Pluricomplex Green function

Remarks

Example: for the unit ball $\Omega = \set{\abs{z} < 1}$ and $w = 0$, $g_\Omega(z, 0) = \log\abs{z}$. Check: $\log\abs{z}$ belongs to $\mathcal{F}_0$. Conversely, let $u \in \mathcal{F}_0$ and let $\xi$ be a unit vector. The function $\lambda \mapsto u(\lambda \xi) - \log\abs{\lambda}$ is subharmonic on $0 < \abs{\lambda} < 1$ and bounded above near $0$, so it extends across $0$. Its boundary values are at most $0$, so $u(\lambda \xi) \le \log\abs{\lambda}$ by the maximum principle. Non-example: $u(z) = \abs{z}^2 - 1$ is negative and plurisubharmonic on the ball but is bounded near $0$, so it fails clause 2. For $n = 1$ and a bounded regular domain, $g_\Omega(z, w)$ is the classical Green function with the sign convention $g_\Omega(z, w) \sim \log\abs{z - w}$ near $w$. Holomorphic maps decrease it: if $f \colon \Omega_1 \to \Omega_2$ is holomorphic, then $g_{\Omega_2}(f(z), f(w)) \le g_{\Omega_1}(z, w)$. If $\Omega$ is bounded, then $g_\Omega(\cdot, w)$ is plurisubharmonic and belongs to $\mathcal{F}_w$.