SHARP WEIGHTED BOUNDS FOR FRACTIONAL INTEGRAL OPERATORS

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1 Journal of Functional Analysis, 259, (2), SHARP WEIGHTED BOUNDS FOR FRACTIONAL INTEGRAL OPERATORS MICHAEL T LACEY, KABE MOEN, CARLOS PÉREZ, AND RODOLFO H TORRES Abstract The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions As an application improved Sobolev inequalities are obtained Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities Introduction Recall that a non-negative locally integrable function, or weight, w is said to belong to the A p class for < p < if it satisfies the condition ( ) ( p [w] Ap sup w(x) dx w(x) dx) p <, where p is the dual exponent of p defined by the equation /p+/p = Muckenhoupt [7] showed that the weights satisfying the A p condition are exactly the weights for which the Hardy-Littlewood maximal function Mf(x) = sup x f(y) dy is bounded on L p (w) Hunt, Muckenhoupt, and Wheeden [2] extended the weighted theory to the study of the Hilbert transform f(y) Hf(x) = pv x y dy They showed that the A p condition also characterizes the L p (w) boundedness of this operator Coifman and Fefferman [3] extended the A p theory to general Calderón- Zygmund operators For example, to operators that are bounded, say on L 2 ( ), and of the form T f(x) = pv f(y)k(x, y) dy, 2 Mathematics Subject Classification 42B2, 42B25 Key words and phrases Maximal operators, fractional integrals, singular integrals, weighted norm inequalities, extrapolation, sharp bounds R

2 2 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES where β K(x, y) c x y n β Bounds on the operators norms in terms of the A p constants of the weights have been investigated as well Buckley [2] showed that for < p <, M satisfies () M L p (w) L p (w) c [w] /(p ) A p and the exponent /(p ) is the best possible A new and rather simple proof of both Muckenhoupt s and Buckley s results was recently given by Lerner [3] The weak-type bound also observed by Buckley [2] is (2) M L p (w) L p, (w) c [w] /p A p For singular integrals operators, however, only partial results are known The interest in sharp weighted norm for singular integral operators is motivated in part by applications in partial differential equations We refer the reader to Astala, Iwaniec, and Saksman []; and Petermichl and Volberg [23] for such applications Petermichl [2], [22] showed that (3) T L p (w) L p (w) c [w] max{,/(p )} A p, where T is either the Hilbert or one of the Riesz transforms in, x j y j R j f(x) = c n pv f(y) dy x y n+ Petermichl s results were obtained for p = 2 using Bellman function methods The general case p 2 then follows by the sharp version of the Rubio de Francia extrapolation theorem given by Dragi cević, Grafakos, Pereyra, and Petermichl [4] We recall that the original proof of the extrapolation theorem was given by Rubio de Francia in [24] and it was not constructive García-Cuerva then gave a constructive proof that can be found in [6, p434] and which has been used to get the sharp version in [4] It is important to remark that so far no proof of the L p version of Petermilch s result is know without invoking extrapolation These are the best known results so far and whether (3) holds for general Calderón-Zygmund operators is not known There are also other estimates for Calderón-Zygmund operators involving weights which have received attention over the years In particular, there is the Muckenhoupt- Wheeden conjecture (4) T f L, (w) c f L (Mw), for arbitrary weight w, and the linear growth conjecture for < p <, (5) T L p (w) L p, (w) c p [w] Ap Both these conjectures remain very difficult open problems Some progress has been recently made by Lerner, Ombrosi and Pérez [4], [5] Motivated by all these estimates, we investigate in this article the sharp weighted bounds for fractional integral operators and the related maximal functions

3 SHARP WEIGHTED BOUNDS 3 For < α < n, the fractional integral operator or Riesz potential I α is defined by f(y) I α f(x) = dy, x y n α while the related fractional maximal operator M α is given by M α f(x) = sup f(y) dy x α/n These operators play an important role in analysis, particularly in the study of differentiability or smoothness properties a functions See the books by Stein [29] or Grafakos [7] for the basic properties of these operators Weighted inequalities for these operators and more general potential operators have been studied in depth See eg the works of Muckenhoupt and Wheeden [8], Sawyer [26], [27], Gabidzashvili and Kokilashvili [5], Sawyer and Wheeden [28], and Pérez [9], [2] Such estimates naturally appear in problems in partial differential equations and quantum mechanics In [8], the authors characterized the weighted strong-type inequality for fractional operators in terms of the so-called condition For < p < n/α and q defined by /q = /p α/n, they showed that for all f, ( ) /q ( ) /p (6) (wt α f) q dx c (wf) p dx, where T α = I α or M α, if and only if w That is, ( ) ( ) q/p [w] Ap,q sup w q dx w p dx < The connection between the constant [w] Ap,q and the operator norms of these fractional operators is the main focus of this article We will obtain the analogous estimates of (), (2), (3), (4), and (5) in the fractional integral case At a formal level, the case α = corresponds to the Calderón-Zygmund case where, as mentioned, some estimates have not been obtained yet Though for α > one deals with positive operators, the corresponding estimates still remain difficult to be proved and we need to use a set of tools different from the ones used in the Calderón- Zygmund situation Our main result, Theorem 26 below, is the sharp bound I α L p (w p ) L q (w q ) c[w] ( α n ) max{, p q } This is the analogous estimate of (3) for fractional integral operators Acknowledgements First author s research supported in part by National Science Foundation under grant DMS 4566 Second and fourth authors research supported in part by the National Science Foundation under grant DMS 8492 Third author s research supported in

