Skip to main content
Springer Nature Link
Log in

Composition Cesàro Operator on the Normal Weight Zygmund Space in High Dimensions

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

Let n > 1 and B be the unit ball in n dimensions complex space Cn. Suppose that φ is a holomorphic self-map of B and ψ ∈ H(B) with ψ(0) = 0. A kind of integral operator, composition Cesàro operator, is defined by

$${T_{\varphi,\psi }}\left( f \right)\left( z \right) = \int_0^1 {f\left[ {\varphi \left( {tz} \right)} \right]R\psi \left( {tz} \right){{{\rm{d}}t} \over t}},\;\;\;\;f \in H\left( B \right),\;\;z \in B.$$

In this paper, the authors characterize the conditions that the composition Cesàro operator Tφ,ψ is bounded or compact on the normal weight Zygmund space \({{\cal Z}_\mu }\left( B \right)\). At the same time, the sufficient and necessary conditions for all cases are given.

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
$34.99 /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. Zhang, X. J., Li, M. and Guan, Y., The equivalent norms and the Gleason’s problem on µ-Zygmund spaces in Cn, J. Math. Anal. Appl., 419, 2014, 185–199.

    Article  MathSciNet  Google Scholar 

  2. Stević, S., On an integral-type operator from Zygmund-type spaces to mixed-norm spaces on the unit ball, Abstr. Appl. Anal., 2010, Article ID 198608, 7 pages.

  3. Li, S. X. and Stević, S., Volterra type operators on Zygmund space, J. Inequal. Appl., 2007, Article ID 32124, 10 pages.

  4. Li, S. X. and Stević, S., Products of Volterra type operator and composition operator from H∞ and Bloch spaces to the Zygmund space, J. Math. Anal. Appl., 345, 2008, 40–52.

    Article  MathSciNet  Google Scholar 

  5. Li, S. X. and Stević, S., Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput., 206(2), 2008, 825–831.

    MathSciNet  MATH  Google Scholar 

  6. Li, S. X. and Stević, S., Integral-type operators from Bloch-type spaces to Zygmund-type spaces, Appl. Math. Comput., 215(2), 2009, 464–473.

    MathSciNet  MATH  Google Scholar 

  7. Li, S. X. and Stević, S., On an integral-type operator from ω-Bloch spaces to µ-Zygmund spaces, Appl. Math. Comput., 215(12), 2010, 4385–4391.

    MathSciNet  MATH  Google Scholar 

  8. Li, S. X. and Stević, S., Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl., 338, 2008, 1282–1295.

    Article  MathSciNet  Google Scholar 

  9. Stević, S., On an integral operator from the Zygmund space to the Bloch type space on the unit ball, Glasgow Math. J., 51, 2009, 275–287.

    Article  MathSciNet  Google Scholar 

  10. Fang, Z. S. and Zhou, Z. H., Extended Cesàro operators from generally weighted Bloch spaces to Zygmund space, J. Math. Anal. Appl., 359, 2009, 499–507.

    Article  MathSciNet  Google Scholar 

  11. Zhu, X., A new characterization of the generalized weighted composition operator from H∞ into the Zygmund space, Math. Ineq. Appl., 18, 2015, 1135–1142.

    MathSciNet  MATH  Google Scholar 

  12. Li, S. X. and Stević, S., Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces, Appl. Math. Comput., 217, 2010, 3144–3154.

    MathSciNet  MATH  Google Scholar 

  13. Liu, Y. M. and Yu, Y. Y., Weighted differentiation composition operators from mixed-norm to Zygmund spaces, Numer. Funct. Anal. Optim., 31, 2010, 936–954.

    Article  MathSciNet  Google Scholar 

  14. Ye, S. L. and Lin, C. S., Composition followed by differentiation on the Zygmund space, Acta Math. Sin., 59, 2016, 11–20 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  15. Ye, S. L. and Hu, Q. X., Weighted composition operators on the Zygmund space, Abstr. Appl. Anal., 2012, Artical ID 462482, 18 pages.

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

    Article  MathSciNet  Google Scholar 

  17. Zhang, J. F. and Xu, H. M., Weighted Cesàro operators on Zygmund type spaces on the unit ball, Acta Math. Sci., 31A(1), 2011, 188–195 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  18. Liang, Y. X., Wang, C. J. and Zhou, Z. H., Weighted composition operators from Zygmund spaces to Bloch spaces on the unit ball, Ann. Polo. Math., 114(2), 2015, 101–114.

    Article  MathSciNet  Google Scholar 

  19. Zhang, X. J. and Xu, S., Weighted differentiation composition operators between normal weight Zygmund spaces and Bloch spaces in the unit ball of Cn for n > 1, Complex Anal. Oper. Theory, 13(3), 2019, 859–878.

    Article  MathSciNet  Google Scholar 

  20. Zhang, X. J. and Li, S. L., The composition operator on the normal weight Zygmund space in high dimensions, Complex Var. and Ellip. Equ., 64(11), 2019, 1932–1953.

    Article  MathSciNet  Google Scholar 

  21. Li, S. L. and Zhang, X. J., Composition operators on the normal weight Zygmund spaces in high dimensions, J. Math. Anal. Appl., 487(2), 2020, 19 pages.

  22. Li, S. L. and Zhang, X. J., Several properties on the normal weight Zygmund space in several complex variables, Acta Math. Sin., 62(5), 2019, 795–808 (in Chinese).

