Matrix weights: On the way to the linear bound

Abstract

In recent years, the attempts to prove sharp bounds for Calderon-Zygmund operators on weighted $L^p$ spaces in terms of the $A_p$-characteristic of the weight has been an im- portant driving force in Harmonic Analysis. After the work of many authors, this culminated with the proof of the conjectured linear bound for p = 2 for all Calderon-Zygmund operators by Tuomas Hyt\”onen in 2010. Recently, the question of the validity of the linear bound for all Calderon-Zygmund operators in the matrix-weighted setting has attracted some interest. In the talk, I want to present the reduction of this question to the case of Haar multipliers and dyadic paraproducts. I also want to talk about the remaining obstacles, some of which have recently been resolved, and focus on the matrix techniques being used. This is joint work with Andrei Stoica