Geometric function theory and non-linear analysis / Tadeusz Iwaniec, Gaven Martin

The reader of this research monograph is supposed to be well prepared, at least on the level of “the competent graduate student" as the authors put it. The book is organized into twenty-three chapters as follows.

1. Introduction and overview, 2. Conformal mappings, 3. Stability of the Möbius group, 4. Sobolev theory and function spaces, 5. The Liouville theorem, 6. Mappings with finite distortion, 7. Continuity, 8. Compactness, 9. Topics from multilinear algebra, 10. Differential forms, 11. Beltrami equations, 12. Riesz transforms, 13. Integral estimates, 14. The Gehring lemma, 15. The governing equations, 16. Topological properties of mappings of bounded distortion, 17. Painlevé’s theorem in space, 18. Even dimensions, 19. Picard and Montel theorems in space, 20. Conformal structures, 21. Uniformly quasiregular mappings, 22. Quasiconformal groups, 23. Analytic continuation for Beltrami systems.

For graduate student readers it would have been helpful if the authors had included exercises with solutions to deepen and underline some key ideas of the text. On the other hand, the lack of exercises may not be a problem since most readers are likely to be specialists in the area of mathematical analysis.

The contents of the book is highly original, because other books do not deal with these topics and also because much of the material was unpublished at the time of the writing of the book. A novel theme is the study of maps with finite distortion. The book is very useful for specialists in the area of multidimensional geometric function theory because it brings under one roof results scattered in journals. The authors display brilliant expertise in this technically very difficult area, much of which is due to the authors and their collaborators. The book addresses many new topics which look very promising, and I think that this book will have an important role in the future development of several areas of mathematical analysis such as nonlinear PDEs, iteration of quasiregular maps, and maps of finite distortion. This book is likely to remain a landmark of the quasiregular mapping theory for many years to come. (Zentralblatt)

Bibliogr. p. 531-546. Index