A new quantitative two weight theorem for the Hardy-Littlewood maximal operator

arXiv:1305.0415v1 [math.CA] 2 May 2013 A NEW QUANTITATIVE TWO WEIGHT THEOREM FOR THE HARDY-LITTLEWOOD MAXIMAL OPERATOR CARLOS P´ EREZ AND EZEQUIEL RELA Abstract. A quantitative two weight theorem for the Hardy-Little- wood maximal operator is proved improving the known ones. As a consequence a new proof of the main results in [HP] and [HPR12] is obtained which avoids the use of the sharp quantitative reverse Holder inequality for A∞proved in those papers. Our results are valid within the context of spaces of homogeneous type without imposing the nonempty annuli condition. 1. Introduction and Main results 1.1. Introduction. The purpose of this note is to present a quantitative two weight theorem for the Hardy-Littlewood maximal operator when the underlying space is a space of homogeneous type S(SHT in the sequel), endowed with a quasimetric ρand a doubling measure µ(see Section 2.1 for the precise definitions). We briefly recall some background on this problem in the euclidean or classical setting, when we are working in Rnand we consider Lebesgue measure and euclidean metric. We also assume that in this classical setting all the maximal operators involved and Apclasses of weights are defined over cubes. Let Mstand for the usual uncentered Hardy- Littlewood maximal operator: Mf(x) = sup Q∋x 1 |Q|ZQ |f|dx. The problem of characterizing the pair of weights for which the maximal operator is bounded between weighted Lebesgue spaces was solved by Sawyer [Saw82]: To be more precise, if 1 < p < ∞we define for any pair of weights w, σ, the (two weight) norm, (1.1) kM(·σ)kLp(w):= sup f∈Lp(σ) kM(fσ)kLp(w) kfkLp(σ) 1991 Mathematics Subject Classification. Primary: 42B25. Secondary: 43A85. Key words and phrases. Two weight theorem, Space of homogeneous type, Muckenhoupt weights, Calder´on-Zygmund, Maximal functions. Both authors are supported by the Spanish Ministry of Science and Innovation grant MTM2012-30748 and by the Junta de Andaluc´ıa, grant FQM-4745. 1 2 CARLOS P´ EREZ AND EZEQUIEL RELA then Sawyer showed that kM(·σ)kLp(w)is finite if and and only if sup QRQ(M(χQσ)pwdx σ(Q)<∞, where the supremum is taken over all the cubes in Rn. A quantitative precise version of this result is the following: if we define [w, σ]Sp:= 1 σ(Q)ZQ M(σχQ)pw dx1/p . then (1.2) kM(·σ)kLp(w)∼p′[w, σ]Sp, where 1 p+1 p′= 1. This result is due to K. Moen and can be found in [Moe09]. However, it is still an open problem to find a characterization more closely related to the Apcondition of Muckenhoupt which is easier to use in applications. Indeed, recall that the two weight Apcondition: sup Q− ZQ w dx− ZQ v−1 p−1dxp−1 <∞ is necessary for the boundedness of Mfrom Lp(v) into Lp(w) (which is clearly equivalent, setting σ=v1−p′, to the two weight problem), but it is not sufficient. Therefore, the general idea is to strengthen the Apcondition to make it sufficient. The first result on this direction is due to Neugebauer [Neu83], proving that, for any r > 1, it is sufficient to consider the following “power bump” for the Apcondition: (1.3) sup Q− ZQ wrdx1 r− ZQ v−r p−1dx(p−1)r <∞ Later, the first author improved this result in [P´er95] by considering a different approach which allows to consider much larger classes weights. The new idea is to replace only the average norm associated to the weight v−1 p−1 in (1.3) by an “stronger” norm which is often called a “bump”. This norm is defined in terms of an appropriate Banach function Xspace satisfying certain special property. This property is related to the Lpboundedness of a natural maximal function related to the space. More precisely, for a given Banach function space X, the local X-average of a measurable function f associated to the cube Qis defined as kfkX,Q = τℓ(Q)(fχQ) X, where τδis the dilation operator τδf(x) = f(δx), δ > 0 and ℓ(Q) stands for the sidelength of the cube Q. The natural maximal operator