Lebesgue and Sobolev Spaces with Variable Exponents The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics 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