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 $f:G\to \mathbb{ℝ}\cup \{-\mathrm{\infty }\}$ be a plurisubharmonic function which is not identically $-\mathrm{\infty }$. The set

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

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.

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.

See also

• Skoda-El Mir theorem

References

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

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