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

https://doi.org/10.4153/CMB-2010-037-3
Canad. Math. Bull. 53(2010), 311-320

  Published:2010-04-06
 Printed: Jun 2010

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.