Skip to main content
SpringerLink
Log in

On Marcinkiewicz integrals associated to compound mappings with rough kernels

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

The L p bounds for the parametric Marcinkiewicz integrals associated to compound mappings, which contain many classical surfaces as model examples, are given, where the kernels of our operators are rather rough on the unit sphere as well as in the radial direction. These results substantially improve and extend some known results.

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Al-Salman, A.: On the L 2 boundedness of parametric Marcinkiewicz integral operator. J. Math. Anal. Appl., 375(2), 745–752 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Al-Qassem, H., Pan, Y.: On certain estimates for Marcinkiewicz integrals and extrapolation. Collect. Math., 60(2), 123–145 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Colzani, L., Taibleson, M., Weiss, G.: Maximal estimates for Cesàro and Riesz means on sphere. Indiana Univ. Math. J., 33, 873–889 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Colzani, L.: Hardy Spaces on Spheres, PhD thesis, Washington University, St. Louis, 1982

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Ding, Y., Fan, D., Pan, Y.: L p-boundedness of Marcinkiewicz integrals with Hardy space function kernel. Acta Math. Sin., Engl. Series, 16, 593–600 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ding, Y., Fan, D., Pan, Y.: On the L p boundedness of Marcinkiewicz integrals. Michigan Math. J., 50, 17–26 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ding, Y., Lu, S. Z., Yabuta, K.: A problem on rough parametric Marcinkiewicz functions. J. Aust. Math. Soc., 72, 13–21 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ding, Y., Xue, Q., Yabuta, K.: Boundedness of the Marcinkiewicz integrals with rough kernel associated to surfaces. Tohoku Math. J., 62, 233–262 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ding, Y., Xue, Q., Yabuta, K.: A remark to the L 2 boundedness of parametric Marcinkiewicz integral. J. Math. Anal. Appl., 387, 691–697 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fan, D., Pan, Y.: Singular integral operators with rough kernels supported by subvarieties. Amer. J. Math., 119, 799–839 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, F., Wu, H.: Multiple singular integrals and Marcinkiewicz integrals with mixed homogeneity along surfaces. J. Inequal. Appl., 2012(189), 1–23 (2012)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Stein, E. M.: Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integral, Princeton University Press, Princeton, N. J., 1993

    Google Scholar 

  19. Sakamoto, M., Yabuta, K.: Boundedness of Marcinkiewicz functions. Studia Math., 135, 103–142 (1999)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China

    Feng Liu & Huo Xiong Wu

Authors
  1. Feng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huo Xiong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huo Xiong Wu.

Additional information

Supported by National Natural Science Foundation of China (Grant Nos. 11071200, 11371295) and NSF of Fujian Province of China (Grant No. 2010J01013)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, F., Wu, H.X. On Marcinkiewicz integrals associated to compound mappings with rough kernels. Acta. Math. Sin.-English Ser. 30, 1210–1230 (2014). https://doi.org/10.1007/s10114-014-3072-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-014-3072-0

Keywords

MR(2010) Subject Classification

Search

Navigation