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Sobolev spaces on domains. (English) Zbl 0893.46024
Teubner-Texte zur Mathematik. 137. Stuttgart: B. G. Teubner. 312 p. (1998).
The book is devoted to the central questions in the study of function spaces – approximation by nicer functions, integral representations, embedding and trace, as well as extension type theorems. Besides, specific questions appear due to noncompactness of considered domains. The main tool is the original mollification-type approach developed by the author in a series of previous papers. The subject of the book is of high importance for the study of differential and integral equations and for their applications.
The chosen style of the material’s presentation combines carefulness of lecture notes for postgraduate students and deep analysis of research monographs. The book is in certain sense self-contained and needs to have only a low level of preliminary knowledge. It is divided into 7 chapters devoted to solving problems of different type.
Chapter 1 has an auxiliary character. It contains the main (fairly standard) definitions and notations surrounding the “Sobolev-spaces area”. Some basic properties of introduced objects are also under discussion.
Chapter 2 (“Approximation by infinitely differentiable functions”) is a conceptual one. The basic tool for the study of Sobolev-type spaces on domains is developed here, namely, mollification, in particular with variable steps. It is applied then to the question of approximation by infinitely differentiable functions. Among the results are those concerning best approximation type among the ones preserving boundary behaviour.
Chapter 3 (“Sobolev’s integral representation”) develops the ideas by S. L. Sobolev to present values of the considered functions in the form of integral of their derivatives multiplied by specially constructed kernels. The author is working in the direction of optimal determination of the kernel. Of course it leeds to certain restrictions on the geometry of domains. Anyway this chapter gives sufficient preparative material for its further applications into extension theorems. Useful are also the constructed Taylor formula and potential type representations.
In Chapter 4 (“Embedding theorems”) the main ideas about embedding on the Sobolev-type spaces for domains are described. The author starts with rather standard results on the one-dimensional case in order to show how geometry of domains begins to play an important role in higher dimensions. Due to the extreme applicability of embedding theorems they are discussed in great detail.
In the following chapter 5 (“Trace theorems”) results which are of opposite type to those from the previous chapter are presented. The traces on subspaces as well as on smooth surfaces of smaller dimensions are considered.
The author constructs and investigates (Chapter 6 “Extension theorems”) an extension operator with minimal norm. It is accompanied by careful analysis of the geometry of the boundary of domains and by a number of two-sided norm inequalities derived.
Chapter 7 (“Comments”) contains historical remarks and bibliographic notes.

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
31B10 Integral representations, integral operators, integral equations methods in higher dimensions