Zero varieties for the Nevanlinna class on all convex domains of finite type (Q2764676)

The authors give a characterization of zero variety of the Nevanlinna class on all convex domains in \(\mathbb{C}^m\) of finite type. Let \(D\) be a bounded smooth convex domain in \(\mathbb{C}^m\) of finite type and \(\rho\) a smooth defining function for \(D\). Denote by \(N(D)\) the Nevanlinna class on \(D\). Let \(X\) be a hypersurface in \(N(D)\) with irreducible decomposition \(\bigcup_kX_k\). Let \(n_k\) be positive integers. Then the effective divisor \(\widehat X=\sum n_kX_k\) is said to satisfy the Blaschke condition if NEWLINE\[NEWLINE\sum_kn_k \int_{X_k} \bigl |\rho (z)\bigr|d\mu_{X_k}(z) <\infty.NEWLINE\]NEWLINE It is well-known that the divisor of any function in \(N(D)\) satisfies the Blaschke condition. After study of \textit{J. Bruna}, \textit{Ph. Charpentier} and \textit{Y. Dupain} [Ann. Math. (2) 147, No. 2, 391-415 (1998; Zbl 0912.32001)], the authors show that the Blaschke condition gives a complete characterization of the divisors which are zero sets of functions in \(N(D)\).NEWLINENEWLINENEWLINEThe main result of this paper is as follows: There exists for any effective divisor \(\widehat X\) in \(D\) satisfying the Blaschke condition a function \(f\) in \(N(D)\) such that \(\widehat X\) is the zero divisor of \(f\).NEWLINENEWLINENEWLINEThe tool for the proof is a non-isotropic \(L^1\) estimate for solutions of Cauchy-Riemann equations on \(D\), which are obtained by estimating suitable kernels of Berndsson-Andersson type. For the proof of this, they use the method essentially due to H. Skoda.