Extrapolation and Factorization of matrix weights

In this paper we prove the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix ${\mathcal{A}}_p$ weights. These results answer longstanding open questions in the study of matrix weights. The proof requires the development of the theory of convex-set valued functions and measurable seminorm functions. In particular, we define a convex-set valued version of the Hardy Littlewood maximal operator and construct an appropriate generalization of the Rubio de Francia iteration algorithm, which is central to the proof of both results in the scalar case.