associated to the space Xis defined as MXf(x) = sup Q:x∈Q kfkX,Q QUANTITATIVE TWO WEIGHT THEOREM 3 and the key property is that the maximal operator MX′is bounded on Lp(Rn) where X′is the associate space to X(see (1.5) below). As a corollary of our main result, Theorem 1.3, we will give a quantitative version of the main result from [P´er95] regarding sufficient conditions for the two weight inequality to hold: Theorem 1.1. Let wand σbe a pair of weights that satisfies the condition (1.4) sup − ZQ w dxkσ1/p′kp X,Q <∞. Suppose, in addition, that the maximal operator associated to the associate space is bounded on Lp(Rn): (1.5) MX′:Lp(Rn)→Lp(Rn). Then there is a finite positive constant Csuch that: kM(·σ)kLp(w)≤C. In this note we give a different result of this type with the hope that it may lead to different, possible better, conditions for the two weight problem for Singular Integral Operators. Most of the interesting examples are obtained when Xis an Orlicz space LΦdefined in term of the Young function Φ (see Section 2.1 for the precise definitions). In this case, the local average with respect to Φ over a cube Q is kfkΦ,Q =kfkΦ,Q,µ = inf λ > 0 : 1 µ(Q)ZQ Φ|f| λdx ≤1 where µis here the Lebesgue measure. The corresponding maximal function is (1.6) MΦf(x) = sup Q:x∈Q kfkΦ,Q. Related to condition (1.4) we introduce here the following quantities. Definition 1.2. Let (S, dµ) be a SHT. Given a ball B⊂ S, a Young function Φ and two weights wand σ, we define the quantity (1.7) Ap(w, σ, B, Φ) := − ZB w dµkσ1/p′kp Φ,B and we say that a pair of weights belong to the Ap,Φclass if [w, σ, Φ]Ap:= sup B Ap(w, σ, B, Φ) <∞, where the sup is taken over all balls in the space. In the particular case of Φ(t) = tp′, this condition corresponds to the classical Apcondition and we use the notation [w, σ]Ap:= sup B− ZB w dµ− ZB σ dµp−1 . 4 CARLOS P´ EREZ AND EZEQUIEL RELA We define now a generalization of the Fuji-Wilson constant of a A∞weight σas introduced in [HP] by means of a Young function Φ: [σ, Φ]Wp:= sup B 1 σ(B)ZB MΦσ1/pχBpdµ Note that the particular choice of Φp(t) := tpreduces to the A∞constant ( see Definition (2.5) from Section 2.1): (1.8) [σ, Φp]Wp= sup B 1 σ(B)ZB M(σχB)dµ = [σ]A∞. 1.2. Main results. Our main purpose in the present note is to address the problem mentioned above within the context of spaces of homogeneous type. In this context, the Hardy–Littlewood maximal operator Mis defined over balls: (1.9) Mf(x) = sup B∋x 1 µ(B)ZB |f|dµ. The Orlicz type maximal operators are defined also with balls and with respect to the measure µin the natural way. Our main result is the following theorem. Theorem 1.3. Let 1< p < ∞and let Φbe any Young function with conjugate function ¯ Φ. Then, for any pair of weights w, σ, there exists a structural constant C > 0such that the (two weight) norm defined in (1.1) satisfies (1.10) kM(·σ)kLp(w)≤Cp′[w, σ, Φ]Ap[σ, ¯ Φ]Wp1/p , We emphasize that (1.10), which is even new in the usual context of Euclidean Spaces, fits into the spirit of the Ap−A∞theorem derived in [HP] and [HPR12]. The main point here is that we have a two weight result with a better condition and with a proof that avoids completely the use of the sharp quantitative reverse H¨older inequality for A∞weights proved in these papers. This property is, of course, of independent interest but it is not used in our results. From this Theorem, we derive several corollaries. First, we have a direct proof of the two weight result derived in [HP] using the [w]A∞constant of Fujii-Wilson (2.5). Corollary 1.4. Under the same hypothesis of Theorem 1.3, we have that there exists a structural constant C > 0such that (1.11) kM(·σ)kLp(w)≤Cp′[w, σ]Ap[σ]A∞1/p . Note that the result in Theorem 1.3 involves two suprema like in Corollary 1.4. It would interesting to find out if there is a version of this result involving only one supremum. There is some evidence that it could be the case, see for example [HP], Theorem 4.3. See