Low Frequency Estimates and Local Energy Decay for Asymptotically Euclidean Laplacians

For Riemannian metrics G on ℝ d which are long range perturbations of the flat one, we prove estimates for (− Δ G − λ −iε)−n as λ → 0, which are uniform with respect to ε, for all n ≤ [d/2] +1 in odd dimension and n ≤ d/2 in even dimension. We also give applications to the time decay of Schrödinger and Wave (or Klein–Gordon) equations.