A global Tb Theorem for compactness and boundedness

TL;DR

This paper establishes a new Tb Theorem that characterizes when Calderon-Zygmund operators are compact or bounded on L^p spaces, without requiring accretivity, and applies to operators like the Double Layer Potential.

Contribution

It introduces a Tb Theorem that characterizes compactness and boundedness of Calderon-Zygmund operators without accretivity assumptions, expanding applicability.

Findings

Characterization of compact Calderon-Zygmund operators on L^p

Conditions for boundedness using non-accretive testing functions

Application to Double Layer Potential operator on various domains

Abstract

We prove a Tb Theorem that characterizes all Calderon-Zygmund operators that extend compactly on L^p(R^n), 1<p<\infty . The result, whose proof does not require the property of accretivity, can be used to prove compactness of the Double Layer Potential operator on a wide class of domains. The study also provides conditions for boundedness of singular integral operators by means of non-accretive testing functions.