Exponentials rarely maximize Fourier extension inequalities for cones

TL;DR

This paper establishes the existence of maximizers for Fourier extension inequalities on cones in certain dimensions, showing that exponential functions rarely maximize these inequalities and characterizing maximizers in low dimensions.

Contribution

It proves the existence and precompactness of maximizers for all scale-invariant Fourier extension inequalities on cones, and characterizes global maximizers in low dimensions.

Findings

Maximizers exist for all valid inequalities on the cone.

Exponential functions rarely maximize these inequalities.

Characterization of maximizers in dimensions 2 and 3.

Abstract

We prove the existence of maximizers and the precompactness of $L^p$-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in $\mathbb R^{1+d}$. In the range for which such inequalities are conjectural, our result is conditional on the boundedness of the extension operator. Global maximizers for the $L^2$ Fourier extension inequality on the cone in $\mathbb R^{1+d}$ have been characterized in the lowest-dimensional cases $d\in\{2,3\}$. We further prove that these functions are critical points for the $L^p$ to $L^q$ Fourier extension inequality if and only if $p = 2$.