Existence domains for holomorphic \(L^ p\) functions (Q1344473)

The author announces the following theorem. Let \(\Omega \Subset U \subset \mathbb{C}^ n\) be domains of holomorphy such that \((U, \Omega)\) is a Runge pair. Then \(\Omega\) is an existence domain for holomorphic \(L^ p\) functions with arbitrary \(p \in (0, + \infty)\). The proof proposed by the author is wrong. The main mistake is on page 210 when the author claims that for any \(v \in \mathbb{N}\) there exists \(k_ v \in \mathbb{N}\) such that \[ \int_ \Omega \left( \sum_{m = 1}^ v \biggl | \bigl[ f_ m(z) \bigr]^{v_ m} \biggr | \right)^ pd \lambda (z) \leq \left( {2k_{v-1} \over k_ v} \right)^ p. \] This is evidently impossible. The reviewer does not know whether the theorem is true.