Regularity of the $p-$Bergman kernel
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929195259527168 |
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| author | Chen, Bo-Yong Xiong, Yuanpu |
| author_facet | Chen, Bo-Yong Xiong, Yuanpu |
| contents | We show that the $p-$Bergman kernel $K_p(z)$ on a bounded domain $Ω$ 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 $K_p(ζ,z)$ for fixed $ζ\in Ω$. Global irregularity of $K_p(ζ,z)$ 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 $Ω$ is positive, we obtain the log-Lipschitz continuity of $p\mapsto{K_p(z)}$ for $1\leq{p}\leq2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_06877 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regularity of the $p-$Bergman kernel Chen, Bo-Yong Xiong, Yuanpu Complex Variables We show that the $p-$Bergman kernel $K_p(z)$ on a bounded domain $Ω$ 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 $K_p(ζ,z)$ for fixed $ζ\in Ω$. Global irregularity of $K_p(ζ,z)$ 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 $Ω$ is positive, we obtain the log-Lipschitz continuity of $p\mapsto{K_p(z)}$ for $1\leq{p}\leq2$. |
| title | Regularity of the $p-$Bergman kernel |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2302.06877 |