Sharp weighted estimates for approximating dyadic operators

Abstract

We give a new proof of the sharp weighted Lp inequality 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 [15] to estimate the oscillation of dyadic operators. The method we use is flexible enough to obtain the sharp one-weight result for other important operators as well as a very sharp two-weight bump type result for T as can be found in [5]. © 2010 American Institute of Mathematical Sciences.