Sharp weighted estimates for classical operators

arXiv:1001.4254v2 [math.CA] 13 Mar 2011 SHARP WEIGHTED ESTIMATES FOR CLASSICAL OPERATORS DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ Date: October 19, 2010. 1991 Mathematics Subject Classification. 42B20, 42B25. Key words and phrases. Apweights, Haar shift operators, singular integral operators, Hilbert transform, Riesz transforms, Beurling-Ahlfors operator, dyadic square function, vector-valued maximal operator. The first author was supported by a grant from the Faculty Research Committee and the Stewart-Dorwart Faculty Development Fund at Trinity College; the first and third authors are supported by grant MTM2009-08934 from the Spanish Ministry of Science and Innovation; the second author is supported by grant MTM2007-60952 from the Spanish Ministry of Science and Innovation and by CSIC PIE 200850I015. 1 2 Abstract. We give a general method based on dyadic Calder´onZygmund theory to prove sharp one and two-weight norm inequalities for some of the classical operators of harmonic analysis: the Hilbert and Riesz transforms, the Beurling-Ahlfors operator, the maximal singular integrals associated to these operators, the dyadic square function and the vector-valued maximal operator. In the one-weight case we prove the sharp dependence on the Apconstant by finding the best value for the exponent α(p) such that kT fkLp(w)≤Cn,T [w]α(p) ApkfkLp(w). For the Hilbert transform, the Riesz transforms and the BeurlingAhlfors operator the sharp value of α(p) was found by Petermichl and Volberg [47,48,49]; their proofs used approximations by the dyadic Haar shift operators, Bellman function techniques, and twoweight norm inequalities. Our proofs again depend on dyadic approximation, but avoid Bellman functions and two-weight norm inequalities. We instead use a recent result due to A. Lerner [34] to estimate the oscillation of dyadic operators. By applying this we get a straightforward proof of the sharp dependence on the Ap constant for any operator that can be approximated by Haar shift operators. In particular, we provide a unified approach for the Hilbert and Riesz transforms, the Beurling-Ahlfors operator (and their corresponding maximal singular integrals), dyadic paraproducts and Haar multipliers. Furthermore, we completely solve the open problem of sharp dependence for the dyadic square functions and vector-valued Hardy-Littlewood maximal function. In the two-weight case we use the very same techniques to prove sharp results in the scale of Apbump conditions. For the singular integrals considered above, we show they map Lp(v) into Lp(u), 1< p < ∞, if the pair (u, v) satisfies sup Q ku1/pkA,Qkv−1/pkB,Q <∞, where ¯ A∈Bp′and ¯ B∈Bpare Orlicz functions. This condition is sharp. Furthermore, this condition characterizes (in the scale of these Apbump conditions) the corresponding two-weight norm inequality for the Hardy-Littlewood maximal operator Mand its dual: i.e., M:Lp(v)−→ Lp(u) and M:Lp′(u1−p′)−→ Lp(v1−p′). Muckenhoupt and Wheeden conjectured that these two inequalities for Mare sufficient for the Hilbert transform to be bounded from Lp(v) into Lp(u). Thus, in the scale of Apbump conditions, we prove their conjecture. We prove similar, sharp two-weight results for the dyadic square function and the vector-valued maximal operator. SHARP WEIGHTED ESTIMATES 3 1. Introduction The problem of proving one and two-weight norm inequalities for the classical operators of harmonic analysis—singular integrals, square functions, maximal operators—has a long and complex history. In the one weight case, the (nearly) universal sufficient and (often) necessary condition for an operator to be bounded on Lp(w) is the Apcondition: given 1 < p < ∞, a weight w(i.e., a non-negative, locally integrable function) is in Apif [w]Ap= sup Q− ZQ w(x)dx− ZQ w(x)1−p′dxp−1 <∞, where the supremum is taken over all cubes in Rnand − RQw(x)dx = |Q|−1RQw(x)dx. For more on one-weight inequalities we refer the reader to [13,18,21]. An important question is to determine the best constant in terms of the Apconstant [w]Ap. More precisely, given an operator T, find the smallest power α(p) such that kTfkLp(w)≤Cn,T [w]α(p) ApkfkLp(w). This problem was first investigated by Buckley [3]. More recently, it has attracted renewed attention because of the work of Astala, Iwaniec and Saksman [1]. They proved that sharp regularity results for solutions to the Beltrami equation hold provided that the Beurling-Ahlfors operator satisfies α(p) = 1 for p > 2. The problem of characterizing the weights that govern the two-weight norm inequalities for classical operators is still open and there are several approaches to finding sufficient conditions on weights for an operator to be bounded from Lp(v) to Lp(u). One approach is to replace the two-weight Apcondition with the Ap“bump” condition: sup Q ku1/pkA,Qkv−1/pkB,Q <∞, where Aand Bare Young functions and the norms are localized Orlicz norms slightly larger than the Lpand Lp′norms. (Precise definitions will be given below.) Sufficient growth conditions on Aand Bare known for many operators