Pluripolar set

{{short description|Analog of a polar set for plurisubharmonic functions}}

In mathematics, in the area of potential theory, a pluripolar set is the analog of a polar set for plurisubharmonic functions.

Definition

Let $G \subset {\mathbb{C}}^n$ and let $f \colon G \to {\mathbb{R}} \cup \{ - \infty \}$ be a plurisubharmonic function which is not identically $-\infty$ . The set

: ${\mathcal{P}} := \{ z \in G \mid f(z) = - \infty \}$

is called a complete pluripolar set. A pluripolar set is any subset of a complete pluripolar set. Pluripolar sets are of Hausdorff dimension at most $2n-2$ and have zero Lebesgue measure.{{cite book |last1=Sibony |first1=Nessim |last2=Schleicher |first2=Dierk |last3=Cuong |first3=Dinh Tien |last4=Brunella |first4=Marco |last5=Bedford |first5=Eric |last6=Abate |first6=Marco |editor1-last=Gentili |editor1-first=Graziano |editor2-last=Patrizio |editor2-first=Giorgio |editor3-last=Guenot |editor3-first=Jacques |title=Holomorphic Dynamical Systems: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 7-12, 2008 |date=2010 |publisher=Springer Science & Business Media |isbn=978-3-642-13170-7 |page=275 |url=https://books.google.com/books?id=M9vorlpkXckC |language=en}}

If $f$ is a holomorphic function then $\log | f |$ is a plurisubharmonic function. The zero set of $f$ is then a pluripolar set if $f$ is not the zero function.

References

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Category:Potential theory

Pluripolar set

Definition

See also

References