Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

TL;DR

This paper demonstrates that Strichartz and dispersive estimates for the Schrödinger equation can hold without loss on manifolds with hyperbolic trapped geodesics of small fractal dimension, contrasting previous results on local smoothing.

Contribution

It establishes lossless Strichartz estimates on manifolds with hyperbolic trapped geodesics, expanding understanding of dispersive properties in such geometric settings.

Findings

Strichartz estimates hold without loss on hyperbolic trapped manifolds.

Dispersive estimates remain valid despite trapping under certain conditions.

Contrast with previous results showing loss in local smoothing effects.

Abstract

Doi proved that the $L^2_t H^{1/2}_x$ local smoothing effect for Schr\"odinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and $L^1\to L^\infty$ dispersive estimates still hold without loss for $e^{it\Delta}$ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.