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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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