The following contents are copied from Demailly's e-book Chapter VII-(2.7)-Complex Analytic and Differential Geometry.
Let $(X, \omega)$ be a compact hermitian manifold, $\operatorname{dim}_{\mathbb{C}} X=n,$ and $E$ a hermitian holomorphic vector bundle over $X$. For any section $u \in \mathscr{C}_{p, q}^{\infty}(X, E)$ we have $\left\langle\left\langle\Delta^{\prime \prime} u, u\right\rangle\right\rangle=$ $\left\|D^{\prime \prime} u\right\|^{2}+\left\|\delta^{\prime \prime} u\right\|^{2}$ and the similar formula for $\Delta_{\tau}^{\prime}$ gives $\left\langle\left\langle\Delta_{\tau}^{\prime} u, u\right\rangle\right\rangle \geqslant 0 .$ Theorem 1.4 implies therefore (2.1) $$ \left\|D^{\prime \prime} u\right\|^{2}+\left\|\delta^{\prime \prime} u\right\|^{2} \geqslant \int_{X}\left(\langle[\mathrm{i} \Theta(E), \Lambda] u, u\rangle+\left\langle T_{\omega} u, u\right\rangle\right) d V $$ This inequality is known as the Bochner-Kodaira-Nakano inequality.
Of course, the above follows from the Bochner-Kodaira-Nakano identity (Demailly's e-book Chapter VII-(1.2)) whose proof depends on a trick of taking the Taylor expansion of corresponding operators. The existence of the Taylor expansion tell us that the smooth metric of the above bundle $E$ is adopted.
My question is: can we still get the same Bochner-Kodaira-Nakano identity or Bochner-Kodaira-Nakano inequality when the vector bundle $E$ is equipped with a singular metric.