Reverse Hölder property for strong weights and general measures

arXiv:1512.01112v2 [math.CA] 4 Dec 2015 REVERSE H ¨ OLDER PROPERTY FOR STRONG WEIGHTS AND GENERAL MEASURES TERESA LUQUE, CARLOS P´ EREZ, AND EZEQUIEL RELA Abstract. We present dimension-free reverse H¨older inequalities for strong A∗ pweights, 1 ≤p < ∞. We also provide a proof for the full range of local integrability of A∗ 1weights. The common ingredient is a multidimensional version of Riesz’s “rising sun” lemma. Our results are valid for any nonnegative Radon measure with no atoms. For p=∞, we also provide a reverse H¨older inequality for certain product measures. As a corollary we derive mixed A∗ p−A∗ ∞weighted estimates. 1. Introduction and Main Results In this article we present several results regarding reverse H¨older inequalities for strong A∗ p(µ) Muckenhoupt weights on Rnfor a general non-atomic Radon measure µ. Before describing these facts, a few words concerning the family of weights considered in here are necessary. The class A∗ p(µ) of strong weights consist of all nonnegative µ-measurable functions on Rnsuch that, for 1 < p < ∞and p′=p/(p−1), satisfy (1.1) [w]A∗ p(µ):= sup R− ZR w dµ− ZR w1−p′dµp−1 <∞, where the supremum is taken over all rectangles R⊂Rnwith sides parallel to the coordinate axes. As usual, we denote by − REf dµ =fE=1 µ(E)REf dµ the average of fover Ewith respect to the measure µ. The limiting case of (1.1), when p= 1, defines the class A∗ 1(µ); that is, the set of weights wsuch that [w]A∗ 1(µ):= sup R− ZR w dµess sup R (w−1)<+∞. This is equivalent to whaving the property Msw(x)≤[w]A∗ 1(µ)w(x)µa.e. x∈Rn. Here Msdenotes the strong maximal function: (1.2) Msf(x) = sup R∋x − ZR |f|dµ, 1991 Mathematics Subject Classification. Primary: 42B25. Secondary: 43A85. Key words and phrases. Reverse H¨older inequality; Muckenhoupt weights; Maximal functions; Multiparameter harmonic analysis. The first author is is supported by the Severo Ochoa Excellence Programme and the second author is supported by the Spanish Government grant MTM2014-53850-P and the Severo Ochoa Excellence Programme. The third author is partially supported by grants UBACyT 20020130100403BA, and PIP (CONICET) 11220110101018. 1 2 TERESA LUQUE, CARLOS P´ EREZ, AND EZEQUIEL RELA where the supremum is taken over all rectangles R⊂Rnwith sides parallel to the coordinate axes containing the point x. Similarly, Mwill denote the Hardy-Littlewood maximal function, namely when the supremum is taken over cubes with sides paralell to the coordinate axes. It follows from H¨older’s inequality and the definitions above that the classes A∗ p(µ) are increasing in p≥1. It is thus natural to define the limiting class A∗ ∞(µ) as (1.3) A∗ ∞(µ) := [ p≥1 A∗ p(µ). If we only consider cubes with sides parallel to the coordinate axes, we obtain the classical Muckenhoupt Ap(µ) classes. Throughout the paper, we will often use the shorthand notations Apand A∗ psince the underlying measure will always be clear from the context. Typically, for w∈Ap, 1 ≤p≤ ∞, one expects an inequality of the form (1.4) − ZQ w1+εdµ ≤C− ZQ w dµ1+ε , valid for any cube Q, where the constant Cmay depend on the Apconstant of the weight, the value of εand on the measure µ. Inequalities like (1.4) are known as reverse H¨older inequalities (RHI) and its study can be traced back to the works of Muckenhoupt [31], Coifman and Fefferman [7], within the context of harmonic analysis, and in