Pluripolar set explained

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 { C }^n and let f \colon G \to { R } \cup \ be a plurisubharmonic function which is not identically -inf ty . The set { l{P}} : = \{ z \in G \mid f(z) = - inf ty \}

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

2 n -2 and have zero Lebesgue measure.^[1]

If

f is a holomorphic function then log | f | is a plurisubharmonic function. The zero set of f is then a pluripolar set.

See also

• Skoda-El Mir theorem

References

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

Notes and References

1. Book: Sibony . Nessim . Schleicher . Dierk . Cuong . Dinh Tien . Brunella . Marco . Bedford . Eric . Abate . Marco . Gentili . Graziano . Patrizio . Giorgio . Guenot . Jacques . Holomorphic Dynamical Systems: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 7-12, 2008 . 2010 . Springer Science & Business Media . 978-3-642-13170-7 . 275 . en.