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Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstractCR manifold of hypersurface type, $n\geq 2$. We prove the following:  (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms,then the complex Green operator $G_q$ exists and is continuous on$L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, wealso establish continuity for $G_0$ and $G_n$. Additionally, the$\bar\partial_b$-equation on $M$ can be solved in $C^\infty(M)$.  (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$-forms, then$G_q$ is a continuous operator on $H^s_{0,q}(M)$ and is therefore globallyregular on $M$ for degrees $q_0\le q\le n-q_0$; and also for the top degrees$q=0$ and $q=n$ in the case $q_0=1$.  We also introduce the notion of a "plurisubharmonic CR manifold" and showthat it generalizes the notion of "plurisubharmonic defining function" for a adomain in $\mathbb C^N$ and implies that $M$ satisfies the weak compactnessproperty.

The Kohn-Laplace equation on abstract CR manifolds: Global regularity