Regularity properties of certain convolution operators in H\"{o}lder spaces

TL;DR

This paper proves a theorem on the H"older regularity of convolution operators related to boundary layer potentials, specifically in the context of $C^{1,1}$ open sets with $C^{0,1}$ densities, extending previous results.

Contribution

It generalizes Miranda's theorem to the limiting case of $C^{1,1}$ boundaries and $C^{0,1}$ densities, providing new regularity results for convolution operators in this setting.

Findings

Established H"older regularity for convolution operators on $C^{1,1}$ boundaries.

Extended the class of densities for which regularity results hold.

Provided tools for analyzing boundary value problems with less smooth data.

Abstract

The aim of this paper is to prove a theorem of C.~Miranda on the H\"older regularity of convolution operators acting on the boundary of an open set in the limiting case in which the open set is of class $C^{1,1}$ and the densities are of class $C^{0,1}$. The convolution operators that we consider are generalizations of those that are associated to layer potential operators, which are a useful tool for the analysis of boundary value problems.