Weighted Bergman Kernel, Directional Lelong Number and John–Nirenberg Exponent

Abstract

Let $\psi$ be a plurisubharmonic function on the closed unit ball and $K_{t\psi}(z)$ the Bergman kernel on the unit ball with respect to the weight $t\psi$. We show that the boundary behavior of $K_{t\psi}(z)$ is determined by certain directional Lelong number of $\psi$ for all $t$ smaller than the John-Nirenberg exponent of $\psi$ associated to certain family of nonisotropic balls, which is always positive.