On the $T1$ theorem for compactness of Calder\'on-Zygmund operators

TL;DR

This paper presents a new, simplified characterization of when Calderón-Zygmund operators are compact on L^2, linking compactness to conditions on T1, T*1, and weak compactness, avoiding complex kernel conditions.

Contribution

It introduces a new formulation of the T1 theorem for compactness, making the criteria more accessible and closely aligned with classical boundedness theorems.

Findings

Calderón-Zygmund operator T is compact iff T1, T*1 are in CMO and T is weakly compact.

The new criterion simplifies previous conditions by avoiding kernel technicalities.

The characterization closely resembles the classical T1 theorem for boundedness.

Abstract

We give a new formulation of the $T1$ theorem for compactness of Calder\'on-Zygmund singular integral operators. In particular, we prove that a Calder\'on-Zygmund operator $T$ is compact on $L^2(\mathbb{R}^n)$ if and only if $T1,T^*1\in \text{CMO}(\mathbb{R}^n)$ and $T$ is weakly compact. Compared to existing compactness criteria, our characterization more closely resembles David and Journ\'e's classical $T1$ theorem for boundedness, avoids technical conditions involving the Calder\'on-Zygmund kernel, and follows from a simpler argument.