Existence of extremals for a Fourier restriction inequality on the one-sheeted hyperboloid

TL;DR

This paper proves the existence of extremal functions for a specific Fourier restriction inequality on the one-sheeted hyperboloid in four-dimensional space, ensuring convergence of extremizing sequences to actual extremizers.

Contribution

It establishes the existence of extremizers for the endpoint Fourier restriction inequality on the hyperboloid, considering symmetries and convergence properties.

Findings

Existence of extremizers for the inequality is proven.

Any extremizing sequence has a convergent subsequence to an extremizer.

Symmetries are taken into account in the analysis.

Abstract

We prove the existence of functions that extremize the endpoint $L^2$ to $L^4$ adjoint Fourier restriction inequality on the one-sheeted hyperboloid in Euclidean space $\mathbb{R}^4$ and that, taking symmetries into consideration, any extremizing sequence has a subsequence that converges to an extremizer.