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

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.

v3: 36 pages, no figures, updated bibliography, typos corrected, simplified proof of b) implies a) in Theorem 2.2, to appear in the journal Integral Equations and Operator Theory

References in corpus (3)

• Commutators in the Two-Weight Setting
• Bloom's Inequality: Commutators in a Two-Weight Setting
• Matrix weighted Poincaré inequalities and applications to degenerate elliptic systems

Cited by in corpus (1)

• Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols