CALDERÓN ZYGMUND THEORY RELATED TO POINCARÉ-SOBOLEV INEQUALITIES, FRACTIONAL INTEGRALS AND SINGULAR INTEGRAL OPERATORS

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1 Function spaces, nonlinear analysis and applications, Lectures notes of the spring lectures in analysis. Editors: Jaroslav Lukes and Lubos Pick, Charles University and Academy of Sciences, (999), 3-94, ISBN: CALDERÓN ZYGMUND THEORY RELATED TO POINCARÉ-SOBOLEV INEUALITIES, FRACTIONAL INTEGRALS AND SINGULAR INTEGRAL OPERATORS Carlos Pérez Universidad de Sevilla Paseky s lecture notes

2 Contents A brief introduction to the A p theory of weights 4 2 L p properties of functions with controlled oscillation 9 2. BMO and Hölder-Lipschitz spaces Poincaré Sobolev Inequalities Fractional Integrals and Poincaré inequalities L p self improving property Proof of the Poincaré-Sobolev inequalities Rearrangements Rearrangements A basic covering lemma Proof of the main Theorem Orlicz maximal functions Orlicz spaces Orlicz maximal functions Fractional integral operators Weak implies strong Step : Discretization of the potential operator Step 2: Replacing dyadic cubes by dyadic Calderón Zygmund cubes Step 3: Patching the pieces together Singular Integral Operators Introduction: Singular Integral Operators and smoothness of functions Weighted L p estimates Banach function spaces Commutators: singular integrals with higher degree of singularity

3 7 Appendix I: The Calderón Zygmund covering lemma 58 8 Appendix II: The A condition of Muckenhoupt 62 9 Appendix III: Exponential self-improving property 67 3

4 Chapter A brief introduction to the A p theory of weights The Hardy-Littlewood maximal function is the operator defined by Mf(x) = sup f(y) dy x where the supremum is taken over all the cubes containing x. By cube we always mean a cube with sides parallel to the axes. There are variations of this definition such as replacing cubes by balls or just considering cubes or balls centered at x but all of them are equivalent with dimensional constants. Maximal functions arise very naturally in analysis, for proving theorems about the existence almost everywhere of limits, for controlling pointiwise important objects such as the Poisson Integrals or for controlling, not pointiwise but at least in average, other basic operators such as singular integral operators. The model example of existence almost everywhere of limits is the Lebesgue differentiation theorem: f(x) = lim r 0 B(x, r) B(x,r) f(y) dy Indeed, as it is well known this follows from the following weak type estimate: Theorem.0. (Hardy Littlewood Wiener) There exists a finite constant C such that for each positive λ the following inequality holds, {x R n : Mf(x) > λ} C f(x) dx (.) λ R n As a consequence of (.) we can derive: Corollary.0.2 Let < p < then there exists a constant C such that (Mf) p dx C f p dx (.2) R n R n 4

5 We say that w is a weight if w is a a.e. postive locally integrable function in R n. If E is any measurable set we denote w(e) = E w. Theorem.0.3 a) There exists a constant C such that for all λ and f w({x R n : Mf(x) > λ}) C f(x) Mw(x)dx. (.3) λ R n b) As a consequence if < p < there exists a constant C such that for all f (Mf) p w(x)dx C R n f p Mw(x)dx. R n (.4) We may think of this inequality as a kind of duality for the (nonlinear) operator. It can also be seen as an antecedent of the A p theory weights. It proved by C. Fefferman and E. Stein in [FS] to derive a vector valued extension of (.0.2). Indeed, for a vector of functions f = {f i } i and for q > 0 the operator M q is defined by ( ) /q M q f(x) = (Mf i (x)) q. i= This operator can also be seen as a generalization of the classical integral of Marcinkiewicz. Then if we denote f(x) q = ( i= f i(x) q ) /q = f(x) l q we have for < p < and < q M q f(x) p dx C f(x) p q dx. (.5) R n R n A corresponding weak type (, ) also holds in the range < p <. These estimates play an important role in Harmonic Analysis, specially in the theory of Littlewood-Paley. To some extent these operators behave more as a singular integral (cf. Chapter 6) than as a maximal function. This is very well reflected in the work [RRT] where M q f, as well many other operators, is seen as a vector valued singular integral operator. Proof: The proof of b) follows from the Marcinkiewicz interpolation theorem since M : L (u) L (v) for arbitrary weights u and v. (see also the proof of (4.6) in Lemma 4.2.2). The proof of a) is based on the classical Vitali covering lemma: Lemma.0.4 (Vitali) Let F = { i } i=,,n be a finite family of cubes (or balls) in R n. Then we can extract from F a sequence of pairwise disjoint cubes F = { j } j=,,m such that m 5 j. F j= 5

6 See [Ma] p.23. Now if we let Ω λ = {x R n : Mf(x) > λ} for a given λ and let K to be any compact subset contained in Ω λ. Let x K then by defintion of the maximal function there is a cube = x containg x such that f(y) dy > λ. (.6) Then K x K x and by compactness we can extract a finite family of cubes F = {} such that K F and where each cube satisfies (.6). Then by the Vitali lemma we can extract a pairwise disjoint family of cubes { j } M j= such that F M j= 5 j. Then C λ w(k) M j= M w(5 j ) λ j= w(5 j ) 5 j M j= f(y) dy C j λ C λ w(5 j ) f(y) dy j j M j= R n f(y) Mw(y)dy. j f(y) Mw(y)dy The main breakthrough in the theory arrived in the early seventies with the discovery by B. Muckenhoupt in [Mu] of the right class of weights for which the Hardy-Littlewood maximal function is bounded on L p (w), p >. This result opened up the possibility of studying operators with higher singularity. Definition.0.5 A weight w belongs to the class A p, < p <, if there is a constant K such that ( ) ( p w(y) dy w(y) dy) p K (.7) for each cube. A weight w belongs to the class A if there is a constant K such that w(y) dy K inf w. (.8) We will denote the infimum of the constants K by [w] Ap. Observe that [w] Ap by Jensen s inequality. Sometimes it is much more convenient to use the following equivalent condition for the A weight: there is a constant K such that for each x Mw(x) K w(x) 6

