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

We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, $\mathrm{\parallel }{e}^{-it\mathrm{\Delta }\mathrm{/}2}f{\mathrm{\parallel }}_{L_t^q{L}_{\mathit{x}}^r({\mathbb{R}}^{1+d})}\le C_{q,r}\parallel f{\parallel }_{L^2({\mathbb{R}}^d)}$, with $2\mathrm{/}q+d\mathrm{/}r=d\mathrm{/}2$, $q,r\ge 2$, and thus $r\le 2d\mathrm{/}(d-2)$ if $d\ge 3$. We show that, in low dimensions $1\le d\le 5$, the thresholds ${\rho }_1=10$, ${\rho }_2=6$, ${\rho }_3=4\sqrt{7}-6\approx 4.583$, ${\rho }_4=2\sqrt{15}-4\approx 3.746$, and ${\rho }_5=10\mathrm{/}3\approx 3.333$ are such that gaussians are stable local maximizers for $2<r<{\rho }_d$, and fail to be local maximizers for ${\rho }_d<r\le 2d\mathrm{/}(d-2)$ (with the conventions there is no upper bound on $r$ when $d\in \{1,2\}$ and that $r=\mathrm{\infty }$ 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.