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Lagrangian distributions on asymptotically Euclidean manifolds

  • Sandro CoriascoEmail author
  • Moritz Doll
  • René Schulz
Article
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Abstract

We develop the notion of Lagrangian distribution on a scattering manifold X. The latter is a manifold with boundary, with the boundary being viewed as points “at infinity.” In analogy with the classical case, a Lagrangian distribution is associated with a submanifold \(\Lambda \) of the compactified cotangent bundle of X. The submanifold \(\Lambda \) is Lagrangian with respect to a symplectic structure induced by the scattering geometry of X. Our analysis relies on the parameterization properties of \(\Lambda \) by means of local phase functions, and the study of the maps which preserve the scattering structure. We study the principal symbol map associating Lagrangian distributions with sections of a line bundle over \(\Lambda \). In particular, we establish the principal symbol short exact sequence.

Keywords

Lagrangian distribution Lagrangian submanifold Scattering calculus \({\text {SG} }\) calculus Principal symbol 

Mathematics Subject Classification

35S30 46F05 53D12 

Notes

Acknowledgements

The second author was supported by the DFG GRK-1463. The third author has been partially supported by the University of Turin “I@UniTO” project “Fourier integral operators, symplectic geometry and analysis on noncompact manifolds” (Resp. S. Coriasco). We wish to thank J. Wunsch for useful discussions. We also wish to thank an anonymous referee for helpful suggestions, improving the overall quality of the paper.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica “G. Peano”Università degli Studi di TorinoTorinoItaly
  2. 2.Institut für AnalysisLeibniz Universität HannoverHannoverGermany

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