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

 

The multiplicative Cousin problem and a zero set for the Nevanlinna class in the polydisc


Author: Sergio E. Zarantonello
Journal: Trans. Amer. Math. Soc. 200 (1974), 291-313
MSC: Primary 32A30
MathSciNet review: 0355092
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Abstract: Let $ \Omega $ be a polydomain in $ {{\mathbf{C}}^n}$, the Nevanlinna class $ N(\Omega )$ consists of all holomorphic functions $ f$ in $ \Omega $ such that $ {\log ^ + }\vert f\vert$ has an $ n$-harmonic majorant in $ \Omega $. Let $ {U^n}$ be the open unit polydisc $ \{ z \in {{\mathbf{C}}^n}:\vert{z_1}\vert < 1, \cdots ,\vert{z_n}\vert < 1\} $. THEOREM 1. Given an open covering $ {({\Omega _\alpha })_{\alpha \in A}}$ of the closure $ {\bar U^n}$ of the polydisc, consisting of polydomains, and for each $ \alpha \in A$ a function $ {f_\alpha } \in N({\Omega _\alpha } \cap {U^n})$ such that for all $ \alpha ,\beta \in A,{f_\alpha }f_\beta ^{ - 1}$ is an invertible element of $ N({\Omega _\alpha } \cap {\Omega _\beta } \cap {U^n})$. There exists a function $ F \in N({U^n})$ such that for all $ \alpha \in A,Ff_\alpha ^{ - 1}$ is an invertible element of $ N({\Omega _\alpha } \cap {U^n})$. This result enables us to find the following sufficient condition for the zero sets of $ N({U^n})$: THEOREM 2. Let $ f$ be a holomorphic function in $ {U^n},n \geqslant 2$. If there exists a constant $ 0 < r < 1$ and a continuous function $ n:[r,1) \to [r,1)$ such that

$\displaystyle \vert{z_n}\vert \leqslant n\left( {\frac{{\vert{z_1}\vert + \cdots + \vert{z_{n - 1}}\vert}}{{n - 1}}} \right)$

for all points $ ({z_1}, \cdots ,{z_n})$ satisfying $ \vert{z_1}\vert > r, \cdots ,\vert{z_n}\vert > r$ and $ f({z_1}, \cdots ,{z_n}) = 0$, then $ f$ has the same zeros as some function $ F \in N({U^n})$. In the above if $ {\overline {\lim } _{\lambda \to 1}}n(x) < 1$, then $ Z(f)$ is a Rudin variety in which case there is a bounded holomorphic function with the same zeros as $ f$.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0355092-4
PII: S 0002-9947(1974)0355092-4
Keywords: Nevanlinna class, polydisc, multiplicative Cousin problem, holomorphic, $ n$-harmonic, Poisson kernel, Poisson integral, Green's function
Article copyright: © Copyright 1974 American Mathematical Society