Hans Triebel, the author of this book, used to amaze the audience in his talks and lectures on selected topics of function spaces quite often with the statement that among his most famous monographs there is one that he has never published. After a short moment of astonishment (which he obviously enjoyed, looking around) the problem was easily resolved: a lot of colleagues use(d) to refer to the book ‘Theory of function spaces I’ which has never existed (in papers this phenomenon almost disappears nowadays thanks to the systematic use of databases, but as far as my experience goes the reference still survives in talks). However, this ‘mistake’ is quite comprehensible in view of the numerous books written by the same author on related subjects, in particular, in view of his later published monographs [Theory of function spaces II. Monographs in Mathematics 84. Basel etc.: Birkhäuser Verlag (1992; Zbl 0763.46025)] and [Theory of function spaces III. Monographs in Mathematics 100. Basel: Birkhäuser (2006; Zbl 1104.46001)]. But when the book ‘Theory of function spaces’ was published in 1983 (at that time separately in the GDR edition [Theory of function spaces. Mathematik und ihre Anwendungen in Physik und Technik, Bd. 38. Leipzig: Akademische Verlagsgesellschaft Geest & Portig K.-G. (1983; Zbl 0546.46028)] and later for Western Europe as [Theory of function spaces. Monographs in Mathematics, Vol. 78. Basel-Boston-Stuttgart: Birkhäuser (1983; Zbl 0546.46027)]), the author was apparently careful enough to hide his intention to continue the series.
Anyway, there is good reason to think of this book as Volume I in a series of (by now) celebrated books on the theory of function spaces treated in a systematic way. It turned out that for many beginners in this part of harmonic analysis this book is still the natural choice to start with. Though its forerunner, Triebel’s monograph [Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library. Vol. 18. Amsterdam-New York-Oxford: North-Holland Publishing Company (1978; Zbl 0387.46032)] already contained a lot of important results for scientists working on PDEs, that book is nowadays more linked with its remarkable contributions to interpolation theory, and less being thought of as the origin of the ‘Theory of function spaces’-series. Also Triebel’s earlier publications [Fourier analysis and function spaces. Selected topics. Teubner-Texte zur Mathematik. Leipzig: BSB B. G. Teubner Verlagsgesellschaft (1977; Zbl 0345.42003)] and [Spaces of Besov-Hardy-Sobolev type. Teubner-Texte zur Mathematik. Leipzig: BSB B. G. Teubner Verlagsgesellschaft (1978; Zbl 0408.46024)] were rather intended for specialists than for beginners in the area. So a comprehensive introduction into the subject was very much welcome when this book appeared. Moreover, the number of monographs devoted especially to the study of function spaces itself was rather limited at that time. So this publication gained a lot of attention and became well known quite soon. A separate Russian edition was published [Theory of function spaces (Teoriya funktsional’nykh prostranst). Transl.from the English. Moskva: Mir (1986; Zbl 0665.46025)] with a considerable appendix written by P.I. Lizorkin, who was one of the most important figures of the Russian school on function spaces at that time. But in spite of its importance the book was soon (and then for decades) out of print, so the present reprint responded to a continuing strong demand. Technically the book is an exact photocopy of the original one with no additions or (layout) modifications. Though not typed in LaT_EXat that time, the quality of the printing artwork is absolutely satisfying up to now.
There is certainly a variety of reasons for the lasting popularity of the text. Without pretending to know or discuss all of them, let me say that the most essential one is surely the great success of the topic itself. There are various concepts to study such function spaces; the most classical one relies on descriptions by differences and derivatives. The presentation in this book is based on the Fourier analytical approach which is then compared with the classical one. Nowadays both these standard approaches are further complemented and enriched by more ‘direct’ (equivalent) characterisations in terms of atoms, molecules, quarks, wavelets etc. But whencesoever one is approaching the subject, the scales of (now generally known as) Besov and Triebel-Lizorkin spaces became an indispensable pillar in the study of PDEs and numerous further applications. Recently the ideas were further developed in different directions, on the one hand pointing towards more and more specific settings adapted to concrete models, on the other hand taking further generalisations and more abstract approaches into account. Both the modification and adaption of well-known spaces and the invention of new ones are actively investigated at the moment. Among the recent monographs dealing with function spaces of the above scales in a rather systematic approach and associated with the book under review, one should certainly mention [Y. S. Han and E. T. Sawyer, “Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces”, Mem.Am.Math.Soc.530 (1994; Zbl 0806.42013)], [D.R. Adams and L.I. Hedberg, Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften 314. Berlin: Springer (1995; Zbl 0834.46021)], [T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations. de Gruyter Series in Nonlinear Analysis and Applications 3. Berlin: de Gruyter (1996; Zbl 0873.35001)], [L.I. Hedberg and Y. Netrusov,“An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation”, Mem.Am.Math.Soc.882 (2007; Zbl 1186.46028)], [W. Yuan, W. Sickel and D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics 2005. Berlin: Springer (2010; Zbl 1207.46002)], in addition to Hans Triebel’s recent books [Fractals and spectra related to Fourier analysis and function spaces. Monographs in Mathematics 91. Basel: Birkhäuser Verlag (1997; Zbl 0898.46030); The structure of functions. Monographs in Mathematics 97. Basel: Birkhäuser (2001; Zbl 0984.46021); Function spaces and wavelets on domains. EMS Tracts in Mathematics 7. Zürich: European Mathematical Society (EMS) (2008; Zbl 1158.46002); Bases in function spaces, sampling, discrepancy, numerical integration. EMS Tracts in Mathematics 11. Zürich: European Mathematical Society (EMS) (2010; Zbl 1202.46002)].
