ) 4diam(Q j ), for every j. That is, nl(q) dist(qj, Ω c ) 4 nl(q). (c) If the boundaries of two cubes Q j and Q k touch, then 4 l(q j) l(q k ) 4. 8

19 (d) For a given Q j, there exists at most 2 n Q k s that touch it. The family of dyadic cubes {Q j } j as in the above theorem is known as a Whitney Decomposition of Ω. ([25]) Remark 3. Let F = {Q j } j R n. Fix 0 < ɛ < 4 be a Whitney decomposition of a proper open subset Ω of and denote by Q k the cube with the same center as Q k but with side length ( + ɛ) times that of Q k.then Q k and Q j touch if and only if Q k and Q j intersect. Consequently, every point of Ω is contained in at most 2 n cubes Q k. That is, χ (+ɛ)q (x) 2 n χ Ω (x) ([25]) k Theorem 4. (Vitali Covering Lemma:)([20]) Let {B j } j τ be a collection of balls in R n. Then there exists an at most countable sub-collection of disjoint balls {B k } such that j τ B j k 5B k. The next covering theorem is due independently to A. Besicovitch and A.P. Morse. Besicovitch Covering Lemma: Theorem 5. (Besicovitch Covering Lemma:)([20]) Let A be a bounded set in R n, and suppose that {B x } x A is a collection of balls such that B x = B(x, r x ), r x > 0. Then there exists an at most countable sub-collection of balls {B j } and a constant C n, depending only on the dimension, such that A j B j and χ Bj (x) C n. The following theorem is a version of classical Besicovitch Covering Therem. Theorem 6. ([30]) Let A be a bounded set in a metric space X and F be a collection of balls centered at points of A. Then, there is a sub-collection G F such that j A B G B 9

20 and that { } 5 G = 5 B : B G is a disjointed family of balls. Moreover, if X carries a doubling measure, then G is countable, and if X = R n, then one can choose G such that χ B (x) C(n) < j for some dimensional constant C(n), where χ E denotes the characteristic function of a set E. Next we state interpolation theorems. They are useful tools in proving the L p boundedness of maximal functions for < p. Interpolation: Theorem 7. (Riesz-Thorin Interpolation, [9]). Let p 0, p, q 0, q, and for 0 < θ < define p and q by p = θ + θ, p 0 p q = θ + θ. q 0 q If T is a linear operator from L p 0 + L p to L q 0 + L q such that and then T f q0 M 0 f p0 for f L p 0 T f q M f p for f L p, T f q M θ 0 M θ f p for f L p. Definition 8. Let (X, µ) be a measure space and let f : X C be a measurable function. We call the function a f : (0, ) [0, ], given by a f (λ) = µ ({x X : f(x) > λ}), the distribution function of f associated with µ. 0

21 Note that the weak inequalities measure the size of the distribution function. The following interpolation theorem which is due to Marcinkiewicz helps to deduce L p boundedness from weak inequalities. It applies to larger class of operators than linear ones. Note that maximal functions are not linear but sublinear. Definition 9. An operator T from a vector space of measurable functions to measurable functions is sublinear if for every pair of measurable functions f and g and for every λ C T (f + g)(x) T f(x) + T g(x), T (λf) = λ T f. Theorem 20. (Marcinkiewicz Interpolation, [9]) Let (X, µ) and (Y, ν) be measure spaces, p 0 < p and let T be a sublinear operator from L p 0 (X, µ) + L p (X, µ) to the measurable functions on Y that is weak (p 0, p 0 ) and weak (p, p ). Then T is strong (p, p) for p 0 < p < p. Definition 2. (Central and noncentral maximal function:) Let µ be a non-negative Borel measure in R n, n. If f L loc (µ), we define the maximal function M µ f(x) = sup f (y)dµ(y) B x µ(b) B where supremum is taken over all balls B containing x. Here B may be assumed open, closed, or containing any µ-measurable part of its boundary. In this definition, x is not necessarily at the center of the ball B. This is called noncentral maximal function of f. The maximal function defined above is called a central maximal function if the supremum is taken over only to the balls with center at x. It is denoted by M c µ. A natural question to ask is: For which µ is M c µ a bounded operator on L p (R n, dµ)? It follows from the Besicovitch Covering lemma and Marcinkiewicz interpolation theorem that M c µ is L p bounded where < p. Theorem 22. ([2], [22], [8]) Let µ be a nonnegative Borel measure in R n, n. Then

22 (a) M c µ is weak type (, ). That is µ ( {x R n : M c µf(x) > λ} ) C λ f L (µ) λ > 0. (b) M c µ is strong type (p, p) for < p. Proof. (a) Let f L loc (R n ) and λ > 0 be arbitrarily chosen, and let E λ = {x R n : Mf(x) > λ}. Let K be a bounded measurable set of R n. For every x E λ K there exists an open ball B x = B(x, r x ) centered at x and radius r x such that f(y) dµ(y) > λ. µ(b x ) B x Then the collection {B x } x Eλ K is an open covering of the set E λ K. Using the Besicovitch Covering Lemma, there exists a sub-collection {B j } j= of {B x } x Eλ K, and a constant C n, depending only on the dimension n, such that Hence, E λ K j B j and χ Bj (x) C n. ( ) ) µ (E λ K µ B j µ(b j ) j j f(y) dµ(y) λ j B j χ Bj (y) f(y) dµ(y) λ j R n j = χ Bj (y) f(y) dµ(y) λ R n C n λ f L (µ). Since this estimate is independent of K, we obtain j j χ Bj for every λ > 0. µ(e λ ) C λ f L (µ) 2

