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.
Similar content being viewed by others
References
B. Beauzamy,Introduction to Banach Spaces and their Geometry, North-Holland, Amsterdam, 1985.
S. Geiß,BMO ψ -spaces and applications to the extrapolation theory, preprint.
V. Mascioni,On weak cotype and weak type in Banach spaces, Note di Matematica8 (1988), 67–110.
A. Pietsch and J. Wenzel,Orthogonal systems and geometry of Banach spaces, in preparation.
G. Pisier,Martingales with values in uniformly convex spaces, Israel Journal of Mathematics20 (1975), 326–350.
G. Pisier,Probabilistic methods in the geometry of Banach spaces, inProbability and Analysis, Lecture Notes in Mathematics, No. 1206, Springer-Verlag, Varenna, Italy, 1985, pp. 167–241.
L. Tzafriri,On the type and cotype of Banach spaces, Israel Journal of Mathematics32 (1979), 32–38.
J. Wenzel,Haar type ideal norms of diagonal operators, preprint.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wenzel, J. Vector-valued walsh-paley martingales and geometry of banach spaces. Isr. J. Math. 97, 29–49 (1997). https://doi.org/10.1007/BF02774024
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02774024