SHARP WEIGHTED ESTIMATES FOR APPROXIMATING DYADIC OPERATORS

We give a new proof of the sharp weighted L-p inequality parallel to T parallel to(Lp(w)) <= C-n,C-T [w](Ap)(max(1,1/p-1)), 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].