4 4 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES part by the Spanish Ministry of Science under research grant MTM Part of the research leading to the results presented in this article was conducted when C Pérez visited the University of Kansas, Lawrence during the academic year and in the spring of 29 Finally, the authors are very grateful to the Centre de Recerca Matemàtica for the invitation to participate in a special research programme in Analysis, held in the spring of 29 where this project was finished 2 Description of the main results We start by observing that to obtain sharp bounds for the strong-type inequalities for I α it is enough to obtain sharp bounds for the weak-type ones This is due to Sawyer s deep results on the characterization of two-weight norm inequalities for I α In fact, he proved in [27] that for two positive locally integrable function v and u, and < p q <, I α : L p (v) L q (u) if and only if u and the function σ = v p [u, σ] Sp,q satisfy the (local) testing conditions sup σ() /p χ I α (χ σ) L q (u) < and [σ, u] Sq,p sup u() /q χ I α (χ u) L p (σ) < Moreover, his proof shows that actually (2) I α L p (v) L q (u) [u, σ] Sp,q + [σ, u] Sq,p On the other hand in his characterization of the weak-type, two-weight inequalities for I α, Sawyer [26] also showed that Combining (2) and (53) it follows that I α L p (v) L q, (u) [σ, u] Sq,p (22) I α L p (v) L q (u) I α L q (u q ) L p, (v p ) + I α L p (v) L q, (u) If we now set u = w q and v = w p, we finally obtain the one-weight estimate (23) I α L p (w p ) L q (w q ) I α L q (w q ) L p, (w p ) + I α L p (w p ) L q, (w q ) We will obtain sharp bounds for the weak-type norms in the right hand side of (23) in two different ways, each of which is of interest on its own Our first approach is based on an off-diagonal extrapolation theorem by Harboure, Macías, and Segovia [] A second one is based in yet another characterization of two-weight norm inequalities for I α in the case p < q, in terms of certain (global) testing condition and which is due to Gabidzashvili and Kokilashvili [5] We present now the extrapolation results The proof follows the original one, except that we carefully track the dependence of the estimates in terms of the constants of the weights

5 SHARP WEIGHTED BOUNDS 5 Theorem 2 Suppose that T is an operator defined on an appropriate class of functions, (eg Cc, or p Lp (w p )) Suppose further that p and q are exponents with p q <, and such that holds for all w A p,q wt f L q ( ) c[w] γ A p,q wf L p ( ) and some γ > Then, q γ max{, p wt f L q ( ) c[w] p q } wf L p ( ) holds for all p and q satisfying < p q < and and all weight w p q = p q, As a consequence we have the following weak extrapolation theorem using an idea from Grafakos and Martell [9] Corollary 22 Suppose that for some p q <, an operator T satisfies the weak-type (p, q ) inequality T f L q, (w q ) c[w] γ A p,q wf L p ( ) for every w A p,q and some γ > Then T also satisfies the weak-type (p, q) inequality, q γ max{, p T f L q, (w q ) c[w] p q } wf L p ( ) for all < p q < that satisfy and all w p q = p q We will use the above corollary to obtain sharp weak bounds in the whole range of exponents for I α As already described, this leads to strong-type estimates too Nevertheless, for a certain range of exponents the strong-type estimates can be obtained in a more direct way without relying on the difficult two-weight results It is not obvious a priori what the analogous of (3) should be for I α A possible guess is (24) w I α f L q ( ) c [w] max{, p q } w f L p ( ) Note that formally, the estimate reduces to (3) when α = suggesting it could be sharp While it is possible to obtain such estimate, simple examples indicate it is not the best one In fact, we will show in this article a direct proof of the following estimate

