Let $(X^{2n},g)$ be a complete non-compact Riemannian manifold which is $\delta$-asymptotically Calabi space in the sense of Definition 5.1. Let $\underline{\delta}\in(0,\delta/10)$ be a constant such that $u$ satisfies

(5.236)

$\displaystyle\begin{split}\Delta_{g}u&=0\\ u&=O(e^{\underline{\delta}\cdot z^{\frac{n}{2}}}),\end{split}$

then there exists $z_{0}>0$, such that for every fixed $k\in\mathbb{Z}_{+}$, we have for all $z\geq z_{0}$,

(5.237)

$$\|\nabla^{k}_{g_{\mathcal{C}^{n}}}\Delta_{g_{\mathcal{C}^{n}}}u(z,\bm{y})\|\leq C_{k}\cdot e^{-\frac{\delta}{2}\cdot z^{\frac{n}{2}}},$$

where $C_{k}$ is a constant depending only on $X$ and $k$.