Bloch type spaces on the unit ball of a Hilbert space

  • Zhenghua XuEmail author


We initiate the study of Bloch type spaces on the unit ball of a Hilbert space. As applications, the Hardy-Littlewood theorem in infinite-dimensional Hilbert spaces and characterizations of some holomorphic function spaces related to the Bloch type space are presented.


Bloch type space Lipschitz space Hardy-Littlewood theorem Hilbert space 


32A18 46E15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    O. Blasco, P. Galindo, A. Miralles: Bloch functions on the unit ball of an infinite dimensional Hilbert space. J. Funct. Anal. 267 (2014), 1188–1204.MathSciNetCrossRefGoogle Scholar
  2. [2]
    H. Chen: Characterizations of-Bloch functions on the unit ball without use of derivative. Sci. China Ser. A 51 (2008), 1965–1981.MathSciNetCrossRefGoogle Scholar
  3. [3]
    S. Chen, S. Ponnusamy, A. Rasila: On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces. Math. Z. 279 (2015), 163–183.MathSciNetCrossRefGoogle Scholar
  4. [4]
    J. Dai, B. Wang: Characterizations of some function spaces associated with Bloch type spaces on the unit ball of Cn. J. Inequal. Appl. 2015 (2015), 10 pages.CrossRefGoogle Scholar
  5. [5]
    F. Deng, C. Ouyang: Bloch spaces on bounded symmetric domains in complex Banach spaces. Sci. China Ser. A 49 (2006), 1625–1632.MathSciNetCrossRefGoogle Scholar
  6. [6]
    I. Graham, G. Kohr: Geometric Function Theory in One and Higher Dimensions. Monographs and Textbooks in Pure and Applied Mathematics 255, Marcel Dekker, New York, 2003.Google Scholar
  7. [7]
    K. T. Hahn: Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem. Canad. J. Math 27 (1975), 446–458.MathSciNetCrossRefGoogle Scholar
  8. [8]
    G. H. Hardy, J. E. Littlewood: Some properties of fractional integrals II. Math. Z. 34 (1932), 403–439.MathSciNetzbMATHGoogle Scholar
  9. [9]
    F. Holland, D. Walsh: Criteria for membership of Bloch space and its subspace, BMOA. Math. Ann. 273 (1986), 317–335.MathSciNetzbMATHGoogle Scholar
  10. [10]
    S. G. Krantz: Lipschitz spaces, smoothness of functions, and approximation theory. Exposition. Math. 1 (1983), 193–260.MathSciNetzbMATHGoogle Scholar
  11. [11]
    S. G. Krantz, D. Ma: Bloch functions on strongly pseudoconvex domains. Indiana Univ. Math. J. 37 (1988), 145–163.MathSciNetCrossRefGoogle Scholar
  12. [12]
    O. Lehto, K. I. Virtanen: Boundary behaviour and normal meromorphic functions. Acta Math. 97 (1957), 47–65.MathSciNetCrossRefGoogle Scholar
  13. [13]
    S. Li, H. Wulan: Characterizations of -Bloch spaces on the unit ball. J. Math. Anal. Appl. 343 (2008), 58–63.MathSciNetCrossRefGoogle Scholar
  14. [14]
    M. Nowak: Bloch space and Möbius invariant Besov spaces on the unit ball of Cn. Complex Variables, Theory Appl. 44 (2001), 1–12.MathSciNetCrossRefGoogle Scholar
  15. [15]
    M. Pavlovic: On the Holland-Walsh characterization of Bloch functions. Proc. Edinb. Math. Soc., II. Ser. 51 (2008), 439–441.MathSciNetzbMATHGoogle Scholar
  16. [16]
    G. Ren, C. Tu: Bloch space in the unit ball of Cn. Proc. Am. Math. Soc. 133 (2005), 719–726.CrossRefGoogle Scholar
  17. [17]
    G. Ren, Z. Xu: Slice Lebesgue measure of quaternions. Adv. Appl. Clifford Algebr. 26 (2016), 399–416.MathSciNetCrossRefGoogle Scholar
  18. [18]
    W. Rudin: Function Theory in the Unit Ball of Cn. Classics in Mathematics, Springer, Berlin, 2008.zbMATHGoogle Scholar
  19. [19]
    R. M. Timoney: Bloch functions in several complex variables I. Bull. Lond. Math. Soc. 12 (1980), 241–267.CrossRefGoogle Scholar
  20. [20]
    R. M. Timoney: Bloch functions in several complex variables II. J. Reine Angew. Math. 319 (1980), 1–22.MathSciNetzbMATHGoogle Scholar
  21. [21]
    W. Yang, C. Ouyang: Exact location of-Bloch spaces in Lp a and Hp of a complex unit ball. Rocky Mountain J. Math. 30 (2000), 1151–1169.MathSciNetCrossRefGoogle Scholar
  22. [22]
    M. Zhang, H. Chen: Equivalent characterizations of-Bloch functions on the unit ball. Acta Math., Sin. Engl. Ser. 27 (2011), 2395–2408.MathSciNetCrossRefGoogle Scholar
  23. [23]
    R. Zhao: A characterization of Bloch-type spaces on the unit ball of Cn. J. Math. Anal. Appl. 330 (2007), 291–297.MathSciNetCrossRefGoogle Scholar
  24. [24]
    K. Zhu: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23 (1993), 1143–1177.MathSciNetCrossRefGoogle Scholar
  25. [25]
    K. Zhu: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics 226, Springer, New York, 2005.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.School of MathematicsHefei University of TechnologyHefeiP.R. China

Personalised recommendations