Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids

Abstract

For $\xi =({\xi }_1,{\xi }_2,\dots ,{\xi }_d)\in {\mathbb{R}}^d$ let $Q(\xi ):={\sum }_{j=1}^d{\sigma }_j{\xi }_j^2$ be a quadratic form with signs ${\sigma }_j\in \{\pm 1\}$ not all equal. Let $S\subset {\mathbb{R}}^{d+1}$ be the hyperbolic paraboloid given by $S=\{(\xi ,\tau )\in {\mathbb{R}}^d\times \mathbb{R}:\tau =Q(\xi )\}$. 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.

7 pages