Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids

Abstract

For $ξ= (ξ_1, ξ_2, \ldots, ξ_d) \in \mathbb{R}^d$ let $Q(ξ) := \sum_{j=1}^d σ_j ξ_j^2$ be a quadratic form with signs $σ_j \in \{\pm1\}$ not all equal. Let $S \subset \mathbb{R}^{d+1}$ be the hyperbolic paraboloid given by $S = \big\{(ξ, τ) \in \mathbb{R}^{d}\times \mathbb{R} \ : \ τ= Q(ξ)\big\}$. 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.