Martingale Transforms and the Geometry Of Banach Spaces

Abstract

This paper is a continuation of [3], which contains a geometrical characterization of the class of Banach spaces having the unconditionality property for martingale differences (see Definition 1.3). Important information about this class, which we denote by UMD, is contained in the work of Maurey, Pisier, Aldous, and others. For example, if B ∈ UMD, then B is superreflexive [5] but there do exist superreflexive spaces that are not in UMD [6]; if 1 < p < ∞, and the Lebesgue-Bochner space ${L}^{P}_{B}(0,1)$ has an unconditional basis, then B ∈ UMD [1]. One of the main objects of study in this paper and [3] is the class MT of Banach spaces B for which B-valued martingale transforms are well-behaved (see Definition 1.2). The geometrical condition introduced in [3] also characterizes the class MT, information about which is of value in the study of B-valued stochastic integrals and B-valued singular integrals. Before recalling the probability background, we shall describe this geometrical condition.