The Tomas–Stein inequality under the effect of symmetries

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