Skip to content

Advertisement

  • Research Article
  • Open Access

Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball

Journal of Inequalities and Applications20062006:61018

https://doi.org/10.1155/JIA/2006/61018

  • Received: 11 October 2005
  • Accepted: 12 February 2006
  • Published:

Abstract

For , let and denote, respectively, the -Bloch and holomorphic -Lipschitz spaces of the open unit ball in . It is known that and are equal as sets when . We prove that these spaces are additionally norm-equivalent, thus extending known results for and the polydisk. As an application, we generalize work by Madigan on the disk by investigating boundedness of the composition operator from to .

Keywords

  • Unit Ball
  • Open Unit
  • Composition Operator
  • Lipschitz Space
  • Open Unit Ball

[123456789101112]

Authors’ Affiliations

(1)
Department of Mathematics, University of California, Riverside, CA 92521, USA
(2)
Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd, 11000, Serbia

References

  1. Choe BR: Projections, the weighted Bergman spaces, and the Bloch space. Proceedings of the American Mathematical Society 1990,108(1):127–136. 10.1090/S0002-9939-1990-0991692-0MATHMathSciNetView ArticleGoogle Scholar
  2. Clahane DD: Composition operators on holomorphic function spaces of several compex variables, M.S. thesis. University of California, Irvine; 2000.Google Scholar
  3. Clahane DD, Stević S, Zhou Z: Composition operators on general Bloch spaces of the polydisk. preprint, 2004, http://arxiv.org/abs/math.CV/0506424Google Scholar
  4. Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Florida; 1995.MATHGoogle Scholar
  5. Duren PL: Theory of H p Spaces, Pure and Applied Mathematics. Volume 38. Academic Press, New York; 1970.Google Scholar
  6. Hardy GH, Littlewood JE: Some properties of fractional integrals. II. Mathematische Zeitschrift 1932,34(1):403–439. 10.1007/BF01180596MathSciNetView ArticleMATHGoogle Scholar
  7. Madigan KM: Composition operators on analytic Lipschitz spaces. Proceedings of the American Mathematical Society 1993,119(2):465–473. 10.1090/S0002-9939-1993-1152987-6MATHMathSciNetView ArticleGoogle Scholar
  8. Rudin W: Function Theory in the Unit Ball of n . In Fundamental Principles of Mathematical Science. Volume 241. Springer, New York; 1980.Google Scholar
  9. 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
  10. Yang W, Ouyang C: Exact location of-Bloch spaces inandof a complex unit ball. The Rocky Mountain Journal of Mathematics 2000,30(3):1151–1169. 10.1216/rmjm/1021477265MATHMathSciNetView ArticleGoogle Scholar
  11. Zhou Z, Zeng H: Composition operators between-Bloch and-Bloch space in the unit ball. Progress in Natural Science. English Edition 2003,13(3):233–236.MATHMathSciNetGoogle Scholar
  12. Zhu K: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Volume 226. Springer, New York; 2005.Google Scholar

Copyright

Advertisement