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Theory of function spaces. (English) Zbl 0546.46027
Monographs in Mathematics, Vol. 78. Basel-Boston-Stuttgart: Birkhäuser Verlag, DM 90.00 (1983).
This book develops a theory of function spaces in which the welll known Besov-Lipschitz, Hölder, Zygmund, Sobolev, Bessel-potential, and Hardy spaces arise as elements of two families \(\{B^ s_{pq}\}\), and \(\{F^ s_{rq}\} (-\infty <s<\infty\), \(0<p\leq\infty \), \(0<q\leq\infty \), \(0<r<\infty)\) of general spaces. \(\{B^ s_{pq}\}\) has been introduced by J. Peetre [C. R. Acad. Sci. Paris Sér. A 264, 281-283 (1967; Zbl 0145.162)] and \(\{F^ s_{rq}\}\) has been introduced by the author [Ark. Mat. 11, 13-64 (1973; Zbl 0255.46026)]. The properties of these spaces are developed in the first part of the book, which is the main part (pp. 1-236). Later on, in the second part (pp. 237-273), the reader gets information about modification like ”homogeneous” counterparts of the ”non homogeneous” spaces \(B^ s_{pq}\) and \(F^ s_{rq}\), as well as weighted and periodic counterparts. Further possibilities are listed in the last chapter of the book. All over the second part detailed proofs are omitted. After an introductory chapter on spaces of entire analytic functions containing the basic inequalities including Fourier multipliers with proofs, the main part of the book begins with chapter two, entitled ’Motivations, principles and historical remarks’. Here, some criteria for ”good spaces” among quasi normed spaces which are continuously embedded in \({\mathcal S}'(R_ n)\) are set up justificating the usefulness of \(B^ s_{pq}\) and \(F^ s_{rq}\). These spaces are firstly defined for functions on the whole \(R_ n\) and the maximal inequality and a Fourier multiplier theorem are proved. Then interpolations, equivalent quasi norms, different representations of the functions in \(B^ s_{pq}\) or \(F^ s_{rq}\), Fourier multipliers, embedding theorems, pointwise multipliers (multiplication operators), restrictions and extensions with respect to half spaces of \(R_ n\), diffeomorphic transformations of variables, and further topics are represented in detail with proofs, motivations, historical notes, and remarks to get a deeper insight. Chapter 3 is concerned with domains \(\Omega\subseteq R_ n\) and the spaces \(B^ s_{pq}\) and \(F^ s_{rq}\) of functions on \(\Omega\). By the ”local coordinate method” the important results from chapter 2 are carried over, Chapter 4, which concludes the main part of the book, leads to modern applications on boundary value problems for regular elliptic differential equations through apriori estimates.
Parts of the results have been earlier published by the author [Fourier analysis and function spaces, Leipzig (1977; Zbl 0345.42003)]; Spaces of Besov-Hardy-Sobolev type, Leipzig (1978; Zbl 0408.46024)]. The present book partly contains revised versions of these former works unifying them with the more recent results. It is written in a concise but well readable style. Since sources and earlier work are thoroughly cited it should be always possible to penetrate through the presented material. This book can be best recommended to researchers and advanced students working on functional analysis or functional analytic methods for partial differential operators or equations.
Reviewer: K.-E.Hellwig

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46E15 Banach spaces of continuous, differentiable or analytic functions
35J40 Boundary value problems for higher-order elliptic equations