## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Integrability of homogeneous polynomials on the unit ball

Piotr Kot

#### Abstract

We construct some measure $\Theta^{\alpha}$ such that if $0<\alpha\leq2n-2$, $\beta=n-\frac{2+\alpha}{2}$ and $E$ is a circular set of type $G_{\delta}$ such that $E\subset\partial\Bbb B^{n}$ and $\Theta^{\alpha}(E)=0$ then there exists $f\in\Bbb O(\Bbb B^{n})\cap L^{2}(\Bbb B^{n})$ such that $E=E^{\beta}(f):=\left\{ z\in \partial B^{n}:\:\int_{\Bbb Dz}\left|f\right|^{2}\chi_{\beta}d\mathfrak{L}^{2}=\infty\right\}$ where $\chi_{s}:\Bbb B^{n}\ni z\longrightarrow\chi_{s}(z)=(1-\left\Vert z\right\Vert ^{2})^{s}$ and $\Bbb D$ denotes the unit disc in $\Bbb C$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 4 (2006), 743-762.

Dates
First available in Project Euclid: 16 January 2007

https://projecteuclid.org/euclid.bbms/1168957350

Digital Object Identifier
doi:10.36045/bbms/1168957350

Mathematical Reviews number (MathSciNet)
MR2300630

Zentralblatt MATH identifier
1124.32002

#### Citation

Kot, Piotr. Integrability of homogeneous polynomials on the unit ball. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 743--762. doi:10.36045/bbms/1168957350. https://projecteuclid.org/euclid.bbms/1168957350