## Abstract and Applied Analysis

### Composition Operators in Hyperbolic Bloch-Type and $F(p,q,s)$ Spaces

#### Abstract

Composition operators ${C}_{\phi }$ from Bloch-type ${\scr B}_{\alpha }$ spaces to $F(p,q,s)$ classes, from $F(p,q,s)$ to ${\scr B}_{\alpha }$, 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

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

#### 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