### Abstract

In several inuential works, Melrose has studied examples of noncompact manifolds M_{0} whose large scale geometry is described by a Lie algebra of vector fields ν ⊂ Γ(M;TM) on a compactification of M_{0} 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 M_{0} 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) |
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Pages (from-to) | 80-87 |

Number of pages | 8 |

Journal | Electronic Research Announcements of the American Mathematical Society |

Volume | 9 |

Issue number | 10 |

DOIs | |

State | Published - Sep 15 2003 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Electronic Research Announcements of the American Mathematical Society*,

*9*(10), 80-87. https://doi.org/10.1090/S1079-6762-03-00114-8