# Perron family

###### Definition.

Let $G\subset{\mathbb{C}}$ be a region, $\partial_{\infty}G$ the extended boundary of $G$ and $S(G)$ the set of subharmonic functions on $G$, then if $f\colon\partial_{\infty}G\to{\mathbb{R}}$ is a continuous function then the set

 ${\mathcal{P}}(f,G):=\{\varphi:\varphi\in S(G)\text{ and }\limsup_{z\to a}% \varphi(z)\leq f(a)\text{ for all a\in\partial_{\infty}G}\},$

is called the Perron family.

One thing to note is the ${\mathcal{P}}(f,G)$ is never empty. This is because $f$ is continuous on $\partial_{\infty}G$ it attains a maximum, say $|f|, then the function $\varphi(z):=-M$ is in ${\mathcal{P}}(f,G)$.

###### Definition.

Let $G\subset{\mathbb{C}}$ be a region and $f\colon\partial_{\infty}G\to{\mathbb{R}}$ be a continuous function then the function $u\colon G\to{\mathbb{R}}$

 $u(z):=\sup\{\phi:\phi\in{\mathcal{P}}(f,G)\},$

is called the Perron function associated with $f$.

Here is the reason for all these definitions.

###### Theorem.

Let $G\subset{\mathbb{C}}$ be a region and suppose $f\colon\partial_{\infty}G\to{\mathbb{R}}$ is a continuous function. If $u\colon G\to{\mathbb{R}}$ is the Perron function associated with $f$, then $u$ is a harmonic function.

Compare this with Rado’s theorem (http://planetmath.org/RadosTheorem) which works with harmonic functions with range in ${\mathbb{R}}^{2}$, but also gives a much stronger statement.

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title Perron family PerronFamily 2013-03-22 14:19:42 2013-03-22 14:19:42 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 31B05 RadosTheorem Perron function