Speaker
Amy Tierney, University of Birmingham
Abstract
Weighted L^2 inequalities for the Fourier extension operator are a topic of considerable current interest, particularly in light of recent progress surrounding the Mizohata-Takeuchi conjecture. One avenue for progress is to find tomographic representations of the extension operator in modulus-square.
Bennett, Gutiérrez, Nakamura and Oliveira proved such a representation for the extension operator for hypersurfaces in terms of a surface-carried Wigner transform. Inspired by their work, we seek Wigner representations of the Fourier extension operator associated with curves in R^n. The focal point of this work is the construction of an appropriate curve-carried Wigner transform yielding representations from which we can prove Mizohata-Takeuchi-type inequalities for curves, including an L^2-Mizohata-Takeuchi inequality and a Sobolev-Mizohata-Takeuchi inequality.
In this talk we will discuss the form the Wigner representation takes in the curve setting, and how it departs from what one might initially expect.