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Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball
Journal of Inequalities and Applications volume 2006, Article number: 61018 (2006)
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.
Choe BR: Projections, the weighted Bergman spaces, and the Bloch space. Proceedings of the American Mathematical Society 1990,108(1):127–136. 10.1090/S0002-9939-1990-0991692-0
Clahane DD: Composition operators on holomorphic function spaces of several compex variables, M.S. thesis. University of California, Irvine; 2000.
Clahane DD, Stević S, Zhou Z: Composition operators on general Bloch spaces of the polydisk. preprint, 2004, http://arxiv.org/abs/math.CV/0506424
Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Florida; 1995.
Duren PL: Theory of Hp Spaces, Pure and Applied Mathematics. Volume 38. Academic Press, New York; 1970.
Hardy GH, Littlewood JE: Some properties of fractional integrals. II. Mathematische Zeitschrift 1932,34(1):403–439. 10.1007/BF01180596
Madigan KM: Composition operators on analytic Lipschitz spaces. Proceedings of the American Mathematical Society 1993,119(2):465–473. 10.1090/S0002-9939-1993-1152987-6
Rudin W: Function Theory in the Unit Ball of ℂn. In Fundamental Principles of Mathematical Science. Volume 241. Springer, New York; 1980.
Stević S: On an integral operator on the unit ball in. Journal of Inequalities and Applications 2005,2005(1):81–88. 10.1155/JIA.2005.81
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/1021477265
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.
Zhu K: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. Volume 226. Springer, New York; 2005.
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Clahane, D.D., Stević, S. Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball. J Inequal Appl 2006, 61018 (2006). https://doi.org/10.1155/JIA/2006/61018
- Unit Ball
- Open Unit
- Composition Operator
- Lipschitz Space
- Open Unit Ball