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Pseudoconvex domains of semiregular type. (English) Zbl 0845.32019
Skoda, Henri (ed.) et al., Contributions to complex analysis and analytic geometry. Based on a colloquium dedicated to Pierre Dolbeault, Paris, France, June 23-26, 1992. Braunschweig: Vieweg. Aspects Math. E 26, 127-161 (1994).
A pseudoconvex domain $$\Omega \subset \mathbb{C}^n$$ of finite type is called semiregular at the point 0 if it satisfies the condition $$m_j (\partial \Omega, 0) = \Delta_{n - j + 1} (\partial \Omega, 0)$$ for all $$j = 2, \dots, n$$. Here $$m_q (\partial \Omega,0)$$ is the $$q$$-th component of the multitype $$M (\partial \Omega, 0)$$ (in the sense of D. Catlin) and $$\Delta_q (\partial \Omega, 0)$$ is the $$q$$-type (in the sense of D’Angelo) of $$\partial \Omega$$ at 0.
In this paper, using the bumping for weight homogeneous domains, the authors prove among other things that $$\Omega$$ is semiregular at 0 if and only if the multiple $$M (\partial \Omega, 0)$$ is an admissible $$n$$-tuple of orders of extendability for $$\partial \Omega$$ at 0. Combined with their previous results [J. Geom. Anal. 3, No. 3, 237-267 (1993; Zbl 0789.32016)], the authors prove the following equalities among the analytic invariants and geometric invariants for domains of semiregular type. \begin{aligned} N (\partial \Omega, 0) & = \tau (\partial \Omega, 0) \\ e (\partial \Omega, 0) & = \sum^n_{i = 1} {1 \over m_i (\partial \Omega, 0)} \\ g (\partial \Omega, 0) & = 2 \sum^n_{i = 1} {1 \over m_i (\partial \Omega, 0)} \\ h (\partial \Omega, 0) & = {1 \over \tau (\partial \Omega, 0)}. \end{aligned} Here $$\tau (\partial \Omega, 0)$$ is the 1-type (in the sense of D’Angelo), $$N (\partial \Omega,0)$$ is the order of pseudoconvex extendability, $$e (\partial \Omega,0)$$ is the exponent of pseudoconvex extendability, $$g (\partial \Omega,0)$$ is the growth exponent for the Banach kernel and $$h (\partial \Omega,0)$$ is the growth exponent for the Bergman metric. The authors also construct a Hölder continuous local holomorphic function with peak at the semiregular boundary point. Furthermore if $$\Omega$$ admits a Stein neighborhood basis, the local peak function can be globalized.
For the entire collection see [Zbl 0811.00006].

MSC:
 32T99 Pseudoconvex domains 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)