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Let $X$ and $Y$ be Banach spaces and let $\Omega$ be a compact Hausdorffspace. 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 integralrepresentation theorems with respect to a vector measure $m:\Sigma\rightarrow\mathcal{L}(X,Y^{**})$, where $\Sigma$ denotes the $\sigma$-algebra of Borelsubsets of $\Omega$. The first theorem extends the classicalBartle-Dunford-Schwartz representation theorem. It is used to prove the secondtheorem, which extends the classical Dinculeanu-Singer representation theorem,also providing to it an alternative simpler proof. For the latter (and themain) result, we build the needed integration theory, relying on a new conceptof the $q$-semivariation, $1\leq q\leq \infty$, of a vector measure$m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$.

The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited