Continuity estimates for Riesz potentials on polygonal boundaries

TL;DR

This paper investigates the continuity properties of Riesz potentials with a specific singular kernel on polygonal domains, demonstrating their mapping from L^2 boundary spaces to fractional Sobolev spaces, especially near corners.

Contribution

It provides the first detailed analysis of Riesz potentials on polygonal boundaries, using Mellin transform techniques to handle corner singularities.

Findings

Riesz potential maps L^2(∂Ω) to H^{1/2}(∂Ω)

Analysis near corners is crucial for understanding boundary behavior

Mellin transform is effective for corner analysis

Abstract

Riesz potentials are well known objects of study in the theory of singular integrals that have been the subject of recent, increased interest from the numerical analysis community due to their connections with fractional Laplace problems and proposed use in certain domain decomposition methods. While the L$^p$-mapping properties of Riesz potentials on flat geometries are well-established, comparable results on rougher geometries for Sobolev spaces are very scarce. In this article, we study the continuity properties of the surface Riesz potential generated by the $1/\sqrt{x}$ singular kernel on a polygonal domain $\Omega \subset \mathbb{R}^2$. We prove that this surface Riesz potential maps L$^{2}(\partial\Omega)$ into H$^{+1/2}(\partial\Omega)$. Our proof is based on a careful analysis of the Riesz potential in the neighbourhood of corners of the domain $\Omega$. The main tool we use for this corner analysis is the Mellin transform which can be seen as a counterpart of the Fourier transform that is adapted to corner geometries.