  We study the positivity properties of Hermitian (or even Finsler) holomorphic
vector bundles in terms of $L^p$-estimates of $\bar\partial$ and
$L^p$-extensions of holomorphic objects. To this end, we introduce four
conditions, called the optimal $L^p$-estimate condition, the multiple coarse
$L^p$-estimate condition, the optimal $L^p$-extension condition, and the
multiple coarse $L^p$-extension condition, for a Hermitian (or Finsler) vector
bundle $(E,h)$. The main result of the present paper is to give a
characterization of the Nakano positivity of $(E,h)$ via the optimal
$L^2$-estimate condition. We also show that $(E,h)$ is Griffiths positive if it
satisfies the multiple coarse $L^p$-estimate condition for some $p>1$, the
optimal $L^p$-extension condition, or the multiple coarse $L^p$-extension
condition for some $p>0$. These results can be roughly viewed as converses of
H\"{o}rmander's $L^2$-estimate of $\bar\partial$
  and Ohsawa-Takegoshi type extension theorems. As an application of the main
result, we get a totally different method to Nakano positivity of direct image
sheaves of twisted relative canonical bundles associated to holomorphic
families of complex manifolds.
