Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids

TL;DR

This paper proves that Gaussian functions do not serve as extremizers for Fourier extension inequalities related to hyperbolic paraboloids, highlighting unique extremization properties in this geometric setting.

Contribution

It establishes that Gaussians are not extremizers for certain Fourier extension inequalities on hyperbolic paraboloids, a novel result in harmonic analysis.

Findings

Gaussians do not extremize the inequalities

Extension inequalities associated with hyperbolic paraboloids are analyzed

The result clarifies extremizer properties in this geometric context

Abstract

For $\xi = (\xi_1, \xi_2, \ldots, \xi_d) \in \mathbb{R}^d$ let $Q(\xi) := \sum_{j=1}^d \sigma_j \xi_j^2$ be a quadratic form with signs $\sigma_j \in \{\pm1\}$ not all equal. Let $S \subset \mathbb{R}^{d+1}$ be the hyperbolic paraboloid given by $S = \big\{(\xi, \tau) \in \mathbb{R}^{d}\times \mathbb{R} \ : \ \tau = Q(\xi)\big\}$. In this note we prove that Gaussians never extremize an $L^p(\mathbb{R}^d) \to L^{q}(\mathbb{R}^{d+1})$ Fourier extension inequality associated to this surface.