Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

Abstract

Let $D$ be a pseudoconvex domain in $\C^k_t\times\Cn_z$ and let $\varphi$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z;(t,z)\in D\}$, let ${\varphi }^t$ be the restriction of $\varphi$ to $D_t$ and denote by $K_t(z,\zeta )$ the Bergman kernel of $D_t$ with the weight function ${\varphi }^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $\varphi =0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting of $\Rn$.