Georgia Institute of Technology College of Sciences

The Hilbert transform is a foundational transform, with deep connections to electrical charge, and analyticity. The `two weight inequality for the Hilbert transform' concerns the most general setting in which the Hilbert transform admits a (weighted) L^2 inequality. We will give a couple of (surprising?) ways that this question arises. And we will indicate the surprise that is behind the recent description of all setting in which the two weight inequality holds.