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{1168957350,
author = {Kot, Piotr},
title = {Integrability of homogeneous polynomials on the unit ball},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {12},
number = {5},
year = {2006},
pages = { 743-762},
language = {en},
url = {http://dml.mathdoc.fr/item/1168957350}
}
Kot, Piotr. Integrability of homogeneous polynomials on the unit ball. Bull. Belg. Math. Soc. Simon Stevin, Tome 12 (2006) no. 5, pp. 743-762. http://gdmltest.u-ga.fr/item/1168957350/