The Hardy-Littlewood maximal type operators between Banach function spaces

The Hardy-Littlewood Maximal Type Operators between Banach Function Spaces MIECZYSŁAW MASTYŁO &CARLOS P´ EREZ ABSTRACT.We investigate variants of the maximal operator and show their applications to study boundedness of the classical Hardy- Littlewood maximal operator between weighted Banach function spaces which satisfy certain geometrical lattice conditions. We prove inequalities for rearrangement of the maximal operators generated by rearrangement invariant spaces. Applying this to the Lorentz spaces, we give new sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp-spaces with different weights. We also prove that under some mild hypotheses these conditions are also necessary. 1. INTRODUCTION Maximal functions have proved to be tools of great importance in analysis and in particular in harmonic analysis. The main example is the Hardy-Littlewood maximal function, which is defined for locally integrable functions f∈L1 loc(Rn) by Mf (x) =sup Q∋x 1 |Q|ZQ |f(y)|dy, x ∈Rn, where the supremum is taken over all the cubes Qcontaining x. Their study not only contains intrinsic interest but also intertwines with the study of singular integral operators, very particularly in the context of weighted norm inequalities (see [7,8,10]). As an example of the central role played by M, we mention the celebrated extrapolation theorem of Rubio de Francia. Indeed, the proof of this theorem is based on appropriate boundedness properties of the Hardy-Littlewood maximal function on weighted Lp-spaces. We refer to [5] for an overview of 883 Indiana University Mathematics Journal c , Vol. 61, No. 3 (2012) 884 MIECZYSŁAW MASTYŁO &CARLOS P´ EREZ the theory and to [9] for a recent improvement of the extrapolation theorem. Therefore the study of weighted estimates with one or two weights for maximal functions will yield a better understanding for the corresponding situation for singular integral operators. The starting point of the modern theory of weighted norm inequalities goes back to the paper of Muckenhoupt [15]. In that paper the inequality (1.1) ZRn(w(x)Mf (x))pdx≤cZRn(v(x)|f(x)|)pdx with w=vwas characterized in terms of the surprisingly simple geometric condition (1.2) sup Q1 |Q|ZQw(y)pdy1/p1 |Q|ZQw(y)−p′dy1/p′ <∞, usually denoted by Ap, although the normalization we will be using in this paper on the weights differs from the usual one in harmonic analysis. The study of the case when the weights are different is of interest for many reasons, and there is a revival of the area mainly because of its relationship with the theory of singular integrals. Sawyer obtained in [22] a characterization by means of the following Spcondition: there is a positive constant csuch that for all cubes Qwe have ZQ(M(v−p′χQ)(y)w(y))pdy≤cZQv−p′(y) dy. We note that in general Sawyer’s Spcondition is not so easy to verify in practice, especially if we compare it with the two weight Apcondition (1.2) which is necessary and sufficient for the weak type boundedness, namely, (1.3) sup λ>0 λpwp({x∈Rn:Mf(x) > λ})≤CZRn(|f (x)|v(x))pdx, as it is well known [7]. However, simple examples show (cf. [7, p. 395]) that this condition is not sufficient for the two weight problem (1.1), namely, we have the strict inclusion Sp⊂Ap. It is therefore natural to try to understand the “gap” between the two conditions. In particular, it would be interesting to obtain sufficient conditions “close” in form to Apdue to its simplicity. The first author who studied this problem was Neugebauer in [16], where he proved that if (w, v) is a couple of weights such that for some r > 1 (1.4) sup Q1 |Q|ZQw(y)pr dy1/pr 1 |Q|ZQv(y)−p′rdy1/p′r <∞, The Hardy-Littlewood Maximal Type Operators 885 where the supremum is taken over all cubes Q, then (1.1) holds. This condition is often called the “power bump” condition. The method in [16] is interesting because it shows that (1.4) is equivalent to “inserting” an Apweight between the given weights wand v. However, it is not sharp enough to obtain better conditions. The two weight problem was further considered in [19] by the second author, who used a different approach that allowed the generation of larger classes of couples of weights (w, v) for which (1.1) holds. These larger classes contain Neugebauer’s condition (1.4) as a particular case. The key idea is to replace the average norm associated to the weight v−1in (1.2) by an “stronger” norm defined in terms of any appropriate Banach function space whose associated space satisfies an appropriate Lpboundedness. The main non-power bump example is given by (1.5) sup Q1 |Q|ZQw(y)pdy1/p kv−1kX,Q <∞, where X=LBis the Orlicz space defined by the Young function Band the average is given by (1.6) kfkX,Q = kfkB,Q =inf λ > 0 : 1 |Q|ZQB|f(y)| λdy≤1. The key point is to assume that the maximal operator corresponding to its associate space X′, namely, MX′, is bounded on Lp(Rn). In the case that X=LBthis is given by the tail condition: (1.7) Z∞ c tp′ B(t)!p−1dt t<∞, and no better than that. Interesting examples are given by B(t) ≈tp′(logt)p′−1+δ, or B(t) ≈tp′(logt)p′−1(log(logt))p′−1+δ, where δ > 0. Applications of these ideas can be found in [4,6]. We refer the reader to [5] for more information about classes of weights and for some other related results. Besides its intrinsic interest, conditions like (1.5) carry some deep information because they are intimately related to some sharp endpoint estimate for singular integral operators. Indeed, it is shown in [18] that as a consequence of a special choice of Bin (1.7), for a given ε > 0, there is c > 0 depending on ε, the dimension and the smoothness of the kernel of Tsuch that (1.8) kTfkL1,∞(w) ≤cε,n,T ZRn|f(x)|ML(log L)ε(w)(x) dx, w ≥0. It was conjectured in that paper that this result would be false in the case ε=0 where ML(log L)εis replaced by M, but only recently this result was shown to be false by Reguera and Thiele in [21] (see also the previous work by Reguera in [20]). 886 MIECZYSŁAW MASTYŁO &CARLOS P´ EREZ The purpose of this paper is to study the two weight problem for Mwithin the context of Banach function spaces. We believe that some of our results may lead to a better understanding of more difficult operators like singular integrals. The paper is organized as follows. In Section 2, we introduce some fundamental notation. In Section 3, we prove the main result on the boundedness of the Hardy-Littlewood maximal operator between Banach function spaces. We involve a general variant of the Hardy-Littlewood maximal MFoperator introduced in [19]. The operator MFis generated by a rearrangement invariant space on Rn. We study the boundedness of this type of operator between Banach function spaces. This problem is motivated by the two weight problem for Mand by (1.8), and seems interesting on its own. In Section 3, we also prove estimates of the rearrangement the Hardy-Littlewood type maximal function generated by Lorentz space. These estimates allow us to provide a sufficient condition on rearrangement invariant spaces Fand Xon Rnwhich ensures that the maximal operator MFis bounded on X. As a consequence we derive that if Fis a rearrangement invariant space with fundamental function ϕF, then MFis bounded on Lp(Rn), 1< p < ∞, provided Z1 0 s−1/p dϕF(s) < ∞. Furthermore, we prove that if ϕFis a submultiplicative function near zero, then the condition above is also necessary for boundedness of MFon Lp(Rn). As an application of our results, we are able to localize a class of couples of weights (w, v) for which Mis bounded from Lp(vp)to