Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

TAUBERIAN CONDITIONS, MUCKENHOUPT WEIGHTS, AND DIFFERENTIATION PROPERTIES OF WEIGHTED BASES PAUL HAGELSTEIN, TERESA LUQUE, AND IOANNIS PARISSIS Abstract. Let Bbe a homothecy invariant collection of convex sets in Rn. Given a measure µ, the associated weighted geometric maximal operator MB,µ is defined by MB,µf(x)∶=sup x∈B∈B 1 µ(B)∫B∣f∣dµ. It is shown that, provided µsatisfies an appropriate doubling condition with respect to Band νis an arbitrary locally finite measure, the maximal operator MB,µ is bounded on Lp(ν)for sufficiently large pif and only if it satisfies a Tauberian condition of the form ν({x∈Rn∶MB,µ(1E)(x)>1 2})≤cµ,ν ν(E). As a consequence of this result we provide an alternative characterization of the class of Muckenhoupt weights A∞,Bfor homothecy invariant Muckenhoupt bases Bconsisting of convex sets. Moreover, it is immediately seen that the strong maximal function MR,µ, defined with respect to a product-doubling measure µ, is bounded on Lp(ν)for some p>1if and only if ν({x∈Rn∶MR,µ(1E)(x)>1 2})≤cµ,ν ν(E) holds for all ν-measurable sets Ein Rn. In addition, we discuss applications in differentiation theory, in particular proving that a µ-weighted homothecy invariant basis of convex sets satisfying appropriate doubling and Tauberian conditions must differentiate L∞(ν). 1. Introduction The study of weighted inequalities for classical operators such as the Hardy-Littlewood maximal function and the Hilbert transform has been an active area of research in harmonic analysis since the seminal paper of Muckenhoupt, [41]. Here, by weighted inequalities we mean the study of the boundedness properties of an operator Ton some weighted Lebesgue space Lp(w), where wis an appropriate non-negative, locally integrable function, that is, a weight. Indeed, the weights wfor which the Hardy-Littlewood maximal function, the Hilbert transform, as well as more general Calderón-Zygmund operators are bounded on Lp(w)were identified in [41] as well as in the subsequent works [7,22]; these investigations led to the definition of the Apclasses of weights; see Definition 3.3. The first quantitative estimate of the operator norm ∥M∥Lp(w) 2010 Mathematics Subject Classification. Primary: 42B25, Secondary: 42B35. Key words and phrases. strong maximal function, Tauberian condition, Muckenhoupt weight. P.H. is partially supported by the Simons Foundation grant 208831. T.L. is supported by the Spanish Ministry of Economy and Competitiveness grant BES-2010-030264. I.P. is supported by the Academy of Finland, grant 138738. 1 arXiv:1304.1015v2 [math.CA] 2 Dec 2013 2 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS on the Ap-constant of the weight was proved in [3] for the Hardy-Littlewood maximal function M. In recent years, the corresponding question concerning the sharp dependence of the norm of a Calderón-Zygmund operator ∥T∥Lp(w)on the Ap-constant of the weight has spurred an overwhelming amount of activity and development of relevant tools. Important representatives of these results include (but are not exhausted to) the work of Petermichl in [44], where the sharp bound for the Hilbert transform is proved, as well as the resolution of the celebrated A2conjecture by Hytönen in [23] where the sharp bound is exhibited for general Calderón-Zygmund operators. Subsequent important developments and simplifications of the proof of the A2theorem can be found in [24] and [35]. The two-weight problem for the Hilbert transform was also a notoriously hard problem, asking for necessary and sufficient conditions on a pair of weights (v, w)so that the Hilbert transform is bounded from Lp(w)to Lp(v). The two-weight inequalities have been intensively investigated in several papers which led to the very recent resolution by Lacey in [32], following previous results by Lacey, Sawyer, Shen and Uriarte-Tuero in [33]. All the results mentioned so far concern the classical or one-parameter theory, where the operators under study commute with one-parameter dilations of Rn. On the other hand, the most basic example of the multi-parameter theory is the strong maximal function MR, namely the maximal average of a function with respect to n-dimensional rectangles with sides parallel to the coordinate axes. This operator is in many senses the prototype for multi-parameter analysis while there are also natural multi-parameter versions of the Hilbert transform as well as of more general singular integral operators. See for example [46]. As the terminology suggests, these operators commute with n-parameter dilations of Rn. A general introduction to multi-parameter harmonic analysis is contained in [14]. The basic weighted theory for the strong maximal function is also pretty well developed in a series of important papers such as [12], [15], [26], [27] and [47]. A natural starting point for a more quantitative multi-parameter weighted theory would be the analogue of Buckley’s theorem for the strong maximal function, namely, a sharp estimate on ∥MR∥Lp(w)in terms of the (strong) Ap-constant of the weight. No such quantitative estimate is currently known, a serious obstruction in carrying over the already described achievements of classical weighted theory to the multi-parameter setting. A possible reason why such a sharp weighted theorem is elusive in the multi-parameter world is the failure of the Besicovitch covering argument for rectangles with arbitrary eccentricities. Indeed, it is an essential fact, underlying many of the sharp quantitative estimates in the classical weighted theory, that the (centered) Hardy-Littlewood maximal operator, defined with respect to a general measure, is always bounded independently of the measure. This fails quite dramatically for the strong maximal function and this is just another manifestation of the failure of the Besicovitch covering argument. See [12]. The relevance of this fact to proving bounds on the