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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].

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