×

Fourier series in weighted Lorentz spaces. (English) Zbl 1351.42006

Summary: The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LogL and more general Lorentz-Zygmund spaces.

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ariño, M., Muckenhoupt, B.: Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for non-increasing functions. Trans. Amer. Math. Soc. 320, 727-735 (1990) · Zbl 0716.42016
[2] Benedetto, J. J., Heinig, H. P.: Weighted Hardy spaces and the Laplace transform. In: Harmonic analysis (Cortona, 1982). Lecture Notes in Mathematics, vol. 992, pp. 240-277, Springer, Berlin (1983) · Zbl 1093.26025
[3] Benedetto, J.J., Heinig, H.P.: Weighted Fourier inequalities: new proofs and generalizations. J. Fourier Anal. Appl. 9, 1-37 (2003) · Zbl 1034.42010 · doi:10.1007/s00041-003-0003-3
[4] Benedetto, J.J., Heinig, H.P., Johnson, R.: Weighted Hardy spaces and the Laplace transform II. Math. Nachr. 132, 29-55 (1987) · Zbl 0626.44002 · doi:10.1002/mana.19871320104
[5] Bennett, C., Rudnick, K.: On Lorentz-Zygmund spaces. Diss. Math. (Rozpr. Mat.) 175, 1-67 (1980) · Zbl 0456.46028
[6] Bennett, C., Sharpley, R.: Interpolation of operators. Pure and Applied Mathematics 129. Academic Press Inc, Boston (1988) · Zbl 0647.46057
[7] Bradley, J.S.: Hardy inequalities with mixed norms. Canad. Math. Bull. 21, 405-408 (1978) · Zbl 0402.26006 · doi:10.4153/CMB-1978-071-7
[8] Carro, M., García del Amo, A., Soria, J.: Weak-type weights and normable Lorentz spaces. Proc. Amer. Math. Soc. 124, 849-857 (1996) · Zbl 0853.42016 · doi:10.1090/S0002-9939-96-03214-5
[9] Jodeit Jr, M., Torchinsky, A.: Inequalities for Fourier transforms. Stud. Math. 37, 245-276 (1971) · Zbl 0224.46037
[10] Maligranda, L.: Weighted inequalities for monotone functions. Collect. Math. 48, 687-700 (1997) · Zbl 0916.26007
[11] Maligranda, L.: Weighted inequalities for quasi-monotone functions. J. Lond. Math. Soc. 57, 363-370 (1998) · Zbl 0923.26016 · doi:10.1112/S0024610798006140
[12] Mastylo, M., Sinnamon, G.: A Calderón couple of down spaces. J. Funct. Anal. 240, 192-225 (2006) · Zbl 1116.46015 · doi:10.1016/j.jfa.2006.05.007
[13] Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. Stud. Math. 96, 145-158 (1990) · Zbl 0705.42014
[14] Sinnamon, G.: The level function in rearrangement invariant spaces. Publ. Mat. 45, 175-198 (2001) · Zbl 0987.46033 · doi:10.5565/PUBLMAT_45101_08
[15] Sinnamon, G.: Embeddings of concave functions and duals of Lorentz spaces. Publ. Mat. 46, 489-515 (2002) · Zbl 1043.46026 · doi:10.5565/PUBLMAT_46202_10
[16] Sinnamon, G.: The Fourier transform in weighted Lorentz spaces. Publ. Mat. 47, 3-29 (2003) · Zbl 1045.42004 · doi:10.5565/PUBLMAT_47103_01
[17] Sinnamon, G.: Transferring monotonicity in weighted norm inequalities. Collect. Math. 54, 181-216 (2003) · Zbl 1093.26025
[18] Sinnamon, G.: Monotonicity in Banach function spaces. NAFSA 8 Nonlinear analysis, function spaces and applications, vol. 8. Czech Academy of Sciences, Prague (2007) · Zbl 1289.26055
[19] Sinnamon, G.: Fourier inequalities and a new Lorentz space. Banach and function spaces II, pp. 145-155, Yokohama Publishers, Yokohama (2008) · Zbl 1268.42040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.