Lineally convex domains of finite type: holomorphic support functions
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 polynomial of degree \(2m\) in \(z\) for any fixed \(\zeta \in W_0\), and there is a radius \(d=d(\varepsilon)>0\), such that for \(\zeta \in W_0\) one has \(\widehat S(\zeta ,\zeta)=0\) and for each unit vector \(t \in T_\zeta^{1,0}\partial D_\zeta \setminus \{0\}\) the following estimate holds for the function \(S_{\zeta,t}(w):= \widehat S(z_{\zeta,t}(w),\zeta)\) as a function of \(w\) on \(| w| <d\): \[ \text{Re} \,S_{\zeta,t}(w) \leq r_{\zeta,t}(w)h_{\zeta, t}(w) - \varepsilon \widehat c \,\sum_{j=2}^{2m} \| P_{\zeta,t}^j\| | w_2| ^j \] where \(h_{\zeta,t} (w) \) is a \(C^\infty\)-function of \((\zeta,t,w)\) of the form \(h_{\zeta,t}(w):= h(\zeta -i w_1n_\zeta +w_2t, \zeta) \), with a positive function \(h\) on \(\{z\in \mathbb{C}^n\,\,:\,\, | z| <d\}\times W_0\), bounded away from \(0\). Here, for a homogeneous polynomial \(p\) we denote by \(\| p\| \) the sum of the absolute values of the coefficients of \(p\). The authors describe the form of the support function \(\widehat S (z,\zeta)\) explicitly.
scientific article; zbMATH DE number 2078207