AI 中文总结
本文构造了与ξ-伯格曼核相关的典范埃尔米特正定核，证明其正定并恢复ξ-伯格曼核为其对角限制，还得到该核的显式公式等性质，揭示了微分泛函选择的影响。
AI 中文摘要
设Ω⊂ℂⁿ，ξ∈ℓ¹。由Bao与Guan引入的ξ-伯格曼核K_{ξ,Ω}通过用序列ξ确定的泛函替代点赋值泛函，推广了经典伯格曼核。尽管该核继承了若干重要的极值性质与多重次调和性质，但它本质上是对角对象，因此缺乏经典伯格曼理论核心的双变量再生核结构。本文的目的是关联一个与ξ-伯格曼核相关的典范埃尔米特正定核，并研究其解析与几何性质。我们的构造基于对应于ξ-赋值泛函的里斯代表元族，更确切地说，我们引入一个埃尔米特核，该核由这些代表元的格拉姆核得到，并证明它是正定的，且对z∈Ω满足B_{ξ,Ω}(z,z)=K_{ξ,Ω}(z)，从而将ξ-伯格曼核作为其对角限制恢复。作为推论，我们证明ξ-伯格曼核在Ω上是实解析的。我们还建立了双全纯变换律，并得到了ξ-伯格曼核用经典伯格曼核的导数表示的形式。此外，我们针对对应于几类序列ξ的上半平面ℍ上的ξ-伯格曼核获得了显式公式，建立了相应的ξ-陆启铿结果，并推导了精确的边界渐近式。这些例子说明了微分泛函的选择如何影响相关核的零点集与边界增长。
英文摘要
Let $Ω\subset \mathbb{C}^{n}$ and $ξ\in \ell^{1}$. The $ξ$-Bergman kernel $K_{ξ, Ω}$, introduced by Bao and Guan, generalizes the classical Bergman kernel by replacing the point evaluation functional with a functional determined by sequence $ξ$. While this kernel inherits several important extremal and plurisubharmonic properties, it is intrinsically an on-diagonal object and therefore lacks the two-variable reproducing kernel structure that lies at the heart of the classical Bergman theory. The purpose of this paper is to associate a canonical Hermitian positive-definite kernel with the $ξ$-Bergman kernel and to investigate its analytic and geometric properties. Our construction is based on the family of Riesz representatives corresponding to the $ξ$-evaluation functionals. More precisely, we introduce a Hermitian kernel obtained as the Gram kernel of these representatives and show that it is positive definite and for $z \in Ω$ satisfies \[ B_{ξ,Ω}(z,z)=K_{ξ,Ω}(z), \] thereby recovering the $ξ$-Bergman kernel as its diagonal restriction. As a consequence, we prove that the $ξ$-Bergman kernel is real analytic on $Ω$. We also establish biholomorphic transformation laws, and obtain a representation of the $ξ$-Bergman kernel in terms of derivatives of the classical Bergman kernel. Furthermore, we obtain explicit formulas for the $ξ$-Bergman kernel on the upper half-plane $\mathbb{H}$ corresponding to several classes of sequences $ξ$, establish corresponding $ξ$-Lu Qi-Keng results, and derive precise boundary asymptotics. These examples illustrate how the choice of the differential functional influences both the zero set and the boundary growth of the associated kernel.
Comments25 pages. Comments are welcome
