STABILITY OF THE BERGMAN KERNEL FUNCTION ON PSEUDOCONVEX DOMAINS IN $C^n$

Let $D \subset C^n$ be a smoothly bounded pseudoconvex domain and let ${\bar{D}_r}_r$ be a family of smooth perturbations of $\bar{D}$ such that $\bar{D} \subset \bar{D}_r$. Let $K_D(z, w)$ be the Bergman kernel function on $D \times D$. Then $lim_{r \to 0} K_{D_r}(z, w) = K_D(z, w)$ locally uniformally on $D \times D$.