# L2 estimate on strongly pseudoconvex complex manifold

Suppose $$(X,g,I)$$ is a Hermitian (non kahler) complex manifold with small torsion, small derivative of torsion and small curvature. Let $$\varphi$$ be smooth PSH function satisfying $$\sqrt{-1}\bar\partial_I\partial_I\varphi\geq\omega_I$$ also assume that $$X:=\{\varphi<1\}$$. My question is that can we solve $$\bar\partial u=v$$, where $$v$$ is a $$(0,1)$$ form, with L2 estimate $$\|u\|_{L^2(e^{-\varphi)}}\leq C\|v\|_{L^2(e^{-\varphi})}.$$

With small torsion, derivative torsion and small curvature. I believe that, for any smooth compact $$(0,1)$$ form $$\phi$$, we are able to prove that $$\|\bar\partial\phi\|_{L^2}+\|\bar\partial^*\phi\|_{L^2}\geq \frac{1}{2}\|\phi\|_{L^2}.$$

Then my understanding for Hormander type estimate is do deal with the density issue. For example, Demailly's notes with complex metric and restrict application to $$(n,1)$$ form. Also Hormander's classical book use three different weights.

I wonder if Hormander's three weights thing can be applied here by using the Strictly PSH function $$\varphi$$.