the work of Gehring [12], within the context of the theory of quasiconformal mappings. Since then, these kind of inequalities have been widely studied in many different situations with several motivations; we refer to [3] and [18] for the applications to elliptic PDE and quasiconformal mappings in the plane. We refer the interested reader to the monographs [11, Chapter 4] and [2, Chapter 6 ] for a more detailed information on these issues. More recently, RHI with good control on the constants have become relevant in the study of sharp bounds for some of the main operators in harmonic analysis such as singular integrals, maximal functions, commutators with BMO functions and others. See, for instance, [24] [25] [16] [6] for an account of this subject. More pre cisely, within this last context, it is particularly interesting to provide a version of such inequalities with a constant Cindependent of the weight. In particular, in [17] the authors proved that, with underlying Lebesgue measure on Rn, we can take C= 2 in the above inequality and the result is valid for any w∈A∞and 0 < ε ≤1 2n+1[w]A∞ , where [w]A∞:= sup Q 1 w(Q)ZQ M(wχQ)dx < ∞ is called the Fujii-Wilson constant. Moreover, this result can be trivially extended to any doubling measures µon Rn; that is a measure µsuch that: µ(2Q)≤Cµµ(Q) for every cube Q. Although the concept of doubling is affected by the family of sets we consider, in the particular case of rectangles the above definition remains the same. See [13, Section 5] for a more complete definition of doubling measures on general basis. REVERSE H ¨ OLDER FOR GENERAL A∗ p(µ) 3 Most of the known proofs of RHI for classical Apweights are based on the Calder´on-Zygmund (C-Z) decomposition lemma applied to the level set of the Hardy-Littlewood maximal function M. This kind of stopping time argument produces a family of maximal cubes with nice properties. But in order to exploit the maximality, one needs to relate the average over some cube Qto the average over the dyadic parent of Q. It the case of doubling measures, the dilation process produces a dependence on the doubling constant. In particular, it forces the dependence on the dimension for the classical situation of the Lebesgue measure. This dependence on the dimension appears on the range of possible values for the exponent in the RHI. In the context of non-doubling measures, [32, Lemma 2.3] presents a characterization of the class A∞=Sp≥1Apin terms of several equivalent properties, and the RHI is among them. The only requirement imposed to the measure µis the “polynomial growth” condition; that is, there exists some 0< α ≤nsuch that, for any x∈Rn, and for any r > 0, (1.5) µ(B(x, r)) ≤Crα. The relevant consequence of this condition is that µdoes not concentrate positive measure on hyperplanes parallel to the coordinate axes of some system of coordinates. By changing variables, we can assume that this conditions is fulfilled for the canonical coordinates). Moreover, this latter condition is in fact a consequence of the absence of atoms (see [29]). In that case, the lack of doubling is solved in [32, Lemma 2.1] by using a suitable version of the Besicovitch’s covering theorem. This result provides a family of quasi-disjoint cubes with controlled average that covers the level set of the maximal function. The overlap is controlled by a dimensional constant B(n), known as the Besicovitch constant. Tracking the constants in [32, Lemma 2.3], the following inequality holds − ZQ w1+εdµ ≤2− ZQ w dµ1+ε for any 0 < ε ≤1 2p+1B(n)[w]Ap,p > 1. Then, in this case, we observe that the RHI depends on the dimension via the Besicovitch constant. In this paper, we focus our attention in the reverse H¨older property for strong weights. In the case of the Lebesgue measure, a simple change of variables produces a RHI for rectangles since it is known for cubes (the details can be found in [9]). However, this argument produces the same range for the exponent, and therefore it will appear a dependence on the dimension. This clashes with the somehow intuitive idea that, in many circumstances, strong weights behave like one dimensional objects; see [22] for further details on this issue. In addition, our purpose here is to investigate (1.4) for arbitrary non-atomic measures for which it is (in general) not possible to apply a change of variables argument. We present a proof of a dimension-free RHI avoiding the use of C–Z type lemmas. We use instead what is known as a multidimensional form of the classical F. Riesz’s “Rising Sun” lemma. The following lemma is from [21], and can be understood as a more precise version of the classic C–Z lemma. 4 TERESA LUQUE, CARLOS P´ EREZ, AND EZEQUIEL RELA Lemma 1.1 (Multidimensional F. Riesz’s lemma).Let R⊂Rnbe a rectangle and let µbe any non-atomic, nonnegative Radon measure on R. Let f∈L1 R(µ)and fR≤λ. Then there is a finite or countable set of pairwise disjoint rectangles {Rj}jfor which − ZRj f dµ =λ, and f(x)≤λfor µ-almost all points x∈R\(∪jRj). We remark that in [21] the above lemma is formulated for absolutely continuous measures. But an inspection of the proof shows that it is sufficient that the measure satisfies µ(L) = 0 for any hyperplane parallel to the coordinate axes. Our first main result is the following. Theorem 1.2. Consider a non-negative, non-atomic Radon measure µ. Let w∈A∗ p,1< p < ∞, and let Rbe a rectangle. Then − ZR w1+εdµ ≤2− ZR w dµ1+ε . for any 0< ε ≤1 2p+2[w]A∗ p . For the particular case of the A∗ 1weights, we also address other questions regarding the sharp local integrability range for the weight w. In dimension 1 (for the Lebesgue measure), it is known that if a weight wis in A∗ 1≡A1 then for any finite interval I⊂Rwe have that (1.6) − ZI w(x)sdx ≤Cs,w − ZI w(x)dxs for all ssuch that 1 < s < [w]A1 [w]A1−1= ([w]A1)′. This result is from [4, Corollary 1] and there is also a sharp estimate on Cs,w (see also [27] and [28]). In higher dimensions the known result is due to Kinnunen. In [19] the author proved the analogue of (1.6) for A∗ 1weights, also with sharp constants. His proof relies strongly on the fact that the Lebesgue measure is a product measure. Therefore, an induction argument on the dimension can be carried out. For the particular case of cubes, the best known result is in [20, Theorem 1.3]. In the case of dyadic A1weights, the sharp result can be found in [30]. For the case of doubling measures in metric spaces, some results are in [1] but without sharp constants. Our second main result is the extension of Kinnunen’s result to general measures: Theorem 1.3. Let µbe a non-atomic Radon measure on Rn. Let w∈A∗ 1. Then − ZR wsdµ ≤s 1−(s−1)([w]A∗ 1−1) − ZR w dµs , for any 1< s < [w]A∗ 1 [w]A∗ 1−1. REVERSE H ¨ OLDER FOR GENERAL A∗ p(µ) 5 We present here a short and simple proof based on the multidimensional Riesz’s lemma to obtain the result for general