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Differential Equations

, Volume 54, Issue 4, pp 468–475 | Cite as

Traces of G-Operators Concentrated on Submanifolds

  • T. I. Dang
Partial Differential Equations
  • 16 Downloads

Abstract

We consider the traces on submanifolds of G-operators generated by pseudodifferential operators and operators of shift along the orbits of a discrete group G. Such operators arise in various problems in differential equations and mathematical physics, for example, in Sobolev problems. We show that the trace of a G-operator on a submanifold is the sum of a pseudodifferential operator on the submanifold and a G-operator concentrated on a sub-submanifold.

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References

  1. 1.
    Novikov, S.P. and Sternin, B.Yu., Traces of elliptic operators on submanifolds and the K-theory, Sov. Math. Dokl., 1966, vol. 7, pp. 1373–1376.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Novikov, S.P. and Sternin, B.Yu., Elliptic operators and submanifolds, Sov. Math. Dokl., 1966, vol. 7, pp. 1508–1512.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Sternin, B.Yu., Elliptic and parabolic problems on manifolds with boundary consisting of components of different dimensions, Tr. Mosk. Mat. Obs., 1966, vol. 15, pp. 346–382.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Antonevich, A.B., Lineinye funktsional’nye uravneniya: Operatornyi podkhod (Linear Functional Equations: Operator Approach), Minsk: Universitetskoe, 1988.Google Scholar
  5. 5.
    Connes, A., C* algèbres et géométrie différentielle, C. R. Seances Acad. Sci. Ser. A, 1980, vol. 290, no. 13, pp. 599–604.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Connes, A., Noncommutative Geometry, San Diego: Academic, 1994.zbMATHGoogle Scholar
  7. 7.
    Kordyukov, Yu.A., Index theory and noncommutative geometry on manifolds with foliation, Russ. Math. Surveys, 2009, vol. 64, no. 2, pp. 273–391.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Nazaikinskii, V.E., Savin, A.Yu., and Sternin, B.Yu., Elliptic theory and noncommutative geometry, in Operator Theory: Advances and Applications, Basel: Birkhäuser, 2008, Vol.183.Google Scholar
  9. 9.
    Savin, A. and Sternin, B., On Traces of Operators Associated with Compact Lie Group Actions, 2015. arXiv: 1612.02286.Google Scholar
  10. 10.
    Loshchenova, D.A., Sobolev problems associated with Lie group actions, Differ. Equations, 2015, vol. 51, no. 8, pp. 1051–1064.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Antonevich, A., Belousov, M., and Lebedev, A., Functional Differential Equations. II. C*-Applications, Parts 1, 2, Numbers 94, 95, in Pitman Monographs and Surveys in Pure and Applied Mathematics, Harlow: Longman, 1998.zbMATHGoogle Scholar
  12. 12.
    Savin, A. and Sternin, B., Elliptic theory for operators associated with diffeomorphisms of smooth manifolds, in Pseudo-Differential Operators, Generalized Functions and Asymptotics. Operator Theory: Advances and Applications, 2013, vol. 231, pp. 1–26.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Savin, A.Yu. and Sternin, B.Yu., Uniformization of nonlocal elliptic operators and KK-theory, Russ. J. Math. Phys., 2013, vol. 20, no. 3, pp. 345–359.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Savin, A.Yu. and Sternin, B.Yu., Index of elliptic operators for diffeomorphisms of manifolds, J. Noncommutative Geom., 2014, vol. 8, no. 3, pp. 695–734.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Luke, G., Pseudodifferential operators on Hilbert bundles, J. Differential Equations, 1972, vol. 12, pp. 566–589.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nazaikinskii, V., Savin, A., Schulze, B.-W., and Sternin, B., Elliptic Theory on Singular Manifolds, Boca Raton: CRC, 2005.CrossRefzbMATHGoogle Scholar
  17. 17.
    Shubin, M.A., Psevdodifferentsial’nye operatory i spektral’naya teoriya (Pseudodifferential operators and Spectral Theory), Moscow: Nauka, 1978.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Peoples’ Friendship University of RussiaMoscowRussia

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