The exceptional sets for functions of the Bergman space in the unit ball

Jakóbczak, Piotr. "The exceptional sets for functions of the Bergman space in the unit ball." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 4.2 (1993): 79-85. <http://eudml.org/doc/244307>.

@article{Jakóbczak1993,
abstract = {Let \( D \) be a domain in \( C^\{2\} \). Given \( w \in C \), set \( D\_\{w\} = \\{ z \in C \mid (z,w) \in D\\} \). If \( f \) is a holomorphic and square-integrable function in \( D \), then the set \( E(D,f) \) of all \( w \) such that \( f ( \cdot, w) \) is not square-integrable in \( D\_\{w\} \) has measure zero. We call this set the exceptional set for \( f \). In this Note we prove that whenever \( 0 < r < 1 \) there exists a holomorphic square-integrable function \( f \) in the unit ball \( B \) in \( C^\{2\} \) such that \( E(B,f) \) is the circle \( C(0, r) = \\{z \in C \mid |z| = r\\} \).},
author = {Jakóbczak, Piotr},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Bergman space; Hartogs domain; Exceptional sets; Bergman space in the unit ball; exceptional sets},
language = {eng},
month = {6},
number = {2},
pages = {79-85},
publisher = {Accademia Nazionale dei Lincei},
title = {The exceptional sets for functions of the Bergman space in the unit ball},
url = {http://eudml.org/doc/244307},
volume = {4},
year = {1993},
}

TY - JOUR
AU - Jakóbczak, Piotr
TI - The exceptional sets for functions of the Bergman space in the unit ball
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1993/6//
PB - Accademia Nazionale dei Lincei
VL - 4
IS - 2
SP - 79
EP - 85
AB - Let \( D \) be a domain in \( C^{2} \). Given \( w \in C \), set \( D_{w} = \{ z \in C \mid (z,w) \in D\} \). If \( f \) is a holomorphic and square-integrable function in \( D \), then the set \( E(D,f) \) of all \( w \) such that \( f ( \cdot, w) \) is not square-integrable in \( D_{w} \) has measure zero. We call this set the exceptional set for \( f \). In this Note we prove that whenever \( 0 < r < 1 \) there exists a holomorphic square-integrable function \( f \) in the unit ball \( B \) in \( C^{2} \) such that \( E(B,f) \) is the circle \( C(0, r) = \{z \in C \mid |z| = r\} \).
LA - eng
KW - Bergman space; Hartogs domain; Exceptional sets; Bergman space in the unit ball; exceptional sets
UR - http://eudml.org/doc/244307
ER -