Let
M
1
{M_1}
and
M
2
{M_2}
be two bounded pseudo-convex domains in
C
n
{{\mathbf {C}}^n}
with smooth boundaries such that
M
¯
1
⊂
M
2
{\overline M _1} \subset {M_2}
. We consider the Cauchy-Riemann operators
∂
¯
\overline \partial
on the annulus
M
=
M
2
∖
M
¯
1
M = {M_2}\backslash {\overline M _1}
. The main result of this paper is the following: Given a
∂
¯
\overline \partial
-closed
(
p
,
q
)
(p,q)
form
α
\alpha
,
0
>
q
>
n
0 > q > n
, which is
C
∞
{C^\infty }
on
M
¯
\overline M
and which is cohomologous to zero on
M
M
, there exists a
(
p
,
q
−
1
)
(p,q - 1)
form
u
u
which is
C
∞
{C^\infty }
on
M
¯
\overline M
such that
∂
¯
u
=
α
\overline \partial u = \alpha
.