7 Observe that it follows from inequality.3 that if w is an A weight then w({x R n : Mf(x) > λ}) C f(x) w(x)dx (.9) λ R n It is not hard to see that this condition is also equivalent to saying that w A. Theorem.0.6 (Muckenhoupt) Let p >. The following conditions are equivalent: a) w A p ; b) there exists a constant K such that for each cube and nonnegative function f ( p f(y) dy) w() K f(y) p w(y)dy c) M : L p (w) L p, (w) d) M : L p (w) L p (w) Conditions a), b) and c) are also equivalent when p =. The main example of A weights are provided by the following lemma of Coifman and Rochberg (cf. [GCRdF] p. 58). Lemma.0.7 Let µ be a positive Borel measure such that Mµ(x) <, a.e. x R n. Then for each 0 δ < the function (Mµ) δ A In fact it is also shown in that paper that any A weight w can be written essentially of this form. Furthermore, it is possible to show that w A p if and only if w = w w p 2 where w and w 2 are A weights. That the last condition was a A p was already known by B. Muckenhoupt, but it was P. Jones who proved the necessity of the factorization theorem with a very difficult proof. The modern approach uses completely different arguments. It is due to J. L. Rubio de Francia and it is presented in [GCRdF] where we remit the reader for more information about the A p theory of weights. Definition.0.8 The A class of weights is defined in a natural way by A = p> A p. 7

8 The A class of weights shares a lot of interesting properties. We present in Appendix 8 several characterizations. Some of them will be used along these notes. Throughout these notes all notation is standard or will be defined as needed. All cubes are assumed to have their sides parallel to the co-ordinate axes. Given a cube, l() will denote the length of its sides and for any r > 0, r will denote the cube with the same center as and such that l(r) = rl(). We will denote the collection of all dyadic cubes by D and by D() to the collection of all dyadic cubes relative to the (non necessarily dyadic) cube. (x, r) will denote the cube centered at x and with side length 2r. By weights we will always mean a a.e. positive, locally integrable functions which are positive on a set of positive measure. Given a Lebesgue measurable set E and a weight w, E will denote the Lebesgue measure of E and w(e) = E w dx. Given < p <, p = p/(p ) will denote the conjugate exponent of p. Finally, C will denote a positive constant whose value may change at each appearance. 8

9 Chapter 2 L p properties of functions with controlled oscillation For a locally integrable function f we define the oscillation of f over a cube as the number Osc(, f) = f(y) f dy inf f(y) c dy c R 2 f(y) f(x) dy dx. Osc(, f) can be seen as the modern way of checking how smooth is a function. For instance, this can be seen in the characterization (2.) given below of the Hölder-Lipschitz spaces and also when considering functions in the Sobolev class, see for instance (2.3) and (2.4). But it goes beyond since R. Coifman and R. Rochberg proved in [CR] that maximal functions themselves have some sort of smoothness. They showed that Osc(, log M µ) is bounded by a dimensional constant provided M µ is finite almost everywhere. What this means is that there is a cancellation phenomenon undergoing. Indeed, this result would not make sense if we remove the average f in the definition of Osc(, f). In this chapter we will study properties of classes of functions which have some control on the oscillation. In particular we will emphasize on L p self-improving properties. Roughly speaking we will show that higher integrability properties of functions belonging to B.M.O. (or Hölder-Lipschitz) and the higher integrability of Sobolev functions are esentially the same phenomenon. The results and techniques we present in this chapter are taken from [FPW] and [MP]. These papers are specially motivated by the work P. Hajlasz and P. Koskela [HaK], whose roots got back to the theory of uasiconformal Mappings as can be seen in [HK], but also by the work of L. Saloff Coste in [SC]. 9

10 2. BMO and Hölder-Lipschitz spaces Recall that a function f is said to satisfy the Hölder condition with exponent α, 0 < α <, if there exists a constant C such that for each x and y f(x) f(y) C x y α. When α = the function is said to satisfy the Lipschitz condition. We will denote them by Λ(α) and it is well known how relevant is this condition in Analysis. The main examples of functions in Λ(α) are given by f(x) = x α, 0 < α. From our point view it is more interesting the following characterization due to Campanato and Morrey: Let 0 < α, then f Λ(α) if and only if f(y) f dy Cl() α, (2.) where C is a constant independent of the cube. Observe that Λ(0) does not yield any interesting space, however if we replace in (2.) α by zero we obtain the celebrated Bounded Mean Oscillation space introduced by F. John and L. Nirenberg in the early sixties in their study of partial differential equations. It is usually denoted by B.M.O.: f B.M.O. if there exists a constant C such that for each cube f f C. The main example is log M(µ)(x). If we take µ to be the point mass at the origin we get log x which is the classical example of unbounded B.M.O. function. In any case they share the following self improving property: Let 0 α and let f be a function satisfying (2.). Then for each < p < there exists a constant c such that ( ) /p f f p c l() α. (2.2) 2.2 Poincaré Sobolev Inequalities Poincaré-Sobolev inequalities play a main role in many places in Analysis. They are crucial tools in many important results for Partial Differential Equations, such as the De Giorgi-Nash-Moser theorem, Harnack s inequalty, properties of the Green functions for second order elliptic equations and existence results for semilinear equations (cf. for 0

11 instance some recent books such as [HKM] or citemaly. It is also important in the theory of uasiconformal mappings as can be seen in the recent theory developed by J. Heinonen and P. Koskela [HK]. Very roughly, this theory shows that if X is a proper metric space, i.e. an space where closed balls are compact, endowed with a mesure which is Ahlfors David regular (hence doubling) then we can develop an analogue of the classical theory of uasiconformal mappings if the space admits, in an appropriate sense, a (, p) type Poincaré inequality. The model example is the classical (, ) Poincaré inequality: f f C l() f, (2.3) where C a constant independent of both and f. In fact there is a more general version ot this estimate for p ( ) /p ( ) /p f f p C l() f p. (2.4) In fact we will show in Lemma 2.3. that they are both equivalent as a first step to a more general result in Theorem An important observation is that this inequality is not only uniform on cubes but also on an appropriate class of functions, which may be a Sobolev class of functions or the Lipschitz class since such functions are differentiable almost everywhere by the Rademacher Stepanov theorem. A second example is a sharp version of inequality (2.3) which turns to be equivalent to the isoperimetric inequality: there exists a constant C such that The version corresponding to p is ( ) /n f f n C l() f. (2.5) ( ) /p ( ) /p f f p C l() f p. (2.6) Here p = pn denotes the Sobolev exponent. Observe that n p = exponent from the isoperimetric inequality. n = n n is the We will show below that indeed both inequalities (2.4) and (2.6) are equivalent. In fact we will stress this relationship by introducing weights in these inequalities. These weighted Poincaré inequalities are basic in the study of the smoothness of the solutions of degenerate elliptic equation (see [FKS] and [HKM]).

