CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Exceptional Sets in Hartogs Domains

  Published:2005-12-01
 Printed: Dec 2005
  • Piotr Kot
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

Abstract

Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$, defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:|z|<\mu(w),w\in H\}$, 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 $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$. For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function $f\in\mathbb{O}(\Omega)$ such that \[ E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}|f(\cdot\,,w)|^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}. \]
Keywords: boundary behaviour of holomorphic functions, exceptional sets boundary behaviour of holomorphic functions, exceptional sets
MSC Classifications: 30B30 show english descriptions Boundary behavior of power series, over-convergence 30B30 - Boundary behavior of power series, over-convergence
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/