On the role of Riesz potentials in Poisson's equation and Sobolev embeddings (Q2788635)

The authors begin by reviewing the literature concerning mapping properties of the Riesz kernel \(\left| x\right| ^{\alpha -N}\) acting by convolution on \(L^{p}(\mathbb{R}^{N})\) for \(0<\alpha <N\) and \(1<p\leq N(\alpha -1)\). They then establish new results for the supercritical case \( p=N/(\alpha -1)\), where the \(L^{p}\) functions are no longer required to have compact support, and study Riesz-type potentials for the case where \(\alpha \geq N\). In particular, they establish continuity estimates for the logarithmic kernel (corresponding to \( \alpha =N\)), regularity results for Poisson's equation, and an approach to the Sobolev embedding theorem in the supercritical exponent based on potential estimates.