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On the Blocki-Zwonek conjectures and beyond
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2015 (English)In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 105, no 4, 371-380 p.Article in journal (Refereed) Published
Abstract [en]

Let \({\Omega}\) be a bounded pseudoconvex domain in \({\mathbb{C}^n}\), and let \({g_{\Omega}(z,a)}\) be the pluricomplex Green function with pole at a in \({\Omega}\). Błocki and Zwonek conjectured that the function given by $$\begin{array}{ll}\alpha = \alpha_{\Omega}, a: (- \infty, 0) \ni t \mapsto \alpha (t) = e^{-2nt} \lambda_n \left( \{z \in \Omega: g_{\Omega}(z, a) < t \} \right)\end{array}$$ is nondecreasing, and that the function given by $$\begin{array}{ll}\beta = \beta_{\Omega}, a: (-\infty, 0) \ni t \to \beta(t)= \log \left(\lambda_n \left(\{z \in \Omega: g_{\Omega}(z,a)< t\}\right)\right)\end{array}$$ is convex. Here \({\lambda_{n}}\) is the Lebesgue measure in \({\mathbb{C}^n}\). In this note we give an affirmative answer to these conjectures when \({\Omega}\) is biholomorphic to a bounded, balanced, and pseudoconvex domain in \({\mathbb{C}^n}\), \({n\geq 1}\). The aim of this note is to consider generalizations of the functions \({\alpha}\), \({\beta}\) defined by the Green function with two poles in \({\mathbb{D}\subset\mathbb{C}}\). We prove that \({\alpha}\) is not nondecreasing, and \({\beta}\) is not convex. By using the product property for pluricomplex Green functions, we then generalize this to n-dimensions. Finally, we end this note by considering two other possibilities generalizing the Błocki–Zwonek conjectures.

Place, publisher, year, edition, pages
Springer, 2015. Vol. 105, no 4, 371-380 p.
Keyword [en]
Bergman kernel, Błocki–Zwonek conjectures, Pluricomplex Green functions
National Category
Mathematical Analysis
URN: urn:nbn:se:umu:diva-106678DOI: 10.1007/s00013-015-0810-1ISI: 000361347100009OAI: diva2:843925
Available from: 2015-08-01 Created: 2015-08-01 Last updated: 2015-10-14Bibliographically approved

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Åhag, Per
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