Regularity of the $p-$Bergman kernel

期刊论文
发表日期：

2024年1月22日

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Bo-Yong Chen,
Yuanpu Xiong

摘要

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

Regularity of the \(p-\)Bergman kernel

摘要