Theory of function spaces II. (English) Zbl 0763.46025

Triebel, Hans

Monographs in Mathematics. 84. Basel etc.: Birkhäuser Verlag. viii, 370 p. (1992).

The book is concerned with the theory of function spaces \(B_{pq}^ s\) and \(F_{pq}^ s\). Although the book is the second part of the author’s book with the same title [for part I see Zbl 0546.46027)], it is self- contained and it gives a comprehensive treatment of these spaces based mainly on the author’s results obtained in the last few years. The characteristic feature of the book is using local means and local methods with applications to pseudo-differential equations.
The book consists of seven chapters: 1. How to measure smoothness, 2. The spaces \(B_{pq}^ s\) and \(F_{pq}^ s\): Definitions and characterizations, 3. Atoms, oscillations, and distinguished representations, 4. Key theorems, 5. Spaces on domains, 6. Mapping properties of pseudodifferential operators, 7. spaces on Riemannian manifolds and Lie groups.
The first chapter deserves special attention. This chapter is both an outstanding historical survey of function spaces treated in the book and a remarkable survey of rather different techniques developed in the last 50 years. It is shown that all these apparently different methods are only different ways of characterizing the same classes of functions. The book can be best recommended to researchers and advanced students working on functional analysis.

Reviewer: B.Opic (Praha)

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

Keywords:

atoms; mapping properties; spaces on Riemannian manifolds and Lie groups; local means; local methods; pseudo-differential equations; oscillations; distinguished representations