Boundary behavior of the pluricomplex Green function
The paper studies the behaviour of the pluricomplex Green function \(g_\Omega(z,w)\) of a bounded domain \(\Omega\subset{\mathbf{C}}^n\) when the pole \(w\) tends to \(w_0\in\partial\Omega\). The main result is as follows. Let \(\Omega\) have a plurisubharmonic peak function \(\rho\) at \(w_0\) (i.e., \(\rho\in PSH(\Omega)\cap C(\overline\Omega)\), \(\rho(w_0)=0\) and \(\rho(z)<0\) for all \(z\in \overline\Omega\setminus\{w_0\}\)) that is Hölder continuous at \(w_0\) and such that \[ \max\{{\log|\rho(z)|\over\log|z-w_0|}: z\in\overline\Omega, \;r\leq|z-w_0|\leq{1\over 2}\} = O(\log \log {1\over r}) \] as \(r\to 0\). Then for any compact set \(K\subset \overline\Omega\setminus\{w_0\}\), \[ \lim_{w\to w_0} g_\Omega(z,w)=0 \] uniformly in \(z\in K\). In particular, this is true when \(\Omega\) is pseudoconvex with smooth boundary and \(w_0\) is a point of finite type.
scientific article; zbMATH DE number 1928883