Fractional maximal functions in metric measure spaces

TL;DR

This paper investigates how fractional maximal operators affect function regularity in metric measure spaces, showing they can enhance Sobolev regularity and produce Hölder continuous functions under certain conditions, with counterexamples provided.

Contribution

It demonstrates the mapping properties of fractional maximal operators in Sobolev and Campanato spaces within metric measure spaces, including regularity improvements and counterexamples.

Findings

Fractional maximal operators improve Sobolev regularity under certain conditions.

They map Campanato spaces to Hölder continuous functions.

Counterexample shows failure of continuity in some spaces.

Abstract

We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to H\"older continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.