On the $p$-Bergman kernel with respect to a functional $ξ$

TL;DR

The paper develops a unified theory of the $p$-Bergman kernel with respect to a functional $\xi$, linking it to the classical $p$-Bergman kernel, the $\xi$-Bergman kernel, and higher-order Bergman kernels. It defines the $p$-Bergman kernel with respect to $\xi$ via minimizing $L^p$ norms under the constraint $(\xiullet f)(z)=1$, establishes basic and regularity properties, and proves that $ frac{}{}\nicefrac{}{} igl( ext{log} K_{\xi,p}(z)igr)$ is (fiberwise) plurisubharmonic, with stronger strictness under certain degree conditions. The work then connects higher-order kernels and $\xi$-kernels by showing $K^{H,p}_{5}(z)= ext{inf}_{\xi\uinoldsymbol{S}_H}K_{\xi,p}(z)$, and proves a Hahn–Banach based and an alternative $p=2$ proof. Finally, it extends Blocki–Zwonek-type results by establishing convexity and monotonicity properties of kernels along sublevel sets of the pluricomplex Green function and derives corollaries about exhaustion, nontriviality, and dimensionality of $A^p$-spaces. Overall, the paper broadens the $L^p$ Bergman theory, offering new tools for comparing generalized Bergman kernels and for examining boundary behavior and regularity.

Abstract

In the present paper, we generalize the notion of the $p$-Bergman kernel and the $ξ$-Bergman kernel to the $p$-Bergman kernel with respect to a functional $ξ$, and establish some properties of the $p$-Bergman kernel with respect to $ξ$. We also study the relations between the $L^p$ versions of higher order Bergman kernels and $ξ$-Bergman kernels, and as applications we give the reproofs and generalizations of some previous results of Blocki and Zwonek about higher order Bergman kernels.

Theorems & Definitions (65)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9: see BlZw20 for $p=2$
• Corollary 1.10: see BlZw20 for $p=2$