Equivalence of estimates on domain and its boundary

Let $\mathrm{\Omega }$ be a pseudoconvex domain in ${\mathbb{C}}^n$ with smooth boundary $b\mathrm{\Omega }$. We define general estimates $(f\text{-}\mathcal{M}{)}_{\mathrm{\Omega }}^k$ and $(f\text{-}\mathcal{M}{)}_{b\mathrm{\Omega }}^k$ on $k$-forms for the complex Laplacian $\mathrm{\square }$ on $\mathrm{\Omega }$ and the Kohn-Laplacian ${\mathrm{\square }}_b$ on $b\mathrm{\Omega }$. For $1\le k\le n-2$,weshowthat$(f$\text{-}$\mathcal M)^k_{b\Omega}$holdsifandonlyif$(f$\text{-}$\mathcal M)^k_{\Omega}$and$(f$\text{-}$\mathcal M)^{n-k-1}_{\Omega}$ hold. Our proof relies on Kohn’s method in [Ann. of Math. (2), 156(1):213–248, 2002].