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