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**An introduction to Sobolev spaces and interpolation spaces.**
*(English)*
Zbl 1126.46001

Lecture Notes of the Unione Matematica Italiana 3. Berlin: Springer (ISBN 978-3-540-71482-8/pbk). xxvi, 218 p. (2007).

This book is based on a set of lecture notes prepared by the author from a graduate course he taught initially in 2000. The main themes are Sobolev spaces and interpolation theory. Brief biographical details of all the mathematicians named in the text as well as appropriate historical remarks and anecdotes are given in footnotes. The author also recalls conversations and correspondence with some of the famous mathematicians he has met in the course of a distinguished career, and includes interesting comments of his, and others, on matters relevant to the subject matter of the book.

The book contains 42 chapters, each intended to contain the amount of material which would be suitable for a graduate lecture. Starting with a brief historical background and introductory chapters on measures, convolutions and distributions, it proceeds to give an excellent coverage of the two main themes. Amongst the topics covered are the following: the Sobolev embedding theorems on \( \mathbb{R}^n\) and on bounded open sets \(\Omega;\) boundary regularity and consequences; extension theorems for sets \(\Omega\) with Lipschitz boundary; traces on the boundary; compactness of embeddings; the Lax–Milgram lemma and applications to boundary-value problems; and spaces of fractional order. On interpolation theory and its applications, the following list gives a flavour of the coverage: the complex method; Peetre’s \(K\) and \(J\) method of real interpolation; the Lions–Peetre reiteration theorem; bilinear and nonlinear interpolation; the Sobolev embedding theorems for Sobolev spaces \(W^{s,p}(\mathbb{R}^n)\) of order \(s \in (0,1)\) and Besov spaces \(B^{s,p}_q(\mathbb{R}^n)\) (the spaces being defined as interpolation spaces); definition and characterization of \(W^{s,p}(\Omega)\) for a bounded open set \(\Omega;\) the space \(BV(\Omega)\) of functions of bounded variation on \(\Omega;\) interpolation spaces as trace spaces; duality and compactness for interpolation spaces.

As well as being an excellent source of material for a graduate course on topics which are of central importance to work on analysis and its applications, this book contains a great deal which will be of interest to the seasoned researcher.

The book contains 42 chapters, each intended to contain the amount of material which would be suitable for a graduate lecture. Starting with a brief historical background and introductory chapters on measures, convolutions and distributions, it proceeds to give an excellent coverage of the two main themes. Amongst the topics covered are the following: the Sobolev embedding theorems on \( \mathbb{R}^n\) and on bounded open sets \(\Omega;\) boundary regularity and consequences; extension theorems for sets \(\Omega\) with Lipschitz boundary; traces on the boundary; compactness of embeddings; the Lax–Milgram lemma and applications to boundary-value problems; and spaces of fractional order. On interpolation theory and its applications, the following list gives a flavour of the coverage: the complex method; Peetre’s \(K\) and \(J\) method of real interpolation; the Lions–Peetre reiteration theorem; bilinear and nonlinear interpolation; the Sobolev embedding theorems for Sobolev spaces \(W^{s,p}(\mathbb{R}^n)\) of order \(s \in (0,1)\) and Besov spaces \(B^{s,p}_q(\mathbb{R}^n)\) (the spaces being defined as interpolation spaces); definition and characterization of \(W^{s,p}(\Omega)\) for a bounded open set \(\Omega;\) the space \(BV(\Omega)\) of functions of bounded variation on \(\Omega;\) interpolation spaces as trace spaces; duality and compactness for interpolation spaces.

As well as being an excellent source of material for a graduate course on topics which are of central importance to work on analysis and its applications, this book contains a great deal which will be of interest to the seasoned researcher.

Reviewer: W. D. Evans (Cardiff)