# Hartogs functions

###### Definition.

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

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

• If $\varphi\in{\mathcal{F}}_{G}$ then $a\varphi_{1}\in{\mathcal{F}}_{G}$ for all $a\geq 0$.

• If $\{\varphi_{k}\}\in{\mathcal{F}}_{G}$ and $\varphi_{1}\geq\varphi_{2}\geq\ldots$, then $\lim_{k\to\infty}\varphi_{k}\in{\mathcal{F}}_{G}$.

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

• If $\varphi\in{\mathcal{F}}_{G}$ and $\hat{\varphi}(w):=\limsup_{w\to z}\varphi(w)$, then $\hat{\varphi}\in{\mathcal{F}}_{G}$

• If $\varphi|_{U}\in{\mathcal{F}}_{U}$ for all $U\subset G$ where $U$ is relatively compact (the closure of $U$ is compact), then $\varphi\in{\mathcal{F}}_{G}$.

These functions are called the Hartogs functions.

It is known that if $n=1$ then the upper semi-continuous 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 HartogsFunctions 2013-03-22 14:29:27 2013-03-22 14:29:27 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 32U05