measures with mild conditions. With this argument we are able to obtain the same optimal range for the exponent s. Moreover, Theorem 5.1 describes a different version of these results with the range sdepending on the norm of the strong maximal operator Mson the dual space, namely Lp′(w1−p′). In fact the case p= 1 can be seen as a limiting case of Theorem 5.1. Finally, we also study reverse H¨older inequalities for strong A∗ ∞weights where the previous approach using the Riesz’s Lemma cannot be extended for general measures. For the particular case of the Lebesgue measure, [14] presents also a different nice approach using Solyanik estimates. This article is organized as follows. In Section 2we prove Theorem 1.2. In Section 3we show that a similar argument can be used to derive Theorem 1.3 and obtain the full range of local integrability for A∗ 1weights. In Section 4we study this problem for A∗ ∞weights. Finally, in Section 5we study different formulations of RHI for A∗ pweights. 2. Dimension-free RHI for A∗ p,1≤p < ∞ In this section we prove Theorem 1.2. We start with the following lemma, valid for A∗ pweights for p∈(1,∞). Lemma 2.1. Let µbe a non-atomic Radon measure µ. Let w∈A∗ p,1< p < ∞. Then, for any rectangle Rand any λ > wR, we have that (2.1) w({x∈R:w(x)> λ})≤2λµ({x∈R:w(x)>1 2p−1[w]A∗ p wR}). Proof. Using H¨older’s inequality with pand its conjugate p′, we have that for every rectangle Rand every f≥0, − ZR f dµp w(R)≤[w]A∗ pZR fpw dµ. In particular, for any µ-measurable set E⊂Rwe can rewrite the last inequality for f≡χE (2.2) µ(E) µ(R)p ≤[w]A∗ p w(E) w(R). For a given rectangle R, define ER={x∈R:w(x)≤1 2p−1[w]A∗ p wR}. Hence, since ERis a µ-measurable subset of R, (2.2) gives µ(ER) µ(R)p ≤[w]A∗ p w(ER) w(R)≤[w]A∗ p wR w(R)µ(ER)1 2p−1[w]A∗ p =1 2p−1 µ(ER) µ(R). Then, (2.3) µ(ER)≤1 2µ(R). 6 TERESA LUQUE, CARLOS P´ EREZ, AND EZEQUIEL RELA Now we apply Lemma 1.1 to the rectangle Rto obtain a countable set of pairwise disjoint rectangles Rj∈Rsatisfying − ZRj w dµ =λ for each j, and w(x)≤λfor µ-a.e. points x∈R\Sj≥1Rj. This decomposition together with (2.3) yields w({x∈R:w(x)> λ})≤w([ j≥1 Rj)≤X j w(Rj) = λX j µ(Rj) ≤2λX j µ({x∈Rj:w(x)>1 2p−1[w]A∗ p wRj}) ≤2λµ({x∈R:w(x)>1 2p−1[w]A∗ p λ}), since wRj=λ. Since λ > wR, this yields (2.1).  We now present the proof of the dimension-free RHI for A∗ pweights. Proof of Theorem 1.2.Define Ωλ:= {x∈R:w(x)> λ}. Then for arbitrary positive εwe have − ZR w(x)εw(x)dµ =ε µ(R)Z∞ 0 λεw(Ωλ)dλ λ =ε µ(R)ZwR 0 λεw(Ωλ)dλ λ+ε µ(R)Z∞ wR λεw(Ωλ)dλ λ =I+II. Observe that I≤(wR)ε+1. To estimate II, we use Lemma 2.1 II =ε µ(R)Z∞ wR λεw(Ωλ)dλ λ ≤2ε µ(R)Z∞ wR λ1+εµ({x∈R:w(x)>1 2p−1[w]A∗ p λ})dλ λ = (2p−1[w]A∗ p)1+ε2ε µ(R)Z∞ wR 2p−1[w]A∗ p λε+1µ(Ωλ)dλ λ ≤(2p−1[w]A∗ p)1+ε2ε 1 + ε− ZR w1+εdµ. Setting 0 < ε ≤1 2p+2[w]A∗ p , we obtain II ≤1 2− ZR w1+εdµ. where we have used that t1/t ≤2 whenever t≥1. Therefore we obtain − ZR w1+εdµ ≤2− ZR w dµ1+ε , which is the desired estimate. REVERSE H ¨ OLDER FOR GENERAL A∗ p(µ) 7  Remark 2.2.Clearly, Lemma 2.1 above does not hold for A∗ 1weights. But any A∗ 1weight wcan be viewed as an A∗ pweight for any p > 1. Therefore we have that wsatisfies a RHI for any exponent εsuch that 0 < ε < 1 2p[w]A∗ p . Since the quantity [w]A∗ pincreases to [w]A∗ 1, we conclude that the same result of Theorem 1.2 is valid for A∗ 1weights with 0 < ε < 1 2η[w]A∗ 1 for any η > 3. 