also the recent work [LM]. As a second consequence of Theorem 1.3, we have the announced quantitative version of Theorem 1.1: QUANTITATIVE TWO WEIGHT THEOREM 5 Corollary 1.5. Under the same hypothesis of Theorem 1.3, we have that there exists a structural constant C > 0such that kM(·σ)kLp(w)≤Cp′[w, σ, Φ]1/p ApkM¯ ΦkLp(Rn) We remark that this approach produces a non-optimal dependence on p, since we have to pay with one p′for using Sawyer’s theorem. However, the ideas from the proof of Theorem 1.3 can be used to derive a direct proof of Corollary 1.5 without the p′factor. We include the proof in the appendix. Finally, for the one weight problem, we recover the known mixed bound. Corollary 1.6. For any Apweight wthe following mixed bound holds: kMkLp(w)≤Cp′[w]Ap[σ]A∞1/p where Cis an structural constant and as usual σ=w1−p′is the dual weight. Remark 1.7.To be able to extend the proofs to this general scenario, we need to use (and prove) suitable versions of classical tools on this subject, such as Calder´on–Zygmund decompositions. We remark that in previous works ([PW01], [SW92]) most of the results are proved under the assumption that the space has non-empty annuli. The main consequence of this property is that in that case the measure µenjoys a reverse doubling property, which is crucial in the proof of Calder´on–Zygmund type lemmas. However, this assumption implies, for instance, that the space has infinite measure and no atoms (i.e. points with positive measure) and therefore constraints the family of spaces under study. Recently, some of those results were proven without this hypothesis, see for example [PS04]. Here we choose to work without the annuli property and therefore we need to adapt the proofs from [PW01]. Hence, we will need to consider separately the cases when the space has finite or infinite measure. An important and useful result on this matter is the following: Lemma 1.8 ([GGKK98]).Let (S, ρ, µ)be a space of homogeneous type. Then Sis bounded if and only if µ(S)<∞. 1.3. Outline. The article is organized as follows. In Section 2we summarize some basic needed results on spaces of homogeneous type and Orlicz spaces. We also include a Calder´on–Zygmund type decomposition lemma. In Section 3we present the proofs of our results. Finally, we include in Section 4an Appendix with a direct proof of a slightly better result than Corollary 1.4. 2. preliminaries In this section we first summarize some basic aspects regarding spaces of homogeneous type and Orlicz spaces. Then, we include a Calder´on– Zygmund (C–Z) decomposition lemma adapted to our purposes. 6 CARLOS P´ EREZ AND EZEQUIEL RELA 2.1. Spaces of homogeneous type. A quasimetric don a set Sis a function d:S × S → [0,∞) which satisfies (1) d(x, y) = 0 if and only if x=y; (2) d(x, y) = d(y, x) for all x, y; (3) there exists a finite constant κ≥1 such that, for all x, y, z ∈ S, d(x, y)≤κ(d(x, z) + d(z, y)). Given x∈ S and r > 0, we define the ball with center xand radius r, B(x, r) := {y∈ S :d(x, y)< r}and we denote its radius rby r(B) and its center xby xB. A space of homogeneous type (S, d, µ) is a set Sendowed with a quasimetric dand a doubling nonnegative Borel measure µsuch that (2.1) µ(B(x, 2r)) ≤Cµ(B(x, r)) Let Cµbe the smallest constant satisfying (2.1). Then Dµ= log2Cµis called the doubling order of µ. It follows that (2.2) µ(B) µ(˜ B)≤C2+log2κ µr(B) r(˜ B)Dµ for all balls ˜ B⊂B. In particular for λ > 1 and Ba ball, we have that (2.3) µ(λB)≤(2λ)Dµµ(B). Here, as usual, λB stands for the dilation of a ball B(x, λr) with λ > 0. Throughout this paper, we will say that a constant c=c(κ, µ)>0 is a structural constant if it depends only on the quasimetric constant κand the doubling constant Cµ. An elementary but important property of the quasimetric is the following. Suppose that we have two balls B1=B(x1, r1) and B2=B(x2, r2) with non empty intersection. Then, (2.4) r1≤r2=⇒B1⊂κ(2κ+ 1)B2. This is usually known as the “engulfing” property