AbstractThis paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$
∂
¯
on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$
E
V
(
Ω
,
E
)
of $${\mathcal {C}}^{\infty }$$
C
∞
-smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$
V
. Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$
C
. We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$
O
(
C
\
K
)
/
O
(
C
)
≅
A
(
K
)
with non-empty compact $$K\subset {\mathbb {R}}$$
K
⊂
R
for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$
∂
¯
:
E
V
(
Ω
,
E
)
→
E
V
(
Ω
,
E
)
for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$
E
V
(
Ω
,
C
)
.
Determination of the Fundamental Solution of the Elliptic Equation of Order Out of the Fundamental Solution of the Cauchy–Riemann Equation
