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

Abstract

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.