From the reviews: "The authors provide a comprehensive survey of the state of the art concerning Lebesgue and Sobolev spaces with variable exponents. ... The book is also having a rich bibliography of 399 entries, a long list of symbols and an index. It will certainly become a standard reference in this field and stimulate further work in this direction." (H. G. Feichtinger, Monatshefte fur Mathematik, Vol. 165 (1), January, 2012) "The book is devoted to Lebesgue and Soboley spaces with variable exponents. ... The present book consists of the introduction and three parts. ... The majority of the results presented in the monograph were obtained by the authors and their collaborators. ... the books is a useful source of unified information on Lebesgue and Soboley spaces with variable exponents." (Alexei Yu. Karlovich, Zentralblatt MATH, Vol. 1222, 2011) "This book consists of three parts of different lengths and intentions, sub-divided into several chapters. There is a nice figure at the very beginning of the monograph explaining the dependencies among the chapters, together with some recommendations on which parts should be used for first reading or when teaching the subject in a graduate course. ... the presentation can thus also be considered as a textbook and extremely useful reference for graduate students and researchers working in related fields ... ." (Dorothee D. Haroske, Mathematical Reviews, January, 2013)

• 1 Introduction.- 2 A framework for function spaces.- 3 Variable exponent Lebesgue spaces.- 4 The maximal operator.- 5 The generalized Muckenhoupt condition*.- 6 Classical operators.- 7 Transfer techniques.- 8 Introduction to Sobolev spaces.- 9. Density of regular functions.- 10. Capacities.- 11 Fine properties of Sobolev functions.- 12 Other spaces of differentiable functions.- 13 Dirichlet energy integral and Laplace equation.- 14 PDEs and fluid dynamics