Resolvent Estimates on Asymptotically Hyperbolic Spaces

TL;DR

This paper extends high energy resolvent estimates on asymptotically hyperbolic spaces to more general metrics, broadening the understanding of spectral properties without requiring evenness or trapping assumptions.

Contribution

It generalizes Vasy's semiclassical estimates to non-even metrics, establishing meromorphic continuation and high energy bounds in broader settings.

Findings

Meromorphic continuation of the resolvent in wider metric classes

High energy estimates in strips without parametrix construction

Extension of the resolvent size matching Guillarmou's results

Abstract

We extend Vasy's results on semiclassical high energy estimates for the meromorphic continuation of the resolvent for asymptotically hyperbolic manifolds to metrics that are not necessarily even. Vasy's method gives the meromorphic continuation of the resolvent and high energy estimates in strips, assuming that the geodesic flow is non-trapping, without having to construct a parametrix. We prove that the size of the strip to which the resolvent can be extended meromorphically is the same obtained by Guillarmou.