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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Boundary interpolation sets for holomorphic functions smooth to the boundary and BMO


Author: Joaquim Bruna
Journal: Trans. Amer. Math. Soc. 264 (1981), 393-409
MSC: Primary 30E05; Secondary 30D60, 42A50
MathSciNet review: 603770
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Abstract: Let $ {A^p}$ denote the class of holomorphic functions on the unit disc whose first $ p$-derivatives belong to the disc algebra. We characterize the boundary interpolation sets for $ {A^p}$, that is, those closed sets $ E \subset T$ such that every function in $ {C^p}(E)$ extends to a function in $ {A^p}$.

We also give a constructive proof of the corresponding result for $ {A^\infty }$ (see [1]).

We show that the structure of these sets is in some sense related to BMO and that this fact can be used to obtain precise estimates of outer functions vanishing on $ E$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0603770-5
PII: S 0002-9947(1981)0603770-5
Keywords: Boundary interpolation sets, Carleson sets, Lipschitz conditions, BMO, BMOA, outer functions
Article copyright: © Copyright 1981 American Mathematical Society