On the two weight inequality for the Hilbert transform
On the two weight inequality for the Hilbert transform

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 realvariable 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 twoweight Poisson inequality, and a pair of testing inequalities on bounded functions. Joint work with Eric Sawyer, ChunYun Shen, and Ignacio UriateTuero.