Dispersive estimates for the wave equation on Riemannian manifolds of bounded curvature

Abstract

We prove space-time dispersive estimates for solutions to the wave equation on compact Riemannian manifolds with bounded curvature tensor, where we assume that the metric tensor is of $W^{1,p}$ regularity for some $p>d$, which ensures that the curvature tensor is well-defined in the weak sense. The estimates are established for the same range of Lebesgue and Sobolev exponents that hold in the case of smooth metrics. Our results are for bounded time intervals, so by finite propagation velocity they hold also on noncompact manifolds under appropriate uniform geometry conditions.