Balanced domains of holomorphy of type \(H^{\infty}\). (English) Zbl 0575.32009

A domain \(D\subset {\mathbb{C}}^ n\) is balanced if \(\lambda D\subset D\) holds for all \(\lambda\in {\mathbb{C}}\) with \(| \lambda | \leq 1\), and the domain D is of type \(H^{\infty}\) if there exists \(f\in H^{\infty}(D)\) for which D is the maximal domain of existence. Here it is shown that a balanced domain \(D\subset {\mathbb{C}}^ n\) is of type \(H^{\infty}\) if and only if \(D=interior(\hat{\bar D})\), where \(\hat{\bar D}\) denotes the polynomial hull of the closure \(\bar D.\) An example is given of a balanced domain of holomorphy \(D\subset {\mathbb{C}}^ 2\) such that \(D=int \bar D\), but interior\((\hat{\bar D})-D\neq \emptyset\).
Reviewer: E.Bedford


32D10 Envelopes of holomorphy
32D05 Domains of holomorphy
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables