Pseudodifferential operators on manifolds with a Lie structure at infinity

Abstract

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 $\Psi_{1,0,\mathcal{V}}^\infty(M_0)$.