Existence of extremals for a Fourier restriction inequality

Abstract

The adjoint Fourier restriction inequality of Tomas and Stein states that the mapping $f\mapsto \widehat{f\sigma }$ is bounded from $L^2\left({\mathbb{S}}^2\right)$ to $L^4\left({ℝ}^3\right)$. We prove that there exist functions that extremize this inequality, and that any extremizing sequence of nonnegative functions has a subsequence that converges to an extremizer.