| Title:
|
Descriptions of exceptional sets in the circles for functions from the Bergman space (English) |
| Author:
|
Jakóbczak, Piotr |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
1997 |
| Pages:
|
633-649 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $E(B,f) = E.$ (English) |
| MSC:
|
32A37 |
| MSC:
|
32H10 |
| MSC:
|
32H99 |
| idZBL:
|
Zbl 0901.32006 |
| idMR:
|
MR1479310 |
| . |
| Date available:
|
2009-09-24T10:09:11Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127384 |
| . |
| Reference:
|
[1] P. Jakóbczak: The exceptional sets for functions from the Bergman space.Portugaliae Mathematica 50, No 1 (1993), 115–128. MR 1300590 |
| Reference:
|
[2] P.Jakóbczak: The exceptional sets for functions of the Bergman space in the unit ball.Rend. Mat. Acc. Lincei s.9, 4 (1993), 79–85. Zbl 0788.46061, MR 1233394 |
| Reference:
|
[3] J.Janas: On a theorem of Lebow and Mlak for several commuting operators.Studia Math. 76 (1983), 249–253. MR 0729105, 10.4064/sm-76-3-249-253 |
| Reference:
|
[4] B.W.Šabat: Introduction to Complex Analysis.Nauka, Moskva, 1969. (Russian) Zbl 0169.09001, MR 0584932 |
| . |
