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Abstract

Let φ be a decreasing positive integrable function on [0, 1]. The space ∧(φ) consists of all measurable functions f on [0, 1] for which
$$\left\| f \right\| = \int\limits_0^1 {\varphi f*dt}< + \infty$$
(1)
f* is the decreasing rearrangement of |f| (see [5]). In particular, if φ(t) = αt α-1 , 0<α≦ 1, we obtain the spaces α . Spaces of this type and more general spaces α,p , Mα,p (which depend on two parameters) have been defined by the author ([4], [5]), see also Calderón [1].

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References

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    A. P. Calderón, Spaces between L l and L 8 and the theorem of Marcinkiewicz. Studia Math. 26 (1966), 273–299.Google Scholar
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    G. G. Lorentz, Some new functional spaces. Ann. of Math. 51 (1950), 37–55.CrossRefGoogle Scholar
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    G. G. Lorentz, Bernstein Polynomials. University of Toronto Press, 1953.Google Scholar
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    W. A. J. Luxemburg, Banach function spaces. Dissertation, Technical Highschool, Delft 1955.Google Scholar
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    B. S. Mitjagin, An interpolation theorem for modular spaces. Mat. Sb. (N. S.) 66 (1965), 473–482.Google Scholar

Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • G. G. Lorentz
    • 1
    • 2
  • T. Shimogaki
    • 1
    • 2
  1. 1.Syracuse University, the University of TexasUSA
  2. 2.Tokyo Institute of TechnologyJapan

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