Sharp A2 inequality for Haar shift operators

As a Corollary to the main result of the paper, we give a new proof of the inequality parallel to T f parallel to(L2(w)) less than or similar to parallel to w parallel to(A2) parallel to f parallel to(L2(w)), where T is either the Hilbert transform (Amer J Math 129(5): 1355-1375, 2007), a Riesz transform (Proc Amer Math Soc 136(4): 1237-1249, 2008), or the Beurling operator (Duke Math J 112(2): 281-305, 2002). The weight w is non-negative, and the linear growth in the A(2) characteristic on the right is sharp. Prior proofs relied strongly on Haar shift operators (CR Acad Sci Paris Ser I Math 330(6): 455-460, 2000) and Bellman function techniques. The new proof uses Haar shifts, and then uses an elegant 'two weight T1 theorem' of Nazarov-Treil-Volberg (Math Res Lett 15(3): 583-597, 2008) to immediately identify relevant Carleson measure estimates, which are in turn verified using an appropriate corona decomposition of the weight w.