MATRIX WEIGHTS, SINGULAR INTEGRALS, JONES FACTORIZATION AND RUBIO DE FRANCIA EXTRAPOLATION (Research on Real, Complex and Functional Analysis from the Perspective of Reproducing Kernel Hilbert Spaces)

Abstract

In this article we give an overview of the problem of finding sharp constants in matrix weighted norm ineqnalities for singular integrals, the so-called matrix A2 conjecture. We begin by reviewing the history of the problem in the scalar case, including ask etch of the proof of the scalar A2 conjecture. We then discuss the original, qnalitative resnlts for singular integrals with matrix weights and the best known quantitative estimates. We give an overview of new results by the author and Bownik, wh o developed ath eory of hannonic analysis on convex set-valned functions. This led to the proof the Jones factorization theorem and the Rnbio de Francia extrapolation theorem for matrix weights, two longstanding problems. Rubio de Francia extrapolation is expected to be ama jor tool in the proof of the matrix A2 conjecture, and we discuss some ideas which may lead to aco mplete solution.