Harmonic and subharmonic function theory on the hyperbolic ball (Q2814152)

The monograph by Manfred Stoll considers the theory of harmonic and subharmonic functions on the hyperbolic ball \(\mathbb B\). With its contents, the book fits into widely understood harmonic analysis.NEWLINENEWLINEAs the title of the monograph says, the emphasis is put on the hyperbolic ball endowed with the metric \(ds=2|dx|\slash(1-|x|^2)\). Nevertheless, some important results are also presented in the corresponding model of the upper half-plane.NEWLINENEWLINEAccording to the authors' words in the Preface, his aim was to write a book which would be ``accessible to large audience''. In my opinion the author achieved his goal. The book is easy to read. It does not require much prerequisites. The author only requires a little knowledge in differential geometry.NEWLINENEWLINEThe contents of the book is as follows: Chapter 1 is about Möbius transformations in \(\mathbb R^n\), Chapter 2 is devoted to self-maps of the unit ball in \(\mathbb{R}^n\). In Chapter 3, a formula for the Laplacian is given (not derived from the metric), and the definitions of the gradient and the measure on \(\mathbb B\) which are invariant under the group of Möbius self-maps are given. In Chapter 4, \(\mathcal H\)-harmonic and \(\mathcal H\)-subharmonic functions are defined, and their properties are investigated. In Chapter 5, the Poisson kernel and the Poisson integral are defined, and relations of them to their real analogs are given. In Chapter 6, the spherical harmonics are considered. In Chapter 7, Hardy and Hardy-Orlicz spaces are discussed, whereas Chapter 8 is devoted to the study of the boundary behavior of the Poisson integral. In Chapter 9, the Riesz decomposition theorem for \(\mathcal H\)-subharmonic function is presented. The last Chapter 10 is devoted to the study of some basic properties of weighted Bergman-type and Dirichlet-type spaces of \(\mathcal H\)-harmonic functions on the unit ball \(\mathbb B\). NEWLINENEWLINENEWLINE Every chapter ends with a list of exercises, which includes not only routine ones, but also more difficult exercises whose solution could be the starting point of research.NEWLINENEWLINETo sum up, the book is very well written (I did not find any misprints). It is easy to read and self-contained, what is a great advantage. I hope that the book will be useful for scientific work in this area, as well as for undergraduate and PhD students. The book can be also used by teachers to prepare lectures on related subjects.