Dimension-free Fourier restriction inequalities
Diogo Oliveira e Silva, Błażej Wróbel
TL;DR
This paper analyzes how Fourier restriction constants to the sphere behave as the dimension grows. It proves a dimension-free endpoint Stein–Tomas inequality for radial functions and an endpoint bound for general functions with an $O(d^{1/2})$ growth, achieved through a uniform two-sided refinement of Stempak's uniform $L^p$ estimates for Bessel functions. It identifies three regions in the $(1/p,1/q)$ plane (A,B,C) with distinct high-dimensional behaviors: in A, the general constant tends to zero; in B, the radial constant remains bounded (and in particular finite at the endpoint $q=p'$) while the nonradial constant may blow up when $q>p'$; in C, both fail. The results highlight the role of radial symmetry and Bessel-function asymptotics in dimension-free restriction phenomena and provide a framework for translating endpoint harmonic-analysis inequalities into dimension-stable estimates.
Abstract
Let ${{\bf R}_{\mathbb{S}^{d-1}}}(p\to q)$ 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 ${\bf R}_{\mathbb{S}^{d-1}} (p\to q;\textrm{rad})$ denote the corresponding constant for radial functions. We investigate the asymptotic behavior of the operator norms ${{\bf R}_{\mathbb{S}^{d-1}}}(p\to q)$ and ${\bf R}_{\mathbb{S}^{d-1}} (p\to q;\textrm{rad})$ 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 $O(d^{1/2})$ dependence. Our methods rely on a uniform two-sided refinement of Stempak's asymptotic $L^p$ estimate of Bessel functions.