Boundedness properties in a family of weighted Morrey spaces with emphasis on power weights
Introduction
We consider a scale of weighted Morrey spaces. The weight w is a nonnegative measurable function. For , , , let , , be the Morrey space formed by the collection of all measurable functions f such that (Here and in what follows w is locally integrable and stands for the integral of w over A and for the Lebesgue measure of A.) We also consider the weak Morrey space , for which Clearly, . Several conditions depending on w will be imposed to and 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 and corresponds to the spaces introduced by Komori and Shirai in [5];
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the case and corresponds to the spaces introduced by N. Samko in [9];
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the case and 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 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 , but only for the case of Samko-type spaces (). For fixed p such weights are beyond . In this paper we consider the same type of weights for the Morrey spaces defined above and results beyond 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 and (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 with . This is particularly helpful for power weights and can be used to identify power weighted spaces with different values of and (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 and are nonnegative we prove that the boundedness of the operator on implies that w must belong to a certain Muckenhoupt class , with . 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 is negative, thus covering the interesting case .
In Section 4 we work in the setting of the extrapolation theorem: if an operator is bounded on for all , we deduce that it is bounded on for 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 , centered at the origin. But it is clear that the same results hold for powers of the form , with .
Section snippets
Preliminary results and significant properties of (power) weighted Morrey spaces
Let with almost everywhere. We say that w is a Muckenhoupt weight in the class for if where the supremum is taken over all Euclidean balls B in . The constant of w is the quantity of the definition. We say that w is in if, for any Euclidean ball B, The constant of w, denoted by , 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 and . It shows that w must belong to a certain Muckenhoupt class of weights. Proposition 3.1 Let , , , and . If M is bounded from to , then .
Proof Let B be a ball of radius r. Define with σ nonnegative to be chosen later. For , we have . If , then . Assuming that M is bounded from
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 and let be a collection of nonnegative measurable pairs of functions. Assume that for every and every we have where C does not depend on the pair 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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