Gluing Semiclassical Resolvent Estimates via Propagation of Singularities

We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application, we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence of trapping which is sufficiently mild in one of several senses. As a corollary, we obtain local exponential decay for the wave propagator and local smoothing for the Schrödinger propagator.