A survey on the boundary behavior of the double layer potential in Schauder spaces in the frame of an abstract approach

Abstract

We provide a summary of the continuity 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 and Schauder spaces on the boundary of a bounded open subset of ${\mathbb{R}}^n$. The purpose is two-fold. On one hand we try present in a single paper all the known continuity results on the topic with the best known exponents in a H\"{o}lder and Schauder space setting and on the other hand we show that many of the properties we present can be deduced by applying results that hold in an abstract setting of metric spaces with a measure that satisfies certain growth conditions that include non-doubling measures as in a series of papers by Garc\'{\i}a-Cuerva and Gatto in the frame of H\"{o}lder spaces and later by the author.