##
**Theory of function spaces II.
Reprint of the 1992 edition.**
*(English)*
Zbl 1235.46003

Modern Birkhäuser Classics. Basel: Birkhäuser (ISBN 978-3-0346-0418-5/pbk; 978-3-0346-0419-2/ebook). viii, 366 p. (2010).

The present book is a complete reprint of the 1992 original published
by Birkhäuser, see [Theory of function spaces II. Monographs in Mathematics 84. Basel etc.: Birkhäuser Verlag (1992; Zbl 0763.46025)]. It is the second volume in
a (by the appearance of this volume recognisable) series of
monographs written by Hans Triebel on this topic. The author was
at that time already a well-known expert in the field of function
spaces with their main applications (and motivation) in the study of
partial differential equations, using Fourier analytical methods as
well as interpolation techniques. As a quite obvious sign of this
generally acknowledged expertise one may surely take the nowadays
standard habit to associate spaces of type \(F^s_{p,q}\) with the names
of Triebel and Lizorkin to honour their essential contributions to the theory.

Triebel’s book ‘Theory of function spaces’ (see [Theory of function spaces. Monographs in Mathematics, Vol. 78. Basel-Boston-Stuttgart: Birkhäuser (1983; Zbl 0546.46027)] as well as its recent reprint [Theory of function spaces. Reprint of the 1983 original. Modern Birkhäuser Classics. Basel: Birkhäuser (2010; Zbl 1235.46002)]) is often regarded as one of the first comprehensive and most essential sources for people working in the theory of function spaces from a systematic and quite general point of view. There are of course various famous forerunners, highly appreciated and carefully collected and listed in that book (and its predecessors, e.g. [H. Triebel, Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library. Vol. 18. Amsterdam-New York-Oxford: North-Holland Publishing Company (1978; Zbl 0387.46032)]). In this respect the original of the present book is also appreciated for its reliable and extensive list of references covering essentially all the closely linked literature at the time of its appearance. (Though nowadays this reads as a rather basic requirement a scientific monograph should satisfy in general, one should keep in mind that this could not always be expected at the time when the original of the book under review was published; scientific achievements and developments in the East and West were not always completely known or acknowledged on the opposite side. But this difficulty does not concern this text at all.)

We now discuss some consequences of the monograph itself seen from a distance of almost twenty years after its publication; for a detailed review written at the time of its publication we refer to [Zbl 0763.46025]. We focus on some arguments to explain the lasting popularity of the text, not only for historic reasons. Technically the book is an exact photocopy of the original one with no additions or (layout) modifications.

A specialty of this second volume on the theory of function spaces is surely its first chapter ‘How to measure smoothness’ where the author introduces different approaches and explains their historic background and motivation in view of applications in a very clear and comprehensible way. This plan is revived and extended in the succeeding volume [Theory of function spaces III. Monographs in Mathematics 100. Basel: Birkhäuser (2006; Zbl 1104.46001)] starting with a chapter under the same title. Both together can be recommended as a stating point for scientists who begin to work in this area or just as a brief overview to grasp the key ideas of the theory, omitting proofs and further technicalities, but providing references and historic comments.

