Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/

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

TeX source (34.4K)
DVI (42.5K)
PostScript (125.1K)

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/.