## 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