发表机构
University of Cologne; Southwest Minzu University（科隆大学; 西南民族大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究加权Bergman核关于参数的实解析依赖性，建立阶乘估计并给出导数公式，证明Gevrey类保持性，适用于多种权函数。
AI 中文摘要
我们研究了在$\mathbb C^n$中固定有界域上加权Bergman核的参数依赖性。我们的主要结果建立了在具有$C^2$边界的伪凸有界域上，与$\delta^t\\,dV$相关的核关于$t\in(-1,\infty)$的实解析依赖性，其中$\delta$是到边界的欧几里得距离。参数导数在$C^\ell(S\times S)$中满足阶乘估计，对每个$S\Subset\Omega$和$\ell\ge0$成立，且在紧参数区间上一致。证明结合了$\bar\partial$的加权$L^2$估计与有界算子的全纯族，并给出了在固定加权Bergman空间中的局部全纯延拓。对于在空间和参数上直到边界都光滑的权函数族，我们还用迭代加权Bergman投影和完全指数Bell多项式表达了所有参数导数。该公式蕴含在直到边界的一致参数估计下，Gevrey类$G^s$（$s\ge1$）的保持性。我们证明了相同的导数公式对权函数$-t\log\delta$也成立。
英文摘要
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 $-t\logδ$.
