A sharp trilinear inequality related to Fourier restriction on the circle

TL;DR

This paper proves a sharp trilinear inequality related to Fourier restriction on the circle, advancing understanding of extremizers and integral estimates involving Bessel functions.

Contribution

It introduces a new sharp trilinear inequality and demonstrates that certain constants are local extremizers for related Fourier restriction inequalities.

Findings

Established a sharp trilinear inequality for the circle

Identified constants as local extremizers of the inequalities

Developed estimates for integrals of sixfold products of Bessel functions

Abstract

In this paper we prove a sharp trilinear inequality which is motivated by a program to obtain the sharp form of the $L^2 - L^6$ Tomas-Stein adjoint restriction inequality on the circle. Our method uses intricate estimates for integrals of sixfold products of Bessel functions developed in a companion paper. We also establish that constants are local extremizers of the Tomas-Stein adjoint restriction inequality as well as of another inequality appearing in the program.