Skip to main content
SpringerLink
Log in

Weighted Multi-Parameter Mixed Hardy Spaces and Their Applications

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

Applying discrete Calderón’s identity, we study weighted multi-parameter mixed Hardy space \(H_{{\rm{mix}}}^p(\omega ,{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}})\). Different from classical multi-parameter Hardy space, this space has characteristics of local Hardy space and Hardy space in different directions, respectively. As applications, we discuss the boundedness on \(H_{{\rm{mix}}}^p(\omega ,{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}})\) of operators in mixed Journé’s class.

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. R. R. Coifman: A real variable characterization of Hp. Stud. Math. 51 (1974), 269–274.

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. D. Cruz-Uribe, J. M. Martell, C. Pérez: Sharp weighted estimates for classical operators. Adv. Math. 229 (2012), 408–441.

    Article  MathSciNet  Google Scholar 

  4. Y. Ding, Y. Han, G. Lu, X. Wu: Boundedness of singular integrals on multiparameter weighted Hardy spaces H pw (ℝn × ℝm). Potential Anal. 37 (2012), 31–56.

    Article  MathSciNet  Google Scholar 

  5. W. Ding, G. Lu, Y. Zhu: Multi-parameter local Hardy spaces. Nonlinear Anal., Theory Methods Appl., Ser. A 184 (2019), 352–380.

    Article  MathSciNet  Google Scholar 

  6. W. Ding, M. Qin, Y. Zhu: The boundedness on mixed Hardy spaces. J. Funct. Spaces 2020 (2020), Article ID 5341674, 12 pages.

  7. W. Ding, Y. Zhu: Mixed Hardy spaces and their applications. Acta Math. Sci., Ser. B, Engl. Ed. 40 (2020), 945–969.

    Article  MathSciNet  Google Scholar 

  8. C. L. Fefferman, E. M. Stein: Hp spaces of several variables. Acta Math. 129 (1972), 137–193.

    Article  MathSciNet  Google Scholar 

  9. R. Fefferman: Strong differentiation with respect to measures. Am. J. Math. 103 (1981), 33–40.

    Article  MathSciNet  Google Scholar 

  10. R. Fefferman: The atomic decomposition of H1 in product spaces. Adv. Math. 55 (1985), 90–100.

    Article  MathSciNet  Google Scholar 

  11. R. Fefferman: Calderón-Zygmund theory for product domains: Hp spaces. Proc. Natl. Acad. Sci. USA 83 (1986), 840–843.

    Article  Google Scholar 

  12. R. Fefferman: Harmonic analysis on product spaces. Ann. Math. (2) 126 (1987), 109–130.

    Article  MathSciNet  Google Scholar 

  13. R. Fefferman, E. M. Stein: Singular integrals on product spaces. Adv. Math. 45 (1982), 117–143.

    Article  MathSciNet  Google Scholar 

  14. G. B. Folland, E. M. Stein: Hardy Spaces on Homogeneous Groups. Mathematical Notes 28. Princeton University Press, Princeton, 1982.

    MATH  Google Scholar 

  15. J. Garcia-Cuerva: Weighted Hp spaces. Diss. Math. 162 (1979), 63 pages.

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

    Book  Google Scholar 

  17. D. Goldberg: A local version of real Hardy spaces. Duke Math. J. 46 (1979), 27–42.

    Article  MathSciNet  Google Scholar 

  18. L. Grafakos: Classical and Modern Fourier Analysis. Pearson/Prentice Hall, Upper Saddle River, 2004.

    MATH  Google Scholar 

  19. R. F. Gundy, E. M. Stein: Hp theory for the poly-disk. Proc. Natl. Acad. Sci. USA 76 (1979), 1026–1029.

    Article  Google Scholar 

  20. R. F. Gundy, R. L. Wheeden: Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh-Paley series. Stud. Math. 49 (1974), 107–124.

    Article  MathSciNet  Google Scholar 

  21. Y. Han, M.-Y. Lee, C.-C. Lin, Y.-C. Lin: Calderón-Zygmund operators on product Hardy spaces. J. Funct. Anal. 258 (2010), 2834–2861.

    Article  MathSciNet  Google Scholar 

  22. Y. Han, C. Lin, G. Lu, Z. Ruan, E. T. Sawyer: Hardy spaces associated with different homogeneities and boundedness of composition operators. Rev. Mat. Iberoam. 29 (2013), 1127–1157.

    Article  MathSciNet  Google Scholar 

  23. Y. Han, G. Lu, Z. Ruan: Boundedness criterion of Journé’s class of singular integrals on multiparameter Hardy spaces. J. Funct. Anal. 264 (2013), 1238–1268.

    Article  MathSciNet  Google Scholar 

  24. Y. Han, G. Lu, Z. Ruan: Boundedness of singular integrals in Journé’s class on weighted multiparameter Hardy spaces. J. Geom. Anal. 24 (2014), 2186–2228.

    Article  MathSciNet  Google Scholar 

  25. Y. Han, G. Lu, K. Zhao: Discrete Calderón’s identity, atomic decomposition and boundedness criterion of operators on multiparameter Hardy spaces. J. Geom. Anal. 20 (2010), 670–689.

    Article  MathSciNet  Google Scholar 

  26. G. H. Hardy: The mean value of the modulus of an analytic function. Proc. Lond. Math. Soc. (2) 14 (1915), 269–277.

    Article  Google Scholar 

  27. J.-L. Journé: Calderón-Zygmund operators on product spaces. Rev. Mat. Iberoam. 1 (1985), 55–91.

    Article  MathSciNet  Google Scholar 

  28. J.-L. Journé: Two problems of Calderón-Zygmund theory on product-spaces. Ann. Inst. Fourier 38 (1988), 111–132.

    Article  MathSciNet  Google Scholar 

  29. R. H. Latter: A characterization of Hp(ℝn) in terms of atoms. Stud. Math. 62 (1978), 93–101.

    Article  MathSciNet  Google Scholar 

  30. G. Lu, Y. Zhu: Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters. Acta Math. Sin., Engl. Ser. 29 (2013), 39–52.

    Article  MathSciNet  Google Scholar 

  31. Z. Ruan: Weighted Hardy spaces in three-parameter case. J. Math. Anal. Appl. 367 (2010), 625–639.

    Article  MathSciNet  Google Scholar 

  32. E. M. Stein, G. Weiss: On the theory of harmonic functions of several variables. I. The theory of Hp-spaces. Acta Math. 103 (1960), 25–62.

    Article  MathSciNet  Google Scholar 

  33. D. Yang, Y. Liang, L. D. Ky: Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics 2182. Springer, Cham, 2017.

    Book  Google Scholar 

  34. D. Yang, S. Yang: Local Hardy spaces of Musielak-Orlicz type and their applications. Sci. China, Math. 55 (2012), 1677–1720.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Sciences, Nantong University, Nantong, 226007, P. R. China

    Wei Ding & Yun Xu

  2. Department of Mathematics, Nantong Normal College, Nantong, 226010, P. R. China

    Yueping Zhu

Authors
  1. Wei Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yun Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yueping Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Ding.

Additional information

Supported by NNSF of China grants (11771223).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ding, W., Xu, Y. & Zhu, Y. Weighted Multi-Parameter Mixed Hardy Spaces and Their Applications. Czech Math J 72, 709–734 (2022). https://doi.org/10.21136/CMJ.2022.0115-21

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2022.0115-21

Keywords

MSC 2020

Search

Navigation