and this has led to a number of conjectures on sharp sufficient conditions. For the history of this approach we refer the reader to [5,7,8,10,11]. In this paper we develop a unified approach to both of these problems and the results we get are sharp. We consider one and two-weight norm inequalities for singular integrals, maximal singular integrals, the 4 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ dyadic paraproduct, the dyadic square function and the vector-valued maximal operator. The results in the one-weight case for singular integrals are not new, but we believe that our proofs are simpler than existing proofs. The remaining theorems, however, are all new. We believe that our approach shows that there is a deep connection between sharp results in the one and two-weight case. Further, key to our approach is that the operators are either dyadic or can be approximated by dyadic operators (e.g., by the Haar shift operators defined below). Thus our results will extend to any operator that can be approximated in this way. Singular integrals. It is conjectured that if Tis any Calder´on-Zygmund singular integral operator, then for any p, 1 < p < ∞, and for any w∈Ap, (1.1) kTfkLp(w)≤CT,n,p [w]max(1,1 p−1) ApkfkLp(w). This inequality is true if Tis the Hilbert transform, a Riesz transform or the Beurling-Ahlfors operator. Theorem 1.1. Given p,1< p < ∞, if Tis the Hilbert transform, a Riesz transform or the Beurling-Ahlfors operator, then for all w∈Ap inequality (1.1)holds. This result was first proved by Petermichl [47,48] and Petermichl and Volberg [49]. For each operator the proof requires several steps. First, it is enough to prove the case p= 2; the other values of pfollow from a version of the Rubio de Francia extrapolation theorem with sharp constants due to Dragiˇcevi´c et al. [12] (Theorem 2.2 below). Second, for each of the above operators the problem is reduced to proving the weighted L2inequality for a corresponding dyadic operator by proving that the given operator can be approximated by integral averages of the dyadic operators (and their analogs defined on translations and dilations of the standard dyadic grid). Finally, the desired inequality was proved for each of these dyadic operators using Bellman function techniques and two-weight norm inequalities. Recently, Lacey, Petermichl and Reguera-Rodriguez [28] gave a proof of the sharp A2constant for a large family of Haar shift operators that includes all of the dyadic operators needed for the above results. Their proof avoids the use of Bellman functions, and instead uses a deep, twoweight “Tb theorem” for Haar shift operators due to Nazarov, Treil and Volberg [40]. We give a different and simpler proof that uses approximation by dyadic Haar shifts but avoids both Bellman functions and two-weight SHARP WEIGHTED ESTIMATES 5 norm inequalities such as the T b theorem. Instead, we use a very interesting decomposition argument based on local mean oscillation recently developed by Lerner [30] to prove the corresponding result for dyadic Haar shifts. Intuitively, this decomposition may be thought of as a version of the Calder´on-Zygmund decomposition of a function, replacing the mean by the median. (We will make this more precise below.) Theorem 1.1 was announced in [6]. Remark 1.2.After this paper was completed we learned of several other related results. First, Vagharsyakhan [52] has shown that in one dimension, all convolution-type Calder´on-Zygmund singular integral operators with sufficiently smooth kernel can be approximated by Haar shifts. Second, Lacey et al. [25] used a deep characterization of the one-weight problem in [45] to prove Theorem 1.1 for all singular integrals with sufficiently smooth kernels. Third, Lerner [35] proved Theorem 1.1 for any convolution-type Calder´on-Zygmund singular integrals provided p≥3 or 1 < p ≤3/2. Finally, Hyt¨onen [24] proved Theorem 1.1 for all singular integrals and all p > 1, thus solving the so-called A2conjecture. His proof is extremely technical: it is based on the approach in [45] and a refinement of the arguments in [28]. A simpler proof of the A2 conjecture based upon the previous three papers appears in [26]. An important advantage of our approach is that it also yields sharp two-weight norm inequalities. To state our result we need a few definitions. A Young function is a function A: [0,∞)→[0,∞) that is continuous, convex and strictly increasing, A(0) = 0 and A(t)/t → ∞ as t→ ∞. Given a cube Qwe define the localized Luxemburg norm by kfkA,Q = inf λ > 0 : − ZQ A|f(x)| λdx ≤1. When A(t) = tp, 1 < p < ∞, we write kfkp,Q =− ZQ |f(x)|pdx1/p . The associate function of Ais the Young function ¯ A(t) = sup s>0 {st −A(s)}. A Young function Asatisfies the Bpcondition if for some c > 0, Z∞ c A(t) tp dt t<∞. 6 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ Important examples of such functions are of the form A(t) = tplog(e+ t)−1−ǫ,ǫ > 0, which have associate functions ¯ A(t)≈tp′log(e+t)p′−1+δ, δ > 0. Theorem 1.3. Given p,1< p < ∞, let Aand Bbe Young functions such that ¯ A∈Bp′and ¯ B∈Bp. Then for any pair of weights (u, v) such that (1.2) sup Q ku1/pkA,Qkv−1/pkB,Q <∞, we have that (1.3) kTfkLp(u)≤CkfkLp(v), where Tis the Hilbert transform, a Riesz transform, or the BeurlingAhlfors operator. Condition (1.2) is referred to as an Apbump condition: when A(t) = tpand B(t) = tp′, we get the two-weight Apcondition. Theorem 1.3 was proved in [5] for the Hilbert transform in the special case that A(t) = tplog(e+t)p−1+δ,δ > 0 (here ¯ A∈Bp′), and for Riesz transforms (indeed, for any Calder´on-Zygmund singular integral) given the additional hypothesis that p > n. Examples (see [7,8]) show that in this particular case these results are sharp, since they are false in general if we take δ= 0 (when ¯ A6∈ Bp′). Theorem 1.3 was proved for the Hilbert transform and general singular integrals when p > n by Lerner [34] by combining his decomposition argument with the arguments in [5]. Two-weight inequalities were first considered by Muckenhoupt [37], who noted that the same proof as in the one-weight case immediately shows that for all p, 1 ≤p < ∞, (u, v)∈Apif and only if the maximal operator satisfies the weak (p, p) inequality. However, Muckenhoupt and Wheeden [38] soon showed that while the two-weight Apcondition is necessary for the strong (p, p) inequality for the maximal operator and the strong and weak type inequalities for the Hilbert transform, it is not sufficient. This led Muckenhoupt and Wheeden to focus not on the structural or geometric properties of Apweights but on their relationship to the maximal operator, in particular, the fact that w∈ Apwas necessary and sufficient for the maximal operator to be bounded on Lp(w) and Lp′(w1−p′). They made the following conjecture that is still open: a sufficient condition for the Hilbert transform to satisfy the strong (p, p) inequality H:Lp(v)→Lp(u), 1 < p < ∞, is that the maximal operator satisfies the pair of inequalities (1.4) M:Lp(v)→Lp(u), M :Lp′(u1−p′)→Lp′(v1−p′). SHARP WEIGHTED ESTIMATES 7 Bump Apconditions were first considered by Neugebauer [41] who showed the following striking result: a pair of weights (u, v) satisfies (1.2) with power bumps A(t) = tr p,A(t) = tr p′for some r > 1 if and only if there exist w∈Apand positive constants c1, c2such that c1u(x)≤w(x)≤c2v(x). From this condition we immediately get a large number of two-weight norm inequalities as corollaries to the analogous one-weight results. In particular, we get the two inequalities (1.4). An immediate question was whether this condition could be weakened and still get that the maximal operator satisfies M:Lp(v)→ Lp(u). This was answered in [43], where it was shown that a sufficient condition for (1.4) was that the pair of weights satisfies (1.2) with ¯ A∈Bp′and ¯ B∈Bp. The centrality of these Bpconditions is shown by the fact that they are sharp within the scale of Orlicz bumps as shown in [43]. This led naturally to the following version of the conjecture of Muckenhoupt and Wheeden: a sufficient condition on the pair of weights (u, v) for any singular integral to satisfy T:Lp(v)→Lp(u) is that (1.2) holds. Progress on this conjecture was made in [11,5,34]. Theorem 1.3 completely solves it for the Hilbert and Riesz transforms and the Beurling-Ahlfors operator, and as we noted above it is the best possible result in the scale of Bpbumps. See [7] for further details and references on this topic. In the past decade, a great deal of attention has been focused on proving that “testing conditions” are necessary and sufficient for twoweight norm inequalities for singular integrals. (See Nazarov, Treil and Volberg [39,53,40] and the recent preprints by Lacey, Sawyer and Uriarte-Tuero [29,30].) More precisely, given a singular integral T, it is conjectured that T:Lp(v)→Lp(u) if and only if for every cube Q, ZQ |T(v1−p′χQ)(x)|pu(x)dx ≤CZQ v(x)1−p′dx ZQ |T(uχQ)(x)|p′v(x)1−p′dx ≤CZQ u(x)dx. The necessity of these conditions is immediate. The best known results are for p= 2; partial results (with additional hypotheses) are known for other values of p. These results are of great interest not only because of the elegance of this conjecture but also because of their connection with Tb-theorems on non-homogeneous spaces (see [53] and the references it contains). Testing conditions and Apbump conditions are not readily comparable: they represent two fundamentally different approaches to the two-weight problem. While both approaches are important, we believe 8 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ that bump conditions have several advantages over testing conditions. First, they are universal, geometric conditions: they are independent of the operators and any pair yields norm inequalities for a range of operators. Second, they are much easier to check than the testing conditions, and it is very easy to construct examples of weights that do and do not satisfy a given bump condition. (For many examples and a general technique for constructing them, see [7].) Third, they are not tied to L2, unlike testing conditions where the transition from p= 2 to all phas proved to be very difficult. (In this regard, we note that in [39] it was claimed—without