The book can be seen as a continuation of the previous book by the same authors [Holomorphic functions in the plane and \(n\)-dimensional space. Transl. from the German. Basel: Birkhäuser (2008; Zbl 1132.30001)]. It is pertinent to say that usually the name ‘holomorphic function’ in \(n\) dimensions refers to the classical notion of holomorphy in the sense of several complex variables, which is not the case here. The functions in both books are those of quaternionic and Clifford analysis (sometimes also the name ‘hypercomplex analysis’ is used), that is, the null solutions of several generalizations of classical complex Cauchy-Riemann (or Wirtinger) operators: such as the Fueter, Dirac, etc. operators. For these functions the adjectives ‘monogenic’, ‘regular’, ‘hyperholomorphic’ (and some others) are frequently used.
What is intended by in the book under review is clearly explained in the preface (page XII): “The main goal of this book is to present applications of hypercomplex analysis connected with boundary value and initial-boundary value problems, appearing in mathematical physics, as well as its applications which arise from hypercomplex Fourier transforms. We are convinced that the methods of quaternionic and Clifford analysis are a natural and clever way to approach higher-dimensions”. As for the readership of the book, one reads on the same page: “The book is written, above all, for graduate students of mathematics and physics, and also for graduate students of disciplines that utilize advanced mathematics”.
The proposed above program of applications is realized both by using the hypercomplex tools developed in the previous book [loc. cit.] and by using those presented in the book under review. The reader finds here the following topics: polynomial systems; operator calculus; decompositions of Bergman-Hodge, Almansi and Fischer type; Riemann-Hilbert problems; hypercomplex Fourier and Radon transforms. The applications of the elaborated tools include: multidimensional conformal and quasiconformal mappings; Helmholtz decomposition and associated boundary value problems; Maxwell equations; Schrödinger equation; Bers-Vekua system; equations of linear elasticity and transmission problems in it; stationary fluid flow problems; the Poisson-Stokes problem; higher-dimensional versions of the Korteweg-deVries and Burgers equations; initial-boundary value problems on the sphere. Both lists are not exhaustive but give a good idea about the contents of the book.
The material covered by the book is mostly not new and is dispersed in many books and papers, thus the book can serve as a good source for a preliminary information and for finding the references for a more elaborated information. I think the book lacks a careful editing which makes the reading rather difficult and confusing sometimes; I give a few examples of this.

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On pages 16–18 in Section 1.2 “Classical function spaces in quaternions” one reads: “Let \(X\), \(Y\) be Banach spaces and \(u_i:Y \to X\). We say that a quaternionic-valued function \(\displaystyle u= \sum_{i=0}^3 u_i e_i\) is in \(X\) if all \(u_i \in X\)”. This is very confusing although one can suppose that if all the four real-valued components \(u_i\) of a quaternion-valued function \(u\) belong to some known class of (real-valued) functions then it is common to say that the quaternion-valued function \(u\) belongs to the class with the same name. The considered classes are: continuous functions, continuously differentiable functions, Hölder functions, etc. For them the usual formulas are written down which means that they are seen as linear spaces. But the linear spaces over quaternionic scalars can be left-linear, right-linear, or two-sided linear; nothing of this is mentioned although it is important to note that the formula defining the quaternion-valued inner product on \(L^2\) is for a quaternionic right- (not left-) linear space.
The case of functions with values in a general Clifford algebra has, additionally, its own peculiarities and deserves more that just Remark 1.2.1. Regretfully, the inaccuracies of this section extend onto the whole book.

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On page 19 the authors note that the usual definition of holomorphy by differentiation does not work in higher-dimensions which should have been made much more precise. What happens with the derivative as the limit of an appropriate quotient for quaternion-valued and, more generally, Clifford-algebra-valued functions can be found in [M. E. Luna-Elizarrarás and the reviewer, Milan J. Math. 79, No. 2, 521–542 (2011; Zbl 1242.30042)].

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The title of Chapter 3 is “Function theoretic function spaces”, and I just wonder what does it mean?

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Page 19. It is useful to note that what is called here “reduced quaternions” are, in fact, paravectors when the quaternions are considered as the Clifford algebra \(\mathcal C \ell (2)\) with two generators.

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Subsections 5.2.1 and 5.2.2 deal with the set \(L^2(G, \mathbb C \mathbb H)\) called complex quaternionic Hilbert space which is, in fact, a right-linear module over \( \mathbb C \mathbb H \) with a \(\mathbb C \mathbb H\)-valued inner product. I believe that such an object is not well-studied, has many peculiarities, and deserves much more explanations.

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On pages 15–16 the bicomplex numbers are briefly introduced but on page 334 they are encountered again under the name “commutative quaternions” with no word that it is the same object. The most recent presentation of bicomplex numbers and their functions can be found in [M. E. Luna-Elizarrarás et al., Bicomplex holomorphic functions. The algebra, geometry and analysis of bicomplex numbers. Cham: Birkhäuser/Springer (2015; Zbl 1345.30002)].