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Riesz potentials are well known objects of study in the theory of singularintegrals that have been the subject of recent, increased interest from thenumerical analysis community due to their connections with fractional Laplaceproblems and proposed use in certain domain decomposition methods. While theL$^p$-mapping properties of Riesz potentials on flat geometries arewell-established, comparable results on rougher geometries for Sobolev spacesare very scarce. In this article, we study the continuity properties of thesurface Riesz potential generated by the $1/\sqrt{x}$ singular kernel on apolygonal domain $\Omega \subset \mathbb{R}^2$. We prove that this surfaceRiesz 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 theneighbourhood of corners of the domain $\Omega$. The main tool we use for thiscorner analysis is the Mellin transform which can be seen as a counterpart ofthe Fourier transform that is adapted to corner geometries.

Continuity estimates for Riesz potentials on polygonal boundaries