proof—that in the specific case they were considering, testing conditions were not sufficient.) Maximal singular integrals. Given a singular integral Twith convolution kernel K, recall that the associated maximal singular integral is defined by T∗f(x) = sup ǫ>0 |Tǫf(x)|= sup ǫ>0Z|y|>ǫ K(y)f(x−y)dy. Somewhat surprisingly, both Theorem 1.1 and Theorem 1.3 remain true if the singular integral is replaced by the associated maximal singular integral. Theorem 1.4. Given p,1< p < ∞, and w∈Ap, then inequality (1.1) holds if Tis replaced by T∗, where Tis the Hilbert transform, a Riesz transform or the Beurling-Ahlfors operator. Similarly, if the pair (u, v) satisfies (1.2), then inequality (1.3)holds if Tis replaced by T∗. In the one-weight case, Theorem 1.4 was proved very recently by Hyt¨onen et al. [23]. Their proof used a very general family of “maximal” dyadic shift operators and a characterization of the two-weight norm inequalities for maximal singular integrals due to Lacey, Sawyer and Uriarte-Tuero [29]. In the two-weight case this result is new. In both the one and two-weight case our approach is to prove the corresponding result for the associated maximal dyadic shift operators. Remark 1.5.Very recently, Lerner [35] has proved Theorem 1.4 in the one-weight case for general Calder´on-Zygmund maximal singular integral operators when p > 3. Dyadic paraproducts and constant Haar multipliers. Let ∆ denote the collection of dyadic cubes in R. We consider two operators defined on the real line. A function bis in dyadic BMO, we write SHARP WEIGHTED ESTIMATES 9 b∈BMOd, if kbk∗,d = sup I∈∆− ZI |b(x)−bI|2dx1/2 <∞, where bI=− RIb(x)dx. Given a dyadic interval I,I+and I−are its right and left halves, and the Haar function hIis defined by hI(x) = |I|−1/2χI−(x)−χI+(x). Define the dyadic paraproduct πbby πbf(x) = X I∈∆ fIhb, hIihI(x). For an overview of the history and properties of the dyadic paraproduct, we refer the reader to Pereyra [42]. Theorem 1.6. Given a function b∈BMOd, and p,1< p < ∞, then for all w∈Ap, kπbfkLp(w)≤Cpkbk∗,d [w]max(1,1 p−1) ApkfkLp(w). Furthermore, given a pair (u, v)that satisfies (1.2), then kπbfkLp(u)≤Ckbk∗,dkfkLp(v). In the one-weight case, Theorem 1.6 was first proved by Beznosova [2] using Bellman function techniques. A different proof that avoided Bellman functions but used two-weight inequalities was given in [23]. Given a sequence α={αI}I∈∆∈ℓ∞, define the constant Haar multiplier Tαby Tαf(x) = X I∈∆ αIhf, hIihI(x). If αI= 1, then Tαis the identity operator. For more on the properties of these operators, see Pereyra [42]. The analog of Theorem 1.6 is true for constant Haar multipliers. Theorem 1.7. Given a sequence α={αI}I∈∆∈ℓ∞, and p,1< p < ∞, then for all w∈Ap, kTαfkLp(w)≤Cpkαkℓ∞[w]max(1,1 p−1) ApkfkLp(w). Furthermore, given a pair (u, v)that satisfies (1.2), then kTαfkLp(u)≤Ckαkℓ∞kfkLp(v). In the special case when αI=±1, Theorem 1.7 was proved by Wittwer [56]. 16 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ (fχQ)∗(λ|Q|)≤1 λ|Q|ZQ |f|pdx1/p .(3.2) Furthermore, (3.3) |mf(Q)| ≤ (fχQ)∗(|Q|/2); in particular, if f∈Lpfor any p > 0, then mf(Q)→0as |Q| → ∞. Proof. Inequality (3.2) follows immediately from (3.1). To prove this inequality, fix α < (fχQ)∗(λ|Q|). Then λ−1/p kfkLp,∞(Q,|Q|−1dx)≥λ−1/p|Q|−1/pα|{x∈Q:|f(x)|> α}|1/p ≥α. Since this is true for all such α, (3.1) follows at once. To prove (3.3) we consider two cases. Suppose mf(Q)≥0. Define m+= sup{β≥mf(Q) : |{x∈Q:f(x)< β}| ≤ |Q|/2}; then 0 ≤mf(Q)≤m+so it will be enough to prove m+≤f∗(|Q|/2). Take any α > 0 such that |{x∈Q:|f(x)|> α}| <|Q|/2. Then |{x∈Q:f(x)≤α}| ≥ |{x∈Q:|f(x)| ≤ α}| >|Q|/2. Hence, for any β > α, |{x∈Q:f(x)< β}| ≥ |{x∈Q:f(x)≤α}| >|Q|/2, and so β≥m+. Since this is true for all β > α, we have that α≥m+, and taking the infimum of all such αwe get that (fχQ)∗(|Q|/2) ≥m+. Finally if mf(Q)<0, define g(x) = −f(x). Then −mf(Q) is a median of gand the previous case yields |mf(Q)|=−mf(Q)≤ g∗(|Q|/2) = f∗(|Q|/2).  Remark 3.2.Inequality (3.1) is central to our proofs as it allows us to use weak (1,1) inequalities directly in our estimates. By way of comparison, in [5] a key technical difficulty resulted from having to use Kolmogorov’s inequality rather than the weak (1,1) inequality for a singular integral. Overcoming this is the reason the results there were limited to log bumps. To state Lerner’s decomposition theorem, we generalize our notation slightly: given a cube Q0, let ∆(Q0) be the collection of dyadic cubes relative to Q0. Given Q∈∆(Q0), Q6=Q0, let b Qbe its dyadic parent: the unique cube in ∆(Q0) containing Qwhose side-length is twice that of Q. SHARP WEIGHTED ESTIMATES 17 Theorem 3.3. ([34]) Given a measurable function fand a cube Q0, for each k≥1there exists a (possibly empty) collection of pairwise disjoint cubes {Qk j} ⊂ ∆(Q0)such that if Ωk=SjQk j, then Ωk+1 ⊂Ωk and |Ωk+1 ∩Qk j| ≤ 1 2|Qk j|. Furthermore, for almost every x∈Q0, |f(x)−mf(Q0)| ≤ 4M# 1 4,Q0f(x) + 4 X k,j ω1 2n+2 (f, b Qk j)χQk j(x). Remark 3.4.If for all jand kwe define Ek j=Qk j\Ωk+1, then the sets Ek jare pairwise disjoint and |Ek j| ≥ 1 2|Qk j|. Remark 3.5.Though it is not