Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols

TL;DR

This paper characterizes the boundedness of matrix-weighted commutators and paraproducts using matrix BMO spaces and Carleson embedding theorems, providing new weighted inequalities and sharp bounds for related operators.

Contribution

It introduces a matrix weighted BMO space and a Carleson embedding theorem to analyze boundedness of commutators and paraproducts in matrix weighted settings, advancing the theory of matrix weighted inequalities.

Findings

Characterization of bounded commutators via matrix weighted BMO.

Development of a matrix weighted Carleson embedding theorem.

Quantitative weighted norm inequalities and sharp L^2 bounds for matrix maximal functions.

Abstract

Let $B$ be a locally integrable matrix function, $W$ a matrix A${}_p$ weight with $1 < p < \infty$, and $T$ be any of the Riesz transforms. We will characterize the boundedness of the commutator $[T, B]$ on $L^p(W)$ in terms of the membership of $B$ in a natural matrix weighted BMO space. To do this, we will characterize the boundedness of dyadic paraproducts on $L^p(W)$ via a new matrix weighted Carleson embedding theorem. Finally, we will use some of the ideas from these proofs to (among other things) obtain quantitative weighted norm inequalities for these operators and also use them to prove sharp $L^2$ bounds for the Christ/Goldberg matrix weighted maximal function associated with matrix A${}_2$ weights.