Let $(\mathcal{C},g_{\mathcal{C}})$ be the model space with a fixed fiber $(Y^{3},h_{0})$. Let $K_{0}\geq 1$ and let $\xi\in C^{2K_{0}}(\mathcal{C})$ satisfy the expansion

(4.123)

$$\xi(z,\bm{y})=\sum\limits_{k=1}^{\infty}\xi_{k}(z)\cdot\varphi_{k}(\bm{y}).$$

In addition, assume that there is some $\eta_{0}\neq 0$ such that for every $0\leq m\leq 2K_{0}$,

(4.124)

$$|\nabla^{m}\xi(z,\bm{y})|=O(e^{\eta_{0}z}),$$

then for every $z\geq 1$ and $k\in\mathbb{Z}_{+}$,

(4.125)

$$|\xi_{k}(z)|\leq\frac{Ce^{\eta_{0}z}}{(\Lambda_{k})^{K_{0}}},$$

where the constant $C>0$ is independent of $k$ and $z$.