Abstract and Applied Analysis

Composition Operators in Hyperbolic Bloch-Type and F p , q , s Spaces

Marko Kotilainen and Fernando Pérez-González

Full-text: Open access

Abstract

Composition operators C φ from Bloch-type α spaces to F p , q , s classes, from F p , q , s to α , and from F p 1 , q 1 , 0 to F p 2 , q 2 , s 2 are considered. The criteria for these operators to be bounded or compact are given. Our study also includes the corresponding hyperbolic spaces.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 156353, 10 pages.

Dates
First available in Project Euclid: 7 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412687018

Digital Object Identifier
doi:10.1155/2014/156353

Mathematical Reviews number (MathSciNet)
MR3198150

Zentralblatt MATH identifier
1259.35180

Citation

Kotilainen, Marko; Pérez-González, Fernando. Composition Operators in Hyperbolic Bloch-Type and $F(p,q,s)$ Spaces. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 156353, 10 pages. doi:10.1155/2014/156353. https://projecteuclid.org/euclid.aaa/1412687018


Export citation

References

  • C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.
  • J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer, New York, NY, USA, 1993.
  • J. M. Anderson, J. Clunie, and Ch. Pommerenke, “On Bloch functions and normal functions,” Journal für die Reine und Angewandte Mathematik, vol. 270, pp. 12–37, 1974.
  • Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin, Germany, 1992.
  • K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, NY, USA, 1990.
  • R. Zhao, “On a general family of function spaces,” Annales Academiæ Scientiarium Fennicæ. Mathematica, no. 105, pp. 1–56, 1996.
  • J. Rättyä, “On some complex function spaces and classes,” Annales Academiæ Scientiarium Fennicæ. Mathematica, no. 124, pp. 1–73, 2001.
  • X. Li, F. Pérez-González, and J. Rättyä, “Composition operators in hyperbolic $Q$-classes,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 31, no. 2, pp. 391–404, 2006.
  • W. Smith and R. Zhao, “Composition operators mapping into the ${Q}_{p}$ spaces,” Analysis. International Mathematical Journal of Analysis and its Applications, vol. 17, no. 2-3, pp. 239–263, 1997.
  • J. Xiao, Holomorphic Q Classes, vol. 1767 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2001.
  • W. Yang, “Composition operators from $F(p,q,s)$ spaces to the $n$th weighted-type spaces on the unit disc,” Applied Mathematics and Computation, vol. 218, no. 4, pp. 1443–1448, 2011.
  • X. Zhu, “Composition operators from Bloch type spaces to $F(p,q,s)$ spaces,” Univerzitet u Nišu. Prirodno-Matematički Fakultet. Filomat, vol. 21, no. 2, pp. 11–20, 2007.
  • S. Ye, “Weighted composition operators from $F(p,q,s)$ into logarithmic Bloch space,” Journal of the Korean Mathematical Society, vol. 45, no. 4, pp. 977–991, 2008.
  • F. Colonna and S. Li, “Weighted composition operators from the minimal Möbius invariant space into the Bloch space,” Mediterranean Journal of Mathematics, vol. 10, no. 1, pp. 395–409, 2013.
  • J. Liu, Z. Lou, and A. K. Sharma, “Weighted differentiation composition operators to Bloch-type spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 151929, 9 pages, 2013.
  • Y. He and L. Jiang, “Composition operators from ${B}^{\alpha }$ to $F(p,q,s)$,” Acta Mathematica Scientia B, vol. 23, no. 2, pp. 252–260, 2003.
  • R. Zhao, “On $\alpha $-Bloch functions and VMOA,” Acta Mathematica Scientia B, vol. 16, no. 3, pp. 349–360, 1996.
  • J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, NY, USA, 1981.
  • J. Arazy, S. D. Fisher, and J. Peetre, “Möbius invariant function spaces,” Journal für die Reine und Angewandte Mathematik, vol. 363, pp. 110–145, 1985.
  • R. Aulaskari, D. A. Stegenga, and J. Xiao, “Some subclasses of BMOA and their characterization in terms of Carleson measures,” The Rocky Mountain Journal of Mathematics, vol. 26, no. 2, pp. 485–506, 1996.
  • D. H. Luecking, “Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives,” American Journal of Mathematics, vol. 107, no. 1, pp. 85–111, 1985.
  • F. Pérez-González and J. Rättyä, “Forelli-Rudin estimates, Carleson measures and $F(p,q,s)$-functions,” Journal of Mathematical Analysis and Applications, vol. 315, no. 2, pp. 394–414, 2006.
  • M. Essén, D. F. Shea, and C. S. Stanton, “A value-distribution criterion for the class \emphL log \emphL, and some related questions,” Annales de l'Institute Fourier, vol. 35, no. 4, pp. 127–150, 1985.
  • C. S. Stanton, “Counting functions and majorization for Jensen measures,” Pacific Journal of Mathematics, vol. 125, no. 2, pp. 459–468, 1986.
  • J. H. Shapiro, “The essential norm of a composition operator,” Annals of Mathematics, vol. 125, no. 2, pp. 375–404, 1987.
  • M. Tjani, “Compact composition operators on Besov spaces,” Transactions of the American Mathematical Society, vol. 355, no. 11, pp. 4683–4698, 2003.
  • S. Yamashita, “Gap series and $\alpha $-Bloch functions,” Yokohama Mathematical Journal, vol. 28, no. 1-2, pp. 31–36, 1980.
  • A. Zygmund, Trigonometric Series, Cambridge University Press, London, UK, 1959.
  • M. Kotilainen, “On composition operators in ${Q}_{K}$ type spaces,” Journal of Function Spaces and Applications, vol. 5, no. 2, pp. 103–122, 2007.
  • J. M. Ortega and J. Fàbrega, “Pointwise multipliers and corona type decomposition in BMOA,” Annales de l'Institute Fourier, vol. 46, no. 1, pp. 111–137, 1996. \endinput