Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates

Abstract

In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic spaces, which was introduced recently by the author. This new method in particular constructs the analytic continuation of the resolvent for even metrics (in the sense of Guillarmou), and gives high energy estimates in strips. The key idea is an extension across the boundary for a problem obtained from the Laplacian shifted by the spectral parameter. The extended problem is non-elliptic -- indeed, on the other side it is related to the Klein-Gordon equation on an asymptotically de Sitter space -- but nonetheless it can be analyzed by methods of Fredholm theory. This method is a special case of a more general approach to the analysis of PDEs which includes, for instance, Kerr-de Sitter and Minkowski type spaces. The present paper is self-contained, and deals with asymptotically hyperbolic spaces without burdening the reader with material only needed for the analysis of the Lorentzian problems considered in the author's original work.