Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

Details Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

2009-07-21

arxiv.org/abs/0907.3545v2

Mathematics

Analysis of PDEs

23 pages. Corrections in the proof of prop 3.9

Scientific paper

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.

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