Strong Factorizations of Operators with Applications to Fourier and Cesaro Transforms

[EN] Consider two continuous linear operators T: X-1 (mu) -> Y-1 (nu) and S: X-2 (mu) -> 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 is an element of 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 Cesar 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.The first author gratefully acknowledge the support of the Ministerio de Economia y Competitividad (project #MTM2015-65888-C4-1-P) and the Junta de Andalucia (project FQM-7276), Spain. The second author was supported by National Science Centre, Poland, project no. 2015/17/B/ST1/00064. The third author acknowledges with thanks the support of the Ministerio de Economia y Competitividad (project MTM2016-77054-C2-1-P), Spain.Delgado Garrido, O.; Mastylo, M.; Sánchez Pérez, EA. (2019). Strong Factorizations of Operators with Applications to Fourier and Cesaro Transforms. The Michigan Mathematical Journal. 68(1):167-192. https://doi.org/10.1307/mmj/1548817532S16719268

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