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April 2019 Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms
O. Delgado, M. Mastyło, E. A. Sánchez Pérez
Michigan Math. J. 68(1): 167-192 (April 2019). DOI: 10.1307/mmj/1548817532

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.

Citation

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O. Delgado. M. Mastyło. E. A. Sánchez Pérez. "Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms." Michigan Math. J. 68 (1) 167 - 192, April 2019. https://doi.org/10.1307/mmj/1548817532

Information

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

zbMATH: 07155462
MathSciNet: MR3934608
Digital Object Identifier: 10.1307/mmj/1548817532

Subjects:
Primary: 46E30, 47B38
Secondary: 43A25, 46B15

Rights: Copyright © 2019 The University of Michigan

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Vol.68 • No. 1 • April 2019
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