operator norms ∥MR∥Lp(w)is revealed by abstract theorems characterizing two-weight norm inequalities in terms of the boundedness of corresponding weighted maximal operators. For example it is implicit in Sawyer’s two-weight norm inequalities for the Hardy-Littlewood maximal function in [48] and for the strong maximal function in [47]. See also [26] and [34]. The strong maximal function with respect to a measure. For the weighted strong maximal function MR,w one thus requires some restriction on the weight wso that MR,w is bounded on TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 3 Lp(w). A sufficient condition is provided by the result of Fefferman, [12], that states that if a weight w∈A∞,R, that is if wis a strong Muckenhoupt weight, then MR,w is bounded on Lp(w). See §3for precise definitions. It is thus no big surprise that many results in multi-parameter weighted theory begin with the assumption that w∈A∞,R. This assumption has also been used in order to derive the Fefferman-Stein inequality for the strong maximal function in [36], [39], [40] and [42]. An apparently weaker substitute for the hypothesis w∈A∞,R, usually referred to as condition (A), is that wsatisfies a Tauberian condition of the form w({x∈Rn∶MR(1E)(x)>1 2})≤cw(E),(A) where E⊂Rnis any measurable set. This condition was introduced in [26] in the context of weighted inequalities for quite general maximal functions. A consequence of our main theorem is however that (A) is equivalent to w∈A∞,R. The previous discussion also motivates the seeking of conditions on a measure µsuch that the strong maximal function MR,µ, defined with respect to the measure µ, is bounded on Lp(µ). With more general results and precise definitions to follow, one of our main theorems implies: Theorem 1.1. Let µbe a Borel non-negative measure which is locally finite and doubling with respect to rectangles with sides parallel to the coordinate axes. Then the operator MR,µ satisfies µ({x∈Rn∶MR,µ(1E)(x)>1 2})≤cµ(E) for every measurable set Eif and only if MR,µ is bounded on Lp(µ)for some p>1. This theorem can be thought of as a testing condition on the operator MR,µ. As it will become apparent, the constant 1 2is not important as it can be replaced by any fixed level γ∈(0,1)in the statement of the theorem. Differentiation with respect to bases of convex sets. A dual point of view and motivation for the investigations in this paper can be given in the language of differentiation theory. Given a collection of convex sets in Rnwhich is invariant under dilations and translations we want to study when the corresponding maximal operator differentiates L∞(Rn). It turns out that boundedness properties of quite general maximal operators can also be characterized in terms of Tauberian conditions in the spirit of (A). Indeed, it is a classical result of Busemann and Feller, [4], that a homothecy invariant basis Bconsisting of open sets differentiates L∞(Rn)if and only if the corresponding maximal operator MBsatisfies a Tauberian condition ∣{x∈Rn∶MB(1E)(x)>γ}∣≤cγ∣E∣, for every γ∈(0,1)and for every measurable set E. See Theorem 4.3 for the details. This point of view is discussed in detail in [19] and taken up in [21]. In the last work it is shown that a homothecy invariant basis consisting of convex sets differentiates L∞(Rn)(with respect to the Lebesgue measure) if and only if it differentiates Lp(Rn)for some sufficiently large p>1. Note here that, lacking the convexity hypothesis on the basis one needs Tauberian conditions at all levels γ∈(0,1). This should be contrasted to the results in [21] as well as in the current paper 4 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS where the convexity assumption allows us to only assume a Tauberian condition at a fixed level, say γ=1 2. It is a natural question whether such results persist under the presence of a weight, or somewhat more generally, a measure µ. More precisely, one seeks conditions on the measure µand the basis Bso that the µ-averages of a function f∈Lp(µ)converge to f µ-almost everywhere. If Bis some abstract basis of convex sets it is in general hard to tell whether Bdifferentiates functions in Lp(µ)with respect to µ, for some p>1. For the basis of rectangles with sides parallel to the coordinate axes this is just a rephrasing of the question: “When is the strong maximal function MR,µ, defined with respect to a measure µ, bounded on Lp(µ)for some p>1?” However our results address the more general case of homothecy invariant bases consisting of convex sets. Our most general theorem has the following form: Theorem 1.2. Let Bbe a homothecy invariant basis consisting of open convex sets and assume that µ, ν are locally finite, non-negative Borel measures on Rn. Assume further that the measure µis doubling with respect to B. Let MB,µ denote the maximal operator with respect to the basis Band the measure µ. Then MB,µ satisfies ν({x∈Rn∶MB,µ(1E)(x)>1 2})≤cν(E) if and only if MB,µ is bounded on Lp(ν)for some p>1. Among other consequences, we get as a corollary a “weighted” version of the Busemann-Feller theorem: Corollary 1.3. Let Bbe a homothecy invariant basis consisting of convex sets and µ, ν be locally finite, non-negative measures on Rn. Assume in addition that µis doubling with respect to B. If ν({x∈Rn∶MB,µ(1E)(x)>1 2})≤cν(E) then Bdifferentiates L∞(ν)with respect to the measure µ. 2. Notations A few words concerning the notation used in this paper are necessary. Due to the technical nature of some of the proofs the notation becomes quite cumbersome but we have tried to be consistent with our choice of symbols. The current paragraph can be used as a guide to the notation and the reader is encouraged to consult it in order to avoid any sort of confusion. We write A≲Bwhenever A≤CB for some constant C>0and A≃Bif A≲Band B≲A. We write A≲nBwhenever the implied constant depends on some parameter n. We omit such dependencies when they are of no importance. In