23 (b) It is clear that Mµf c f L (µ) for every f L (µ). This implies that Mµf c L (µ) f L (µ). Thus the centered Hardy-Littlewood maximal operator Mµ c is both weak type (, ) and strong type (, ). Therefore, by the Marcinkiewicz interpolation theorem, it follows that the centered Hardy-Littlewood maximal operator Mµ c is strong type (p, p) for < p. Remark 23. The above theorem is not necessarily true for noncentral maximal function M µ.the following theorem reveals this fact: Theorem 24. ([4])Let µ be a nonnegative Borel measure in R n, n. (a) For n =, M µ is weak type (, ) and strong type (p, p) for < p. (b) For n = 2, there is a measure µ for which M µ doesn t map L (µ) into weak L (µ). The situation is the same if the balls in the definition of M µ are replaced by squares parallel to the axes. Proof. (a) Without loss of generality, we may assume that 0 f L (µ) and let λ > 0. Let E λ = {x : M µ f(x) > λ}. Then for every x E λ there exists an interval I x x such that 0 < µ(i x ) < f(y)dµ(y). λ I x Let F be the set of these intervals. Clearly µ(i) < f(y)dµ(y) for every I F. λ That is µ(i) is bounded for every I F. Next we extract a family of disjoint intervals from F in the following way: Having chosen I,..., I j, j let I j be an interval in F disjoint from I,..., I j, if any, such that 2µ(I j ) > sup{µ(i) : I F, I disjoint from I,..., I j }. 3 I

24 The disjoint condition is void for j =. This selection either stops at some j or gives an infinite sequence. Note that the collection {I, I 2, I 3,..., I j,...} doesn t necessarily cover E λ. Then µ(i j ) f(y)dµ(y) f(y)dµ(y) <. (.7) λ I j λ R n j j So µ(i j ) 0 as n in the case when there are infinitely many disjoint intervals I j s. Now we enlarge each I j. Let a j and b j denote the left and right end points of I j respectively. Define, a j = inf{x : x a j and µ((x, a j ]) < 2µ(I j )} and b j = sup{y : y b j and µ([b j, y)) < 2µ(I j )} and let Ij = (a j, b j). Then Ij I j and µ(ij ) 5µ(I j ). Let I F and I I j for every j. Then I must intersect some I j. If I j is the first one which intersects I then µ(i) < 2µ(I j ) because of the selection process of I j s. But then I Ij, and therefore E λ j I j. this implies that µ(e λ ) 5 j µ(i j ) 5 λ R n f(y)dµ(y). Clearly M µ f L (µ) f L (µ). Therefore, part (a) follows from Marcinkiewicz interpolation theorem. (b) (Two-dimensional counter example:) Consider the standard Gaussian measure µ where dµ(x, y) = e (x2 +y 2 )/2 dxdy. Let the maximal function M µ be defined with disks. Then M µ is not weak (, ).( See [4], [5]) Proof. Here we take as µ the standard Gaussian measure dµ(x, y) = e (x2 +y 2 )/2 dxdy. Consider that the non-centered maximal functions M µ are defined with disks. By a simple limiting argument, a weak type (, ) estimate for M µ would imply a weak L estimate 4

25 for the maximal function M µ λ of a finite measure λ, which it is enough to disprove. Let us take λ as a unit mass at (0, a + ), a > 0 large. Consider a unit disk B s centered at (s, a + ), s <. Observe that (x s) 2 + [y (a + )] 2 = implies that y > a + (x s)2. 2 So, the ball B s {(x, y) : y > a + (x s)2 2 }. Using also the fact that x e t2 /2 dt x e x2 /2 (.8) for large x, we obtain µ(b s ) C a e a2 /2 dx e y2 /2 dy C a+x 2 /2 a e ax2 /2 dx C a a e a2 /2. e (a+x2 /2) 2 /2 dx Here the constant C denote various positive constants. Hence, M µ λ Ca ae a2 /2 in the set {(x, y) : x <, a < y < a + 2}. This set has µ-measure at least Ca e a2 /2 because of the relation (.8). As a, this disproves the weak (, ) estimate. Remark 25. If µ satisfies the doubling condition, then M µ is always of weak type (, ). The following theorem reveals this fact: Theorem 26. Let µ be nonnegative Borel measure in R n satisfying the doubling condition.. Then the noncentral maximal function M µ satisfies the weak (, ) inequality. Proof. Let E λ = {x R n : M µ f(x) > λ}, λ > 0. Then for every x E λ there exists a ball B x containing x such that µ(b x ) < λ B x f (y)dµ(y). 5

