The Michigan Mathematical Journal

Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms

O. Delgado, M. Mastyło, and E. A. Sánchez Pérez

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Abstract

Consider two continuous linear operators T:X1(μ)Y1(ν) and S:X2(μ)Y2(ν) between Banach function spaces related to different σ-finite measures μ and ν. 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 fX1(μ). 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(μ) 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.

Article information

Source
Michigan Math. J., Volume 68, Issue 1 (2019), 167-192.

Dates
Received: 27 March 2017
Revised: 14 September 2018
First available in Project Euclid: 30 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1548817532

Digital Object Identifier
doi:10.1307/mmj/1548817532

Mathematical Reviews number (MathSciNet)
MR3934608

Subjects
Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B38: Operators on function spaces (general)
Secondary: 46B15: Summability and bases [See also 46A35] 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

Citation

Delgado, O.; Mastyło, M.; Sánchez Pérez, E. A. Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms. Michigan Math. J. 68 (2019), no. 1, 167--192. doi:10.1307/mmj/1548817532. https://projecteuclid.org/euclid.mmj/1548817532


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