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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] P. L. Butzer and H. Berens, Semi-groups of operators and approximation, Springer-Verlag, New York, 1967. MR 0230022 (37:5588)
  • [3] A. P. Calderón, Spaces between $ {L^1}$ and $ {L^\infty }$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273-299. MR 0203444 (34:3295)
  • [4] M. Cwikel, $ K$-divisibility of the $ K$-functional and Calderón couples, Ark. Mat. 22 (1984), 39-62. MR 735877 (85m:46074)
  • [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 $ \Lambda $, Abstract Spaces and Approximation (P. L. Butzer and B. Sz. Nagy, Eds.), ISNM 10, Birkhäuser Verlag, Basel, 1969, pp. 94-98. MR 0257774 (41:2423)
  • [7] -, Interpolation theorems for operators in function spaces, J. Functional Anal. 2 (1968), 31-51. MR 0257775 (41:2424)
  • [8] B. S. Mitjagin, An interpolation theorem for modular spaces, Mat. Sb. 66 (1965), 473-482. (Russian) MR 0177299 (31:1562)

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