Book reviews: "Wojbor A. Woyczynski, Geometry and martingales in Banach spaces", CRC Press, Boca Raton, FL, 2019, ISBN 978-1-138-61637-0/hbk; 978-0-4298-6883-2/ebook, xiii+315 p.

Abstract

The study of Banach space valued random variables is tightly connected with the geometric properties of the underlying space. In particular, martingale theory is essential in the study of Radon-Nikody´m property, finite tree property and super- reflexivity, and of the local properties of Banach spaces. The UMD spaces (meaning Banach spaces X for which X-valued martingale differences are unconditionally con- vergent in Lp(X), 1 < p < ∞) provide the correct framework for the development of the harmonic analysis for vector-valued functions. This is masterly illustrated in two recent books: G. Pisier, Martingales in Banach spaces, Cambridge University Press, Cambridge, 2016, and T. Hyt¨onen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach spaces. Vol. I. Martingales and Littlewood-Paley theory, Springer 2016;