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Strong factorizations of operators with applications to Fourier and Cesàro transforms. (English) Zbl 1445.47020

Summary: 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.

MSC:

47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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References:

[1] J. M. Ash, S. Tikhonov, and J. Tung, Wiener’s positive Fourier coefficients theorem in variants of \(L^p\) spaces, Michigan Math. J. 59 (2010), no. 1, 143-151. · Zbl 1204.42012
[2] J. J. Benedetto and H. P. Heinig, Weighted Fourier inequalities: new proofs and generalizations, J. Fourier Anal. Appl. 9 (2003), no. 1, 1-37. · Zbl 1034.42010
[3] J. M. Calabuig, O. Delgado, and E. A. Sánchez Pérez, Generalized perfect spaces, Indag. Math. (N.S.) 19 (2008), 359-378. · Zbl 1177.46018
[4] D. Cruz-Uribe, J. M. Martell, and C. Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2012), no. 1, 408-441. · Zbl 1236.42010
[5] O. Delgado and E. A. Sánchez Pérez, Summability properties for multiplication operators on Banach function spaces, Integral Equations Operator Theory 66 (2010), 197-214. · Zbl 1206.46032
[6] O. Delgado and E. A. Sánchez Pérez, Strong factorizations between couples of operators on Banach function spaces, J. Convex Anal. 20 (2013), 599-616. · Zbl 1286.46031
[7] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, 1934.
[8] C. N. Kellogg, An extension of the Hausdorff-Young theorem, Michigan Math. J. 18 (1971), 121-127. · Zbl 0197.38702
[9] P. Kolwicz, K. Leśnik, and L. Maligranda, Pointwise multipliers of Calderón-Lozanovskii spaces, Math. Nachr. 286 (2013), no. 8, 876-907. · Zbl 1288.46023
[10] P. Kolwicz, K. Leśnik, and L. Maligranda, Pointwise products of some Banach function spaces and factorization, J. Funct. Anal. 266 (2014), no. 2, 616-659. · Zbl 1308.46039
[11] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, vol. II, Springer-Verlag, Berlin, 1979. · Zbl 0403.46022
[12] L. Maligranda and L. E. Persson, Generalized duality of some Banach function spaces, Indag. Math. (N.S.) 51 (1989), 323-338. · Zbl 0704.46018
[13] H. Mhaskar and S. Tikhonov, Wiener type theorems for Jacobi series with nonnegative coefficients, Proc. Amer. Math. Soc. 140 (2012), no. 3, 977-986. · Zbl 1250.42084
[14] C. Pérez, Sharp \(L^p\)-weighted Sobolev inequalities, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 3, 809-824. · Zbl 0820.42008
[15] A. Pietsch, Operator ideals, North Holland, Amsterdam, 1980.
[16] E. A. Sánchez Pérez, Factorization theorems for multiplication operators on Banach function spaces, Integral Equations Operator Theory 80 (2014), no. 1, 117-135.
[17] A. R. Schep, Products and factors of Banach function spaces, Positivity 14 (2010), no. 2, 301-319. · Zbl 1216.46028
[18] A. C. Zaanen, Integration, second edition, North Holland, Amsterdam, 1967. · Zbl 0175.05002
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