Vector-valued walsh-paley martingales and geometry of banach spaces

Abstract

The concept of Rademacher typep (1≤p≤2) plays an important role in the local theory of Banach spaces. In [3] Mascioni considers a weakening of this concept and shows that for a Banach spaceX weak Rademacher typep implies Rademacher typer for allr<p. As with Rademacher typep and weak Rademacher typep, we introduce the concept of Haar typep and weak Haar typep by replacing the Rademacher functions by the Haar functions in the respective definitions. We show that weak Haar typep implies Haar typer for allr<p. This solves a problem left open by Pisier [5]. The method is to compare Haar type ideal norms related to different index sets.