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 suchthat ${\\mathcal{F}}_{G}$ 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 $\\{\\varphi_{k}\\}\\in{\\mathcal{F}}_{G}$ andthe 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 closureof $U$ 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 functionsare 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.

单词	HartogsFunctions
释义	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 suchthat ${\\mathcal{F}}_{G}$ 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 $\\{\\varphi_{k}\\}\\in{\\mathcal{F}}_{G}$ andthe 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 closureof $U$ 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 functionsare 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.
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