12 The main example holds for any w A 2 : ( /2 ( /2 f f w) 2 C l() f w) 2. (2.7) w() w() To run the Moser iterative technique the following was derived in [FKS]: there exists a tiny positive constant ɛ for which ( /(2+ɛ) ( /2 f f w) 2+ɛ C l() f w) 2 (2.8) w() w() We will show below that these examples are particular cases of a more general situation. 2.3 Fractional Integrals and Poincaré inequalities It is well known that there is an intimate relationship between smoothness and fractional integration. This is best reflected in the following estimate: let f be a, say C function, on a cube, then there is a universal constant C such that for each x f(x) f C I ( f χ )(x). (2.9) Here I α, 0 < α < n, denotes the fractional integral or Riesz potentials of order α and it is defined by f(y) I α f(x) = n α dy. x y R n We will prove below the well known fact that (2.9) implies (2.3). However what it is new is that in fact these two estimates are equivalent. In fact there is no need to work with the gradient as the following characterization shows. Lemma 2.3. Let α > 0, f a locally integrable function and let g be a nonnegative function. Then the following conditions are equivalent: a) There exists a constant C such that for all cubes : f f C l()α b) There exists a constant C such that for all cubes : f(x) f C I α (gχ )(x) a.e. x (2.0) g, 2

13 c) there is a positive constant C such that for every weight w and for all cubes : f f w C l()α g M(wχ ) d)let p, then there exists a constant C such that for all cubes : ( ) /p ( ) /p f f p C l() α g p. The key part is the classical integral representation formula of Sobolev type (2.0). This formula is also available in different and more general contexts thanks to the work of [FLW2] which we adapt to our situation. Proof: a) b) It is in this implication where we adapt the ideas from following [FLW2]. Fix = 0 and let x and consider the unique one-way inifinite chain of dyadic cubes { k } k=0, k+ k such that {x} = k=0 k. See Appendix I. For each k we let f k = k k f. Then by the Lebesgue differentiation theorem we have a.e. x that f(x) f = lim k f k f = (f k f k+ ) C 2 n k=0 f f k 2 n k+ k+ k=0 l( k ) α k k g = C 2 n k=0 g(y) k=0 k k=0 k f f k l( k ) α χ k k (y) dy. We now use that k (x, l( k )). Observe that the natural estimate k=0 l( k ) α k χ k (y) l( k) α χ k (y) y (x,l(k )) x α n χ (y) (x,l(k )) is not sufficient to sum up the series, but we can proceed as follows: let ɛ be such that 0 < ɛ < n α, then l( k ) α c χ k k (y) = l( k ) n α ɛ l( k ) χ c (y) ɛ k y x n α ɛ l( k ) χ (y), ɛ k k=0 since x, y k. Now pick the integer k 0 such that 2 k 0 l() y x. Then c y x n α ɛ k=0 l( k ) ɛ χ k (y) = k=0 c y x n α ɛ l() ( + ɛ 2ɛ + 2 2ɛ k0ɛ ) 3 y x n α

14 this shows that a.e. for x f(x) f C ɛ,n g(y) dy. y x n α b) c) Indeed by assumption and Fubini s g(y) f(x) f w(x)dx c n α dy w(x)dx = x y w(x) n α dx g(y) dy x y For the inner integral we do the following: if the cube = (z, r) is centered at z and with radius r we have for each y that (y, 2r). Then if we let r 0 = 2r, r k = r k /2, and A(y, k) = (y, r k ) \ (y, r k ), k =, 2, we have (z,r) C w(x) n α dx x y r α k r n k=0 k (y,r k ) (y,2r) w(x) n α dx = x y w(x)dx C Mw(y) k= k= A(y,k) w(x) x y n α dx r α 2 kα = C l()α Mw(y) c) d) The case p = follows by taking w. The case p > follows by a duality argument combined with a good weighted estimate. This is a very fruitful idea as can be seen in the following Chapters. Indeed, we can write ( ) /p f f p = sup{ (f f )h } where the supremum is taken over all functions h such that h p =. Taking one of these h s we have by hypothesis that (f f )h f f h C l() α g M(hχ ) C l() α ( d) a) Just take p = C l() α ( g p ) /p ( g p ) /p ( (M(hχ )) p ) /p ) /p ( h p = C l() α g p ) /p 4

15 2.4 L p self improving property In this section we will show that Lemma 2.3. can be pushed further to get higher L p integrability. The methods we will use are completely different and more related to the Calderón-Zygmund theory. We will be working in a more general framework. We consider functionals a a : (0, ) denotes the family of all cubes from R n. Our model example is associated to the fractional average a() = l()α dν, (2.) where 0 α and ν is a nonnegative measure. Some interesting cases are provided when the measure ν is absolutely continous, with dν = g dx, with g A as we shall see below (for instance a polynomial weight). Motivated by the theory developed in [HK] we can consider a more general case ( ) /p ν(λ) a() = l() α, (2.2) with λ. Observe that these functionals are not necessarily radial, i.e., a need not be of the form a() = ϕ( ) where ϕ(0, ) (0, ) which under certain restrictions on ϕ have appeared in the literature as generalizations of the Lipschitz spaces [Ja]. Now let f be a locally integrable function on R n and let a be a functional. Assume that f satisfies the following estimate f(y) f dy a() (2.3) uniformly on. Our goal is to find a condition on a such that (2.3) implies a L q self improving property of the form ( /q f(y) f dy) q C a(). (2.4) We impose the following discrete condition on the functional a relative to a locally integrable weight function w. To get such an improvement we must impose a mild discrete type condition which reflects the underlying geometry of the space. We expect in the first case that the exponent q will lie on a certain generally open range (, r) where r = r(a) is an exponent which depends upon a mild condition on the functional a. To be more precise we shall assume the following condition. Definition 2.4. Let 0 r < and let w be a weight. We say that the functional a satisfies the (weighted) D r condition if there exists a finite constant c such that for each 5

