Strichartz estimates for the Schr\"odinger equation on compact manifolds with nonpositive sectional curvature

TL;DR

This paper improves Strichartz estimates for Schr"odinger equations on compact manifolds with nonpositive curvature, achieving a logarithmic gain over universal bounds by leveraging geometric properties and advanced kernel estimates.

Contribution

It refines existing Strichartz estimates on such manifolds, obtaining no-loss estimates on logarithmic time intervals for all admissible pairs, extending prior results.

Findings

Achieved logarithmic time interval estimates for Schr"odinger solutions.

Extended universal Strichartz bounds with a log-gain.

Utilized geometric assumptions to improve kernel estimates.

Abstract

We obtain improved Strichartz estimates for solutions of the Schr\"odinger equation on compact manifolds with nonpositive sectional curvatures which are related to the classical universal results of Burq, G\'erard and Tzvetkov [11]. More explicitly, we are able refine the arguments in the recent work of Blair and the authors [3] to obtain no-loss $L^p_tL^{q}_{x}$-estimates on intervals of length $\log \lambda\cdot \lambda^{-1} $ for all {\em admissible} pairs $(p,q)$ when the initial data have frequencies comparable to $\lambda$, which, given the role of the Ehrenfest time, is the natural analog in this setting of the universal results in [11]. We achieve this log-gain over the universal estimates by applying the Keel-Tao theorem along with improved global kernel estimates for microlocalized operators which exploit the geometric assumptions.