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)


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
Full Text: DOI