Skip to main content
SpringerLink
Log in

On the boundedness of periodic pseudo-differential operators

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper we investigate the \(L^p\)-boundedness of certain classes of periodic pseudo-differential operators. The operators considered arise from the study of symbols on \({\mathbb {T}}^n\times {\mathbb {Z}}^n\) with limited regularity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agranovich, M.S.: Spectral properties of elliptic pseudodifferential operators on a closed curve. Funct. Anal. Appl. 13, 279–281 (1971)

    Article  MATH  Google Scholar 

  2. Ashino, R., Nagase, M., Vaillancourt, R.: Pseudodifferential operators on \(L^p(\mathbb{R}^n)\) spaces. Cubo 6(3), 91–129 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Calderón, A., Vaillancourt, R.: On the boundedness of pseudo-differential operators. J. Math. Soc. Jpn. 23, 374–378 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calderón, A., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Natl. Acad. Sci. USA 69, 1185–1187 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cardona, D.: Weak type (1, 1) bounds for a class of periodic pseudo-differential operators. J. Pseudodiffer. Oper. Appl. 5(4), 507–515 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cardona, D.: Hölder-Besov boundedness for periodic pseudo-differential operators. J. Pseudodiffer. Oper. Appl. 8(1), 13–34 (2017). doi:10.1007/s11868-016-0174-8

    Article  MATH  Google Scholar 

  7. Cardona, D.: Besov continuity for multipliers defined on compact Lie groups. Palest. J. Math. 5(2), 35–44 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Delgado, J.: \(L^p\) bounds for Pseudo-differential operators on the torus. Oper. Theory Adv. Appl. 231, 103–116 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Delgado, J. Ruzhansky, M.: \(L^p\)-bounds for Pseudo-differential operators on compact Lie groups. arXiv:1605.07027

  10. Duoandikoetxea, J.: Fourier Analysis. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  11. Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  12. Fefferman, C.: \(L^p-\)bounds for pseudo-differential operators. Isr. J. Math. 14, 413–417 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fischer, V.: Hörmander condition for Fourier multipliers on compact Lie groups. arXiv:1610.06348

  14. Ghaemi, M.B., Nabizadeh, M.E.: A study on the inverse of pseudo-differential operators on \({\mathbb{S}}^{1}\). J. Pseudodiffer. Oper. Appl. 7(4), 511–517 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghaemi, M.B., Jamalpour, B.M., Nabizadeh, M.E.: A study on pseudo-differential operators on \({\mathbb{S}}^{1}\) and \({\mathbb{Z}}\). J. Pseudodiffer. Oper. Appl. 7(2), 237–247 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghaemi, M.B., Nabizadeh, M.E., Jamalpour, B.M.: A study on the adjoint of pseudo-differential operators on \({\mathbb{S}}^{1}\) and \({\mathbb{Z}}\). J. Pseudodiffer. Oper. Appl. 6(2), 197203 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Hörmander, L.: Pseudo-differential Operators and Hypo-elliptic equations. In: Proceedings of the Symposium on Singular Integrals, vol. 10, pp. 138–183. American Mathematical Society (1967)

  18. Hörmander, L.: The Analysis of the Linear Partial Differential Operators, vol. III. Springer, Berlin (1985)

    MATH  Google Scholar 

  19. McLean, W.M.: Local and global description of periodic pseudo-differential operators. Math. Nachr. 150, 151–161 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Molahajloo, S., Wong, M.W.: Pseudo-differential operators on \({\mathbb{S}}^{1}\). In: Rodino, L., Wong, M.W. (eds) New Developments in Pseudo-differential Operators, pp. 297–306. (2008)

  21. Molahajloo, S., Wong, M.W.: Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on \({\mathbb{S}}^{1}\). J. Pseudodiffer. Oper. Appl. 1, 183–205 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nagase, M.: On some classes of \(L^p\)-bounded pseudodifferential operators. Osaka J. Math. 23(2), 425440 (1986)

    MathSciNet  Google Scholar 

  23. Ruzhansky, M., Turunen, V.: Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics. Birkhaüser-Verlag, Basel (2010)

    Book  MATH  Google Scholar 

  24. Ruzhansky, M., Turunen, V.: Quantization of Pseudo-Differential Operators on the torus. J. Fourier Anal. Appl. 16, 943–982 (2010). (Birkhäuser Verlag, Basel)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ruzhansky, M., Wirth, J.: \(L^p\) Fourier multipliers on compact Lie groups. Math. Z. 280, 621–642 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stein, E.: Harmonic Analysis. Princeton University Press, Princeton (1993)

    Google Scholar 

  27. Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  28. Wang, L.: Pseudo-differential Operators with Rough Coefficients. Thesis (Ph.D.) McMaster University (Canada). ProQuest LLC, Ann Arbor, MI (1997). ISBN:978–0612–30120–7

  29. Wong, M.W.: An Introduction to Pseudo-Differential Operators, 2nd edn. World Scientific, Singapore (1999)

    Book  MATH  Google Scholar 

  30. Wong, M.W.: Discrete Fourier Analysis. Birkhäuser, Basel (2011)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank the anonymous referee for his/her remarks which helped to improve the manuscript. This project was partially supported by Universidad de Los Andes, Mathematics Department, Bogotá-Colombia.

Author information

Authors and Affiliations

  1. Department of Mathematics, Universidad de Los Andes, Bogotá, Colombia

    Duván Cardona

Authors
  1. Duván Cardona
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Duván Cardona.

Additional information

Communicated by A. Constantin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cardona, D. On the boundedness of periodic pseudo-differential operators. Monatsh Math 185, 189–206 (2018). https://doi.org/10.1007/s00605-017-1029-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-017-1029-y

Keywords

Mathematics Subject Classification

Search

Navigation