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

Abstract

In this paper, we prove the existence of (global) solutions of the Poincaré-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., $L_{(1,1)}^2({\mathbb{C}}^n,e^{-|z{|}^2})$. The novelty of this paper is to apply a weighted $L^2$ version of Poincaré Lemma for $2$-forms, and then apply Hörmander's $L^2$ solutions for Cauchy-Riemann equations. In the both cases, the same weight $e^{-|z{|}^2}$ is used.

References in corpus (1)

• Hardy-Sobolev type inequalities and their applications