We focus on $w^{1\bar{1}}=\gamma_{1}+\gamma_{3}$, where $\gamma_{1},\gamma_{3}$ admit the Green’s representation (4.11). We split the integral on $S$ into the short distance contribution from $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}$ and the long distance contribution from $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|\gtrsim\frac{1}{2}\text{ or }|\eta_{2}-\eta^{\prime}_{2}|\gtrsim\frac{1}{2}\}$.

The short distance contribution to the integral $w^{1\bar{1}}$ is

$$-\pi A^{1/2}\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}.$$

Applying Lemma 4.2, we can replace the periodic Newtontian potential $\gamma$ by the ordinary Newtonian potential, so the short distance contribution is replaced by

$$-\pi A^{1/2}\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}-\frac{1}{8\pi^{2}|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime},$$

at a cost of a smooth error of order $O(A^{-1/4})$. The measure $\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}$ is equal to $2\frac{|z_{1}^{\prime}|^{2}}{Aa^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})$, so the above expression is

$$\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}-\frac{1}{8\pi^{2}|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}\frac{-2\pi|z_{1}^{\prime}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}).$$

We may assume the submanifold $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}$ is graphical, so Lemma 4.15 applies after scaling. Thus the short distance contribution to $w^{1\bar{1}}$ is

$$-\frac{1}{4\pi R}\frac{-2\pi|z_{1}^{\prime}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}+O(A^{-1/4})=\frac{|z_{1}^{\prime}|^{2}}{2RA^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}+O(A^{-1/4}),$$

where the complex coordinates $z_{i}^{\prime}$ are computed at $\vec{h}(\xi_{2})\in S$. But the factor $\frac{|z_{1}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}\bar{z}_{j}}$ varies slowly, so we may as well compute it at $(\eta_{1},\eta_{2},\mu)$.

The long distance contribution to $w^{1\bar{1}}$ is $O(A^{-1/4}\max(1,\log(A^{-1/4}\varrho)))$ by following the same steps as in Section 4.2, 4.3, using Lemma 4.1. Combining the two contributions,

$$|w^{1\bar{1}}-\frac{|z_{1}|^{2}}{2RA^{1/2}a^{i\bar{j}}z_{i}\bar{z}_{j}}|\leq CA^{-1/4}\max(1,\log(A^{-1/4}\varrho))).$$

The cases of $w^{p\bar{q}}$ and $v=Aa^{p\bar{q}}w^{p\bar{q}}$ are similar. ∎