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

By Fedor Nazarov, Sergei Treil, Alexander Volberg. 2010

85 citations recorded by the source corpus.

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".

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/f2a4c952e7d9869adf6eb930dbd03e288ceea6d4

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In Applied Mathematics

Citations in the corpus

The sharp weighted bound for general Calderón-Zygmund operators (2010, background): “d by the proof of a T(1) theorem — this time, the one for nonhomogeneous spaces due to Nazarov, Treil and Volberg [19]. (A variant of the same proof, from a more recent Nazarov–Treil–Volberg preprint [18], is also behind the reduction of Pérez, Treil and Volberg [21].) Recall that the basic philosop”
The sharp weighted bound for general Calderon-Zygmund operators (2010, background): “(A variant of the same proof, from a more recent Nazarov–Treil–Volberg preprint [18], is also behind the reduction of Pérez, Treil and Volberg [21].”
The two‐weight inequality for the Hilbert transform with general measures (2018, unknown): “1] and so-called ‘pivotal’ [16] and ‘energy’ [10] hypotheses.”
A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure (2008, methodology): “Corresponding results for even the Hilbert transform have only recently been obtained [Nazarov et al. 2010; Lacey et al. 2011] and even then only for p = 2; evidently these are much harder to obtain.”