Density, Duality and Preduality in Grand Variable Exponent Lebesgue and Morrey Spaces

TL;DR

This paper investigates the structural properties, duality, and preduality of grand variable exponent Lebesgue and Morrey spaces over spaces of homogeneous type, revealing new characterizations and equivalences.

Contribution

It establishes the coincidence of closures of bounded functions and classical Morrey space in grand variable exponent Morrey spaces and provides new duality and preduality results.

Findings

Closure of bounded functions equals closure of classical Morrey space in finite measure spaces.

Two different characterizations of the closure class are provided.

Duality and preduality of grand variable exponent Lebesgue spaces are established on quasi-metric spaces.

Abstract

In this note some structural properties of grand variable exponent Lebesgue/ Morrey spaces over spaces of homogeneous type are obtained. In particular, it is proved that the closure of the class of bounded functions and the closure of classical Morrey space in grand variable exponent Morrey space coincide if the measure of the underlying space is finite. Moreover, we get two different characterizations of this class. Further, duality and preduality of grand variable exponent Lebesghue space defined on quasi-metric measure spaces with sigma-finite measure are obtained, which is new even when underlying measure is finite.