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Journal of the London Mathematical Society Advance Access originally published online on October 23, 2009
Journal of the London Mathematical Society 2009 80(3):798-814; doi:10.1112/jlms/jdp051
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© 2009 London Mathematical Society

Extrapolation of vector-valued rearrangement operators

Stefan Geiss

Department of Mathematics and Statistics
University of Jyväskylä
P.O. Box 35 (MaD)
FIN-40014
Finland
geiss@maths.jyu.fi

Paul F. X. Müller

Department of Analysis
J. Kepler University
A-4040 Linz
Austria

Given an injective map {tau}:D -> D between the dyadic intervals of the unit interval [0, 1), we study extrapolation properties of the induced rearrangement operator of the Haar system IdX {otimes} Tp, {tau}: LFormula([0,1)) -> LFormula ([0,1)), where X is a Banach space and LFormula the subspace of mean zero random variables. If X is a UMD-space, then we prove that the property that IdX {otimes} Tp, {tau} is an isomorphism for some 1 < p != 2 < {infty} extrapolates across the entire scale of LFormula-spaces with 1 < q < {infty}. By contrast, if only IdX {otimes} Tp, {tau} is bounded and not its inverse, then we prove one-sided extrapolation theorems and provide examples showing that this is best possible.

2000 Mathematics Subject Classification 46B07, 46B70, 47B37.

The first author was supported by the project Stochastic and Harmonic Analysis, Interactions and Applications of the Academy of Finland. The second author was supported by the FWF-project P 20166-N18.

Received August 29, 2007; revised May 27, 2009; published online October 23, 2009.


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