Pluripolar set
In mathematics, in the area of potential theory, a pluripolar set is the analog of a polar set for plurisubharmonic functions.
Definition
Let [math]\displaystyle{ G \subset {\mathbb{C}}^n }[/math] and let [math]\displaystyle{ f \colon G \to {\mathbb{R}} \cup \{ - \infty \} }[/math] be a plurisubharmonic function which is not identically [math]\displaystyle{ -\infty }[/math]. The set
- [math]\displaystyle{ {\mathcal{P}} := \{ z \in G \mid f(z) = - \infty \} }[/math]
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 [math]\displaystyle{ 2n-2 }[/math] and have zero Lebesgue measure.[1]
If [math]\displaystyle{ f }[/math] is a holomorphic function then [math]\displaystyle{ \log | f | }[/math] is a plurisubharmonic function. The zero set of [math]\displaystyle{ f }[/math] is then a pluripolar set.
See also
- Skoda-El Mir theorem
References
- ↑ Sibony, Nessim; Schleicher, Dierk; Cuong, Dinh Tien; Brunella, Marco; Bedford, Eric; Abate, Marco (2010). Gentili, Graziano; Patrizio, Giorgio; Guenot, Jacques. eds (in en). Holomorphic Dynamical Systems: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 7-12, 2008. Springer Science & Business Media. p. 275. ISBN 978-3-642-13170-7. https://books.google.com/books?id=M9vorlpkXckC.
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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