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arXiv · 1201.4319

Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, I

Michael T. Lacey·Eric T. Sawyer·Chun-Yen Shen·Ignacio Uriarte-Tuero
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Abstract

The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two L^2 to weak-L^2 inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions.

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Michael T. Lacey, Eric T. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero. 2012-01-20. Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, I. https://doi.org/10.1215/00127094-2826690

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