## The Michigan Mathematical Journal

### Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms

#### Abstract

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.

#### Article information

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

Dates
Revised: 14 September 2018
First available in Project Euclid: 30 January 2019

https://projecteuclid.org/euclid.mmj/1548817532

Digital Object Identifier
doi:10.1307/mmj/1548817532

Mathematical Reviews number (MathSciNet)
MR3934608

#### 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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