Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms

Abstract

Consider two continuous linear operators T : X1(μ) ! Y1( ) and S : X2(μ) ! Y2( ) between Banach function spaces related to different -finite measures μ and . 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 2 X1(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 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(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.