On the role of riesz potentials in poisson's equation and sobolev embeddings

In this paper, we show how standard techniques can be used to obtain new galmost h-Lipschitz estimates for the classical Riesz potentials acting on Lp-spaces in the supercritical exponent. Whereas similar results are known to hold for Riesz potentials acting on Lp(.) for. RN a bounded domain (and also on Sobolev, Sobolev-Orlicz functions), our results concern the mapping properties of the Riesz potentials on all of RN. 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 Riesztype 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 fs equation, as well as an alternative proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.