this paper we use several differentiation bases which are basically collections of open sets in Rn. We use the symbol Bto denote a generic basis consisting of convex sets, the symbol Gto denote a generic basis consisting of rectangles, the symbol Rfor the basis of all rectangles with sides parallel to the coordinate axes, the symbol bfor the collection of all Euclidean balls and the symbol Qfor the collection of all cubes in Rnwith sides parallel to the coordinate axes. TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 5 For a rectangle R∈Rwe denote by DRthe mesh of “dyadic rectangles” associated to R. The “dyadic children” of Rare produced by dividing each side of Rinto two equal parts while the dyadic parent of Ris the rectangle R(1)whose sidelengths are double the corresponding sidelengths of Rand shares exactly one corner with R. Thus every R⊂Rnhas exactly 2ndyadic children and is contained in a unique dyadic parent. For a dyadic rectangle Rwe write R(1)for the parent of Rand R(j)for the ancestor of Rwhich is jgenerations “before” R. For more details see the discussion before Proposition 6.11. The measures µ, ν that appear in this paper are always assumed to be locally finite, non-negative Borel measures. The symbol νis used to denote such a measure in the underlying space. Thus we prove bounds on Lp(ν). We use the symbol µto denote a measure which is additionally assumed to be doubling with respect to some differentiation basis B. The doubling constant of µwith respect to Bis denoted by ∆µ,B. We omit these indices when the definition of the underling measure or basis is clear from the context. The measure µis typically used in order to define some maximal operator MB,µ. Thus our main questions concern the mapping property MB,µ ∶Lp(ν)→Lp(ν). We use the symbol wto denote a non-negative locally integrable function, that is, a weight. In this case we write MB,w for the weighted maximal function with respect to Band w. Some words are also necessary concerning the dilations that we use in the paper. There are three kinds of dilations of a set Ewith respect to some parameter c>0. If the set Ehas a natural center of symmetry then cE denotes the dilates of Eby a factor c, with respect to its center. If Bis a convex set we write cB even if Bis not symmetric with respect to some point. In this case the homothecy center is taken to be the center of the John ellipsoid of B. See § 5.2. We write dilcE∶={cx ∶x∈E}, that is, for the dilation with respect to 0. Finally we write c∗R whenever Ris a “dyadic rectangle” to denote the dilation of Rby the factor c, with respect to the corner shared by Rand its parent R(1). We also consider translations of sets; for σ∈Rnand E⊂Rnwe set τσE∶={x+σ∶x∈E}. Finally, almost all the logarithms that appear in this paper are base-2logarithms; we still just write log tfor log2t. 3. Definitions and overview of known results 3.1. Bases of open sets and maximal operators. By a basis Bwe mean a collection of bounded open sets in Rn. The differentiation properties of a basis Bare determined by the boundedness properties of the corresponding maximal function, acting on locally integrable functions fas MBf(x)∶=sup B∈B B∋x 1 ∣B∣∫B∣f(y)∣dy, if x∈∪B∈BBand MBf(x)∶=0otherwise. Particular attention will be given to two special bases of open sets; namely the basis Q, consisting of all n-dimensional cubes with sides parallel to the coordinate axes, and the basis Rconsisting of all rectangles with sides parallel to the coordinate axes. The corresponding maximal operators are the Hardy-Littlewood maximal function MQand 6 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS the strong maximal function MR. We are interested in Lp-bounds for the maximal functions MB of the type ∥MBf∥Lp(Rn)≲B,p,n ∥f∥Lp(Rn),1<p≤+∞,(3.1) together with appropriate endpoint bounds as p→1+. The existence of such bounds cannot be guaranteed in the generality of Bdiscussed above. Indeed, if Bis the family of all rectangles in Rn, allowing all rotations, dilations and translations, then MBis called the universal maximal function which is known to be unbounded for any p<+∞; see [20]. Note, however, that this operator restricted to radial functions is bounded on Lp(Rn)for p>n. See [6] and [10]. On the other hand, the maximal operators defined with respect to the bases Rand Qare well understood, with the corresponding sharp endpoint bounds being ∣{x∈Rn∶MQf(x)>λ}∣≲n∫Rn∣f(x)∣ λdx, λ >0, ∣{x∈Rn∶MRf(x)>λ}∣≲n∫RnΦn(∣f(x)∣ λ)dx, λ >0. Here Φn(t)∶=t(1+(log+t)n−1)and log+t∶=max(log t, 0). The first weak inequality above is the classical maximal theorem of Hardy and Littlewood; see for example [51]. The second distributional inequality is the strong maximal theorem of Jessen, Marcinkiewicz and Zygmund from [28]. See also [9] for a geometric approach to the same result. By interpolation, the previous endpoint bounds imply (3.1) for both Qand R. 3.2. Weights associated to bases. We say that wis a weight associated to the basis Bif wis a non-negative locally integrable function on Rnand w(B)∶=∫Bw(x)dx <+∞for every B∈B. The weighted analogue of estimate (3.1) takes the form ∥MBf∥Lp(w)≲B,p,n ∥f∥Lp(w),1<p≤+∞.