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 $(f\text{-}\mathcal{M}{)}_{\Omega }^k$ and $(f\text{-}\mathcal{M}{)}_{b\Omega }^k$ on $k$-forms for the complex Laplacian $\square$ on $\Omega$ and the Kohn-Laplacian ${\square }_b$ on $b\Omega$. For $1\le k\le n-2$, we show that $(f\text{-}\mathcal{M}{)}_{b\Omega }^k$ holds if and only if $(f\text{-}\mathcal{M}{)}_{\Omega }^k$ and $(f\text{-}\mathcal{M}{)}_{\Omega }^{n-k-1}$ hold. Our proof relies on Kohn's method in [Ann. of Math. (2), 156(1):213--248, 2002].

final version

References in corpus (2)

• A general method of weights in the d-bar-Neumann problem
• Global hypoellipticity of the Kohn-Laplacian ${\square }_b$ on pseudoconvex CR manifolds