© 2017, Springer International Publishing. Let X and Y be Banach spaces and let ? be a compact Hausdorff space. In 1973, Swartz, in his by now classical theorem, characterized the absolute summability of an operator U from C(? , X) to Y in terms of its associated operator U# and of its representing measure m. We study the interplay between U, U#, and m in the context of absolutely (r, q)-summing operators, considering the spaces Cp(? , X) of p-continuous functions on ? , 1 ? p? ?, instead of C(? , X) = C?(? , X). This encompasses the Swartz theorem together with its existing extensions on absolutely (r, q)-summing operators, providing, among others, an improvement even to the Swartz theorem. Counterexamples are exhibited to indicate sharpness of our results.