and follows directly from the quasitriangular property of the quasimetric. In a general space of homogeneous type, the balls B(x, r) are not necessarily open, but by a theorem of Macias and Segovia [MS79], there is a continuous quasimetric d′which is equivalent to d(i.e., there are positive constants c1and c2such that c1d′(x, y)≤d(x, y)≤c2d′(x, y) for all x, y ∈ S) for which every ball is open. We always assume that the quasimetric dis continuous and that balls are open. We will adopt the usual notation: if νis a measure and Eis a measurable set, ν(E) denotes the ν-measure of E. Also, if fis a measurable function on (S, d, µ) and Eis a measurable set, we will use the notation f(E) := REf(x)dµ. We also will denote the µ-average of fover a ball Bas fB= − RBfdµ. We recall that a weight w(any non negative measurable function) QUANTITATIVE TWO WEIGHT THEOREM 7 satisfies the Apcondition for 1 < p < ∞if [w]Ap:= sup B− ZB w dµ− ZB w−1 p−1dµp−1 , where the supremum is taken over all the balls in S. The A∞class is defined in the natural way by A∞:= Sp>1Ap This class of weights can also be characterized by means of an appropriate constant. In fact, there are various different definitions of this constant, all of them equivalent in the sense that they define the same class of weights. Perhaps the more classical and known definition is the following due to Hruˇsˇcev [Hru84] (see also [GCRdF85]): [w]exp A∞:= sup B− ZB w dµexp − ZB log w−1dµ. However, in [HP] the authors use a “new” A∞constant (which was originally introduced implicitly by Fujii in [Fuj78] and later by Wilson in [Wil87]), which seems to be better suited. For any w∈A∞, we define (2.5) [w]A∞:= [w]W A∞:= sup B 1 w(B)ZB M(wχB)dµ, where Mis the usual Hardy–Littlewood maximal operator. When the underlying space is Rd, it is easy to see that [w]A∞≤c[w]exp A∞for some structural c > 0. In fact, it is shown in [HP] that there are examples showing that [w]A∞is much smaller than [w]exp A∞The same line of ideas yields the inequality in this wider scenario. See the recent work of Beznosova and Reznikov [BR] for a comprehensive and thorough study of these different A∞constants. We also refer the reader to the forthcoming work of Duoandikoetxea, Martin-Reyes and Ombrosi [DMRO13] for a discussion regarding different definitions of A∞classes. 2.2. Orlicz spaces. We recall here some basic definitions and facts about Orlicz spaces. A function Φ : [0,∞)→[0,∞) is called a Young function if it is continuous, convex, increasing and satisfies Φ(0) = 0 and Φ(t)→ ∞ as t→ ∞. For Orlicz spaces, we are usually only concerned about the behaviour of Young functions for tlarge. The space LΦis a Banach function space with the Luxemburg norm kfkΦ=kfkΦ,µ = inf λ > 0 : ZS Φ(|f| λ)dµ ≤1. Each Young function Φ has an associated complementary Young function ¯ Φ satisfying t≤Φ−1(t)¯ Φ−1(t)≤2t for all t > 0. The function ¯ Φ is called the conjugate of Φ, and the space L¯ Φ is called the conjugate space of LΦ. For example, if Φ(t) = tpfor 1 < p < ∞, then ¯ Φ(t) = tp′, p′=p/(p−1), and the conjugate space of Lp(µ) is Lp′(µ). 8 CARLOS P´ EREZ AND EZEQUIEL RELA A very important property of Orlicz spaces is the generalized H¨older inequality (2.6) ZS |fg|dµ ≤2kfkΦkgk¯ Φ. Now we introduce local versions of Luxemburg norms. If Φ is a Young function, let kfkΦ,B =kfkΦ,B,µ = inf λ > 0 : 1 µ(B)ZB Φ|f| λdµ ≤1. Furthermore, the local version of the generalized H¨older inequality (2.6) is (2.7) 1 µ(B)ZB fg dµ ≤2kfkΦ,Bkgk¯ Φ,B. Recall the definition of the maximal type operators MΦfrom (1.6): (2.8) MΦf(x) = sup B:x∈B kfkΦ,B. An important fact related to this sort of operator is that its boundedness is related to the so called Bpcondition. For any positive function Φ (not necessarily a Young function), we have that kMΦkp Lp(S)≤cµ,κ αp(Φ), where αp(Φ) is the following tail condition (2.9) αp(Φ) = Z∞ 1 Φ(t) tp dt t<∞. It is worth noting that in the recent article [LL] the authors define the appropriate analogue of the Bpcondition in order to characterize the boundedness of the strong Orlicz-type maximal function defined over rectangles both in the linear and multilinear cases. Recent developments and improvements can also be found in [MP], where the authors addressed the problem of studying the maximal operator between Banach function spaces. 