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

created by spector on 07 Apr 2014
modified on 12 Oct 2015

[BibTeX]

Accepted Paper

Inserted: 7 apr 2014
Last Updated: 12 oct 2015

Journal: Indiana Univ. Math. J.
Pages: 21
Year: 2014

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.