Descriptions of exceptional sets in the circles for functions from the Bergman space
Czechoslovak Mathematical Journal, Tome 47 (1997) no. 4, pp. 633-649
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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 $0$,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 $E(B,f) = E.$
@article{CMJ_1997__47_4_a5,
author = {Jak\'obczak, Piotr},
title = {Descriptions of exceptional sets in the circles for functions from the {Bergman} space},
journal = {Czechoslovak Mathematical Journal},
pages = {633--649},
publisher = {mathdoc},
volume = {47},
number = {4},
year = {1997},
mrnumber = {1479310},
zbl = {0901.32006},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_1997__47_4_a5/}
}
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 SP - 633 EP - 649 VL - 47 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/CMJ_1997__47_4_a5/ LA - en ID - CMJ_1997__47_4_a5 ER -
Jakóbczak, Piotr. Descriptions of exceptional sets in the circles for functions from the Bergman space. Czechoslovak Mathematical Journal, Tome 47 (1997) no. 4, pp. 633-649. http://geodesic.mathdoc.fr/item/CMJ_1997__47_4_a5/