Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid

TL;DR

This paper proves that extremizers do not exist for certain Fourier restriction inequalities on the hyperboloid in 3 and 4 dimensions, using Foschi's method.

Contribution

It establishes the nonexistence of extremizers for the $L^2$ to $L^p$ adjoint restriction inequalities on the hyperboloid in specific dimensions and cases.

Findings

Extremizers do not exist for the inequalities studied.

The proof uses Foschi's method.

Results apply to dimensions 3 and 4 with even integer p.

Abstract

We study the problem of existence of extremizers for the $L^2$ to $L^p$ adjoint Fourier restriction inequalities on the hyperboloid in dimensions 3 and 4, in which cases $p$ is an even integer. We will use the method developed by Foschi to show that extremizers do not exist.