26 Let F be the collection of such balls B x s. Then by the Vitali-type covering lemma due to N. Wiener (Theorem 4), there exists an at most countable sub-collection of disjoint balls {B k } such that E λ B F B k 5B k. Then µ(e λ ) k µ(5b k ) < C λ k B k f (y)dµ(y) = C λ k B k f (y)dµ(y) C λ f L (µ). We saw that, for n =, M µ always maps L (dµ) into L, (dµ), no matter what µ is. In R n, and if µ is a doubling measure, then M µ is of weak type (, ). For n = 2, we saw that the maximal operator associated with the measure dµ(x) = e x 2 /2 dx does not have the same boundedness property. So, there are two questions that arise from these observations: () Is there any non-doubling measure µ in R n, n >, such that M µ is weak (, )? (2) How can we know whether or not a measure µ provides a weak type (, ) operator? For certain kinds of measures, a weaker hypothesis than doubling implies that M µ is weak (, ). The following theorem characterizes the measures µ for which M µ is bounded from L (dµ) to L, (dµ). This answers the second question. Theorem 27. ([53]) Let µ be a rotation invariant and strictly increasing measure on R n which is finite on compact sets. The following assertions are equivalent: (a) M µ : L (dµ) L, (dµ) is bounded. (b) There exists a constant C such that for all r 0a, µ ({a < x < a + 3r/2}) Cµ ({a + r/2 < x < a + 2r}). 6

27 (c) µ is a doubling measure away from the origin, that is, µ (B(x 0, 2s)) Cµ (B(x 0, s)) for all s x 0 /4, with C independent of s and x 0. Example 28. dµ(x) = ( + x α ) dx is doubling away from the origin. So, M µ is of weak type (, ). But if α n, dµ is not a doubling measure. This example provides an affirmative answer to the first question. For the case of a measure dµ(x) = g( x )dx with g monotonic, (b) has even simpler statement: Corollary 29. (A. M. Vargas [53]) Let dµ(x) = g( x )dx be a measure in R n with g monotonic and strictly positive on (0, ). Then M µ is of weak type (, ) if and only if there are some constants c k > 0, k Z and C > 0 such that c k g(r) Cc k for 2 k r 2 k+..4 Non-doubling Measures We are interested in measures weaker than doubling measures. In this section, we introduce non-doubling measures with some examples. Definition 30. We say that a Borel measure µ on a metric space (X, d) satisfies a growth condition if there exists a constant C > 0 and N > 0 such that µ (B(x, r)) Cr N, x X, r > 0. (.9) This inequality is known as a growth condition for µ. Example 3. The following are some examples of non-doubling measures: 7

28 (i) The Lebesgue measure in R n. (ii) The gaussian measure dµ(x) = e x 2 dx in R n. (iii) In general, dµ(x) = w(x)dx in R n with a bounded density w. (iv) Let Q = [, ] [, ] R 2 and I = Q R. Then dµ = χ Q (x, y)dxdy + χ I (x)dx is a non-doubling measure in R 2. In fact, if B is the disk centered at (x, y) Q, y > 0, of radius y, then µ(b) = πy 2 while µ(2b) y. Note that some doubling measures are non-doubling as well. For example, the Lebesgue measure in R n is doubling and non-doubling as well. But every doubling measure is not necessarily non-doubling. The following examples reveals this fact: Example 32. Consider the doubling measure dµ(x) = 2xdx in (R,. ). Let B be a ball with center y > 0 and radius r in R. Then µ(b) = y+r y r 2xdx = 4yr. Clearly µ(b) Cr N is false for any pair of constants C and N because µ(b) can be made as large as possible by taking y > 0 sufficiently large. In this sense, the non-doubling measures are weaker than the doubling measure. 8

29 Chapter 2 Estimates for Riesz Potentials by Maximal Function 2. Introduction In this chapter we generalize the Hedberg type inequalities to a metric apace (X, d, µ) endowed with a Radon measure µ satisfying the following growth condition: For every ball B(x, r) there exists a constant C independent of x and r such that µ(b(x, r)) Cr N, (2.) This allows, in particular, non-doubling measures. Sometimes we shall refer to condition (2.) by saying that the measure µ is N-dimensional. Hedberg inequalities are point-wise estimates of potentials in terms of maximal functions. In the sequel, we will define modified maximal functions which are bounded operators on L p (X, µ) for < p and apply these maximal functions to obtain exponential inequalities involving Riesz potentials. It was V. I. Yudovich [56] in 96 who first announced these estimate. N. S. Trudinger [5] in 967, J. A. Hempel, G. R. Morris and N. S. Trudinger [34] in 970, and R.S. Strichartz [44] in 972 provide generalization and extension of such inequality. The latest known development with correct limiting exponent is due to L. I. Hedberg [28] in

30 2.2 Generalization of the Hedberg Inequalities Definition 33. Let (X, d, µ) be a metric measure space, and let 0 < α < N. The fractional integral I α associated to the measure µ satisfying the growth condition (2.) is defined, for appropriate function f on X as I α f(x) = X f(y) dµ(y). d(x, y) N α This definition makes sense when f is bounded and has bounded support because the function f is locally integrable. This follows from the lemmas that follow the d(x, y) N α next definition. Definition 34. Let (X, d) be a metric space, µ be a Borel measure on X and f be a locally integrable function on X. The maximal function of f, M(f), is defined by M(f)(x) = sup r>0 f(y) dµ(y). µ(b(x, r)) B(x,r) The fractional maximal function of f is defined for 0 < α < N by M α (f)(x) = sup r>0 µ(b(x, r)) N α N B(x,r) f(y) dµ(y). Theorem 35. In an Euclidean space R n there exists a constant C such that M α (f) CI α (f) for all f 0 where f(x y)dy I α f(x) =. R y n n α Then, M α maps L p to L q whenever I α does. Proof. Without loss of generality, we may assume that f 0. For y t and 0 α < n it follows that ( ) n α t = t n α y α n. y 20

