The Fourier restriction problem for biradial functions

Abstract

We determine the exact strong-type Fourier restriction range for the unit sphere $\mathbb{S}^{d-1}$ on the class of $O(d-k)\times O(k)$-invariant functions, for every $d\geq4$ and $2\leq k\leq d-2$. This includes the critical estimates beyond the biradial Stein--Tomas endpoint left open by Mandel and Oliveira e Silva. The proof combines a weighted $L^2$ estimate for extension from a circular arc with the full two-dimensional restriction theorem. An appendix gives a direct proof of the classically conjectured range for axially symmetric functions on the whole sphere.