On the compactness of bi-parameter singular integrals

Abstract

We establish a new $T1$ theorem for the compactness of bi-parameter Calderón-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak compactness/CMO property, and $T1,T^{t}1,$ $T_{t}1,T_t^{t}1\in \text{CMO}({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$, then $T$ is compact on $L^2({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$. We also obtain endpoint compactness results for these operators and use them to deduce the necessity of most of our hypotheses. In particular, our conditions characterize the simultaneous $L^2({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$-compactness of a bi-parameter CZO and its partial transpose. Our assumptions improve upon previously known sufficient conditions, and our proof, which is shorter and simpler than earlier arguments, utilizes a new abstract compactness criterion for partially localized operators on tensor products of Hilbert spaces.

22 pages