On the Błocki-Zwonek conjectures. (English) Zbl 1343.32028
Let \(g_\Omega(z,a)\) be the pluricomplex Green function of a bounded pseudoconvex domain \(\Omega\) with pole at a point \(a\in\Omega\). It was conjectured by Błocki and Zwonek that the logarithm of the Lebesgue measure of the sublevel set \(\{z:g_\Omega(z,a)<t\}\) is a convex function of \(t\). The authors prove this in the case when \(\Omega\) is biholomorphic to the unit ball or polydisk.
Reviewer: Alexandr Yu. Rashkovsky (Stavanger)
MSC:
| 32U35 | Plurisubharmonic extremal functions, pluricomplex Green functions |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
References:
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