16 cube and any family of pairwise disjoint dyadic subcubes of, a(p ) r w(p ) C r a() r w(). (2.5) P Observe that the condition is of local nature and it reminds a bit the Carleson condition. It is convenient to denote the best constant C by a. Observe that a and that by Hölder s inequality, the family {D r } is decreasing as r increases, that is, if r < s, D s D r. Hence for a functional a D r we can define the optimal exponent to which it belogs, namely r a = sup{r : a D r }. For instance if we consider the fractional functional a() = l()α we have that r a = n and furthermore a satisfies the (unweighted) D n α r a. Indeed if is an arbitrary family of disjoint cubes we have: a(p ) n n α P = ( dν(y)) n n α ( dν(y)) n n α P P P ( dν(y)) n n α dν, P P = a() n n α (2.6) If we consider instead the functional (2.2) with λ > it is possible to show (see [FPW]) that with α > 0, p < n/α then a D t for < t < pn/(n αp) and hence r a = pn/(n αp). However, it does not satisfy D ra. On the other hand this is good enough to recover the strong type estimate in part b) of Theorem 2.4.2, but not part a), the optimal weak type endpoint estimate. Observe that pn/(n αp) is a more generally Sobolev exponent involving the index α. It can also be shown that for particular ν a is not in D n n α +ɛ for all ɛ > 0 for a general measure ν. Last observation shows that the class D r does not share in general a self improving or openness property of the sort D r D r+ɛ, which may be of interest to get optimal endpoint estimates (see Theorem 2.4.2). In some cases this openness property holds. For instance consider a() = α/n w(y) dy, (2.7) with w A. Then one can shows that a D n n α +ɛ where ɛ depends upon the A constant of w. 6

17 On the other hand side there are examples of functionals which satisfy the D r condition for all r. Indeed, this the case of any nondecreasing nonradial case, namely: a(p ) a(), P. In fact it belongs to r> D r. An example of this situation is given by a() = π(y) dy, (2.8) where π is a polynomial or more generality any weight satisfying the reverse Hölder property of order infinity. Nondecreasing functionals are special as can be seen in Appendix III. Indeed if f satisfies (2.3) with nondecraesing a then f is at least of exponential type including the John and Nirenberg classical theorem. In fact it is shown in [MP2] (see also Appendix III) that the space B.M.O., or more generally the spaces defined by these nondecreasing functionals, can be seen as limiting of certain appropriate generalized Trudinger inequalities. In next result we will use the following notation g L r, () = sup t>0 ( ) /r {x : g(x) > t} t, for the normalized Marcinkiewicz norm. Similarly, whenever a weight w is involved we have ( ) /r w({x : g(x) > t}) g L r, (,w) = sup t. t>0 w() Theorem Let w be an A weight and let a be a functional satisfying the (weighted) D r condition (2.5) for some r > 0. Let f be a locally integrable function such that f f a(), (2.9) for every cube. Then a) There exists a constant C such that for all the cubes in R n f f L r, (,w) C a(). (2.20) b) Let p with 0 < p < r. Then, as a consequence of a) there exists a constant C such that for all the cubes in R n we have ( /p f(y) f w(y)dy) p C a(). (2.2) w() Remark We emphasize that the natural exponent p = r is not attained in the strong inequality (2.2). In fact it is an open question whether the D r condition is sufficient to assure the strong L r result. Under certain extra condition the answer is yes as the next corollary shows where a very special selfimproving functional is considered. 7

18 Remark It is not hard to see that we can replace the Lebesgue mesure by any doubling measure µ. However, it is much more interesting the fact that we can go beyond and consider more general measures which are nondoubling. Indeed it has been first shown in [MMNO] how to work with measures that do not see hyperplanes parallel to the axes. In particular this is the case of mesures µ satisfying the growth condition µ() C l() α for some positive α. These measures have a growing interest since they arise in the study of the Cauchy transform along curves and its generalizations (see the recent papers [NTV] [Tor]). The A p (µ) theory of weights has been developed in [OP] with applications in the spirit of Theorem and to singular integrals. Remark It is interesting to compare our situation with that in the context of the Campanato-Morrey spaces. These spaces are defined as those functions f satisfying (2.9) with the functional a() = δ, where δ is positive. It is well known that this condition is equivalent to requiring that f C δ. for all cubes. This same equivalence holds when the L norm is replaced by the L p norm. This means that the cancellation does not play any role in these spaces. Indeed there are examples by Piccinini showing that these spaces do not have the self improving properties as derived above. Corollary Let f and g be locally integrable functions and let g be an A weight. Let 0 < α < n, and suppose that f satisfies the following inequality for all cubes: f f C l()α g. (2.22) Then with r = n, n α f f C l()α g. (2.23) L r, () If furthermore the function g also satisfies the A condition then ( ) /r f f r C l()α g. (2.24) Proof of Theorem 2.4.2: Part b) follows from Kolmogorov s inequality: let 0 < q < r we have that for nonnegative g ( /q g(x) wdx) q r ( w() r q )/q g L r, (,w). (2.25) See [GCRdF] p. 485 for instance. We just need to apply part a) to this inequality to g = f f. To prove part a) we fix a cube. Observe that we may assume that f = 0. Hence we have to prove that r w({x : f(x) > t}) t w() 8 C r a() r,

19 with C independent of, and t. We will try to get an appropriate good λ inequality (2.28). Good λ inequalities is an important tool discovered by D. L. Burkholder and R. F. Gundy in [BG] (see also [Tor]) Now, for each t > 0, we let Ω t = {x : Mf(x) > t} where M will denote in this proof the dyadic Hardy Littlewood maximal function relative to (See Appendix I). Then by the Lebesgue differentiation theorem We will asume first that t > a(). Hence {x : f(x) > t} Ω t. t > a() and we can consider the Calderón Zygmund covering lemma of f relative to for these values of t. This yields a collection of dyadic subcubes of, { i }, maximal with respect to inclusion, satisfying Ω t = i i and t < i f i f 2 n t (2.26) for each i. Now let q > a big enough number that will be chosen in a moment. Since Ω q t Ω t, we have w(ω q t ) = w(ω q t Ω t ) = w({x i : Mf(x) > q t}) = i = i w({x i : M(fχ i )(x) > qt}) where the last equation follows by the maximality of each of the cubes i. Indeed, for any of these i s we have Mf(x) = max{ sup f, sup f } P :x P D() P P P :x P D() P P P i P i = sup P :x P D() P i P P f = M(fχ i )(x), since by the maximality of the cubes i when P is dyadic (relative to ) containing i then f t. P On the other hand for arbitrary x, f(x) f(x) f i + f i f(x) f i + i P 9 i f f(x) f i + 2 n t

