The Stein-Tomas inequality under the effect of symmetries

TL;DR

This paper establishes new Fourier restriction estimates for symmetric functions on the sphere, proves the existence of maximizers for the endpoint Stein--Tomas inequality in this class, and demonstrates the sharpness of the exponent range.

Contribution

It introduces novel restriction estimates for symmetric functions and confirms the existence of maximizers, advancing understanding of the Stein--Tomas inequality under symmetries.

Findings

New restriction estimates for $O(d-k) imes O(k)$-symmetric functions.

Existence of maximizers for the endpoint Stein--Tomas inequality in the symmetric class.

Sharpness of the Lebesgue exponent range in the estimates.

Abstract

We prove new Fourier restriction estimates to the unit sphere $S^{d-1}$ on the class of $O(d-k)\times O(k)$-symmetric functions, for every $d\geq 4$ and $2\leq k\leq d-2$. As an application, we establish the existence of maximizers for the endpoint Stein--Tomas inequality within that class. Moreover, we construct examples showing that the range of Lebesgue exponents in our estimates is sharp.