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.

References

↑ 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.

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

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Pluripolar set

Definition

See also

References