Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

In [Do], Doi proved that the ${L^{2}_{t}H^{1/2}_{x}}In [Do], Doi proved that the L²_tH^1/2_x{L^{2}_{t}H^{1/2}_{x}} local smoothing effect for Schr?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 L ¹ → 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.