Pointwise estimates of the Bergman kernel with an exponential weight on the unit ball

TL;DR

This paper derives pointwise estimates for the Bergman kernel on the unit ball with an exponential weight, revealing decay properties related to the Riemannian distance induced by the weight function.

Contribution

It provides the first explicit exponential decay estimate for the weighted Bergman kernel with a specific exponential weight on the unit ball.

Findings

Bergman kernel satisfies a specific exponential decay estimate.

The estimate involves the Riemannian distance induced by the weight function.

The result generalizes previous work by Christ on similar kernels.

Abstract

We consider the weighted Bergman space $A^2_\psi(\Bn)$ of all holomorphic functions on $\Bn$ square integrable with respect to a particular exponential weight measure $e^{-{\psi}} dV$ on $\Bn$, where \begin{align*} \psi(z):=\frac{1}{1-|z|^2}. \end{align*} We prove the following estimate for the Bergman kernel $K_\psi(z,w)$ of $A^2_\psi(\Bn)$: \begin{align*} |K_\psi(z,w)|^2\le C\frac{e^{\psi(z)+\psi(w)}}{{\rm Vol}(B_\psi(z,1)){\rm Vol}(B_\psi(w, 1))}e^{-\varepsilon d_\psi(z,w)}, \quad z, w\in\Bn, \end{align*} where $d_\psi$ is the Riemannian distance induced by the potential function $\psi$ and $B_\psi(z,1)$ is the $d_\psi$-ball of center $z$ and radius $1$. The result is motivated by Christ \cite{Chr}.