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 Mat. Sb. (N.S.), 1974, Volume 95(137), Number 4(12), Pages 560–587 (Mi msb3769)

Differential and pseudodifferential operators in spaces of almost periodic functions

M. A. Shubin

Abstract: We construct a Sobolev-type space of almost periodic functions, in which we study differential and pseudodifferential operators. We show that the usual theorem on the regularity of solutions of elliptic equations is not true in these spaces, and we present modified variants of this theorem. We also study the question of the defect indices of differential operators in the space of Besicovitch almost periodic functions, and prove a Liouville-type theorem in this space.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Sbornik, 1974, 24:4, 547–573

Bibliographic databases:

UDC: 517.43
MSC: Primary 35J30, 35D05, 35S05, 47F05, 47G05, 42A84; Secondary 46E35

Citation: M. A. Shubin, “Differential and pseudodifferential operators in spaces of almost periodic functions”, Mat. Sb. (N.S.), 95(137):4(12) (1974), 560–587; Math. USSR-Sb., 24:4 (1974), 547–573

Citation in format AMSBIB
\Bibitem{Shu74} \by M.~A.~Shubin \paper Differential and pseudodifferential operators in spaces of almost periodic functions \jour Mat. Sb. (N.S.) \yr 1974 \vol 95(137) \issue 4(12) \pages 560--587 \mathnet{http://mi.mathnet.ru/msb3769} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=358445} \zmath{https://zbmath.org/?q=an:0311.47022} \transl \jour Math. USSR-Sb. \yr 1974 \vol 24 \issue 4 \pages 547--573 \crossref{https://doi.org/10.1070/SM1974v024n04ABEH001923} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. A. Shubin, “Almost-periodic elliptic operators and von Neumann algebras”, Funct. Anal. Appl., 9:1 (1975), 88–89
2. V. I. Bezyaev, “Asymptotics of the density of states of hypoelliptic almost periodic operators”, Math. USSR-Sb., 34:4 (1978), 425–447
3. B. V. Fedosov, M. A. Shubin, “The index of random elliptic operators. II”, Math. USSR-Sb., 35:1 (1979), 131–156
4. M. A. Shubin, “Almost periodic functions and partial differential operators”, Russian Math. Surveys, 33:2 (1978), 1–52
5. M. A. Shubin, “The spectral theory and the index of elliptic operators with almost periodic coefficients”, Russian Math. Surveys, 34:2 (1979), 109–157
6. A. S. Mishchenko, “Banach algebras, pseudodifferential operators, and their application to $K$-theory”, Russian Math. Surveys, 34:6 (1979), 77–91
7. P. E. Dedik, M. A. Shubin, “Random pseudodifferential operators and the stabilization of solutions of parabolic equations with random coefficients”, Math. USSR-Sb., 41:1 (1982), 33–52
8. Bruno Scarpellini, Pierre-A. Vuillermot, “Smooth manifolds for semilinear wave equations on 2: On the existence of almost-periodic breathers”, Journal of Differential Equations, 77:1 (1989), 123
9. Pierre-A Vuillermot, “Almost-periodic attractors for a class of nonautonomous reaction-diffusion equations on N. I. Global stabilization processes”, Journal of Differential Equations, 94:2 (1991), 228
10. Pierre-A. Vuillermot, “Almost-periodic attractors for a class of nonautonomous reaction-diffusion equations on N—III. Center curves and liapounov stability”, Nonlinear Analysis: Theory, Methods & Applications, 22:5 (1994), 533
11. Alessandro Oliaro, Luigi Rodino, Patrik Wahlberg, “Almost periodic pseudodifferential operators and Gevrey classes”, Annali di Matematica, 2011
12. Patrik Wahlberg, “Representations of almost periodic pseudodifferential operators and applications in spectral theory”, J. Pseudo-Differ. Oper. Appl, 2011
13. Desheng Ji, Chuanyi Zhang, “Pseudo almost periodic weak solutions of a semilinear elliptic equation”, Adv Diff Equ, 2014:1 (2014), 46
14. N. Bodendorfer, “Quantum reduction to Bianchi I models in loop quantum gravity”, Phys. Rev. D, 91:8 (2015)
15. Paolo Boggiatto, Carmen Fernández, Antonio Galbis, “Gabor systems and almost periodic functions”, Applied and Computational Harmonic Analysis, 2015
16. M. Braverman, V. M. Bukhshtaber, M. Gromov, V. Ivrii, Yu. A. Kordyukov, P. Kuchment, V. Mazya, S. P. Novikov, T. Sunada, L. Fridlender, A. G. Khovanskii, “Mikhail Aleksandrovich Shubin (nekrolog)”, UMN, 75:6(456) (2020), 162–170
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