Boundedness properties in a family of weighted Morrey spaces with emphasis on power weights

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Abstract

We define a scale of weighted Morrey spaces which contains different versions of weighted spaces appearing in the literature. This allows us to obtain weighted estimates for operators in a unified way. In general, we obtain results for weights of the form |x|αw(x) with wAp and nonnegative α. We study particularly some properties of power-weighted spaces and in the case of the Hardy-Littlewood maximal operator our results for such spaces are sharp. By using extrapolation techniques the results are given in abstract form in such a way that they are automatically obtained for many operators.

Introduction

We consider a scale of weighted Morrey spaces. The weight w is a nonnegative measurable function. For 0<p<, λ1R, λ2R, let Lp,λ(w), λ=(λ1,λ2), be the Morrey space formed by the collection of all measurable functions f such thatfLp,λ(w):=supxRn,r>0(1rλ1w(B(x,r))λ2/nB(x,r)|f|pw)1/psupxRn,r>0(1|B(x,r)|λ1/nw(B(x,r))λ2/nB(x,r)|f|pw)1/p<. (Here and in what follows w is locally integrable and w(A) stands for the integral of w over A and |A| for the Lebesgue measure of A.) We also consider the weak Morrey space WLp,λ(w), for whichfWLp,λ(w):=supxRn,r>0,t>0(tpw({yB(x,r):|f(y)|>t})|B(x,r)|λ1/nw(B(x,r))λ2/n)1/p<. Clearly, Lp,λ(w)WLp,λ(w). Several conditions depending on w will be imposed to λ1 and λ2 so that the involved Morrey spaces are not reduced to the zero function.

This general formulation encompasses three interesting special cases of weighted Morrey spaces that have been considered in the literature, namely,

  • the case λ1=0 and 0<λ2<n corresponds to the spaces introduced by Komori and Shirai in [5];

  • the case 0<λ1<n and λ2=0 corresponds to the spaces introduced by N. Samko in [9];

  • the case λ1<0 and λ2=n corresponds to the spaces considered by Poelhuis and Torchinsky in [7], built as the weighted versions of the spaces in [10]. Actually, their definition considers a function of r, not necessarily a power of r. An even more general definition of this type is in [3].

There are many papers devoted to the study of the boundedness of operators in weighted Morrey spaces, mainly defined in the way of the first two cases above, with some generalizations. In [1] we studied the extension of inequalities on weighted Lebesgue spaces with Ap weights to Morrey spaces of the first two types mentioned in the previous paragraph (see also [8] for the first type), using techniques which are related to the extrapolation results in the Lebesgue setting. This provides immediately the boundedness on weighted Morrey spaces of a variety of operators. The results in [1] were extended in [2] to weights of the form |x|αw(x), but only for the case of Samko-type spaces (λ2=0). For fixed p such weights are beyond Ap. In this paper we consider the same type of weights for the Morrey spaces defined above and results beyond Ap are thus obtained for all of them. In particular, we emphasize the case of power weights for which the results are optimal.

In Section 2 we discuss several properties of the spaces in the family and, in particular, we give conditions on λ1 and λ2 (depending on the weight) which guarantee the existence of nontrivial functions in the weighted Morrey space. For power weights we fully describe such conditions (Proposition 2.10). An interesting observation (see Proposition 2.1) is the possibility of restricting the definition of the norm in (1.1) to special balls, namely, to B(x,r) with r|x|/4. This is particularly helpful for power weights and can be used to identify power weighted spaces with different values of λ1 and λ2 (for instance, in (2.13) below Komori-Shirai and N. Samko type spaces are identified).

In Section 3 we deal with the Hardy-Littlewood maximal operator. When λ1 and λ2 are nonnegative we prove that the boundedness of the operator on Lp,λ(w) implies that w must belong to a certain Muckenhoupt class Aq, with q=q(n,p,λ)>p. In the case of power weights with positive exponent this class is sharp. The sharp range for power weights is also obtained in the cases in which λ1 is negative, thus covering the interesting case λ2=n.

In Section 4 we work in the setting of the extrapolation theorem: if an operator is bounded on Lp0(w) for all wAp0, we deduce that it is bounded on Lp,λ(|x|αw) for wAp and α in a range depending on λ and n. This result immediately applies to many operators as can be seen in [1, Section 5]. We also show a variant of the extrapolation theorem with a weaker assumption that is useful for many other operators.

We are using power weights of the form |x|β, centered at the origin. But it is clear that the same results hold for powers of the form |xx0|β, with x0Rn.

Section snippets

Preliminary results and significant properties of (power) weighted Morrey spaces

Let wL1loc(Rn) with w>0 almost everywhere. We say that w is a Muckenhoupt weight in the class Ap for 1<p< if[w]ApsupBw(B)|B|(w1p(B)|B|)p1<, where the supremum is taken over all Euclidean balls B in Rn. The Ap constant of w is the quantity [w]Ap of the definition. We say that w is in A1 if, for any Euclidean ball B,w(B)|B|cw(x) for almost all xB. The A1 constant of w, denoted by [w]A1, is the smallest constant c for which the inequality holds.

We say that a nonnegative locally

The Hardy-Littlewood maximal operator

In this section we deal with the Hardy-Littlewood maximal operator. First we obtain a necessary condition in the case of nonnegative λ1 and λ2. It shows that w must belong to a certain Muckenhoupt class of weights.

Proposition 3.1

Let 1p<, λ10, λ20, and 0<λ1+λ2<n. If M is bounded from Lp,λ(w) to WLp,λ(w), then wAnp+λ1nλ2.

Proof

Let B be a ball of radius r. Define f=σχB with σ nonnegative to be chosen later. For xB, we have Mf(x)σ(B)/|B|. If t<σ(B)/|B|, then B={xB:Mf(x)>t}. Assuming that M is bounded from Lp,λ

Extrapolation techniques

In this section we use extrapolation techniques as in [1], [2] to obtain abstract results in the class of weighted Morrey spaces defined in this paper. They can subsequently be applied to a variety of operators, their vector-valued extensions, weak-type estimates, etc.

Theorem 4.1

Let 1p0< and let F be a collection of nonnegative measurable pairs of functions. Assume that for every (f,g)F and every uAp0 we havegLp0(u)CfLp0(u), where C does not depend on the pair (f,g) and it depends on u only in

Acknowledgements

The work of the first author was supported by the grants MTM2017-82160-C2-2-P of the Ministerio de Economía y Competitividad (Spain) and FEDER, and IT1247-19 of the Basque Government. He would like to thank Yoshihiro Sawano for sharing with him the paper [6] before it was published.

Part of this work was carried out during several visits of the second author to the University of the Basque Country (UPV/EHU). He would like to acknowledge the partial financial support of the analysis group of the

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