On the Poincare-Lelong equation in ${\mathbb{C}}^n$

In this paper, we prove the existence of (global) solutions of the Poincar'e-Lelong equation $\mathrm{\partial }\bar{\text{p}}u=f$, where $f$ is a $d$-closed $(1,1)$ form and is in the weighted Hilbert space with Gaussian measure, i.e., $L_{(1,1)}^2({\mathbb{C}}^n,e^{-\mathrm{\mid }z{\mathrm{\mid }}^2})$. The novelty of this paper is to apply a weighted $L^2$ version of Poincar'e Lemma for $2$-forms, and then apply H"{o}rmander’s $L^2$ solutions for Cauchy-Riemann equations. In the both cases, the same weight $e^{-\mathrm{\mid }z{\mathrm{\mid }}^2}$ is used.