Sharp $ A_2$ Inequality for Haar Shift Operators

Abstract

As a corollary to our main theorem we give a new proof of the result that the norm of the Hilbert transform on L^2(w) has norm bounded by a the A_2 characteristic of a weight to the first power, a theorem of one of us. This new proof begins as the prior proofs do, by passing to Haar shifts. Then, we apply a deep two-weight T1 theorem of Nazarov-Treil-Volberg, to reduce the matter to checking a certain carleson measure condition. This condition is checked with a corona decomposition of the weight. Prior proofs of this type have used Bellman functions, while this proof is flexible enough to address all Haar shifts at the same time.