On the $T1$ theorem for compactness of Calderón-Zygmund operators

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