follows: $$|G(y,\eta)-G(0,\eta)|\le |G(y,\eta)|+|G(0,\eta)|$$ $$|G(0,\eta)|\le C\log |y|.$$ $$\begin{aligned}
|G(y,\eta)|&\le &\frac 1{2\pi}\big |\log |y-\eta |\big |+\frac 1{2\pi}\log \frac 87+\frac 1{2\pi}\big |\log |\frac{\epsilon_k^{-2}y}{|y|^2}-\eta |\big |\\
&\le & C(\log |y|+\big |\log |y-\eta |\big |)\end{aligned}$$ where the last inequality holds because $$|y-\eta |\le |\frac{\epsilon_k^{-2}y}{|y|^2}-\eta |<C|y|,$$ which implies $$\big |\log |\frac{\epsilon_k^{-2}y}{|y|^2}-\eta | \big |\le C(\big |\log |y-\eta |\big |+\log |y|).$$ The second case of (\[1020e5\]) (when $\eta\in \Sigma_2$) is proved.

For $\eta\in \Sigma_3$, we first consider when $|\eta|>2|y|$. In this case $$|\log \frac{|\eta |}{|\eta -y|}|=|\log |\frac{\eta}{|\eta |}-\frac{y}{|\eta |}||\le C\frac{|y|}{|\eta |}.$$ For the second term, since $\eta,y\in \Omega_k$ and $|\eta |>2|y|$, we have $|y||\eta |<\frac 12 \epsilon_k^{-2}$, consequently $$\bigg |\log |\frac{y}{|y|}-\frac{|y|\eta }{\epsilon_k^{-2}}| \bigg |\le C|y||\eta |\epsilon_k^2\le C\frac{|y|}{|\eta |}.$$ So (\[1020e5\]) is proved in this case. Now we consider $\frac{|y|}2\le |\eta |\le 2|y|$ and $|\eta -y|\ge \frac{|y|}2$. For the first term we have $$\bigg |\log \frac{|\eta |}{|y-\eta |}\bigg |\le \log 4\le C\frac{|y|}{|\eta |}.$$ For the second term, we want to show $$\label{1027e5}
\bigg |\log |\frac{y}{|y|}-\frac{|y|\eta }{\epsilon_k^{-2}}| \bigg |\le C\le C\frac{|y|}{|\eta |}.$$ If either $|y|\le \frac{15}{16}\epsilon_k^{-1}$ or $|\eta |\le \frac{15}{16}\epsilon_k^{-1}$ we have $$|\frac{y}{|y|}-\frac{|y|\eta}{\epsilon_k^{-2}}|\ge \frac{1}{16},$$ therefore (\[1027e5\]) obviously holds. For $\frac{15}{16}\epsilon_k^{-1}<|y|, |\eta |\le \epsilon_k^{-1}$, using $|y-\eta |>\frac 12|y|$ we obtain easily $$\bigg |\frac{y}{|y|}-\frac{|y|\eta}{\epsilon_k^{-2}}\bigg |\ge \frac 38.$$ Therefore (\[1020e5\]) is proved 