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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the extension of $ H\sp{p}$-functions in polydiscs


Author: P. S. Chee
Journal: Proc. Amer. Math. Soc. 88 (1983), 270-274
MSC: Primary 32A35; Secondary 32D15, 46J15
MathSciNet review: 695257
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Abstract: For $ N = 2\;{\text{or}}\;3$ it is shown that if $ E$ is the zero set of a holomorphic function in $ {U^N}$ satisfying the separation condition of Alexander [1], viz., there exist $ r \in (0,1)$ and $ \delta > 0$ such that $ \left\vert {\alpha - \beta } \right\vert \geqslant \delta $ whenever $ (z',\alpha ,z'') \ne (z',\beta ,z'')$ are both in $ ({Q^{k - 1}} \times U \times {Q^{N - k}}) \cap E$, where $ Q = \{ \lambda \in {\mathbf{C}}:r < \left\vert \lambda \right\vert < 1\} $, then (a) $ E$ is the zero set of some $ F \in {H^\infty }({U^N})$, and (b) $ 0 < p \leqslant \infty $, every $ g \in H(E)$ such that $ {\left\vert g \right\vert^p}$ has a pluriharmonic majorant on $ E$ extends to a $ G \in {H^p}({U^N})$. This generalizes earlier results of the author [3] and Zarantonello [9].


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0695257-6
PII: S 0002-9939(1983)0695257-6
Keywords: Polydiscs, Hardy spaces, zero sets, extensions of $ {H^p}$-functions
Article copyright: © Copyright 1983 American Mathematical Society