Hartogs functions
Definition.
Let $G\subset {\u2102}^{n}$ be an open set and let ${\mathcal{F}}_{G}$ be the smallest class of functions^{} on $G$ to $\mathbb{R}\cup \{\mathrm{\infty}\}$ that contains all of the functions $z\mapsto \mathrm{log}f(z)$ where $f$ is holomorphic on $G$ and such that ${\mathcal{F}}_{G}$ is closed with respect to the following conditions:

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If ${\phi}_{1},{\phi}_{2}\in {\mathcal{F}}_{G}$, then ${\phi}_{1}+{\phi}_{2}\in {\mathcal{F}}_{G}$.

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If $\phi \in {\mathcal{F}}_{G}$ then $a{\phi}_{1}\in {\mathcal{F}}_{G}$ for all $a\ge 0$.

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If $\{{\phi}_{k}\}\in {\mathcal{F}}_{G}$ and ${\phi}_{1}\ge {\phi}_{2}\ge \mathrm{\dots}$, then ${lim}_{k\to \mathrm{\infty}}{\phi}_{k}\in {\mathcal{F}}_{G}$.

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If $\{{\phi}_{k}\}\in {\mathcal{F}}_{G}$ and the sequence is uniformly bounded above on compact sets, then ${sup}_{k}{\phi}_{k}\in {\mathcal{F}}_{G}$.

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If $\phi \in {\mathcal{F}}_{G}$ and $\widehat{\phi}(w):={lim\; sup}_{w\to z}\phi (w)$, then $\widehat{\phi}\in {\mathcal{F}}_{G}$

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If ${\phi }_{U}\in {\mathcal{F}}_{U}$ for all $U\subset G$ where $U$ is relatively compact (the closure of $U$ is compact), then $\phi \in {\mathcal{F}}_{G}$.
These functions are called the Hartogs functions.
It is known that if $n=1$ then the upper semicontinuous Hartogs functions are precisely the subharmonic functions on $G$.
Theorem (H. Bremerman).
All plurisubharmonic functions^{} are Hartogs functions if $G$ is a domain of holomorphy.
References
 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title  Hartogs functions 

Canonical name  HartogsFunctions 
Date of creation  20130322 14:29:27 
Last modified on  20130322 14:29:27 
Owner  jirka (4157) 
Last modified by  jirka (4157) 
Numerical id  7 
Author  jirka (4157) 
Entry type  Definition 
Classification  msc 32U05 