Equivalence of estimates on domain and its boundary

Abstract

Let $\Omega$ be a pseudoconvex domain in $\mathbb C^n$ with smooth boundary $b\Omega$. We define general estimates $(f\text{-}\mathcal M)^k_{\Omega}$ and $(f\text{-}\mathcal M)^k_{b\Omega}$ on $k$-forms for the complex Laplacian $\Box$ on $\Omega$ and the Kohn-Laplacian $\Box_b$ on $b\Omega$. For $1\le k\le n-2$, we show that $(f\text{-}\mathcal M)^k_{b\Omega}$ holds if and only if $(f\text{-}\mathcal M)^k_{\Omega}$ 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].