Some Characterizations of Weakly Uniformly Perfect Sets
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911128384176128 |
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| author | Zheng, Zhiyuan |
| author_facet | Zheng, Zhiyuan |
| contents | In this paper, the concept of weakly uniform perfectness is considered. As an analogue of the theory of uniform perfectness, we obtain the relationships between weakly uniform perfectness and Bergman kernel, Poincaré metric and Hausdorff content. In particular, for a bounded domain $Ω\subset \mathbb{C}$, we show that the uniform perfectness of $\partial Ω$ is equivalent to $K_Ω(z) \gtrsim δ(z)^{-2}$, where $K_Ω(z)$ is the Bergman kernel of $Ω$ and $δ(z)$ denotes the boundary distance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Characterizations of Weakly Uniformly Perfect Sets Zheng, Zhiyuan Complex Variables In this paper, the concept of weakly uniform perfectness is considered. As an analogue of the theory of uniform perfectness, we obtain the relationships between weakly uniform perfectness and Bergman kernel, Poincaré metric and Hausdorff content. In particular, for a bounded domain $Ω\subset \mathbb{C}$, we show that the uniform perfectness of $\partial Ω$ is equivalent to $K_Ω(z) \gtrsim δ(z)^{-2}$, where $K_Ω(z)$ is the Bergman kernel of $Ω$ and $δ(z)$ denotes the boundary distance. |
| title | Some Characterizations of Weakly Uniformly Perfect Sets |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2402.09235 |