Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates

Understand

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.