Let
M
M
be a compact, pseudoconvex-oriented,
(
2
n
+
1
)
(2n+1)
-dimensional, abstract CR manifold of hypersurface type,
n
≥
2
n\geq 2
. We prove the following:
(i) If
M
M
admits a strictly CR-plurisubharmonic function on
(
0
,
q
0
)
(0,q_0)
-forms, then the complex Green operator
G
q
G_q
exists and is continuous on
L
0
,
q
2
(
M
)
L^2_{0,q}(M)
for degrees
q
0
≤
q
≤
n
−
q
0
q_0\le q\le n-q_0
. In the case that
q
0
=
1
q_0=1
, we also establish continuity for
G
0
G_0
and
G
n
G_n
. Additionally, the
∂
¯
b
\bar {\partial }_{b}
-equation on
M
M
can be solved in
C
∞
(
M
)
C^\infty (M)
.
(ii) If
M
M
satisfies “a weak compactness property” on
(
0
,
q
0
)
(0,q_0)
-forms, then
G
q
G_q
is a continuous operator on
H
0
,
q
s
(
M
)
H^s_{0,q}(M)
and is therefore globally regular on
M
M
for degrees
q
0
≤
q
≤
n
−
q
0
q_0\le q\le n-q_0
; and also for the top degrees
q
=
0
q=0
and
q
=
n
q=n
in the case
q
0
=
1
q_0=1
.
We also introduce the notion of a “plurisubharmonic CR manifold” and show that it generalizes the notion of “plurisubharmonic defining function” for a domain in
C
N
\mathbb {C}^N
and implies that
M
M
satisfies the weak compactness property.