## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Exceptional sets with a weight in a unit ball

Piotr Kot

#### Abstract

For a given number $s>-1$ and a multiindex $\alpha\in\Bbb N^{n}$ we give a proof of the following equality: $\int_{\left\Vert z\right\Vert <R}z^{\alpha}\overline{z^{\alpha}}\left(R^{2}-\left\Vert z\right\Vert ^{2}\right)^{s}dz=\frac{\pi^{n}\alpha!R^{2(s+|\alpha|+n)}}{\prod_{i=1}^{|\alpha|+n}(s+i)}.$ As a result we receive different properties of the sets defined by the following formula $E^{s}(f)=\left\{ z\in\partial\Bbb B^{n}:\:\int_{|\lambda|<1}\left|f(\lambda z)\right|^{2}\left(1-|\lambda|^{2}\right)^{s}d\mathfrak{L}^{2}=\infty\right\}$ for the holomorphic function $f\in\Bbb O(\Bbb B^{n})$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 1 (2006), 43-53.

Dates
First available in Project Euclid: 19 May 2006

https://projecteuclid.org/euclid.bbms/1148059331

Digital Object Identifier
doi:10.36045/bbms/1148059331

Mathematical Reviews number (MathSciNet)
MR2245976

Zentralblatt MATH identifier
1127.32004

Subjects
Primary: 30B30: Boundary behavior of power series, over-convergence

#### Citation

Kot, Piotr. Exceptional sets with a weight in a unit ball. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 1, 43--53. doi:10.36045/bbms/1148059331. https://projecteuclid.org/euclid.bbms/1148059331