$L^2$ estimates of Poincar\'e-Lelong equations on convex domains in $\mathbb{C}^n$

Abstract

In this paper, we prove the existence of solutions of the Poincar\'e-Lelong equation $\sqrt{-1}\partial\bar{\partial}u=f$ on a strictly convex bounded domain $\Omega\subset\mathbb{C}^n$ $(n\geq1)$, where $f$ is a $d$-closed $(1,1)$ form and is in the weighted Hilbert space $L^2_{(1,1)}(\Omega,e^{-\varphi})$. The novelty of this paper is to apply a weighted $L^2$ version of Poincar\'e Lemma for real $2$-forms, and then apply H\"{o}rmander's $L^2$ solutions for Cauchy-Riemann equations.