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Algebras of pseudodifferential operators on complete manifolds
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Algebras of pseudodifferential operators on complete manifolds
Bernd Ammann, Robert Lauter, and Victor Nistor
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 {\em
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).
Copyright 2003 American Mathematical Society
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Article Info
- ERA Amer. Math. Soc. 09 (2003), pp. 80-87
- Publisher Identifier: S 1079-6762(03)00114-8
- 2000 Mathematics Subject Classification. Primary 58J40; Secondary 58H05, 65R20
- Key words and phrases. Differential operator, pseudodifferential operator, principal symbol, conormal distribution, Riemannian manifold, Lie algebra, exponential map
- Received by editors April 24, 2003
- Posted on September 15, 2003
- Communicated by Michael E. Taylor
- Comments (When Available)
Bernd Ammann
Universität Hamburg, Fachbereich 11--Mathematik, Bundesstrasse 55, D-20146 Hamburg, Germany
E-mail address: ammann@berndammann.de
Robert Lauter
Universität Mainz, Fachbereich 17--Mathematik, D-55099 Mainz, Germany
E-mail address: lauter@mathematik.uni-mainz.de, lauterr@web.de
Victor Nistor
Mathematics Department, Pennsylvania State University, University Park, PA 16802
E-mail address: nistor@math.psu.edu
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/.
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