Gaussians Do Not Always Maximize Mixed-Norm Strichartz Inequalities for the Schrödinger Equation

Abstract: We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, $$|e^{{-itΔ/2}f|<em>{L_t^qL</em>{\boldsymbol{x}}^{r(\mathbb{R}^{1+d})}\le}} C_{q,r}\lVert f\rVert_{L<sup>2(\mathbb{R}<sup>d)}$$, with $2/q+d/r=d/2$, $q,r\geq 2$, and thus $r\leq 2d/(d-2)$ if $d\geq 3$. We show that, in low dimensions $1\leq d\leq 5$, the thresholds $ρ_1=10$, $ρ_2=6$, $ρ_3=4\sqrt{7}-6\approx 4.583$, $ρ_4=2\sqrt{15}-4\approx 3.746$, and $ρ_5=10/3\approx 3.333$ are such that gaussians are stable local maximizers for $2&lt;r&lt;ρ_d$, and fail to be local maximizers for $ρ_d&lt;r\leq 2d/(d-2)$ (with the conventions there is no upper bound on $r$ when $d\in{1,2}$ and that $r=\infty$ is excluded when $d=2$). In the cases $(q,r,d)\in{(6,6,1),(8,4,1),(4,4,2)}$, we establish global stability inequalities with effective stability constants. Both proofs hinge on spectral gaps which we compute exactly.