SpringerLink
Log in

Calderón–Zygmund Operators and Commutators on Weak Musielak–Orlicz Hardy Spaces

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

In this paper, we study the boundedness of \(\delta \)-Calderón–Zygmund operators and their commutators from weak Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces. To obtain the boundedness of a commutator of \(\delta \)-Calderón–Zygmund operator generated by a locally integrable function, we introduce a non-trivial subspace of \(BMO(\mathbb {R}^n)\) and prove that the commutator of a \(\delta \)-Calderón–Zygmund operator is bounded from a weak Musielak–Orlicz Hardy space to a weak Musielak–Orlicz space if the locally integrable function belongs to such a subspace of \(BMO(\mathbb {R}^n)\).

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 includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Álvarez, J., Bagby, R.J., Kurtz, D.S., Pérez, C.: Weighted estimates for commutators of linear operators. Stud. Math. 104(2), 195–209 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonami, A., Cao, J., Ky, L.D., Liu, L., Yang, D., Yuan, W.: Multiplication between Hardy spaces and their dual spaces. J. Math. Pures Appl. 9(131), 130–170 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonami, A., Grellier, S., Ky, L.D.: Paraproducts and products of functions in \({BMO}({\mathbb{R} }^{n})\) and \( {H}^1({\mathbb{R} }^{n})\) through wavelets. J. Math. Pures Appl. 97(3), 230–241 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carro, M.J., Li, H., Soria, J., Sun, Q.: Calderón-Zygmund operators and commutators on weighted Lorentz spaces. J. Geom. Anal. 31(9), 8979–8990 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang, D.-C., Fu, Z., Yang, D., Yang, S.: Real-variable characterizations of Musielak-Orlicz-Hardy spaces associated with Schrödinger operators on domains. Math. Methods Appl. Sci. 39(3), 533–569 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103(3), 611–635 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duoandikoetxea, J.: Fourier analysis, volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, Translated and revised from the 1995 Spanish original by David Cruz-Uribe (2001)

  8. Fan, X., He, J., Li, B., Yang, D.: Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces. Sci. China Math. 60(11), 2093–2154 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fu, X., Ma, T., Yang, D.: Real-variable characterizations of Musielak-Orlicz Hardy spaces on spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math. 45(1), 343–410 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fu, Z., Gong, S., Lu, S., Yuan, W.: Weighted multilinear Hardy operators and commutators. Forum Math. 27(5), 2825–2851 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fu, Z., Liu, Z., Lu, S.: Commutators of weighted Hardy operators in \(\mathbb{R} ^n\). Proc. Am. Math. Soc. 137(10), 3319–3328 (2009)

    Article  MATH  Google Scholar 

  12. Grafakos, L.: Modern Fourier analysis, volume 250 of Graduate Texts in Mathematics. Springer, New York, third edition, (2014)

  13. He, L., Long, L., **e, G., Yang, D.: Doob’s maximal inequality and Burkholder-Davis-Gundy’s inequality on Musielak-Orlicz spaces. Math. Z. 303(3), 53 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  14. Holmes, I., Lacey, M.T., Wick, B.D.: Commutators in the two-weight setting. Math. Ann. 367(1–2), 51–80 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Holmes, I., Wick, B.D.: Two weight inequalities for iterated commutators with Calderón-Zygmund operators. J. Operator Theory 79(1), 33–54 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Huy, D.Q., Ky, L.D.: Weighted Hardy space estimates for commutators of Calderón-Zygmund operators. Vietnam J. Math. 49(4), 1065–1077 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jiang, R., Yang, D.: New Orlicz-Hardy spaces associated with divergence form elliptic operators. J. Funct. Anal. 258(4), 1167–1224 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiang, R., Yang, D.: Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Commun. Contemp. Math. 13(2), 331–373 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiang, R., Yang, D.: Predual spaces of Banach completions of Orlicz-Hardy spaces associated with operators. J. Fourier Anal. Appl. 17(1), 1–35 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ky, L.D.: Bilinear decompositions and commutators of singular integral operators. Trans. Am. Math. Soc. 365(6), 2931–2958 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ky, L.D.: New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equ. Oper. Theory 78(1), 115–150 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, B., Fan, X., Yang, D.: Littlewood-Paley characterizations of anisotropic Hardy spaces of Musielak-Orlicz type. Taiwanese J. Math. 19(1), 279–314 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, B., Liao, M., Li, B.: Boundedness of Marcinkiewicz integrals with rough kernels on Musielak-Orlicz Hardy spaces. J. Inequal. Appl., 228, 14 (2017)

  24. Liang, Y., Huang, J., Yang, D.: New real-variable characterizations of Musielak-Orlicz Hardy spaces. J. Math. Anal. Appl. 395(1), 413–428 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liang, Y., Ky, L.D., Yang, D.: Weighted endpoint estimates for commutators of Calderón-Zygmund operators. Proc. Am. Math. Soc. 144(12), 5171–5181 (2016)

    Article  MATH  Google Scholar 

  26. Liang, Y., Yang, D.: Musielak-Orlicz Campanato spaces and applications. J. Math. Anal. Appl. 406(1), 307–322 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liang, Y., Yang, D., Jiang, R.: Weak Musielak-Orlicz Hardy spaces and applications. Math. Nachr. 289(5–6), 634–677 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, H. P .: The weak \(H^p\) spaces on homogeneous groups. In Harmonic analysis (Tian**, 1988), volume 1494 of Lecture Notes in Math., pp. 113–118. Springer, Berlin (1991)

  29. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Hardy’s inequality in Musielak-Orlicz-Sobolev spaces. Hiroshima Math. J. 44(2), 139–155 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Duality of non-homogeneous central Herz-Morrey-Musielak-Orlicz spaces. Potential Anal. 47(4), 447–460 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Quek, T., Yang, D.: Calderón-Zygmund-type operators on weighted weak Hardy spaces over \({ R}^n\). Acta Math. Sin. 16(1), 141–160 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shi, S., Fu, Z., Lu, S.: On the compactness of commutators of Hardy operators. Pac. J. Math. 307(1), 239–256 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, D., Liang, Y., Ky, L.D.: Real-variable theory of Musielak-Orlicz Hardy spaces. Lecture Notes in Mathematics, vol. 2182. Springer, Cham (2017)

  34. Yang, D., Yang, S.: Musielak-Orlicz-Hardy spaces associated with operators and their applications. J. Geom. Anal. 24(1), 495–570 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yang, D., Yuan, W., Zhang, Y.: Bilinear decomposition and divergence-curl estimates on products related to local Hardy spaces and their dual spaces. J. Funct. Anal. 280(2), 74 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yang, D., Yuan, W., Zhuo, C.: Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces. Rev. Mat. Complut. 27(1), 93–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the referees very much for elaborate and valuable suggestions which helped to improve the paper.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tian**, China

    **ao Wang & Wenchang Sun

Authors
  1. **ao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenchang Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenchang Sun.

Ethics declarations

Conflict of interest

The authors state that there is no conflict of interest.

Additional information

Communicated by Saeid Maghsoudi.

Publisher's Note

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

This work was partially supported by the National Natural Science Foundation of China (12171250, U21A20426 and 12271267).

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

Wang, X., Sun, W. Calderón–Zygmund Operators and Commutators on Weak Musielak–Orlicz Hardy Spaces. Bull. Iran. Math. Soc. 49, 20 (2023). https://doi.org/10.1007/s41980-023-00772-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s41980-023-00772-w

Keywords

Mathematics Subject Classification

Search

Navigation