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Weighted Sobolev spaces. (English) Zbl 0455.46034
Teubner-Texte zur Mathematik, Bd. 31. Leipzig: BSB B. G. Teubner Verlagsgesellschaft. 152 S. M 16.00 (1980).
In this book, the well known Czech author provides an introductory survey on Sobolev spaces with weights, thus deepening one aspect of his, O. John’s, and S. Fučík’s excellent previous book [Function spaces. Prague: Publishing House Czech. Acad. Sci. (1977; Zbl. 364.46022)]. The book under review is divided into three parts: After motivating the usefulness of weighted spaces by means of several examples from the theory of boundary value problems with singularities or degeneracies, unbounded domains, domains with irregu-lar boundary etc., the first (main) chapter deals with power type weights which give already a sufficiently complete insight into typical phenomena of the theory. After that, Chapter II is dedicated to general weights, while the final Chapter III provides some applications involving both power type and general weights.
This book can be recommended without any restriction. By saturating and enriching the text by a great deal of motivating remarks and interesting examples, the author succeeded marvelously in avoiding a pure accumulation of technical results. Moreover, the amount of both results and references remains within a reasonable size such that this book should be read with great pleasure not only by the specialist but also by any student who wants to get for the first time in touch with the theory of weighted spaces. Finally, a further nontrivial advantage lies in its very reasonable price.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations