Classes of kernels and continuity properties of the double layer potential in H\"{o}lder spaces

TL;DR

This paper investigates the regularizing effects of the double layer potential boundary operator for nonhomogeneous elliptic PDEs in Hölder spaces, using kernel estimates and operator theory.

Contribution

It establishes regularization properties of the boundary integral operator for nonhomogeneous elliptic operators in Hölder spaces, extending previous results to more general operators.

Findings

Proves regularizing properties of the boundary integral operator in Hölder spaces.

Develops estimates on the maximal function of the kernel's tangential gradient.

Generalizes existing results for integral operators on differentiable manifolds.

Abstract

We prove the validity of regularizing properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in H\"{o}lder spaces by exploiting an estimate on the maximal function of the tangential gradient with respect to the first variable of the kernel of the double layer potential and by exploiting specific imbedding and multiplication properties in certain classes of integral operators and a generalization of a result for integral operators on differentiable manifolds.