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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

##### MSC:
 32D10 Envelopes of holomorphy 32D05 Domains of holomorphy 32A35 $$H^p$$-spaces, Nevanlinna spaces of functions in several complex variables