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Zero varieties for the Nevanlinna class in convex domains of finite type in $$\mathbb{C}^n$$. (English) Zbl 0912.32001
Let $$\Omega$$ be a smoothly bounded domain in $$\mathbb{C}^n$$, $$\Omega= \{\rho<0\}$$, $$\partial\Omega= \{\rho=0\}$$ and $$\nabla\rho\neq 0$$ on $$\partial \Omega$$ and $$N(\Omega)$$ denotes the Nevanlinna class of the functions analytic in $$\Omega$$. For any $$f\in N(\Omega)$$, the zero-variety $$X= \{f=0\}$$ satisfies the Blaschke condition, i.e. if $$X_k$$ are the irreducible components of $$X$$ and $$n_k$$ the corresponding multiplicities of $$f$$, then $\sum_k n_k\int_{X_k} \bigl| \rho(z) \bigr| d\mu/X_k (z)<+ \infty, \tag{B}$ where $$d\mu_{X_k}$$ is the Euclidean measure on the regular part of $$X_k$$.
Now, let $$\Omega$$ be a convex domain in $$\mathbb{C}^n$$.
The authors solve the problem whether in dimension $$n>1$$ the Blaschke condition (B) is sufficient for the divisor $$\widehat X=\{X_k,n_k\}$$ to be defined by a function $$f\in N(\Omega)$$. The main result is: If $$\Omega$$ is a convex domain of finite strict type, every divisor $$\widehat X$$ in $$\Omega$$ satisfying the Blaschke condition can be defined by a function $$f\in N(\Omega)$$. The proof consists of the following steps: To each divisor $$\widehat X$$ is associated a (1,1) current $$\Theta$$, and the nonisotropic estimates for the current $$\Theta$$ are obtained, and the corresponding $$\partial$$-equation are solved.
Reviewer: A.Klíč (Praha)

##### MSC:
 32A10 Holomorphic functions of several complex variables 32A17 Special families of functions of several complex variables 32C25 Analytic subsets and submanifolds 32C30 Integration on analytic sets and spaces, currents
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