Gaussians Do Not Always Maximize Mixed-Norm Strichartz Inequalities for the Schr\"odinger Equation

Abstract

We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schr\"odinger equation, $\|e^{-it\Delta/2}f\|_{L_t^qL_{\boldsymbol{x}}^r(\mathbb{R}^{1+d})}\le C_{q,r}\lVert f\rVert_{L^2(\mathbb{R}^d)}$, 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 $\rho_1=10$, $\rho_2=6$, $\rho_3=4\sqrt{7}-6\approx 4.583$, $\rho_4=2\sqrt{15}-4\approx 3.746$, and $\rho_5=10/3\approx 3.333$ are such that gaussians are stable local maximizers for $2<r<\rho_d$, and fail to be local maximizers for $\rho_d<r\leq 2d/(d-2)$ (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.