Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids

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.