Perron family

Definition.

Let $G\subset ℂ$ be a region, ${\partial }_{\mathrm{\infty }}G$the extended boundary of $G$ and $S(G)$ the set of subharmonic functionson $G$, thenif $f:{\partial }_{\mathrm{\infty }}G\to ℝ$ is a continuousfunction then the set

$$𝒫(f,G):=\{\phi :\phi \in S(G)\text{ and }\underset{z\to a}{lim\; sup}\phi (z)\le f(a)\text{ for all }a\in {\partial }_{\mathrm{\infty }}G\},$$

is called the Perron family.

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

Definition.

Let $G\subset ℂ$ be a region and$f:{\partial }_{\mathrm{\infty }}G\to ℝ$ be a continuousfunction then the function $u:G\to ℝ$

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

is called the Perron function associated with $f$.

Here is the reason for all these definitions.

Theorem.

Let $G\mathrm{\subset }\mathrm{C}$ be a region and suppose $f\mathrm{:}{\mathrm{\partial }}_{\mathrm{\infty }}\mathit{}G\mathrm{\to }\mathrm{R}$ is a continuous function.If $u\mathrm{:}G\mathrm{\to }\mathrm{R}$ is the Perron function associatedwith $f$, then $u$ is a harmonic function.

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