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

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 ${\log ^ + }|f|$ has an $n$-harmonic majorant in $\Omega$. Let ${U^n}$ be the open unit polydisc $\{ z \in {{\mathbf {C}}^n}:|{z_1}| < 1, \cdots ,|{z_n}| < 1\}$. 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 \[ |{z_n}| \leqslant n\left ( {\frac {{|{z_1}| + \cdots + |{z_{n - 1}}|}}{{n - 1}}} \right )\] for all points $({z_1}, \cdots ,{z_n})$ satisfying $|{z_1}| > r, \cdots ,|{z_n}| > r$ 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 $f$.