Sharp weighted estimates for approximating dyadic operators

We give a new proof of the sharp weighted Lᵖ inequality|\|T\||Lᵖ(w) n,T[w]Aₚmax(1,1/p-1),where T is the Hilbert transform, a Riesz transform, theBeurling-Ahlfors operator or any operator that can be approximatedby Haar shift operators. Our proof avoids the Bellman functiontechnique and two weight norm inequalities. We use instead a recentresult due to A. Lerner [15] to estimate theoscillation of dyadic operators. The method we use is flexible enough to obtain the sharp one-weightresult for other important operators as well as a very sharptwo-weight bump type result for T as can be found in[5].