Weighted Bergman kernel, directional Lelong number and John-Nirenberg exponent

TL;DR

This paper investigates how the boundary behavior of weighted Bergman kernels on the unit ball is governed by directional Lelong numbers of a plurisubharmonic function, linking complex analysis and geometric measure theory.

Contribution

It establishes a connection between the boundary behavior of weighted Bergman kernels and directional Lelong numbers, extending understanding of plurisubharmonic functions in complex analysis.

Findings

Boundary behavior determined by directional Lelong number

Applicable for weights with t smaller than John-Nirenberg exponent

Provides a positive lower bound for the 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.