The theory of Lebesgue and Sobolev spaces with variable integrability is experiencing a steady expansion, and is the subject of much vigorous research by functional analysts, function-space analysts and specialists in nonlinear analysis. These spaces have attracted attention not only because of their intrinsic mathematical importance as natural, interesting examples of non-rearrangement-invariant function spaces but also in view of their applications, which include the mathematical modeling of electrorheological fluids and image restoration. The main focus of this book is to provide a solid functional-analytic background for the study of differential operators on spaces with variable integrability. It includes some novel stability phenomena which the authors have recently discovered. At the present time, this is the only book which focuses systematically on differential operators on spaces with variable integrability. The authors present a concise, natural introduction to the basic material and steadily move toward differential operators on these spaces, leading the reader quickly to current research topics.

Preface vii

1. Preliminaries 1

1.1 The Geometry of Banach Spaces 1

1.2 Spaces with Variable Exponent 10

2. Sobolev Spaces with Variable Exponent 41

2.1 Definition and Functional-analytic Properties 41

2.2 Sobolev Embeddings 42

2.3 Compact Embeddings 46

      2.3.1 Potential estimates and the class plog(Ω) 52

2.4 Riesz Potentials 56

2.5 Poincare-type Inequalities 64

      2.5.1 A remark on the geometry of the domain 64

      2.5.2 Poincare's inequalities 66

2.6 Embeddings 70

2.7 Hölder Spaces with Variable Exponents 71

2.8 Compact Embeddings Revisited 73

3. The p(·)-Laplacian 79

3.1 Preliminaries 79

3.2 The p(·)-Laplacian 82

3.3 Stability with Respect to Integrability 89

4. Eigenvalues 97

4.1 The Derivative of the Modular 97

4.2 Compactness and Eigenvalues 100

4.3 Modular Eigenvalues 102

4.4 Stability with Respect to the Exponent 114

4.5 Convergence Properties of the Eigenfunctions 117

      4.5.1 Alternative Rayleigh quotients 144

5. Approximation on Lp Spaces 147

5.1 s-numbers and n-widths 147

      5.1.1 s-numbers 147

      5.1.2 n-widths 156

5.2 A Sobolev Embedding 158

      5.2.1 The case when p is a step-function 163

      5.2.2 The case of strongly log-Hölder-continuous exponent 166

5.3 Integral Operators 175

      5.3.1 The case when p is a step-function 182

      5.3.2 The case of strongly log-Hölder-continuous p 185

Bibliography 197

Author Index 203

Subject Index 205

Notation Index 207

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