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Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function spaceover 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 associatedto $X$. We show some of its properties, and the relevant fact that every$q$-summing operator $T$ defined on $X$ can be continuously (strongly) extendedto $S_{X_p}^{\,q}(\xi)$. This result turns out to be a mixture of the Pietschand Maurey-Rosenthal factorization theorems, which provide (strong)factorizations for $q$-summing operators through $L^q$-spaces when $1 \le q \lep$. Thus, our result completes the picture, showing what happens in thecomplementary case $1\le p\le q$, opening the door to the study of themultilinear versions of $q$-summing operators also in these cases.

Strong extensions for $q$-summing operators acting in $p$-convex Banach function spaces for $1 \le p \le q$