Description of simple exceptional sets in the unit ball

Kot, Piotr

Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 55-63.

Résumé

For $ z\in \partial B^n$, the boundary of the unit ball in $\mathbb{C}^n$, let $\Lambda (z)=\lbrace \lambda \:|\lambda |\le 1\rbrace $. If $ f\in \mathbb{O}(B^n)$ then we call $E(f)=\lbrace z\in \partial B^n\:\int _{\Lambda (z)}|f(z)|^2\mathrm{d}\Lambda (z)=\infty \rbrace $ the exceptional set for $f$. In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a $G_\delta $ and $F_\sigma $ subset of the projective $(n-1)$-dimensional space $\mathbb{P}^{n-1}=\mathbb{P}(\mathbb{C}^n)$ then there exists a holomorphic function $f$ in the unit ball $B^n$ so that $E(f)=E$.

Détail

Classification : 30B30, 32A40
Keywords: boundary behavior of power series; exceptional set