Boundedness of Commutators and H ¹ -BMO Duality in the Two Matrix Weighted Setting

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.