## Abstract

We define and study an algebra ψ_{1,0,ν} ^{∞}(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 ν 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 ψ_{1,0,ν} ^{∞}(M_{0}). We also consider the algebra Diff _{ν}*(M_{0}) of differential operators on M_{0} generated by ν and C^{∞} (M), and show that ψ_{1,0,ν}^{∞}(M_{0}) is a microlocalization of Diff_{ν}*(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 ψ_{1,0,ν} ^{∞}(M_{0}).

Original language | English (US) |
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Pages (from-to) | 717-747 |

Number of pages | 31 |

Journal | Annals of Mathematics |

Volume | 165 |

Issue number | 3 |

DOIs | |

State | Published - May 2007 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty