Let $V$ be a normal analytic space of pure dimension $n$. A plurisubharmonic (psh) function $\varphi$ on $V$ is an upper semicontinuous function on $V$ with values in $\mathbb{R}\cup\{-\infty\}$, which is not locally $-\infty$, and extends to a psh function in some local embedding $V\to\mathbb{C}^{N}$. The function $\varphi$ is strongly psh (resp. ${\mathcal{C}}^{0}$, resp. ${\mathcal{C}}^{\infty}$) iff it extends to a strongly psh function (resp. ${\mathcal{C}}^{0}$, resp. ${\mathcal{C}}^{\infty}$) in some local embedding. A continuous function is psh iff its restriction to $V^{reg}$ is so [FN]. A bounded psh function on $V^{reg}$ extends to $V$.

A pluriharmonic function on $V$ is a real valued continuous function on $V$ $f$ on $V$ such that one of the following equivalent conditions holds:

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  $f$ is locally the real part of a holomorphic function.

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  Given a local embedding $V\to\mathbb{C}^{N}$, $f$ extends locally to a pluriharmonic function on $\mathbb{C}^{N}$.

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  $f|_{V^{reg}}$ is pluriharmonic.