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A note on function spaces generated by Rademacher series

  • Guillermo P. Curbera (a1)
Abstract

Let X be a rearrangement invariant function space on [0,1] in which the Rademacher functions (rn) generate a subspace isomorphic to ℓ2. We consider the space Λ(R, X) of measurable functions f such that fgX for every function g=∑bnrn where (bn)∈ℓ2. We show that if X satisfies certain conditions on the fundamental function and on certain interpolation indices then the space Λ(R, X) is not order isomorphic to a rearrangement invariant space. The result includes the spaces Lp, q and certain classes of Orlicz and Lorentz spaces. We also study the cases X = Lexp and X = Lψ2 for ψ2) = exp(t2) – 1.

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Copyright
References
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1. Bennett, C. and Sharpley, R., Interpolation of operators (Academic Press, Inc., Boston, 1988).
2. Boyd, D. W., The Hilbert transform on rearrangement-invariant spaces, Canad. J. Math. 19 (1967), 599616.
3. Curbera, G. P., Banach space properties of L 1 of a vector measure, Proc. Amer. Math. Soc. 123 (1995), 37973806.
4. Krasnoselskii, M. A. and Rutickii, Ya. B., Convex functions and Orlicz spaces (Noordhoff, Groningen, 1961).
5. Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces, vol. II (Springer-Verlag, Berlin, New York, 1979).
6. Rodin, V. A. and Semenov, E. M., Rademacher series in symmetric spaces, Anal. Math. 1 (1975), 207222.
7. Sharpley, R., Spaces Λx(X) and interpolation, J. Funct. Anal. 11 (1972), 479513.
8. Zippin, M., Interpolation of operators of weak type between rearrangement-invariant function spaces, J. Funct. Anal. 7 (1971), 267284.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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