This is an announcement for the paper “Strong factorizations of operators with applications to Fourier and Cesáro transforms” by O. Delgadohttps://arxiv.org/find/math/1/au:+Delgado_O/0/1/0/all/0/1, M. Mastylohttps://arxiv.org/find/math/1/au:+Mastylo_M/0/1/0/all/0/1, E.A. Sanchez-Perezhttps://arxiv.org/find/math/1/au:+Sanchez_Perez_E/0/1/0/all/0/1.
Abstract: Consider two continuous linear operators $T: X_1(\mu)\rightarrow Y_1(\nu)$ and $S: X_2(\mu)\rightarrow Y_2(\nu)$ between Banach function spaces related to different $\sigma$-finite measures $\mu$ and $\nu$. We characterize by means of weighted norm inequalities 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 operator, or the Ces`aro operator. Our aim is to study the case when the map $T$ is besides injective. Then we say that it is a representing operator ---in the sense that it allows to represent each elements 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. Some examples and applications involving recent results on the Hausdorff-Young and the Hardy-Littlewood inequalities for operators on weighted Banach function spaces are also provided.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.02260