A Characterization of Pluriharmonic Functions via $p$-Bergman Kernels and A Forelli-type Result

Abstract

Given a plurisubharmonic function $φ>-\infty$ on the unit ball $\mathbb{B}^n$ and a constant $0<p\leq2$, Guan-Zhou's optimal $L^p$ extension theorem yields a sharp lower bound for the weighted $p$-Bergman kernel. We show that the equality holds at some point if and only if $φ$ is pluriharmonic on $\mathbb{B}^n$. This provides an analogue of the equality part of Suita's conjecture. As an application, we prove a Forelli-type result. Let $φ$ be a real-valued function on $\mathbb{B}^n$. If every slice of $φ$ through the origin is harmonic and if $φ$ is plurisubharmonic in a neighborhood of the origin, then $φ$ is pluriharmonic on $\mathbb{B}^n$.