| Title:
|
A formula for the Bloch norm of a $C^1$-function on the unit ball of $\Bbb C^n$ (English) |
| Author:
|
Pavlović, Miroslav |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
1039-1043 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a $C^1$-function $f$ on the unit ball $\mathbb B \subset \mathbb C ^n$ we define the Bloch norm by $\|f\|_\mathfrak B=\sup \|\tilde df\|,$ where $\tilde df$ is the invariant derivative of $f,$ and then show that $$ \|f\|_\mathfrak B= \sup _{z,w\in {\mathbb B} \atop z\neq w} (1-|z|^2)^{1/2}(1-|w|^2)^{1/2}\frac {|f(z)-f(w)|}{|w-P_wz-s_wQ_wz|}.$$ (English) |
| Keyword:
|
Bloch norm |
| Keyword:
|
Möbius transformation |
| MSC:
|
30D45 |
| MSC:
|
32A18 |
| MSC:
|
32A37 |
| MSC:
|
46E15 |
| idZBL:
|
Zbl 1174.32003 |
| idMR:
|
MR2471163 |
| . |
| Date available:
|
2010-07-21T08:08:40Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140437 |
| . |
| Reference:
|
[1] Holland, F., Walsh, D.: Criteria for membership of Bloch space and its subspace, BMOA.Math. Ann. 273 (1986), 317-335. Zbl 0561.30025, MR 0817885, 10.1007/BF01451410 |
| Reference:
|
[2] Nowak, M.: Bloch space and Möbius invariant Besov spaces on the unit ball of {${\mathbb C}^n$}.Complex Variables Theory Appl. 44 (2001), 1-12. MR 1826712, 10.1080/17476930108815339 |
| Reference:
|
[3] Pavlovi'c, M.: On the Holland-Walsh characterization of Bloch functions.Proc. Edinb. Math. Soc. 51 (2008), 439-441. MR 2465917, 10.1017/S0013091506001076 |
| Reference:
|
[4] Rudin, W.: Function Theory in the Unit Ball of {$C^n$}.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York (1980). MR 0601594 |
| . |
