On the role of Riesz potentials in Poisson's equation and Sobolev embeddings Rahul GargDaniel Spector 31B1031A3035B6546E35Riesz potentialslogarithmic potentialPoisson's equationSobolev embedding 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 $\Omega\subset\mathbb{R}^N$ a bounded domain (and also Sobolev, Sobolev-Orlicz functions), our results concern the mapping properties of the Riesz potentials on all of $\mathbb{R}^N$. 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. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5706 10.1512/iumj.2015.64.5706 en Indiana Univ. Math. J. 64 (2015) 1697 - 1719 state-of-the-art mathematics http://iumj.org/access/