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A remark on Bourgain algebras on the disk


Authors: Pratibha G. Ghatage, Shun Hua Sun and De Chao Zheng
Journal: Proc. Amer. Math. Soc. 114 (1992), 395-398
MSC: Primary 46J10; Secondary 30D55, 30H05, 46J15
DOI: https://doi.org/10.1090/S0002-9939-1992-1065947-X
MathSciNet review: 1065947
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Abstract: It is shown that the Bourgain algebra $ {X_b}$ of the space $ X = {H^\infty }$ considered as a subalgebra of $ \mathcal{U} = \operatorname{alg} \{ {H^\infty },{\overline H ^\infty }\} $ is $ {H^\infty }(\mathbb{D}) + UC(\mathbb{D})$ where $ UC(\mathbb{D})$ is the algebra of uniformly continuous functions on the open unit disk $ \mathbb{D}$. This uses and extends a recent result of Cima-Janson-Yale on the Bourgain algebra of $ {H^\infty }$ on $ \partial \mathbb{D}$. Further, $ {({X_b})_b} = {X_b}$.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1065947-X
Keywords: Bourgain algebra, zero set, thin Blaschke products
Article copyright: © Copyright 1992 American Mathematical Society