20 and then for q > 2 n w(ω q t) w(e i ), where E i = {x i : M((f f i )χ i )(x) > (q 2 n )t}. Let ɛ > 0 to be chosen in a moment. We split the family { i } in two: (i) i I if f f i < ɛ t i i or (ii) i II if Then w(ω q t ) i f f i ɛ t i i w(e i ) i I i w(e i ) + i II w(e i ) = I + II. For I we use that M is of weak type (, ) (with constant one) to control the unweighted part: ɛ E i f f (q 2 n i )t i i (q 2 n ) i. Recall that the A condition is characterized by w(e) w() c ( E )ρ, for some c and ρ and independent of and E. Therefore we get for each i I for some positive δ. Then w(e i ) C ɛ δ w( i ) I C ɛ ρ i I w( i ) C ɛ ρ w( i i ) = C ɛ ρ w(ω t ). For II we use the hypothesis on a (2.5) II i II w( i ) i II i II ( ) r f f i w( i ) t ɛ i i a( t r ɛ r i ) r w( i ) a r t r ɛ r a()r w(). Combining all these estimates we have that for q > 2 n, t > a() (qt) r w(ω qt) w() ɛρ q r w(ω t) (q 2 n ) tr w() + a r q r a() r. (2.27) ɛ r However, if we choose ɛ a we have the same inequality holds for t a(). In conclusion we have the following inequality: 20

21 For all q > 2 n, t > 0 and 0 < ɛ a we have (qt) r w(ω qt) w() ɛρ q r w(ω t) (q 2 n ) tr w() + a r q r a() r. (2.28) ɛ r To conclude we use a standard good λ method. For N > 0 we let ϕ(n) = sup t r w(ω t) 0<t<N w(), which is finite since is bounded by N r. Since ϕ is increasing (2.28) clearly implies ϕ(n) ϕ(nq) We now conclude by chosing ɛ small enough such that ɛρ q r q 2 n ϕ(n) + a r q r ɛ r a() r. ɛ ρ q r (q 2 n ) <, and letting N. We conclude this section by pointing out that as a corollary of the proof of above theorem we can deduce a key estimate relating a function and the action of the sharp maximal operator M # on it. This is perhaps the best way of showing the relationship between a function and its smoothness. Recall that the sharp maximal operator M # is an operator introduced by C. Fefferman and E. Stein and it is defined as follows: which is equivalent to sup x M # f(x) = sup inf x c f(y) c dy f(y) f dy sup f(y) f(x) dydx x 2 where as usual the supremum is taken over all the cubes containing x. In order to derive a global result we need to consider spaces on which we can make the global Calderón Zygmund covering lemma(cf. Appendix I): It is easy to see that p L p CZ. CZ = {f L loc(r n ) : f 0 when R n } Theorem Let 0 < p < and let w be a A weight. Then a) Suppose that f be an integrable function on a cube of R n. Then there exists a constant C independent of f such that f f p wdx C [w] p A (M # f)p wdx (2.29) b) Suppose that f CZ. Then there exists a constant C independent of f such that R n f(x) p w(x)dx C [w] p A 2 R n (M # f(x)) p w(x)dx. (2.30)

22 2.5 Proof of the Poincaré-Sobolev inequalities The purpose of this section is to prove the Poincaré-Sobolev that are in presented Section 2.2. The first part of next result is classical (see [EG]). However, our point of view is different and can be extended to other situations such as the case of subelliptic operators [FPW] [MP]. The second part is not that well known. Recall that p = pn denotes the Sobolev exponent of p. n p 22

23 Theorem 2.5. Let p < n. a) Let f be Lipschitz function, then ( ) /p ( ) /p f f p C l() f p. (2.3) b) Let w A p, and let f be a Lipschitz function, then there exists a positive constant δ depending upon the A p constant of w such that ( ) /p(n +δ) ( /p f f p(n +δ) w C l() f w) p (2.32) w() w() Proof/Discussion: a) Define the following functional ( ) /p a(, f) = l() f p, as a function of both and f. Observe that a satisfies the (unweighted) D p condition (see the calculation 2.6 from previous section) with constant equal one, in particular is uniform on f. Hence, we may apply Theorem and get the weak type estimate f f L p, (,w) C a(, f), with constant C independent of and f. What we need to do is to pass from this weak type inequality to the corresponding strong type estimate. To accomplish this, we use some ideas taken from R. Long and F. Nie in [LN]. We remit [FPW] for details and generalizations. The key point is to have a gradient type functional on the right hand side. We will illustrate this in the proof of a (global) sharp two weight isoperimetric inequality Theorem 5.. from Chapter 5. b) We start with the (, ) Poincaré inequality f f C l() f. By the A p condition using part b) of Theorem.0.6 we have that ( /p f f C l() f w) p = a(, f). w() P. Koskela informed me that Long-Nie method is somehow hidden in [Maz], although not in the way presented here; also it has been rediscovered by some other authors [BCLSC] 23

24 Also by part b) of Theorem.0.6 we have that ( ) p P C w(p ) w() for any subcube P or what is the same l(p ) l() c ( ) /np w(p ). w() By the openness property of the A p condition (8.7) there is small ɛ such that w A p ɛ. Then w satisfies ( ) /n(p ɛ) l(p ) w(p ) l() c. w() Now let q = λp with λ > such that with λ = n(p ɛ) = p q = pλ ( p ɛ p n ) = λ = n + δ for some δ > 0. Then we have that l(p ) l() ( ) /q w(p ) c w() ( ) /p w(p ). w() Using this property it is easy to check that a(, f) satisfies the D q condition with a constant independent of and f. As in part a) we can pass from the weak type estimate which follows from Theorem to obtain the desired result ( /q f f w) q C a(, f). (2.33) w() MEJORAR ESTE TROZO UE SE INCORPORA DESPUES: These results are not knew, we know give applications to derive new results. In particular we improve this theorem at the endpoint p = in two ways. We also improve the result given in p. 308 [HKM] since we are able to reach both endpoints: p = and for the Sobolev exponent by p = pn, p < n. The result is even new in the euclidean setting. n p Theorem Also let w be a weight function such that w A. Then, we have ( ) /p ( /p f f p wdx C l() f wdx) p (2.34) w() w() with C independent of f,. 24

25 Observe that this result sharpen the previous theorem when p =. Proof: We follow here [LP2] where the result is given in a more general setting and for higher order derivatives. Let w A and we begin again by using the (, ) Poincaré inequality f f C l() f Now, since A A p we have that the right hand side is less or equal than l() ( /p f w) p = b(, f) = b() w() It will enough to check the D p condition and for which it is enough to check l( ) l( 2 ) ( ) /p w( ) C w( 2 ) ( ) /p w( ) w( 2 ) for all cubes, 2 such that 2. But this is equivalent to ( ) n l( ) C w( ) l( 2 ) w( 2 ) since n = p p. Now by the doubling property it will be enough to check and this follows from the fact that w A. 2 C w( ) w( 2 ) 25

