Integral operators in Hölder spaces on upper Ahlfors regular sets

Abstract

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 present a unified approach to some results 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. We prove continuity statements in the frame of (generalized) Hölder spaces upon variation both of the density functions on $Y$ and of the off-diagonal potential kernel and $T1$ theorems that generalize corresponding results of J. García-Cuerva and A. E. Gatto in case $X=Y$ for kernels that include the standard ones.