Skip to main content

The Integral Weight and Superposition Operators Between Bloch-Type Spaces


Using the notion of the integral weight, we characterize all entire functions that transform a Bloch-type space \({\mathcal {B}}^{\mu _1}\) into another space of the same kind \({\mathcal {B}}^{\mu _2}\) by superposition for very general weights \(\mu _1\) and \(\mu _2\), satisfying a growth condition.

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


  1. 1.

    Álvarez, V., Márquez, M.A., Vukotić, D.: Superposition operators between the Bloch space and Bergman spaces. Arkiv Mat. 42(2), 205–216 (2004)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Attele, K.: Toeplitz and Hankel operators on Bergman spaces. Hokkaido Math. J. 21, 279–293 (1992)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bierstedt, K., Summers, W.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. (Ser. A) 54, 70–79 (1993)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bierstedt, K., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Studia Math. 127, 70–79 (1998)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bonet, J., Vukotić, D.: Superposition operators between weighted Banach spaces of analytic functions of controlled growth. Monatshefte Math. 170(3–4), 311–323 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Boyd, C., Rueda, P.: Holomorphic superposition operators between Banach function spaces. Aust. Math. Soc. 96(2), 186–197 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Brown, L., Shields, A.L.: Multipliers and cyclic vectors in the Bloch space. Mich. Math. J. 38(1), 141–146 (1991)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Buckley, S.M., Vukotić, D.: Superposition Operators and the Order and Type of Entire Functions. Recent Advances in Operator-Related Function Theory Contemporary Mathematics, vol. 393, pp. 51–57. American Mathematical Society, Providence, RI (2006)

    Book  Google Scholar 

  9. 9.

    Buckley, S.M., Vukotić, D.: Univalent interpolation in Besov space and superposition into Bergman spaces. Potential Anal. 29, 1–16 (2008)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Buckley, S.M., Fernández, J.L., Vukotić, D.: Superposition operators on Dirichlet type spaces. Rep. Univ. Jyväs. 83, 41–61 (2001)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Cámera, G., Giménez, J.: Nonlinear superposition operators acting on Bergman spaces. Compos. Math. 93, 23–35 (1994)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Castillo, R.E., Ramos Fernández, J.C., Salazar, M.: Bounded superposition operators between Bloch–Orlicz and \(\alpha \)-Bloch spaces. Appl. Math. Comput. 218, 3441–3450 (2011)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Girela, D., Márquez, M.A.: Superposition operators between \(Q_p\) spaces and Hardy spaces. J. Math. Anal. Appl. 364, 463–472 (2010)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Levin, B.Ya.: Lectures on Entire Functions, Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1996)

    Book  Google Scholar 

  15. 15.

    Ramos Fernández, J.C.: Bounded superposition operators between weighted Banach spaces of analytic functions. Appl. Math. Comput. 219, 4942–4949 (2013)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Ramos-Fernández, J.C.: Composition operators on Bloch–Orlicz type spaces. Appl. Math. Comput. 217, 3392–3402 (2010)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Ramos-Fernández, J.C.: Logarithmic Bloch spaces and their weighted composition operators. Rend. Circ. Mat. Palermo 65(1), 159–174 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Stević, S.: On new Bloch-type spaces. Appl. Math. Comput. 215(2), 841–849 (2009)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Xiong, C.: Superposition operators between \(Q_p\) spaces and Bloch-type spaces. Complex Var. Theory Appl. 50(12), 935–938 (2005)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Xu, W.: Superposition operators on Bloch-type spaces. Comput. Methods Funct. Theory 7(2), 501–507 (2007)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Ye, S.: Weighted composition operator on the logarithmic Bloch space. Bull. Korean Math. Soc. 47(3), 527–540 (2010)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Zhu, K.: Bloch type spaces of analytic functions. Rocky Mountain J. Math. 23, 1143–1177 (1993)

    MathSciNet  Article  Google Scholar 

Download references


The authors wish to express their sincere gratitude to the anonymous referee for his/her useful comments.

Author information



Corresponding author

Correspondence to Julio C. Ramos-Fernández.

Additional information

Publisher's Note

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

Communicated by Dr. Saminathan Ponnusamy.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Malavé-Malavé, R.J., Ramos-Fernández, J.C. The Integral Weight and Superposition Operators Between Bloch-Type Spaces. Bull. Malays. Math. Sci. Soc. 43, 3035–3047 (2020).

Download citation


  • Bloch-type spaces
  • Superposition operator
  • Entire function

Mathematics Subject Classification

  • 47H30
  • 30D45
  • 30H05