Gaussians rarely extremize adjoint Fourier restriction inequalities for paraboloids

Abstract:

It was proved independently by Foschi and by Hundertmark and Zharnitsky that Gaussians extremize the adjoint Fourier restriction inequality for $L^2$ functions on the paraboloid in the two lowest-dimensional cases. Here we prove that Gaussians are critical points for the $L^p$ to $L^q$ adjoint Fourier restriction inequalities if and only if $p=2$. Also, Gaussians are critial points for the $L^2$ to $L^r_t L^q_x$ Strichartz inequalities for all admissible pairs $(r,q) \in (1,\infty )^2$.