6 6 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES Theorem 23 Let < p < n/α and q be defined by the equations /q = /p α/n and q /p = α/n, and let w A p,q Then, (25) wi α f L q ( ) c [w] Ap,q wf L p ( ) The pair (p, q ) in the above theorem could be seen as the replacement of the L 2 case when α = That is, it yields a linear growth on the weight constant However, unlike the case α =, one can check that starting from this point (p, q ), extrapolation and duality give sharp estimates for a reduced set of exponents See (48) below To get the full range we use first Corollary 22 to obtain sharp estimates for the weak-type (p, q) inequality for I α We have the following result Theorem 24 Suppose that p < n/α and that q satisfies /q = /p α/n Then (26) I α f L q, (w q ) c [w] α n w f L p ( ) Furthermore, the exponent α is sharp n We will also present a second proof of Theorem 24 for p > without using extrapolation Remark 25 Once again, the estimate in the above weak-type result should be contrasted with the case α = and the linear growth conjecture for a Calderón- Zygmund operator T Namely, T L p (w) L p, (w) c p [w] Ap Such results have remained elusive so far For the best available result see [5] The extrapolation proof of Theorem 24 will also show that for any weight u the weak-type inequality I α f L (n/α), (u) c f L ((Mu) α n ) holds For α = the analogous version of this inequality is the Muckenhoupt-Wheeden conjecture T f L, (w) c f L (Mw), which is an open problem As a consequence of the weak-type estimate (26) we obtain the sharp bounds indicated by examples Theorem 26 Let < p < n/α and q be defined by the equation /q = /p α/n, and let w Then, (27) I α L p (w p ) L q (w q ) c [w] ( α n ) max{, p q } Furthermore this estimate is sharp Another consequence of (26) is a Sobolev-type estimate We obtain this when we use the fact that weak-type inequalities implies strong-type inequalities when a gradient operator is involved We have the following result based on the ideas of Long and Nie [6] See also Hajlasz []

7 SHARP WEIGHTED BOUNDS 7 Theorem 27 Let p and let w with q satisfying /p /q = /n Then, for any Lipschitz function f with compact support, ( ) /q /p (28) ( f(x) w(x)) q dx c [w] /n ( f(x) w(x)) dx) R (R p n n Remark 28 We note that this estimate is better than what the strong bound on I in Theorem 26 gives In fact, for f sufficiently smooth and compactly supported, we have the estimate f(x) ci ( f )(x) Hence, if we applied Theorem 26 we obtain the estimate fw L q c [w] /n max{,p /q} fw L p However, Theorem 27 gives a better growth in terms of the weight, simply [w] /n This is a better growth in the range < p < min(2n, n) (ie p /q > ) where the estimate (27) only gives [w] p /(qn ) Note also that (28) includes the case p =, which cannot be obtained using Theorem 26 We also find the sharp constant for M α in the full range of exponents Theorem 29 Suppose α < n, < p < n/α and q is defined by the relationship /q = /p α/n If w, then p q (29) wm α f L q c[w] ( α n ) wf L p Furthermore, the exponent p ( α ) is sharp q n Note one more time that formally replacing α = the estimates clearly generalize the result in [2] Remark 2 We also note that there is a weak-type estimate for M α For p and /q = /p α/n, standard covering methods give (2) M α L p (w p ) L q, (w q ) c [w] /q See for instance the book by Garcia-Cuerva and Rubio de Francia [6, pp ], for the estimate in the case α = Remark 2 Continuing with the formal comparison with the case α =, it would be interesting to know if the analog of (23) also holds for Calderón-Zygmund singular integrals Namely, (2) T L p (w) L p (w) T L p (w p ) L p, (w p ) + T L p (w) L p, (w) This estimate, if true, may be beyond reach with the current available techniques

8 8 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES The rest of the paper is organized as follows We separate the proofs of the main results in different sections which are essentially independent of each other In Section 3 we collect some additional definitions and the proof of the version of the extrapolation result Theorem 2 We repeat the proof of such result from [] for the convenience of the reader, but also to show that the constant we need can indeed be tracked through the computations A faithful reader familiar with the extrapolation result may skip the details, move directly to the following sections of the article, and come back later to Section 3 to verify our claims Section 4 contains the proof of of Theorem 23 The proof of Corollary 22 and the two proofs of the weak-type result for I α, Theorem 24, are in Section 5 The proof of Theorem 26 as a corollary of Theorem 24 is in this section too The proof of the result for the fractional maximal function, Theorem 29, is presented in Section 6 In Section 7 we present the examples for the sharpness in Theorems 24, 26, and 29 Finally, in Section 8 we present the proof of the application to Sobolev-type inequalities 3 Constants in the off-diagonal extrapolation theorem For a Lebesgue measurable set E, E will denote its Lebesgue measure and w(e) = w(x) dx will denote its weighted measure We will be working on weighted versions E of the classical L p spaces, L p (w), and also on the weak-type ones, L p, (w), defined in the usual way with the Lebesgue measure dx replaced by the measure w dx Often, however, it will be convenient to viewed the weight not as a measure but as a multiplier For example f L p (w p ) if ( ) /p fw L p = ( f(x) w(x)) p dx < This is more convenient when dealing with the condition already defined in the introduction Recall, that for < p q <, we say w if ( ) ( ) q/p (3) [w] Ap,q sup w q dx w p dx < Also, for q < we define the class A,q to be the weights w that satisfy, ( ) (32) w q dx c inf wq Here [w] A,q will denote the smallest constant c that satisfies (32) Notice that w if and only if w q A +q/p with (33) [w] Ap,q = [w q ] A+q/p In particular, [w] Ap,q We also note for later use that (34) [w ] Aq,p = [w] p /q The term cube will always refer to a cube in with sides parallel to the axis A multiple r of a cube is a cube with the same center of and side-length r times as

