  Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space
over a $\sigma$-finite measure $\mu$. We combine the structure of the spaces
$L^p(\mu)$ and $L^q(\xi)$ for constructing the new space $S_{X_p}^{\,q}(\xi)$,
where $\xi$ is a probability Radon measure on a certain compact set associated
to $X$. We show some of its properties, and the relevant fact that every
$q$-summing operator $T$ defined on $X$ can be continuously (strongly) extended
to $S_{X_p}^{\,q}(\xi)$. This result turns out to be a mixture of the Pietsch
and Maurey-Rosenthal factorization theorems, which provide (strong)
factorizations for $q$-summing operators through $L^q$-spaces when $1 \le q \le
p$. Thus, our result completes the picture, showing what happens in the
complementary case $1\le p\le q$, opening the door to the study of the
multilinear versions of $q$-summing operators also in these cases.
