We consider a non-trapping n-dimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes. The rates of decay seen in the asymptotic expansion are related to the resonances of a natural asymptotically hyperbolic problem on the "northern cap" of the compactification. For small perturbations of Minkowski space that fit into our framework, our asymptotic expansions yield a rate of decay that improves on the Klainerman-Sobolev estimates.

We consider three problems for the Helmholtz equation in interior and exterior domains in R d , (d = 2, 3): the exterior Dirichlet-to-Neumann and Neumannto-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.

We consider manifolds with conic singularites that are isometric to R n 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.

In this paper we construct a parametrix for the forward fundamental solution of the wave and KleinGordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of (2 − λ)u = f and to obtain a uniform L p estimate for a family of bump functions traveling to infinity.

We show the existence of the full compound asymptotics of solutions to the scalar wave equation on long-range non-trapping Lorentzian manifolds modeled on the radial compactification of Minkowski space. In particular, we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions of the inhomogeneous equation. In two appendices we show how these results apply to certain spacetimes whose null infinity is modeled on that of the Kerr family. In these cases the leading order logarithmic term in our asymptotic expansions at null infinity is shown to be nonzero.