Let $(\mathcal{C}^{n},g_{\mathcal{C}^{n}})$ be a Calabi model space with $\dim_{\mathbb{C}}(\mathcal{C}^{n})=n$. Define a constant

(5.181)

$$\delta_{b}\equiv 2\underline{\lambda}^{\frac{1}{2}}n^{-\frac{1}{2}}>0$$

where $\underline{\lambda}>0$ is given by (5.14). If $u$ is a harmonic function outside a compact set in $\mathcal{C}^{n}$ satisfying

(5.182)

$$|u(z,\bm{y})|=O(e^{\delta\cdot z^{\frac{n}{2}}})$$

for some $\delta\in(0,\delta_{b})$ as $z\rightarrow\infty$. Then $u$ can be decomposed as

(5.183)

$$u(z,\bm{y})=L(z)+h(z,\bm{y})$$

with the following properties:

1. (1)

   $L(z)=\kappa_{0}\cdot z+c_{0}$ for some $\kappa_{0},c_{0}\in\mathbb{R}$.

2. (2)

   $h(z,\bm{y})$ is harmonic and for any $k\in\mathbb{N}$, there is some $C_{k}>0$ such that

   (5.184)

   $$|\nabla^{k}h(z,\bm{y})|\leq C_{k}\cdot e^{-\underline{\delta}\cdot z^{\frac{n}{2}}}$$

   for all $\underline{\delta}\in(0,\delta_{b})$, as $z\to+\infty$.