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$ K$-divisibility and a theorem of Lorentz and Shimogaki

Authors: Colin Bennett and Robert Sharpley
Journal: Proc. Amer. Math. Soc. 96 (1986), 585-592
MSC: Primary 46M35; Secondary 46E30
MathSciNet review: 826485
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Abstract: The Brudnyi-Krugljak theorem on the $ K$-divisibility of Gagliardo couples is derived by elementary means from earlier results of Lorentz-Shimogaki on equimeasurable rearrangements of measurable functions. A slightly stronger form of Calderón's theorem describing the Hardy-Littlewood-Pólya relation in terms of substochastic operators (which itself generalizes the classical Hardy-Littlewood-Pólya result for substochastic matrices) is obtained.

References [Enhancements On Off] (What's this?)

  • [1] Ju. A. Brudnyi and N. Ja. Krugljak, Real interpolation functors, Soviet Math. Dokl. 23 (1981), 5-8.
  • [2] Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York Inc., New York, 1967. MR 0230022
  • [3] A.-P. Calderón, Spaces between 𝐿¹ and 𝐿^{∞} and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. MR 0203444
  • [4] Michael Cwikel, 𝐾-divisibility of the 𝐾-functional and Calderón couples, Ark. Mat. 22 (1984), no. 1, 39–62. MR 735877,
  • [5] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1967.
  • [6] G. G. Lorentz and T. Shimogaki, Interpolation theorems for spaces 𝑐𝑙𝑎𝑚, Abstract Spaces and Approximation (Proc. Conf., Oberwolfach, 1968) Birkhäuser, Basel, 1969, pp. 94–98. MR 0257774
  • [7] G. G. Lorentz and T. Shimogaki, Interpolation theorems for operators in function spaces, J. Functional Analysis 2 (1968), 31–51. MR 0257775
  • [8] B. S. Mitjagin, An interpolation theorem for modular spaces, Mat. Sb. (N.S.) 66 (108) (1965), 473–482 (Russian). MR 0177299

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Article copyright: © Copyright 1986 American Mathematical Society