Consider two continuous linear operators and between Banach function spaces related to different -finite measures and . By means of weighted norm inequalities we characterize when can be strongly factored through , that is, when there exist functions and such that for all . For the case of spaces with Schauder basis, our characterization can be improved, as we show when is, for instance, the Fourier or Cesàro operator. Our aim is to study the case where the map 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 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.
O. Delgado. M. Mastyło. E. A. Sánchez Pérez. "Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms." Michigan Math. J. 68 (1) 167 - 192, April 2019. https://doi.org/10.1307/mmj/1548817532