Boundedness of commutators and H${}^1$-BMO duality in the two matrix weighted setting

TL;DR

This paper characterizes the boundedness of matrix-weighted commutators with Riesz transforms via a two matrix weighted BMO space, establishes duality with a matrix weighted H^1 space, and extends classical results to the matrix setting.

Contribution

It introduces a natural two matrix weighted BMO space, characterizes commutator boundedness in this setting, and connects it with matrix weighted H^1 duality, extending classical harmonic analysis results.

Findings

Characterization of commutator boundedness in matrix weighted setting

Identification of BMO as dual of H^1 in this context

Extension of Bloom's theorem and Buckley's condition to matrices

Abstract

In this paper we characterize the two matrix weighted boundedness of commutators with any of the Riesz transforms (when both are matrix A${}_p$ weights) in terms of a natural two matrix weighted BMO space. Furthermore, we identify this BMO space when $p = 2$ as the dual of a natural two matrix weighted H${}^1$ space, and use our commutator result to provide a converse to Bloom's matrix A${}_2$ theorem, which as a very special case proves Buckley's summation condition for matrix A${}_2$ weights. Finally, we use our results to prove a matrix weighted John-Nirenberg inequality, and we also briefly discuss the challenging question of extending our results to the matrix weighted vector BMO setting.