In several inuential works, Melrose has studied examples of noncompact manifolds M0 whose large scale geometry is described by a Lie algebra of vector fields ν ⊂ 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 (Formula Presented) 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 (Formula Presented) is a "microlocalization" of the algebra (Formula Presented) 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).
|Original language||English (US)|
|Number of pages||8|
|Journal||Electronic Research Announcements of the American Mathematical Society|
|State||Published - Sep 15 2003|
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