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A note on UMD spaces and transference in vector-valued function spaces*

Published online by Cambridge University Press:  20 January 2009

N. H. Asmar ,
B. P. Kelly  and
S. Montgomery-Smith
N. H. Asmar
Affiliation:
Department of Mathematics University of Missouri-Columbia Columbia, Missouri 65211, U.S.A.
B. P. Kelly
Affiliation:
Department of Mathematics University of Missouri-Columbia Columbia, Missouri 65211, U.S.A.
S. Montgomery-Smith
Affiliation:
Department of Mathematics University of Missouri-Columbia Columbia, Missouri 65211, U.S.A.
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Abstract

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A Banach space X is called an HT space if the Hilbert transform is bounded from Lp(X) into Lp(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in Lp(X), 1 < p < ∞. Berkson, Gillespie and Muhly [5] showed that X ∈ HT ⇒ X ∈ ACF. In this note, we will show that X ∈ ACF ⇒ X ∈ UMD, thus providing a new proof of Bourgain's result X ∈ HT ⇒ X ∈ UMD.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 3 , October 1996 , pp. 485 - 490
Copyright
Copyright © Edinburgh Mathematical Society 1996

Footnotes

*

The work of the first and third authors was partially funded by NSF grants. The second and third authors' work was partially funded by the University of Missouri Research Board.

References

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