31 Therefore, (ν n t n ) n α n y t f(x y) dy = (ν n t n ) n α n ν n α n n y t I α (f)(x). f(x y) y n+α t n α dy Now taking supremum over all t > 0 yields M α (f)(x) CI α (f)(x) for every x and for every f 0. Therefore, M α (f) CI α (f) for every f 0. Next, suppose that I α maps L p to L q. This means that I α (f) L q C f L p f L p. Then for any f L p, ( M α (f) L p = M α (f)(x) p dx R ( n CI α (f)(x) p dx R n = C I α (f) L p ) p ) p C f L q. Therefore, M α maps L p to L q whenever I α does. Now we state the following two classical theorems from Harmonic Analysis: Theorem 36 (Hardy-Littlewood-Weiner Maximal Theorem). For every < p M : L p (R n, R) L p (R n, R) is continuous. That is Mf L p (R n,r) C f L p (R n,r) For p = the Hardy-Littlewood maximal operator M is weak type (,). 2

32 Theorem 37 ( Hardy-Littlewood-Sobolev Fractional Integral Theorem ). If < p < q <, q = p α n and 0 < α < n then I α : L p (R n, R) L q (R n, R) is continuous. That is, there exists a constant C > 0 such that for every f L p (R n, R) I α f L q (R n,r) C f L p (R n,r). These two classical theorems stated above are linked by an inequality due to L. R. Hedberg [28] which states that I α f(x) A p,q (α)[mf(x)] p/q f L p (R n,r) p/q, x R n (2.2) for every f L p (R n, R) where p < q <, q = p α n and the constant A p,q(α) is independent of f. This was first introduced by L. R. Hedberg in 972. D. R. Adams in 975 [2] extends the Hedberg inequality for a fractional maximal operator. This is summarized in the following theorem: Theorem 38. [2] Let α > 0, < p < n α, α q be such that r = p α n + αp nq. Then there exists a constant C > 0 (depending on the previous parameters) such that for all positive functions f we have I α f(x) CM n/p (f)(x) αp αp n M0 (f)(x) n. This yields I α (f) L r C M n/p (f) αp n L q f αp n L (2.3) p Proof. For f 0, set I α (f)(x) = f(x y) y n+α dy R n = f(y) x y n+α dy R n =: I + II, 22

33 where, I = II = x y δ x y >δ f(y) dy, x y n α f(y) dy x y n α For every k Z define a k (x) = {y : 2 k δ x y < 2 k+ δ}. Then, I = k= k= a k (x) f(y) x y n+α dy 2 k δ x y <2 k+ δ (2 k δ) n+α k= f(y) x y n+α dy f(y)dy. x y <2 k+ δ (2 k δ) n+α (2 k+ δ) n M 0 f(x). k= = Cδ α M 0 (f)(x), since 0 < α < n. Similarly, 23

34 II = = k=0 k=0 a k (x) f(y) x y n+α dy 2 k δ x y <2 k+ δ (2 k δ) α n k=0 x y <2 k+ δ f(y) x y n+α dy f(y) dy (2 k δ) α n (2 k+ δ) n n p (2 k+ δ) (n n p ) k=0 = 2 n n p δ α n p k=0 Cδ α n p M n p (f)(x). 2 k(α n p ) M n p (f)(x) x y <2 k+ δ f(y) dy Thus, I α (f)(x) C To minimize this expression, we choose ( δ α M 0 (f)(x) + δ α n p M n p (f)(x) ). Then we obtain, δ = [ M n p (f) M 0 (f) ] p n. αp I α (f)(x) CM n p (f)αp n M0 (f) n. We have r = p α n + αp nq, which yields rαp r(n αp) + =. nq pn ( ) nq This means that rαp, pn is a conjugate pair. Using Hölder s inequality for this r(n αp) conjugate pair, we get 24

35 I α (f) r L r C C M n R n M n = C M n p p (f) rαp n (f)rαp n p rαp n (f) L q αp r( (x)m0 (f) n ) (x)dx L nq M αp r( rαp 0(f) M 0(f) αp r( n ) L. p n ) pn L r(n αp) Therefore, I α (f) L r C M n p αp n (f) L q M 0(f) αp ( n ) L. p We note that M 0 is the classical maximal function M which maps L p to itself. Hence, I α (f) L r C M n p αp n (f) αp n Lq f ( ) L. p Note that since M n/p f C(M n f p ) /p the inequality (2.3) becomes the familiar Sobolev inequality when q =. M. Martin and P. Szeptycky in 997 [32] give a generalization of the Hedberg inequality. We summarize this in the following theorem. Let k : R n R be defined by k(tx) = t κn k(x), x R n \ {0}, 0 < κ < and t (0, ). The convolution operator T associated with k is defined as T f(x) = k f(x) = k(y)f(x y)dy, x R n. R n The maximal function associated to k is defined as Mf(x) = sup f(x y) dy, x R n t>0 vol(tx) where tx X ={x R n \ {0} : k(x) } {0} tx ={tx : x X}, t (0, ). It is assumed that vol.(tx) 0. We observe that T and M become the classical Riesz potential I α and the classical maximal function M respectively by taking k(x) = x α n where 0 < α < n. The following theorem is the generalization of the Hedberg inequality: 25

