PrefaceIn the past few years the subject of variable exponent spaces has undergone a vast development. Nevertheless, the standard reference is still the article by Kováčik and Rákosník from 1991. This paper covers only basic properties, such as reflexivity, separability, duality and first results concerning embeddings and density of smooth functions. In particular, the boundedness of the maximal operator, proved by Diening in 2002, and its consequences are missing.Naturally, progress on more advanced properties is scattered in a large number of articles. The need to introduce students and colleagues to the It has been our goal to make the book accessible to graduate students as well as a valuable resource for researchers. We present the basic and advanced theory of function spaces with variable exponents and applications to partial differential equations. Not only do we summarize much of the existing literature but we also present new results of our most recent research, including unifying approaches generated while writing the book.Writing such a book would not have been possible without various sources of support. We thank our universities for their hospitality and the Academy of Finland and the DFG research unit "Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis" for financial support. We also wish to express our appreciation of our fellow researchers whose results are presented and ask for understanding for the lapses, omissions and misattributions that may have entered the text. Thanks are also in order to Springer Verlag for their cooperation and assistance in publishing the book.We thank our friends, colleagues and especially our families for their continuous support and patience during the preparation of this book. v vi PrefaceFinally, we hope that you find this book useful in your journey into the world of variable exponent Lebesgue and Sobolev spaces.

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In this article we introduce Triebel-Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Using it we derive molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition we prove optimal trace theorem in the variable indices case.

In this article we introduce Besov spaces with variable smoothness and integrability indices. We prove independence of the choice of basis functions, as well as several other basic properties. We also give Sobolev-type embeddings, and show that our scale contains variable order Hölder-Zygmund spaces as special cases. We provide an alternative characterization of the Besov space using approximations by analytic functions.

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