I'm reading 《Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry》 by Simon Donaldson & Sun Song. In their Section 2, named Complex differential geometry: the Hormander technique, they mention the following elliptic estimate and Sobolev inequality for sections of holomorphic line bundles:
$(U^n,J,\omega)$ is a complex manifold, $u_*\in U$ and $D\subset U$ is an open neighborhood of $u_*$. $\Lambda\to U$ is a $C^{\infty}$ Hermitian line bundle with curvature $-\sqrt{-1}\omega$. Fix $p>2n$.
Then for any smooth section $\tau$ of $\Lambda$ over a neighborhood of $\bar{D}$ we have
the elliptic estimate: $\|\tau\|_{L_1^p\left(D_0\right)} \leqslant C\left(\|\bar{\partial} \tau\|_{L^p(D)}+\|\tau\|_{L^2(D)}\right)$.
and
the Sobolev inequality:$\left|\tau\left(u_*\right)\right| \leqslant C\|\tau\|_{L_1^p\left(D_0\right)}$.
I don't know where can I find the proof of these two formulas? And I'm not familiar with the Hormander technique mentioned here, can anyone give me some reference? Thanks in advance.