Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres

TL;DR

This paper proves that constant functions are the unique real-valued maximizers for certain Fourier restriction inequalities on low-dimensional spheres, extending previous results to higher dimensions and complex-valued functions.

Contribution

It establishes the uniqueness of real-valued maximizers and characterizes complex-valued maximizers for adjoint Fourier restriction inequalities on spheres in dimensions 3 to 7.

Findings

Constant functions are the unique real-valued maximizers.

Complex-valued maximizers are characterized as nonnegative maximizers times a character.

The proof employs tools from probability, Lie theory, and special functions.

Abstract

We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in a companion paper. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao to arbitrary dimensions $d\geq 2$ and general even exponents.