  We study conditions of H\"ormander's $L^2$-estimate and the Ohsawa-Takegoshi
extension theorem. Introducing a twisted version of H\"ormander-type condition,
we show a converse of H\"ormander $L^2$-estimate under some regularity
assumptions on an $n$-dimensional domain. This result is a partial
generalization of the 1-dimensional result obtained by Berndtsson. We also
define new positivity notions for vector bundles with singular Hermitian
metrics by using these conditions. We investigate these positivity notions and
compare them with classical positivity notions.
