  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.