(3.2) The corresponding endpoint bounds as p→1+are also of great interest and are typically harder (and stronger) than their Lp-analogues (3.2). Again, for the bases Qand R, estimates (3.2) are much better understood and the validity of (3.2) for any 1<p<+∞is characterized by the membership of wto the classes of Muckenhoupt weights Ap,B: Definition 3.3. We say that a weight wbelongs to the class Ap,B,1<p<+∞, if [w]Ap,B∶=sup B∈B(1 ∣B∣∫B w(y)dy)( 1 ∣B∣∫B w(y)1−p′dy)p−1<+∞. Here and throughout the paper p′denotes the dual exponent of p, that is 1 p+1 p′=1. For the limiting case p=1the class A1,Bis defined to be the set of weights wsuch that [w]A1,B∶=sup B∈B(1 ∣B∣∫B w(y)dy)ess sup B(w−1)<+∞. TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 7 This is equivalent to whaving the property MBw(x)≤[w]A1,B⋅w(x),a.e. x∈Rn. It follows from Hölder’s inequality and the definitions above that for all 1≤p<q<+∞we have that Ap,B⊂Aq,B, that is, the classes Ap,Bare increasing in p≥1. It is thus natural to define the limiting class A∞,Bas A∞,B∶=⋃ p>1 Ap,B=⋃ p≥1 Ap,B. For the special bases Q,Rwe use the shorthand notation Ap∶=Ap,Qand A∗ p∶=Ap,R. Remark 3.4. For a general basis B, as considered here, there is really no “obvious” definition of the class A∞,B. For the basis Qmany definitions have appeared in the literature and they are all equivalent to each other. See for example [16]. This remains true for the basis R. However, for a general basis B, these different definitions may define different classes of weights. We adhere to the definition of the class A∞,Bas the union of the classes Ap,∞for notational simplicity mostly, keeping in mind that for the bases Qand Rthe definition above coincides with all the standard definitions that appear in the literature. For the bases Qand R, the boundedness properties of the corresponding maximal operators on weighted Lebesgue spaces are well known. This is completely classical and due to Muckenhoupt for the basis Q; see [41]. The theorem of Muckenhoupt extends without difficulty to the basis R whenever p>1; see for example [1]. We summarize these results below. Theorem 3.5. The following statements are true. (i) Let Bbe either Qor Rand 1<p<+∞. Then MB∶Lp(w)→Lp(w)if and only if MB∶Lp(w)→Lp,∞(w), if and only if w∈Ap,B. (ii) For Q:MQ∶L1(w)→L1,∞(w)if and only if w∈A1,Q. (iii) For R: If w∈A1,Rthen w({x∈Rn∶MRf(x)>λ})≲n,w ∫RnΦn(∣f(x)∣ λ)w(x)dx, λ >0.(3.6) Thus, the boundedness properties of MBon Lp(w),1<p<+∞, are completely characterized for the bases Q,R, and the same is true for the weighted endpoint estimate MQ∶L1(w)→ L1,∞(w). However, the endpoint estimate (3.6) for the strong maximal function is not so transparent. Indeed, the presence of the logarithmic terms on the right hand side of the (3.6) results in the condition A1,Rbeing sufficient, but not necessary, for the validity of (3.6). A necessary condition for (3.6), which is weaker than w∈A1,R, appears in [1] but the authors show that it is not sufficient. On the other hand, the weighted endpoint estimate (3.6) has been characterized in [17] in terms of a certain covering property for rectangles. See also [31, Theorem 4.3.1] for a detailed proof of this fact. Similar characterizations of the boundedness properties of maximal operators on general Lp(µ)-spaces in terms of covering properties are contained for example in [26, Theorem 2.2] and [16, Lemma IV.6.11], while the approach goes back to [8] and [9]. There is however no characterization in the spirit of the Muckenhoupt Ap,R-classes, of the weights w such that (3.6) holds. 8 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS 3.3. Maximal operators with respect to measures. Let µbe a non-negative measure on Rn, finite on compact sets, and let Bbe a basis. For f∈L1 loc(µ)we write MB,µf(x)∶=sup B∈B B∋x µ(B)>0 1 µ(B)∫B∣f(y)∣dµ(y), if x∈∪B∈BBand MB,µf(x)∶=0if x∉∪B∈BB. If dµ(x)=w(x)dx for some weight wassociated to the basis Bwe just write MB,w for MB,µ and this operator will be called the weighted maximal operator with respect to w. The boundedness properties of MB,µ are much harder than the corresponding properties of the unweighted maximal operator, with definitive information only for special cases of bases Band measures µ. Again, we mainly restrict our attention to the case that Bis either Qor R. As in the unweighted case, the one-parameter operator MQ,µ is easier to analyze than the operator MR,µ. However, even in the one-parameter case, there is no complete characterization of the measures µfor which MQ,µ is bounded on Lp(Rn, µ). Below we give a brief overview of the known results for the bases Qor Rand refer the interested reader to the monograph [30] for further details and proofs. 3.3.1. A special one-dimensional result. In dimension n=1, let µbe any non-negative Borel measure. We have that MQ,µ ∶L1(µ)→L1,∞(µ)and by interpolation MQ,µ ∶Lp(µ)→Lp(µ)for all 1<p≤+∞. This result is very special to one dimension since the proof depends on a covering lemma for intervals of the real line. Observe that there is essentially no restriction on the measure µ. See for example [49] for the details of this result. 3.3.2. The centered, one-parameter maximal function with respect to a measure µ.A common variation of MQ,µ is the centered weighted Hardy-Littlewood maximal function, given as Mc Q,µf(x)∶=sup r>0 1 µ(Q(x, r))∫Q(x,r)∣f(y)∣dµ(y), where Q(x, r)denotes the cube with sides parallel to the coordinate axes and sidelength r>0, centered at x∈Rn. Then for any non-negative Borel measure µwe have that Mc Q,µ ∶L1(Rn)→ L1,∞(Rn)and thus, by interpolation, Mc Q,µ ∶Lp(Rn)→Lp(Rn)for all 1<p≤+∞. The proof of this result depends on the Besicovitch covering lemma and it remains valid whenever the Besicovitch argument goes through. Thus, the condition that the maximal function defined above is centered is essential. For example, it was shown in [49] that if γis the Gaussian measure in R2then the non-centered weighted maximal operator MQ,γ does not map L1to L1,∞. The second essential hypothesis, hidden in the definition of Mc Q,µ, is that it is a one-parameter maximal operator, that is, we average with respect to a one-parameter family of cubes. Here one could replace cubes by Euclidean balls or more general “balls”, given by translations and one-parameter dilations of a convex set in Rnsymmetric about the origin. On the other hand, emphasizing the need for the one-parameter hypothesis mentioned previously, the boundedness fails for the weighted strong maximal function, even in its centered version. The reason is that the family Ris an n-parameter family of sets for which the Besicovitch covering TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 9 is not valid. See for example [12] for an example of a locally finite measure µfor which Mc R,µ is unbounded on Lp(µ)for all p<∞. 