Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids

Abstract

For ξ = ( ξ 1 , ξ 2 , … , ξ d ) ∈ R d \xi = (\xi _1, \xi _2, \ldots , \xi _d) \in \mathbb {R}^d let Q ( ξ ) ≔ ∑ j = 1 d σ j ξ j 2 Q(\xi ) \colonequals \sum _{j=1}^d \sigma _j \xi _j^2 be a quadratic form with signs σ j ∈ { ± 1 } \sigma _j \in \{\pm 1\} not all equal. Let S ⊂ R d + 1 S \subset \mathbb {R}^{d+1} be the hyperbolic paraboloid given by S = { ( ξ , τ ) ∈ R d × R : τ = Q ( ξ ) } 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 ( R d ) → L q ( R d + 1 ) L^p(\mathbb {R}^d) \to L^{q}(\mathbb {R}^{d+1}) Fourier extension inequality associated to this surface.