9 SHARP WEIGHTED BOUNDS 9 large By D we denote the collection of all dyadic cubes in That is, the collection of all cubes with lower-felt corner 2 l m and side-length 2 l with l Z and m Z n As usual, B(x, r) will denote the Euclidean ball in centered at the point x and with radius r To prove Theorem 2 we will need the sharp version of the Rubio de Francia algorithm given by García-Cuerva The proof can be found in the article [4] Lemma 3 Suppose that r > r, v A r, and g is a non-negative function in L (r/r ) (v) Then, there exists a function G such that () G g, (2) G L (r/r ) (v) 2 g L (r/r ) (v), (3) Gv A r with [Gv] Ar c [v] Ar Proof of Theorem 2 First suppose w and p < p, which implies q > q Then, ( ) /q ( ) q q T f q w q = ( T f q ) q/q w q q R ( n ) = T f q gw q q for some non-negative g L (q/q ) (w q ) with g L (q/q ) (w q ) = Now, let r = + q/p and r = + q /p Since p > p we have r > r Furthermore, by the relationship p q = p q, we have q/q = r/r Hence by Lemma 3 and using that w q A r, there exists G with G g, G L (r/r ) (w q ) 2, Gwq A r, and [Gw q ] Ar c [w q ] Ar = c [w] Ap,q Also, since Gw q A r then (Gw q ) /q A p,q since, ( ) ( ) q /p [(Gw q ) /q ] Ap,q = sup = sup ( = [Gw q ] Ar Then, we can proceed with ( ) /q T f q w q = ( (G /q w q/q ) q (G /q w q/q ) p ) ( ) q /p Gw q (Gw q ) p /q T f q gw q R ( n T f q Gw q ) q ) q

10 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES = ( where we have used the relationship T f q (G /q w q/q ) q ) q c [G /q w q/q ] γ A p,q ( f p (G /q w q/q ) p ( = c [Gw q ] γ A r f p w p G p /q w R ( n ) /p ( c [w] γ f p w p R ( n ) /p c [w] γ f p w p, p q = p q q/(p/p ) (r/r) G For the case < p < p, and hence q < q, notice that we can write ( ) /p ( ) /p f p w p = ( fw p p ) p/p w p Since p/p <, there exists a function g satisfying g p/(p p) w p = such that ( ) /p ( ) /p f p w p = fw p p gw p, ) p ) p w q ) (p p )/pp see [8, p 335] Let h = g p /p, r = + p /q and r = + p /q, so that r > r Notice that p q = p q implies r/r = p /p, which in turn yields (35) Hence, (r/r) h p p ( r r ) = p p p w p = g p/(p p) w p = Observe that w p A r, so by Lemma 3 we obtain a function H such that H h, H L (r/r ) (w p ) 2, and Hw p A r with [Hw p ] Ar c [w p ] Ar = c [w] p /q

11 SHARP WEIGHTED BOUNDS Now, for Hw p A r we claim that (Hw p ) /p Ap,q with [(Hw p ) /p ]Ap,q = [Hw p ] q /p A r Indeed, ( [(Hw p ) /p ]Ap,q = sup = sup ( = [Hw p ] q /p A r )( (H /p w p /p ) q (H /p w p /p ) p )( ) q (Hw p ) q /p /p Hw p ) q /p Finally expressing g in terms of h and using (35), working backwards we have ( ) /p ( ) /p f p w p = f p p/p h w p (p ) R ( n ) /p f p p/p H w p (p ) = [(Hw p ) /p ] γ ( ) /p A p,q [(Hw p ) /p ] γ f p (H /p w p /p ) p A p,q ( ) /q c [(Hw p ) /p ] γ T f q (H /p w p /p ) q A p,q ( ) /q ( ) q q /qq c [(Hw p ) /p ] γ T f q w q (r/r) H w p A p,q ( ) /q c [(Hw p ) /p ] γ T f q w q A p,q In the second to last inequality we have used Hölder s inequality for exponents less than one, ie, if < s < then fg L f L s g L s, where as usual s = s/(s ) See [7, p ] for more details Thus we have shown, ( ) /q ( ) /p T f q w q c [(Hw p ) /p ] γ A p f p w p,q From here we have T c [(Hw p ) /p ] γ A p,q = c [Hw p ] γ q p A r c [w p ] γ q p A +p /q = c [w] γ q p p q This proves the theorem

12 2 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES 4 Proofs of strong-type results using extrapolation We will need to use the following weighted versions of M α For α < n, let Mα,νf(x) c = sup f(y) dν, x ν( x ) α/n x where the supremum is over all cubes x with center x A dyadic version of M α was first introduced by Sawyer in [25] This maximal function will be an effective tool in obtained the estimates for I α The following lemma will be used in the proofs of Theorems 23 and 29 Lemma 4 Let α < n and ν be a positive Borel measure Then, M c α,νf L q (ν) c f L p (ν) for all < p q < that satisfy /p /q = α/n Furthermore, the constant c is independent of ν (it depends only on the dimension and p) The proof of Lemma 4 can be obtained by interpolation In fact, the strong (n/α, ) inequality follows directly from Hölder s inequality, while a weak-(, (n/α) ) estimate is a consequence of the Besicovich covering lemma Proof of Theorem 23 The equation q /p = α/n along with the fact that /p /q = α/n yields p = 2 α/n α/n (α/n) 2 + We want to show the linear estimate and (4) wi α f L q c [w] Ap,q wf L p Notice that (4) is equivalent to q = 2 α/n α/n (42) I α (fσ) L q (u) c [w] Ap,q f L p (σ), where u = w q and σ = w p Moreover, by duality, showing (42) is equivalent to prove ( ) /p ( ) /q (43) I α (fσ)gu dx c [w] Ap,q f p σ dx g q u dx for all f and g non-negative bounded functions with compact support We first discretize the operator I α as follows Given a non-negative function f, I α f(x) = f(y) dy k Z 2 k < x y 2 x y k n α c χ (x) f(y) dy l() n α k D x y l() l()=2 k

