Sharp weighted estimates for classical operators

TL;DR

This paper provides a new, Bellman function-free proof of sharp one-weight $L^p$ inequalities for classical operators like the Hilbert and Riesz transforms, using Lerner's oscillation estimate, and extends results to related harmonic analysis operators.

Contribution

It introduces a flexible proof technique for sharp weighted inequalities that avoids traditional Bellman function methods and applies to a broad class of harmonic analysis operators.

Findings

Established sharp one-weight $L^p$ inequalities for classical operators.

Extended sharp inequalities to maximal singular integrals, dyadic square functions, and paraproducts.

Derived a sharp two-weight bump condition for these operators.

Abstract

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. 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. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions and paraproducts, and the vector-valued maximal operator of C. Fefferman-Stein. Also we can derive a very sharp two-weight bump type condition for $T$.