Dimension-free Fourier restriction inequalities

Let $R_{S^{d-1}}(p\to q)$ denote the best constant for the $L^p\left(R^d\right)\to L^q\left(S^{d-1}\right)$ Fourier restriction inequality to the unit sphere $S^{d-1}$, and let $R_{S^{d-1}}(p\to q;rad)$ denote the corresponding constant for radial functions. We investigate the asymptotic behavior of the operator norms $R_{S^{d-1}}(p\to q)$ and $R_{S^{d-1}}(p\to q;rad)$ 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 $O\left(d^{1/2}\right)$ dependence. Our methods rely on a uniform two-sided refinement of Stempak's asymptotic $L^p$ estimate of Bessel functions.