  In this paper we develop a general, systematic, microlocal framework for the
Fredholm analysis of non-elliptic problems, including high energy (or
semiclassical) estimates, which is stable under perturbations. This framework
is relatively simple given modern microlocal analysis, and only takes a bit
over a dozen pages after the statement of notation. It resides on a compact
manifold without boundary, hence in the standard setting of microlocal
analysis, including semiclassical analysis.
  The rest of the paper is devoted to applications. Many natural applications
arise in the setting of non-Riemannian b-metrics in the context of Melrose's
b-structures. These include asymptotically Minkowski metrics, asymptotically de
Sitter-type metrics on a blow-up of the natural compactification and Kerr-de
Sitter-type metrics.
  The simplest application, however, is to provide a new approach to analysis
on Riemannian or Lorentzian (or indeed, possibly of other signature)
conformally compact spaces (such as asymptotically hyperbolic or de Sitter
spaces). The results include, in particular, a new construction of the
meromorphic extension of the resolvent of the Laplacian in the Riemannian case,
as well as high energy estimates for the spectral parameter in strips of the
complex plane.
  The appendix written by Dyatlov relates his analysis of resonances on exact
Kerr-de Sitter space (which then was used to analyze the wave equation in that
setting) to the more general method described here.