36 Theorem 39. [32] Suppose that p < ( κ). Then T f(x) A (Mf(x)) ( κ)p f ( κ)p p, (2.4) for every f L p (R n ) and almost all x R n, with A = A(k, p) = [ ] κ κp κ ( κ)p vol(x). (2.5) Moreover the inequality (2.4) is sharp. Note that the Hedberg inequality (2.2) follows from (2.4) by taking q = p ( κ). The best value of A p.q in (2.2) follows from (2.5) which is A p,q = q [ pq q + p vol.(b )] p +q n. (2.6) q p p Theorem 39 thus provides an improvement and generalization of the Hedberg inequality (2.2). The improvement amounts to determining the best value of the constant A p,q (α), and the generalization is achieved by replacing the kernel function k α with a broader class of homogeneous kernels, and providing the sharp form of the inequalities derived for these kernels. M. Martin and P. Szeptycki [33] in 2004 completely characterize the kernel function on R n with the property that the associated convolution operators are controlled by certain maximal operators, in a way similar to Hedberg s inequality (2.2). Let X and Y be two metric measure spaces equipped with positive Borel measure. Let K : X Y [, ] be a measurable function on X Y. Let T K denote the integral operator associated to K given by T K f(x) = Y K(x, y)f(y)dy, x X and defined for measurable function f : Y R such that the integral above exists for almost all x X. Let Σ = {(x, y) X Y : K(x, y) = } and assume that Σ has measure 0 with respect to the product measure on the product measure space X Y. Now for x X and 0 < t < define the set Ω[x, t] = {y Y : 26

37 K(x, y) t} and let ω : (0, ) : [0, ] be the function defined by ω(x, t) = measure Ω[x, t]. That is ω(x,.) : (0, ) [0, ] is the distribution function corresponding to the measurable function K(x,.) : Y R. We always assume that ω(x, t) <. For every measurable function f : Y R we associate its maximal function m K on X by setting m K (x) = sup t>0 f(y)dy, x X. ω(x, t) Ω[x,t] We set f(y) dy = 0 if Ω[x, t] = 0. Now we state the theorem due to M. Martin Ω[x,t] ω(x, t) and P. Szeptycki [33] which provides Hedberg s inequality in its most general form. Theorem 40. [33] Suppose < κ <. The following two statements are equivalent: (i) If p < and q = p + κ, then for every x X there exists a positive constant A p,q (K, x) such that T K f(x) A p,q (K, x)[m K f(x)] p/q q f p, (2.7) for every f L p (Y, R). (ii) There exists a measurable function λ : [0, ] such that ω K (x, t) λ(x)t κ, x X, 0 < t <. (2.8) Whenever (i) or (ii) is true, and λ(x) in (2.8) is defined as a possible value of A p,q (K, x) in (2.7) is given by A p,q (K, x) = λ(x) = sup t κ ω(x, t), x X, (2.9) t>0 q [ pq q + p.λ(x)] p +q. (2.0) q p p Moreover, at points x X where (2.8) is an equality for all t (0, ), the value of A p,q (K, x) in (2.0) is the best constant in inequality (2.7). 27

38 The above discussion is a short description of the improvement and generalization of Hedberg s inequality. Next follows the generalization of the Hedberg type inequalities presented in Chapter 3 of [4] to a metric measure space endowed with a Radon measure satisfying the growth condition (2.). For this we start with the following two useful lemmas. Lemma 4. For every γ > 0 where C is a constant. Proof. Suppose N γ. condition (2.) we obtain B(x,r) d(x, y) N γ dµ(y) Crγ, Then d(x, y) r implies that B(x,r) Next suppose that γ < N and let d(x, y) N γ dµ(y) rγ N Cr N = Cr γ. d(x, y) N γ rγ N. Using the B k (x) = {y : 2 k r d(x, y) < 2 k r}, k =, 2, 3,... Then B(x, r) = k=0 B k(x) a disjoint union. Also note that B k B(x, 2 k r) for every k = 0,, 2... Now using these remarks and the condition (2.), we obtain B(x,r) d(x, y) dµ(y) = dµ(y) N γ k=0 B k d(x, y) N γ (2 k r) µ(b(x, N γ 2 k r)) k=0 k=0 2 (k+)(n γ) r N γ C(2 k r) N ( ) 2 N =C r γ = Cr γ 2 γ We also have 28

