Fractional Sobolev Spaces and Inequalities

'Although introduced in the mid-1950s, the fractional spaces Ws p (Ω) have attracted much recent interest because of their many applications, in probability, continuum mechanics, mathematical biology and elsewhere. This monograph is a concise account of the fractional Banach spaces Ws p (Ω) and their embedding theorems, and of inequalities such as the fractional analogues of those attributed to Hardy and Rellich in the case of Sobolev spaces. While classical and fractional Sobolev spaces have much in common, it is noted that they differ in important respects, and results are derived from the basic definitions, rather than being inferred from the theory of other function spaces with which Ws p (Ω) can be identified, but only when the boundary of Ω is sufficiently regular. The text is a useful guide to the classical and emerging literature.' John Toland, University of Bath 'A comprehensive treatise of the theory of fractional Sobolev spaces, defined either on the ambient Euclidean space, or on its generic open subset, and the related inequalities. Striking dissimilarities are discovered in comparison to the classical theory. Contemporary challenging problems are tackled, such as the questions concerning the geometry of underlying domains, the effect of symmetrization techniques, the impact of interpolation, relations to Besov spaces, questions of compactness of embeddings, and more. The book will be of great use to graduate students and researchers of a wide array of scientific interests ranging from partial differential equations and calculus of variations, to approximation and interpolation theory, to the theory of function spaces and related areas. All of the material is accessible through real-variable methods. The only prerequisites, namely basic knowledge of measure theory and Lebesgue integration, will therefore be met by any standard graduate course in real analysis.' Luboš Pick, Charles University, Prague