On the 𝑇1 theorem for compactness of Calderón-Zygmund operators

We give a new formulation of the T 1 T1 theorem for compactness of Calderón-Zygmund singular integral operators. We prove that a Calderón-Zygmund operator T T is compact on L 2 ( R n ) L^{2}(\mathbb {R}^{n}) if and only if T 1 , T ∗ 1 ∈ C M O ( R n ) T1,T^{*}1\in CMO(\mathbb {R}^{n}) and T T is weakly compact. Compared to existing compactness criteria, our characterization more closely resembles David and Journé’s classical T 1 T1 theorem for boundedness and follows from a simpler argument.