Existence of Extremals for a Fourier Restriction Inequality on the One-Sheeted Hyperboloid

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