Taiwanese Journal of Mathematics

GENERALIZED INTEGRATION OPERATORS BETWEEN BLOCH-TYPE SPACES AND $F(p,q,s)$ SPACES

Zhong Hua He and Guangfu Cao

Full-text: Open access

Abstract

Let $H(\mathbb{D})$ denote the space of all holomorphic functions on the unit disk $\mathbb{D}$ of $\mathbb{C}$. Let $\varphi$ be a holomorphic self-map of $\mathbb{D}$, $n$ be a positive integer and $g\in H(\mathbb{D})$. In this paper, we investigate the boundedness and compactness of a generalized integration operator $$ I^{(n)}_{g,\varphi}f(z) = \int^z_0 f^{(n)}(\varphi(\zeta)) g(\zeta) d\zeta,\ \ z \in \mathbb{D},$$ between Bloch-type spaces and $F(p,q,s)$ spaces.

Article information

Source
Taiwanese J. Math., Volume 17, Number 4 (2013), 1211-1225.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706113

Digital Object Identifier
doi:10.11650/tjm.17.2013.2658

Mathematical Reviews number (MathSciNet)
MR3085507

Zentralblatt MATH identifier
1295.47046

Subjects
Primary: 47G10: Integral operators [See also 45P05] 30H05: Bounded analytic functions

Keywords
generalized integration operator Bloch-type space $F(p,q,s)$ space

Citation

He, Zhong Hua; Cao, Guangfu. GENERALIZED INTEGRATION OPERATORS BETWEEN BLOCH-TYPE SPACES AND $F(p,q,s)$ SPACES. Taiwanese J. Math. 17 (2013), no. 4, 1211--1225. doi:10.11650/tjm.17.2013.2658. https://projecteuclid.org/euclid.twjm/1499706113


Export citation

References

  • A. Aleman and A. G. Siskakis, An integral operator on $H^p$, Complex Variables Theory Appl., 28(2) (1995), 149-158.
  • A. Aleman and J. A. Cima, An integral operator on Hp and Hardy's inequality, J. Anal. Math., 85 (2001), 157-176.
  • J. Arazy, S. D. Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math., 363 (1985), 110-145.
  • C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. xii+388 pp. ISBN: 0-8493-8492-3.
  • R. A. Hibschweiler and N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math., 35(3) (2005), 843-855.
  • S. Li and S. Stević, Integral type operators from mixed-norm spaces to $\alpha$-Bloch spaces, Integral Transforms Spec. Funct., 18(7) (2007), 485-493.
  • S. Li and S. Stević, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl., 338(2) (2008), 1282-1295.
  • S. Li and S. Stević, Products of composition and integral type operators from $H^{\infty}$ to the Bloch space, Complex Var. Elliptic Equ., 53(5) (2008), 463-474.
  • S. Li and S. Stević, Products of Volterra type operator and composition operator from $H^{\infty}$ and Bloch spaces to Zygmund spaces, J. Math. Anal. Appl., 345(1) (2008), 40-52.
  • S. Li and S. Stević, Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. Appl., 349(12) (2009), 596-610.
  • S. Ohno, Products of composition and differentiation between Hardy spaces, Bull. Austral. Math. Soc., 73(2) (2006), 235-243.
  • F. Pérez-Gonzalez and J. Rattya, Forelli-Rudin estimates, Carleson measures and $F$($p,q,s$)-functions, J. Math. Anal. Appl., 315(2) (2006), 394-414.
  • Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschrankter mittlerer Oszillation (German), Comment. Math. Helv., 52(4) (1977), 591-602.
  • W. Ramey and D. Ullrich, Bounded mean oscillation of Bloch pull-backs, Math. Ann., 291(4) (1991), 591-606.
  • J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. xvi+223 pp. ISBN: 0-387-94067-7.
  • S. D. Sharma and A. Sharma, Generalized integration operators from Bloch type spaces to weighted BMOA spaces, Demonstratio Math., 44(2) (2011), 373-390.
  • A. L. Shields and D. L. Williams, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc., 162 (1971), 287-302.
  • S. Stević, On $\alpha$-Bloch spaces with Hadamard gaps, Abstr. Appl. Anal., 2007, Art. ID 39176, 2007, 8 pp.
  • S. Stević, Generalized composition operators from logarithmic Bloch spaces to mixed-norm spaces, Util. Math., 77 (2008), 167-172.
  • S. Stević, A. K. Sharma and S. D. Sharma, Generalized integration operators from the space of integral transforms into Bloch-type spaces, J. Comput. Anal. Appl., 14(6) (2012), 1139-1147.
  • M. Tjani, Compact Composition Operators on Some Moebius Invariant Banach Spaces, Thesis (PhD.), Michigan State University, 1996, 68 pp. ISBN: 978-0591-27288-8.
  • S. Yamashita, Gap series and -Bloch functions, Yokohama Math. J., 28(1-2) (1980), 31-36.
  • W. Yang, Generalized weighted composition operators from the $F(p,q,s)$ space to the Bloch-type space, Appl. Math. Comput., 218(9) (2012), 4967-4972.
  • W. Yang, Composition operators from $F(p,q,s)$ spaces to the nth weighted-type spaces on the unit disc, Appl. Math. Comput., 218(4) (2011), 1443-1448.
  • R. Zhao, Composition operators from Bloch type spaces to Hardy and Besov spaces, J. Math. Anal. Appl., 233(2) (1999), 749-766.
  • R. Zhao, On a General Family of Function Spaces, Ann. Acad. Sci. Fenn. Math. Diss., No. 105, 1996, 56 pp.
  • K. Zhu, Operator Theory in Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 139, Marcel Dekker, Inc., New York, 1990. xii+258 pp. ISBN: 0-8247-8411-1.
  • K. Zhu, Bloch type spaces of analytic functions, Rochy Mountain J. Math., 23 (1993), 1143-1177.
  • X. Zhu, Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces, Indian J. Math., 49(2) (2007), 139-150.
  • X. Zhu, Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces, Integral Transforms Spec. Funct., 18(3-4) (2007), 223-231.
  • X. Zhu, Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces, J. Korean Math. Soc., 46(6) (2009), 1219-1232.
  • A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959.