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Elliptic partial differential equations and quasiconformal mappings in the plane. (English) Zbl 1182.30001

Princeton Mathematical Series 48. Princeton, NJ: Princeton University Press (ISBN 978-0-691-13777-3/hbk). xvi, 677 p. (2009).
This book (AIM) deals with a field of classical analysis where plane quasiconformal (qc) maps interact with several other currently active research areas. These areas include geometric function theory, PDEs, harmonic analysis, holomorphic dynamics, geometry, Teichmüller spaces. Anybody, who wishes to study these topics and related areas of geometric function theory, should be aware of the recent Kühnau handbook [R. Kühnau, Handbook of complex analysis: geometric function theory. Volume. I. (Amsterdam): North Holland. (2002; Zbl 1057.30001), Volume. II. (Amsterdam): Elsevier/North Holland. (2005; Zbl 1056.30002)], a most valuable collection of surveys. In particular, the surveys [F.W. Gehring, Handbook of complex analysis: geometric function theory. Volume 2. Amsterdam: Elsevier/North Holland. 1–29 (2005; Zbl 1078.30014)], [U. Srebro, E. Yakubov, Handbook of complex analysis: geometric function theory. Volume 2. Amsterdam: Elsevier/North Holland. 555-597 (2005; Zbl 1078.30010)], and [C. Andreian-Cazacu, Handbook of complex analysis: geometric function theory. Volume 2. Amsterdam: Elsevier/North Holland. 687-753 (2005; Zbl 1083.30019)] may be recommended to the readers of AIM. This last survey contains a very comprehensive and useful bibliography and lists the existing monographs on qc maps. Some of the pioneers of the field were L. V. Ahlfors (e.g. [Lectures on quasiconformal mappings. Princeton, N.J.-Toronto-New York-London: D. Van Nostrand Company. (1966; Zbl 0138.06002)]), L. Bers (e.g. [Theory of pseudo-analytic functions, New York: New York University, Institute for Mathematics and Mechanics, III, 187 p. (1953; Zbl 0051.31603)]), and I. N. Vekua (e.g. [Generalized analytic functions. (Russian) Moskva: Nauka. 512 p. (1988; Zbl 0698.47036)]).
The readers of this opus magnum should be well prepared in order to absorb the presented material. The readers should be familiar at least with measure theory, topology, real, complex, functional and harmonic analysis, operator theory, PDEs. Operator theoretic methods (Beltrami, Beurling-Ahlfors, Cauchy, Fourier, Riesz, operators/transforms, Calderon-Zygmund theory, maximal functions etc etc) are used throughout the text. The selection of the material reflects the authors’ own significant contributions to qc maps and the reader can observe their inspiration and excitement. The emphasis is on some most modern results of the field, and also some new results are given.
By definition, the partials of a plane qc map are required to be locally of class \(L^p\) with \(p=2\,.\) It was B. Bojarski who first proved that the partials, in fact, belong locally to the class \(L^p\) for some \(p>2\,.\) This milestone result was published by Bojarski in his thesis [Mat. Sb. N. Ser. 43(85), 451–503 (1957; Zbl 0084.30401)] written under the supervision of Vekua (the English translation of the thesis is available on the www-page http://urn.fi/URN:ISBN:978-951-39-3486-6). What remained an open problem for many years was the question about the best possible integrability exponent \(p(K)>2\) in terms of the maximal dilatation \(K\) of a \(K\)-qc mapping. This natural question was solved by K. Astala [Acta Math. 173, No. 1, 37–60 (1994; Zbl 0815.30015)] in 1993. Apparently one of the highlights of the book is Chapter 13 with a description of Astala’s sharp version of Bojarski’s \(L^p\)-integrability result. Some of the consequences are discussed in later chapters. The book consists of 21 chapters.
This book is essentially a research monograph and it covers a surprising amount of material presented in a somewhat condensed style. The authors say in the introduction that some parts of the book might be used for graduate level lecture series and give a list of suggested material for that purpose. This may not be an easy book for an average student, but for a dedicated and well-prepared reader this book provides perhaps the most accessible gateway to these topics of current research. The book is maybe best suited to researchers of analysis for whom the authors have given many open problems throughout the text.
No doubt this book will have an important role for the qc mapping theory within the next few years. Every mathematics graduate library should have a copy of this book.

MSC:

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
30C62 Quasiconformal mappings in the complex plane
35J60 Nonlinear elliptic equations
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
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