Two-weight inequality for the Hilbert transform: A real variable characterization, II

Michael T. Lacey, Eric T. Sawyer, Chun Shen, Ignacio Uriarte-Tuero

Type: Article

Publication Date: 2014-12-01

Citations: 113

DOI: https://doi.org/10.1215/00127094-2826799

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.

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