Strong factorizations of operators with applications to Fourier and Cesáro transforms
Consider two continuous linear operators $T\colon X_1(μ)\to Y_1(ν)$ and $S\colon X_2(μ)\to Y_2(ν)$ between Banach function spaces related to different $σ$-finite measures $μ$ and $ν$. 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 $f\in X_1(μ)$. 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 Cesàro 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 $X(μ)$ 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.