3. Full range of local integrability for strong A∗ 1weights In this section we show how to apply Lemma 1.1 to prove the full range of local integrability for A∗ 1weights. The key is to obtain a sort of selfimproving property for the operator Msdefined in (1.2). Proof of Theorem 1.3:Set Ωt:= {x∈R:Msw(x)≥t}. Then for any arbitrary positive εwe have − ZR (Msw)εw dx ≤ε µ(R)Z∞ 0 tε−1w(Ωt)dt =ε µ(R)ZwR 0 tε−1w(Ωt)dt +ε µ(R)Z∞ wR tε−1w(Ωt)dt ≤(wR)ε+1 +ε µ(R)Z∞ wR tε−1w(Ωt)dt. To estimate the last integral, we use Lemma 1.1 to obtain a collection of disjoint rectangles {Rj}contained in Rsuch that − ZRj w dx =tand w(x)≤tfor a.e. x∈R\ ∪jRj. Set E:= R\SjRj. Then, w(Ωt) = w(Ωt∩ ∪jRj) + w(Ωt∩E) ≤tX j |Rj|+t|Ωt∩E|. Note that for any x∈ ∪jRj, we have that Msw(x)≥t, and therefore we obtain Pj|Rj| ≤ |Ωt∩(∪jRj)|. Hence, w(Ωt)≤t|Ωt∩(∪jRj)|+t|Ωt∩E| ≤t|Ωt|. And then ε µ(R)Z∞ wR tε−1w(Ωt)dt ≤ε µ(R)Z∞ wR tε|Ωt|dt ≤ε 1 + ε− ZR (Msw)1+εdµ ≤ε[w]A∗ 1 1 + ε− ZR (Msw)εw dµ. Collecting all estimates, we have that (3.1) − ZR (Msw)εw dµ ≤(wR)ε+1 +ε[w]A∗ 1 1 + ε− ZR (Msw)εw dµ. 8 TERESA LUQUE, CARLOS P´ EREZ, AND EZEQUIEL RELA Setting 0 < ε < 1 [w]A∗ 1−1, (3.1) yields − ZR (Msw)εw dµ ≤1 + ε 1−ε([w]A∗ 1−1) − ZR w dµ1+ε . To finish, we take 1 < s < [w]A∗ 1 [w]A∗ 1−1and let ε=s−1. Then − ZR wsdµ ≤ − ZR (Msw)(s−1)w dµ ≤s 1−(s−1)([w]A∗ 1−1) − ZR w dµs , which is the desired estimate.  4. The case of A∗ ∞weights Until now, we have been focused on A∗ pweights with 1 ≤p < ∞. The aim of this section is to investigate a quantitative reverse H¨older property for the A∗ ∞class in terms of its constant. First, we remark here that in this case there are several possible definitions of [w]A∗ ∞. Apart from the natural definition (1.3), a classical definition of the A∗ ∞constant is the one obtained by taking the limit in the A∗ pcondition: (4.1) [w]exp A∗ ∞:= sup R1 µ(R)ZR w dµexp 1 µ(R)ZR log w−1dµ<∞ where the supremum is taken over all rectangles R∈Rnwith sides parallel to the coordinate axes. See [15] for more details on this definition. However, the current tendency is to use a different A∞constant (implicitly introduced by Fujii in [10]), which seems to be better suited: (4.2) [w]A∗ ∞:= sup R 1 w(R)ZR Ms(wχR)dµ < ∞. If the measure µis doubling, definitions (4.1),(4.2) and (1.3) define the same class of weights. However, for general measures some extra conditions need to be imposed to establish the equivalence. For further details in the case of A∞weights, see [32, Remark 2.4]. Below, we consider separately the cases of dimension n= 1 and n > 1. 4.1. A∞for the line. In this case clearly there is no difference between cubic and rectangular weights and both definitions are equivalent when µ is doubling. It makes sense also in this one-dimensional case to use the centered maximal function Mcinstead of Min definition (4.2): [w]c A∞:= sup I 1 w(I)ZI Mc(wχI)dµ. Note that this other definition is again equivalent to the others whenever µ is doubling. However, as we remark before, when the underlying measure µis non doubling, the equivalence is not clear. It can be shown, as in [16, Proposition 2.2], that [w]A∞≤cn[w]exp A∞. This inequality relies on the fact that Mis bounded on Lp(µ) for any measure µ. Also It is obvious that [w]c A∞≤[w]A∞but the finiteness of [w]c A∞does not characterize A∞. Indeed in [32, p. 2021] there is an example of a weight wwhich is not in REVERSE H ¨ OLDER FOR GENERAL A∗ p(µ) 9 A∞satisfying that Mcw.wfor µ-a.e. x∈R. In other words, the centered maximal operator is too small to characterize