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

Consider two continuous linear operators $T\text{\hspace{0.17em}⁣}:X_1(\mu )\to Y_1(\nu )$ and $S\text{\hspace{0.17em}⁣}:X_2(\mu )\to Y_2(\nu )$ 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.