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

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 Calderón-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".

40 pages. This article was written in 2005 and subsequently lost. In view of the importance of it for Lacey-Sawyer-Uriarte-Tuero's paper arXiv:1001.4043 31 Jan 2010, we present it to the attention of the reader.

References in corpus (1)

• Noncommutative Perron-Frobenius-Ruelle theorem, two weight Hilbert transform, and almost periodicity

Cited by in corpus (19)

• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II
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• Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture
• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• On $A_2$ conjecture and corona decomposition of weights
• The sharp weighted bound for general Calderon-Zygmund operators
• Characterization of a Two Weight Inequality for Multilinear Fractional Maximal Operators
• A geometric condition, necessity of energy, and two weight boundedness of fractional Riesz transforms
• A two weight theorem for fractional singular integrals with an energy side condition and quasicube testing
• Energy conditions and twisted localizations of operators
• A sharp estimate of weighted dyadic shifts of complexity 0 and 1
• A two weight theorem for fractional singular integrals in higher dimension
• Failure of necessity of the energy condition
• A restricted weak type inequality with application to a Tp theorem and optimal cancellation conditions for CZO's
• Restricted testing for the Hardy-Littlewood maximal function
• The theta bump condition for product fractional integrals
• T1 testing implies Tp polynomial testing: optimal cancellation conditions for CZO's
• Two weight norm inequalities for the bilinear fractional integrals
• Spectral perturbation theory and the two weights problem