Estimates for the variable order Riesz potential with applications

TL;DR

This paper investigates weak-type estimates and exponential integrability for the variable order Riesz potential, applying these results to variable exponent Sobolev spaces and extending exponential integrability to domains with complex geometries.

Contribution

It introduces new exponential integrability results for variable order Riesz potentials, improving previous results and applying them to complex domain geometries.

Findings

Improved exponential integrability results for variable exponent Sobolev spaces

Extension of integrability results to John domains and domains with outward cusps

Enhanced understanding of Riesz potentials in variable order settings

Abstract

We study weak-type estimates and exponential integrability for the variable order Riesz potential. As an application we prove an exponential integrability result with respect to the Hausdorff content for functions from variable exponent Sobolev spaces. In particular, the earlier exponential integrability results are improved to a corresponding one with respect to the Choque integral whenever John domains are considered. Moreover, new exponential integrability results also for domains with outward cusps are obtained.