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Extrapolation of vector-valued rearrangement operators. (English) Zbl 1190.46016

Given an injective map \({\tau}:\mathcal{D} \to \mathcal{D}\) between the dyadic intervals of the unit interval \([0, 1)\), the authors study extrapolation properties of the induced rearrangement operator of the Haar system Id\(_X{\otimes}T_{p, {\tau}}: L_{X,0}^p([0,1)) \to L_X^p ([0,1))\), where \(X\) is a Banach space and \(L_{X,0}^p\) the subspace of mean zero random variables.
In contrast to the situation for vector-valued martingale transforms where continuity for some \(p \in (1,\infty)\) implies continuity for all \(q \in (1,\infty)\), it is shown that a general extrapolation result for rearrangement operators does not hold. In particular, there exists a permutation \(\tau_0\) such that, for \(p \in (1,2)\) and \(X= \ell_p\), continuity holds for \(p\), but not for \(q \in (p,2]\).
However, if \(X\) is a UMD-space, then the authors prove that the property that \(\text{Id}_X{\otimes}T_{p, {\tau}}\) is an isomorphism for some \(1 < p\neq 2 < {\infty}\) extrapolates across the entire scale of \(L_X^q\)-spaces with \(1 < q < {\infty}\). By contrast, if only \(\text{Id}_X{\otimes}T_{p, {\tau}}\) is bounded and not its inverse, then they prove one-sided extrapolation theorems. In particular, for any Banach space \(X\) and any permutation \(\tau\) with \(|\tau(I)|=|I|\), boundedness for some \(p \in (1,2)\) implies boundedness for all \(q \in (1,p)\).

MSC:

46B07 Local theory of Banach spaces
46B70 Interpolation between normed linear spaces
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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