In the above context, given any constants $k_{-},k_{+}\in\mathbb{R}$ with

(3.347)

$$k_{-}-k_{+}=k,$$

there exists a unique global Green’s current $G_{P}$ for $P$ in $Q$ such that the following properties hold:

1. (1)

   $G_{P}$ is of the form

   (3.348)

   $$G_{P}=\psi(z)\wedge dz.$$

   Moreover, for each $z\in\mathbb{R}$, $\psi(z)$ is a closed real $(1,1)$-current on $D$.

2. (2)

   For any nonnegative integer $k\in\mathbb{N}$ and for any $\delta\in(0,10^{-2})$,

   (3.349)

   $\displaystyle\begin{cases}|\nabla^{k}(\psi(z)-(k_{-}z)\cdot\omega_{D})|=O(e^{(1-\delta)\sqrt{\lambda_{1}}z}),&z\rightarrow-\infty,\\ |\nabla^{k}(\psi(z)-(k_{+}z)\cdot\omega_{D})|=O(e^{-(1-\delta)\sqrt{\lambda_{1}}z}),&z\rightarrow\infty,\end{cases}$

   where $\lambda_{1}>0$ is the first eigenvalue of the Hodge Laplacian acting on closed real $(1,1)$-forms on $D$.