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Normed domains of holomorphy. (English) Zbl 1196.32009

The author first shows that any bounded, pseudoconvex, Runge domain \(\Omega \subseteq \mathbb C^{n}\) is convex with respect to the family of bounded holomorphic functions. Next, he says that a domain \(\Omega \subseteq \mathbb C^{n}\) is of type \(HL^{p}\), \(1\leq p\leq \infty\), if there is a holomorphic function \(f \in L^{p}(\Omega)\), which cannot be analytically continued to any larger domain, while if there exists a strictly larger domain \(\Omega\) to which every holomorphic \(L^{p}\) function on \(\Omega\) extends, then \(\Omega\) is of type \(EL^{p}\). He furnishes several sufficient conditions under which \(\Omega\) is of one of the above mentioned types. For example, if \(\Omega \subseteq \mathbb C\) is bounded and the interior of its closure, it is of type \(HL^{p}\). Some other cases where an \( \Omega \subseteq \mathbb C^{n}\) is of type \(HL^{p}\) are: if it is bounded and convex, or if it is bounded, strongly pseudoconvex and with \(\mathcal C ^{2}\)-boundary, or if it is a bounded analytic polyhedron, or if it is a pseudoconvex complete circular domain, or if it is bounded, pseudoconvex with a Stein neighborhood basis. Other properties of \(HL^{p}\) and \(EL^{p}\) domains are investigated and the relation to \(\overline{\partial}\)-problem is studied. The problem of finding characterizations for these types of domains is still open.

MSC:

32D15 Continuation of analytic objects in several complex variables
32D05 Domains of holomorphy
32E10 Stein spaces
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A38 Algebras of holomorphic functions of several complex variables
32A50 Harmonic analysis of several complex variables
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
32F17 Other notions of convexity in relation to several complex variables
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References:

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