New Local T1 Theorems on non-homogeneous spaces

TL;DR

This paper introduces new local T1 theorems for Calderón-Zygmund operators on non-homogeneous spaces, providing criteria for boundedness and compactness on L^p spaces with measures of power growth, without relying on random grids.

Contribution

It develops novel local T1 theorems that characterize operator boundedness and compactness on non-homogeneous spaces using countable testing functions, expanding classical theory.

Findings

Characterization of boundedness of Calderón-Zygmund operators on non-homogeneous spaces.

Criteria for compactness of operators, including the Cauchy integral, on L^p spaces.

Description of measures for which the Cauchy integral is compact on L^p( ext{complex plane}, ext{measure}).

Abstract

We develop new local $T1$ theorems to characterize Calder\'on-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 $L^p(\mathbb C,\mu)$.