On the Role of Riesz Potentials in Poisson\'s Equation and Sobolev Embeddings

Abstract

In this paper, we show how standard techniques can be used to obtain new "almost"-Lipschitz estimates for the classical Riesz potentials acting on L-P-spaces in the supercritical exponent. Whereas similar results are known to hold for Riesz potentials acting on L-P(Omega) for Omega subset of R-N a bounded domain (and also on Sobolev, Sobolev-Orlicz functions), our results concern the mapping properties of the Riesz potentials on all of R-N. Additionally, we introduce and prove sharp estimates on the modulus of continuity for a family of Riesz-type potentials. In particular, through a new representation via these Riesz-type potentials, we establish analagous results for the logarithmic potential. As applications of these continuity estimates, we deduce new regularity estimates for distributional solutions to Poisson\'s equation, as well as an alternative proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.