Algebras of pseudodifferential operators on complete manifolds

Ammann, Bernd¹ ; Lauter, Robert² ; Nistor, Victor³

¹ Universität Hamburg, Fachbereich 11–Mathematik, Bundesstrasse 55, D-20146 Hamburg, Germany
² Universität Mainz, Fachbereich 17–Mathematik, D-55099 Mainz, Germany
³ Mathematics Department, Pennsylvania State University, University Park, PA 16802

Electronic research announcements of the American Mathematical Society, Tome 09 (2003), pp. 80-87.

Résumé

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 $\textrm {Diff}^{*}_{\mathcal V}(M)$ 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).

Détail

DOI : 10.1090/S1079-6762-03-00114-8

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