Equivalent Characterizations and Pointwise Multipliers of Normal Weight Dirichlet Space on the Unit Ball

Abstract

Let \(\mu \) be a normal function on [0, 1) and \(p>0\). In this paper, the authors give several equivalent characterizations of the functions in the normal weight Dirichlet type space \(D_{\mu }^{p}(B)\) on the unit ball B in \(\mathbf{C}^{n}\). At the same time, the authors describe the boundedness of a class of integral operator T from the normal weight Lebesgue space \(L_{\mu }^{p}(B)\) to the abstract measure Lebesgue space \(L_{\mu ,\psi }^{p}(B)\). As an application of the previous results, the authors discuss the pointwise multipliers on \(D_{\mu }^{p}(B)\) or from \(D_{\mu }^{p}(B)\) to \({\mathcal {B}}_{\nu _{p}}(B)\).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Chen, H., Gauthier, P.: Composition operators on \(\mu \)-Bloch spaces. Can. J. Math. 61, 50–75 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Zhang, X., Li, J.: Weighted composition operators between \(\mu \)-Bloch spaces on the unit ball of \({ C}^{n}\). Acta Math. Sci. 29A, 573–583 (2009)

    Google Scholar 

  3. 3.

    Dai, J.: Composition operators on Zygmund spaces of the unit ball. J. Math. Anal. Appl. 394, 696–705 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Zhang, X., Li, S.: The composition operator on the normal weight Zygmund space in high dimensions. Complex Var. Elli. Equ. 64, 1932–1953 (2019)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Zhang, M., Xu, W.: Composition operators on \(\alpha \)-Bloch spaces of the unit ball. Acta Math. Sin. 23, 1991–2002 (2007)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Liu, J., Li, J., Zhang, X.: Characterizations of composition operators between Bloch type spaces on the unit ball again. Acta Math. Sin. 50(3), 711–720 (2007)

    MATH  Google Scholar 

  7. 7.

    Taylor, G.: Multipliers on \(D_{\alpha }\). Trans. Am. Math. Soc. 123, 229–240 (1966)

    Google Scholar 

  8. 8.

    Stegenga, D.: Multipliers of the Dirichlet space. Ill. J. Math. 24, 113–139 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Hu, P., Shi, J.: Multipliers on Dirichlet type spaces. Acta Math. Sin. 17, 263–272 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Zhang, X.: The pointwise multipliers of Bloch type space \({\beta ^{p}}\) and Dirichlet type space \( D_{q}\) on the unit ball of \(\bf {C^n}\). J. Math. Anal. Appl. 285, 376–386 (2003)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Axler, S., Shields, A.: Univalent multipliers of the Dieichlet space. Mich. Math. J. 32, 65–80 (1985)

    Article  MATH  Google Scholar 

  12. 12.

    Zhu, K.: Multipliers of BMO in the Bergman metric with applications to Toeplitz operators. J. Funct. Anal. 87, 31–50 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Zhang, X., Wang, M.: The pointwise multipliers of p-Bloch space in the unit ball of \({ C}^{n}\). J. Math. Study 34, 158–163 (2001)

    MathSciNet  Google Scholar 

  14. 14.

    Guo, Y., Shang, Q., Zhang, X.: The pointwise multiplier on the normal weight Zygmund space in the unit ball. Acta Math. Sci. 38A, 1041–1048 (2018)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Hu, Z.: Composition operators between Bloch-type spaces in the polydisc. Sci. China 48A(supp), 268–282 (2005)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Zhang, X., Xiao, J.: Weighted composition operators between \(\mu \)-Bloch spaces on the unit ball. Sci. China 48A, 1349–1368 (2005)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zhang, X., Xi, L., Fan, H., Li, J.: Atomic decomposition of \(\mu \)-Bergman space in \({ C}^{n}\). Acta Math. Sci. 34B, 779–789 (2014)

    Article  Google Scholar 

  18. 18.

    Li, S., Zhang, X., Xu, S.: The compact composition operator on the \(\mu \)-Bergman Space in the unit ball. Acta Math. Sci. 37B(2), 425–438 (2017)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Zhao, Y., Zhang, X.: On an integral-type operator from Dirichlet spaces to Zygmund type spaces on the unit ball. Acta Math. Sci. 37A, 217–227 (2017)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer (GTM 226), New York (2005)

    Google Scholar 

  21. 21.

    Tang, P., Zhang, X., Lv, R.: Equivalent characterization of several quantities on holomorphic function spaces. Acta Math. Sci. 39A, 1291–1299 (2019)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Rudin, W.: Function Theory in the Unit Ball of \({{ C}^{n}}\). Springer, New York (1980)

    Google Scholar 

  23. 23.

    Zhang, X., He, C., Cao, F.: The equivalent norms of F(p, q, s) space in \({ C}^{n}\). J. Math. Anal. Appl. 401, 601–610 (2013)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Zhang, X., Li, S., Shang, Q., Guo, Y.: An integral estimate and the equivalent norms on F(p, q, s, k) spaces in the unit ball. Acta Math. Sci. 38B, 1861–1880 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Zhang, X., Li, M., Guan, Y.: The equivalent norms and the Gleason’s problem on \(\mu \)-Zygmund spaces in \({ C}^{n}\). J. Math. Anal. Appl. 419, 185–199 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Zhang, X., Fan, H., Xi, L., Li, J.: Characterizations and differentiation composition operators of \(\mu \)-Bergman space in \({ C}^{n}\). Chin. Ann. Math. 35A, 741–756 (2014)

    Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their useful suggestions!

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xuejun Zhang.

Additional information

Publisher's Note

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

The research is supported by the National Natural Science Foundation of China (No. 11571104).

Communicated by Irene Sabadini, Michael Shapiro and Daniele Struppa.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tan, M., Zhang, X. Equivalent Characterizations and Pointwise Multipliers of Normal Weight Dirichlet Space on the Unit Ball. Complex Anal. Oper. Theory 15, 24 (2021). https://doi.org/10.1007/s11785-020-01075-2

Download citation

Keywords

  • Normal weight Dirichlet space
  • Equivalent characterization
  • Pointwise multiplier
  • Boundedness
  • Compactness

Mathematics Subject Classification

  • 32A36
  • 47B33