The topic of the book is Sobolev spaces on metric measure spaces. The main focus is laid on the so-called Newtonian spaces introduced by N. Shanmugalingam, who is one of the authors. Analysis on metric spaces (and Sobolev spaces defined on them) is a research topic that started in the 1990s.
On the back cover of the book it is stated that “[t]he New Mathematical Monographs are dedicated to books containing an in-depth discussion of a substantial area of mathematics. They bring the reader to the forefront of research by presenting a synthesis of the key results, while also acknowledging the wider mathematical context. As well as being detailed, they are readable and contain the motivational material necessary for those entering a field. For established researchers they are a valuable resource”.
In my opinion, the book under review is a prime example of such a monograph. I believe that the book serves both clienteles very well: the ones entering the field as well as researchers knowledgeable already in the field who want to use it as a reference work. My feeling is that the monograph is also very suitable as a source for a lecture course. The tone of the book and the fact that it is basically self-contained makes reading very enjoyable. Researchers will appreciate that the monograph contains many important results in the field together with their proofs as well as an extensive bibliography.
When I was still in Jyväskylä, we had a seminar on (a yet incomplete draft of) the book where the participants were either PhD students or postdoctoral researchers. Each week, someone would present between 10 and 20 pages of the book. I believe the seminar was a big success.
Let me give short summaries of the individual chapters.
After the introductory chapter, the second chapter reviews basic functional analysis. Measure theory is the main content of the third chapter; it also includes differentiation of measures and a treatment of the Hardy-Littlewood maximal function. Chapter 4 deals with Lipschitz functions. The part about curves is a preparation for the following chapter, which looks at the modulus – an (outer) measure on curve families. Having the modulus at hand, the subsequent chapter introduces weak upper gradients, which are a substitute for the norm of the gradient. The introduction motivates by first looking at classical Sobolev spaces and how their elements behave on curves. A lot of results in this part are such that if functions \(u_i\) with weak upper gradients \(\rho_i\) are given, and if a transformation on the \(u_i\) is executed (for example, two of them are added or multiplied or some sort of truncation is applied), then a weak upper gradient of the transformed function based on the \(\rho_i\) is given.
After all this introductory material, the reader is now prepared for the central definition of the book: the Newtonian space (as it is the main Sobolev space in the book, the authors prefer to refer to it as Sobolev space). Roughly speaking, the Newtonian space \(N^{1,p}\) consists of \(p\)-integrable functions that have a \(p\)-weak upper gradient that is also \(p\)-integrable. As the theory of real-valued and Banach-space-valued Sobolev functions is very similar, the authors look at Banach-space-valued mappings. They also point out how to treat the situation where the target is a metric space. Basic properties of the Sobolev spaces are investigated, for example, it is shown that they are Banach spaces.
Poincaré inequalities are the main topic of the next two chapters. The inequalities push the control that weak upper gradients exercise on their respective functions on curves to a control of the functions on the space. Spaces supporting the right kind of Poincaré inequalities resemble Euclidean spaces in many ways. Many consequences of the Poincaré inequalities are treated.
In the chapter “Other definitions of Sobolev-type spaces”, variants of Sobolev spaces that are not based on weak upper gradients are considered. Further, results concerning the equality of these different concepts are proved.
The topics of the following chapters are Gromov-Hausdorff convergence and self-improvement of the Poincaré inequality.
The Gromov-Hausdorff convergence of metric measure spaces is considered in different flavours. Under some weak assumptions on the spaces, the limits of sequences of doubling metric spaces supporting a Poincaré inequality are doubling and satisfy a Poincaré inequality as well.
The Keith-Zhong results concerning the self-improvement property of Poincaré inequalities (in complete spaces equipped with a doubling measure) is the content of the next chapter.
The penultimate chapter looks at parts of J. Cheeger’s seminal paper about differentiable structures. The existence of such structures is proven.
The final chapter treats three topics. The first part looks at quasiconformal mappings. Their study was one of the main reasons to develop the theory treated in the book. Examples of spaces supporting a Poincaré inequality are the content of the second topic. The last part looks at applications and further research directions.
The book is really at the forefront of research, and so there are quite a few preprints cited. Some of them have now been published, hence I can update the bibliography:
[9]: [L. Ambrosio et al., Ann. Probab. 43, No. 1, 339–404 (2015;Zbl 1307.49044)].
[10]: To appear in J. Geom. Anal.
[11]: [L. Ambrosio et al., J. Eur. Math. Soc. (JEMS) 17, No. 8, 1817–1853 (2015;Zbl 1331.28005)].
[81]: There is a new 2nd revised edition [L. C. Evans andR. F. Gariepy, Measure theory and fine properties of functions. 2nd revised ed. Boca Raton, FL: CRC Press (2015;Zbl 1310.28001)].
[166]: [R. Korte et al., Calc. Var. Partial Differ. Equ. 54, No. 2, 1393–1424 (2015;Zbl 1327.31026)].
Let me give some further references.
Related to [71] concerning the openendedness of Orlicz-Poincaré inequalities: [N. DeJarnette, J. Math. Anal. Appl. 423, No. 1, 358–376 (2015;Zbl 1333.46034)].
Connected to J. Gong’s work is [A. Schioppa, Ann. Acad. Sci. Fenn., Math. 39, No. 1, 275–304 (2014;Zbl 1296.53090)]. At the time of writing of this review, there are further preprints of A. Schioppa on this topic. With respect to Cheeger’s differentiability structures, I would also like to mention [D. Bate, J. Am. Math. Soc. 28, No. 2, 421–482 (2015;Zbl 1307.30097)].
In the spirit of [270] is [N. Aïssaoui, Abstr. Appl. Anal. 2004, No. 1, 1–26 (2004;Zbl 1081.46025)]. The reader who is interested in this sort of generalization of Sobolev spaces might also be interested in the following papers concerning Newtonian Lorentz spaces: [Ş. Costea andM. Miranda Jr., Ill. J. Math. 56, No. 2, 579–616 (2012;Zbl 1319.31015)], [A. Ranjbar-Motlagh, Stud. Math. 191, No. 1, 1–9 (2009;Zbl 1176.26006)]. Further works on this topic are [A.S. {Romanov}, Sib. Mat. Zh. 49, No. 5, 1148–1157 (2008); translation in Sib. Math. J. 49, No. 5, 911–918 (2008;Zbl 1224.46067)], [K. Wildrick andT. Zürcher, Math. Z. 270, No. 1–2, 103–131 (2012;Zbl 1242.46047)]. For Banach function spaces, see [M. Mocanu, Complex Var. Elliptic Equ. 55, No. 1–3, 253–267 (2010;Zbl 1191.46030)].
Concerning non-embeddings, there is a survey in French byP. Pansu [Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2006–2007. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 25, 159–176 (2007;Zbl 1170.46304)].

Reviewer: Thomas Zürcher (Coventry)