Closed Range for $\bar{\partial }$ and ${\bar{\partial }}_b$ on Bounded Hypersurfaces in Stein Manifolds

Abstract

We define weak $Z(q)$, a generalization of $Z(q)$ on bounded domains $\Omega$ in a Stein manifold $M^n$ that suffices to prove closed range of $\bar{\partial }$. Under the hypothesis of weak $Z(q)$, we also show (i) that harmonic $(0,q)$-forms are trivial and (ii) if $\partial \Omega$ satisfies weak $Z(q)$ and weak $Z(n-1-q)$, then $\dbar_b$ has closed range on $(0,q)$-forms on $\partial \Omega$. We provide examples to show that our condition contains examples that are excluded from $(q-1)$-pseudoconvexity and the authors' previous notion of weak $Z(q)$.

29 pages