  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 $Z(q)$.
