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Remark on Zero Sets of Holomorphic Functions in Convex Domains of Finite Type

Open Access article
 Printed: Jun 2010
  • MichaƂ Jasiczak,
    Faculty of Mathematics and Computer Science, A. Mickiewicz University, 61-614 Poznan, Poland and
    Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw, Poland
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We prove that if the $(1,1)$-current of integration on an analytic subvariety $V\subset D$ satisfies the uniform Blaschke condition, then $V$ is the zero set of a holomorphic function $f$ such that $\log |f|$ is a function of bounded mean oscillation in $bD$. The domain $D$ is assumed to be smoothly bounded and of finite d'Angelo type. The proof amounts to non-isotropic estimates for a solution to the $\overline{\partial}$-equation for Carleson measures.
MSC Classifications: 32A60, 32A35, 32F18 show english descriptions Zero sets of holomorphic functions
$H^p$-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15]
Finite-type conditions
32A60 - Zero sets of holomorphic functions
32A35 - $H^p$-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15]
32F18 - Finite-type conditions

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