Algebras of Pseudodifferential Operators on Complete Manifolds

In several influential works, Melrose has studied examples of noncompact manifolds M0 whose large scale geometry is described by a Lie algebra of vector fields V ⊂ Γ(M ;TM) on a compactification of M0 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 Ψ1,0,V (M0) of pseudodifferential operators canonically associated to a manifold M0 with a Lie structure at infinity V ⊂ Γ(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 Ψ1,0,V (M0) is a “microlocalization” of the algebra DiffV(M) of differential operators with smooth coefficients onM generated by V and C∞(M). This proves a conjecture of Melrose (see his ICM 90 proceedings paper).