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A note on function spaces generated by Rademacher series

Published online by Cambridge University Press:  20 January 2009

Guillermo P. Curbera
Affiliation:
Facultad De Matemáticas, Universidad de Sevilla, Aptdo (P.O. Box) 1160, Sevilla 41080, SpainE-mail address:curbera@cica.es
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

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