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Circled domains of holomorphy of type \(H^{\infty}\). (English) Zbl 0579.32015

Summary: 1) Given a balanced domain of holomorphy \(D:=\{z\in {\mathbb{C}}^ n\); \(h(z)<1\}\), where h is a non-zero homogeneous function (i.e. \(h(tz)=| t| h(z)\), \(t\in {\mathbb{C}}\), \(z\in {\mathbb{C}}^ n)\) plurisubharmonic in \({\mathbb{C}}^ n\), the following conditions are equivalent:
(i) D is a maximal domain of existence of a bounded holomorphic function (shortly: D is of type \(H^{\infty});\)
(ii) \(D=int\cap_{Q\in F}D_ Q\), where F is a family of homogeneous polynomials and \(D_ Q:=\{| Q| <1\};\)
(iii) D is a maximal domain of existence of a bounded holomorphic function such that for every multiindex \(\alpha \in {\mathbb{Z}}^ n_+\) there exists \(f_{\alpha}\in \phi (\bar D)\) with \(f_{\alpha}=D^{\alpha}f\) on D.
2) Given a bounded balanced domain of holomorphy \(D=\{h<1\}\) the following conditions are equivalent:
(a) D is of type \(H^{\infty}\); (b) the set of discontinuities of h \(N:=\{a\in {\mathbb{C}}^ n;\quad h\quad is\quad discontinuous\quad at\quad a\}\) is pluripolar; (c) \(h=(\psi_{\bar D})^*\), where \(\psi_{\bar D}:=\sup \{| Q|^{1/\deg Q}\); Q is a homogeneous polynomial with deg \(Q\geq 1\) and \(\| Q\|_ D=1\}.\)
3) If \(D\subset {\mathbb{C}}^ n\) is a Reinhardt domain of holomorphy (we do not assume that \(0\in D)\), then the following conditions are equivalent:
(a) D is of type \(H^{\infty};\)
(b) There exists a subset A of \({\mathbb{Z}}^ n\) such that \(D=int\cap_{\alpha \in A}D_{\alpha}\), where \(D_{\alpha}:=\{z\in {\mathbb{C}}^ n\); \(| z^{\alpha}| \leq \sup_{w\in D}| w^{\alpha}| \};\)
(c) D is of rational type (in the sense of M. Jarnicki and P. Pflug [Ann. Pol. Math. 46 (1985), to appear].

MSC:

32D05 Domains of holomorphy
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A10 Holomorphic functions of several complex variables
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32U05 Plurisubharmonic functions and generalizations
32A05 Power series, series of functions of several complex variables
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