explicit in [34], it follows at once from the proof that we can replace M# 1 4,Q0by the corresponding dyadic operator M#,d 1 4,Q0, where M#,d λ,Q f(x) = sup x∈Q′∈∆(Q) ωλ(f, Q′). Intuitively, one may think of the cubes {Qk j}as being the analog of the Calder´on-Zygmund cubes for the function f−mf(Q0) but defined with respect to the median instead of the mean. The cubes Qk jare maximal dyadic cubes with respect to a dyadic local sharp maximal operator. The terms on the right-hand side of the above inequality then play a role like that of the good and bad parts of the Calder´onZygmund decomposition. A key difference, of course, is that while the Calder´on-Zygmund decomposition is done at one “scale,” the above theorem requires that we estimate the local mean oscillation of fat all scales. 4. The Haar shift operators To prove Theorems 1.1 and 1.3 we need to prove the corresponding inequalities for certain dyadic operators that can be used to approximate the Hilbert transform, the Riesz transforms and the BeurlingAhlfors operator. We follow the approach used in [28] and consider simultaneously a family of dyadic operators—the Haar shift operators— that contains all the operators we are interested in. Let ∆ be the set of dyadic cubes in Rn. For our arguments we properly need to consider the sets ∆s,t,s∈Rn,t > 0, of translations and dilations of dyadic cubes. However, it will be immediate that all of our arguments for dyadic cubes extend to these more general families, so without loss of generality we will restrict ourselves to dyadic cubes. We define a Haar function on a cube Q∈∆ to be a function hQsuch that 18 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ (a) supp(hQ)⊂Q; (b) if Q′∈∆ and Q′(Q, then hQis constant on Q′; (c)khQk∞≤ |Q|−1/2; (d)RQhQ(x)dx = 0. Given an integer τ≥0, a Haar shift operator of index τis an operator of the form Hτf(x) = X Q∈∆X Q′,Q′′ ∈∆(Q) 2−τn|Q|≤|Q′|,|Q′′| aQ′,Q′′ hf, hQ′ihQ′′ (x), where aQ′,Q′′ is a constant such that |aQ′,Q′′ | ≤ C|Q′| |Q| |Q′′| |Q|1/2 . We say that Hτis a CZ Haar shift operator if it is bounded on L2. An important example of a Haar shift operator when n= 1 is the Haar shift (also known as the dyadic Hilbert transform) Hd, defined by Hdf(x) = X I∈∆ hf, hIihI−(x)−hI+(x), where, as before, given a dyadic interval I,I+and I−are its right and left halves, and hI(x) = |I|−1/2χI−(x)−χI+(x). Clearly hIis a Haar function on Iand one can write Hdas a Haar shift operator of index τ= 1 with aI′,I′′ =±1 for I′=I,I′′ =I± and aI′,I′′ = 0 otherwise. These are the operators used by Petermichl [46,47] to approximate the Hilbert transform. More precisely, she used the family of operators Hd s,t,s∈R,t > 0, which are defined as above but with the dyadic grid replaced by its translation by sand dilation by t. The Hilbert transform is then the limit of integral averages of these operators, so norm inequalities for Hfollow from norm inequalities for Hd s,t by Fatou’s lemma and Minkowski’s inequality. Similar approximations hold for the Riesz transforms and Beurling-Ahlfors operator, and we refer the reader to [48,49] for more details. To apply Theorem 3.3 to the Haar shift operators we need two lemmas. The first is simply that CZ Haar shift operators satisfy a weak (1,1) inequality. The proof of this is known but we could not find it in the literature, except when τ= 0—in this case Hτis a constant Haar multiplier and the proof is given in [42]. Therefore, we provide a brief SHARP WEIGHTED ESTIMATES 19 sketch of the details. Here and below we will use the following notation: given an integer τ≥0 and a dyadic cube Q, let Qτdenote its τ-th generation “ancestor”: that is, the unique dyadic cube Qτcontaining Qsuch that |Qτ|= 2τn|Q|. Lemma 4.1. Given an integer τ≥0, there exists a constant Cτ,n such that for every t > 0, |{x∈Rn:|Hτf(x)|> t}| ≤ Cτ,n tZRn |f(x)|dx. Proof. Fix t > 0 and form the Calder´on-Zygmund decomposition of fat height t. Decompose fas the sum of the good and bad parts, g+b. The estimate for gis standard. For b, since Lebesgue measure is doubling, it suffices to show that the set |{x∈Rn\(∪jQτ j) : |Hτb(x)|> t/2}| has measure 0. Fix jand x∈Rn\Qτ j; then we would be done if we could show that Hτbj(x) = 0. Fix a term aQ′,Q′′ hbj, hQ′ihQ′′ (x) in the sum defining Hτbj(x). If this is non-zero, then hQ′′ (x)6= 0, so Q′′ ∩Rn\Qτ j6= Ø. Since Q′′ ⊂Q,Q∩Rn\Qτ j6= Ø. On the other hand, since RQjbj(x)dx = 0, hbj, hQ′i 6= 0 only if Q′⊂Qj, which in turn implies that Q⊂Qτ j, a contradiction.  Our second lemma is a key estimate that is a sharper variant of a result known for Calder´on-Zygmund singular integrals (see [27]) and whose proof is similar. For completeness we include the details. Lemma 4.2. Given τ≥0, let Hτbe a CZ Haar shift operator. Fix λ, 0< λ < 1. Then for any function f, every dyadic cube Q0, and every x∈Q0, ωλ(Hτf, Q0)≤Cτ,n λ− ZQτ 0 |f(x)|dx, M#,d λ,Q0(Hτf)(x)≤Cτ,n λMdf(x). Proof. It suffices to prove the first inequality; the second follows immediately from the definition of M#,d λ,Q0. Fix Q0and write Hτas the sum of two operators: Hτf(x) = Hτ(fχQτ 0)(x) + Hτ(fχRn\Qτ 0)(x). We claim the second term is constant for all x∈Q0. Let Qbe any dyadic cube. Then the corresponding term in the sum defining 20 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ Hτ(fχRn\Qτ 0)(x) is (4.1) X Q′,Q′′ ∈∆(Q) 2−τn|Q|≤|Q′|,|Q′′| aQ′,Q′′ hfχRn\Qτ 0, hQ′ihQ′′ (x). We may assume that Q′′ ∩Q06= Ø (otherwise we get a zero term); since Q′′ ⊂Q, this implies that Q∩Qτ 06= Ø. Similarly, we have Q∩(Rn\Qτ 0)6= Ø. Therefore, Qτ 0(Q, so |Q0|<2−τn|Q| ≤ |Q′′|. Hence, Q0(Q′′ and hQ′′ is constant on Q0. Thus, (4.1) does not depend on xand so is constant on Q0. Denote this constant by Hτf(Q0); then |{x∈Q0:|Hτf(x)−Hτf(Q0)|> t}| =|{x∈Q0:|Hτ(fχQτ 0)(x)|> t}|. Since Hτis a CZ Haar shift operator it is weak (1,1). Therefore, by inequality (3.1), ωλ(Hτf, Q0)≤(Hτf−Hτf(Q0))χQ0∗(λ|Q0|) ≤λ−1kHτ(fχQτ 0)kL1,∞(Q0,|Q0|−1dx)≤Cτ,n λ− ZQτ 0 |f(x)|dx.  5. Singular integrals, paraproducts and Haar multipliers In this section we prove Theorems 1.1,1.3,1.6 and 1.7. The principal results are the first two for singular integrals; the results for paraproducts and constant Haar multipliers are variations of these and we will only sketch the changes. We will also indicate the technical obstacles in attempting to apply our results to general Calder´on-Zygmund singular integrals. One weight inequalities: proof of Theorem 1.1.As we discussed in the previous section, to prove Theorem 1.1 it will suffice to establish the analogous result for Haar shift operators. Theorem 5.1. Given an integer τ≥0and a CZ Haar shift operator Hτ, and given p,1< p < ∞, then for any w∈Ap, kHτfkLp(w)≤Cτ,n,p [w]max(1,1 p−1) ApkfkLp(w). Proof of Theorem 5.1.By Theorem 2.2 it will suffice to prove that kHτfkL2(w)≤Cτ,n[w]A2kfkL2(w). Fix w∈A2and fix f. By a standard approximation argument we may assume without loss of generality that fis bounded and has compact SHARP WEIGHTED ESTIMATES 21 support. Let Rn j, 1 ≤j≤2n, denote the n-dimensional quadrants in Rn: that is, the sets I±×I±× · · · × I±where I+= [0,∞) and I−= (−∞,0). For each j, 1 ≤j≤2n, and for each N > 0 let QN,j be the dyadic cube adjacent to the origin of side length 2Nthat is contained in Rn j. Since QN,j ∈∆, ∆(QN,j)⊂∆. Because Hτis a CZ shift operator, it is bounded on L2. Thus, since f∈L2, by (3.3) and (3.2), mHτf(QN,j)→ 0 as N→ ∞. Therefore, by Fatou’s lemma and Minkowski’s inequality, kHτfkL2(w)≤lim inf N→∞ 2n X j=1 ZQN,j |Hτf(x)−mHτf(QN,j)|2w(x)dx!1/2 . Hence, it will suffice to prove that each term in the sum on the right is bounded by Cτ,n[w]A2kfkL2(w). Fix jand let QN=QN,j. By Theorem 3.3 and Lemma 4.2, for every x∈QNwe have that |Hτf(x)−mHτf(QN)|(5.1) ≤4M#,d 1 4,QN(Hτf)(x) + 4 X j,k ω1 2n+2 (Hτf, b Qk j)χQk j(x) ≤Cτ,n Mf(x) + Cτ,n X j,k − ZPk j |f(x)|dx!χQk j(x) =Cτ,n Mf(x) + Cτ,n F(x), where Pk j= ( b Qk j)τ. We get the desired estimate for the first term from Theorem 2.1 with p= 2: kMfkL2(QN,w)≤ kMfkL2(w)≤Cn[w]A2kfkL2(w). To estimate Fwe use duality. Fix a non-negative function h∈L2(w) with khkL2(w)= 1; then by Remark 3.4 and Lemma 2.3 we have that ZQN F(x)h(x)w(x)dx =Cτ,n X j,k − ZPk j |f(x)|dx ZQk j w(x)h(x)dx ≤2·2(τ+1)nX j,k w(Pk j) |Pk j| w−1(Pk j) |Pk j||Ek j| ×1 w−1(Pk j)ZPk j |f(x)|w(x)w(x)−1dx ×1 w(Qk j)ZQk j h(x)w(x)dx 22 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ ≤Cτ,n [w]A2X j,k ZEk j Md w−1(fw)(x)Md wh(x)dx ≤Cτ,n [w]A2ZRn Md w−1(fw)(x)Md wh(x)dx ≤Cτ,n [w]A2ZRn Md w−1(fw)(x)2w(x)−1dx1/2 ×ZRn Md wh(x)2w(x)dx1/2 ≤Cτ,n [w]A2ZRn |f(x)w(x)|2w(x)−1dx1/2 ×ZRn h(x)2w(x)dx1/2 =Cτ,n [w]A2ZRn |f(x)|2w(x)dx1/2 . If we take the supremum over all such functions h, we conclude that kFkL2(QN,w)≤Cτ,n [w]A2kfkL2(w). Combining our estimates we have that ZQN |Hτf(x)−mHτf(QN)|2w(x)dx1/2 ≤Cτ,n [w]A2kfkL2(w), and this completes the proof.  Two weight inequalities: Proof of Theorem 1.3.To prove Theorem 1.3 it will suffice to establish the corresponding result for the Haar shift operators. We record this as a separate result. Theorem 5.2. Given an integer τ≥0, let Hτbe a CZ Haar shift operator. Given p,1< p < ∞, and let Aand Bbe Young functions such that ¯ A∈Bp′and ¯ B∈Bp. Then for any pair (u, v)such that (1.2) holds we have that kHτfkLp(u)≤CkfkLp(v). Proof of Theorem 5.2.The proof is very similar to the proof of Theorem 5.1 replacing the A2estimates with an argument from the second half of the proof of the main theorem in [5]; therefore, we omit many of the details. SHARP WEIGHTED ESTIMATES 23 We argue as in the one-weight case and with the same notation; it will suffice to prove ZQN |Hτf(x)−mHτf(QN)|pu(x)dx1/p ≤CkfkLp(v) and we use (5.1). The estimate of the term containing the maximal operator is straightforward: by Theorem 2.5 we have M:Lp(v)→ Lp(u) since the pair (u, v) satisfies (1.2). Therefore, by duality (this time with respect to Lebesgue measure) it is enough to show that for every non-negative h∈Lp′(Rn) with khkLp′= 1, I=ZQN F(x)u(x)1/p h(x)dx ≤CkfkLp(v). We apply Remark 3.4 and the generalized H¨older’s inequality to get I≤CX j,k − ZPk j |f(x)|dx− ZQk j u(x)1/ph(x)dx |Ek j| ≤CX j,k kfv1/pk¯ B,P k jkv−1/pkB,P k jku1/pkA,Qk jkhk¯ A,Qk j|Ek j|. By convexity, ku1/pkA,Qk j≤2n(τ+1)ku1/pkA,P k j, so since the pair (u, v) satisfies (1.2), I≤CX j,k ZEk j M¯ B(fv1/p)(x)M¯ Ah(x)dx ≤CZRn M¯ B(fv1/p)(x)M¯ Ah(x)dx. Since ¯ A∈Bp′and ¯ B∈Bp, by Theorem 2.4,M¯ Bis bounded on Lpand M¯ Ais bounded in Lp′. The desired estimate now follows by H¨older’s inequality.  