Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables
Abstract
We introduce a class of plurisubharmonic weights on \mathbb C^n and develop a theory of the associated weighted Fock spaces F_φ^p. The curvature assumptions are imposed on a \mathcal C^2-regularization at bounded distance from the original weight. Thus the original weight need not be smooth or strictly plurisubharmonic, and, even when it is of class \mathcal C^2, its complex Hessian eigenvalues need not be uniformly comparable. This provides a counterpart of Christ's doubling theory in several complex variables. We establish a global upper estimate with decay away from the diagonal and a uniform lower estimate near the diagonal for the weighted Bergman kernel. These estimates yield L^p bounds for the kernel functions, boundedness of the Bergman projection, and duality and complex interpolation for the associated Fock spaces. For a certain class of weights beyond the uniformly controlled curvature setting, we construct an integral solution operator for the \bar\partial equation and establish scale-adapted weighted L^p estimates for 1\leq p\leq\infty. As an application, for all 1\leq p,q<\infty, we characterize the boundedness and compactness of Hankel operators from F_φ^p to L_φ^q with possibly unbounded symbols in terms of weighted IDA spaces.
Related
- Semigroup of composition operators on the Fock space of several complex variables
- Parameter Dependence of Weighted Bergman Kernels Beyond Smoothness
- Compact linear combination of composition operators on weighted Bergman-Orlicz spaces induced by doubling weights
- Quantum harmonic analysis and the Differential operators on weighted Bergman spaces of bounded symmetric domains
- Weighted Cesàro type operators on the weighted Bergman spaces in the unit ball
BibTeX
@misc{liu2026bergman,
title = {{Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables}},
author = {Guijun Liu and Xiaofeng Wang and Zhicheng Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26810},
eprint = {2609.26810},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26810},
}
RIS
TY - UNPB
TI - Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables
AU - Guijun Liu
AU - Xiaofeng Wang
AU - Zhicheng Zeng
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.26810
UR - https://arxiv.org/abs/2609.26810
AB - We introduce a class of plurisubharmonic weights on \mathbb C^n and develop a theory of the associated weighted Fock spaces F_φ^p. The curvature assumptions are imposed on a \mathcal C^2-regularization at bounded distance from the original weight. Thus the original weight need not be smooth or strictly plurisubharmonic, and, even when it is of class \mathcal C^2, its complex Hessian eigenvalues need not be uniformly comparable. This provides a counterpart of Christ's doubling theory in several complex variables. We establish a global upper estimate with decay away from the diagonal and a uniform lower estimate near the diagonal for the weighted Bergman kernel. These estimates yield L^p bounds for the kernel functions, boundedness of the Bergman projection, and duality and complex interpolation for the associated Fock spaces. For a certain class of weights beyond the uniformly controlled curvature setting, we construct an integral solution operator for the \bar\partial equation and establish scale-adapted weighted L^p estimates for 1\leq p\leq\infty. As an application, for all 1\leq p,q<\infty, we characterize the boundedness and compactness of Hankel operators from F_φ^p to L_φ^q with possibly unbounded symbols in terms of weighted IDA spaces.
ER -