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.