On the role of Riesz potentials in Poisson's equation and Sobolev embeddings

Abstract

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.

21 pages

References in corpus (1)

• On regularity of solutions to Poisson's equation

Cited by in corpus (4)

• Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions
• On regularity of solutions to Poisson's equation
• A trace inequality for solenoidal charges
• Sharp pointwise gradient estimates for Riesz potentials with a bounded density