SpringerLink
Log in

On the Commutators of Marcinkiewicz Integrals with Rough Kernels in Weighted Lebesgue Spaces

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

This paper is devoted to studying the boundedness and compactness of commutators for Marcinkiewicz integrals with rough kernels in weighted Lebesgue spaces. The characterized theorems on the boundedness and compactness for such commutators are established, respectively. We improve and extend the previous results. Meanwhile, the quantitative weighted bounds for the commutator of Marcinkiewicz integral is also obtained.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. A. Al-Salman, H. Al-Qassem, L. C. Chen and Y. Pan, Lp bounds for the function of Marcinkiewicz, Math. Res. Lett., 9 (2002), 697–700.

    MathSciNet  MATH  Google Scholar 

  2. A. Benedek, A. P. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 356–365.

    Article  MathSciNet  Google Scholar 

  3. A. Bényi, J. M. Martell, K. Moen, E. Stachura, and R. H. Torres, Boundedness results for commutators with BMO functions via weighted estimates: a comprehensive approach, ar**v:1710.08515.

  4. A. Bényi and R. H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc., 141 (2013), 3609–3621.

    Article  MathSciNet  Google Scholar 

  5. A. Clop and V. Cruz, Weighted estimates for Beltrami equations, Ann. Acad. Sci. Fenn. Math., 38 (2013), 91–113.

    Article  MathSciNet  Google Scholar 

  6. L. Chaffee, P. Chen, Y. Han, R. H. Torres and L. A. Ward, Characterization of compactness of commutators of bilinear singular integral operators, Proc. Amer. Math. Soc., 146 (2018), 3943–3953.

    Article  MathSciNet  Google Scholar 

  7. L. Chaffee and R. H. Torres, Characterization of compactness of the commutators of bilinear fractional integral operators, Potential Anal., 43 (2015), 481–494.

    Article  MathSciNet  Google Scholar 

  8. D. Chen and S. Lu, Lp boundedness for commutators of parabolic Littlewood-Paley operators with rough kernel, Math. Nachr., 284 (2011), 973–986.

    Article  MathSciNet  Google Scholar 

  9. Y. Chen and Y. Ding, Compactness characterization of commutators for Littlewood- Paley operators, Kodai Math. J., 32 (2009), 256–323.

    MathSciNet  MATH  Google Scholar 

  10. Y. Chen and Y. Ding, Commutators of Littlewood-Paley operators, Sci. China Ser. A, 52 (2009), 2493–2505.

    Article  MathSciNet  Google Scholar 

  11. Y. Chen, Y. Ding and X. Wang, Compactness of commutators of Riesz potential on Morrey spaces, Potential Anal., 30 (2009), 301–313.

    Article  MathSciNet  Google Scholar 

  12. Y. Chen, Y. Ding and X. Wang, Compactness for commutators of Marcinkiewicz integral on Morrey spaces, Taiwanese J. Math., 15 (2011), 633–658.

    MATH  Google Scholar 

  13. J. Chen and G. Hu, Compact commutators of Rough singular integral operators, Canad. Math. Bull., 58 (2015), 19–29.

    Article  MathSciNet  Google Scholar 

  14. J. Chen and G. Hu, Completely continuious commutators of Marcinkiewicz integral, Publ. Math. Debrecen., 90 (2017), 11–32.

    Article  Google Scholar 

  15. R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables. Ann. of Math. (2), 103 (1976), 611–635.

    Article  MathSciNet  Google Scholar 

  16. Y. Ding, Weighted boundedness for commutators of fractional integral operators with rough kernels, J. Bei**g Normal Univ., 32 (1996), 157–161.

    MATH  Google Scholar 

  17. Y. Ding, D. Fan and Y. Pan, Lp-boundedness of Marcinkiewicz integrals with Hardy space function kernel, Acta Math. Sin. (Engl. Ser.), 16 (2000), 593–600.

    Article  Google Scholar 

  18. Y. Ding, D. Fan and Y. Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana Univ. Math. J., 48 (1999), 1037–1055.

    MathSciNet  MATH  Google Scholar 

  19. Y. Ding and S. Lu, Weighted norm inequalities for fractional integral operator with rough kernel, Canad. J. Math., 50 (1998), 29–39.

    MathSciNet  MATH  Google Scholar 

  20. Y. Ding, S. Lu and K. Yabuta, On commutators of Marcinkiewicz integrals with rough kernel, J. Math. Anal. Appl., 275 (2002), 60–68.