39 Lemma 42. For every γ > 0 where C is a constant. X\B(x,r) d(x, y) N+γ dµ(y) Cr γ, Proof. Let B k (x) = {y : 2 k r d(x, y) < 2 k+ r}, k = 0,, 2,.... Then B(x, r) = k=0 B k(x). Then X\B(x,r) d(x, y) dµ(y) = N+γ C =C k=0 B k (x) dµ(y) d(x, y) N+γ µ ( B(x, 2 k+ r) ) k=0 (2 k r) N+γ (2 k+ r) N (2 k r) N+γ 2 γk r γ = Cr γ k=0 k=0 Lemma 43. Let µ be a measure on a metric space (X, d) which satisfies the growth condition (2.) and let f be a function on X which is either a nonnegative measurable function or f L loc (X). Let 0 < α < N. Then, (a) d(x,y)<δ f(y) d(x, y) dµ(y) = (N α) ( δ ) f(y)dµ(y) N α 0 B(x,r) dr+ rn α+ δ f(y)dµ(y). N α B(x,δ) (b) log d(x,y)<δ d(x, y) f(y)dµ(y) = ( δ ) dr ( ) f(y)dµ(y) 0 B(x,r) r + f(y)dµ(y) log( B(x,δ) δ ) (c) d(x,y) δ f(y) d(x, y) dµ(y) = (N α) ( ) f(y)dµ(y) N α δ B(x,r) dr rn α+ δ f(y)dµ(y). N α B(x,δ) 29

40 Proof. (a) Using Fubini, we get δ ( ) (N α) f(y)dµ(y) 0 B(x,r) r ( δ = (N α) f(y) (b) Using Fubini, we get ( δ = = = 0 B(x,δ) B(x,δ) B(x,δ) (c) Using Fubini, we get ( (N α) =(N α) =(N α) = B(x,δ) δ B(x,δ) B(x,δ) = = B(x,δ) B(x,δ) B(x,δ) ( f(y) f(y)dµ(y) B(x,r) ( δ ( log log d(x,y) f(y)dµ(y) B(x,r) ( δ 0 dr r N α+ [ N α. f(y)dµ(y) + δn α N α+ dr dr d(x,y) r N α+ d(x, y) N α f(y)dµ(y) d(x, y) N α δ N α ) dr r ) dr f(y)dµ(y) r d(x, y) log δ δ N α ) f(y)dµ(y) d(x, y) f(y)dµ(y) log δ ) r ) f(y)dµ(y) + (N α) r N α B(x,δ) c ] δ N α+ dr B(x,δ) B(x,δ) B(x,δ) c f(y)dµ(y) + (N α) f(y)dµ(y) d(x, y) N α. ) dµ(y) ) dµ(y) f(y)dµ(y) f(y)dµ(y). ( B(x,δ) c d(x,y) ) dr f(y)dµ(y) r N α+ [ N α. r N α ] f(y)dµ(y) d(x,y) Proposition 44. Let µ be a measure on a metric space (X, d) which satisfies the growth condition (2.). For 0 < α < N, p <, there exists a constant A = A(α, p, N) such that for any measurable function f 0 and x X (a) I α f(x) A f αp/n αp p Mf(x) N p < N α. 30

41 (b) I αθ f(x) A (I α f(x)) θ Mf(x) θ 0 < θ <. (c) I αθ f(x) AM α f(x) θ Mf(x) θ 0 < θ <. Proof. (a) We claim that () d(x,y)<δ (2) d(x,y) δ f(y)dµ(y) d(x, y) N α Aδα Mf(x). f(y)dµ(y) d(x, y) N α Aδα N/p f L p. Proof of (). Using the part (a) of Lemma 43, we obtain f(y)dµ(y) δ ( ) =(N α) f(y)dµ(y) dr + d(x,y)<δ d(x, y) N α 0 B(x,r) rn α+ (N α)mf(x) (N α)mf(x) =(N α)mf(x) δ 0 δ 0 δ 0 δ N α B(x,δ) µ(b(x, r)) dr + Mf(x)µ(B(x, δ)) r N α+ δn α cr N r + N α+ Mf(x)CδN δn α r α dr + Cδ α Mf(x) =C( N α α )Mf(x)δα + Cδ α Mf(x) ( ) N α =CMf(x)δ α + α =Aδ α Mf(x). p Proof of (2). We observe that (N α) p = N + inequality and Lemma 42, we obtain f(y)dµ(y) d(x, y) N α d(x,y) δ f p ( d(x,y) δ f p ( d(x,y) δ = f p Cδ ( N p α) =Aδ α N/p f p. f(y)dµ(y) ( N ) p p α. Now, using Hölder s p ) p p dµ(y) d(x, y) (N α) p p dµ(y) d(x, y) N+( N p α) p p ) p p 3

42 Now, using the claims () and (2) we obtain f(y) dµ(y) I α f(x) = X d(x, y) N α f(y) dµ(y) = d(x, y) + N α Then by taking δ = δ(x) = Proof of (b): d(x,y)<δ d(x,y) δ Aδ α M(f)(x) + Aδ α N/p f p. ( ) p/n f p we obtain, Mf(x) f(y) dµ(y) d(x, y) N α ( ) αp/n ( ) p N f p f p N (α p ) I α f(x) A + A f p M(f)(x) M(f)(x) =A f αp/n p M(f)(x) αp/n + A f αp/n p M(f)(x) αp/n =C f αp/n p M(f)(x) αp/n Note that 0 < αθ < N for 0 < α < N and 0 < θ <. So we can replace α by αθ in claim () of part (a) in Proposition 44 and get f(y)dµ(y) d(x, y) N αθ Aδαθ M(f)(x). (2.) Note that d(x,y)<δ So, d(x, y) δ implies that ( ) α αθ d(x, y) N αθ = (N α) + (α αθ). Now, using the inequalities (2.) and (2.2), we obtain f(y)dµ(y) I αθ f(x) = X d(x, y) N αθ f(y)dµ(y) = d(x,y) δ d(x, y) + N αθ δ αθ α f(y)dµ(y) d(x, y) + N α d(x,y) δ ( ) α αθ = δ αθ α. (2.2) δ d(x,y)<δ A ( δ αθ α I α f(x) + δ αθ Mf(x) ) 32 f(y)dµ(y) d(x, y) N αθ d(x,y)<δ f(y)dµ(y) d(x, y) N αθ

