The Stein–Tomas inequality under the effect of symmetries

Abstract

We prove new Fourier restriction estimates to the unit sphere {mathbb{S}^{d-1}} on the class of O(d − k) × O(k)-symmetric functions, for every d ≥ 4 and 2 ≤ k ≤ 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.