On the role of Riesz potentials in Poisson's equation and Sobolev embeddings

TL;DR

This paper investigates the mapping properties of Riesz potentials on L^p spaces, deriving new regularity and continuity estimates that enhance understanding of Poisson's equation solutions and Sobolev embeddings.

Contribution

It introduces novel 'almost' Lipschitz continuity estimates for Riesz potentials in the supercritical regime, extending previous regularity results and providing a new proof of the supercritical Sobolev embedding theorem.

Findings

New 'almost' Lipschitz continuity estimates for Riesz potentials

Enhanced regularity results for solutions to Poisson's equation

A new proof of the supercritical Sobolev embedding theorem

Abstract

In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.