We work in $U_{\beta}^{1}$ for a fixed $\beta$. We have

(7.132)

$$\phi_{t,\bf{I}_{-}}(\bm{x})=\frac{T}{d_{2}}\log|f_{2}(\bm{x})|+\pi_{\mathcal{N}}^{*}\phi_{t}(\bm{x})$$

and

(7.133)

$$(\pi_{\beta}^{\mathcal{N}})^{*}\phi_{-}(\bm{x})=\phi_{t}(\bm{y})-\frac{T}{d_{2}}\log r_{-}(\bm{y})$$

where $\bm{y}=\pi_{\beta}^{\mathcal{N}}(\bm{x})$. By definition it is easy to see that $\bm{y}-\bm{x}$ is of order $\underline{\epsilon}_{T^{2}}$ in the coordinates in $v_{2},\zeta_{3},w_{2},\cdots,w_{n-1}$. By our choice of $T$ in terms of $t$ we have

(7.134)

$$-\log|r_{-}(\bm{y})|=d_{1}\log|t|-\log|f_{2}(\bm{y})|.$$

Then by Lemma 7.6, and use weighed Schauder estimates as above we get the conclusion.

∎