43 ( ) /α Iα f(x) Now we choose δ = and obtain, Mf(x) f(y) dµ(y) I αθ f(x) = = A d(x, y) n αθ ( ) θ Iα f(x) I α f(x) + A M(f)(x) X A (I α f(x)) θ M(f)(x) θ ( ) θ Iα f(x) M α (f)(x) M(f)(x) Proof of (c): We claim that d(x,y) δ f(y)dµ(y) d(x, y) N αθ Aδαθ α M α f(x). Proof of the Claim: d(x,y) δ f(y)dµ(y) d(x, y) N αθ f(y)dµ(y) = d(x,y) δ d(x, y) (N α)+(α αθ) δ αθ α δ α N f(y)dµ(y) d(x,y) δ = δ αθ α δ α N µ(b(x, δ)) N α N Aδ αθ α M α (f)(x). µ(b(x, δ)) N α N d(x,y) δ f(y)dµ(y) We also have, Now taking d(x,y)<δ we obtain the desired inequality. f(y)dµ(y) d(x, y) N αθ Aδαθ M(f)(x). δ = ( ) /α Mα (f)(x) M(f)(x) 33

44 2.3 Modified Centered Maximal Function Definition 45. Set (X, d) be a metric space endowed with a Radon measure µ such that µ(b) > 0 for all ball B with positive radius. Such balls are called non-degenerate. We define the k times modified centered Hardy-Littlewood maximal operator as follows: M k f(x) = sup r>0 f(y) dµ(y). µ (B(x, kr)) B(x,r) Lemma 46. Let µ be a Radon measure which satisfies the growth condition (2.).Then for any measurable function f on X d(x,y)<δ f(y)dµ(y) d(x, y) N α Aδα M k f(x). Proof. Without loss of generality, we may assume that f 0. By part (a) of lemma 43, f(y)dµ(y) d(x,y)<δ d(x, y) N α δ ( ) = (N α) f(y)dµ(y) dr + 0 B(x,r) rn α+ δ δ N α B(x,δ) f(y)dµ(y). = (N α) µ (B(x, kr)) M k f(x). dr + 0 rn α+ δ µ (B(x, kδ)) M kf(x). N α δ (N α)ck N M k f(x) r N o N α + dr + δ N α CkN δ N M k f(x). ( ) N α = Ck N δ α M k f(x) +. α = Aδ α M k f(x). Lemma 47. Let µ be a Radon measure which satisfies the growth condition (2.).Then d(x,y) δ f(y)dµ(y) d(x, y) N α Aδα N/p f L p (X,µ). 34

45 p Proof. Note that (N α) p = N + Lemma 4, we get d(x,y) δ ( ) N p α ( f(y)dµ(y) d(x, y) f N α L p (X,µ) d(x,y) δ ( f L p (X,µ) p.. Using the Hölder s inequality and p d(x,y) δ f L p (X,µ)Cδ ( N p α) = Aδ α N/p f L p (X,µ). ) p p dµ(y) d(x, y) (N α)( p p ) dµ(y) d(x, y) N+(N/p α) p p ) p p. Proposition 48. For 0 < α < N, p <, there exists a constant A = A(α, p, N) such that for any measurable function f 0 and x X where p < N α. I α f(x) A f αp/n p M k f(x) αp/n Proof. The proof follows from the above two lemmas and by taking δ = δ(x) = ( ) p/n f p. M k f(x) Lemma 49. Let (X, d, µ) be a metric measure space and let W N i= B(x i, r i ). Then there exists a set S {, 2, 3,..., N} such that (a) the balls {B(x i, r i ) : i S} are disjoint, and (b) W i S B(x i, 3r i ). Proof. Without loss of generality, we can reorder the balls B i = B(x i, r i ) in such a way that r r 2 r 3... r N. Let i = and consider the ball B(x i, r i ). We discard all B j that intersect B i. Let i 2 be the smallest index among the remaining balls. That is B(x i2, r i2 ) is the largest ball disjoint from B(x i, r i ). Now we discard all B j with j > i 2 that intersect 35

