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

TL;DR

This paper investigates conditions under which bounded holomorphic functions on analytic sets in pseudoconvex domains can be extended to larger domains without increasing their norm, providing new extension domains and norm-preserving extensions.

Contribution

It establishes the existence of pseudoconvex extension domains for analytic sets and introduces a norm-preserving extension method for certain cases, advancing the theory of holomorphic function extension.

Findings

Existence of pseudoconvex domains containing V with norm-preserving extensions.

Construction of specific extension domains for particular analytic sets.

Norm preservation for holomorphic functions on V when Ω is an operhedron.

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 $\Omega$.