# Strong factorizations of operators with applications to Fourier and Ces\'aro transforms

## TL;DR

This paper characterizes when one linear operator between Banach function spaces can be factored through another, with applications to Fourier and Cesàro transforms, using weighted norm inequalities and studying injective operators.

## Contribution

It provides a new characterization of strong factorizations of operators via weighted norm inequalities, especially for spaces with Schauder bases, including Fourier and Cesàro operators.

## Key findings

- Characterization of operator factorization through weighted inequalities
- Improved results for spaces with Schauder bases, including Fourier and Cesàro operators
- Applications to Hausdorff-Young and Hardy-Littlewood inequalities

## Abstract

Consider two continuous linear operators $T\colon X_1(\mu)\to Y_1(\nu)$ and $S\colon X_2(\mu)\to Y_2(\nu)$ between Banach function spaces related to different $\sigma$-finite measures $\mu$ and $\nu$. We characterize by means of weighted norm inequalities when $T$ can be strongly factored through $S$, that is, when there exist functions $g$ and $h$ such that $T(f)=gS(hf)$ for all $f\in X_1(\mu)$. For the case of spaces with Schauder basis our characterization can be improved, as we show when $S$ is for instance the Fourier operator, or the Ces\`aro operator. Our aim is to study the case when the map $T$ is besides injective. Then we say that it is a~representing operator ---in the sense that it allows to represent each elements of the Banach function space $X(\mu)$ by a~sequence of generalized Fourier coefficients---, providing a complete characterization of these maps in terms of weighted norm inequalities. Some examples and applications involving recent results on the Hausdorff-Young and the Hardy-Littlewood inequalities for operators on weighted Banach function spaces are also provided.