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

Abstract

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

arXiv admin note: text overlap with arXiv:1909.10871

References in corpus (1)

• Hardy-Sobolev type inequalities and their applications