Vector–valued Walsh–Paley martingales and geometry of Banach spaces
نویسنده
چکیده
Abstract The concept of Rademacher type p (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 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 [5]. The method is to compare Haar type ideal norms related to different index sets.
منابع مشابه
Lower estimates of random unconditional constants of Walsh-Paley martingales with values in Banach spaces
for all n = 1, 2, ... and all martingales {Mk}0 ⊂ L X p with values in X . It turns out that this definition is equivalent to the modified one if we replace ”for all 1 < p < ∞” by ”for some 1 < p < ∞”, and ”for all martingales” by ”for all Walsh-Paley-martingales” (see [3] for a survey). Motivated by these definitions we investigate Banach spaces X by means of the sequences {RUMDn(X)}n=1 wherea...
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