Characterizations of variable fractional Haj{\l}asz-Sobolev spaces

TL;DR

This paper introduces variable fractional Haj{ }asz-Sobolev spaces on spaces of homogeneous type with variable exponents, providing multiple characterizations using maximal functions to deepen understanding of their structure.

Contribution

The paper develops the theory of variable fractional Haj{ }asz-Sobolev spaces on spaces of homogeneous type, offering new characterizations via maximal functions.

Findings

Established various characterizations of the spaces

Connected the spaces with maximal function techniques

Extended the theory to variable exponent settings

Abstract

Let (X, \r{ho},\mu) be a space of homogeneous type, a variable exponent satisfying the globally log-Holder continuous condition. In this article, the author introduce the variable fractional Sobolev spaces on X via Haj{\l}asz gradient. Using various maximal functions, several characterizations of this space are established.