D. E. Edmunds was Professor of Mathematics at Sussex University for twenty-six years; now Emeritus. He has co-authored nine books and written more than 230 papers. Honours include the LMS Pólya Prize, the Czech Academy of Sciences Bolzano Medal and the Medal for Mathematics of the Czech Mathematical Society. W. D. Evans was a Professor in the School of Mathematics, Cardiff University for more than 30 years and is now Emeritus there. In 2011 he was elected Fellow of the Learned Society of Wales. His publications include eight co-authored books and about 200 research papers.

• 1. Preliminaries; 2. Classical Sobolev spaces; 3. Fractional Sobolev spaces; 4. Eigenvalues of the fractional p-Laplacian; 5. Classical (local) Hardy inequalities; 6. Fractional analogues; 7. Classical and fractional inequalities of Rellich type; References; Symbol index; Author index; Index of terms.

'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

D. E. Edmunds was Professor of Mathematics at Sussex University for twenty-six years; now Emeritus. He has co-authored nine books and written more than 230 papers. Honours include the LMS Pólya Prize, the Czech Academy of Sciences Bolzano Medal and the Medal for Mathematics of the Czech Mathematical Society. W. D. Evans was a Professor in the School of Mathematics, Cardiff University for more than 30 years and is now Emeritus there. In 2011 he was elected Fellow of the Learned Society of Wales. His publications include eight co-authored books and about 200 research papers.

• 1. Preliminaries; 2. Classical Sobolev spaces; 3. Fractional Sobolev spaces; 4. Eigenvalues of the fractional p-Laplacian; 5. Classical (local) Hardy inequalities; 6. Fractional analogues; 7. Classical and fractional inequalities of Rellich type; References; Symbol index; Author index; Index of terms.

'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

D. E. Edmunds was Professor of Mathematics at Sussex University for twenty-six years; now Emeritus. He has co-authored nine books and written more than 230 papers. Honours include the LMS Pólya Prize, the Czech Academy of Sciences Bolzano Medal and the Medal for Mathematics of the Czech Mathematical Society. W. D. Evans was a Professor in the School of Mathematics, Cardiff University for more than 30 years and is now Emeritus there. In 2011 he was elected Fellow of the Learned Society of Wales. His publications include eight co-authored books and about 200 research papers.

• 1. Preliminaries; 2. Classical Sobolev spaces; 3. Fractional Sobolev spaces; 4. Eigenvalues of the fractional p-Laplacian; 5. Classical (local) Hardy inequalities; 6. Fractional analogues; 7. Classical and fractional inequalities of Rellich type; References; Symbol index; Author index; Index of terms.

'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