AI 中文总结
研究平面区域上伯格曼函数的边界渐近行为，证明\(\partial\Omega\)一致完美的充要条件，建立弱一致完美边界\(K_{\Omega}^{(1)}\)的上下界，结合伯格曼核容量上界，得到Zalcman型区域伯格曼度量和距离的最优增长率，扩展细化前人工作。
AI 中文摘要
我们研究平面区域上伯格曼函数的边界渐近行为。证明了\(\partial\Omega\)是一致完美的当且仅当\(K_{\Omega}^{(1)}(w)\asymp \delta_{\Omega}(w)^{-4}\)。对于弱一致完美边界，建立了\(K_{\Omega}^{(1)}\)的上下界以刻画弱一致完美性。结合伯格曼核的容量上界，得到了Zalcman型区域上伯格曼度量和伯格曼距离的最优增长率，如幂型情形\(d_{\Omega}\gtrsim \log\log |z|^{-1}\)，对数型情形\(d_{\Omega}\gtrsim (\log |z|^{-1})/(\log\log |z|^{-1})\)。结果扩展并细化了陈和熊 - 郑的早期工作。
英文摘要
We study the boundary asymptotic behavior of Bergman functions on planar domains. Motivated by Chen's question on the equivalence between uniform perfectness of the boundary and the sharp growth rates of the Bergman kernel and the Bergman metric, we focus on the second part of the question concerning the Bergman metric. We prove that $\partialΩ$ is uniformly perfect if and only if $K_Ω^{(1)}(w)\asymp δ_Ω(w)^{-4}$. We also find that under suitable weak uniform perfectness conditions, there exist sequences of points along which $b_Ω(w_n)=o(δ_Ω(w_n)^{-1})$, providing partial evidence toward an affirmative answer. Our method relies on sharp lower and upper bounds for $K_Ω^{(1)}$ and $K_Ω$. As an application, we obtain corresponding lower bounds for the Bergman distance on certain planar domains.