26 Chapter 3 Rearrangements The main purpose of this chapter to improve the main result in the previous chapter by means of a new technique based on rearrangements. This method yields not only yields stronger but also provide new direct proofs avoiding the use the classical good-λ inequality of Burkholder and Gundy as in described in the prevoius chapter. The natural question that we consider is the following: Is it possible to relax the L norm in (2.9) to derive the same self-improving property? In this Chapter, we study this issue and weaken the hypothesis to obtain such self-improving property. More precisely, we will show that the L norm can be replaced by any L q quasi-norm with 0 < q < (See Corollaries and below). Even further, we will show that the left hand side of (2.9) can be replaced by a weaker expression which is defined in terms of non-increasing rearrangements (see below for the precise definition). Our main result is the following. Theorem Let ω be an A weight and let a be a functional satisfying the D r (ω) condition (2.5) for some r > 0. Suppose that f is a measurable function such that for any cube ( ) ( ) inf (f α)χ λ cλ a(), 0 < λ <. (3.) α Then the same consequences of Theorem holds namely, there exists a constant c such that for any cube f f L r, (,ω) c a(). (3.2) where Recall that the non-increasing rearrangement f of a measurable function f is defined by is the distribution function of f. f (t) = inf { λ > 0 : µ f (λ) < t }, t > 0, µ f (λ) = {x R n : f(x) > λ}, λ > 0, 26

27 An important fact is that R n f p dx = or more generally if E is any measurable function: E f p dx = 0 E 0 f (t) p dt, f (t) p dt, If f is only a measurable function and if is a cube then we define the following quantity: (fχ ) (λ ), 0 < λ. We can think of this expression as another way of averaging the function. Indeed for any p > 0, and 0 < λ Hence we have the following. (fχ ) (λ ) ( /p f dx) p. λ Corollary Let f, a and ω as in Theorem Suppose that for some 0 < p < and for any cube ( /p inf f α dx) p a() α Then there is a geometric constant c such that for any cube f f L r, (,ω) c a(). Using that for ay measurable function f, and any 0 < δ < ( /p f dx) p c p f. L, () Corollary If as above we suppose that for any cube f f a(), L, () then there is a geometric constant c such that for any cube f f L r, (,ω) c a(). 27

28 Our proof of Theorem is essentially based on the following two ingredients. The first one is a relation between rearrangements and oscillations of a function f. This technique goes back to [BDVS], and it was further developed in [L, L2]. We also use some ideas from the works [J, S]. The second ingredient is an appropriate covering lemma of Calderón-Zygmund type. In the context of the standard Euclidean space R n with doubling measure such covering lemmas can be obtained simply by application of the usual Calderón-Zygmund lemma to characteristic functions (see, e.g., [BDVS, L]. In the case of R n with non-doubling measure a covering lemma of such type has been recently proved in [MMNO]. Our covering lemma, in the context of spaces of homogeneous type, is presented in Section 3 below. The main result of this paper can be applied to improve the results of [MP2, OP]. In [MP2], an exponential self-improving property was established assuming (2.9) with a functional a satisfying the so-called T p condition, stronger than the D r condition. In Chapter 9, we will improve this result by relaxing the initial assumption (2.9) as in Theorem In [OP], a non-homogeneous variant of Theorem was proved in the context of R n with any (non-doubling) measure. We will show that this result can be also proved under similar relaxed assumptions. We also obtain an analogue of the main results of [L, L2] concerning the so-called local sharp maximal function, in the context of spaces of homogeneous type. 3. Rearrangements We mention here some simple properties of rearrangements. It follows easily from the definition that µ { x R n : f(x) > f µ(t) } t and µ { x R n : f(x) f µ(t) } t. Next, for any 0 < λ <, (f + g) µ(t) f µ(λt) + g µ(( λ)t). (3.3) For any measurable set E with finite positive measure µ we have, inf E f (fχ E ) µ(µ(e)). (3.4) Finally we recall that the (weighted normalized) weak L r norm is defined by ( ) /r ω({x E : g(x) > λ}) g = sup L λ r, (E,ω) λ>0 ω(e) or, equivalently, g L r, (E,ω) = ( t ) /r. sup (gχ E ) ω(t) 0<t ω(e) ω(e) We will need the following two propositions about local rearrangements. 28

29 Proposition 3.. For any measurable function f, any weight ω, and each measurable set E R n with 0 < ω(e) <, and for any 0 < λ < : ( ) ( ) ( ) ( ) ( ) ( ) fχe ω λω(e) 2 inf (f c)χe c ω λω(e) + fχe ω ( λ)ω(e). Proof: Essentially the same was proved in [L] in a less general context. The proof of this proposition follows the same lines, and we shall recall it only for the sake of completeness. Using (3.3) and (3.4), for any constant α we get α inf x E ( f(x) α + f(x) ) ( ( f α + f )χe ) ω ( ω(e) ) ( (f α)χ E ) ω( λω(e) ) + ( fχe ) ω( ( λ)ω(e) ). From this and from the estimate ( ) ( ) ( ) ( ) fχe ω λω(e) (f α)χe ω λω(e) + α we get immediately the required inequality. Proposition 3..2 Let ω A. For any cube and any measurable function f, and for any 0 < λ < : ( ( ) ( ) ( ) fχ ω λω() fχ ( λ ) c )/ε, where c and ε are A -constants of ω. Proof: It follows from the definitions of A and of the rearrangement that for any δ > 0, ω { x : f(x) > ( ) ( λ fχ µ ( c )/ε ) + δ } ( { ( ) ( µ x : f(x) > fχ µ ( λ c c )/ε ) + δ } ) ε ω() < λω(), which is equivalent to that ( ( ) ( ) ( ) fχ ω λω() fχ ( λ ) µ c )/ε + δ. Letting δ 0 yields the required inequality. A relevant class of weights is given by the A class of Muckenhoupt: if there are positive constants c, ε such that ( ) ε E ω(e) c ω() (3.5) for every cube and every measurable set E. 29

