Two matrix weighted inequalities for commutators with fractional integral operators

In this paper we prove two matrix weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a matrix symbol. More precisely, we extend the recent results of the second author, Pott, and Treil on two matrix weighted norm inequalities for commutators of Calderon-Zygmund operators and multiplication by a matrix symbol to the fractional integral operator setting. In particular, we completely extend the fractional Bloom theory of Holmes, Rahm, and Spencer to the two matrix weighted setting with a matrix symbol and also provide new two matrix weighted norm inequalities for commutators of fractional integral operators with a matrix symbol and two arbitrary matrix weights. These results are new even in the scalar one weighted setting with a scalar symbol.