Apart from this introductory chapter, volume II is nowadays generally considered as the appropriate reference when dealing with ‘direct’ characterisations of Besov spaces \(B^s_{p,q}\) and Triebel-Lizorkin spaces \(F^s_{p,q}\) like local means and atomic decompositions. The celebrated papers of Frazier and Jawerth on atomic characterisations, [M. Frazier and B. Jawerth, “A discrete transform and decompositions of distribution spaces”, J. Funct. Anal. 93, No. 1, 34–170 (1990; Zbl 0716.46031)] and [M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces. Regional Conference Series in Mathematics 79. Providence, RI: American Mathematical Society (1991; Zbl 0757.42006)] had just appeared, so the nowadays unquestioned strength of these direct, local methods and their far-reaching consequences were still little known to the scientific community, but at their breakthrough. Triebel’s central results in this direction, the fundamental Theorems 2.4.1 (for \(F\)-spaces) and 2.5.1 (for \(B\)-spaces) closely rely on his paper [H. Triebel,“Characterizations of Besov-Hardy-Sobolev spaces: A unified approach”, J. Approximation Theory 52, No. 2, 162–203 (1988; Zbl 0644.46017)], which was known to a rather small group of specialists only. However, they turned out to be the basis (in the Triebel approach) for the – by now more famous – characterisations by local means and atomic descriptions. Triebel developed both concepts in a parallel (dual) way; this can be observed even more strongly in his later books where the setting is extended to more general (e.g., weighted, anisotropic, fractal) situations, admitting less smooth building blocks (atoms), too. Today one has further exquisite tools available like wavelet representations, quarkonial decompositions and a variety of particularly developed, perfectly adapted methods to tackle, say, PDEs in ‘nasty’ domains in a very efficient way. But still the rather qualitatively (i.e., less explicitly) described atoms or local means have their advantages in many respects, dealing with traces and extensions, multipliers and diffeomorphisms, as exemplified in Chapter 4, as well as in the study of pseudodifferential operators, presented in Chapter 6. Moreover, the connections between several approximation procedures, descriptions by differences as well as atomic characterisations still lead to many interesting papers and books, see, for instance, [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)] in addition to Triebel’s Theory of function spaces III cited above.

There are at least two further topics considered in this monograph which gained some special importance later on: There is a longer part devoted to spaces on Riemannian manifolds and Lie groups, Chapter 7, which became the starting point for many investigations and developments of spaces on manifolds, concerning also atomic decompositions, embedding results and further observations.

Finally, the rather short section on Morrey-Campanato spaces as it can be found in Chapter 5 quite recently experiences a remarkable renaissance: it emerged that the study of the Navier-Stokes equation benefits a lot from a more detailed knowledge of spaces of Morrey type. This close link has recently led to a systematic approach to spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type, including also a comparison with the quite old Morrey-Campanato approach. The new results are surprisingly sharp and promising, which might explain that Morrey type spaces enjoy a lot of attention at the moment. One of the so far latest monographs dealing with this subject is [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)]. This is an exciting part of harmonic analysis attractive for a lot of very active groups working in this field and – most likely – keeping the (reprint of the) present book on a prominent place on their bookshelves.

Triebel’s book ‘Theory of function spaces’ (see [Theory of function spaces. Monographs in Mathematics, Vol. 78. Basel-Boston-Stuttgart: Birkhäuser (1983; Zbl 0546.46027)] as well as its recent reprint [Theory of function spaces. Reprint of the 1983 original. Modern Birkhäuser Classics. Basel: Birkhäuser (2010; Zbl 1235.46002)]) is often regarded as one of the first comprehensive and most essential sources for people working in the theory of function spaces from a systematic and quite general point of view. There are of course various famous forerunners, highly appreciated and carefully collected and listed in that book (and its predecessors, e.g. [H. Triebel, Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library. Vol. 18. Amsterdam-New York-Oxford: North-Holland Publishing Company (1978; Zbl 0387.46032)]). In this respect the original of the present book is also appreciated for its reliable and extensive list of references covering essentially all the closely linked literature at the time of its appearance. (Though nowadays this reads as a rather basic requirement a scientific monograph should satisfy in general, one should keep in mind that this could not always be expected at the time when the original of the book under review was published; scientific achievements and developments in the East and West were not always completely known or acknowledged on the opposite side. But this difficulty does not concern this text at all.)

We now discuss some consequences of the monograph itself seen from a distance of almost twenty years after its publication; for a detailed review written at the time of its publication we refer to [Zbl 0763.46025]. We focus on some arguments to explain the lasting popularity of the text, not only for historic reasons. Technically the book is an exact photocopy of the original one with no additions or (layout) modifications.

