In this book a new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and HölderZygmund spaces on the one hand and MorreyCampanato spaces on the other. MorreyCampanato spaces extend the notion of functions of bounded mean oscillation. These spaces play an important role in the theory of linear and nonlinear PDEs. Chapters 13 deal with local smoothness spaces in Euclidean \(n\)space based on the MorreyCampanato refinement of the Lebesgue spaces. The presented approach relies on wavelet decompositions. This is applied in Chapter 4 to GagliardoNirenberg inequalities. Chapter 5 deals with linear and nonlinear heat equations in global and local function spaces. The obtained assertions about function spaces and nonlinear heat equations are used in Chapter 6 to study NavierStokes equations. The book is addressed to graduate students and mathematicians with a working knowledge of basic elements of (global) function spaces and an interest in applications to nonlinear PDEs with heat and NavierStokes equations as prototypes. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and researchers interested in applications to nonlinear PDEs with heat and NavierStokes equations as prototypes. Table of Contents  Global and local spaces
 Local spaces: Properties
 MorreyCampanato spaces
 GagliardoNirenberg inequalities
 Heat equations
 NavierStokes equations
 Bibliography
 Symbols
 Index
