Resumen de Extrapolation of vector-valued rearrangement operators

Stefan Geiss, Paul F. X. Müller

• Given an injective map �Ñ :D �¨Dbetween the dyadic intervals of the unit interval [0, 1), we study extrapolation properties of the induced rearrangement operator of the Haar system IdX . Tp,�Ñ :

  Lp X,0([0, 1)) �¨ Lp X([0, 1)), where X is a Banach space and Lp X,0 the subspace of mean zero random variables. If X is a UMD-space, then we prove that the property that IdX . Tp,�Ñ is an isomorphism for some 1 < p = 2 < �� extrapolates across the entire scale of Lq X-spaces with 1 < q < ��. By contrast, if only IdX . Tp,�Ñ is bounded and not its inverse, then we prove one-sided extrapolation theorems and provide examples showing that this is best possible