Bergman functions on weakly uniformly perfect domains II

We study the boundary asymptotic behavior of Bergman functions on planar domains. We prove that $\mathrm{\partial }\mathrm{\Omega }$ is uniformly perfect if and only if $K_{\mathrm{\Omega }}^{(1)}(w)\asymp {\delta }_{\mathrm{\Omega }}(w{)}^{-4}$. For weakly uniformly perfect boundaries, we establish lower and upper bounds for $K_{\mathrm{\Omega }}^{(1)}$, which characterize weak uniform perfectness. Combining these estimates with capacity upper bounds for the Bergman kernel, we derive optimal growth rates for the Bergman metric and Bergman distance on Zalcman-type domains; in particular, we obtain $d_{\mathrm{\Omega }}\gtrsim \log \log \mathrm{\mid }z{\mathrm{\mid }}^{-1}$ in the power-type case and $d_{\mathrm{\Omega }}\gtrsim (\log \mathrm{\mid }z{\mathrm{\mid }}^{-1})\mathrm{/}(\log \log \mathrm{\mid }z{\mathrm{\mid }}^{-1})$ in the logarithmic-type case. Our results extend and refine earlier work of Chen and Xiong–Zheng.