Stability of sharp Fourier restriction

Abstract

The classical paper of R. Strichartz [Duke Math. J., 1977] has been very influential for the study of partial differential equations. This is the birth of what became known as 'Strichartz estimates', which can be equivalently seen as Fourier (adjoint) restriction inequalities over certain quadratic surfaces: paraboloids, cones, spheres and hyperboloids. In the first part of this talk I will briefly give an overview of what is currently known in the theme of `sharp Fourier restriction' theory, that is, the search for extremizers of these inequalities. For instance, in the case of the sphere, it is conjectured that the constant functions should be the extremizers for the $L^2 o L^q$ adjoint Fourier restriction, but this has only been proved in some cases. I will then discuss how some of these extremizers remain stable under perturbations of the underlying Lebesgue measure, by exploring the structure of the associated Helmholtz equation. This is based on a recent joint work with Negro and Oliveira e Silva (https://arxiv.org/abs/2108.03412).