General Calder´on-Zygmund singular integrals. Key to the proofs of Theorems 5.1 and 5.2 are the sharp estimates for the local mean oscillation in Lemma 4.2. If we were to try to extend these proofs to an arbitrary Calder´on-Zygmund singular integral T, then we would have to estimate the local mean oscillation by a sum (see [27]): ωλ(Tf, Q)≤C ∞ X i=0 2−i− Z2iQ |f(x)|dx. If we use this estimate in the proof of Theorem 5.1, then we still get that Tis bounded (since the sum is bounded by 2 infx∈QMf(x)), but we get an additional factor of [w]A2. The proof of Theorem 5.2 24 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ can be modified to handle this sum, but to get convergence you need the additional assumption that p > n. This is the approach used by Lerner [34]. This (seemingly artificial) restriction p > n also appears in [5]. It would be very interesting to find a refinement of Theorem 3.3 that would let us remove this restriction. Alternatively, it is tempting to conjecture that the estimate above could be improved by replacing 2−iby 2−(n+ǫ)i, which would be sufficient to adapt the proofs in both the one and two-weight case. However, it is not clear that such an inequality is true, even for singular integrals with smooth kernels. Dyadic paraproducts and Haar multipliers: Proof of Theorems 1.6 and 1.7.The proof of Theorem 1.6 is essentially identical to the proof of the corresponding results for singular integrals once we prove the analog of Lemma 4.2: ωλ(πbf, Q)≤Ckbk∗,d λ− ZQ |f(x)|dx. The proof follows as before. The dyadic paraproduct is a local operator, since for any I∈∆, hIis constant on proper dyadic sub-intervals of I, and so, given a fixed dyadic interval I0,πb(fχR\I0) is constant on I0. Furthermore, πbis bounded on Lpand satisfies a weak (1,1) inequality: for every t > 0, |{x∈R:|πbf(x)|> t}| ≤ Ckbk∗,d tZR |f(x)|dx. For a proof, see Pereyra [42]. Theorem 1.7 is actually a special case of Theorems 5.1 and 5.2, since the constant Haar multipliers are clearly Haar shift operators of index τ= 0. The dependence on kαkℓ∞follows at once by linearity. 6. Maximal singular integrals In this section we prove Theorem 1.4. To do so, we will follow the approach used by Hyt¨onen et al. [23] and actually prove the corresponding result for a family of “maximal” dyadic shift operators. The underlying dyadic operators are a generalization of the Haar shift operators defined in Section 4. As noted in [23], the results for the maximal singular integrals associated to the Hilbert transform, the Riesz transforms and the Beurling-Ahlfors operator are gotten by the same approximation arguments as we discussed above. We begin by defining the appropriate shift operators. To distinguish them from the operators defined above, we will refer to them as generalized Haar shift operators. (In [23] they are simply referred to as SHARP WEIGHTED ESTIMATES 25 Haar shift operators, but our change in terminology should not cause any confusion.) We say that an operator Tis a generalized Haar shift operator of index τ≥0 if Tf =X Q∈∆ hf, gQiγQ, where the functions γQare such that: (a) supp(γQ)⊂Q; (b) if Q′∈∆ and Q′⊂Qwith |Q′| ≤ 2−τ n |Q|, then γQis constant on Q′; (c)kγQk∞≤ |Q|−1/2. The functions gQalso have these properties. Finally we assume that the functions γQ, gQare such that Textends to a bounded operator on L2. Together, these hypotheses imply that Tis of weak-type (1,1) (see [23]). Examples of generalized Haar shift operators include the dyadic paraproducts and their adjoints. Associated with a generalized Haar shift operator Tis the maximal Haar shift operator T∗f= sup ǫ>0 |Tǫf|= sup ǫ>0X Q∈∆ |Q|≥ǫn hf, gQiγQ, We again have that T∗is bounded on L2and is of weak-type (1,1) (see [23]). Our main result for maximal Haar shift operators is the following. Theorem 6.1. Let Tbe a generalized Haar shift operator of index τ≥0, and let T∗be the corresponding maximal Haar shift operator. Then, for every p,1< p < ∞, and for all w∈Ap, kT∗fkLp(w)≤Cτ,n,p[w]max(1,1 p−1) ApkfkLp(w). Furthermore, if the pair of weights (u, v)satisfies (1.2), then kT∗fkLp(u)≤CkfkLp(v). The proof of Theorem 6.1 is very much the same as the proofs of Theorem 5.1 and 5.2, so we will only describe the differences between the two arguments. First, if fis bounded and has compact support, supǫ>0|mTǫf(Q)| → 0 as |Q| → ∞. Indeed, by (3.3) and (3.2), sup ǫ>0 |mTǫf(Q)| ≤ 21/p sup ǫ>01 |Q|ZQ |Tǫf|2dx1/2 ≤ |Q|−1/2kT∗fkL2, 32 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ ≤CZRn M¯ B(fv1/p)(x)pdx ≤Ckfkp Lp(v). Combining these two estimates we get the desired inequality. Now suppose that p > 2. In this case the proof is very similar to the proof of Theorem 5.2 and we highlight the changes. To use duality with respect to Lebesgue measure, fix a non-negative function h∈L(p/2)′(Rn) with khkL(p/2)′= 1. Then (1.7) gives ZQN F(x)u(x)2/p h(x)dx ≤CX j,k − Zb Qk j |f(x)|dx!2 − ZQk j u(x)2/ph(x)dx |Ek