Gaussians Rarely Extremize Adjoint Fourier Restriction Inequalities For Paraboloids

TL;DR

This paper investigates the extremizing properties of Gaussian functions for adjoint Fourier restriction inequalities on paraboloids, revealing that Gaussians are only critical points when p=2 and for certain Strichartz inequalities.

Contribution

It proves that Gaussians are critical points for these inequalities only in specific cases, clarifying their role in the extremization problem.

Findings

Gaussians are critical points only when p=2.

Gaussians are critical points for all admissible Strichartz pairs.

Gaussians rarely extremize the inequalities outside these cases.

Abstract

It was proved independently by Foschi and Hundertmark, Zharnitsky that Gaussians extremize the adjoint Fourier restriction inequality for L^2 functions on the paraboloid in the two lowest-dimesional 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,infinity)^2.