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

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}.