Lp(wp). In particular, in the case of classical Lorentz space Ls,1with 1 < s < ∞, we recover the result proved in [19] in a different way. 2. NOTATION AND PRELIMINARIES In this paper we shall use the standard notation and terminology in the theory of Banach lattices. A Banach function space on a measure space (S, µ) is defined to be a Banach space Xwhich is a subspace of L0=L(µ) (the topological linear space of all equivalence classes of the real Lebesgue measurable functions equipped with the topology of convergence in measure) such that there exists u∈Xwith u > 0 almost everywhere and if |f| ≤ |g|almost everywhere, where g∈Xand f∈L0, then f∈Xand kfkX≤ kgkX. Let Xbe a Banach lattice on (S, µ) and w∈L0(µ) be a weight, i.e., with w > 0 almost everywhere on S. We define the weighted Banach function space X(w) by kxkX(w) = kxwkX. The Banach function space Xis said to be order continuous if for every x∈Xand every sequence (xn)such that 0 ≤xn≤ |x| and xn↓0 almost everywhere, it holds that kxnkX→0. We say that Xsatisfies the Fatou property whenever for any xn∈Xand x∈L0such that xn→x almost everywhere and supnkxnkX<∞, we have that x∈Xand kxkX≤ liminfnkxnkX. The Hardy-Littlewood Maximal Type Operators 887 The K¨ othe dual space X′of Xis a collection of all elements y∈L0such that kykX′=supZS |xy|dµ:kxkX≤1<∞. The space X′equipped with the norm k · kX′is a Banach function space with the Fatou property. It is well known that X=X′′ with equality of norms whenever X satisfies the Fatou property. Given f∈L0, its distribution function is defined by µf(λ) =µ({x∈S:|f(x)|> λ}), and its decreasing rearrangement is defined by f∗(t) =f∗ µ(t) :=inf{λ≥0 : µf(λ) ≤t} for t≥0. A Banach function space (X, k · kX)on (S, µ) is called admissible provided χA∈Xfor every measurable set Aof finite measure. In what follows, if Xis a Banach function space on Rn(i.e., on (Rn, µ), where µis the Lebesgue measure), we write ϕX(|A|):= kχAk, where χAdenotes the characteristic function of a measurable set Aand |A|:=µ(A). Here ϕXis said to satisfy the ∆2-condition provided there exists a constant C > 0 such that ϕX(|2Q|)≤CϕX(|Q|)for all cubes Qwith finite Lebesgue measure. Important classes of admissible Banach function spaces Xare rearrangement invariant (r.i.) spaces. Recall that Xis called an r.i. space provided µf=µg, f∈Ximplies g∈X,kfkX= kgkX. Clearly kχAkXdepends only on µ(A). The function ϕX(t) := kχAkX, where µ(A) =t, 0 ≤t≤µ(S), is called the fundamental function of X. Examples of r.i. Banach spaces are Marcinkiewicz and Lorentz spaces on (S, µ). Let I:=[0, µ(S)) and ϕ:I→[0,∞)be a quasi-concave function, that is, ϕ(0)=0 and ϕ(t) > 0 for all 0 < t ∈Iand both ϕand t֏ϕ∗(t) :=t/ϕ(t) are non-decreasing functions on I. We note that a quasi-concave function ϕis equivalent to its least concave majorant ˜ ϕ; more exactly, 1 2˜ ϕ(t) ≤ϕ(t) ≤˜ ϕ(t) for all t∈I. The Marcinkiewicz space M(ϕ) is the r.i. space of all f∈L0(µ) such that kfkM(ϕ) :=sup t∈I 1 ϕ(t) Zt 0 f∗(s) ds < ∞. If ϕ:I→[0,∞)is a non-decreasing concave function, with ϕ(0)=0, the Lorentz space Λ(ϕ) consists of all f∈L0such that kfkΛ(ϕ) =ZIf∗(s) dϕ(s) :=ϕ(0+)f∗(0+)+ZIf∗(s)ϕ′(s) ds < ∞, 888 MIECZYSŁAW MASTYŁO &CARLOS P´ EREZ where ϕ′denotes the derivative of ϕ(which exists almost everywhere). Notice here that the fundamental functions of these spaces are ϕΛ(ϕ) =ϕ and ϕM(ϕ) =ϕ∗. In the sequel we will need the well-known fact that if Xis an r.i. Banach space on Rnwith the fundamental function ϕ, then ϕis quasi-concave and the following continuous inclusions hold (see [11, Theorems II.5.5 and II.5.7] or [3, Theorem II.5.13]): Λ(˜ ϕ) ֓X֓M(ϕ∗). 