Quantitative estimates for the Green function and an application to the Bergman metric
Let \(D\Subset\mathbb{C}^n\) be a bounded pseudoconvex domain. There is a bounded plurisubharmonic exhaustion function \(\rho: D\to [-1,0)\), such that for some number \(0<\alpha<1\) and constants \(C_1,C_2>0\) the inequality \[ C_2{\delta^\alpha}_D(z)\leq |\rho(z)|\leq C_1{\delta^\alpha}_D(z) \] holds everywhere on \(D\). Here \(\delta_D(\cdot)\) denotes the boundary distance in \(D\). It is proved that for any fixed number \(0<t\ll 1\) there exists a constant \(\widetilde{C}_t\gg 1\), such that for all compact subsets \(K\subset D\) and all \(w\in D\) satisfying \[ \delta_D(w)< \widetilde{C}_t^{-1}\delta_D(K) \] the pluricomplex Green function \(G_D\) of \(D\) can be estimated by \[ \sup_{K}\bigl|G_D(\cdot,w)\bigr|\quad\leq\widetilde{C}_t\left(\left(\frac{\delta_D^{1-2nt}(w)}{\delta_D(K)}\right)^{\alpha /n} \left(\log\frac{2R_D}{\delta_D(w)}\right)^{1/n}+\delta_D^{t\alpha}(w)\bigl|\log\delta_D(w)\bigr|\right) \] where \(R_D\) denotes the diameter of \(D\). The authors show two applications of the last inequality to questions about the boundary behavior of the Bergman differential metric \(B_D\) of \(D\).
scientific article; zbMATH DE number 1478820
scientific article