Two weight inequality for the Hilbert transform

Series: 
Analysis Seminar
Wednesday, January 25, 2012 - 2:00pm
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
Georgia Tech
Organizer: 
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. Joint work with Eric Sawyer, Chun-Yun Shen, and Ignacio Uriate-Tuero.