Strong Factorizations of Operators with Applications to Fourier and Cesaro Transforms

Abstract

[EN] Consider two continuous linear operators T: X-1 (mu) -> Y-1 (nu) and S: X-2 (mu) -> 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 is an element of 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 Cesar 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.