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arXiv:math/9508207 [math.FA]AbstractReferencesReviewsResources

Vector-valued Walsh-Paley martingales and geometry of Banach spaces

Joerg Wenzel

Published 1995-08-01Version 1

The concept of Rademacher type $p$ ($1\leq p\leq2$) plays an important role in the local theory of Banach spaces. In \cite{mas88} Mascioni considers a weakening of this concept and shows that for a Banach space $X$ weak Rademacher type $p$ implies Rademacher type $r$ for all $r<p$. As with Rademacher type $p$ and weak Rademacher type $p$, we introduce the concept of Haar type $p$ and weak Haar type $p$ by replacing the Rademacher functions by the Haar functions in the respective definitions. We show that weak Haar type $p$ implies Haar type $r$ for all $r<p$. This solves a problem left open by Pisier \cite{pis75}. The method is to compare Haar type ideal norms related to different index sets.

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