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

TL;DR

This paper establishes the existence of solutions to the Poincaré-Lelong equation in complex Euclidean space using weighted L^2 techniques and Hörmander's methods, expanding the understanding of such equations with Gaussian weights.

Contribution

It introduces a weighted L^2 Poincaré lemma for 2-forms and applies Hörmander's L^2 solutions to the Cauchy-Riemann equations in the context of the Poincaré-Lelong equation.

Findings

Existence of solutions in weighted Hilbert spaces with Gaussian measure.

Application of weighted L^2 Poincaré lemma for 2-forms.

Use of Hörmander's L^2 techniques for solving the Poincaré-Lelong equation.

Abstract

In this paper, we prove the existence of (global) solutions of the Poincar\'e-Lelong equation $\partial\overline{\p}u=f$, where $f$ is a $d$-closed $(1,1)$ form and is in the weighted Hilbert space with Gaussian measure, i.e., $L^2_{(1,1)}(\mathbb{C}^n,e^{-|z|^2})$. 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 $e^{-|z|^2}$ is used.