We now add a few comments on the content of the monograph itself seen from a distance of almost thirty years after its publication; for a detailed review written at the time of its appearance we refer to the Zentralblatt review of the original edition in [Zbl 0546.46027].
This book is divided into two parts each consisting of several chapters. Part I, ‘Function spaces and elliptic differential equations’ contains the fundamentals on spaces of Besov type \(B^s_{p,q}\) and Triebel-Lizorkin type \(F^s_{p,q}\) both on domains and on \(\mathbb{R}^n\). Their definition and a comprehensive collection of features with detailed proofs can be found in Chapter 2, the by far longest chapter covering more than half of the whole text. Without devaluating other parts, this can certainly be regarded as the core of the monograph. The quality and clarity of the arguments presented there emerged over the years: they have served many times as proto-types for later, adapted proofs (and still do so). Though the results were partly known before, the author developed a streamlined approach which covers both the (more familiar) Banach space situation as well as its counterpart for the quasi-Banach space setting (which has recently proved to be more important in various applications). This concerns first of all a large number of equivalent characterisations and descriptions (which then pave the way to the identification of different scales, or to verify their diversity). In addition to results on embeddings, multipliers and interpolation special attention is paid to traces and extensions, also in view of the succeeding chapters devoted to function spaces on domains (Chapter 3) and regular elliptic differential operators (Chapter 4). The central Chapter 2 is preceded by an introductory, more outlined chapter on spaces of entire analytic functions which remains of independent interest, but in the present book it seems to be intended to mainly serve as a source of necessary background material. Part II, called by the author rather indeterminately ‘Further types of function spaces’, is of a different structure: A number of parallel or distinct approaches is collected, briefly introduced and discussed, but with few details only. Usually the main outcome is formulated and some ideas of the proofs are given, occasionally complemented by some remarks. Reviewing this from today’s perspective it emerges that these ‘further types of function spaces’ were developed quite differently in extent and time. This is mainly due to the necessity to return to some ideas and expand them in view of up-to-date applications. Surely it also depends on the point of view which model seems to be the most reasonable and promising one that deserves to be studied in detail. As an example one can take the topics of weighted spaces which enjoy a strong renaissance today, whereas spaces of homogeneous type have always attracted a lot of attention; also periodic spaces are far more than just ‘another type’. This list can easily be extended. However, to the author’s excuse (if necessary at all), Triebel admits already in the introduction that ‘the aim of the book (is not) to give an exhaustive treatment of the theory of function spaces in the widest sense of the word’, but the primary interest is to demonstrate ‘the power of Fourier analysis […] in connection with the above-mentioned spaces’.
At the end of the day, this book can still be recommended for people who start their work in function spaces of rather general type and who have some working knowledge in functional analysis, in particular, in the theory of distributions, basic Fourier analysis and (classical) PDEs. One could surely complain that a rather historically-oriented, descriptive access to the topic is missing, but the author himself may have realized it and compensated for this lack in the succeeding volumes II and III ([Zbl 0763.46025] and [Zbl 1104.46001]) with detailed introductory chapters, both entitled ‘How to measure smoothness’ (recommendable both for beginners and specialists). Having this close connection in mind, the present monograph can rightly claim to be considered as the first volume of Triebel’s series of monographs dealing with the ‘Theory of function spaces’ (though volume I never existed).