13 c D SHARP WEIGHTED BOUNDS 3 χ (x) α/n 3 f dy where the last inequality holds because if x, then B(x, l()) 3 One immediately gets then α/n fσ dx gu dx I α (fσ)gu dx c D 3 The next crucial step is to pass to a more convenient sum where the family of dyadic cubes is replaced by an appropriate subset formed by a family of Calderón-Zygmund dyadic cubes We combine ideas from the work of Sawyer and Wheeden in [28, pp ], together with some techniques from [2] (see also [9]) Fix a > 2 n Since g is bounded with compact support, for each k Z, one can construct a collection { k,j } j of pairwise disjoint maximal dyadic cubes (maximal with respect to inclusion) with the property that a k < gu dx k,j k,j By maximality the above also gives gu dx 2 n a k k,j k,j Although the maximal cubes in the whole family { k,j } k,j are disjoint in j for each fixed k, they may not be disjoint for different k s If we define for each k the collection C k = { D : a k < gu dx a k+ }, then each dyadic cube belongs to only one C k or gu vanishes on it Moreover, each C k has to be contained in one of the maximal cubes k,j and verifies for all k,j gu dx a k+ a gu dx k,j k,j From these properties and the fact that for any dyadic cube, fσ dx c α 3 α/n fσ dx, D, α/n one easily deduces as in [28] that α/n fσ dx gu dx a c α D Notice also that, 3 [w] Ap,q 3 = sup u() k,j k,j α/n k,j ( ) α/n σ() <, fσ dx gu dx 3 k,j k,j

14 4 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES so we can estimate I α (fσ)gu dx c k,j = c k,j k,j α/n k,j σ(5 k,j ) α/n fσ dx gu dx 3 k,j k,j fσ dx 3 k,j u(3 k,j ) k,j gu dx (44) u(3 ( ) α/n k,j) σ(5k,j ) k,j k,j k,j c [w] Ap,q fσ dx gu dx σ(5 k,j ) α/n k,j, 3 k,j u(3 k,j ) k,j k,j where we have set up things to use, in a moment, certain centered maximal functions Before we do so, we need one last property about the Calderón-Zygmund cubes k,j We need to pass to a disjoint collection of sets E k,j each of which retains a substantial portion of the mass of the corresponding cube k,j Define the sets E k,j = k,j {x : a k < M d (gu) a k+ }, where M d is the dyadic maximal function The family {E k,j } k,j is pairwise disjoint for all j and k Moreover, suppose that for some point x k,j it happens that M d (gu)(x) > a k+ By the maximality of k,j, this implies that there exist some dyadic cube such that x k,j and so that the average of gu over is larger than a k+ It must also hold then that M d (guχ k,j )(x) > a k+ But {M d (guχ k,j ) > a k+ } gu dx 2n k,j a k+ k,j a It follows that Recalling now that = u n n α σ = u q (45) k,j E k,j = since E k,j ( 2n a ) k,j n n α σ E k,j u q q q q, we can use Hölder s inequality to write n n α σ q u(e k,j ) /q σ(ek,j ) /q, n n α = With (45) we go back to the string of inequalities to estimate I α (fσ) gu dx Using the discrete version of Hölder s inequality, we can estimate in (44) ( ( ) q /q c [w] Ap,q fσ dx σ(e k,j)) σ(5 k,j ) α/n k,j 3 k,j

15 SHARP WEIGHTED BOUNDS 5 ( ) q /q gu dx u(e k,j) u(3 k,j ) k,j k,j ( ) /q ( ) /q c [w] Ap,q (Mα,σf) c q σ dx (Mug) c q u dx k,j E k,j k,j E k,j ) /q ( ) /q c [w] Ap,q (Mα,σf) (R c q σ dx (Mug) c q u dx n R ) n /p ( ) /q c [w] Ap,q f (R p σ dx g q u dx n Here we have denoted by Mu c = M,u, c the centered maximal function with respect to the measure u We have also used in the last step Lemma 4, which gives the boundedness of Mu c and Mα,σ c with operator norms independent of the corresponding measure We obtain then the desired linear estimate (46) wi α f L q c [w] Ap,q wf L p From this last estimate we can extrapolate (Theorem 2) to get, (47) wi α f L q c [w] max{,( α/n)p /q} wf L p for all < p < q < with /p /q = α/n Moreover, a simple duality argument gives then (48) I α L p (w p ) L q (w q ) c[w] min{max( α n, p q ),max(,( α n ) p q )} This is sharp for p /q (, α/n] [n/(n α), ) We obtain the right estimate in the full range of exponents in the next section The sharpness will be obtained in Section 7 5 Proof of the weak-type results and sharp bounds for the full range of exponents We start with the weak-type version of the extrapolation theorem Proof of Corollary 22 Note that Theorem 2 does not require T to be linear We can simply apply then the result to the operator T λ f = λχ { T f >λ} Fix λ >, then wt λ f L q = λw q ({x : T f(x) > λ}) /q T f L q, (w q ) c[w] γ A p,q wf L p,

