Regularity of the pluricomplex Green function with several poles (Q2758105)

For a bounded domain \(\Omega\) in \(\mathbb C^n\), points \(p^1,\dots,p^k\in \Omega\), and positive numbers \(\mu_1,\dots,\mu_k\), the pluricomplex Green function \(g\) with logaritmic poles at \(p^i\) of weight \(\mu_i\) is defined as the supremum over the class of all \(v\in\text{PSH}(\Omega)\) such that \(v<0\) and \(\limsup_{z\to p^i} (v(z)-\mu_i\log| z-p^i| )<\infty\) for all \(i=1,\dots,k\). It is well known that for hyperconvex domains \(g\) tends to \(0\) at the boundary. The author proves that if \(\Omega\) is hyperconvex, then \(g\) is a continuous function defined on the set \(\{(z,p^1,\dots,p^k,\mu_1,\dots,\mu_k)\in \bar \Omega\times\Omega^k\times \mathbb R_+^k\,;\, z\neq p^i\neq p^j \text{ if } i\neq j \}\). The main result of the paper is the following: If \(\Omega\) is of class \(C^{2,1}\) and stricty pseudoconvex, then \(g\in C^{1,1}(\Omega\setminus\{p^1,\dots,p^k\})\), and there exists a positive constant \(C\) only depending on \(\Omega\), \(p_1,\dots,p^k\), and \(\mu_1,\dots,\mu_k\), such that \(| \nabla^2g(z)| \leq C/\min_i| z-p^i| ^2\) for all \(z\in \Omega\setminus\{p^1,\dots,p^k\}\). Here \(\nabla^2g\) is the Hessian of \(g\), which is defined at almost every point of \(\Omega\). If one drops the regularity assumtion of the boundary but assumes that \(\limsup_{z\to \partial \Omega} | g(z)| /\text{dist}(z,\partial \Omega)<\infty\), then the estimate \(| \nabla g(z)| \leq C/\min_i| z-p^i| \) for the first order derivatives holds.