On several problems in p-Bergman theory

TL;DR

This paper advances p-Bergman theory by solving an open problem, analyzing kernel regularity, asymptotic behavior, and characterizing integrable holomorphic functions, thereby deepening understanding of p-Bergman spaces.

Contribution

It provides a solution to Chen-Zhang's problem, establishes Hölder continuity of the p-Bergman kernel, and characterizes $L^p$-integrable holomorphic functions.

Findings

Solved Chen-Zhang's problem on p-Bergman metric.

Proved Hölder continuity of the off-diagonal p-Bergman kernel.

Analyzed asymptotic behavior of the kernel's maximizer as p approaches 1.

Abstract

In this paper, we first answer Chen-Zhang's problem on $p$-Bergman metric proposed in \cite{CZ22}. Second, we prove the off-diagonal p-Bergman kernel function $K_p(z,w)$ is H\"older continuous of order (1-$\varepsilon$) about the second component when $p>1$ for any $\varepsilon>0$, which improves the corresponding result of Chen-Zhang. Moreover, we prove the asymptotic behavior of the maximizer of $p$-Bergman kernel as $p\rightarrow 1^-$. Finally, we give a characterization of a class of holomorphic functions on $\mathbb{B}^1$ to be $L^p$-integrable.