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

TL;DR

This paper proves that extremizers for the Tomas-Stein inequality on the 1D sphere are smooth functions, using analysis of the related Euler-Lagrange equation to establish regularity.

Contribution

It demonstrates the smoothness of extremizers for the Tomas-Stein inequality on the circle, a novel result in harmonic analysis.

Findings

Extremizers are smooth functions.

Euler-Lagrange equation analysis is key to establishing regularity.

Advances understanding of extremal functions in harmonic analysis.

Abstract

We prove that the extremizers to the Tomas-Stein inequality for the one dimension sphere are smooth. This is achieved by studying the associated Euler-Lagrange equation.