Bounds for the Hilbert transform with matrix A2 weights

Let W   denote a matrix A2A2 weight. In this paper, we implement a scalar argument using the square function to deduce related bounds for vector-valued functions in L2(W)L2(W). These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on L2(W)L2(W). Our proof shortens the original argument by Treil and Volberg and improves the dependence on the A2A2 characteristic. In particular, we prove that:‖T‖L2(W)→L2(W)≲[W]A232log[W]A2, where T is either the Hilbert transform or a Haar multiplier.