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On a class of nonlocal elliptic problems. (English. Russian original) Zbl 0798.58074
Russ. Acad. Sci., Dokl., Math. 45, No. 2, 317-321 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 323, No. 2, 245-249 (1992).
The authors define a class of nonlocal elliptic problems, motivated by several examples, consider sets $$\mathbb{A}(m,l,r)$$ of operators of the form $${\mathcal A} = \widehat{A} + \widehat{\phi}_{11} + \widehat{\phi}_{12} + \widehat{\phi}_{21} + \widehat{\phi}_{22}$$, $$\widehat{A}$$ a $$\psi$$ DO of order $$r$$ and the $$\widehat{\phi}_{ij}$$ certain Fourier integral operators. An element $${\mathcal A} \in {\mathbb{A}}(m,l,r)$$ is said to be elliptic if $$\widehat{A}$$ and $$1 + \widehat{A}^{-1} \circ \widehat{\phi}_{11} + \widehat{A}^{-1} \circ \widehat{\phi}_{12} + \widehat{A}^{-1} \circ \widehat{\phi}_{21} + \widehat{A}^{-1} \circ \widehat{\phi}_{22}$$ is elliptic. The authors assert that under certain additional conditions an elliptic operator is Fredholm. Proofs are not given but several examples.
##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 58J05 Elliptic equations on manifolds, general theory 35J45 Systems of elliptic equations, general (MSC2000)