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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 $\varphi$.

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