30 3.2 A basic covering lemma In this section we prove a covering lemma of Calderón-Zygmund which is nothing but a special case of the usual Calderón-Zygmund classical lemma for a cube. We assume that ω is dyadic doubling: where is the ancestor of. ω( ) d ω() Lemma 3.2. Let ω be a doubling weight with doubling dyadic constant d. Then for any 0 < λ <, any cube and any set E with ω(e) λω(), there is a countable family of pairwise disjoint dyadic cubes { i } i from with E i i, except for a ω-null set such that (i) ω( i E) dλω( i ); (ii) ω( i E) λω( i ); (iii) ω(e) d λ i ω( i). Proof: Items (i) and (ii) follow from the weighted version of the Calderón-Zygmund for cubes (see 7.0.5): If f wdx < λ ω() there exists mutually disjoint dyadic cubes { i } i (dyadic with respect to ) contained in such that λ f ωdx dλ ω() and f(x) λ for almost x \ i i. Letting f = χ E we get (i) and (ii). Observe that since 0 < λ <, w( \ i i ) = 0 and hence E i i, except for a ω-null set. Part (iii) is immediate. 3.3 Proof of the main Theorem Adapting ideas from [L, L2] we prove in this section the main theorem of this Chapter Theorem Proof: 30

31 The goal is to prove the following estimate: there are constants c, λ with 0 < λ < c < such that for any cube and for any 0 < t < ω() ( ω() (fχ ) ω(t) c a() t ) r + (fχ ) (λ ). (3.6) First we will prove this estimate for small values of t. The key result in the proof is the following estimate for the rearrangement of f: there exists a constant λ, 0 < λ < such that for any 0 < t < λ 2 ω() ( ω() (fχ ) ω(t) c a() t ) r + (fχ ) ω(2t). (3.7) Throughout the proof d is doubling constant and c and ε are the constants from the A -condition (3.5) of ω. For any t we consider the set E = E t E = {x : f(x) (fχ ) ω(t)}. We can assume that (fχ ) ω(t) > (fχ ) ω(2t), since otherwise (3.7) is trivial. Observe that t ω(e) 2t. Set λ = λ and λ = ( 4d c )/ε and let 0 < t < λ ω(). Then we have that 2 ω(e) λ ω(), and we can apply Lemma 3.2. to the set E and number λ. Hence, we get a countable family of pairwise disjoint balls { i } satisfying (i)-(iii) of Lemma Thus combining item (ii) and Proposition 3.. we have, (fχ ) ω(t) inf f(x) inf inf f(x) x E i x E i ( ) inf fχe i i ω( ω(e i ) ) ( ) inf fχi i ω( λ ω( i ) ) inf i [ ( ) 2 inf (f α)χi α ω( λ ω( i ) ) + ( fχ i ) ω( ( λ )ω( i ) )]. Applying item (i) together with Proposition 3..2 we obtain for each i ( ) ( inf (f α)χi α ω λ ω( i ) ) ( ) ( inf (f α)χi λ i ) α c λ a( i ), where in the last step we have used the basic initial hypothesis (3.) to the ball i. Hence, ( (fχ ) ω(t) inf 2c λ a( i ) + ( ) fχ i i ω( ( λ )ω( i ) )). 3

32 Since the balls i are dyadic, disjoint and contained in we have by the D r condition (2.5) a( i ) r ω( i ) a r a() r ω(). i Let M a big enough number to be chosen soon. We split the family of cubes as follows: i I if ( ω() ) a( i ) M /r r a a() (3.8) t and i II if it satisfies the opposite inequality. The proof below shows that the family I is not empty. We claim that ω( i ) ( dλ )t, (3.9) M where we choose M > dλ. Indeed, by the D r condition i I ω( i ) t M, i II and hence (3.9) follows from the item (iii) of Lemma 3.2. and our choice of λ. Therefore we have ( ω() (fχ ) ω(t) 2c λ M /r a a() t Set now for i I By (3.9), ) r ( ) ( + inf fχi i I ω ( λ )ω( i ) ). E i = { x i : f(x) ( fχ i ) ω( ( λ )ω( i ) )}. ω( i I E i ) = i I ω(e i ) ( λ ) i I ω( i ) ( λ )( dλ )t > 2t, M if we choose for instance M > 2dλ and 0 < λ <. +4d Thus, ( ) ( inf fχi i I ω ( λ )ω( i ) ) inf i I (fχ ) ω( ω( i I E i ) ) (fχ ) ω(2t), inf f(x) = x E i and hence (3.7) is proved for the values 0 < t < λ 2 ω(). Now, for one of these values of t, there is k =,, such that λ λ ω() t < 2k+ 2 ω(), k 32 inf f(x) x i I E i

33 and iterating (3.7) yields (fχ ) ω(t) ( ω() ca() t ( ω() c a() t ) r ) r k (2 j ) r + (fχ ) ω(2 k+ ω()) j=0 + (fχ ) ω(λ ω()). (3.0) Next, we observe that (3.0) trivially holds for t λ ω(), and hence this formula holds for any 0 < t < ω(). Applying Proposition 3..2 to the second term on the right-hand side of (3.0), we obtain (3.6). To finish the proof we observe that if α is any number, f α also satisfies the initial assumption (3.) and hence (3.6) holds for f α as well: Recalling that ( we have by multiplying ( ω() ((f α)χ ) ω(t) c a() t g L r, (,ω) = t ω() ) r ) r + ((f α)χ ) (λ ). ( t ) /r, sup (gχ ) ω(t) 0<t ω() ω() and taking the supremum over 0 < t < ω() that f α L r, (,ω) c a() + ((f α)χ ) (λ ). Taking the infimum over all α and using again (3.) combined with the D r condition we have inf α f α L r, (,ω) c a() + inf α ((f α)χ ) (λ ) c a() + c λ a() c a(). The proof is now complete. 33

34 Chapter 4 Orlicz maximal functions 4. Orlicz spaces To derive some sharp weighted estimates in Chapters (5) and (6) we need to consider certain maximal functions defined in terms of Orlicz spaces. In this section we recall some basic definitions and facts about Orlicz spaces, referring to [RR] and [KJF] for a complete account. A function B : [0, ) [0, ) is called a Young function if it is continuous, convex, increasing and satisfies B(0) = 0 and B(t) as t. It follows that B(t)/t is increasing. Some relevant examples are B(t) = t p (log( + t)) α, p <, α > 0 or B(t) = Φ(t) = exp t p, p <. For Orlicz norms we are usually only concerned about the behavior of Young functions for t large. Given two functions B and C, we write B(t) C(t) if B(t)/C(t) is bounded and bounded below for t c > 0. By definition, the Orlicz space L B consists of all measurable functions f such that B( f R λ ) dµ < n for some positive λ. The space L B is a Banach function space with the Luxemburg norm defined by f B = inf{λ > 0 : B( f ) dµ }. R λ n Each Young function B has an associated complementary Young function B satisfying t B (t) B (t) 2 t (4.) for all t > 0. The function B is called the conjugate of B, and the space L B is called the conjugate space of L B. For example, if B(t) = t p for < p <, then B(t) = t p, p = p/(p ), and the conjugate space of L p (µ) is L p (µ). Another example 34

