Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains (Q913106)

The main results of the paper are the following two theorems: (A) Let \(\Omega \subset {\mathbb{C}}^ 2\) have a transverse symmetry and let \(\phi\) be normalized by the symmetry. Let g be a Kähler metric on \(\Omega\) whose Kähler form is \((i/2)\partial {\bar \partial} \log (- 1/\phi)\) near \(\partial \Omega\) and let \(\Delta_{\phi}\) be the Laplace- Beltrami operator for the metric g. If \(u\in C^{\infty}({\bar \Omega})\) satisfies \(\Delta_{\phi}u=0\), then u is plurisubharmonic. (B) For each \(n\geq 2\) there is a smooth radial defining function \(\phi\) for \(B^{n+1}\) (where \(B^{n+1}:=\{z\in {\mathbb{C}}^{n+1} |\) \(| z|^ 2<1\})\) so that \(\log (-1/\phi)\) is strictly plurisubharmonic in all \(B^{n+1}\), and such that \(\Delta_{\phi}u=0\) has a solution \(u\in C^{\infty}(\overline{B^{n+1}})\) which is not plurisubharmonic.