Norm Preserving Extensions of Holomorphic Functions Defined on Varieties in ${\mathbb C}^n$

Abstract: If $V$ is an analytic set in a pseudoconvex domain $\Omega$, we show there is always a pseudoconvex domain $G \subseteq \Omega$ that contains $V$ and has the property that every bounded holomorphic function on $V$ extends to a bounded holomorphic function on $G$ with the same norm. We find such a $G$ for some particular analytic sets. When $\Omega$ is an operhedron we show there is a norm on holomorphic functions on $V$ that can always be preserved by extensions to $\Omega$.