Dimension-free Fourier restriction inequalities

Let ${\mathbf{R}}_{{\mathbb{S}}^{d-1}}(p\to q)$ denote the best constant for the $L^p({\mathbb{R}}^d)\to L^q({\mathbb{S}}^{d-1})$ Fourier restriction inequality to the unit sphere ${\mathbb{S}}^{d-1}$, and let ${\mathbf{R}}_{{\mathbb{S}}^{d-1}}(p\to q;\text{rad})$ denote the corresponding constant for radial functions. We investigate the asymptotic behavior of the operator norms ${\mathbf{R}}_{{\mathbb{S}}^{d-1}}(p\to q)$ and ${\mathbf{R}}_{{\mathbb{S}}^{d-1}}(p\to q;\text{rad})$ as the dimension $d$ tends to infinity. We further establish a dimension-free endpoint Stein-Tomas inequality for radial functions, together with the corresponding estimate for general functions which we prove with an $O(d^{1\mathrm{/}2})$ dependence. Our methods rely on a uniform two-sided refinement of Stempak’s asymptotic $L^p$ estimate of Bessel functions.