Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)
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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, described in Sect. 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis.
Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose’s b-structures. These include asymptotically de Sitter-type metrics on a blow-up of the natural compactification, Kerr-de Sitter-type metrics, as well as asymptotically Minkowski metrics.
The simplest application is 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), including 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. These results are also available in a follow-up paper which is more expository in nature (Vasy in Uhlmann, G. (ed.) Inverse Problems and Applications. Inside Out II, 2012).
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.
Mathematics Subject Classification35L05 35P25 58J47 83C57
A.V. is very grateful to Maciej Zworski, Richard Melrose, Semyon Dyatlov, Mihalis Dafermos, Gunther Uhlmann, Jared Wunsch, Rafe Mazzeo, Kiril Datchev, Colin Guillarmou, Andrew Hassell, Dean Baskin and Peter Hintz for very helpful discussions, for their enthusiasm for this project and for carefully reading parts of this manuscript. Special thanks are due to Semyon Dyatlov in this regard who noticed an incomplete argument in an earlier version of this paper in holomorphy considerations, and to Mihalis Dafermos, who urged the author to supply details to the argument at the end of Sect. 6, which resulted in the addition of Sect. 3.3, as well as the part of Sect. 3.2 and Sect. 7 covering the complex absorption, to the main body of the argument, and that of Sect. 2.7 for stability considerations. A.V. is also very grateful to the three anonymous referees whose detailed reports improved the manuscript significantly.
A.V. gratefully acknowledges partial support from the National Science Foundation under grants number DMS-0801226 and DMS-1068742 and from a Chambers Fellowship at Stanford University, as well as the hospitality of Mathematical Sciences Research Institute in Berkeley. S.D. is grateful for partial support from the National Science Foundation under grant number DMS-0654436.
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