Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T20:20:47.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Exceptional Sets of Slices for Functions From the Bergman Space in the Ball

Published online by Cambridge University Press:  20 November 2018

Piotr Jakóbczak
Piotr Jakóbczak*
Affiliation:
Politechnika Krakowska Instytut Matematyki ul. Warszawska 24 31-155 Kraków Poland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ${{B}_{N}}$ be the unit ball in ${{\mathbb{C}}^{N}}$ and let $f$ be a function holomorphic and ${{L}^{2}}$-integrable in ${{B}_{N}}$. Denote by $E\left( {{B}_{N}},\,f \right)$ the set of all slices of the form $\Pi \,=\,L\,\cap \,{{B}_{N}}$, where $L$ is a complex one-dimensional subspace of ${{\mathbb{C}}^{N}}$, for which $f{{|}_{\Pi }}$ is not ${{L}^{2}}$-integrable (with respect to the Lebesgue measure on L). Call this set the exceptional set for $f$. We give a characterization of exceptional sets which are closed in the natural topology of slices.

Keywords

32A3732A22
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 2 , 01 June 2001 , pp. 150 - 159
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Jakóbczak, P., The exceptional sets for functions from the Bergman space. Portugal. Math. (1) 50 (1993), 115–128.Google Scholar
[2] Jakóbczak, P., The exceptional sets for holomorphic functions in Hartogs domains. Complex Variables (Aachen) 32 (1997), 89–97.Google Scholar
[3] Jakóbczak, P., Highly nonintegrable functions in the unit ball. Israel J. Math. 97 (1997), 175–181.Google Scholar
[4] Wojtaszczyk, P., On highly nonintegrable functions and homogeneous polynomials. Ann. Polon. Math. 65 (1997), 245–251.Google Scholar