Sharp weighted estimates for approximating dyadic operators

TL;DR

This paper presents a new proof for sharp weighted L^2 inequalities for certain dyadic operators, including the Hilbert and Riesz transforms, using oscillation estimates instead of Bellman functions.

Contribution

It introduces a novel proof technique that bypasses Bellman functions and two-weight inequalities, leveraging recent oscillation estimates for dyadic operators.

Findings

Established sharp weighted L^2 bounds for key operators

Provided a Bellman-function-free proof method

Extended results to operators approximable by Haar shifts

Abstract

We give a new proof of the sharp weighted $L^2$ 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.