Descriptions of exceptional sets in the circles for functions from the Bergman space

Jakóbczak, Piotr. "Descriptions of exceptional sets in the circles for functions from the Bergman space." Czechoslovak Mathematical Journal 47.4 (1997): 633-649. <http://eudml.org/doc/30389>.

@article{Jakóbczak1997,
abstract = {Let $D$ be a domain in $\mathbb \{C\}^2$. For $w \in \mathbb \{C\} $, let $D_w = \lbrace z \in \mathbb \{C\} \mid (z,w) \in D \rbrace $. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E(D,f)$ of all $w$ such that $f(.,w)$ is not square-integrable in $D_w$ is of measure zero. We call this set the exceptional set for $f$. In this note we prove that for every $0<r<1$,and every $G_\delta $-subset $E$ of the circle $C(0,r) = \lbrace z \in \mathbb \{C\} \mid | z | =r \rbrace $,there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in $\mathbb \{C\}^2$ such that $E(B,f) = E.$},
author = {Jakóbczak, Piotr},
journal = {Czechoslovak Mathematical Journal},
keywords = {holomorphic and square-integrable function; Bergman space},
language = {eng},
number = {4},
pages = {633-649},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Descriptions of exceptional sets in the circles for functions from the Bergman space},
url = {http://eudml.org/doc/30389},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Jakóbczak, Piotr
TI - Descriptions of exceptional sets in the circles for functions from the Bergman space
JO - Czechoslovak Mathematical Journal
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 4
SP - 633
EP - 649
AB - Let $D$ be a domain in $\mathbb {C}^2$. For $w \in \mathbb {C} $, let $D_w = \lbrace z \in \mathbb {C} \mid (z,w) \in D \rbrace $. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E(D,f)$ of all $w$ such that $f(.,w)$ is not square-integrable in $D_w$ is of measure zero. We call this set the exceptional set for $f$. In this note we prove that for every $0<r<1$,and every $G_\delta $-subset $E$ of the circle $C(0,r) = \lbrace z \in \mathbb {C} \mid | z | =r \rbrace $,there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in $\mathbb {C}^2$ such that $E(B,f) = E.$
LA - eng
KW - holomorphic and square-integrable function; Bergman space
UR - http://eudml.org/doc/30389
ER -