Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures

TL;DR

This paper establishes necessary and sufficient conditions for the boundedness of the Hilbert transform with two weights, using nonhomogeneous harmonic analysis techniques and a stopping time argument, contributing to the understanding of two-weight inequalities.

Contribution

It provides a natural characterization of two-weight boundedness for the Hilbert transform, utilizing nonhomogeneous analysis and paraproducts, and relates to recent developments by Lacey, Sawyer, and Uriarte-Tuero.

Findings

Characterization of two-weight boundedness for the Hilbert transform.

Introduction of a natural weight condition that is both necessary and sufficient.

Application of nonhomogeneous harmonic analysis methods to the problem.

Abstract

This article was written in 2005 and subsequently lost (at least by the third author). Recently it resurfaced due to one of the colleagues to whom a hard copy has been sent in 2005. We consider here a problem of finding necessary and sufficient conditions for the boundedness of two weight Calder\'on-Zygmund operators. We give such necessary and sufficient conditions in very natural terms, if the operator is the Hilbert transform, and the weights satisfy some very natural condition. The condition on weights was lifted in a recent paper of Michael Lacey, Eric Sawyer and Ignacio Uriarte-Tuero: "A characterization of the two weight norm inequality for the Hilbert transform", arXiv:1001.4043 [math.CA] 31 January 2010. The paper of Lacey--Sawyer-Uriarte-Tuero alliviated the "pivotal" condition used in a present article and replaced it by the very interesting and correct energy condition, which, unlike the "pivotal" condition turned out to be also necessary. The paper of Lacey-Sawyer-Uriarte-Tuero used the present article in its main aspect. The thrust of the present article is to use the methods of nonhomogeneous Harmonoc Analysis together with a several paraproducts arising from a certain stopping time argument. In view of the importance of the present article for Lacey--Sawyer-Uriarte-Tuero's paper arXiv:1001.4043 [math.CA] 31 January 2010, we present it to the attention of the reader. Drawing no parallels, "Darwin spent 1838-1859 getting ready to publish "On the Origin of Species" without actually publishing it, only brooding over beaks of finches".