On Hausdorff content maximal operator and Riesz potential for non-measurable functions

TL;DR

This paper extends the theory of Riesz potentials and maximal operators to non-measurable functions by defining them via Choquet integrals with respect to Hausdorff content, establishing boundedness results and generalizing previous findings.

Contribution

It introduces Riesz potentials for non-measurable functions using Choquet integrals and extends boundedness results for these operators and maximal operators to this broader context.

Findings

Established boundedness of Riesz potentials for non-measurable functions.

Extended earlier results to functions not necessarily Lebesgue measurable.

Generalized maximal operator results to non-measurable functions.

Abstract

We introduce Riesz potentials for non-Lebesgue measurable functions by taking the integrals in the sense of Choquet with respect to Hausdorff content and prove boundedness results for these operators. Some earlier results are recovered or extended now using integrals taken in the sense of Choquet with respect to Hausdorff content. Some earlier results also for maximal operators are considered, but now for non-measurable functions.