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

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.