Skip to main content
Log in

Littlewood–Paley Functions Under Sharp Kernel Conditions

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

A Correction to this article was published on 26 February 2024

This article has been updated

Abstract

We prove \(L^p\)-estimates for Littlewood–Paley functions under sharp kernel conditions without assuming compactness of support by applying extrapolation arguments.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

Change history

References

  1. Al-Salman, A., Al-Qassem, H., Cheng, L.C., Pan, Y.: \(L^p\) bounds for the function of Marcinkiewicz. Math. Res. Lett. 9, 697–700 (2002)

    Article  MathSciNet  Google Scholar 

  2. Al-Salman, A., Pan, Y.: Singular integrals with rough kernels in \(L\log L(S^{n-1})\). J. London Math. Soc. (2) 66, 153–174 (2002)

    Article  MathSciNet  Google Scholar 

  3. Benedek, A., Calderón, A.P., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Nat. Acad. Sci. USA 48, 356–365 (1962)

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  4. Cheng, L.C.: On Littlewood-Paley functions. Proc. Am. Math. Soc. 135, 3241–3247 (2007)

    Article  MathSciNet  Google Scholar 

  5. Coifman, R.R., Meyer, Y.: Au delà des opérateurs pseudo-différentiels, Astérisque no. 57, Soc. Math. France (1978)

  6. Ding, Y., Sato, S.: Littlewood-Paley functions on homogeneous groups. Forum Math. 28, 43–55 (2016)

    Article  MathSciNet  Google Scholar 

  7. Duoandikoetxea, J.: Sharp \(L^p\) boundedness for a class of square functions. Rev. Mat. Comput. 26, 535–548 (2013)

    Article  Google Scholar 

  8. Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541–561 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  9. Fan, D., Sato, S.: Remarks on Littlewood-Paley functions and singular integrals. J. Math. Soc. Japan 54, 565–585 (2002)

    Article  MathSciNet  Google Scholar 

  10. Hörmander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104, 93–140 (1960)

    Article  MathSciNet  Google Scholar 

  11. Marcinkiewicz, J.: Sur quelques intégrales du type de Dini. Annales de la Société Polonaise 17, 42–50 (1938)

    Google Scholar 

  12. Sato, S.: Remarks on square functions in the Littlewood-Paley theory. Bull. Austral. Math. Soc. 58, 199–211 (1998)

    Article  MathSciNet  Google Scholar 

  13. Sato, S.: Estimates for Littlewood-Paley functions and extrapolation. Integr. Equ. Oper. Theory 62, 429–440 (2008)

    Article  MathSciNet  Google Scholar 

  14. Sato, S.: Estimates for singular integrals and extrapolation. Stud. Math. 192, 219–233 (2009)

    Article  MathSciNet  Google Scholar 

  15. Sato, S.: Boundedness of Littlewood-Paley operators relative to non-isotropic dilations. Czech. Math. J. 69, 337–351 (2019)

    Article  MathSciNet  Google Scholar 

  16. Sato, S.: Sobolev spaces with non-isotropic dilations and square functions of Marcinkiewicz type. Stud. Math. 267, 295–320 (2022)

    Article  MathSciNet  Google Scholar 

  17. Sato, S.: Sobolev spaces and functions of Marcinkiewicz type with repeated averaging operations over spheres. Partial Differ. Equ. Appl. 3, 66 (2022). https://doi.org/10.1007/s42985-022-00203-1

    Article  MathSciNet  Google Scholar 

  18. Stein, E.M.: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans. Am. Math. Soc. 88, 430–466 (1958)

    Article  MathSciNet  Google Scholar 

  19. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    Google Scholar 

  20. Walsh, T.: On the function of Marcinkiewicz. Stud. Math. 44, 203–217 (1972)

    Article  MathSciNet  Google Scholar 

  21. Zygmund, A.: Trigonometric Series, 2nd edn. Cambridge University Press, Cambridge (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Contributions

Shuichi Sato proved \(L^p\)-estimates for Littlewood-Paley functions under sharp kernel conditions without assuming compactness of support by applying extrapolation arguments.

Corresponding author

Correspondence to Shuichi Sato.

Ethics declarations

Conflict of interest

None of the authors has competing interests related to the work described in the paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sato, S. Littlewood–Paley Functions Under Sharp Kernel Conditions. Mediterr. J. Math. 21, 33 (2024). https://doi.org/10.1007/s00009-023-02569-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02569-x

Keywords

Mathematics Subject Classification

Navigation