3.3.3. The non-centered, one-parameter maximal function with respect to a doubling measure. Let µbe a non-negative Borel measure. The following definition is standard. Definition 3.7. The measure µis called doubling if there is a constant ∆µ>0such that, for every cube Q=Q(x, r)⊆Rnwe have µ(2Q)≤∆µµ(Q), where 2Q=Q(x, 2r). It is an easy observation that for µdoubling, the non-centered weighted maximal operator MQ,µ is pointwise equivalent to its centered version, that is, MQ,µf(x)≃Mc Q,µf(x), where the implicit constants depend only on ∆µ. It follows from the discussion in §3.3.2 that the maximal operator MQ,µ with respect to a doubling measure µmaps L1(µ)to L1,∞(µ)and Lp(µ)to Lp(µ)for all 1<p≤∞. Here, the doubling hypothesis cannot be removed. Indeed, the example from [49] mentioned above shows that there exists a non-doubling measure, in particular the Gaussian measure in R2, such that MQ,γ does not map L1to L1,∞. Most of the results in the literature that study MB,µ for non-doubling measures concern the basis bconsisting of all Euclidean balls in Rnand radial measures. For example it is shown in [52] that if µis rotationally invariant and assigns positive measure to all open sets then Mb,µ ∶L1(µ)→L1,∞(µ)if and only if µis doubling away from the origin. In [50] the authors provide a sufficient condition on a radial measure µso that Mb,µ satisfies certain weak type inequalities close to L1(µ)which in turn imply that Mb,µ ∶Lp(µ)→Lp(µ). An example of a radial measure µsuch that Mb,µ is unbounded on all Lp(µ)-spaces, p<+∞, can be found in [25]. Note that in the non-doubling case, the operators MQ,µ and Mb,µ can behave quite differently, unlike the doubling case. For example, if µis a product of non-negative one-dimensional Borel measures then obviously MQ,µ ≤MR,µ. By the one-dimensional result mentioned in §3.3.1 and the methods from [5] we get that MR,µ, and a fortiori MQ,µ, is bounded on Lp(µ)and satisfies the endpoint estimate µ({x∈Rn∶MQ,µf(x)>λ})≤µ({x∈Rn∶MR,µf(x)>λ})≲n∫RnΦn(∣f(x)∣ λ)dµ(x). One such product measure is the Gaussian measure γon Rnfor which we can thus conclude γ({x∈Rn∶MQ,γf(x)>λ})≲n∫Rn∣f(x)∣ λ(1+(log+∣f(x)∣ λ)n−1)γ(x)dx.(3.8) On the other hand it was shown in [50] that γ({x∈Rn∶Mb,γf(x)>λ})≲n∫Rn∣f(x)∣ λ(1+(log+∣f(x)∣ λ)n+1 2)γ(x)dx 16 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS 5.3. Properties of general doubling measures. The doubling condition has some important consequences in that the measure is “homogeneously” distributed in the space. We summarize these properties in the proposition below. We note that these properties are classical and refer the reader to [51, §8.6] for more details. Proposition 5.6. Let µbe a (not identically zero) locally finite, non-negative Borel measure. Assume that µis doubling with respect to some family Kconsisting of all the homothetic copies of a fixed rectangle. The following properties are satisfied. (i) We have µ(U)>0for every open set U⊂Rn. (ii) Let R∈Kand DRbe the dyadic grid generated by R. There exists a constant γµ>1, depending only on the doubling constant of µand the dimension nsuch that µ(R)≤ γ−m µµ(R(m)), where R(m)is the ancestor of R,mgenerations higher. In particular µ(Rn)=+∞. (iii) The maximal operator MK,µ is of weak type (1,1)and strong type (p, p)for all 1<p≤∞, with respect to µ, and the operator norms depend only on the doubling constant of the measure µ, the exponent pand the dimension n. Also the centered maximal operator Mc K,µ satisfies the same bounds. (iv) If Bis a convex set in Rnwe have µ(∂B)=0where ∂B ∶=¯ B∖Bis the boundary of B. Proof. The proof of (i) can be found for example in [51, §8.6]. For (ii) let R(1)be the dyadic parent of Rand let {Rj}2n j=1denote the dyadic children of R(1)and suppose that R=R1. Then µ(R(1))=2n ∑ j=1 µ(Rj)=µ(R1)+2n ∑ j=2 µ(Rj)≥(1+(2n−1)δ−1 µ)µ(R1), where δµ>1is the doubling constant of µ. Let γµ=1+(2n−1)δ−1 µ>1. Since Ris mgenerations inside R(m)we iterate to get µ(R)≤γ−m µµ(R(m))as desired. For (iii) observe that MK,µ is essentially the Hardy-Littlewood maximal operator with respect to a doubling measure and the result is classical. Since the measure µis doubling the operators MK,µ, Mc K,µ are pointwise comparable and satisfy the same bounds. Finally for (iv) let us fix the convex set Band x∈∂B. Let Hbe a supporting hyperplane of Bthrough xand let H−be the open half-space defined by Hso that H−∩B=∅. Let R∈K, centered at xand sR be the rectangle with the same center as Rand sides s<1times the corresponding sides of R. So sR is an homothetic copy of R. Consider the 4nsubrectangles Rs,j produced by dividing each side of sR into four equal parts. Now at least one of these Rs,j’s is contained in the open half space H−. Let us call this rectangle R′and observe that it is of the form R′=z+1 4sR ⊂sR and R′∩B=∅. We can then estimate µ(∂B ∩sR)=µ(∂B ∩sR ∩R′)+µ(∂B ∩sR ∖R′) =µ(∂B ∩sR ∖R′)≤µ(sR)−µ(R′) ≤µ(sR)−1 δ2 µ µ(sR)≤cµ(sR), TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 17 with c<1. Applying (iii) for the centered operator Mc K,µ we see that 1>c≥µ(∂B ∩sR)/µ(sR)→1∂B, µ-almost everywhere as s→0+, which implies that µ(∂B)=0. 6. Tauberian conditions for bases of rectangles We now turn our attention to the maximal function MG,µ defined with respect to a non-negative measure µwhich is finite on compact sets and a homothecy invariant basis Gin Rnconsisting of rectangles. Observe that we do not assume the rectangles in Gto have sides parallel to the coordinate axes but one possible choice of Gis the basis R. Our main objective is to find a characterization of the measures µsuch that MG,µ ∶Lp(ν)→ Lp(ν)for some p>1in terms of a mixed µ, ν-Tauberian condition. The Tauberian condition (Aµ B,γ,ν) now takes the form ν({x∈Rn∶MG,µ(1E)(x)>γ})≤cµ G,γ,νν(E).