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

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.