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On the Closures of Dirichlet Type Spaces in the Bloch Space

  • Guanlong Bao
  • Nihat Gökhan Göğüş
Article

Abstract

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.

Keywords

The Bloch space Dirichlet type spaces Closure Interpolating Blaschke product 

Mathematics Subject Classification

30H30 30J10 46E15 

Notes

Acknowledgements

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.

References

  1. 1.
    Anderson, J., Clunie, J., Pommerenke, Ch.: On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12–37 (1974)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arazy, J., Fisher, S., Peetre, J.: Möbius invariant function spaces. J. Reine Angew. Math. 363, 110–145 (1985)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bao, G., Wulan, H.: Hankel matrices acting on Dirichlet spaces. J. Math. Anal. Appl. 409, 228–235 (2014)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carleson, L.: Interpolation by bounded analytic functions and the corona problem. Ann. of Math. 76, 547–559 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duren, P.: Theory of \(H^p\) Spaces. Academic Press, New York (1970)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Garnett, J.: Bounded Analytic Functions. Springer, New York (2007)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lou, Z.: Composition operators on Bloch type spaces. Analysis (Munich) 23, 81–95 (2003)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rochberg, R., Wu, Z.: A new characterization of Dirichlet type spaces and applications. Ill. J. Math. 37, 101–122 (1993)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Stegenga, D.: Multipliers of the Dirichlet space. Ill. J. Math. 24, 113–139 (1980)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tjani, M.: Distance of a Bloch function to the little Bloch space. Bull. Aust. Math. Soc. 74, 101–119 (2006)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Zhu, K.: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23, 1143–1177 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhu, K.: Operator Theory in Function Spaces. American Mathematical Society, Providence (2007)CrossRefzbMATHGoogle 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

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