Tuomas P. Hytönen
Published 2013-12-03, updated 2019-11-18Version 3
The two-weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures $\sigma$ and $w$ on $\mathbb R$. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two-weight problem by Lacey, Sawyer, Shen and Uriarte-Tuero. Our characterization is in terms of Sawyer-type testing conditions and a variant of the two-weight $A_2$ condition, where $\sigma$ and $w$ are integrated over complementary intervals only. A key novely of the proof is a two-weight inequality for the Poisson integral with `holes'.
Comments: The last version of the article submitted to the publisher (but without journal style file)
Journal: Proc. Lond. Math. Soc. (3) 117 (2018), no. 3, 483-526
A Two-weight inequality between $L^p(\ell^2)$ and $L^p$
Convergence of Singular integrals with general measures
Local $Tb$ theorem with $L^2$ testing conditions and general measures: Calderón-Zygmund operators
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