(Aµ G,γ,ν) Our second main result gives a characterization of the boundedness of MG,µ on Lp(ν)in terms of the Tauberian condition (Aµ G,γ,ν), whenever the measure µis doubling with respect to G. Note that for the measure νwe only assume that it is non-negative and locally finite. Theorem 6.1. Let Gbe a homothecy invariant basis consisting of rectangles and µ, ν be two non-negative measures on Rn, finite on compact sets. Assume that µis doubling with respect to G. The following are equivalent: (i) The measures µ, ν satisfy the Tauberian condition (Aµ G,γ,ν)with respect to some fixed level γ∈(0,1). (ii) There exists 1<po=po(cµ G,γ,ν, γ, µ)<+∞such that MG,µ ∶Lp(ν)→Lp(ν)for all p>po. The previous theorem has an interesting corollary whenever µ≡ν. In this special case our main theorem concerns the boundedness of the operator MG,µ on Lp(µ), for sufficiently large p>1and µdoubling with respect to G. As discussed in § 3.3 this scenario is very well understood for the basis Q. Indeed, we already know that for a doubling measure µthe operator MQ,µ is of weak type (1,1)and thus of strong type (p, p)for all p>1. Thus both (i) and (ii) of this theorem are always satisfied for Qand µ≡ν. However, for G=Rand µproduct-doubling we get a new characterization of the measures µsuch that MR,µ is bounded on Lp(µ), for sufficiently large p>1. When µ≡νthe mixed Tauberian condition becomes: µ({x∈Rn∶MG,µ(1E)(x)>γ})≤cµ G,γ,µµ(E).(Aµ G,γ,µ) We then have: Corollary 6.2. Let Gbe a homothecy invariant basis consisting of rectangles and µbe a nonnegative measure on Rn, finite on compact sets. Assume that µis doubling with respect to G. The following are equivalent: 18 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS (i) The measure µsatisfies the Tauberian condition (Aµ G,γ,µ)with respect to some fixed level γ∈(0,1). (ii) There exists 1<po=po(cµ G,γ,ν, γ, µ)<+∞such that MG,µ ∶Lp(µ)→Lp(µ)for all p>po. 6.1. Proof of Theorem 6.1.In this subsection we give the details of the proof of Theorem 6.1. First of all observe that if MG,µ ∶Lp(ν)→Lp(ν)then trivially (Aµ G,γ,ν) is satisfied for every γ∈(0,1). For the rest of this section we will thus assume that (Aµ G,γ,ν) holds for some γ∈(0,1). Let β∈(γ, 1). Any such choice of βwill work equally well but for definitiveness we can take βto be the arithmetic mean of γand 1. The hypothesis implies that ν({x∈Rn∶MG,µ(1E)(x)≥β})≤cν(E)for all measurable sets E⊆Rn.(6.3) Here c=cµ G,γ,ν but we suppress these dependencies for the sake of simplicity. We will need the following notation introduced in [21]. For every measurable set E⊂Rnwe define H0 β(E)∶=E and for k≥1Hk β(E)∶={x∈Rn∶MG,µ(1Hk−1 β(E))(x)≥β}. With these definitions at hand it is not difficult to check the following basic properties. Let k, k′≥0be non-negative integers and A, B measurable subsets of Rn. Then H1 β(Hk β(A))=Hk+1 β(A),(6.4) A⊆B⇒Hk β(A)⊆Hk β(B),(6.5) If k′≤kthen Hk′ β(A)⊆Hk β(A).(6.6) (Aµ G,γ,µ) implies (6.3) which in turn implies that ν(Hk β(A))≤ckν(A).(6.7) The properties above will be used in several parts of the proof with no particular mention. The following lemma is the heart of the proof of Theorem 6.1. Lemma 6.8. Let µbe a doubling measure with respect to G, with doubling constant ∆µ, and E be a measurable set in Rn. Suppose that for some α∈(0, β)and R∈Gwe have 1 µ(R)∫R1Edµ =α. Then R⊂Hkα,β β(E)where kα,β ∶=⌈−log(β α) log β⌉⌈2+log+(β∆µ)) log(1/β)⌉+1. Here we denote by ⌈x⌉the smallest positive integer which is no less than x. Before giving the proof of the lemma let us see how we can use it to conclude the proof of Theorem 6.1. By restricted weak type interpolation it suffices to show that for every 0<λ<1 and every measurable set E⊂Rnwe have the estimate ν({x∈Rn∶MG,µ(1E)(x)>λ})≤C λpoν(E)(6.9) for some po>1and some constant C>0, independent of λand E. Estimate (6.9) above is the claim that the sublinear operator MG,µ is of restricted weak type (po, po)with respect to the TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 19 measure ν, for some po>1. Now we have ν({x∈Rn∶MG,µ(1E)(x)>λ})≤ν({x∈Rn∶λ<MG,µ(1E)(x)<β}) +ν({x∈Rn∶MG,µ(1E)(x)≥β})(6.10) ≤ν({x∈Rn∶λ<MG,µ(1E)(x)<β})+c λpoν(E), by (6.3), for all po>0. In order to estimate the first summand let Eλ,β ∶={λ<MG,µ(1E)(x)<β}. For every x∈Eλ,β there exists Rx∈Gand λ<α<βwith Rx∋x, µ(Rx)>0and µ(Rx∩E) µ(Rx)=α. By Lemma 6.8 we get that Rx⊂Hkα,β β(E). Now observe that kα,β is a nonincreasing function of α. Thus for all α>λwe have that kα,β ≤kλ,β which by (6.6) implies that Hkα,β β(E)⊆Hkλ,β β(E). Combining these observations we get that Eλ,β ⊆⋃ x∈Eλ,β Rx⊆Hkλ,β β(E). Using (6.7) we now see that ν(Eλ,β)≤ν(Hkλ,β β(E))≤ckλ,β ν(E). By the explicit expression for kλ,β observe that we can write kλ,β ≤log(β λ) log 1 β ηβ,µ +1 with ηβ,µ ≥2, depending only on βand µ. Thus ckλ,β ≤ccηβ,µ log 1 λ/log 1 β≤c λpo=cµ G,γ,ν λpo, with po=ηβ,µ log cµ G,γ,ν/log(1/β)>0. Remember that βis completely determined by the level γin hypothesis (Aµ G,γ,µ) so that po=po(cµ G,γ,ν, γ, µ). Together with (6.10) this completes the proof of (6.9) and thus of Theorem 6.1. For the proof of Lemma 6.8 we will need an intermediate result. For this we introduce a final piece of notation. If R∈Gthen there is a natural “dyadic system of rectangles” associated to R which we will denote by DR. This system has the properties (i) We have that R∈DR⊆G. (ii) Every S∈DRhas a unique dyadic parent S(1)and 2ndyadic children. Furthermore, each corner of a rectangle S∈DRis shared by Sand exactly one of its dyadic children. (iii) If V, S ∈DRthen V∩S∈{∅, V, S}. 