On the tangential gradient of the kernel of the double layer potential

TL;DR

This paper analyzes the tangential gradient of the double layer potential kernel for elliptic operators, establishing continuity properties in Hölder spaces that extend known results for Laplace and Helmholtz operators.

Contribution

It provides new estimates for the maximal function of the tangential gradient, generalizing previous boundary regularity results to broader elliptic operators.

Findings

Maximal function estimates for the tangential gradient

Continuity of double layer potential in Hölder spaces

Extension of results from Laplace and Helmholtz to general elliptic operators

Abstract

In this paper we consider an elliptic operator with constant coefficients and we estimate the maximal function of the tangential gradient of the kernel of the double layer potential with respect to its first variable. As a consequence, we deduce the validity of a continuity property of the double layer potential in H\"{o}lder spaces on the boundary that extends previous results for the Laplace operator and for the Helmholtz operator.