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

Abstract

The concept of Rademacher type $p$ ($1\le p\le 2$) 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.