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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)