2.3. Calder´on–Zygmund decomposition for spaces of homogeneous type. The following lemma is a classical result in the theory, regarding a decomposition of a generic level set of the Hardy–Littlewood maximal function M. Some variants can be found in [AM84] for Mand in [Aim85] for the centered maximal function Mc. In this latter case, the proof is straightforward. We include here a detailed proof for the general case of Mwhere some extra subtleties are needed. Lemma 2.1 (Calder´on–Zygmund decomposition).Let Bbe a fixed ball and let fbe a bounded nonnegative measurable function. Let Mbe the usual non centered Hardy–Littlewood maximal function. Define the set Ωλas (2.10) Ωλ={x∈B:Mf(x)> λ}, QUANTITATIVE TWO WEIGHT THEOREM 9 Let λ > 0be such that λ≥− RBf dµ. If Ωλis non-empty, then given η > 1, there exists a countable family {Bi}of pairwise disjoint balls such that, for θ= 4κ2+κ, i) ∪iBi⊂Ωλ⊂ ∪iθBi, ii) For all i, λ < 1 µ(Bi)ZBi fdµ. iii) If Bis any ball such that Bi⊂Bfor some iand r(B)≥ηr(Bi), we have that (2.11) 1 µ(ηB)ZηB fdµ ≤λ. Proof. Define, for each x∈Ωλ, the following set: Rλ x=r > 0 : − ZB f dµ > λ, x ∈B=B(y, r), which is clearly non-empty. The key here is to prove that Rλ xis bounded. If the whole space is bounded, there is nothing to prove. In the case of unbounded spaces, we argue as follows. Since the space is of infinite measure (recall Lemma 1.8), and clearly S=Sr>0B(x, r), we have that µ(B(x, r)) goes to +∞when r→ ∞ for any x∈ S. Therefore, for K=κ(2κ+ 1), we can choose r1such that the ball B1=B(x, r1) satisfies the inequality µ(B1)≥2(2K)DµkfkL1 λ Suppose now that sup Rλ x= +∞. Then we can choose a ball B2=B(y, r2) for some ysuch that x∈B2,− RB2f dµ > λ and r2> r1. Now, by the engulfing property (2.4) we obtain that B1⊂KB2. The doubling condition (2.3) yields µ(B1)≤µ(KB2)≤(2k)Dµµ(B2) Then we obtain that 2kfkL1 λ≤µ(B2)<kfkL1 λ which is a contradiction. We conclude that, in any case, for any x∈Ωλ, we have that sup Rλ x<∞. Now fix η > 1. If x∈Ωλ, there is a ball Bxcontaining x, whose radius r(Bx) satisfies sup Rλ x η< r(Bx)≤sup Rλ x, and for which − RBxf dµ > λ. Thus the ball Bxsatisfies ii) and iii). Also note that Ωλ=Sx∈ΩλBx. Picking a Vitali type subcover of {Bx}x∈Ωλas in [SW92], Lemma 3.3, we obtain a family of pairwise disjoint balls {Bi} ⊂ {Bx}x∈Ωλsatisfying i). Therefore {Bi}satisfies i), ii) and iii).  We will need another important lemma, in order to handle simultaneously decompositions of level sets at different scales. 16 CARLOS P´ EREZ AND EZEQUIEL RELA [P´er95] Carlos P´erez. On sufficient conditions for the boundedness of the Hardy- Littlewood maximal operator between weighted Lp-spaces with different weights. Proc. London Math. Soc. (3), 71(1):135–157, 1995. [PS04] Gladis Pradolini and Oscar Salinas. Maximal operators on spaces of homogeneous type. Proc. Amer. Math. Soc., 132(2):435–441 (electronic), 2004. [PW01] Carlos P´erez and Richard L. Wheeden. Uncertainty principle estimates for vector fields. J. Funct. Anal., 181(1):146–188, 2001. [Saw82] Eric T. Sawyer. A characterization of a two-weight norm inequality for maximal operators. Studia Math., 75(1):1–11, 1982. [SW92] E. Sawyer and R. L. Wheeden. Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer. J. Math., 114(4):813–874, 1992. [Wil87] J. Michael Wilson. Weighted inequalities for the dyadic square function without dyadic A∞.Duke Math. J., 55(1):19–50, 1987. Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, 41080 Sevilla, Spain E-mail address:[email protected] Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, 41080 Sevilla, Spain E-mail address:[email protected]