Equivalence of estimates on domain and its boundary
Tran Vu Khanh · arXiv (Cornell University) · 2017
Let $Ω$ be a pseudoconvex domain in $\mathbb C^n$ with smooth boundary $bΩ$. We define general estimates $(f\text{-}\mathcal M)^k_Ω$ and $(f\text{-}\mathcal M)^k_{bΩ}$ on $k$-forms for the complex Laplacian $\Box$ on $Ω$ and the Kohn-Laplacian $\Box_b$ on $bΩ$. For $1\le k\le n-2$, we show that $(f\text{-}\mathcal M)^k_{bΩ}$ holds if and only if $(f\text{-}\mathcal M)^k_Ω$ and $(f\text{-}\mathcal M)^{n-k-1}_Ω$ hold. Our proof relies on Kohn's method in [Ann. of Math. (2), 156(1):213--248, 2002].