Matrix weights, singular integrals, Jones factorization and Rubio de Francia extrapolation

In this article we give an overview of the problem of finding sharp constants in matrix weighted norm inequalities for singular integrals, the so-called matrix A2 conjecture. We begin by reviewing the history of the problem in the scalar case, including a sketch of the proof of the scalar A2 conjecture. We then discuss the original, qualitative results for singular integrals with matrix weights and the best known quantitative estimates. We give an overview of new results by the author and Bownik, who developed a theory of harmonic analysis on convex set-valued functions. This led to the proof the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix weights, two longstanding problems. Rubio de Francia extrapolation was expected to be a major tool in the proof of the matrix A2 conjecture; however, this conjecture was very recently proved false. We discuss this problem.