TAILIEUCHUNG - Đề tài "Pseudodifferential operators on manifolds with a Lie structure at infinity "

We define and study an algebra Ψ∞ (M0 ) of pseudodifferential opera1,0,V tors canonically associated to a noncompact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M0 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∞ (M0 ). | Annals of Mathematics Pseudodifferential operators on manifolds with a Lie structure at infinity By Bernd Ammann Robert Lauter and Victor Nistor Annals of Mathematics 165 2007 717 747 Pseudodifferential operators on manifolds with a Lie structure at infinity By Bernd Ammann Robert Lauter and Victor Nistor Abstract We define and study an algebra T 0 V M0 of pseudodifferential operators canonically associated to a noncompact Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M0 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 T 0 V Mo . We also consider the algebra DiffV M0 of differential operators on M0 generated by V and C M and show that 1 O V M0 is a microlocalization of DiffV M0 . Our construction solves a problem posed by Melrose in 1990. Finally we introduce and study semi-classical and suspended versions of the algebra T 0 V M0 . Contents Introduction 1. Manifolds with a Lie structure at infinity 2. Kohn-Nirenberg quantization and pseudodifferential operators 3. The product 4. Properties of T 0 V M0 5. Group actions and semi-classical limits References Introduction Let M0 g0 be a complete noncompact Riemannian manifold. It is a fundamental problem to study the geometric operators on M0. As in the compact case pseudodifferential operators provide a powerful tool for that purpose provided that the geometry at infinity is taken into account. One needs however to restrict to suitable classes of noncompact manifolds. Ammann was partially supported by the European Contract Human Potential Program Research Training Networks HPRN-CT-2000-00101 and HPRN-CT-1999-00118 Nistor was partially supported by the NSF Grants DMS-9971951 and DMS-0200808. 718 B. AMMANN R. LAUTER AND V. NISTOR Let M be a compact manifold with corners such that Mo M dM and assume that the geometry at infinity of Mo is .

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