Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids

Abstract:

For $\xi = (\xi _1, \xi _2, \ldots , \xi _d) \in \mathbb {R}^d$ let $Q(\xi ) \colonequals \sum _{j=1}^d \sigma _j \xi _j^2$ be a quadratic form with signs $\sigma _j \in \{\pm 1\}$ 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.