Dimension-free Fourier restriction inequalities

TL;DR

This paper studies how Fourier restriction inequalities behave as the dimension grows, establishing dimension-free bounds for radial functions and asymptotic estimates using Bessel function analysis.

Contribution

It introduces a dimension-free endpoint Stein-Tomas inequality for radial functions and provides asymptotic behavior of restriction constants in high dimensions.

Findings

Dimension-free endpoint Stein-Tomas inequality for radial functions

Asymptotic analysis of restriction constants as dimension increases

An $O(d^{1/2})$ dependence estimate for general functions

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.