A Canonical Positive Definite Kernel Associated with the $\xi$-Bergman Kernel

Abstract

Let $\Omega \subset {\mathbb{C}}^n$ and $\xi \in {\ell }^1$. The $\xi$-Bergman kernel $K_{\xi ,\Omega }$, introduced by Bao and Guan, generalizes the classical Bergman kernel by replacing the point evaluation functional with a functional determined by sequence $\xi$. 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 $\xi$-Bergman kernel and to investigate its analytic and geometric properties. Our construction is based on the family of Riesz representatives corresponding to the $\xi$-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 $z\in \Omega$ satisfies \[ B_{ξ,Ω}(z,z)=K_{ξ,Ω}(z), \] thereby recovering the $\xi$-Bergman kernel as its diagonal restriction. As a consequence, we prove that the $\xi$-Bergman kernel is real analytic on $\Omega$. We also establish biholomorphic transformation laws, and obtain a representation of the $\xi$-Bergman kernel in terms of derivatives of the classical Bergman kernel. Furthermore, we obtain explicit formulas for the $\xi$-Bergman kernel on the upper half-plane $\mathbb{H}$ corresponding to several classes of sequences $\xi$, establish corresponding $\xi$-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.

25 pages. Comments are welcome