# Perron family

###### Definition.

Let $G\subset \u2102$ be a region, ${\partial}_{\mathrm{\infty}}G$
the extended boundary of $G$ and $S(G)$ the set of subharmonic functions^{}
on $G$, then
if $f:{\partial}_{\mathrm{\infty}}G\to \mathbb{R}$ is a continuous
function^{} then the set

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

is called the Perron family.

One thing to note is the $\mathcal{P}(f,G)$ 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 $\mathcal{P}(f,G)$.

###### Definition.

Let $G\subset \u2102$ be a region and $f:{\partial}_{\mathrm{\infty}}G\to \mathbb{R}$ be a continuous function then the function $u:G\to \mathbb{R}$

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

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 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 |
---|---|

Canonical name | PerronFamily |

Date of creation | 2013-03-22 14:19:42 |

Last modified on | 2013-03-22 14:19:42 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 31B05 |

Related topic | RadosTheorem |

Defines | Perron function |