| Title:
|
Description of simple exceptional sets in the unit ball (English) |
| Author:
|
Kot, Piotr |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
55-63 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For $ z\in \partial B^n$, the boundary of the unit ball in $\mathbb{C}^n$, let $\Lambda (z)=\lbrace \lambda \:|\lambda |\le 1\rbrace $. If $ f\in \mathbb{O}(B^n)$ then we call $E(f)=\lbrace z\in \partial B^n\:\int _{\Lambda (z)}|f(z)|^2\mathrm{d}\Lambda (z)=\infty \rbrace $ the exceptional set for $f$. In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a $G_\delta $ and $F_\sigma $ subset of the projective $(n-1)$-dimensional space $\mathbb{P}^{n-1}=\mathbb{P}(\mathbb{C}^n)$ then there exists a holomorphic function $f$ in the unit ball $B^n$ so that $E(f)=E$. (English) |
| Keyword:
|
boundary behavior of power series |
| Keyword:
|
exceptional set |
| MSC:
|
30B30 |
| MSC:
|
32A40 |
| idZBL:
|
Zbl 1052.30006 |
| idMR:
|
MR2040218 |
| . |
| Date available:
|
2009-09-24T11:09:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127863 |
| . |
| Reference:
|
[1] J. Globevink: Holomorphic functions which are highly nonintegrable at the boundary.Israel J. Math (to appear). MR 1749678 |
| Reference:
|
[2] J. Globevnik and E. L. Stout: Highly noncontinuable functions on convex domains.Bull. Sci. Math. 104 (1980), 417–439. MR 0602409 |
| Reference:
|
[3] J. Globevnik and E. L. Stout: Holomorphic functions with highly noncontinuable boundary behavior.J. Anal. Math. 41 (1982), 211–216. MR 0687952 |
| Reference:
|
[4] J. Siciak: Highly noncontinuable functions on polynomially convex sets.Zeszyty Naukowe Uniwersytetu Jagiellonskiego 25 (1985), 95–107. Zbl 0585.32012, MR 0837828 |
| Reference:
|
[5] W. Rudin: Function Theory in the Unit Ball of $ \mathbb{C}^{n} $.Springer, New York, 1980. MR 0601594 |
| Reference:
|
[6] P. Wojtaszczyk: On highly nonintegrable functions and homogeneous polynomials.Ann. Pol. Math. 65 (1997), 245–251. Zbl 0872.32001, MR 1441179, 10.4064/ap-65-3-245-251 |
| . |
