Existence of Extremals for a Fourier Restriction Inequality

TL;DR

This paper proves the existence of extremal functions for a key Fourier restriction inequality on the sphere, showing that extremizing sequences converge to actual extremizers.

Contribution

It establishes the existence of extremizers for the Tomas-Stein Fourier restriction inequality on the sphere, a problem previously unresolved.

Findings

Existence of extremizers for the inequality.

Convergence of extremizing sequences to extremizers.

Extension of Fourier restriction theory.

Abstract

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