Bulletin of the
Korean Mathematical Society BKMS

Let $\Omega$ be a bounded, uniformly totally pseudoconvex domain in $\mathbb{C}^2$ with the smooth boundary $b\Omega$. Assuming that $\Omega$ satisfies the negative $\bar\partial$ property. Let $M$ be a positive, finite area divisor of $\Omega$. In this paper, we will prove that: if $\Omega$ admits a maximal type $F$ and the $\rm \check{C}$eck cohomology class of the second order vanishes in $\Omega$, there is a bounded holomorphic function in $\Omega$ such that its zero set is $M$. The proof is based on the method given by Shaw \cite{Sha89}.

Keywords: pseudoconvex domains, Poincar\'e-Lelong equation, zero set, finite area, $\bar{\partial}_b$-operator, Henkin solution

MSC numbers: 32W05, 32W10, 32W50, 32A26, 32A35, 32A60, 32F18, 32T25, 32U40