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Journal of Convex Analysis 12 (2005), No. 1, 159--172
Copyright Heldermann Verlag 2005

Strong Martingale Type and Uniform Smoothness

Jörg Wenzel
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

We introduce stronger versions of the usual notions of martingale type p ≤ 2 and cotype q ≥ 2 of a Banach space X and show that these concepts are equivalent to uniform p-smoothness and q-convexity, respectively. All these are metric concepts, so they depend on the particular norm in X.
These concepts allow us to get some more insight into the fine line between X being isomorphic to a uniformly p-smooth space or being uniformly p-smooth itself.
Instead of looking at Banach spaces, we consider linear operators between Banach spaces right away. The situation of a Banach space X can be rediscovered from this by considering the identity map of X.

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