Resolvent Estimates and Local Decay of Waves on Conic Manifolds

We consider manifolds with conic singularites that are isometric to R outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonancefree region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author [23] to establish a “very weak” Huygens’ principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schrödinger equation.