Triebel, Hans Theory of function spaces II. (English) Zbl 0763.46025 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) Cited in 6 ReviewsCited in 426 Documents 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 Citations:Zbl 0546.46027 PDF BibTeX XML Cite \textit{H. Triebel}, Theory of function spaces II. Basel etc.: Birkhäuser Verlag (1992; Zbl 0763.46025) OpenURL