VECTOR-VALUED WALSH-PALEY MARTINGALES AND GEOMETRY OF BANACH-SPACES

The concept of Rademacher type p (1 less than or equal to p less than or equal to 2) plays an important role in the local theory of Banach s paces. In [3] Mascioni considers a weakening of this concept and shows that for a Banach space X weak Rademacher type p implies Rademacher t ype r for all r < p. As with Rademacher type p and weak Rademacher typ e 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 respec tive definitions. We show that weak Haar type p implies Haar type r fo r all r < p. This solves a problem left open by Pisier [5]. The method is to compare Haar type ideal norms related to different index sets.