3. MAIN RESULTS Let Xbe a Banach function space on Rn. Throughout the paper Xloc denotes the space of all f∈L0such that f χQ∈Xfor every cube Qin Rn. Following [19], for every f∈Xloc, we define the X-average by kfkX,Q = kτℓ(Q)(fχQ)kX. Here τδwith δ > 0 is the dilatation operator defined for all f∈L0by τδf(x) = f(δx), and ℓ(Q) is the side-length of the cube Q. We define the maximal operator MXassociated to the space Xby MXf(x) =sup Q∋x kfkX,Q, f ∈Xloc, where the supremum is taken over all the cubes Qcontaining x. We observe that in the case of the Orlicz space Lϕon Rn, the maximal operator Mϕ:=MLϕis given by Mϕf(x) =sup Q∋x kfχQkϕ,Q, where kfkϕ,Q :=infλ > 0 : 1 |Q|ZQϕ|f(x)| λdx≤1. In [1] it was proved that there exists a constant cn>0 such that (Mϕf)∗(t) ≤cnkf∗kϕ,(0,t), t > 0. It is easy to see that the maximal operator Mp,q :=MLp,q associated to the Lorentz Lp,q on Rnwith 1 < p, q < ∞is given by Mp,qf(x) =sup Q∋x 1 |Q|1/p kfχQkp,q. The Hardy-Littlewood Maximal Type Operators 889 This operator was studied in [2,12,23]. It was proved in [2], using interpolation, that if 1 ≤q≤p, then there exists a constant c > 0 such that (Mp,qf)∗(t) ≤c t1/p Zt 0 f∗(s)qsq/p−1ds1/q , t > 0. We extend the above-mentioned rearrangement estimates to the more general setting of maximal operators and show applications to study boundedness of the Hardy-Littlewood maximal operator. For formulation of the first result on boundedness of the maximal operator, some additional notation and definitions are required. Given a Banach function space Xand 1 <p<∞, we define the p-concavification Xpto be a quasi-Banach function space of all x∈L0(µ) such that |x|1/p ∈X, equipped with the quasinorm kxkXp= k |x|1/p kp X. The Banach function space Xis said to be p-convex if there exists a constant C > 0 such that for any x1,...,xn∈X, we have   n X k=1 |xk|p1/p  X≤Cn X k=1 xk p X1/p. The least Cis denoted by M(p)(X). Note that if Xis p-convex, then for x∈Xpand x1,...,xn∈Xpwith |x| ≤ Pn k=1|xk|, we have kxkXp≤  n X k=1 (|xk|1/p)p1/p  Xp ≤M(p)(X)p n X k=1 |xk|1/p  p X =M(p)(X)p n X k=1 kxkkXp. This shows that the lattice norm k · k∗defined on Xpby kxk∗:=infnn X k=1 kxkkXp:|x| ≤ n X k=1 |xk|, n ∈No, x ∈Xp, satisfies M(p)(X)−pk · kXp≤ k · k∗≤ k · kXp. In particular, this implies a well-known fact: if a Banach function space Xon (S, µ) is p-convex (1< p < ∞), then there exists a Banach function space Yon (S, µ) such that x∈Xif and only if |x|p∈Yand M(p)(X)−1kxkX≤ |x|p 1/p Y≤ kxkX. This fact will be used below in the proof without any references. 890 MIECZYSŁAW MASTYŁO &CARLOS P´ EREZ A Banach lattice Xis said to satisfy a lower p-estimate, 1 < p < ∞, if there exists a constant Csuch that n X i=1 xi p E1/p ≤C   n X i=1 xi  E, for every finite set of pairwise disjoint elements {x1,...,xn}in X(see [14, 1.f.4]). The least Cis denoted by ℓ(p)(X). The following theorem is an extension of the main result, Theorem 1.2, from [19]. Theorem 3.1. Let E,X, and Ybe Banach function spaces over Rnwhere E,X are admissible, and let (w, v) be a couple of weights on Rn. If the maximal operator ME′associated to E′is bounded on X,Xsatisfies a lower p-estimate, Yis p-convex for some 1< p < ∞,ϕXsatisfies the ∆2-condition, and sup Q∋Q ϕY(w)(|Q|) ϕX(|Q|)kv−1kE,Q <∞, then the Hardy-Littlewood maximal operator Mis bounded from X(v) into Y(w). Proof. Our hypothesis that X(w) satisfies a lower p-estimate implies that X(w) is order continuous and so a separable space. Thus the set of bounded functions with compact support is dense in X(w). Fix f∈X(w)band a > 2n. For each integer k, we set Ωk= {x∈Rn:ak< Mf(x)}, Dk=(x∈Rn:Mdf(x) > ak 4n). Applying the classical Calder´ on-Zygmund decomposition [7, p. 137], we conclude that there is a family of maximal non-overlapping dyadic cubes {Qk,j}such that Ωk⊂Sj3Qk,j,Dk=SjQk,j, and ak 4n<1 |Qk,j|ZQk,j f (y) dy≤ak 2n. Without loss of generality, we may assume that the M(p)(Y) =1 and so there exists a Banach function space ˜ Yon Rnsuch that y∈Yif and only if |x|p∈˜ Y and kxkY= k |x|pk1/p ˜ Y. Combining the above yields that for some universal The Hardy-Littlewood Maximal Type Operators 891 constants Cj(j=1,...,4), we have kMfkY (w) = ((Mf)w)p 1/p ˜ Y=  X k ((Mf)w)pχΩk\Ωk+1  1/p ˜ Y ≤C1  X k akpwpχΩk\Ωk+1  1/p ˜ Y ≤C2   X k,j 1 |Qk,j|ZQk,j |f(y)|dyp wpχ3Qk,j     1/p ˜ Y ≤C3   X k,j 1 |3Qk,j|Z3Qk,j |f(y)|v(y)v(y)−1dyp wpχ3Qk,j     1/p ˜ Y ≤C4X k,j 1 |3Qk,j|Z3Qk,j |f(y)|v(y)v(y)−1dyp χ3Qk,j  p Y(w)1/p . For any integers k,j, we put Ek,j =Qk,j \Qk,j ∩Dk+1, and by [19, Lemma 4.2] there exists a constant β > 0 such that |Qk,j|< β|Ek,j|for each k,j. The ∆2-condition for ϕXgives that there exists a constant Cβ>0 such that for all k,j ϕX(|3Qk,j|)≤CβϕX(|Ek,j|)=CβkχEk,j kX. For all cubes Q, g ∈Eloc and h∈E′ loc, we have 1 |Q|ZQg(y)h(y) dy≤ kgkX,Q khkX′,Q. Thus the combination of the inequalities shown above with our hypotheses on ME′ and Ximplies that the following estimates hold with some universal constants ˜ Cj, j=1,2,3: kMfkY (w) ≤˜ C1X k,j  fv p E′,3Qk,j  v−1 p E,3Qk,j  χ3Qk,j  p Y(w)1/p ≤˜ C2X k,j  fv p E′,3Qk,j  v−1 p E,3Qk,j ϕY (w)(|3Qk,j|) ϕX(|3Qk,j|)p ϕX(|3Qk,j|)p1/p ≤˜ C2CβX k,j  ME′(fv)χEk,j  p X1/p ≤˜ C2Cβℓ(p)(X)ksup k,j ME′(fv)χEk,j kX ≤˜ C3kME′(fv)kX≤˜ C3kfkX(v). ❐ 898 MIECZYSŁAW MASTYŁO &CARLOS P´ EREZ Theorem 3.10. If Fis an r.i. space on Rnwith γF<∞, then the following formulas hold: γF=infp > 1 : MFis bounded on Lp(Rn) =infp > 1 : Z1 0 ϕF(s)p s2ds < ∞ =infp > 1 : Z1 0 s−1/p dϕF(s) < ∞. To state the next result, we need the K¨ othe duality formulas for Lorentz and Marcinkiewicz spaces (cf. [3,11]). For the sake of completeness we show these formulas below. If (S, µ) is an atomless measure space, then for any concave function ψ:[0, µ(S)) →[0,∞)with ψ(0)=0, the following duality formulas with equality of norms hold true: Λ(ψ)′=M(ψ) and M(ψ)′=Λ(ψ). An immediate consequence of the above Corollary 3.7, Theorem 3.6, and the duality formula M(ϕ∗)′=Λ(g ϕ∗)is the following theorem. Theorem 3.11. Let 1< p < ∞, and let ϕ:[0,∞)→[0,∞)be a quasi-concave function such that R1 0s−1/p dϕ∗(s) < ∞. Suppose that (w, v) is a couple of weights on Rnsuch that sup Q∈Q 1 |Q|ZQw(y)pdy1/p kv−1kM(ϕ∗),Q <∞. Then the maximal Hardy-Littlewood operator Mis bounded from Lp(vp)to Lp(wp). We conclude with the following remark: it follows from Theorem 3.11 that in the class of r.i. spaces Xon Rnwith the same fundamental function ϕsatisfying the condition R1 0s−1/p dϕ∗(s) < ∞, the largest class of couples of weights (w, v) on Rnwhich satisfy the condition sup Q∈Q 1 |Q|ZQw(y)pdy1/p kv−1kX,Q <∞ is generated when Xis the largest r.i. space with the fundamental function ϕ, i.e., when Xis the Marcinkiewicz space M(ϕ∗). Acknowledgements. The first author was partially supported by the National Science Centre (NCN), Poland, grant no. 2011/01/B/ST1/06243. The second author was supported by the Spanish Ministry of Science and Innovation grant MTM2009-08934 and by the Junta de Andaluc´ ıa, grant FQM-4745. We thank the referee for very careful reading of the paper. The Hardy-Littlewood Maximal Type Operators 899 REFERENCES [1] R.J. BAGBY and J.D. PARSONS,Orlicz spaces and rearranged maximal functions, Math. 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MR607898 (84j:35077) KEY WORDS AND PHRASES: maximal operators, rearrangement estimate. 2000 MATHEMATICS SUBJECT CLASSIFICATION: 42B20, 42B25, 46E30. Received: May 15, 2011; revised: December 9, 2011. MIECZYSŁAW MASTYŁO: Faculty of Mathematics and Computer Science Adam Mickiewicz University Pozna´ n, and Institute of Mathematics Polish Academy of Sciences (Pozna´ n branch) Umultowska 87 61-614 Pozna´ n, Poland E-MAIL:[email protected] CARLOS P´ EREZ: Departamento de A´ alisis Matem´ atico Facultad de Matem´ aticas, Universidad de Sevilla 41080 Sevilla, Spain E-MAIL:[email protected]