Regularity of the $p-$Bergman kernel

TL;DR

This paper investigates the regularity properties of the $p$-Bergman kernel on bounded domains, establishing local $C^{1,1}$ regularity for $p\\geq 1$ and exploring irregularities and continuity aspects related to the kernel.

Contribution

It proves local $C^{1,1}$ regularity of the $p$-Bergman kernel for $p\geq 1$, analyzes irregularities for large $p$, and studies the kernel's continuity in $p$ under certain conditions.

Findings

$p$-Bergman kernel is locally $C^{1,1}$ for $p\geq 1$

Global irregularity occurs for large $p$ in some domains

Log-Lipschitz continuity of $K_p(z)$ in $p$ for $1\leq p\leq 2$

Abstract

We show that the $p-$Bergman kernel $K_p(z)$ on a bounded domain $\Omega$ is of locally $C^{1,1}$ for $p\geq1$.The proof is based on the locally Lipschitz continuity of the off-diagonal $p-$Bergman kernel $K_p(\zeta,z)$ for fixed $\zeta\in \Omega$. Global irregularity of $K_p(\zeta,z)$ is presented for some smooth strongly pseudoconvex domains when $p\gg 1$. As an application of the local $C^{1,1}-$regularity, an upper estimate for the Levi form of $\log K_p(z)$ for $1<p<2$ is provided. Under the condition that the hyperconvexity index of $\Omega$ is positive, we obtain the log-Lipschitz continuity of $p\mapsto{K_p(z)}$ for $1\leq{p}\leq2$.