  This paper is dedicated to the question of surjectivity of the Cauchy-Riemann
operator on spaces $\mathcal{EV}(\Omega,E)$ of $\mathcal{C}^{\infty}$-smooth
vector-valued functions whose growth on strips along the real axis with holes
$K$ is induced by a family of continuous weights $\mathcal{V}$. Vector-valued
means that these functions have values in a locally convex Hausdorff space $E$
over $\mathbb{C}$. We characterise the weights $\mathcal{V}$ which give a
counterpart of the Grothendieck-K\"othe-Silva duality
$\mathcal{O}(\mathbb{C}\setminus K)/\mathcal{O}(\mathbb{C})\cong\mathscr{A}(K)$
with non-empty compact $K\subset\mathbb{R}$ for weighted holomorphic functions.
We use this duality to prove that the kernel
$\operatorname{ker}\overline{\partial}$ of the Cauchy-Riemann operator
$\overline{\partial}$ in
$\mathcal{EV}(\Omega):=\mathcal{EV}(\Omega,\mathbb{C})$ has the property
$(\Omega)$ of Vogt. Then an application of the splitting theory of Vogt for
Fr\'{e}chet spaces and of Bonet and Doma\'nski for (PLS)-spaces in combination
with some previous results on the surjectivity of the Cauchy-Riemann operator
$\overline{\partial}\colon\mathcal{EV}(\Omega)\to\mathcal{EV}(\Omega)$ yields
the surjectivity of the Cauchy-Riemann operator on $\mathcal{EV}(\Omega,E)$ if
$E:=F_{b}'$ with some Fr\'{e}chet space $F$ satisfying the condition $(DN)$ or
if $E$ is an ultrabornological (PLS)-space having the property $(PA)$. This
solves the smooth (holomorphic, distributional) parameter dependence problem
for the Cauchy-Riemann operator on $\mathcal{EV}(\Omega)$.
