On the two weight inequality for the Hilbert transform

On the two weight inequality for the Hilbert transform

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Michael Lacey, Georgia Institute of Technology
Fine Hall 314

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 $L2$ to $L2$ inequality holds if and only if two $L2$ to weak-$L2$ 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. Joint work with Eric Sawyer, Chun-Yun Shen, and Ignacio Uriate-Tuero.