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Description of simple exceptional sets in the unit ball. (English) Zbl 1052.30006

Summary: For \( z\in \partial B^n\), the boundary of the unit ball in \(\mathbb C^n\), let \(\Lambda (z)=\{\lambda:| \lambda | \leq 1\}\). If \(f \in \mathbb O (B^n)\) then we call \(E (f) = \{z \in \partial B^n : \int _{\Lambda (z)} | f (z)| ^2 \,d \Lambda (z) = \infty \}\) 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 \(E(f)=E\).

MSC:

30B30 Boundary behavior of power series in one complex variable; over-convergence
32A40 Boundary behavior of holomorphic functions of several complex variables
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References:

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