Schatten classes and commutator in the two weight setting, I. Hilbert transform

Abstract

We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1<p<\infty$. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.