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

Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space over a $σ$-finite measure $μ$. We combine the structure of the spaces $L^p(μ)$ and $L^q(ξ)$ for constructing the new space $S_{X_p}^{\,q}(ξ)$, where $ξ$ 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 $S_{X_p}^{\,q}(ξ)$. 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.

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