Exponentials rarely maximize Fourier extension inequalities for cones

TL;DR

The paper establishes the existence of maximizers and precompactness for scale-invariant Fourier extension inequalities on the cone in $\mathbb{R}^{1+d}$, under a boundedness assumption on the extension operator. Through a two-stage Penrose-transform analysis, it shows that Foschi-type $\mathfrak F$-functions are critical points only when $p=2$, and, in subcritical $p<2$ and supercritical $p>2$, they cannot maximize the inequality. A novel two-stage frequency localization (dyadic annuli and sectors) plus spatial profile decompositions are developed to prove the existence of maximizers, with a detailed treatment of the cone’s symmetry group. The results connect sharp restriction theory on the cone with classical maximizer classifications in low dimensions and highlight the special role of the Strichartz/$L^2$ case, offering a robust framework for sharp cone restriction problems.

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 $p = 2$.

Theorems & Definitions (51)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Remark 2.1
• Remark 2.2
• Proposition 2.1
• proof
• Remark 2.3: see GN20
• Lemma 2.2
• proof