Sobolev spaces. 2nd ed.

*(English)*Zbl 1098.46001
Pure and Applied Mathematics 140. New York, NY: Academic Press (ISBN 0-12-044143-8/hbk). xiii, 305 p. (2003).

R. A. Adams’ treatise [“Sobolev Spaces” (Pure Appl. Math. 65, Academic Press) (1975; Zbl 0314.46030)] has become the quintessential reference work in this area. The present second edition, prepared jointly with J. Fournier, is similar but not entirely identical to the first edition.

There are eight chapters, as in the first edition, namely: 1. Preliminaries; 2. The Lebesgue spaces \(L^p(\Omega)\); 3. The Sobolev spaces \(W^{m,p}(\Omega)\); 4. The Sobolev imbedding theorem; 5. Interpolation, extension, and approximation theorems; 6. Compact imbeddings of Sobolev spaces; 7. Fractional order spaces; 8. Orlicz spaces and Orlicz–Sobolev spaces.

The most noticeable changes have occurred in Chapters 2, 4, 5, and 7. Chapter 2 now contains a treatment of weak \(L^p\)-spaces, weak type operators and the Marcinkiewicz interpolation theorem. The proofs of the various versions of the Sobolev embedding theorem have been streamlined; as a result, Chapters 4 and 5 have switched their position. Some results in the current Chapter 5 have been strengthened in that, for example, the assumption of the uniform cone condition has been replaced by the mere cone condition.

Chapter 7 has been completely rewritten. It is now based on the real interpolation method that is presented in detail. The authors introduce and study Besov spaces as well as Triebel–Lizorkin spaces and Bessel potential spaces, their relations and their emdeddings. The remaining chapters have undergone only minor changes.

Without doubt, the new edition will serve generations of mathematicians as a reference tool, just as the first edition has over the past 30 years.

There are eight chapters, as in the first edition, namely: 1. Preliminaries; 2. The Lebesgue spaces \(L^p(\Omega)\); 3. The Sobolev spaces \(W^{m,p}(\Omega)\); 4. The Sobolev imbedding theorem; 5. Interpolation, extension, and approximation theorems; 6. Compact imbeddings of Sobolev spaces; 7. Fractional order spaces; 8. Orlicz spaces and Orlicz–Sobolev spaces.

The most noticeable changes have occurred in Chapters 2, 4, 5, and 7. Chapter 2 now contains a treatment of weak \(L^p\)-spaces, weak type operators and the Marcinkiewicz interpolation theorem. The proofs of the various versions of the Sobolev embedding theorem have been streamlined; as a result, Chapters 4 and 5 have switched their position. Some results in the current Chapter 5 have been strengthened in that, for example, the assumption of the uniform cone condition has been replaced by the mere cone condition.

Chapter 7 has been completely rewritten. It is now based on the real interpolation method that is presented in detail. The authors introduce and study Besov spaces as well as Triebel–Lizorkin spaces and Bessel potential spaces, their relations and their emdeddings. The remaining chapters have undergone only minor changes.

Without doubt, the new edition will serve generations of mathematicians as a reference tool, just as the first edition has over the past 30 years.

Reviewer: Dirk Werner (Berlin)