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

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

2012-01-20

arxiv.org/abs/1201.4319v6

Mathematics

Classical Analysis and ODEs

43 pages. v2 citations tweaked. v3 citations tweaked again. v4 a few typos corrected v5: conclusion and proof of Lemma 4.8 cha

Scientific paper

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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