Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

Nicolas Burq; Colin Guillarmou; Andrew Hassell

doi:10.1007/s00039-010-0076-5

Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

Nicolas Burq^*, Colin Guillarmou, Andrew Hassell

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

39 Citations (Scopus)

Abstract

In [Do], Doi proved that the L_t ²H_x ^1/2 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 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.

Original language	English
Pages (from-to)	627-656
Number of pages	30
Journal	Geometric and Functional Analysis
Volume	20
Issue number	3
DOIs	https://doi.org/10.1007/s00039-010-0076-5
Publication status	Published - 2010

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title = "Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics",

abstract = "In [Do], Doi proved that the Lt 2Hx 1/2 local smoothing effect for Schr{\"o}dinger equations 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 eitΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.",

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AU - Burq, Nicolas

AU - Guillarmou, Colin

AU - Hassell, Andrew

PY - 2010

Y1 - 2010

N2 - In [Do], Doi proved that the Lt 2Hx 1/2 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 L1 → L∞ dispersive estimates still hold without loss for eitΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.

AB - In [Do], Doi proved that the Lt 2Hx 1/2 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 L1 → L∞ dispersive estimates still hold without loss for eitΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.

KW - Schrödinger equation

KW - Strichartz estimates

KW - hyperbolic trapped set

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U2 - 10.1007/s00039-010-0076-5

DO - 10.1007/s00039-010-0076-5

M3 - Article

SN - 1016-443X

VL - 20

SP - 627

EP - 656

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 3

ER -

Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

Abstract

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