Skip to content


  • Research Article
  • Open Access

Volterra-Type Operators on Zygmund Spaces

Journal of Inequalities and Applications20072007:032124

  • Received: 26 November 2006
  • Accepted: 4 March 2007
  • Published:


The boundedness and the compactness of the two integral operators ; , where is an analytic function on the open unit disk in the complex plane, on the Zygmund space are studied.


  • Analytic Function
  • Complex Plane
  • Integral Operator
  • Unit Disk
  • Open Unit


Authors’ Affiliations

Department of Mathematics, Shantou University, Shantou, Guang Dong, 515063, China
Department of Mathematics, Jia Ying University, Meizhou, Guang Dong, 514015, China
Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 35/I, Beograd, 11000, Serbia


  1. Duren PL: Theory of H p Spaces, Pure and Applied Mathematics. Volume 38. Academic Press, New York, NY, USA; 1970:xii+258.Google Scholar
  2. Pommerenke Ch: Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Commentarii Mathematici Helvetici 1977,52(4):591–602.MathSciNetView ArticleMATHGoogle Scholar
  3. Aleman A, Siskakis AG: An integral operator on . Complex Variables. Theory and Application 1995,28(2):149–158. 10.1080/17476939508814844MathSciNetView ArticleMATHGoogle Scholar
  4. Aleman A, Siskakis AG: Integration operators on Bergman spaces. Indiana University Mathematics Journal 1997,46(2):337–356.MathSciNetView ArticleMATHGoogle Scholar
  5. Aleman A, Cima JA: An integral operator on and Hardy's inequality. Journal d'Analyse Mathématique 2001, 85: 157–176.MathSciNetView ArticleMATHGoogle Scholar
  6. Benke G, Chang D-C: A note on weighted Bergman spaces and the Cesàro operator. Nagoya Mathematical Journal 2000, 159: 25–43.MathSciNetMATHGoogle Scholar
  7. Chang D-C, Gilbert R, Tie J: Bergman projection and weighted holomorphic functions. In Reproducing Kernel Spaces and Applications, Oper. Theory Adv. Appl.. Volume 143. Birkhäuser, Basel, Switzerland; 2003:147–169.View ArticleGoogle Scholar
  8. Chang D-C, Stević S: The generalized Cesàro operator on the unit polydisk. Taiwanese Journal of Mathematics 2003,7(2):293–308.MathSciNetMATHGoogle Scholar
  9. Hu Z: Extended Cesàro operators on mixed norm spaces. Proceedings of the American Mathematical Society 2003,131(7):2171–2179. 10.1090/S0002-9939-02-06777-1MathSciNetView ArticleMATHGoogle Scholar
  10. Hu Z: Extended Cesáro operators on the Bloch space in the unit ball of . Acta Mathematica Scientia. Series B. English Edition 2003,23(4):561–566.MathSciNetMATHGoogle Scholar
  11. Li S: Riemann-Stieltjes operators from spaces to -Bloch spaces on the unit ball. Journal of Inequalities and Applications 2006, 2006: 14 pages.View ArticleGoogle Scholar
  12. Li S, Stević S: Reimann-Stieltjies type integral operators on the unit ball in . Complex Variables Elliptic Functions 2007., 2:Google Scholar
  13. Siskakis AG, Zhao R: A Volterra type operator on spaces of analytic functions. In Function Spaces (Edwardsville, IL, 1998), Contemp. Math.. Volume 232. American Mathematical Society, Providence, RI, USA; 1999:299–311.Google Scholar
  14. Stević S: Cesàro averaging operators. Mathematische Nachrichten 2003,248/249(1):185–189. 10.1002/mana.200310013View ArticleGoogle Scholar
  15. Stević S: On an integral operator on the unit ball in . Journal of Inequalities and Applications 2005,2005(1):81–88. 10.1155/JIA.2005.81MATHGoogle Scholar
  16. Stević S: Boundedness and compactness of an integral operator on a weighted space on the polydisc. Indian Journal of Pure and Applied Mathematics 2006,37(6):343–355.MathSciNetMATHGoogle Scholar
  17. Tamrazov PM: Contour and solid structure properties of holomorphic functions of a complex variable. Russian Mathematical Surveys 1973,28(1):141–1731. 10.1070/RM1973v028n01ABEH001398MathSciNetView ArticleMATHGoogle Scholar
  18. Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, Fla, USA; 1995:xii+388.Google Scholar
  19. Madigan K, Matheson A: Compact composition operators on the Bloch space. Transactions of the American Mathematical Society 1995,347(7):2679–2687. 10.2307/2154848MathSciNetView ArticleMATHGoogle Scholar


© S. Li and S. Stević 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.