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

DOI:
https://doi.org/10.1090/S1079-6762-03-00114-8

Published electronically:
September 15, 2003

MathSciNet review:
2029468

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In several influential works, Melrose has studied examples of non-compact manifolds whose large scale geometry is described by a Lie algebra of vector fields on a *compactification* of to a manifold with corners . 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 of pseudodifferential operators canonically associated to a manifold with a Lie structure at infinity . 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 is a ``microlocalization'' of the algebra of differential operators with smooth coefficients on generated by and . 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

Email:
ammann@berndammann.de

**Robert Lauter**

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

Email:
lauter@mathematik.uni-mainz.de, lauterr@web.de

**Victor Nistor**

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

Email:
nistor@math.psu.edu

DOI:
https://doi.org/10.1090/S1079-6762-03-00114-8

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 http://www.math.psu.edu/nistor/.

Communicated by:
Michael E. Taylor

Article copyright:
© Copyright 2003
American Mathematical Society