  Let $X$ and $Y$ be Banach spaces and let $\Omega$ be a compact Hausdorff
space. Denote by $\mathcal{C}_{p}(\Omega,X)$ the space of $p$-continous
$X$-valued functions, $1\leq p\leq \infty$. For operators
$S\in\mathcal{L}(\mathcal{C}(\Omega),\mathcal{L}(X,Y))$ and
$U\in\mathcal{L}(\mathcal{C}_{p}(\Omega,X),Y)$, we establish integral
representation theorems with respect to a vector measure $m:\Sigma\rightarrow
\mathcal{L}(X,Y^{**})$, where $\Sigma$ denotes the $\sigma$-algebra of Borel
subsets of $\Omega$. The first theorem extends the classical
Bartle-Dunford-Schwartz representation theorem. It is used to prove the second
theorem, which extends the classical Dinculeanu-Singer representation theorem,
also providing to it an alternative simpler proof. For the latter (and the
main) result, we build the needed integration theory, relying on a new concept
of the $q$-semivariation, $1\leq q\leq \infty$, of a vector measure
$m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$.
