Vladimir Maz'ya is one of the foremost authorities in the subject of Sobolev spaces. However, many of his papers are published in Soviet publications with very limited circulation. As a result he has sometimes suffered the misfortune of seeing his results rediscovered or attributed to others. With the appearance of the book under review this state of affairs should belong to the past.
In the present book the author has collected and rewritten the results of many years of research by himself and his collaborators. His has added introductory material and results due to others, but most of the contents of the book are due to the author himself. Naturally there is a very little overlap with other existing books on Sobolev spaces.
Generally speaking, this book is not the right choice for someone who is just trying to learn a few simple facts about Sobolev spaces. The author's taste is for completeness. He treats every conceivable aspect of his problems, which make the book rather overwhelming for the general reader.
On the other hand, this makes the book all the more valuable as a work of reference. It is a treasure house, for example, for someone who is looking for a weird domain as a counterexample to some theorem, and for many others. Every good library should have it.
(From the review of L. I. Hedberg)
The book is written for the expert, and a pretty determined one at that. On first inspection it looks like a rather dreary catalogue of hard technical facts. However the reader who is prepared to roll up his sleeves and expend a lot of time and energy on mastering the intricate analysis will be amply rewarded for the effort. For it is a treasure-house of important results produced by a master of his craft. It should best be thought of as an encyclopedia of results and techniques. As such, it is to be recommended as an indispensable tool for anyone with interests in function spaces, inequalities and differential equations.
(from the review of W. D. Evans)