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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 859–878 | Cite as

Weighted Differentiation Composition Operators Between Normal Weight Zygmund Spaces and Bloch Spaces in the Unit Ball of \(\hbox {C}^{\mathrm{n}}\) for \(\hbox {n}>1\)

  • Xuejun ZhangEmail author
  • Si Xu
Article

Abstract

Let \(\mu \) be a normal functions on [0, 1), and H(B) be the space of all holomorphic functions on the unit ball B of \(\mathbf C^{n}\). Let \(\varphi \) be a nonconstant holomorphic self-map on B, and \(\psi \) be a holomorphic function on B. The weighted differentiation composition operator \(\psi D_{\varphi }\) is defined on the space H(B) by \(\psi D_{\varphi }(f)=\psi (Rf)\circ \varphi \), for all \(f\in H(B)\). In this paper, the authors characterize the boundedness and compactness of the weighted differentiation composition operator \(\psi D_{\varphi }\) from the normal weight Zygmund space \(Z_{\mu }(B)\) to the normal weight Bloch space \(\beta _{\mu }(B)\) for \(n>1\). As a consequence of the main results, the authors give the briefly sufficient and necessary conditions that the differentiation composition operator \( D_{\varphi }\) is compact from \(Z_{\mu }(B)\) to \(\beta _{\mu }(B)\) for \(\displaystyle {\mu (r)=(1-r)^{s}\log ^{t}\frac{e}{1-r}}\).

Keywords

Weighted differentiation composition operator Unit ball Normal weight Zygmund space Normal weight Bloch space Boundedness Compactness 

Mathematics Subject Classification

32A37 

References

  1. 1.
    Li, S., Stević, S.: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 338, 1282–1295 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Zhu, X.: A new characterization of the generalized weighted composition operator from \(H^{\infty }\) into the Zygmund space. Math. Inequal. Appl. 18, 1135–1142 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Li, S., Stević, S.: Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Appl. Math. Comput. 217, 3144–3154 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Liu, Y., Yu, Y.: Weighted differentiation composition operators from mixed-norm to Zygmund spaces. Numer. Funct. Anal. Optim. 31, 936–954 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ye, S., Lin, C.: Composition followed by differentiation on the Zygmund space. Acta Math. Sin. 59, 11–20 (2016). (in Chinese)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ye, S., Hu, Q.: Weighted composition operators on the Zygmund space. Abstr. Appl. Anal. (2012) Art. ID 462482Google Scholar
  7. 7.
    Han, X., Xu, H.: Composition operators between Besov spaces and Zygmund spaces. J. Math. Study 42, 310–319 (2009). (in Chinese) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Madigan, K., Matheson, A.: Compact composition operators on the Bloch space. Trans. Am. Math. Soc. 347, 2679–2687 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhao, R.: Composition operators from Bloch type spaces to Hardy and Besov spaces. J. Math. Anal. Appl. 233, 749–766 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Xiao, J.: Composition operators associated with Bloch type spaces. Complex Var. Theory Appl. 46, 109–121 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Zhang, X.: The composition operators and weighted composition operators on p-Bloch spaces. Chin. Ann. Math. 24A(6), 711–720 (2003). (in Chinese)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Madigan, K.: Composition operators on analytic Lipschitz spaces. Proc. Am. Math. Soc. 119, 465–473 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ohno, S., Zhao, R.: Weighted composition operators on the Bloch space. Bull. Aust. Math. Soc. 63, 177–185 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yoneda, R.: The composition operators on weighted Bloch space. Arch. Math. 78, 310–317 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dai, J.: Composition operators on Zygmund spaces of the unit ball. J. Math. Anal. Appl. 394, 696–705 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shi, J., Luo, L.: Composition operators on the Bloch space of several complex variables. Acta Math. Sin. (English series) 16, 85–98 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhou, Z., Zeng, H.: Composition operators between p-Bloch space and q-Bloch space in the unit ball. Prog. Nat. Sci. 13(3), 233–236 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Zhang, M., Xu, W.: Composition operators on Bloch spaces of the unit ball. Acta Math. Sin. (English series) 23(11), 1991–2002 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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). (in Chinese)zbMATHGoogle Scholar
  20. 20.
    Zhang, X., Xiao, J.: Weighted composition operators between \(\mu \)-Bloch spaces on the unit ball. Sci. China 48A(10), 1349–1368 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, H., Gauthier, P.: Composition operators on \(\mu \)-Bloch spaces. Can. J. Math. 61, 50–75 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, X., Li, J.: Weighted composition operators between \(\mu \)-Bloch spaces on the unit ball of \({ C}^{n}\). Acta Math. Sci. 29A(3), 573–583 (2009). (in Chinese)zbMATHGoogle Scholar
  23. 23.
    Stević, S.: Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput. 211, 222–233 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Liu, G.: Weighted differentiation composition operators from mixed-norm spaces to Bloch-type spaces. Chin. Q. J. Math. 30(2), 236–243 (2015)zbMATHGoogle Scholar
  25. 25.
    Qu, H., Liu, Y., Cheng, S.: Weighted differentiation composition operators from logarithmic Bloch spaces to Zygmund-type spaces. Abstr. Appl. Anal. Article ID 832713, 14 p (2014)Google Scholar
  26. 26.
    Liu, J., Lou, Z., Sharma, A.: Weighted differentiation composition operators to Bloch-type spaces. Abstr. Appl. Anal. Article ID 151929, 9 p (2013)Google Scholar
  27. 27.
    Waleed, A.: Weighted differentiation composition operators from Navanlinna classes to weighted-type spaces. J. Math. Anal. 8(1), 163–175 (2017)MathSciNetGoogle Scholar
  28. 28.
    Zhou, J., Zhu, X.: Product of differentiation and composition operators on the logarithmic Bloch space. J. Inequ. Appl. 543, 1–12 (2014)MathSciNetGoogle Scholar
  29. 29.
    Zhang, X., Fan, H., Xi, L., Li, J.: Characterizations and differentiation composition operators of \(\mu \)-Bergman space in \({{ C}^{n}}\). Chin. Ann. Math. 35A(6), 741–756 (2014). (in Chinese)zbMATHGoogle Scholar
  30. 30.
    Stević, S.: On an integral-type operator from Zygmund-type spaces to mixed-norm spaces on the unit ball. Abstr. Appl. Anal. Article ID 198608, 7 p (2010)Google Scholar
  31. 31.
    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(1), 185–199 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hu, Z.: Composition operators between Bloch-type spaces in the polydisc. Sci. China 48A((supp)), 268–282 (2005)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer (GTM 226), New York (2005)zbMATHGoogle Scholar
  34. 34.
    Rudin, W.: Function Theory in the Unit Ball of \({{ C}^{n}}\). Springer, New York (1980)CrossRefzbMATHGoogle Scholar
  35. 35.
    Chen, Z., Liu, Y.: Schwarz–pick estimates for bounded holomorphic functions in the unit ball of \({{ C}^{n}} \). Acta Math. Sin. 26B(5), 901–908 (2010)CrossRefzbMATHGoogle Scholar
  36. 36.
    Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  37. 37.
    Zhang, X., Xi, L., Fan, H., Li, J.: Atomic decomposition of \(\mu \)-Bergman space in \({ C}^{n}\). Acta Math. Sci. 34B(3), 779–789 (2014)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsHunan Normal UniversityChangshaChina

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