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 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.[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

References

  1. ↑ Patrizio, Marco Abate ... [et al.] ; editors, Graziano Gentili, Jacques Guenot, Giorgio (2010). Holomorphic dynamical systems : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 7-12, 2008. Berlin: Springer. p. 275. ISBN 9783642131707.

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