20 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS We leave the details of the dyadic construction above to the interested reader. We now define the dyadic weighted maximal function with respect to DRand µas MDR,µf(x)∶=sup S∈DR S∋x µ(S)>0 1 µ(S)∫S∣f(y)∣dµ(y), x ∈Rn. The dyadic maximal function just defined satisfies all the desired bounds: Proposition 6.11. Let µbe a locally finite non-negative measure. We have that MDR,µ ∶L1(µ)→ L1,∞(µ). We conclude that the family {R∶R∈DR, R ∋x, µ(R)>0}differentiates L1 loc(µ). Note that there is no doubling assumption on the measure µin this proposition. Indeed, the proof amounts to selecting the maximal “dyadic rectangles” S∈DR∩[0,2N)nsuch that 1 µ(S)∫S∣f(y)∣dµ(y)>λand noting that they are disjoint. One then lets N→+∞. An identical argument works for “dyadic rectangles” contained in the other quadrants of Rn. We leave the details to the interested reader. Lemma 6.12. Let µ, E and Rbe as in the hypothesis of Lemma 6.8 above. Then there exists a non-negative integer Nsuch that µ(R∩HN+2 β(E))≥1 βµ(E∩R). Proof. We perform a Calderón-Zygmund decomposition of 1E∩Rat level βwith respect to the dyadic grid DR. Namely, let {Sj}j⊂DRbe the collection of “dyadic rectangles” which are maximal among the S∈DRthat satisfy 1 µ(S)∫S 1E∩R(y)dµ(y)>β. Observe that µ(S)>0for all rectangles Sby Proposition 5.6. Furthermore µ(E∩R)/µ(R)<βso that every dyadic rectangle Sas above is contained in a maximal dyadic rectangle. This selection algorithm together with the hypothesis µ(R∩E)/µ(R)=α<βallows us to choose a µ-a.e. disjoint family {Sj}j⊂DRsuch that ⋃ j Sj⊆R, Sj≠Rfor all j, {x∈Rn∶MDR,µ(1E∩R)(x)>β}=⋃ j Sj,(6.13) 1 µ(Sj)∫Sj 1E∩Rdµ >β, 1E∩R≤1∪jSjµ-a.e. in R.(6.14) For any constant c>1we let c∗Sjdenote the rectangle containing Sjthat has sidelength ctimes the sidelength of Sjand has a common corner with Sjand S(1) j. With this notation we have S(1) j=2∗Sjwhile the doubling hypothesis for µimplies that µ(S(1) j)≤∆µµ(Sj)for every j. TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 21 For each jwe set Sj,0∶=Sj. Suppose we have defined Sj,0⊂⋯⊂Sj,k for some k≥0. We define Sj,k+1to be a rectangle of the form cj,k+1∗Sj, where cj,k+1>1is chosen so that Sj,k ⊂Sj,k+1and µ(Sj,k+1) µ(Sj,k)=1 β>1.(6.15) Observe that such a choice is always possible since the function f(c)∶=µ(c∗Sj,k)/µ(Sj,k)satisfies f(1)=1,f(c)→+∞as c→+∞and by (iv) of Proposition 5.6 it is continuous on [1,+∞). For k≥0we now set Ek∶=⋃ j Sj,k. Observe that for k≥0we have Ek+1⊂{x∈Rn∶MG,µ(1Ek)(x)≥β}.(6.16) Indeed if x∈Ek+1then x∈Sj0,k+1for some j0. We estimate MG,µ(1Ek)(x)=sup S∈G S∋x µ(S∩Ek) µ(S)≥µ(Sj0,k+1∩⋃jSj,k) µ(Sj0,k+1)≥µ(Sj0,k) µ(Sj0,k+1)=β, by (6.15). Next we claim that for every k≥0we have Ek⊂Hk+1 β(E).(6.17) For k=0this is an immediate consequence of (6.13) since E0=⋃ j Sj={x∈Rn∶MDR,µ(1E∩R)(x)>β} ⊆{x∈Rn∶MG,µ(1E)(x)≥β}=H1 β(E). Assume now that (6.17) is valid for some k≥0. By (6.16), the inductive hypothesis and properties (6.4),(6.5) we get that Ek+1⊂{x∈Rn∶MG,µ(1Ek)(x)≥β}=H1 β(Ek)⊆H1 β(Hk+1 β(E))=Hk+2 β(E), which proves the claim. Now let Nbe the smallest non-negative integer such that β−(N+1)≥∆µ, where ∆µis the doubling constant of the measure µ. It follows that S(1) j⊆Sj,N+1 (6.18) for every j. Indeed, assume for the sake of contradiction that Sj,N+1⊊S(1) j. Then the doubling property of µimplies that µ(Sj,N+1)<µ(S(1) j).Thus ∆µ≥µ(S(1) j) µ(Sj)>µ(Sj,N+1) µ(Sj)=β−(N+1) which contradicts the choice of N. 22 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS Now (6.18) implies that for every jwe have µ(Sj,N ) µ(S(1) j)≥µ(Sj,N ) µ(Sj,N+1)=β and we can conclude that for every j µ(EN∩S(1) j) µ(S(1) j)=µ(⋃νSν,N ∩S(1) j) µ(S(1) j)≥µ(Sj,N ∩S(1) j) µ(S(1) j)≥min (1,µ(Sj,N ) µ(S(1) j))≥β. Hence ⋃ j S(1) j⊆{x∈R∶MG,µ(1EN)(x)≥β}.(6.19) Let {S(1) jk}kdenote the maximal elements of {S(1) j}j. Then the sets {S(1) jk}kare µ-a.e. pairwise disjoint and ⋃kS(1) jk=⋃jS(1) j. Note that all S(1) jk’s are contained in Rsince for all jwe have Sj⫋R. We also have that we have S(1) jk≠Smfor any k, m. Indeed, if S(1) jk=Smfor some k, m then we would have S(1) jk⫋S(1) mwhich is impossible because of the maximality of the S(1) jk’s among the S(1) m’s. Thus none of the S(1) jkwere selected in the Calderón-Zygmund decomposition so that µ(S(1) jk∩E∩R)≤βµ(S(1) jk) and hence µ(S(1) jk∩E)≤βµ(S(1) jk)for all ksince S(1) jk⊆Rfor all k. Using the last estimate and (6.19) we now have µ({x∈R∶MG,µ(1EN)(x)≥β})≥µ(⋃ j S(1) j)=µ(⋃ k S(1) kj) =∑ k µ(S(1) kj)≥1 β∑ k µ(E∩S(1) kj) =1 βµ(E∩⋃ k S(1) kj)=1 βµ(E∩⋃ j S(1) j) ≥1 βµ(E∩⋃ j Sj). Now (6.14) implies that 1E∩R≤1R∩∪jSjalmost everywhere so that µ(E∩R)≤µ(R∩∪jSj). Thus the previous estimate reads µ({x∈R∶MG,µ(1EN)(x)≥β})≥1 βµ(E∩R) which by (6.17) implies that µ(R∩HN+2 β(E))≥β−1µ(E∩R)as desired.  We can now conclude the proof of Lemma 6.8. TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 23 Proof of Lemma 6.8.By the hypothesis of the lemma there exists α∈(0, β)and R∈Gwith µ(E∩R)/µ(R)=α. Let jobe the smallest positive integer such that β−joα≥β. Such an integer obviously exists since β<1. There are two possibilities. case 1: We have that µ(R∩Hj(N+2) β(E))<βµ(R)for j=0,...,jo−1. Then we claim that we have µ(R∩Hk(N+2) β(E))≥1 βkµ(R∩E)for all k=1,...,jo.(6.20) We will prove (6.20) by induction on k. Indeed, the case k=1is just Lemma 6.12. Assume that (6.20) is true for some 1≤k≤jo−1. Then, since µ(R∩Hk(N+2) β(E))<βµ(R)we can apply Lemma 6.12 for the rectangle Rand the set Hk(N+2) β(E)in place of Eto conclude that µ(R∩HN+2 β(Hk(N+2) β(E)))≥1 βµ(Hk(N+2) β(E)∩R)≥1 β(1 β)kµ(R∩E)=(1 β)k+1µ(R∩E). However this is just (6.20) for k+1since HN+2 β(Hk(N+2) β(E))=H(k+1)(N+2) β(E). Now by (6.20) for k=jowe get that 1 µ(R)µ(R∩Hjo(N+2) β(E))≥(1 β)joµ(R∩E) µ(R)=β−joα≥β by the choice of jo. This implies that R⊆Hjo(N+2)+1 β(E). case 2: We have that µ(R∩Hj(N+2) β(E))≥βµ(R)for some j∈{0,...,jo−1}. In fact, by the hypothesis we necessarily have that j≥1in this case. Then 1 µ(R)µ(R∩Hj(N+2) β(E))≥β which implies that R⊆{x∈Rn∶MG,µ(1Hj(N+2) β(E))(x)≥β}=Hj(N+2)+1 β(E). Observe that in either one of the complementary cases considered above we can conclude that R⊆Hjo(N+2)+1 β(E). This proves the lemma with kα,β =jo(N+2)+1. It remains to estimate kα,β. This can be easily done by going back to the way the integers Nand jowere chosen. For Nremember that it is the smallest non-negative integer such that (1/β)N+1≥∆µ. If 1/β≥∆µ then the choice N=0will do. If 1/β<∆µthen we get that Nis the smallest positive integer which is greater or equal to log(β∆µ)/log(1/β). Thus the choice N∶=⌈log+(β∆µ) log(1/β)⌉ covers both cases. Likewise, jois the smallest integer such that β−jo≥β/αor jois the smallest integer greater than log(β/α)/log(1/β). Thus we can choose jo∶=⌈log(β α) log 1 β⌉. 