Extrapolation of vector-valued rearrangement operators. II (Q2890308)

The article under review has its predecessors in the author's colloberations with [\textit{S.\ Geiss} and \textit{V.\ Pillwein}, Stud. Math.\ 171, No. 2, 197--205 (2005; Zbl 1092.46014)] (scalar-valued extrapolation) and [\textit{S.\ Geiss}, J. Lond. Math. Soc., II. Ser. 80, No. 3, 798--814 (2009; Zbl 1190.46016)], which is the first part of this paper (extrapolation in vector-valued \(L^p\) spaces). Now the author considers vector-valued extrapolation of rearrangement operators acting on the Haar system in vector-valued \(H^p\) spaces.NEWLINENEWLINEFrom the author's abstract: ``We determine the extrapolation law of rearrangement operators acting on the Haar system in vector valued \(H^p\) spaces: If \( 0<q \leq p <2 , \) then, NEWLINE\[NEWLINE \| T_{\tau, q } \otimes \text{Id}_X \| _{q}^{\frac{q}{2-q}} \leq A(p,q) \| T_{\tau, p } \otimes \text{Id}_X \| _{p}^{\frac{p}{2-p}} . NEWLINE\]NEWLINE For a fixed Banach space \(X, \) the extrapolation range \( 0<q \leq p <2 \) is optimal. If, however, there exists \( 1 < p_0 < \infty , \) so that NEWLINE\[NEWLINE \| T_{\tau, p_0 } \otimes \text{Id}_E \|_{L^{p_0}_E} < \infty \quad\text{for each UMD space E,} NEWLINE\]NEWLINE then, for any \(1 < p < \infty \), NEWLINE\[NEWLINE \| T_{\tau, p } \otimes \text{Id}_E \|_{L^{p}_E} < \infty, NEWLINE\]NEWLINE for any UMD space \(E.\) (The value \(p_0 =2\) is \textit{not} excluded.) We characterize Hilbert spaces in terms of vector-valued rearrangement operators. If NEWLINE\[NEWLINE \|T_{\tau,2} \otimes \text{Id}_X \|_{L_Y^2} <\infty \quad \text{and} \quad \|T_{\tau,p} \|_{L^p}=\infty \quad \text{(for \(1 <p \not= 2 <\infty\)),} NEWLINE\]NEWLINE then \(X\) is isomorphic to a Hilbert space.''NEWLINENEWLINEThe main ingredients of the proof are extrapolation by factoring, consideration of the Carleson measure and comparing integral estimates for maximal functions. The latter idea is based on \textit{K.\ Smela's} article [Stud. Math.\ 189, No. 2, 189--199 (2008; Zbl 1169.42013)].