New Local T1 Theorems on non-homogeneous spaces

Abstract

We develop new local $T1$ theorems to characterize Calderón-Zygmund operators that extend boundedly or compactly on $L^p({\mathbb{R}}^n,\mu )$ with $\mu$ a measure of power growth. The results, whose proofs do not require random grids, allow the use of a countable collection of testing functions. As a corollary, we describe the measures $\mu$ of the complex plane for which the Cauchy integral defines a compact operator on $L^p(\mathbb{C},\mu )$.