46 B i2. Let B i3 be the first of the remaining balls. We proceed as long as possible. The process stops after a finite number of steps, say t, and gives S = {i, i 2,..., i t }. Part (a) follows from the way the balls have been constructed. Next we show that part (b) holds. For this let x W. Then x B(x j, r j ) for some j {, 2, 3,..., N}. If x B(x j, r j ) is one of B(x ik, r ik ) for some i k S then we are done. Suppose B(x j, r j ) B(x ik, r ik ) for every i k S. Then B(x j, r j ) is one of the balls B j discarded at some stage. That is B(x j, r j ) B(x ik, r ik ) φ for some i k S with r ik r j. Then B(x j, r j ) B(x ik, 3r ik ). That is x B(x ik, 3r ik ) for some i k S. This proves part (b). Theorem 50. Let (X, d) be a metric space endowed with a Radon measure µ. Then the modified 3-times Hardy Littlewood maximal operator M 3 defined by is weak-(,). Namely, for every f L (X, µ). M 3 f(x) = sup r>0 µ ({x : M 3 f(x) > λ}) λ f(y) dµ(y) µ(b(x, 3r)) B(x,r) B(x,r) f(y) dµ(y) Proof. Let E λ = {x : M 3 f(x) > λ}. We first prove that the set E λ is open. Choose x 0 E λ. Then there exists r > 0 such that f(y) dµ(y) > λ. µ(b(x 0, 3r)) B(x 0,r) By the absolute continuity of the integral, there exists a compact set K B(x 0, r) such that f(y) dµ(y) > λ. µ(b(x 0, 3r)) K 36

47 If we take δ > 0 sufficiently small, then for any y satisfying y x 0 < δ, it holds that K B(y, r) and that λ < µ(b(y, 3r)) Therefore the set E λ is an open set. K f(y) dµ(y) f(y) dµ(y). µ(b(y, 3r)) B(y,r) Let K be a compact set and K B(x, r) such that µ (B(x, r) \ K) < δ. On the other hand, given ɛ > 0 there exists δ > 0 such that for every measurable set E µ(e) < δ implies that f(y)dµ(y) < ɛ. Therefore, B(x,r)\K E f(y)dµ(y) < ɛ. (2.3) Since K B(x, r) there exists η > 0 such that d (K, B(x, r) c ) > η. Pick a z X such that d(z, x) < η. Now we claim that 2 K B(z, r η ) B(x, r). 2 Note that K n= B(x, r ) and K is compact. So there exists a natural number N n such that K B(x, r ). Without loss of generality, we may choose η sufficiently small N so that N < η. Thus, K B(x, r N ) B(z, r η ) and therefore our claim follows. 2 Now, using these inclusions and the inequality (2.3), it follows that M 3 f(z) µ (B(z, 3(r η ) 2 )) η f(y) dµ(y) B(z,r 2 ) [ f(y)dµ(y) f(y)dµ(y) + µ (B(x, 3r)) K B(x,r) = f(y) dµ(y) + µ (B(x, 3r)) B(x,r)\K µ (B(x, 3r)) ɛ > µ (B(x, 3r)) + λ. B(x,r) B(x,r) ] f(y) dµ(y) f(y) dµ(y) This is true for every ɛ > 0. Therefore M 3 f(z) > λ. This proves that M 3 is lower semicontinuous and therefore the set E λ is open. Finally, we prove that µ is weak-(,). Because 37

48 µ is Radon measure and E λ is open µ(e λ ) = sup{µ(k) : K E, Kis compact}. Let K E λ and K be compact. Then for every x K there exists r x > 0 such that f(y) dµ(y) > λ. µ (B(x, 3r x )) B(x,r x) Then the collection {B(x, r x ) : x K} is an open covering of K. So there exists a finite subcover B(x j, r j ) j =, 2, 3... N which also cover K. That is K j j= B(x j, r j ). By above lemma there exists S {, 2, 3,..., N} such that the collection of the ball {B(x t, r t ) : t S} is disjoint and K t S B(x t, 3r t ). Then µ(k) µ (B(x t, 3r t )) t S f(y) dµ(y) λ t S B(x t,r t) λ f L (X,µ). With stronger hypothesis on the metric measure space (X, µ) a little more can be said. The following theorem on nonhomogeneous space is due to Y. Terasawa. Theorem 5. Let X be a metric space possessing a non-degenerate Radon measure µ such that µ(b(x, r)) is continuous in the variable r > 0 when x X is fixed. Then M k f(x) = sup r>0 f(y) dµ(y) is weak-(, ) bounded with constant when k 2. B(x,r) µ (B(x, kr)) Namely, µ ({x : M k f(x) > λ}) f(y) dµ(y) λ for any f L (X, µ) when k X

49 Remark 52. Obviously, M k : L (X, µ) L (X, µ).we know from above theorem that M k (k 2) is weak-(, ). Therefore by the Marcinkiewicz interpolation theorem it follows that M k : L p (X, µ) L p (X, µ) is continuous for k 2 and < p, that is, there exists a constant C = C(p, k) such that M k (f) L p (X,µ) C f L p (X,µ). Theorem 53. Let X be a metric space, 0 < α < N, < p < q <, and /q = /p α/n where N is the dimension of the growth condition (2.). Then there exists a constant A = A(p, k) such that I α (f) L q (X,µ) A f L p (X,µ). ( Proof. We observe that q αp ) = p. Then by using Proposition 48 and the Remark 52 N we obtain, I α (f) q L = q X X =A q I α (f)(x) q dµ(x) A f αp/n p M k f(x) αp/n q dµ(x) X =A q f αpq N L p f qαp/n p X =A q f αpq N L M kf p p L p CA q αp Nq f L p f p L p =CA q f q p L p f p L p =CA f q L p. M k f(x) q( αp/n) dµ(x) M k (f)(x) p dµ(x) Therefore the theorem follows. Proposition 54. Let (X, d) be a metric space and µ be a Radon measure on X with µ (B(x, r)) Cr N, for every r > 0, x X, p <, αp = N. 39