On the Closures of Dirichlet Type Spaces in the Bloch Space



In this paper, via high order derivatives and via embedding derivatives of Bloch type functions into Lebesgue spaces, we characterize the closures of Dirichlet type spaces in the Bloch space. We obtain the inclusion relation between the closures and the little Bloch space. We consider the separability of the closures as Banach spaces. A criterion for an interpolating Blaschke product to be in the closures is given. We also consider the relation between the closures and the space of bounded analytic functions.


The Bloch space Dirichlet type spaces Closure Interpolating Blaschke product 

Mathematics Subject Classification

30H30 30J10 46E15 



The authors thank the referee and the editor for providing useful suggestions. The work was done while G. Bao was at Sabanci University from 01 February 2016 to 31 January 2017. It is his pleasure to acknowledge the excellent working environment provided to him there.


  1. 1.
    Anderson, J., Clunie, J., Pommerenke, Ch.: On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12–37 (1974)MathSciNetMATHGoogle Scholar
  2. 2.
    Arazy, J., Fisher, S., Peetre, J.: Möbius invariant function spaces. J. Reine Angew. Math. 363, 110–145 (1985)MathSciNetMATHGoogle Scholar
  3. 3.
    Aulaskari, R., Zhao, R.: Composition operators and closures of some Möbius invariant spaces in the Bloch space. Math. Scand. 107, 139–149 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bao, G., Pau, J.: Boundary multipliers of a family of Möbius invariant function spaces. Ann. Acad. Sci. Fenn. Math. 41, 199–220 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bao, G., Wulan, H.: Hankel matrices acting on Dirichlet spaces. J. Math. Anal. Appl. 409, 228–235 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bao, G., Yang, J.: The Libera operator on Dirichlet spaces. Bull. Iran. Math. Soc. 41, 1511–1517 (2015)MathSciNetGoogle Scholar
  7. 7.
    Carleson, L.: An interpolation problem for bounded analytic functions. Am. J. Math. 80, 921–930 (1958)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Carleson, L.: Interpolation by bounded analytic functions and the corona problem. Ann. of Math. 76, 547–559 (1962)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Duren, P.: Theory of \(H^p\) Spaces. Academic Press, New York (1970)MATHGoogle Scholar
  10. 10.
    Galanopoulos, P., Monreal Galán, N., Pau, J.: Closure of Hardy spaces in the Bloch space. J. Math. Anal. Appl. 429, 1214–1221 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Garnett, J.: Bounded Analytic Functions. Springer, New York (2007)MATHGoogle Scholar
  12. 12.
    Ghatage, P., Zheng, D.: Analytic functions of bounded mean oscillation and the Bloch space. Integral Equ. Oper. Theory 17, 501–515 (1993)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Girela, D., Peláez, J.A., Pérez-González, F., Rättyä, J.: Carleson measures for the Bloch space. Integral Equ. Oper. Theory 61, 511–547 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Girela, D., Peláez, J.A., Vukotić, D.: Integrability of the derivative of a Blaschke product. Proc. Edinb. Math. Soc. 50, 673–687 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kaptanoğlu, H.T., Tülü, S.: Weighted Bloch, Lipschitz, Zygmund, Bers, and growth spaces of the ball: Bergman projections and characterizations. Taiwan. J. Math. 15, 101–127 (2011)MathSciNetMATHGoogle Scholar
  16. 16.
    Lou, Z.: Composition operators on Bloch type spaces. Analysis (Munich) 23, 81–95 (2003)MathSciNetMATHGoogle Scholar
  17. 17.
    Monreal Galán, N., Nicolau, A.: The closure of the Hardy space in the Bloch norm. Algebra i Anal. 22, 75–81 (2010). (translation in St. Petersburg Math. J. 22, 55–59 (2011)) MathSciNetGoogle Scholar
  18. 18.
    Nicolau, A., Xiao, J.: Bounded functions in Möbius invariant Dirichlet spaces. J. Funct. Anal. 150, 383–425 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pau, J., Peláez, J.A.: On the zeros of functions in Dirichlet-type spaces. Trans. Am. Math. Soc. 363, 1981–2002 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pau, J., Zhao, R.: Carleson measures, Riemann–Stieltjes and multiplication operators on a general family of function spaces. Integral Equ. Oper. Theory 78, 483–514 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Pérez-González, F., Rättyä, J.: Forelli–Rudin estimates, Carleson measures and \(F(p, q, s)\)-functions. J. Math. Anal. Appl. 315, 394–414 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rochberg, R., Wu, Z.: A new characterization of Dirichlet type spaces and applications. Ill. J. Math. 37, 101–122 (1993)MathSciNetMATHGoogle Scholar
  23. 23.
    Stegenga, D.: Multipliers of the Dirichlet space. Ill. J. Math. 24, 113–139 (1980)MathSciNetMATHGoogle Scholar
  24. 24.
    Tjani, M.: Distance of a Bloch function to the little Bloch space. Bull. Aust. Math. Soc. 74, 101–119 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Xiao, J.: Holomorphic \({\cal{Q}}\) Classes. Springer, Berlin (2001). (LNM 1767) CrossRefGoogle Scholar
  26. 26.
    Xiao, J.: Geometric \(\cal{Q}_p\) Functions. Birkhäuser Verlag, Basel (2006)Google Scholar
  27. 27.
    Zhao, R.: Distances from Bloch functions to some Möbius invariant spaces. Ann. Acad. Sci. Fenn. Math. 33, 303–313 (2008)MathSciNetMATHGoogle Scholar
  28. 28.
    Zhu, K.: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23, 1143–1177 (1993)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zhu, K.: Operator Theory in Function Spaces. American Mathematical Society, Providence (2007)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsShantou UniversityShantouChina
  2. 2.Faculty of Engineering and Natural SciencesSabanci UniversityTuzla, IstanbulTurkey

Personalised recommendations