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Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

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Abstract

In [Do], Doi proved that the \({L^{2}_{t}H^{1/2}_{x}}\) local smoothing effect for Schrödinger equations 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 → L ∞ dispersive estimates still hold without loss for eitΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Bât. 425, Université Paris-Sud 11, F-91405, Orsay Cedex, France

    Nicolas Burq

  2. Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, F-75230, Paris Cedex 05, France

    Colin Guillarmou

  3. Department of Mathematics, Australian National University, Canberra, ACT, 0200, Australia

    Andrew Hassell

Authors
  1. Nicolas Burq
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  2. Colin Guillarmou
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  3. Andrew Hassell
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Correspondence to Nicolas Burq.

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Burq, N., Guillarmou, C. & Hassell, A. Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics. Geom. Funct. Anal. 20, 627–656 (2010). https://doi.org/10.1007/s00039-010-0076-5

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  • DOI: https://doi.org/10.1007/s00039-010-0076-5

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