Algebras of pseudodifferential operators on complete manifolds

Abstract

In several influential works, Melrose has studied examples of non-compact manifolds M 0 M_0 whose large scale geometry is described by a Lie algebra of vector fields V ⊂ Γ ( M ; T M ) \mathcal V \subset \Gamma (M;TM) on a compactification of M 0 M_0 to a manifold with corners M 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 Ψ 1 , 0 , V ∞ ( M 0 ) \Psi _{1,0,\mathcal V}^\infty (M_0) of pseudodifferential operators canonically associated to a manifold M 0 M_0 with a Lie structure at infinity V ⊂ Γ ( M ; T M ) \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 Ψ 1 , 0 , V ∞ ( M 0 ) \Psi _{1,0,\mathcal V}^\infty (M_0) is a “microlocalization” of the algebra Diff V ∗ ( M ) \textrm {Diff}^{*}_{\mathcal V}(M) of differential operators with smooth coefficients on M M generated by V \mathcal V and C ∞ ( M ) \mathcal {C}^\infty (M) . This proves a conjecture of Melrose (see his ICM 90 proceedings paper).