For any $p\in(0,\frac{n}{n-k})$, there is a uniform constant $C=C(n,p,\omega_{X})>0$ such that

$$\|u\|_{L^{p}(\omega_{X}^{n})}\leq C,$$

with $\sup_{X}u=0$ and $\lambda[\omega_{u}]\in\Gamma_{k}$, $\omega_{u}=\omega_{X}+i\partial\bar{\partial}u$.