Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II

TL;DR

This paper confirms a conjecture about the two weight inequality for the Hilbert transform, showing it holds if and only if certain Poisson and testing conditions are met, thus completing the local characterization.

Contribution

It proves the local two weight inequality for the Hilbert transform, advancing the understanding of measure conditions for boundedness.

Findings

Verification of the Nazarov--Treil--Volberg conjecture.

Characterization of the Hilbert transform boundedness via Poisson A2 and testing conditions.

Completion of the local problem in two weight inequalities.

Abstract

A conjecture of Nazarov--Treil--Volberg on the two weight inequality for the Hilbert transform is verified. Given two non-negative Borel measures u and w on the real line, the Hilbert transform $H_u$ maps $L^2(u)$ to $L^2(w)$ if and only if the pair of measures of satisfy a Poisson $A_2$ condition, and dual collections of testing conditions, uniformly over all intervals. This strengthens a prior characterization of Lacey-Sawyer-Shen-Uriate-Tuero arxiv:1201.4319. The latter paper includes a `Global to Local' reduction. This article solves the Local problem.