Suppose that $\Omega$ is a pseudoconvex domain, for $f\in L^{ 2 }_{ (0,p) }(\Omega )$,and ${ \bar { \partial } f }=0$. Use $L^2$ method to show that there exists solution $u\in L^{ 2 }_{ (0,p-1) }(\Omega )$, such that ${\bar{\partial}}u=f$.
For the situation $f\in L^{ 2 }_{ (0,1) }(\Omega )$, the problem is solved via contributing an inequality as $c||g||^2\le||T^*g||^2+||Sg||^2$, where $T$ and $S$ are both the operator $\bar{\partial}$, $T^*$ is the adjoint operator of $T$. Then we can use functional analysis theory solve the problem.
Is there similar inequality like that in the situation of $(0,1)$-form?
