Pseudodifferential operators on manifolds with a Lie structure at infinity

Bernd Ammann¹ ; Robert Lauter² ; Victor Nistor³

¹L'Institut Élie Cartan, Université Henri Poincaré , 54506 Vandoeuvre-lès-Nancy, France
²Institut für Mathematik, Universität Mainz, 55099 Mainz, Germany
³Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States

Annals of mathematics, Tome 165 (2007) no. 3, pp. 717-747

Résumé

We define and study an algebra $\Psi_{1,0,\mathcal{V}}^\infty(M_0)$ of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold $M_0$ whose geometry at infinity is described by a Lie algebra of vector fields $\mathcal{V}$ on a compactification $M$ of $M_0$ to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to $\Psi_{1,0,\mathcal{V}}^\infty(M_0)$. We also consider the algebra $Diff^{*}_{\mathcal{v}}(M_0)$ of differential operators on $M_0$ generated by $\mathcal{V}$ and $\mathcal{C}^{\infty}(M)$, and show that $\Psi_{1,0,\mathcal{V}}^\infty(M_0)$ is a microlocalization of $Diff^{*}_{\mathcal{V}}(M_0)$. Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and “suspended” versions of the algebra $\Psi_{1,0,\mathcal{V}}^\infty(M_0)$.

Détail

DOI : 10.4007/annals.2007.165.717