Assume that $(X^{4},g)$ is $\delta$-asymptotically Calabi. Let $\hat{\delta}\in(0,\delta/10)$ such that $u$ satisfies

(4.173)

$\displaystyle\begin{split}\Delta_{g}u&=0\\ u&=O(e^{\hat{\delta}z}),\end{split}$

then for every fixed $k\in\mathbb{Z}_{+}$, let $\bm{x}_{0}\in[T_{0}(k),+\infty)\times Y^{3}$ with $T_{0}(k)\geq 100^{k^{3}}>0$ and denote $z_{0}\equiv z(\bm{x}_{0})$, we have

(4.174)

$$\|\nabla^{k}\Delta_{g_{\mathcal{C}}}u(\bm{x}_{0})\|\leq C(k,g)\cdot e^{-\frac{\delta z_{0}}{2}}.$$