Parameter Dependence of Weighted Bergman Kernels Beyond Smoothness

Abstract

We study the parameter dependence of weighted Bergman kernels on fixed bounded domains in $\mathbb C^n$. Our main result establishes real-analytic dependence on $t\in(-1,\infty)$ for the kernels associated with $δ^t\,dV$ on bounded pseudoconvex domains with $C^2$ boundary, where $δ$ is the Euclidean distance to the boundary. The parameter derivatives satisfy factorial estimates in $C^\ell(S\times S)$ for every $S\SubsetΩ$ and $\ell\ge0$, uniformly on compact parameter intervals. The proof combines weighted $L^2$ estimates for $\bar\partial$ with a holomorphic family of bounded operators and gives a local holomorphic extension with values in a fixed weighted Bergman space. For weight families smooth jointly in space and parameter up to the boundary, we also express all parameter derivatives in terms of iterated weighted Bergman projections and complete exponential Bell polynomials. This formula implies preservation of the Gevrey class $G^s$, $s\ge1$, under uniform parameter estimates up to the boundary. We show that the same derivative formula holds for the weights $-t\logδ$.