Hartogs functions

Definition.

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

• If ${\phi }_1,{\phi }_2\in {ℱ}_G$, then${\phi }_1+{\phi }_2\in {ℱ}_G$.

• If $\phi \in {ℱ}_G$ then$a{\phi }_1\in {ℱ}_G$ for all $a\ge 0$.

• If $\{{\phi }_k\}\in {ℱ}_G$ and${\phi }_1\ge {\phi }_2\ge \mathrm{\dots }$,then ${lim}_{k\to \mathrm{\infty }}{\phi }_k\in {ℱ}_G$.

• If $\{{\phi }_k\}\in {ℱ}_G$ andthe sequence is uniformly bounded above on compact sets, then${sup}_k{\phi }_k\in {ℱ}_G$.

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

• If ${\phi |}_U\in {ℱ}_U$ for all$U\subset G$ where $U$ is relatively compact (the closureof $U$ is compact), then $\phi \in {ℱ}_G$.

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.