24 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS We set kα,β ∶=jo(N+2)+1=⌈log(β α) log 1 β⌉(⌈log+(β∆µ) log(1/β)⌉+2)+1 =⌈log(β α) log 1 β⌉⌈2+log+(β∆µ) log(1/β)⌉+1. Of course, any integer greater than the kα,β above will also do since the sets Hk β(E)are increasing in k. 7. An extension to bases of convex sets The purpose of this section is to provide an extension of Theorem 6.1 to the case that the Tauberian condition is given with respect to a homothecy invariant basis Bconsisting of convex sets: ν({x∈Rn∶MB,µ(1E)(x)>γ})≤cµ B,γ,νν(E).(Aµ B,γ,ν) As in the previous section where the basis Gconsisted of rectangles, we will need to assume the doubling property of the measure µwith respect to the basis B. The main theorem of this section is the following. Theorem 7.1. Let Bbe a homothecy invariant basis consisting of convex sets and µ, ν be two non-negative measures on Rn, finite on compact sets. Assume that µis doubling with respect to B. The following are equivalent: (i) The measures µ, ν satisfy the Tauberian condition (Aµ B,γ,ν)with respect to some fixed level γ∈(0,1). (ii) There exists 1<po=po(cµ B,γ,ν, n, γ, µ)<+∞ such that MB,µ ∶Lp(ν)→Lp(ν)for all p>po. The general strategy of the proof is the following. Assuming that (Aµ B,γ,ν) is satisfied for some level γ∈(0,1)we will show that the maximal operator MGB,µ also satisfies a Tauberian condition with respect to every level α∈(γ, 1). We will then use Theorem 6.1 to conclude that MGB,µ is bounded on some Lp(ν)-space, for sufficiently large p. According to Lemma 5.5 the operators MGB,µ,MB,µ are pointwise comparable so this will complete the proof of Theorem 7.1. 7.1. The Tauberian condition for MGB,µ.In the subsection we will show that (Aµ B,γ,ν) implies a Tauberian condition for the operator MGB,µ. This is the content of: Lemma 7.2. Suppose that MB,µ satisfies the Tauberian condition (Aµ B,γ,ν)for some fixed level γ∈(0,1). Then for all α∈(γ, 1)the operator MGB,µ satisfies a Tauberian condition with respect to α: ν({x∈Rn∶MB,µ(1E)(x)>α})≤cµ GB,α,νν(E)(Aµ GB,α,ν) where cµ GB,α,ν depends on cµ B,γ,ν, the measures µ, ν, the dimension n,γand α. TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 25 The proof of this lemma is the most crucial step towards Theorem 7.1 and its proof will be explained through several intermediate steps in this section. We will adopt the definitions and notation of § 5.1, namely for every B∈Bwe consider the associated rectangle RBwhere B⊂RB⊂n3 2Band GB={RB∶B∈B}forms a homothecy invariant basis. Our basic assumption is that µis doubling with respect to Bwith doubling constant ∆µ,B. By Lemma 5.5 this implies that µis also doubling with respect to GB, with doubling constant ∆µ,GB. All these notions and constants will be fixed throughout this section so we will just write ∆µ∶=∆µ,Band δµ∶=∆µ,GB. We now fix a convex set Band its associated rectangle R=RB⊃Band work locally inside R. By using a bijection T∶Rn→Rnwe always have [0,1]n=Q=T(R)and then we set K∶=T(B)⊂Q. By considering the pushforward of µ, that is the measure defined as µT(E)∶= µ(T−1E)for every measurable set E, we readily see that the measure µTis doubling with respect to the basis TB∶={T(B)∶B∈B}with doubling constant ∆µ. Also the measure µTis doubling with respect to the basis TGwith doubling constant δµ. Using these invariances we can and will henceforth assume that R=Qand Bis a convex set inside Q. We will use the same notation µ for the measure µT. This will hopefully create no confusion as all our estimates will only depend on the doubling constants which are the same for both measures. The following lemma is the heart of the matter when it comes to the proof of Lemma 7.2. Lemma 7.3. Let Kbe a convex set contained in the unit cube Q=[0,1]nand µbe a doubling measure which is doubling with respect to Euclidean cubes in Rnwith doubling constant δµ. For every >0we have the estimate µ({x∈Rn∖K∶0≤dist(x, K)<})≤vµ(Q), where v≤9δ4+⌈log(34√n)⌉ µ(log 1 )−1. We immediately get the following corollary. Corollary 7.4. Let mbe a positive integer and {Qj}jdenote the dyadic cubes of sidelength 2−m, contained in Qand disjoint from K. Then µ(⋃ j Qj)+µ(K)≥ξmµ(Q), where ξm=1−v√n2−m→1as m→+∞and vis as in Lemma 7.3. Proof. Suppose that x∈Q∖Kand dist(x, K)≥√n2−m. Then since the cubes Qjhave diameter less than √n2−mwe have that x∈Qjfor some j. Thus Q∖(⋃ j Qj∪K)⊂{x∈Q∖K∶0≤dist(x, K)<√n2−m}. Using Lemma 7.3 and the previous inclusion we conclude µ(⋃ j Qj)+µ(K)≥(1−v√n2−m)µ(Q) which is the desired estimate with ξm=1−v√n2−m. 32 P. HAGELSTEIN, T. LUQUE, AND I. PARISSIS bp p Q L `p zp K 2−k Hp H− pH+ p Figure 1. A figure for the proof of Lemma 7.3 so that diam Sp≥1 5dist(zp, K)>1 8dist(zp, K). Thus the Whitney cube Spsatisfies diam Sp≤dist(zp, K)=2−j≤8 diam Sp. By (7.12) we also get that diam Sp>1 8dist(zp, K)=1 8∣zp−p∣. Remember that x∈Aksatisfies ∣x−p∣≤2−k≤2−j. An easy calculation now verifies that x∈34√nSp, where we remember that the dilation is taken with respect to the center of Sp. We have actually shown that for every x∈Akand every 2−j∈[2−k,1], there exists a Whitney cube Sxsuch that x∈34√nSxand 1 82−j<diam Sx≤2−j Let Cjdenote the Whitney cubes such that 1 82−j<diam S≤2−j. Observe that for different j’s the collections C4jare disjoint. We can write for every positive integer j∈[0, k/4) Ak⊂⋃ S∈C4j 34√nS, and thus µ(Ak)≤δ⌈log 34√n⌉ µµ(⋃ S∈C4j S). TAUBERIAN CONDITIONS AND MUCKENHOUPT WEIGHTS 33 Summing in j∈[0, k/4)yields ⌊k/4⌋µ(Ak)≤δ⌈log 34√n⌉ µ ⌊k/4⌋ ∑ j=0 µ(⋃ S∈C4j S), with ⌊x⌋denoting the largest integer less or equal to x. Now all the Whitney cubes Sthat appear on the right hand side of the last display satisfy diam(S)≤1and dist(S, K)≤4, and thus they are all contained in 11Q⊂L. 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