35 that will be used frecuently is B(t) t p (log t) ɛ, < p <, ɛ > 0 with complementary function B(t) t p (log t) (p )(+ɛ) (cf. [O]). A very important property of Orlicz (or, more generally, Banach function spaces) is the generalized Hölder inequality R n fg dµ f B g B. (4.2) We will always assume that the Young function B satisfies the doubling condition B(2t) C B(t). If B is doubling then B (t) B(t)/t almost everywhere. L B is a rearrangement invariant space with fundamental function (see [BS]) given by ϕ B (t) = ϕ (t) = BL B (µ) B ( (4.3) ). t In particular if E is a measurable subset of X, then χ = E B,µ B ( (4.4) ). µ(e) 4.2 Orlicz maximal functions We begin the section by defining a class of Young functions that will play a key role in Chapters 5 and 6. We need to define an appropriate maximal in terms of Orlicz norm. Given a Young function B and a cube, the mean Luxembourg norm of a measurable function f on is defined by f B, = inf { λ > 0 : B ( ) } f dx. (4.5) λ When B(t) = t log( + t) p +δ, this norm is also denoted by L(log L) p +δ,. Given a Young function B, define the maximal function c M B f(x) = sup f B,. x Definition 4.2. Let p <. A nonnegative function B(t), t > 0, satisfies the B p condition if there is a constant c > 0 such that B(t) dt ( ) t p p dt t p t B(t) t <. c Simple examples of functions which satisfy B p are t p δ and t p (log( + t)) δ, both when δ > 0. The interest of this class of Young functions can be seen from the following lemma taken from [P]. It is used in this paper to derive sharp two weight norm inequalities for the Hardy-Littlewood maximal function. 35

36 Lemma Let < p <. Then the following statements are equivalent. a) B is a Young doubling function satisfying the B p condition; b) there is a constant c such that M B f(y) p dy c R n f(y) p dy R n (4.6) for all nonnegative, locally integrable functions f; c) there is a constant c such that for all nonnegative functions f and u ; Mf(y) p dy c R [M n p B(u)(y)] Rn f(y) p dy, (4.7) u(y) p Proof: To prove b) we will use the classical approach (cf. for instance [GCRdF] Ch. 2.). Observe that we may assume that f 0. For t > 0 we let Ω t = {x R n : M B f(x) > t}. We claim that Ω t C R n B ( ) f(y) dy (4.8) t Assume the claim for the moment. Using (4.8) together with the fact that M B is bounded on L we write f as f = f + f 2, where f (x) = f(x) if f(x) > t, and f 2 (x) = 0 otherwise. Then M B f(x) M B f (x) + M B f 2 (x) M B f (x) + t. and it follows immediately that 2 ( ) f(x) Ω t C B dx t x:f(x)>t/2 Hence, combining this with the change of variable t = f(y) yield s M B f(y) p dy = p t p {y R n : M B f(y) > t} dt R n 0 t C 0 since B B p. t p {y R n :f(y)>t/2} = C ( ) f(y) B t R n f(y) p dy /2 dy dt t = C B(t) t p R n dt t = C 2f(y) 0 ( ) f(y) dt t p B t t dy = R n f(y) p dy, 36

37 The proof of the claim follows easily from the Vitali covering lemma.0.4. Indeed, assuming that Ω t is not empty let K be any compact contained in Ω t. Let x K, then by defintion of the maximal function there is a cube = x containg x such that or what is the same f B, > t, (4.9) B( f t ) >. Then K x K x and by compactness we can extract a finite family of cubes F = {} such that K F and where each cube satisfies (4.9). Then by Vitali s lemma we can extract a pairwise disjoint cubes { j } M j= such that F M j= 5 j. Then by definition of f B, M M K C j C B( f j t ) C B( f R t ). n j= This proves that a) implies b). Let us assume that c) holds. Observe that (4.7) is equivalent to M(fg)(y) p dy c f(y) p dy, R [M n p B(g)(y)] R n for all nonnegative functions f, g. Then c) follows immediately from (4.6) after an application of the generalized Hölder s inequality (4.2) j= M(fg)(y) M B f(y)m Bg(y) y R n. To prove that c) implies a) we test (4.7) with f = u = χ (0,) where (0, ) is the cube centered at the origin and with radius : Mf(y) p dy C. (4.0) R [M n p B(f)(y)] On the other hand a computation using (4.4) shows that for large x that M B(f)(x) B ( x n ) and we can continue (4.0) with C Mf(y) p dy C R [M n p B(f)(y)] y >c y np B ( dy = n )p y = C c dr r np B ( rn ) p r B(t) dt r n c t p t. 37

38 This shows that c) a). We conclude the Chapter by proving a weighted inequality dual to the classical Fefferman Stein inequality Mf(y) p w(y)dy c f(y) p Mw(y)dy. R n R n The main interest follows from the fact that its natural dual inequality, thinking as if M were linear and selfadjoint, would be Mf(y) p Mw(y) p dy c f(y) p w(y) p dy. R n R n However this estimate is false in general as can be shown by taking f = w positive and integrable. However we the following sharp result. Corollary Mf(y) p [M [p ]+ (w)(y)] p dy c f(y) p w(y) p dy. (4.) R n R n The proof is a consequence of (4.7), choosing B such that B t p (log t) p +δ, combined with Lemma 5.4. below. See [P] for details. 38

39 Chapter 5 Fractional integral operators We have already discussed in Chapter 2 the relationship of a function f and its gradient via the fractional integral. Indeed, recall that if f is a sufficiently smooth function, then there is a universal constant C such that for each cube and for each x f(x) f C I ( f χ )(x) (5.) If we further assume that f has compact support then this inequality clearly shows that for each x R n f(x) C I ( f )(x). (5.2) Recall that I α, 0 < α < n, denotes the fractional integral of order α on R n or Riesz potentials defined by f(y) I α f(x) = n α dy. x y 5. Weak implies strong R n The main point of this section is to illustrate the fact that a weak type inequality with a gradient (or gradient like) operator on the right hand side implies the corresponding strong type inequality. This is never the case in the standard interpolation theory. As we mentioned in Section 2.5 the basic ideas are taken from [LN]. Theorem 5.. (weighted isoperimetric inequality) Let n > and let µ be any nonnegative measure. Then there is a constant C such that for any Lipschitz function f with compact support ( R n f(x) n ) /n dµ C This result and generalizations can be found in [FPW2]. f(x) (Mµ(x)) /n dx. (5.3) R n 39