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

We consider the weighted Bergman space $A_{\psi }^2(\text{Bn})$ of all holomorphic functions on $\text{Bn}$ square integrable with respect to a particular exponential weight measure $e^{-\psi }dV$ on $\text{Bn}$, where \begin{align*} ψ(z):=\frac{1}{1-|z|^2}. \end{align*} We prove the following estimate for the Bergman kernel $K_{\psi }(z,w)$ of $A_{\psi }^2(\text{Bn})$: \begin{align*} |K_ψ(z,w)|^2\le C\frac{e^{ψ(z)+ψ(w)}}{{\rm Vol}(B_ψ(z,1)){\rm Vol}(B_ψ(w, 1))}e^{-\varepsilon d_ψ(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}.