Lineally convex domains of finite type: holomorphic support functions
type: Item
instance of: scholarly article
title: Lineally convex domains of finite type: holomorphic support functions
zbMATH Open document ID: 1066.32016
author: K. Diederich
author: J. E. Fornæss
published in: Manuscripta Mathematica
publication date: 2004-07-07T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/46643
review text: The aim of the present article is the construction of a smooth family of optimal holomorphic support functions on lineally convex domains. This continues earlier work on holomorphic support functions on convex domains of finite type in \(\mathbb{C}^n\), see the review in [Math. Z. 230, No. 1, 145--164 (1999; Zbl 1045.32016)]. Let \(D\subset \subset \mathbb{C}^n\) be a domain, \(0 \in \partial D\). Assume that there is an open neighborhood \(W\) of \(0\) such that \(\partial D \cap W\) is \(C^\infty\)-smooth and lineally convex, in the sense that for any \(\zeta \in (\partial D )\cap W\) the holomorphic tangent space \(\zeta + T_\zeta^{1,0}\partial D\) does not intersect \(D\cap W\). Assume further that any \(\zeta \in (\partial D )\cap W\) is of finite type \(\leq 2m\), and on \(W\) there exists a defining function \(r\) for \(D\), such that \(\partial D_\eta =\{z \in W: r(z)=\eta\}\) is also lineally convex and of finite type, for \(-\eta_0\leq \eta \leq \eta_0\), with a fixed suitably small number \(\eta_0>0\). Let \(W_0=\{| r| <\eta_0\}\) and \(\partial D_z:= \{\zeta \in W_0: r(\zeta)=r(z)\}\) for any \(z\in W_0\). For \(\zeta \in W_0\) let \(n_\zeta\) denote the exterior unit normal to \(\partial D_\zeta\) at \(\zeta\), and for \(t \in T_\zeta^{1,0}\partial D_\zeta \setminus \{0\}\) let \[ A_{\zeta, t} = \{z=z_{\zeta,t} (w) := \zeta - iw_1 n_\zeta +w_2t,\text{ with } w = (w_1,w_2) \in \mathbb{C}^2\} \] and \[ D_{\zeta,t} =\{ w\in \mathbb{C}^2: z_{\zeta,t}(w) \in W_0, \;r_{\zeta,t} (w):=r(z_{\zeta,t} (w)) - r(\zeta)<0\} \] Put \[ P^j_{\zeta,t} (w) := \sum_{k+\ell=j} \frac{1}{k!\ell!} \;\frac{\partial^j r_{\zeta,t}(0)}{\partial w_2^k \partial \bar w_2^\ell} w_2^k\bar w_2^\ell \] for \(2\leq j \leq 2m\). Then the main result of the authors can be stated as follows: Theorem. For a suitable constant \(\widehat c >0\) and any \(\varepsilon>0\) small enough there is a function \(\widehat S (z,\zeta) \in C^\infty (\mathbf{C}^n \times W_0)\) which is a holomorphic polyno…
reviewed by: Gregor Herbort
Mathematics Subject Classification ID: 32F17
Mathematics Subject Classification ID: 32T25
Mathematics Subject Classification ID: 32T40
zbMATH DE Number: 2078207
zbMATH Keywords: finite type domains
zbMATH Keywords: support functions
MaRDI profile type: Publication
OpenAlex ID: W2009005801
DOI: 10.1007/S00229-003-0418-9
description: scientific article; zbMATH DE number 2078207