Martingales in Banach spaces (Q2814191)

This long-awaited book, whose preliminary versions have circulated in the community for quite a while, deals with the fruitful applications of (mostly discrete-time) martingales to the geometry of Banach spaces. Its author has been a key figure in the development of this theory over several decades.NEWLINENEWLINEThe main themes of the book are three classes of Banach spaces that are either defined, or can be characterized, in terms of martingale properties: the Radon-Nikodým property (RNP), unconditionality of martingale differences (UMD), and uniform convexity (together with its equivalent reincarnations as uniform smoothness and super-reflexivity). In addition, there is an extensive discussion of interpolation of Banach spaces and its applications to so-called strong \(p\)-variation norms. Faithful to the title, the related notions of type and cotype, which involve independent random variables rather than general martingales, are only briefly mentioned in this book. The author intends to dedicate an entire second volume to martingales in non-commutative \(L^p\)-spaces, and accordingly the present work merely outlines the key notions and results in this direction.NEWLINENEWLINEOn the side of martingale theory, there are lengthy excursions to harmonic analysis of Banach space-valued functions, illustrating the rich interplay of these topics. These analytic excursions are mainly directed towards the theory of different Hardy spaces of harmonic, holomorphic, as well as real-variable functions. Among other things, the Radon-Nikodým property is used to characterize the dual spaces of both the Lebesgue-Bochner spaces \(L^p(B)\) and the Hardy space \(H^1(B)\) in terms of \(B^*\)-valued function spaces, but useful alternative descriptions of these duals are also provided in the absence of RNP in terms of \(B^*\)-valued measures or weakly\(^*\) measurable functions.NEWLINENEWLINEThe UMD property is discussed in two chapters, first from a pure martingale perspective, and then in connection with the Hilbert transform and Littlewood-Paley theory. In this direction, the book goes as far as Bourgain's vector-valued Marcinkiewicz multiplier theorem on \(L^p(\mathbb T;B)\), but does not develop the analogous results on the line nor in several variables. For these, the reader is referred to [\textit{T. Hytönen} et al., Analysis in Banach spaces. Volume I. Martingales and Littlewood-Paley theory. Cham: Springer (2016; Zbl 1366.46001)]. It is curious that, 30--40 years after the creation of the main body of the theory of UMD spaces, its first two detailed treatments in a book form finally appeared in the same year, but the reviewer fully acknowledges that the book under review was the very first one by a margin of several months. Both books have chosen to favour S. Petermichl's more recent argument over D. L. Burkholder's original one to deduce the boundedness of the Hilbert transform from the UMD property.NEWLINENEWLINEUniform convexity and smoothness are first connected to martingale inequalities in parallel treatments in one chapter, and then to super-reflexivity and to each other in a second one. The discussion of super-reflexivity stands out from the rest of the book in terms of inherently more abstract concepts that it involves, while most of the rest may be regarded as relatively concrete.NEWLINENEWLINEThe different topics covered by the 14 chapters can be studied largely independently, and may appeal to a variety of audiences. Besides the usual table of contents, the book has a nice 15-page ``description of contents'' in prose form to help the reader get oriented.