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

TL;DR

This paper introduces a new space combining $L^p$ and $L^q$ structures for $p$-convex Banach function spaces and proves that $q$-summing operators can be strongly extended to this space, generalizing existing factorization theorems.

Contribution

It constructs the space $S_{X_p}^{q}(\xi)$ and proves strong extension of $q$-summing operators, extending factorization results to the case $1 \\le p \\le q$.

Findings

Every $q$-summing operator on $X$ extends to $S_{X_p}^{q}(\xi)$.

The space $S_{X_p}^{q}(\xi)$ combines $L^p$ and $L^q$ structures.

The result generalizes Pietsch and Maurey-Rosenthal theorems.

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.