Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols

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.

v2, 38 pages, minor changes made (including a shorter proof of (b) implies (a) in Theorem 1.3), to appear in the Journal of the London Mathematical Society

References in corpus (5)

• Commutators in the Two-Weight Setting
• Bloom's Inequality: Commutators in a Two-Weight Setting
• Matrix weighted Poincaré inequalities and applications to degenerate elliptic systems
• Well-Localized Operators on Matrix Weighted $L^2$ Spaces
• Boundedness of commutators and H${}^1$-BMO duality in the two matrix weighted setting

Cited by in corpus (1)

• Dyadic harmonic analysis and weighted inequalities: the sparse revolution