A specialty of this second volume on the theory of function spaces is surely its first chapter ‘How to measure smoothness’ where the author introduces different approaches and explains their historic background and motivation in view of applications in a very clear and comprehensible way. This plan is revived and extended in the succeeding volume [Theory of function spaces III. Monographs in Mathematics 100. Basel: Birkhäuser (2006; Zbl 1104.46001)] starting with a chapter under the same title. Both together can be recommended as a stating point for scientists who begin to work in this area or just as a brief overview to grasp the key ideas of the theory, omitting proofs and further technicalities, but providing references and historic comments.

Apart from this introductory chapter, volume II is nowadays generally considered as the appropriate reference when dealing with ‘direct’ characterisations of Besov spaces \(B^s_{p,q}\) and Triebel-Lizorkin spaces \(F^s_{p,q}\) like local means and atomic decompositions. The celebrated papers of Frazier and Jawerth on atomic characterisations, [M. Frazier and B. Jawerth, “A discrete transform and decompositions of distribution spaces”, J. Funct. Anal. 93, No. 1, 34–170 (1990; Zbl 0716.46031)] and [M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces. Regional Conference Series in Mathematics 79. Providence, RI: American Mathematical Society (1991; Zbl 0757.42006)] had just appeared, so the nowadays unquestioned strength of these direct, local methods and their far-reaching consequences were still little known to the scientific community, but at their breakthrough. Triebel’s central results in this direction, the fundamental Theorems 2.4.1 (for \(F\)-spaces) and 2.5.1 (for \(B\)-spaces) closely rely on his paper [H. Triebel,“Characterizations of Besov-Hardy-Sobolev spaces: A unified approach”, J. Approximation Theory 52, No. 2, 162–203 (1988; Zbl 0644.46017)], which was known to a rather small group of specialists only. However, they turned out to be the basis (in the Triebel approach) for the – by now more famous – characterisations by local means and atomic descriptions. Triebel developed both concepts in a parallel (dual) way; this can be observed even more strongly in his later books where the setting is extended to more general (e.g., weighted, anisotropic, fractal) situations, admitting less smooth building blocks (atoms), too. Today one has further exquisite tools available like wavelet representations, quarkonial decompositions and a variety of particularly developed, perfectly adapted methods to tackle, say, PDEs in ‘nasty’ domains in a very efficient way. But still the rather qualitatively (i.e., less explicitly) described atoms or local means have their advantages in many respects, dealing with traces and extensions, multipliers and diffeomorphisms, as exemplified in Chapter 4, as well as in the study of pseudodifferential operators, presented in Chapter 6. Moreover, the connections between several approximation procedures, descriptions by differences as well as atomic characterisations still lead to many interesting papers and books, see, for instance, [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)] in addition to Triebel’s Theory of function spaces III cited above.

There are at least two further topics considered in this monograph which gained some special importance later on: There is a longer part devoted to spaces on Riemannian manifolds and Lie groups, Chapter 7, which became the starting point for many investigations and developments of spaces on manifolds, concerning also atomic decompositions, embedding results and further observations.

Finally, the rather short section on Morrey-Campanato spaces as it can be found in Chapter 5 quite recently experiences a remarkable renaissance: it emerged that the study of the Navier-Stokes equation benefits a lot from a more detailed knowledge of spaces of Morrey type. This close link has recently led to a systematic approach to spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type, including also a comparison with the quite old Morrey-Campanato approach. The new results are surprisingly sharp and promising, which might explain that Morrey type spaces enjoy a lot of attention at the moment. One of the so far latest monographs dealing with this subject is [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)]. This is an exciting part of harmonic analysis attractive for a lot of very active groups working in this field and – most likely – keeping the (reprint of the) present book on a prominent place on their bookshelves.

Reviewer: Dorothee Haroske (Jena)

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

42B35 | Function spaces arising in harmonic analysis |

01A75 | Collected or selected works; reprintings or translations of classics |