Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces

TL;DR

This paper develops a microlocal framework for analyzing non-elliptic problems on manifolds, with applications to asymptotically hyperbolic and Kerr-de Sitter spaces, including resolvent extension and high energy estimates.

Contribution

It introduces a systematic, stable microlocal analysis framework for non-elliptic problems applicable to various geometric settings, including Kerr-de Sitter spaces.

Findings

New construction of meromorphic resolvent extension for Laplacian

High energy estimates for spectral parameters

Unified microlocal approach for diverse geometric contexts

Abstract

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.