New characterizations of Haj{\l}asz-Sobolev type spaces with variable exponent on metric measure spaces

TL;DR

This paper introduces and characterizes variable exponent Haj{ extl}asz-Sobolev spaces on metric measure spaces, providing new descriptions via maximal functions and establishing their equivalence, thus advancing the understanding of variable smoothness in analysis.

Contribution

It offers new characterizations and equivalences of Haj{ extl}asz-Sobolev spaces with variable exponents on metric measure spaces, extending existing theories.

Findings

New classes of functions controlled by measure and variable exponent p(.)

Multiple descriptions of these classes via maximal functions

Equivalence between different characterizations established

Abstract

In this article, we introduce classes of functions whose increment is controlled by the measure of a ball containing the corresponding points and a nonnegative function p(.) that is summable with respect to measure. These classes of functions can be considered as spaces with variable smoothness depending on the structure of the measure in a neighborhood of a given point. Moreover, we present several descriptions generalized classes of variable exponent Haj{\l}asz-Sobolev type on metric measure spaces by various maximal functions and we establish the equivalence between them.