Massimo Lanza de Cristoforis
Università degli Studi di Padova, Padova, Italy

Volume and layer potentials are integrals on a subset $Y$ of the Euclidean space ${\mathbb{R}}^n$ that depend on a variable in a subset $X$ of ${\mathbb{R}}^n$. Here we follow an abstract approach by assuming that $X$ and $Y$ are subsets of a metric space $M$ and that $Y$ is equipped with a measure $\nu$ that satisfies upper Ahlfors growth conditions that include non-doubling measures as done by J. García-Cuerva and A.E. Gatto in a series of papers in case $X=Y$ and for standard kernels, and we prove a necessary and sufficient condition on a singular kernel for an integral operator to be bounded in Hölder spaces.

Then we present some application to layer potentials that are associated to the fundamental solution of an arbitrary constant coefficient second order elliptic operator with real principal coefficients.