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, $\Vert e^{-it\Delta /2}f{\Vert }_{L_t^q{L}_{\mathbf{x}}^r({\mathbb{R}}^{1+d})}\le C_{q,r}\Vert f{\Vert }_{L^2({\mathbb{R}}^d)}$, with $2/q+d/r=d/2$, $q,r\ge 2$, and thus $r\le 2d/(d-2)$ 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 ${\rho }_5=10/3\approx 3.333$ are such that gaussians are stable local maximizers for $2<r<{\rho }_d$, and fail to be local maximizers for ${\rho }_d<r\le 2d/(d-2)$ (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.

34 pages, 1 figure