Open Access

Riemann-Stieltjes operators from spaces to -Bloch spaces on the unit ball

Journal of Inequalities and Applications20062006:27874

DOI: 10.1155/JIA/2006/27874

Received: 5 December 2005

Accepted: 19 April 2006

Published: 28 August 2006


Let denote the space of all holomorphic functions on the unit ball . We investigate the following integral operators: , , , , where , and is the radial derivative of . The operator can be considered as an extension of the Cesàro operator on the unit disk. The boundedness of two classes of Riemann-Stieltjes operators from general function space , which includes Hardy space, Bergman space, space, BMOA space, and Bloch space, to -Bloch space in the unit ball is discussed in this paper.


Authors’ Affiliations

Department of Mathematics, JiaYing University
Department of Mathematics, Shantou University


  1. Aleman A, Cima JA: An integral operator onand Hardy's inequality. Journal d'Analyse Mathématique 2001, 85: 157–176.MATHMathSciNetView ArticleGoogle Scholar
  2. Aleman A, Siskakis AG: An integral operator on. Complex Variables. Theory and Application 1995,28(2):149–158. 10.1080/17476939508814844MATHMathSciNetView ArticleGoogle Scholar
  3. Aleman A, Siskakis AG: Integration operators on Bergman spaces. Indiana University Mathematics Journal 1997,46(2):337–356.MATHMathSciNetView ArticleGoogle Scholar
  4. Hardy GH, Littlewood JE: Some properties of fractional integrals. II. Mathematische Zeitschrift 1932,34(1):403–439. 10.1007/BF01180596MathSciNetView ArticleMATHGoogle Scholar
  5. Hu ZJ: Extended Cesàro operators on mixed norm spaces. Proceedings of the American Mathematical Society 2003,131(7):2171–2179. 10.1090/S0002-9939-02-06777-1MATHMathSciNetView ArticleGoogle Scholar
  6. Hu ZJ: Extended Cesáro operators on the Bloch space in the unit ball of. Acta Mathematica Scientia. Series B. English Edition 2003,23(4):561–566.MATHMathSciNetGoogle Scholar
  7. Hu ZJ: Extended Cesàro operators on Bergman spaces. Journal of Mathematical Analysis and Applications 2004,296(2):435–454. 10.1016/j.jmaa.2004.01.045MATHMathSciNetView ArticleGoogle Scholar
  8. Miao J: The Cesàro operator is bounded onfor. Proceedings of the American Mathematical Society 1992,116(4):1077–1079.MATHMathSciNetGoogle Scholar
  9. Ouyang C, Yang W, Zhao R: Möbius invariantspaces associated with the Green's function on the unit ball of. Pacific Journal of Mathematics 1998,182(1):69–99. 10.2140/pjm.1998.182.69MathSciNetView ArticleMATHGoogle Scholar
  10. Pommerenke C: Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Commentarii Mathematici Helvetici 1977,52(4):591–602.MATHMathSciNetView ArticleGoogle Scholar
  11. Shi J-H, Ren G-B: Boundedness of the Cesàro operator on mixed norm spaces. Proceedings of the American Mathematical Society 1998,126(12):3553–3560. 10.1090/S0002-9939-98-04514-6MATHMathSciNetView ArticleGoogle Scholar
  12. Siskakis AG: Composition semigroups and the Cesàro operator on. Journal of the London Mathematical Society. Second Series 1987,36(1):153–164. 10.1112/jlms/s2-36.1.153MATHMathSciNetView ArticleGoogle Scholar
  13. Siskakis AG: The Cesàro operator is bounded on. Proceedings of the American Mathematical Society 1990,110(2):461–462.MATHMathSciNetGoogle Scholar
  14. 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, Rhode Island; 1999:299–311.Google 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.2005.81MATHMathSciNetGoogle Scholar
  16. Xiao J: Cesàro-type operators on Hardy, BMOA and Bloch spaces. Archiv der Mathematik 1997,68(5):398–406. 10.1007/s000130050072MATHMathSciNetView ArticleGoogle Scholar
  17. Xiao J: Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball. Journal of the London Mathematical Society. Second Series 2004,70(1):199–214. 10.1112/S0024610704005484MATHMathSciNetView ArticleGoogle Scholar
  18. Yoneda R: Pointwise multipliers fromto. Complex Variables. Theory and Application 2004,49(14):1045–1061.MATHMathSciNetView ArticleGoogle Scholar
  19. Zhang XJ: Multipliers on some holomorphic function spaces. Chinese Annals of Mathematics. Series A 2005,26(4):477–486.MATHMathSciNetGoogle Scholar
  20. Zhao R: On a general family of function spaces. Annales Academiae Scientiarum Fennicae. Mathematica. Dissertationes 1996, (105):56.
  21. Zhu K: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Volume 226. Springer, New York; 2005:x+271.MATHGoogle Scholar


© Li 2006

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