多复变量加权Fock空间上的Bergman核与Hankel算子
Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables
- School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在多复变量加权Fock空间中，通过正则化曲率假设建立Bergman核估计，并构造\bar\partial解算子，进而用加权IDA空间刻画Hankel算子的有界性与紧性。
AI中文摘要：
我们在\mathbb C^n上引入一类多次调和权，并发展相应的加权Fock空间F_φ^p的理论。曲率假设施加在距原权有界距离的\mathcal C^2正则化上。因此，原权不必是光滑或严格多次调和的，即使它属于\mathcal C^2类，其复Hessian特征值也不必一致可比。这提供了Christ加倍理论在多复变量中的对应物。我们建立了加权Bergman核的全局上界估计（在对角线外衰减）和对角线附近的均匀下界估计。这些估计导出核函数的L^p界、Bergman投影的有界性，以及相应Fock空间的对偶与复插值。对于超出一致受控曲率设置的某类权，我们构造了\bar\partial方程的积分解算子，并建立了1≤p≤∞的尺度自适应加权L^p估计。作为应用，对于所有1≤p,q<∞，我们利用加权IDA空间刻画了从F_φ^p到L_φ^q的Hankel算子（可能具有无界符号）的有界性与紧性。
英文摘要：
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.
