Advertisement

Besov continuity for pseudo-differential operators on compact homogeneous manifolds

  • Duván Cardona
Article
  • 61 Downloads

Abstract

In this paper we study the Besov continuity of pseudo-differential operators on compact homogeneous manifolds \(M=G/K\). We use the global quantization of these operators in terms of the representation theory of compact homogeneous manifolds.

Keywords

Besov space Compact homogeneous manifold Pseudo-differential operators Global analysis 

Mathematics Subject Classification

Primary 19K56 Secondary 58J20 43A65 

Notes

Acknowledgements

The author is indebted with Alexander Cardona for helpful comments on an earlier draft of this paper. This project was supported by Faculty of Sciences of Universidad de los Andes, Proyecto: Una clase de operadores pseudo-diferenciales en espacios de Besov. 2016-1, Periodo intersemestral.

References

  1. 1.
    Akylzhanov, R., Nursultanov, E., Ruzhansky, M.: Hardy–Littlewood–Paley type inequalities on compact Lie groups. Math. Notes 100, 287–290 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Akylzhanov, R., Ruzhansky, M.: Fourier multipliers and group von Neumann algebras. C. R. Math. Acad. Sci. Paris 354, 766–770 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Akylzhanov, R., Nursultanov, E., Ruzhansky, M.: Hardy–Littlewood inequalities and Fourier multipliers on SU(2). Stud. Math. 234, 1–29 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourdaud, G.: \(L^p\)-estimates for certain non-regular pseudo-differential operators. Commun. Partial Differ. Eqa. 7, 1023–1033 (1982)CrossRefGoogle Scholar
  5. 5.
    Cardona, D.: Besov continuity for Multipliers defined on compact Lie groups. Palest. J. Math. 5(2), 35–44 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cardona, D.: Besov continuity of pseudo-differential operators on compact Lie groups revisited. C. R. Math. Acad. Sci. Paris 355(5), 533–537 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cardona, D.: Nuclear pseudo-differential operators in Besov spaces on compact Lie groups. J. Fourier Anal. Appl. (2017). doi: 10.1007/s00041-016-9512-8 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cardona, D.: Multipliers for Besov spaces on graded Lie groups. C. R. Math. Acad. Sci. Paris. 355(4), 400–405 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cohen, A.: Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam (2003)zbMATHGoogle Scholar
  10. 10.
    Delgado, J., Ruzhansky, M.: \(L^p\)-nuclearity, traces, and Grothendieck–Lidskii formula on compact Lie groups. J. Math. Pures Appl. (9) 102(1), 153–172 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dasgupta, A., Ruzhansky, M.: The Gohberg Lemma, compactness, and essential spectrum of operators on compact Lie groups. J. Anal. Math. 128, 179–190 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Delgado, J., Ruzhansky, M.: Schatten classes on compact manifolds: Kernel conditions. J. Funct. Anal. 267(3), 772–798 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Delgado, J., Ruzhansky, M.: Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds. C. R. Acad. Sci. Paris. Ser. I. 352, 779–784 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Delgado, J., Ruzhansky, M.: \(L^{p}\)-bounds for pseudo-differential operators on compact Lie groups. J. Inst. Math. Jussieu, to appear, arXiv:1605.07027, doi: 10.1017/S1474748017000123
  15. 15.
    Ebert, S., Wirth, J.: Diffusive wavelets on groups and homogeneous spaces. Proc. R. Soc. Edinburgh Sect. A 141(3), 497520 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Feichtinger, H., Führ, H., Pesenson, I.: Geometric space-frequency analysis on manifolds. arXiv:1512.08668
  17. 17.
    Fischer, V.: Intrinsic pseudo-differential calculi on any compact Lie group. J. Funct. Anal 268(11), 34043477 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gibbons, G.: Operateurs pseudo-differentiels et espaces de Besov. C. R. Acad. Sci. Paris, Serie A 286, 895–897 (1978)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hairer, M., Labbé, C.: The reconstruction theorem in Besov spaces. arXiv:1609.04543
  20. 20.
    Hörmander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104, 93–140 (1960)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nursultanov, E., Ruzhansky, M., Tikhonov, S.: Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds. Ann. Sc. Norm. Super Pisa Cl. Sci. XVI, 981–1017 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Nursultanov, E., Ruzhansky, M., Tikhonov, S.: Nikolskii inequality and functional classes on compact Lie groups. Funct. Anal. Appl. 49, 226–229 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Park, B.: On the boundedness of Pseudo-differential operators on Triebel-Lizorkin and Besov spaces. arXiv:1602.08811
  24. 24.
    Ruzhansky, M., Turunen, V.: Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics. Birkhäuser-Verlag, Basel (2010)CrossRefGoogle Scholar
  25. 25.
    Ruzhansky, M., Turunen, V.: Sharp Garding inequality on compact Lie groups. J. Funct. Anal. 260, 2881–2901 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ruzhansky, M., Turunen, V.: Global quantization of pseudo-differential operators on compact Lie groups. Int. Math. Res. Not. 2013(11), 2439–2496 (2012)CrossRefGoogle Scholar
  27. 27.
    Ruzhansky, M., Turunen, V., Wirth, J.: Hormander class of pseudo-differential operators on compact Lie groups and global hypoellipticity. J. Fourier Anal. Appl. 20, 476–499 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ruzhansky, M., Wirth, J.: Global functional calculus for operators on compact Lie groups. J. Funct. Anal. 267(1), 144–172 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ruzhansky, M., Wirth, J.: \(L^p\) Fourier multipliers on compact Lie groups. Math. Z. 280, 621–642 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsPontificia Universidad JaverianaBogotáColombia

Personalised recommendations