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

Abstract

The Tomas-Stein inequality or the adjoint Fourier restriction inequality for the sphere $S^1$ states that the mapping $f\mapsto \widehat{f\sigma }$ 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 $x\in S^1$.

arXiv comment38 pages. A mistake in proving an inequality on the old page 26 is corrected

References in corpus (2)

• Quasiextremals for a Radon-like transform
• Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation