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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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

2. D. W. Boyd , The Hilbert transform on rearrangement-invariant spaces, Canad. J. Math. 19 (1967), 599616.

3. G. P. Curbera , Banach space properties of L1 of a vector measure, Proc. Amer. Math. Soc. 123 (1995), 37973806.

5. J. Lindenstrauss and L. Tzafriri , Classical Banach Spaces, vol. II (Springer-Verlag, Berlin, New York, 1979).

6. V. A. Rodin and E. M. Semenov , Rademacher series in symmetric spaces, Anal. Math. 1 (1975), 207222.

7. R. Sharpley , Spaces Λx(X) and interpolation, J. Funct. Anal. 11 (1972), 479513.

8. M. Zippin , 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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