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arXiv · 1511.02926

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

Joshua Isralowitz
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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.

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Joshua Isralowitz. 2017-07-11. Boundedness of commutators and H${}^1$-BMO duality in the two matrix weighted setting. https://arxiv.org/abs/1511.02926

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