A∞. The following result in this section shows that in fact [w]A∞characterizes A∞. Theorem 4.1. Let µbe any non atomic Radon measure on Rand let wbe a weight such that [w]A∞<∞. Then it satisfies the following RHI. For any 0< ε < 1 4[w]A∞−1and for any interval I, we have that (4.3) − ZI w1+εdµ ≤2− ZI w dµ1+ε . Remark 4.2.Using the characterization from [32, Lemma 2.3], we deduce from this theorem that w∈A∞. Proof. We use an specific stopping time argument adapted to the µ-dyadic grid for a given interval I. We begin with a similar idea as in [29, p. 536], where a proof of John-Nirenberg’s inequality for non-atomic measures in the real line is presented. We sketch here the construction. The first generation G1(I) of the dyadic grid consists of the two disjoint subintervals I+,I− of Isatisfying µ(I+) = µ(I−) = µ(I)/2. The second generation G2(I) is G1(I+)∪G1(I−). Next generations are defined recursively. Since the measure has no atoms, we can take closed intervals sharing the endpoints. Let Dµ I be the family of all the dyadic intervals generated with this procedure. A collection of nested intervals from this grid will be called a chain. More precisely, a chain Cwill be of the form C={Ji}i∈Nsuch that Ji∈Gi(I), and Ji+1 ⊂Jifor all i≥1. If we define C∞:= TJ∈C Jas the limit set of the chain C, we have that C∞could be a single point or a closed interval of positive length. In any case, we clearly have that µ(C∞) = 0. We will say that those limit sets C∞ of positive length are removable. Since we are in the real line, there are at most a countable many of them and the whole union is also a µ-null set. We denote by Rthe set of all chains with removable limits. If we define (4.4) E:= I\[ C∈R C∞, we conclude that µ(I) = µ(E) and, in addition, for any x∈E, there exists a chain of nested intervals shrinking to x. Therefore the grid Dµ Iforms a differential basis on E. Moreover, the dyadic structure of the basis guarantees the Vitali covering property (see [8, Ch.1] ) and therefore this basis differentiates L1(E). Associated to this grid we define a dyadic maximal operator as follows. For any x∈E, MDµ If(x) = sup J∈Dµ I − ZJ |f|dµ, By a standard differentiation argument, we have that this maximal function satisfies that f≤MDµ If,f≥0, almost everywhere on E. Now the proof of the main inequality (4.3) follows the same steps as in [17, Lemma 2.2]. First, we prove the following inequality for the maximal 16 TERESA LUQUE, CARLOS P´ EREZ, AND EZEQUIEL RELA for any 1< s < kMskLp′(σ) kMskLp′(σ)−1. Proof. It follows from the same argument as in the proof of Theorem 1.3, inequality (3.1), that for any arbitrary positive εwe have − ZR (Msw)εw dx ≤(wR)ε+1 +ε 1 + ε− ZR (Msw)ε+1dx. Following [23], by H¨older’s inequality and the trivial bound w≤Mswwe obtain ZR (Msw)ε+1 dx =ZR (Msw)ε pw1/p (Msw)1+ ε p′w−1/p dx ≤ZR (Msw)εwdx1/p ZR (Msw)p′+εw1−p′dx1/p′ ≤ kMskLp′(σ)ZR (Msw)εwdx In the last inequality we use that kMskp1 Lp1(µ)≤ kMskp2 Lp2(µ)for p1≥p2. Hence, (5.5) 1 |R|ZR (Msw)εwdx ≤(wR)ε+1 +ε ε+ 1kMskLp′(σ) 1 |R|ZR (Msw)εwdx and letting ε=1 2kMkLp′(σ) we get 1 |R|ZR (Msw)εwdx ≤2 (wR)ε+1 This last estimate clearly yields (5.3). Simliarly, setting 0 < ε < 1 kMkLp′(σ)−1, we have that − ZR (Msw)εw dµ ≤1 + ε 1−ε(kMkLp′(σ)−1) − ZR w dµ1+ε , which yields (5.4)  In a similar way as in Corollary 4.3, we can derive also an alternative version of the open property for A∗ pclasses; more precisely, if w∈A∗ pand r(w) = 1 + 1 2kMskLp(w)then w∈A∗ p−εwhere (5.6) ε=p−1 r(σ)′=p−1 1 + 2kMskLp′(σ) . Now we want to collect all the estimates for the RHI. We have Theorem 1.2 and Theorem 5.1. In addition, we also have Theorem 4.5 or Theorem 4.6 since we are considering µas the Lebesgue measure which is both doubling and a product measure. Then, we have that any w∈A∗ psatisfies a RHI with exponent 1 + εfor any REVERSE H ¨ OLDER FOR GENERAL A∗ p(µ) 17 0≤ε < max (1 2p+2[w]A∗ p ,1 2kMskLp′(σ) ,1 2n+1[w]A∗ ∞−1) Which of these estimates is better, will depend on what is the best bound for the weighted norm of the strong maximal function. It is clear that we have kMskLp′(σ)≤cnpn[w]n A∗ p. But it is not known in general if the exponent on the constant of the weight can be smaller than n. Note for example that in the case of strong product weights, [26, Theorem 3.7] shows that 2kMskLp′(σ)≤2p+2[w]A∗ pand therefore Theorem 5.1 would provide a better result than Theorem 1.2 in this case. Moreover in this particular case, [26, Theorem 3.1] also assures the next sharp result: (5.7) kMsfkLp,∞(w).n,p [w] 1 p−1(1−1 np ) A∗ pkfkLp(w). This combined with Theorem 5.1 yield the multiparameter version of (5.2). However, we do not recover the sharp result in this particular case of product weights. References [1] Daniel Aalto and Lauri Berkovits. Asymptotical stability of Muckenhoupt weights through Gurov-Reshetnyak classes. Trans. Amer. Math. Soc., 364(12):6671–6687, 2012. [2] Ravi P. Agarwal, Shusen Ding, and Craig Nolder. Inequalities for differential forms. Springer, New York, 2009. [3] Kari Astala, Tadeusz Iwaniec, and Gaven Martin. Elliptic partial differential equations and quasiconformal mappings in the plane, volume 48 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2009. [4] B. Bojarski, C. Sbordone, and I. Wik. The Muckenhoupt class A1(R). Studia Math., 101(2):155–163, 1992. [5] Stephen M. Buckley. Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Amer. Math. Soc., 340(1):253–272, 1993. [6] Daewon Chung, M. Cristina Pereyra, and Carlos Perez. Sharp bounds for general commutators on weighted Lebesgue spaces. Trans. Amer. Math. Soc., 364(3):1163– 1177, 2012. [7] Ronald R. Coifman and C. Fefferman. Weighted norm inequalities for maximal functions and singular integrals. Studia Math., 51:241–250, 1974. [8] Miguel de Guzm´an. Differentiation of integrals in Rn. Springer-Verlag, Berlin, 1975. With appendices by Antonio C´ordoba, and Robert Fefferman, and two by Roberto Moriy´on, Lecture Notes in Mathematics, Vol. 481. [9] Javier Duoandikoetxea, Francisco Mart´ın-Reyes, and Sheldy Ombrosi. On the A∞ conditions for general bases. 2013. DOI: 10.1007/s00209-015-1572-y. [10] Nobuhiko Fujii. Weighted bounded mean oscillation and singular integrals. Math. Japon., 22(5):529–534, 1977/78. [11] Jos´e Garc´ıa-Cuerva and Jos´e L. Rubio de Francia. Weighted norm inequalities and related topics, volume 116 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1985. [12] F. W. Gehring. The Lp-integrability of the partial derivatives of a quasiconformal mapping. Acta Math., 130:265–277, 1973. [13] Paul Hagelstein, Teresa Luque, and Ioannis Parissis. Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases. Trans. Amer. Math. Soc., 367(11):7999–8032, 2015. [14] Paul Hagelstein and Ioannis Parissis. Weighted solyanik estimates for the strong maximal function. Preprint, arXiv:1410.3402 (2015). 18 TERESA LUQUE, CARLOS P´ EREZ, AND EZEQUIEL RELA [15] Sergei V. Hruˇsˇcev. A description of weights satisfying the A∞condition of Muckenhoupt. Proc. Amer. Math. Soc., 90(2):253–257, 1984. [16] Tuomas Hyt¨onen and Carlos P´erez. Sharp weighted bounds involving A∞.Anal. PDE, 6(4):777–818, 2013. [17] Tuomas Hyt¨onen, Carlos P´erez, and Ezequiel Rela. Sharp Reverse H¨older property for A∞weights on spaces of homogeneous type. J. Funct. Anal., 263(12):3883–3899, 2012. [18] Tadeusz Iwaniec and Gaven Martin. Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001. [19] Juha Kinnunen. Sharp results on reverse H¨older inequalities. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, (95):34, 1994. [20] Juha Kinnunen. A stability result on Muckenhoupt’s weights. Publ. Mat., 42(1):153– 163, 1998. [21] A. A. Korenovskyy, A. K. Lerner, and A. M. Stokolos. On a multidimensional form of F. Riesz’s “rising sun” lemma. Proc. Amer. Math. Soc., 133(5):1437–1440, 2005. [22] Douglas S. Kurtz. Littlewood-Paley and multiplier theorems on weighted Lpspaces. Trans. Amer. Math. Soc., 259(1):235–254, 1980. [23] Andrei K. Lerner and Sheldy Ombrosi. An extrapolation theorem with applications to weighted estimates for singular integrals. J. Funct. Anal., 262(10):4475–4487, 2012. [24] Andrei K. Lerner, Sheldy Ombrosi, and Carlos P´erez. Sharp A1bounds for Calder´on- Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden. Int. Math. Res. Not. IMRN, (6):Art. ID rnm161, 11, 2008. [25] Andrei K. Lerner, Sheldy Ombrosi, and Carlos P´erez. Weak type estimates for singular integrals related to a dual problem of Muckenhoupt-Wheeden. J. Fourier Anal. Appl., 15(3):394–403, 2009. [26] Teresa Luque and Ioannis Parissis. Sharp weighted norm inequalities for multiparameter operators. (preprint). [27] N. A. Malaksiano. On exact inclusions of Gehring classes in Muckenhoupt classes. Mat. Zametki, 70(5):742–750, 2001. [28] Nikolay Aleksandrovich Malaksiano. The precise embeddings of one-dimensional Muckenhoupt classes in Gehring classes. Acta Sci. Math. (Szeged), 68(1-2):237–248, 2002. [29] J. Mateu, P. Mattila, A. Nicolau, and J. Orobitg. BMO for nondoubling measures. Duke Math. J., 102(3):533–565, 2000. [30] Antonios D. Melas. A sharp Lpinequality for dyadic A1weights in Rn.Bull. London Math. Soc., 37(6):919–926, 2005. [31] Benjamin Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165:207–226, 1972. [32] Joan Orobitg and Carlos P´erez. Apweights for nondoubling measures in Rnand applications. Trans. Amer. Math. Soc., 354(5):2013–2033 (electronic), 2002. [33] Carlos P´erez and Ezequiel Rela. A new quantitative two weight theorem for the Hardy-Littlewood maximal operator. Proc. Amer. Math. Soc., 143:641–655, 2015. [34] Peter Sj¨ogren. A remark on the maximal function for measures in Rn.Amer. J. Math., 105(5):1231–1233, 1983. [35] Peter Sj¨ogren and Fernando Soria. Sharp estimates for the non-centered maximal operator associated to Gaussian and other radial measures. Adv. Math., 181(2):251– 275, 2004. REVERSE H ¨ OLDER FOR GENERAL A∗ p(µ) 19 Instituto de Ciencias Matem´ aticas CSIC-UAM-UC3M-UCM, C/ Nicol´ as Cabrera, 13-15, 28049 Madrid, Spain E-mail address:[email protected] Department of Mathematics, University of the Basque Country, Ikerbasque and BCAM, 48080, Spain E-mail address:[email protected] Departamento de Matem´ atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pabell´ on I, Buenos Aires 1428 Capital Federal Argentina E-mail address:erel[email protected]