Strong factorizations of operators with applications to Fourier and Cesàro transforms. (English) Zbl 1445.47020

Summary: Consider two continuous linear operators \(T\colon X_1(\mu)\to Y_1(\nu)\) and \(S\colon X_2(\mu)\to Y_2(\nu)\) 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.


47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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