Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

2009/07/21 by Nicolas Burq, Colin Guillarmou, Burq, Nicolas +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.0907.3545

23 pages. Corrections in the proof of prop 3.9

arxiv created 2011/03/09 · arxiv updated 2011/03/10

Abstract

Doi proved that the L²_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 L¹→ L^∞ dispersive estimates still hold without loss for e^itΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.

Related

• Strichartz estimates without loss outside two strictly convex obstacles
• Strichartz estimates on asymptotically hyperbolic manifolds
• Dispersive Estimates for Manifolds with one Trapped Orbit
• Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds
• Resolvent and Spectral Measure on Non-Trapping Asymptotically Hyperbolic Manifolds III: Global-in-Time Strichartz Estimates without Loss