Bounds for Calder\'on-Zygmund operators with matrix $A_2$ weights

TL;DR

This paper establishes bounds for Calderón-Zygmund operators with matrix A2 weights, showing their norms depend on the A2 characteristic similarly to dyadic martingale transforms, using Bellman functions and Hytönen's representation theorem.

Contribution

It extends weighted norm inequalities to matrix-weighted Calderón-Zygmund operators, linking their bounds to dyadic martingale transforms and providing new proofs for special cases.

Findings

Norm dependence on A2 characteristic matches martingale transform case

Reduction of matrix-weighted operator bounds to dyadic martingales

Proved matrix-weighted Carleson Embedding Theorem

Abstract

It is well-known that dyadic martingale transforms are a good model for Calder\'on-Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that, if $W$ is an $A_2$ matrix weight, then the weighted $L^2$-norm of a Calder\'on-Zygmund operator with cancellation has the same dependence on the $A_2$ characteristic of $W$ as the weighted $L^2$-norm of the martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calder\'on-Zygmund operators on the $A_2$ characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calder\'on-Zygmund operators with even kernel. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hyt\"onen to extend the result to general Calder\'on-Zygmund operators.