Skip to main content
SpringerLink
Log in

Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

We consider Littlewood-Paley functions associated with a non-isotropic dilation group on ℝn. We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted Lp spaces, 1 < p < ∞, with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. A. P. Calderón: Inequalities for the maximal function relative to a metric. Stud. Math. 57 (1976), 297–306.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. P. Calderón, A. Torchinsky: Parabolic maximal functions associated with a distribution. Adv. Math. 16 (1975), 1–64.

    Article  MathSciNet  MATH  Google Scholar 

  4. O. N. Capri: On an inequality in the theory of parabolic H p spaces. Rev. Unión Mat. Argent. 32 (1985), 17–28.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. J. Duoandikoetxea: Weighted norm inequalities for homogeneous singular integrals. Trans. Am. Math. Soc. 336 (1993), 869–880.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Duoandikoetxea, E. Seijo: Weighted inequalities for rough square functions through extrapolation. Stud. Math. 149 (2002), 239–252.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Garcia-Cuerva, J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116, Notas de Matemática 104, North-Holland, Amsterdam, 1985.

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Riviere: Singular integrals and multiplier operators. Ark. Mat. 9 (1971), 243–278.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. L. Rubio de Francia: Factorization theory and A p weights. Am. J. Math. 106 (1984), 533–547.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Sato: Estimates for Littlewood-Paley functions and extrapolation. Integral Equations Oper. Theory 62 (2008), 429–440.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Sato: Estimates for singular integrals along surfaces of revolution. J. Aust. Math. Soc. 86 (2009), 413–430.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Sato: Littlewood-Paley equivalence and homogeneous Fourier multipliers. Integral Equations Oper. Theory 87 (2017), 15–44.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton University Press, Princeton, 1970.

    MATH  Google Scholar 

  21. E. M. Stein, S. Wainger: Problems in harmonic analysis related to curvature. Bull. Am. Math. Soc. 84 (1978), 1239–1295.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Kanazawa University, Kanazawa, 920-1192, Japan

    Shuichi Sato

Authors
  1. Shuichi Sato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuichi Sato.

Additional information

The author is partly supported by Grant-in-Aid for Scientific Research (C) No. 25400130, Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sato, S. Boundedness of Littlewood-Paley operators relative to non-isotropic dilations. Czech Math J 69, 337–351 (2019). https://doi.org/10.21136/CMJ.2018.0313-17

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2018.0313-17

Keywords

MSC 2010

Search

Navigation