Bounds for Calderón–Zygmund operators with matrix A2 weights

Abstract

It is well-known that dyadic martingale transforms are a good model for Calderón–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 A2 matrix weight, then the weighted L2-norm of a Calderón–Zygmund operator with cancellation has the same dependence on the A2 characteristic of W as the weighted L2-norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderón–Zygmund operators on the A2 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 Calderón–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 Bellman 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. Hytönen to extend the result to general Calderón–Zygmund operators.