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

TL;DR

This paper characterizes when the commutator with the Hilbert transform belongs to Schatten classes in a two-weight setting, linking it to a new weighted Besov space and extending dyadic paraproduct results.

Contribution

It introduces a new weighted Besov space characterization for commutator Schatten class membership and extends dyadic paraproduct Schatten class results to the two-weight Hilbert transform setting.

Findings

Characterization of Hilbert--Schmidt class membership via a new weighted Besov space.

Schatten class $S_p$ results for dyadic paraproducts with $1< p < olinebreak \infty$.

Discussion of challenges in extending results to the full Schatten class range.

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.