Bounds for Calderon-Zygmund operators with matrix A2 weights

It is well-known that dyadic martingale transforms are a good model for Calder6n-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 Calderon Zygmund operator with cancellation has the same dependence on the A(2) characteristic of W as the weighted L-2-norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderon 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 Calder6n-Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses a Hellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytonen to extend the result to general Calderon Zygmund operators. (C) 2017 Published by Elsevier Masson SAS.