2017/03/07 by Delgado, O., Mastylo, M., Sanchez-Perez, E. A.
#43A25 #46B15 #46E30 #47B38 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1703.02260
Consider two continuous linear operators T\colon X1(μ)→ Y1(ν) and S\colon X2(μ)→ Y2(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∈ 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 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.