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.

MSC:

 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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References:

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