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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, p. 371-380Article 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, p. 371-380
##### Keyword [en]
Bergman kernel, Błocki–Zwonek conjectures, Pluricomplex Green functions
##### National Category
Mathematical Analysis
##### Identifiers
ISI: 000361347100009OAI: oai:DiVA.org:umu-106678DiVA, id: diva2:843925
Available from: 2015-08-01 Created: 2015-08-01 Last updated: 2017-12-04Bibliographically approved

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CiteExportLink to record
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