Sharp weighted estimates for approximating dyadic operators

Abstract

We give a new proof of the sharp weighted L2 inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where T is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.