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Algebras of pseudodifferential operators on complete manifolds

Authors: Bernd Ammann, Robert Lauter and Victor Nistor
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 80-87
MSC (2000): Primary 58J40; Secondary 58H05, 65R20
Published electronically: September 15, 2003
MathSciNet review: 2029468
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Abstract: In several influential works, Melrose has studied examples of non-compact manifolds $M_0$ whose large scale geometry is described by a Lie algebra of vector fields $\mathcal V \subset \Gamma(M;TM)$ on a compactification of $M_0$ to a manifold with corners $M$. The geometry of these manifolds--called ``manifolds with a Lie structure at infinity''--was studied from an axiomatic point of view in a previous paper of ours. In this paper, we define and study an algebra $\Psi_{1,0,\mathcal V}^\infty(M_0)$ of pseudodifferential operators canonically associated to a manifold $M_0$ with a Lie structure at infinity $\mathcal V \subset \Gamma(M;TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to the algebras that we introduce. In particular, the algebra $\Psi_{1,0,\mathcal V}^\infty(M_0)$ is a ``microlocalization'' of the algebra ${\rm Diff}^{*}_{\mathcal V}(M)$ of differential operators with smooth coefficients on $M$ generated by $\mathcal V$ and $\mathcal{C}^\infty(M)$. This proves a conjecture of Melrose (see his ICM 90 proceedings paper).

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Additional Information

Bernd Ammann
Affiliation: Universität Hamburg, Fachbereich 11–Mathematik, Bundesstrasse 55, D-20146 Hamburg, Germany

Robert Lauter
Affiliation: Universität Mainz, Fachbereich 17–Mathematik, D-55099 Mainz, Germany

Victor Nistor
Affiliation: Mathematics Department, Pennsylvania State University, University Park, PA 16802

Keywords: Differential operator, pseudodifferential operator, principal symbol, conormal distribution, Riemannian manifold, Lie algebra, exponential map
Received by editor(s): April 24, 2003
Published electronically: September 15, 2003
Additional Notes: Ammann was partially supported by the European Contract Human Potential Program, Research Training Networks HPRN-CT-2000-00101 and HPRN-CT-1999-00118; Nistor was partially supported by NSF Grants DMS 99-1981 and DMS 02-00808. Manuscripts available from
Communicated by: Michael E. Taylor
Article copyright: © Copyright 2003 American Mathematical Society

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