Regularity properties of certain convolution operators in Hölder spaces

TL;DR

The paper extends Miranda's classical result on the Hölder regularity of boundary convolution (layer potential) operators to the limiting case where the domain is $C^{1,1}$ and the boundary densities are $C^{0,1}$. It introduces and leverages the $\omega_1$-Hölder framework to obtain bilinear, continuous maps from odd, positively homogeneous kernels of degree $-(n-1)$ and boundary densities to Hölder extensions inside and outside the domain. A key technical component is a new sufficient condition for $\omega_1$-Hölder continuity of $C^1$ functions on $C^1$ boundaries, built via tubular neighborhoods and a smooth vector field, which then underpins the Miranda-type estimates in the limiting case. The results enhance the understanding of boundary integral operators in potential theory and provide robust continuity properties essential for boundary value analyses in rougher geometries.

Abstract

The aim of this paper is to prove a theorem of C.~Miranda on the Hölder 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.

Theorems & Definitions (9)

• Remark 2.2
• Lemma 2.5
• Lemma 3.1
• Lemma 3.3
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• Proposition 3.13
• Theorem 4.1