### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Electronic Research Announcements of the American Mathematical Society*,

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

}

*Electronic Research Announcements of the American Mathematical Society*, vol. 9, no. 10, pp. 80-87. https://doi.org/10.1090/S1079-6762-03-00114-8

**Algebras of pseudodifferential operators on complete manifolds.** / Ammann, Bernd; Lauter, Robert; Nistor, Victor.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Algebras of pseudodifferential operators on complete manifolds

AU - Ammann, Bernd

AU - Lauter, Robert

AU - Nistor, Victor

PY - 2003/9/15

Y1 - 2003/9/15

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=2942655482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2942655482&partnerID=8YFLogxK

U2 - 10.1090/S1079-6762-03-00114-8

DO - 10.1090/S1079-6762-03-00114-8

M3 - Article

AN - SCOPUS:2942655482

VL - 9

SP - 80

EP - 87

JO - Electronic Research Announcements in Mathematical Sciences

JF - Electronic Research Announcements in Mathematical Sciences

SN - 1935-9179

IS - 10

ER -