Zhuo Liu, Xujun Zhang
We establish a Skoda-type $L^2$ division theorem for $L^2$-optimal pairs, using a technique that combines a new Bochner-type inequality derived from the $L^2$-optimal conditions and Skoda's basic inequality. As applications, we provide some new characterizations of domains of holomorphy.
This paper contains 6 sections, 19 theorems, 72 equations.
Theorem 1.1
Let $D$ be a domain in $\mathbb{C}^n$, $\varphi$ an upper semi-continuous function on $D$ and $g=(g_1,\cdots,g_p)\in{\mathcal{O}}(D)^{\oplus p}$. Assume that the pair $(D,\varphi)$ is $L^2$-optimal. Set $\varepsilon>0$ and $m=\min\{n,p-1\}$. Then for any holomorphic $(n,0)$-form $f$ with there exist holomorphic $(n,0)$-forms $(h_1,\cdots,h_p)$ on $D$ such that $\sum_{j=1}^{p} h_j g_j=f$ and
