Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, and let $(L,h)$ be a Hermitian holomorphic line bundle on $X$. Let $\Theta(L,h)$ denote the curvature form of the Chern connection of $(L,h)$, and assume that $i\Theta(L,h)$ is a positive real $(1,1)$-form. For each integer $m\ge 1$, let $h^m$ be the Hermitian metric induced on $L^m$, let $A^{n,q}(X,L^m)$ denote the space of smooth $L^m$-valued $(n,q)$-forms on $X$, and let the $L^2$ inner products and norms on these spaces be those induced by $\omega$ and $h^m$. For every integer $q\ge 1$, there exist an integer $m_0\ge 1$ and a constant $C_q>0$ such that for every integer $m\ge m_0$ and every $\alpha\in A^{n,q}(X,L^m)$,

\begin{align*} \|\alpha\|_{L^2}^2\le C_q\left(\|\bar\partial_{L^m}\alpha\|_{L^2}^2+\|\bar\partial_{L^m}^*\alpha\|_{L^2}^2\right). \end{align*}

Here $\bar\partial_{L^m}:A^{n,q}(X,L^m)\to A^{n,q+1}(X,L^m)$ is the Dolbeault operator induced by the holomorphic structure on $L^m$, and $\bar\partial_{L^m}^*:A^{n,q}(X,L^m)\to A^{n,q-1}(X,L^m)$ denotes its formal adjoint with respect to the corresponding $L^2$ inner products.