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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
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Abstract

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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

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[4] Wojtaszczyk, P., On highly nonintegrable functions and homogeneous polynomials. Ann. Polon. Math. 65 (1997), 245–251.Google Scholar