j| ≤CX j,k kfv1/pk2 ¯ B, b Qk jkv−1/pk2 B, b Qk jku2/pkA,Qk jkhk¯ A,Qk j|Ek j| ≤CX j,k ZEk j M¯ B(fv1/p)(x)2M¯ Ah(x)dx ≤CZRn M¯ B(fv1/p)(x)2M¯ Ah(x)dx ≤CkM¯ B(fv1/p)(x)k2 LpkM¯ AhkL(p/2)′ ≤Ckfk2 Lp(v), where we have used H¨older’s inequality, Theorem 2.4 and the fact that ¯ A∈B(p/2)′and ¯ B∈Bp. The desired estimate follows at once if we take the supremum over all such functions h. 8. The vector-valued maximal operator Our two results for the vector-valued maximal operator are exact parallels of our results for the dyadic square function. Formally, the change only requires replacing “2” by “q”, 1 < q < ∞, and in fact, the proofs do adapt readily as we will sketch below. As with singular integral operators, in order to prove sharp results for vector-valued maximal operator, we need to consider a dyadic operator. Recall that the dyadic maximal operator is defined by Mdf(x) = sup Q∈∆ Q∋x − ZQ |f(y)|dy. SHARP WEIGHTED ESTIMATES 33 Given q > 1 and f={fi}, define the dyadic vector valued maximal operator by Md qf(x) = ∞ X i=1 Mdfi(x)q!1/q . By an argument that goes back to C. Fefferman and Stein [14] (see also [50] and [18]), the maximal operator can be approximated by the dyadic maximal operator and the analogous operator defined on all translates of the dyadic grid. Therefore, by a straightforward argument using Fatou’s lemma and Minkowski’s inequality, to prove weighted norm inequalities for the vector-valued maximal operator it suffices to prove them for Md q. (For the details of this argument, see [7].) Again like the dyadic square function, the key estimate for the dyadic vector-valued maximal operator is to control the local mean oscillation of (Md qf)q. Lemma 8.1. Fix λ,0< λ < 1, and q,1< q < ∞. Then for any function f={fi}, every dyadic cube Q0, and every x∈Q0, ωλ((Md qf)q, Q0)≤Cn,q λq− ZQ0 kf(x)kℓqdxq , M#,d λ,Q0((Md qf)q)(x)≤Cn,q λqMd(kf(·)kℓq)(x)q. Proof. The second estimate again follows from the first. To prove the first, fix Q0. Then for every x∈Q0and every i≥1, we observe that Mdfi(x) = max  Md(fiχQ0)(x),sup Q∈∆ Q0⊂Q − ZQ |fi(y)|dy . The second term on the right is constant; using this we define K0=  ∞ X i=1  sup Q∈∆ Q0⊂Q − ZQ |fi(y)|dy  q  1/q . For x∈Q0,Md qf(x)q≥Kq 0. We also have the following elementary inequality: for every a, b ≥0, 0 ≤max(a, b)−b≤a. Combining these facts we get that 0≤Md qf(x)q−Kq 0≤ ∞ X i=1 Md(fiχQ0)(x)q=Md q(fχQ0)(x)q. 34 DAVID CRUZ-URIBE, SFO, JOS´ E MAR´ IA MARTELL, AND CARLOS P´ EREZ Since the vector-valued maximal operator is weak (1,1) (see [14]), for any t > 0, |{x∈Q0:|Md qf(x)q−Kq 0|> t}| ≤ |{x∈Q0:Md q(fχQ0)(x)> t1/q}| ≤ Cn,q t1/q ZQ0 kf(x)kℓqdx. Therefore, by (3.1) with p= 1/q, ωλ((Md qf)q, Q0) ≤((Md qf)q−Kq 0)χQ0∗(λ|Q0|)≤Cn,q λq− ZQ0 kf(x)kℓqdxq .  One weight inequalities: Proof of Theorem 1.12.As we noted above, the proof is very similar to the proof of Theorem 1.8, and so we briefly sketch the key details. By Theorem 2.2 it will suffice to prove it for the special case when p=q+ 1. For this value of pwe have that (p/q)′=pand 1 −p′=−1/q. As before, fix w∈Apand QN; we will show that ZQN |Md qf(x)q−m(Md qf)q(QN)|p/qw(x)dxq/p ≤Cn,q[w]ApZRn kf(x)kp ℓqw(x)dxq/p . By Theorem 3.3 and Lemma 8.1, for every x∈QN, |Md qf(x)q−m(Md qf)q(QN)|(8.1) ≤Cn,q M(kf(·)kℓq)(x)q+Cn,q X j,k − Zb Qk j kf(x)kℓqdx!q χQk j(x) =Cn,q M(kf(·)kℓq)(x)q+Cn,q F(x). To estimate the first term we use Theorem 2.1. The estimate for Fuses duality: fix a non-negative function h∈Lp(w) with khkLp(w)=1 (recall that (p/q)′=p). Then, proceeding as before, we use the definition of Ap=Aq+1 to show that ZQN F(x)h(x)w(x)dx ≤Cn[w]ApZRn Md w−1/q (kf(·)kℓqw1/q)(x)qw(x)−1/pMd wh(x)w(x)1/pdx. SHARP WEIGHTED ESTIMATES 35 Finally, we use H¨older’s inequality, Theorem 2.3 and then take the supremum over all such functions hto get the desired estimate. To prove that the exponent max 1 q,1 p−1is the best possible, we consider two cases. If p≤q+ 1, then the exponent is 1 p−1, which is the same as the sharp exponent for the scalar maximal function. Therefore, the examples given by Buckley [3] immediately adapt to the vector-valued maximal operator. If p > q+1, then we can argue exactly as we did for the dyadic square function, replacing the exponent 1/2 by 1/q. Therefore, to show that the exponent 1/q is sharp we need to show that there exists a vectorvalued function f={fi}such that kMqfkp≥cp1/qkfkp. But such a function is given by Stein [51, p. 75]. Two weight inequalities: Proof of Theorem 1.13.The proof is again nearly the same as the proof of Theorem 1.10 for the dyadic square function, so we only sketch the highlights. Fix p, 1 < p < ∞; then it suffices to show that ZQN |Md qf(x)q−m(Md qf)q(Qn)|p/qu(x)dx ≤CZRn kf(x)kp ℓqv(x)dx. We use (8.1). We estimate the term involving Musing Theorem 2.5 and the fact that (u, v) satisfies (1.8) when 1 < p ≤qor (1.10) when p > q. To estimate Fwe consider two cases. Suppose first that 1 < p ≤q, then ZQN F(x)p/q u(x)dx ≤X j,k − Zb Qk j kf(x)kℓqdx!p u(Qk j), and this term is estimated exactly as before. Combining these two estimates we get the desired inequality. 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