Exceptional sets with a weight in a unit ball

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 $f\in\Bbb O(\Bbb B^{n})$.