16 6 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES with constant independent of λ Hence by Theorem 2 if w, T λ maps L q (w q ) L p (w p ) for all /p /q = /p /q and with bound wt λ f L q with c independent of λ Hence, T f L q, (w q ) = sup wt λ f L q λ> q γ max{, p c [w] p q } fw L p q γ max{, p c [w] p q } fw L p Proof of Theorem 24 First Proof (valid for p ) We apply Corollary 22 with p =, q = n/(n α) = (n/α), and u = w q Actually, we are going to prove a better estimate, namely (5) I α f L q, (u) c f L ((Mu) /q ) for any weight u From this estimate, and since by (32) the A,(n/α) condition for w is equivalent to M(u) [w] A,(n/α) u, we can deduce I α f L q, (u) c [w] α/n A,(n/α) fw L The weak extrapolation Corollary 22 with γ = α/n gives the right estimate In order to prove (5), we note that L q, (u) is equivalent to a norm since q > Hence, we may use Minkowski s integral inequality as follows (52) I α f L q, (u) c q f(y) y α n L q, (u) dy We can finally calculate the inner norm by y α n L q, (w q ) = sup λu({x : x y α n > λ}) /q λ> = (sup t> = cmu(y) /q t n u({x : x y < t}))/q Once again, the sharpness of the exponent α/n will be shown with an example in Section 7 Second Proof (valid for p > only) We need to recall another characterization of the weak-type inequality for I α for two weights This characterization is due to Gabidzashvili and Kokilashvili [5] and establishes that for < p < q <, the two-weight weak type inequality, (53) I α L p (v) L q, (u) <

17 hods if and only if ( (54) sup u(x) dx ) /q ( SHARP WEIGHTED BOUNDS 7 ( /n + x x ) (α n)p v(x) p ) /p dx < where x denotes the center of the cube We will refer to (54) as the global testing condition, given its global character when compared to the local testing conditions of Sawyer We will use the notation [u, v] Glo(p,q) = sup ( u(x) dx ) /q ( It follows from the proof in [5] (see also [28]) that ( /n + x x ) (α n)p v(x) p (55) I α L p (v) L q, (u) [u, v] Glo(p,q) ) /p dx We now need a reverse doubling property satisfied by w q when w class (see [28] for precise definitions) Lemma 5 Let w, then for any cube we have the estimate (56) wq dx 2 wq dx c[w] for an absolute constant c Proof Let E Our goal is to show that ( ) q E (57) [w] A p,q E wq dx wq dx Applying this with E = will prove the Lemma We can estimate 2 E = w E w [ E wq dx] /q [ w q dx] /q [ E wq dx] /q [ E w p dx] /p (q < p ) [ E = wq dx] /q [ wq dx] /q [ w p dx] /p wq dx [ E wq dx] /q[w] /q wq A dx p,q The proof is complete

18 8 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES We now claim that in the case u = w q and v = w p the constant in the global testing condition and the constant of w are comparable: (58) [w q, w p ] Glo(p,q) [w] ( α/n) Proof of (58) Observe that p ( α/n) = + p /q One of the inequalities in (58) is clear For the other we estimate ( ) /q ( ) /p w(x) q dx ( /n + x x ) (α n)p w(x) p( p ) dx ( ) [ /q ] /p c w q 2 j p ( α/n) w p [ ( = c j= wq 2 j wq j= [ ( c[w] /q j= ) p /q ( wq 2 j wq [ c[w] /q ( c[w] j= c[w] α/n 2 j wq 2 j ) p /q] /p ) p j/q 2 j ) p /q ] /p 2 j wp ] /p 2 j Note that the next to last line follows from (56) and an immediate inductive argument In the last line, we just use the equality /q + /p = α/n To conclude the second proof of Theorem 24 we use (55) I α L p (w p ) L q, (w q ) [w p, w q ] Glo(p,q) [w] α/n We conclude this section by verifying that (23) and (26) yield Theorem 26 Indeed I α L p (w p ) L q (w q ) I α L p (w p ) L q, (w q ) + I α L q (w q ) L p, (w p ) (59) [w] α n + [w ] α n A q,p since [w ] Aq,p = [w] p /q and since [w] Ap,q [w] ( α n ) max{, p q }

