Celebratio Mathematica — Burkholder

Complete Bibliography

Works connected to Anatole Beck

Filter the Bibliography List

year	title	people

			clear

D. L. Burkholder: “Martingale transforms and the geometry of Banach spaces,” pp. 35–50 in Probability in Banach spaces, III (Medford, MA, 4–16 August 1980). Edited by A. Beck. Lecture Notes in Mathematics 860. Springer (Berlin), 1981. MR 647954 Zbl 0471.60012 incollection

This paper is a continuation of [1980], which contains a geometrical characterization of the class of Banach spaces having the unconditionality property for martingale differences. Important information about this class, which we denote by UMD, is contained in the work of Maurey, Pisier, Aldous, and others. For example, if \( B \in \) UMD, then \( B \) is superreflexive [Maurey 1975] but there do exist superreflexive spaces that are not in UMD [Pisier 1975]; if \( 1 < p < \infty \), and the Lebesgue–Bochner space has an unconditional basis, then \( B \in \) UMD [Aldous 1979]. One of the main objects of study in this paper and [1980] is the class MT of Banach spaces \( B \) for which \( B \)-valued martingale transforms are well-behaved. The geometrical condition introduced in [1980] 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.