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Fourier-Maslov integral operators associated with a bundle, and an extension of the algebra of pseudodifferential operators. (English. Russian original) Zbl 0798.58075
Russ. Acad. Sci., Dokl., Math. 45, No. 2, 252-256 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 323, No. 1, 25-27 (1992).
Let \(E \to B\) be a smooth bundle with smooth section \(i: B\to E\), \(E\), \(B\) closed. The authors define submanifolds \(X_{ij} \subset E \times E\), consider their conormal bundles \(L_{ij}\) which are Lagrangian submanifolds, define symbol classes \(\text{Smbl}_{ij}(m,k,l)\), Fourier integral operators \(\widehat{\phi}_{ij} \in \text{Op}_{ij}\) and consider operators of the form \({\mathcal A} = \widehat{A} + \widehat{\phi}_{11} + \widehat{\phi}_{12} + \widehat{\phi}_{21} + \widehat{\phi}_{22}\). Main theorem. The set of all these operators \(= \oplus \mathbb{A}(m,l,r)\) is an algebra with involution and an extension of the algebra of pseudodifferential operators.
MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J05 Elliptic equations on manifolds, general theory
35J45 Systems of elliptic equations, general (MSC2000)
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