19 SHARP WEIGHTED BOUNDS 9 6 Proof of the sharp bounds for the fractional maximal function Proof of Theorem 29 First notice that M α Mα c where Mα c is the centered version Let x, a cube centered at x, u = w q, σ = w p and r = + q/p Noticing that p /q( α/n) = r /q, we proceed as in [3] to obtain ( ) p f dy 3 nr /q [w] p /q( α/n) /q( α/n) f α/n u() σ(3) α/n σ σ dy ( ) r c [w] p /q( α/n) /q M c u() α,σ(f/σ) q/r dy Taking the supremum over all cubes centered at x we have the pointwise estimate M c αf(x) c [w] p /q( α/n) M c u{m c α,σ(f/σ) q/r u }(x) r /q Using the fact that M u : L r (u) L r (u) with operator norm independent of u combined with Lemma 4, we get w M α f L q c M c αf L q (u) which is the desired estimate c [w] p /q( α/n) M c u{m c α,σ(f/σ) q/r u } r /q L r (u) c [w] p /q( α/n) fw L p, 7 Examples We will use the power weights considered in [2] to show that Theorems 24, 26, and 29 are sharp Suppose again < α < n with Let w δ (x) = x (n δ)/p p q = α n so that w δ, with [w δ ] Ap,q = [w q δ ] A +q/p δ q/p Then, if f δ (x) = x δ n χ B, where B is the unit ball in, we have For x B, M α f δ (x) w δ f δ L p δ /p C x n α B(, x ) and so we have w q δ M αf δ (x) q dx δ q B f δ (y) dy x δ n+α, δ x (δ n+α)q x (n δ) q p dx δ q

20 2 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES It follows that (7) δ /q p q c w δ Mf δ L q c [w δ ] ( α n ) w δ f δ L p δ ( α n ) δ /p = δ /q, showing Theorem 29 is sharp Next we show that the same example can be used to show that the exponent in Theorem 26 is sharp Assume first that p /q We simply observe that, pointwise, M α CI α for some universal constant C Then using the same w δ and f δ as above and the estimate in Theorem 26 we arrive at the estimate in equation (7) with M α replaced by I α, showing sharpness The case when p /q immediately follows by the duality arguments described after the proof of Theorem 23 Finally, we show that the exponent α/n in the estimate (72) I α f L q, (w q ) c [w] α/n fw L p from Theorem 24 is sharp for p By (33) (73) I α f L q, (w q ) c [w q ] α/n A +q/p fw L p, and if we let u = w q, (74) I α f L q, (u) c [u] α/n A +q/p f L p (u p/q ) Assume now that u A Then (74) yields (75) I α f L q, (u) c [u] α/n A f L p (u p/q ) Since p = pα, this is equivalent to q n (76) I α (u α n f) L q, (u) c [u] α/n A f L p (u) We now prove that (76) is sharp Let u(x) = x δ n with < δ < Then standard computations shows that (77) [u] A δ Consider the function f = χ B, where B is again the unit ball, we can compute its norm to be ( ) /p (78) f L p (u) = u(b) /p = c δ Let < x δ < be a parameter whose value will be chosen soon We have ( I α (u α/n y (δ )α/n f) L q, (u) sup λ u{ x < x δ : dy > λ} λ> x y α/n B ) /q

21 SHARP WEIGHTED BOUNDS 2 ( sup λ u{ x < x δ : λ> B\B(, x ) sup λ λ> = sup λ λ> c α,n 2δ ( u{ x < x δ : ( u{ x < x δ : c α,n B\B(, x ) ) y (δ )α/n /q dy > λ} x y α/n y (δ )α/n (2 y ) α/n dy > λ} ) /q δ ( x δα/n ) > λ} ( u{ x < x δ : c α,n δ ( x δα/n ) > c ) /q α,n 2δ } = c α,n 2δ u(b(, x δ)) /q if x δ = ( 2 )n/αδ It now follows that for < δ <, (79) I α (u α/n f) L q, (u) c ( x δ δ δ δ ) /q = c δ ( ) /q δ Finally, combining (77), (78), (79), and using that = α, we have that (75) is q p n sharp 8 Proof of the Sobolev-type estimate Proof of Theorem 27 Since f(x) ci ( f )(x) we can use Theorem 24 to obtain (8) f L q, (w q ) c[w] /n fw L p From this weak-type estimate we can pass to a strong one with the procedure that follows We use the so-called truncation method from [6] Given a non-negative function g and λ > we define its truncation about λ, τ λ g, to be g(x) λ τ λ g(x) = min{g, 2λ} min{g, λ} = g(x) λ λ < g(x) 2λ λ g(x) > 2λ A well-know fact about Lipschitz functions is that they are preserved by absolute values and truncations Define Ω k = {x : 2 k < f(x) 2 k+ } and let u = w q Then, ( ) ( /q ) /q ( f(x) w(x)) q dx f(x) q u(x) dx k {2 k+ < f(x) 2 k+2 } ( ) /q c 2 kq u(ω k+ ) k ( ) /p c 2 kp u(ω k+ ) p/q k ) /q

22 22 M T LACEY, K MOEN, C PÉREZ, AND R H TORRES Notice that if x Ω k+, then τ 2 k f (x) = 2 k > 2 k and hence Ω k+ {x : τ 2 k f (x) > 2 k } Furthermore, notice that τ 2 k( f ) = f χ Ωk f χ Ωk, ae Continuing and using the weak-type estimate (8) we have ( ) /p f L q (w q ) c (2 k u({x : τ 2 k f (x) > 2 k }) /q ) p k c [w] /n ( k Ω k ( τ 2 k f (x) w(x)) p dx ) /p c [w] /n ( ( f(x) w(x)) p dx) /p, since p < q and the sets Ω k are disjoint This finishes the proof of the theorem References [] K Astala, T Iwaniec, and E Saksman, Beltrami operators in the plane, Duke Math J, 7 (2), no, [2] S Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans Amer Math Soc, 34 (993), no, [3] R Coifman and C Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math, 5 (974), [4] O Dragi cević, L Grafakos, C Pereyra, and S Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebegue spaces Publ Math 49 (25), no, 73-9 [5] M Gabidzashvili and V Kokilashvili, Two weight weak type inequalities for fractional type integrals, Ceskoslovenska Akademie Ved, 45 (989), - [6] J García-Cuerva and JL Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland Math Studies 6, North Holland, Amsterdam, 985 [7] L Grafakos, Classical Fourier Analysis, Springer-Verlag, Graduate Texts in Mathematics 249, Second Edition 28 [8] L Grafakos, Modern Fourier Analysis, Springer-Verlag, Graduate Texts in Mathematics 25, Second Edition 28 [9] L Grafakos and JM Martell, Extrapolation of weighted norm inequalities for multivariable operators and applications, J Geom Anal 4 (24), 9-46 [] E Harboure, R Macías, and C Segovia, Extrapolation results for classes of weights, Amer J Math, (988), [] P Hajlasz, Sobolev inequalities, truncation method, and John domains, Papers on analysis: A volume dedicated to Olli Martio on the occasion of his 6th birthday, Report Univ Jyvskyl Dep Math Stat, 83, p 9 26; Univ Jyvskyl, Jyvskyl, 2 [2] R Hunt, B Muckenhoupt, and R Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans Amer Math Soc, 76 (973), [3] A Lerner, An elementary approach to several results on the Hardy-Littlewood maximal operator, Proc AMS 36 (28), no 8, [4] A Lerner, S Ombrosi, and C Pérez, Sharp A bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden, Int Math Res Notices (28) Vol 28

23 SHARP WEIGHTED BOUNDS 23 [5] A Lerner, S Ombrosi and C Pérez, A bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden Math Res Lett 6, to appear (29) [6] R Long and F Nie, Weighted Sobolev inequalities and eigenvalue estimate of Schrödinger operators, Lecture Notes in Math, 494 (99), 3-4 [7] B Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans Amer Math Soc, 65 (972), [8] B Muckenhoupt and R Wheeden, Weighted norm inequalities for fractional integrals, Trans Amer Math Soc, 92 (974), [9] C Pérez, Two weight inequalities for potential and fractional type maximal operators, Indiana Univ Math J 43, No 2 (994), [2] C Pérez, Sharp L p -weighted Sobolev inequalities, Ann Inst Fourier 45 (995), no 3, -6 [2] S Petermichl, The sharp bound for the Hilbert transform in weighted Lebesgue spaces in terms of the classical A p characteristic, Amer J Math 29 (27), no 5, [22] S Petermichl, The sharp weighted bound for the Riesz transforms, The sharp weighted bound for the Riesz transforms Proc Amer Math Soc 36 (28), no 4, [23] S Petermichl and A Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math J, 2 (22), no 2, [24] J L Rubio de Francia, Factorization theory and A p weights, Amer J Math 6 (984), no 3, [25] E Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math 75 (982), - [26] E Sawyer A two weight weak type inequality for fractional integrals, Trans Amer Math Soc 28 (984), no, [27] E Sawyer, A characterization of a two-weight norm inequality for fractional and poisson integrals, Trans Amer Math Soc 38 (988), no 2, [28] E Sawyer and R Wheeden, Weighted inequlities for fractional integrals on euclidean and homogeneous spaces, Amer J Math 4 (992), [29] E Stein, Singular Integrals and Differentiability Properties of Functions, Princton Univ Press, Princeton, New Jersey, 97 Michael T Lacey, Department of Mathematics, Georgia Institute of Technology, Atlanta GA 3332, USA address: lacey@mathgatechedu Kabe Moen, Department of Mathematics, University of Kansas, 45 Snow Hall 46 Jayhawk Blvd, Lawrence, Kansas , USA address: moen@mathkuedu Carlos Pérez, Departamento De Análisis Matemático, Facultad de Matemáticas, Universidad De Sevilla, 48 Sevilla, Spain address: carlosperez@uses Rodolfo H Torres, Department of Mathematics, University of Kansas, 45 Snow Hall 46 Jayhawk Blvd, Lawrence, Kansas , USA address: torres@mathkuedu