Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms

Abstract

Consider two continuous linear operators $T:X_1(\mu )\to Y_1(\nu )$ and $S:X_2(\mu )\to Y_2(\nu )$ between Banach function spaces related to different $\sigma$-finite measures $\mu$ and $\nu$. By means of weighted norm inequalities we characterize 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 or Cesàro operator. Our aim is to study the case where the map $T$ is besides injective. Then we say that it is a representing operator—in the sense that it allows us to represent each element 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. We also provide some examples and applications involving recent results on the Hausdorff–Young and the Hardy–Littlewood inequalities for operators on weighted Banach function spaces.