Proposition 5.14.15 (Kiselman's principle)BB26476C13FA

Proposition 5.14.15 (Kiselman's principle).BB26476C13FA Let $\Omega \subseteq \mathbb {C}^ m\times \mathbb {C}^ n$ be a pseudoconvex domain. Assume that for each $z\in \mathbb {C}^ m$, the set

has the form $E+\mathrm{i}\mathbb {R}^ n$, where $E\subseteq \mathbb {R}^ n$ is a subset. Let $\varphi \in \mathrm{PSH}(\Omega )$, assume that $\varphi $ is independent of the imaginary part of the variable in $\mathbb {C}^ n$. Let $\Omega '$ be the projection of $\Omega $ to $\mathbb {C}^ m$. Define $\psi \colon \Omega '\rightarrow [-\infty ,\infty )$ as follows:

Then either $\psi \equiv -\infty $ or $\psi \in \mathrm{PSH}(\Omega ')$.