Weighted Fourier extension estimates are intimately connected to a wide range of problems of geometric and discrete nature. The main goal of the talk is to present a certain "ray bundle representation" of the Fourier extension operator in terms of the Wigner transform to investigate such estimates.

In joint work with Bennett, Gutierrez and Nakamura, we show how Sobolev estimates for the Wigner transform can be converted into "tomographic bounds" for the Fourier extension operator, which implies a variant of the (recently shown by H. Cairo to be false) Mizohata-Takeuchi conjecture. Together with Bez and the previous three authors, we employed our phase-space approach to study the orthonormal systems version of the Mizohata-Takeuchi conjecture, which allowed us, in particular, to give a direct proof of the orthonormal Strichartz estimates of Frank and Sabin in dimension 1. If time allows, we will make a further connection between our results and Flandrin's conjecture in signal processing through the study of certain singular integral operators similar to those studied by Lacey, Lie, Muscalu, Tao and Thiele.