Global and local maximizers for some Fourier extension estimates on the sphere

TL;DR

This paper investigates maximizers for Fourier extension inequalities on the sphere, showing constant functions are local and sometimes global maximizers, with new results in low dimensions and novel Bessel function norm hierarchies.

Contribution

It establishes that constant functions are local maximizers for certain Fourier extension estimates and extends the range of dimensions where they are global maximizers, introducing new Bessel function norm hierarchies.

Findings

Constant functions are local maximizers for specific Fourier extension estimates.

In dimensions 2 and 3, the range of exponents for which constant functions are global maximizers is expanded.

New hierarchies between weighted norms of Bessel functions are developed.

Abstract

In this note, we study maximizers for Fourier extension inequalities on the sphere. We prove that constant functions are local maximizers for the $L^p(\mathbb{S}^{d-1})$ to $L^p(\mathbb{R}^d)$ Fourier extension estimates in the same range of exponents $p$ for which they are global maximizers for the $L^2(\mathbb{S}^{d-1})$ to $L^p_{rad}L^2_{ang}(\mathbb{R}^d)$ mixed-norm Fourier extension inequalities. Moreover, in the case of low dimensions, we improve the range of exponents for which constant functions are known to be the unique global maximizers for the $L^2(\mathbb{S}^{d-1})$ to $L^p_{rad}L^2_{ang}(\mathbb{R}^d)$ mixed-norm Fourier extension estimate on the sphere, covering, for the case of dimensions $d=2,3$, the entire Stein-Tomas range. This is achieved by establishing novel hierarchies between certain weighted norms of Bessel functions.