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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)
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.
References
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 â„‚n. 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
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DOI: https://doi.org/10.1155/JIA/2006/61018