On existence of extremizers for the Tomas-Stein inequality for $S^1$

TL;DR

This paper proves the existence of extremizers for the Tomas-Stein inequality on the circle $S^1$ and characterizes their symmetry properties, advancing understanding of Fourier restriction phenomena.

Contribution

It establishes the existence of extremizers for the Tomas-Stein inequality on $S^1$ and shows they are symmetric in magnitude.

Findings

Existence of extremizers for the inequality.

Extremizers satisfy a symmetry condition $|f(-x)|=|f(x)|$.

Provides insight into the structure of extremizers.

Abstract

The Tomas-Stein inequality or the adjoint Fourier restriction inequality for the sphere $S^1$ states that the mapping $f\mapsto \hat{f\sigma}$ is bounded from $L^2(S^1)$ to $L^6(\mathbb{R}^2)$. We prove that there exists an extremizer for this inequality. We also prove that any extremizer satisfies $|f(-x)|=|f(x)|$ for almost every $x\in S^1$.