Parameter Dependence of Weighted Bergman Kernels Beyond Smoothness
Abstract
We study the parameter dependence of weighted Bergman kernels on fixed bounded domains in ℂn. Our main result establishes real-analytic dependence on t∈(−1,∞) for the kernels associated with δt dV on bounded pseudoconvex domains with C2 boundary, where δ is the Euclidean distance to the boundary. The parameter derivatives satisfy factorial estimates in Cℓ(S× S) for every SSubsetΩ and ℓ≥0, uniformly on compact parameter intervals. The proof combines weighted L2 estimates for ∂̅ 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 Gs, s≥1, under uniform parameter estimates up to the boundary. We show that the same derivative formula holds for the weights −tlogδ.
Related
- Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables
- Sharp weighted norm estimates for the positive Bergman operator: the two-parameter case
- Comparison of the Bergman and Szegő kernels on arbitrary bounded pseudoconvex domains with C2 boundary
- A Characterization of Pluriharmonic Functions via p-Bergman Kernels and A Forelli-type Result
- Norm of the generalized Hilbert operator on weighted Bergman spaces
BibTeX
@misc{marinescu2026parameter,
title = {{Parameter Dependence of Weighted Bergman Kernels Beyond Smoothness}},
author = {George Marinescu and Xu Xing},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24258},
eprint = {2609.24258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24258},
}
RIS
TY - UNPB
TI - Parameter Dependence of Weighted Bergman Kernels Beyond Smoothness
AU - George Marinescu
AU - Xu Xing
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24258
UR - https://arxiv.org/abs/2609.24258
AB - We study the parameter dependence of weighted Bergman kernels on fixed bounded domains in ℂ^n. Our main result establishes real-analytic dependence on t∈(−1,∞) 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^ℓ(S× S) for every SSubsetΩ and ℓ≥0, uniformly on compact parameter intervals. The proof combines weighted L^2 estimates for ∂̅ 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≥1, under uniform parameter estimates up to the boundary. We show that the same derivative formula holds for the weights −tlogδ.
ER -