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Exceptional Sets in Hartogs Domains

Published online by Cambridge University Press:  20 November 2018

Piotr Kot
Piotr Kot*
Affiliation:
Politechnika Krakowska, Instytut Matematyki, ul. Warszawska 24, 31-155 Kraków, Poland email: pkot@usk.pk.edu.pl
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Abstract

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Assume that $\Omega$ is a Hartogs domain in ${{\mathbb{C}}^{1+n}}$, defined as $\Omega =\left\{ \left( z,w \right)\,\in \,{{\mathbb{C}}^{1+n}}\,:\left| z \right|\,<\,\mu \left( w \right),w\,\in H \right\}$, where $H$ is an open set in ${{\mathbb{C}}^{n}}$ and $\mu$ is a continuous function with positive values in $H$ such that –ln $\mu$ is a strongly plurisubharmonic function in $H$. Let ${{\Omega }_{w}}=\Omega \cap \left( \mathbb{C}\times \left\{ w \right\} \right)$. For a given set $E$ contained in $H$ of the type ${{G}_{\delta }}$ we construct a holomorphic function $f\in \mathbb{O}\left( \Omega \right)$ such that

$$E=\left\{ w\in {{\mathbb{C}}^{n}}:\int\limits_{{{\Omega }_{w}}}{{{\left| f\left( \cdot ,w \right) \right|}^{2}}d{{\mathfrak{L}}^{2}}=\infty } \right\}.$$

Keywords

30B30boundary behaviour of holomorphic functionsexceptional sets
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005 , pp. 580 - 586
Copyright
Copyright © Canadian Mathematical Society 2005

References

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