This is a tremendous monograph. The title “Concrete Functional Calculus” is explained by the authors as follows. They consider “existence and smoothness questions for some concrete nonlinear operators acting on some concrete Banach spaces of functions”. The book has little overlap with any previous book except for the lecture notes by the same authors [Differentiability of six operators on nonsmooth functions and \(p\)-variation. With the collaboration of Jinghua Qian. Lecture Notes in Mathematics 1703.Berlin:Springer (1999; Zbl 0973.46033)] and [An introduction to \(p\)-variation and Young integrals. With emphasis on sample functions of stochastic processes. MaPhySto.Lecture Notes 1.Aarhus:Univ.of Aarhus, Department of Mathematical Sciences (1998; Zbl 0937.28001)].
To exemplify the situation, the authors ask how to define \(\int f\, dg\) for two given functions \(f,g\), defined on (subsets of) the real line and taking values in some Banach lattices. This is a subject with a long history, and it is intimately connected with the variability of the functions involved, their \(\Phi\)- and, more specifically, their \(p\)-variation. The fundamental introductory chapters are Chapter 2 (Definitions and Basic Properties of Extended Riemann-Stieltjes Integrals) and Chapter 3 (\(\Phi\)-variation and \(p\)-variation, Inequality for Integrals), each of which covers most of the known (and some new) theory, and each chapter interesting on its own. These two chapters cover about 200 pages. Detailed historical comments highlight the evolution and interrelations of the material.
However, the subject of this book is much wider. Chapter 8 (Two Function Composition) studies the composition map \(TC: (F,G) \longrightarrow F\circ G\), where \(G : S\to U\) and \(F: U \to Y\) are given on (subsets of) Banach spaces \(U,Y\). Again, the mapping properties of \(TC\) are determined by the variability of the functions \(F\) and \(G\). Emphasis is put on differentiability properties of \(TC\), and the introductory Chapter 5 (Derivatives and Analyticity in Normed Spaces) gives an account of the required prerequisites.
A specific instance of \(TC\) is for a fixed function \(F\), yielding a function of \(G\). This leads to (autonomous) Nemytskii operators, which are studied in Chapters 6 (Nemytskii Operators on Some Function Spaces) and 7 (Nemytskii Operators on \( L^{p}\) Spaces). The standard reference for Nemytskii operators is [J. Appell and P. B. Zabrejko, “Nonlinear superposition operators” (Cambridge Tracts in Mathematics 95; Cambridge University Press) (1990; Zbl 0701.47041)], but this does not cover functions having finite \(p\)-variation, a subject which did not get attention previously.
Other topics in this book are Chapter 9 (Product Integrals) and Chapter 10 (Nonlinear Differential and Integral Equations) as well as Chapter 11 (Fourier Series), for functions of bounded \(p\)-variation, or, more generally, of bounded \(\Phi\)-variation.
A special role plays the final Chapter 12 (Stochastic Processes and \(\Phi\)-Variation). It gives (among other things) a concise introduction to the theory of semimartingales, Markov processes and Itô integrals, and the authors study sample path properties of such processes, specifically their \(p\)-variation. Of particular interest is § 12.5, treating sample path properties of Markov processes, based on recent work by [M. Manstavičius, Ann.Probab.32, No. 3A, 2053–2066 (2004; Zbl 1052.60058)], and § 12.7, in which the asymptotic properties of empirical processes are analysed.
This book will be a standard reference on results related to functions of bounded \(\Phi\)- or \(p\)-variation, in particular, within the “rough path theory” as developed by T. Lyons and others, see, e.g.,the recent monograph [P. K. Friz and N. B. Victoir, “Multidimensional stochastic processes as rough paths. Theory and applications” (Cambridge Studies in Advanced Mathematics 120; Cambridge University Press) (2010; Zbl 1193.60053)].
Many of the results are final, but there are some topics that “leave openings for research”. As the authors claim, the monograph should be accessible to graduate students with a background in real analysis and probability theory; the preface includes a guide to the reader mentioning the dependencies.