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 $\Bbb C$.
Citation
Piotr Kot. "Integrability of homogeneous polynomials on the unit ball." Bull. Belg. Math. Soc. Simon Stevin 13 (4) 743 - 762, December 2006. https://doi.org/10.36045/bbms/1168957350
Information