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http://dx.doi.org/10.4134/BKMS.b160429

A NOTE ON ZEROS OF BOUNDED HOLOMORPHIC FUNCTIONS IN WEAKLY PSEUDOCONVEX DOMAINS IN ℂ2  

Ha, Ly Kim (Faculty of Mathematics and Computer Science University of Science Vietnam National University HoChiMinh City (VNU-HCM))
Publication Information
Bulletin of the Korean Mathematical Society / v.54, no.3, 2017 , pp. 993-1002 More about this Journal
Abstract
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 ${\check{C}}eck$ 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 [27].
Keywords
pseudoconvex domains; $Poincar{\acute{e}}$-Lelong equation; zero set; finite area; ${\bar{\partial}}_b$-operator; Henkin solution;
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