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

We consider the weighted Bergman space A²_ψ() of all holomorphic functions on square integrable with respect to a particular exponential weight measure e^-ψ dV on , where {align*} ψ(z):=1/1-|z|^2. {align*} We prove the following estimate for the Bergman kernel K_ψ(z,w) of A²_ψ(): {align*} |K_ψ(z,w)|^2≤ C{eψ(z)+ψ(w)}{{ Vol}(B_ψ(z,1)){ Vol}(B_ψ(w, 1))}e-ε d_ψ(z,w), z, w∈, {align*} where d_ψ is the Riemannian distance induced by the potential function ψ and B_ψ(z,1) is the d_ψ-ball of center z and radius $1$. The result is motivated by Christ {Chr}.