    Article  MathSciNet  Google Scholar 

  21. L. Grafakos and A. Stefanov, Lp bounds for singular integrals and maximal singular integrals with rough kernels, Indiana Univ. Math. J., 47 (1998), 455–469.

    MATH  Google Scholar 

  22. W. Guo, J. He, H. Wu and D. Yang, Characterizations of the compactness of commutators associated with Lipschitz functions, ar**v:1801.06064v2.

  23. W. Guo, H. Wu and D. Yang, A revisit on the compactness of commutators, ar**v:1712.08292v1.

  24. G. Hu, Weighted complete continuity for the commutator of Marcinkiewicz integral, Ann. Acad. Sci. Fenn. Math., 44 (2019), 459–484.

    Article  MathSciNet  Google Scholar 

  25. G. Hu, M. Qu, Quantitative weighted Lp bounds for the Marcinkiewicz integral, Math. Inequal. Appl., 22 (2019), 885–899.

    MathSciNet  MATH  Google Scholar 

  26. G. Hu and D. Yan, On the commutator of the Marcinkiewicz integral, J. Math. Anal. Appl., 283 (2003), 351–361.

    Article  MathSciNet  Google Scholar 

  27. T. P. Hytönen, The Holmes-Wick theorem on two-weight bounds for higher order commutators revisited, Arch. Math. (Basel), 107 (2016), 389–395.

    Article  Google Scholar 

  28. S. G. Krantz and S.-Y. Li, Boundedness and compactness of integral operators on spaces of homogeneous type and applications. II, J. Math. Anal. Appl., 258 (2001), 642–657.

    MathSciNet  MATH  Google Scholar 

  29. A. K. Lerner, S. Ombrosi and I. P. Rivera-Ríos, Commutators of singular integrals revisited, Bull. Lond. Math. Soc., 51 (2019), 107–119.

    Article  MathSciNet  Google Scholar 

  30. S. Mao, Y. Sawano and H. Wu, On the compactness of operators for rough Marcinkiewicz integral operators, Taiwanese J. Math., 19 (2015), 1777–1793.

    MATH  Google Scholar 

  31. J. Marcinkiewicz, Sur quelques intégrales du type de Dini, Ann. Polon. Math., 17 (1938), 42–50.

    MATH  Google Scholar 

  32. N. G. Meyers, Mean oscillation over cubes and Hölder continuity, Proc. Amer. Math. Soc., 15 (1964), 717–721.

    MathSciNet  MATH  Google Scholar 

  33. T. Nogayama and Y. Sawano, Compactness of the commutators generated by Lipschitz functions and fractional integral operators, Mat. Zametki, 102 (2017), 749–760.

    Article  Google Scholar 

  34. Z. Si, L. Wang and Y. Jiang, Fractional type Marcinkiewicz integral on Hardy spaces, J. Math. Res. Exposition, 31 (2011), 233–241.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral, Colloq. Math., 60/61 (1990), 235–243.

    MATH  Google Scholar 

  37. A. Uchiyama, On the compactness of operators of Hankel type, Tôhoku Math. J. (2), 30 (1978), 163–171.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. S. Wang, The compactness of the commutator of fractional integral operator, Chin. Ann. Math (A), 8 (1987), 475–482.

    MathSciNet  Google Scholar 

  40. H. Wu, On Marcinkiewicz integral operators with rough kernels, Integral Equations Operator Theory, 52 (2005), 285–298.

    Article  MathSciNet  Google Scholar 

  41. H. Wu, Lp bounds for Marcinkiewicz integrals associated to surfaces of revolution, J. Math. Anal. Appl., 321 (2006), 811–827.

    Article  MathSciNet  Google Scholar 

  42. A. Zygmund, Trigonomatric Series, 3rd ed., Cambridge Univ. Press (Cambridge, 2002).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistic, Minnan Normal University, Zhangzhou, 363000, China

    Y.-M. Wen

  2. School of Mathematical Sciences, **amen University, **amen, 361005, China

    H.-X. Wu

Authors
  1. Y.-M. Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H.-X. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H.-X. Wu.

Additional information

Supported by the NNSF of China (Nos. 11771358, 11871101).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wen, YM., Wu, HX. On the Commutators of Marcinkiewicz Integrals with Rough Kernels in Weighted Lebesgue Spaces. Anal Math 46, 619–638 (2020). https://doi.org/10.1007/s10476-020-0053-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-020-0053-7

Key words and phrases

Mathematics Subject Classification

Search

Navigation