Local Morrey and Campanato Spaces on Quasimetric Measure Spaces

Krzysztof Stempak,Xiangxing Tao

doi:10.1155/2014/172486

Abstract

We define and investigate generalized local Morrey spaces and generalized local Campanato spaces, within a context of a general quasimetric measure space. The locality is manifested here by a restriction to a subfamily of involved balls. The structural properties of these spaces and the maximal operators associated to them are studied. In numerous remarks, we relate the developed theory, mostly in the “global” case, to the cases existing in the literature. We also suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets ofRn.

Highlights

If (X, d) is a quasimetric space, Td, the topology in X induced by d, is canonically defined by declaring G ⊂ X to be open, that is, G ∈ Td, if and only if, for every x ∈ G, there exists r > 0 such that B(x, r) ⊂ G
If (X, d) is a given quasimetric space, X is considered as a topological space equipped with the topology Td
By defining and investigating generalized local Morrey and Campanato spaces on quasimetric measure spaces, we adapt the general approach to these spaces presented by Nakai [2] and extend the concept of locality introduced in [4]

Summary

Introduction

A quasimetric on a nonempty set X is a mapping d : X×X → [0, ∞) which satisfies the following conditions:. If (X, d) is a given quasimetric space, X is considered as a topological space equipped with the (metrizable) topology Td. It may happen that a ball in X is not a Borel set (i.e., it does not belong to the Borel σalgebra generated by Td), see, for instance, [1] as an example. Taking closed balls makes no difference with respect to the theory based on open balls, when μ has the property that μ(∂B) = 0, for every ball B, where ∂B = B \ B; this happens, for instance, when dμ(x) = w(x)dx, where w ≥ 0 and dx denotes Lebesgue measure on Rn. In general, the two alternative ways may give different outcomes. The general notion of local maximal operators was introduced in [4] and some objects associated to them, mostly the BMO spaces, were investigated there in the setting of measure metric spaces. When the situation is specified to the Euclidean setting of Rn, we shall consider either the metric d(2) induced by the norm ‖ ⋅ ‖2 or d(∞) induced by ‖ ⋅ ‖∞

Generalized Local Maximal Operators

Local Morrey and Campanato Spaces

Boundedness of Operators on Local Morrey Spaces