Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

Nicolas Burq (1) , Colin Guillarmou (2) , Andrew Hassell (3)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 JAD - Laboratoire Jean Alexandre Dieudonné
3 Department of Mathematics

Doi proved that the $L^2_t H^{1/2}_x$ local smoothing effect for Schrödinger 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.

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Burq, Colin Guillarmou, Andrew Hassell. Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics. Geometric And Functional Analysis, 2010, 20 (3), pp.627-656. ⟨hal-00405720v2⟩