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arXiv · 1312.0843

The two-weight inequality for the Hilbert transform with general measures

Tuomas P. Hytönen
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Abstract

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'.

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BibTeXRIS

Tuomas P. Hytönen. 2013-12-03. The two-weight inequality for the Hilbert transform with general measures. https://doi.org/10.1112/plms.12136

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