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

TL;DR

This paper proves that the logarithm of the weighted Bergman kernel's diagonal is plurisubharmonic in pseudoconvex domains, generalizing previous results and extending to Robin functions and real settings.

Contribution

It generalizes recent results on subharmonicity of the Bergman kernel to higher dimensions and weighted cases, including Robin functions and real domain analogs.

Findings

             

         

         

Abstract

Let $D$ be a pseudoconvex domain in $\C^k_t\times\Cn_z$ and let $\phi$ 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 $\phi^t$ be the restriction of $\phi$ to $D_t$ and denote by $K_t(z,\zeta)$ the Bergman kernel of $D_t$ with the weight function $\phi^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $\phi=0$) 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 $\Rn$.