Skip to main content
SpringerLink
Log in

On rough singular integrals related to homogeneous mappings

  • Published:
Collectanea Mathematica Aims and scope Submit manuscript

Abstract

In this paper, we consider the singular integrals related to homogeneous mappings as well as the corresponding maximal truncated singular integrals. Under the rather weak size conditions on the integral kernels both on the unit sphere and in the radial direction, the \(L^p\) bounds for such operators are given, which essentially improve and generalize 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-Qassem, H.M., Ali, M.: \(L^p\) boundedness for singular integral operators with \(L(\log ^+L)^2\) kernels on product spaces. Kyungpook Math. J. 46, 377–387 (2006)

    MATH  MathSciNet  Google Scholar 

  2. Al-Qassem, H.M., Ali, M.: Singular integrals related to homogeneous mapping with rough kernels on product spaces. Tamkang J. Math. 39, 165–176 (2008)

    MathSciNet  Google Scholar 

  3. Al-Qassem, H.M., Al-Salman, A., Pan, Y.: Singular integrals associated to homogeneous mappings with rough kernels. Hokkaido Math. J. 33, 551–569 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheng, L.C.: Singular ingegrals related to homogeneous mappings. Mich. Math. J. 47, 407–416 (2000)

    Article  MATH  Google Scholar 

  6. Coifman, R., Weiss, G.: Extension of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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

  8. Connett, W.C.: Singular integral near \(L^1\). Harmonic analysis in Euclidean space (Proc. Sympos. Pure Math., WilliamsColl., Williamstown, Mass., 1978), Part 1, pp. 163–165, Proc. Sympos. Pure Math. 35, Part, Amer. Math. Soc., Providence, RI (1979)

  9. Calderón, A.P., Zygmund, A.: On singular integral. Am. J. Math. 78, 289–309 (1956)

    Article  MATH  Google Scholar 

  10. Ding, Y., Xue, Q., Yabuta, Y.: On singular integral integral operators with rough kernel along surfaces. Inter. Equa. Oper. Theory 68, 151–161 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. 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 

  12. Fefferman, R.: A note on singular integrals. Proc. Am. Math. Soc. 74, 266–270 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fan, D., Guo, K., Pan, Y.: \(L^p\) estimates for singular integrals associated to homogemeous surfaces. J. Reine Angew. Math. 542, 1–22 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1983)

    MATH  Google Scholar 

  16. Liu, F.: On singular integrals associated to surfaces. Tohoku Math. J. 66(1), 1–14 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Liu, F., Wu, H.: Parabolic singular integrals with rough kernels and extrapolation (in Chinese). Sci. Sin. Math. 44, 519–534 (2014)

    Article  Google Scholar 

  18. Namazi, J.: A singular integral. Proc. Am. Math. Soc. 96, 201–219 (1986)

    Article  MathSciNet  Google Scholar 

  19. Ricci, F., Stein, E.M.: Harmonic analysis on nilpotent groups and singular integrals I: osicllatory integrals. J. Funct. Anal. 73, 179–194 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ricci, F., Weiss, G.: A claracterization of \(H^1(\sum _{n-1})\). Proc. Synposia Pure Math. I 35, 289–294 (1979)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality and oscillatory integral. Princeton Math. Ser. 43, Monographs in Harmonic Analysis, III. Princeton University Press, Princeton (1993)

Download references

Acknowledgments

The authors would like to thank the referee for his/her invaluable comments and suggestions.

Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, People’s Republic of China

    Feng Liu

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Suzhen Mao & Huoxiong Wu

Authors
  1. Feng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suzhen Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huoxiong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suzhen Mao.

Additional information

The first author is supported by Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (No. 2015RCJJ053) and Research Award Fund for Outstanding Young Scientists of Shandong Province (No. BS2015SF012). The second and third authors are supported by the NNSF of China (Nos. 11371295, 11471041) and the NSF of Fujian Province of China (No. 2015J01025).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, F., Mao, S. & Wu, H. On rough singular integrals related to homogeneous mappings. Collect. Math. 67, 113–132 (2016). https://doi.org/10.1007/s13348-015-0155-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13348-015-0155-x

Keywords

Mathematics Subject Classification

Search

Navigation