    MATH  Google Scholar 

  23. Guo, Y. T., Shang, Q. L. and Zhang, X. J., The pointwise multiplier on the normal weight Zygmund space in the unit ball, Acta Math. Sci., 38A(6), 2018, 1041–1048 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  24. Zhao, Y. H. and Zhang, X. J., On an integral-type operator from Dirichlet spaces to Zygmund type spaces on the unit ball, Acta Math. Sci., 37A(2), 2017, 217–227 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  25. Zhao, Y. H. and Zhang, X. J., Integral-type operators on Zygmund type spaces on the unit ball, Math. Adv. (China), 45(5), 2016, 755–766 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  26. Long, J. R., Qiu, C. H. and Wu, P. C., Weighted composition followed and proceeded by differentiation operators from Zygmund spaces to Bloch-type spaces, J. of Ineq. and Appl., 2014, 152, 12 pages.

  27. Dai, J. N. and Ouyang C. H., Composition operators from Zygmund spaces to α-Bloch spaces in the unit ball, J. of Wuhan Uni. (Natur. Sci. Ed.), 56(4), 2010, 961–968 (in Chinese).

    MathSciNet  Google Scholar 

  28. Siskakis, A. G., Composition semigroups and the Cesàro operator on Hp, J. London Math. Soc., 36(2), 1987, 153–164.

    Article  MathSciNet  Google Scholar 

  29. Miao, J., The Cesàro operator is bounded on Hp for 0 < p < 1, Proc. Amer. Math. Soc., 116, 1992, 1077–1079.

    MathSciNet  MATH  Google Scholar 

  30. Shi, J. H. and Ren, G. B., Boundedness of the Cesàro operator on mixed norm spaces, Proc. Amer. Math. Soc., 126, 1998, 3553–3560.

    Article  MathSciNet  Google Scholar 

  31. Xiao, J., Cesàro operators on Hardy, BMOA and Bloch spaces, Arch. Math., 68, 1997, 398–406.

    Article  MathSciNet  Google Scholar 

  32. Xiao, J. and Tan, H., p-Bergman spaces, α-Bloch spaces, little α-Bloch spaces and Cesàro means, Chin. Ann. Math., 19A(2), 1998, 187–196 (in Chinese).

    Google Scholar 

  33. Aleman, A. and Siskakis, A. G., An Integral operator on Hp, Complex Variables, 28, 1995, 149–158.

    MathSciNet  MATH  Google Scholar 

  34. Hu, Z. J., Extended Cesàro operators on the Bloch space in the ball of Cn, Acta Math. Sci., 23B(4), 2003, 561–566.

    Article  Google Scholar 

  35. Hu, Z. J., Extended Cesàro operators on mixed norm spaces, Proc. Amer. Math. Soc., 131(7), 2003, 2171–2179.

    Article  MathSciNet  Google Scholar 

  36. Aleman, A. and Siskakis, A. G., Integration operators on Bergman spaces, Indiana Uni. Math. J., 46, 1997, 337–356.

    MathSciNet  MATH  Google Scholar 

  37. Zhang, X. J., Weighted Cesàro operators on Dirichlet type spaces and Bloch type spaces of Cn, Chin. Ann. Math., 26A(1), 2005, 139–150 (in Chinese).

    MATH  Google Scholar 

  38. Stević, S., On a new operator from H∞ to the Bloch type spaces on the unit ball, Util. Math., 77, 2008, 257–263.

    MathSciNet  MATH  Google Scholar 

  39. Stević, S. and Ueki, S., Integral-type operator acting between weighted-type spaces on the unit ball, Appl. Math. Comput., 215(7), 2009, 2464–2471.

    MathSciNet  MATH  Google Scholar 

  40. Stević, S., On operator \(P_\varphi g\) from the logarithmic Bloch-type spaces to the mixed-norm spaces on the unit ball, Appl. Math. Comput., 215(12), 2010, 4248–4255.

    MathSciNet  MATH  Google Scholar 

  41. Stević, S., On some integral-type operator between a general space and Bloch type spaces, Appl. Math. Comput., 218(6), 2011, 2600–2618.

    MathSciNet  MATH  Google Scholar 

  42. Zhang, X. J. and Chu Y. M., Compact Cesàro operator from spaces H(p, q, u) to H(p, q, v), Acta Math. Appl. Sin. (English Series), 22(3), 2006, 437–442.

    Article  MathSciNet  Google Scholar 

  43. Chen, H. H. and Gauthier, P. H., Composition operators on µ-Bloch spaces, Canad. J. Math., 61, 2009, 50–75.

    Article  MathSciNet  Google Scholar 

  44. Zhang, X. J. and Li, J. X., Weighted composition operators between µ-Bloch spaces on the unit ball of Cn, Acta Math. Sci., 29A, 2009, 573–583 (in Chinese).

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

The authors thank the referees for their useful suggestions!

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Hunan Normal University, Changsha, 410006, China

    Si Xu, Xuejun Zhang & Shenlian Li

Authors
  1. Si Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuejun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shenlian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuejun Zhang.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11571104) and the Hunan Provincial Innovation Foundation for Postgraduate (No. CX2018B286).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, S., Zhang, X. & Li, S. Composition Cesàro Operator on the Normal Weight Zygmund Space in High Dimensions. Chin. Ann. Math. Ser. B 42, 69–84 (2021). https://doi.org/10.1007/s11401-021-0245-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-021-0245-x

Keywords

2000 MR Subject Classification

Search

Navigation