Michael V. Ruzhansky & Ville Turunen 
Pseudo-Differential Operators and Symmetries 
Background Analysis and Advanced Topics

This monograph is devoted to the development of the theory of pseudo-di?erential n operators on spaces with symmetries. Such spaces are the Euclidean space R , the n torus T, compact Lie groups and compact homogeneous spaces. The book consists of several parts. One of our aims has been not only to present new results on pseudo-di?erential operators but also to show parallels between di?erent approaches to pseudo-di?erential operators on di?erent spaces. Moreover, we tried to present the material in a self-contained way to make it accessible for readers approaching the material for the ?rst time. However, di?erent spaces on which we develop the theory of pseudo-di?er- tial operators require di?erent backgrounds. Thus, while operators on the – clidean space in Chapter 2 rely on the well-known Euclidean Fourier analysis, pseudo-di?erentialoperatorsonthetorusandmoregeneral Liegroupsin Chapters 4 and 10 require certain backgrounds in discrete analysis and in the representation theory of compact Lie groups, which we therefore present in Chapter 3 and in Part III, respectively. Moreover, anyonewhowishestoworkwithpseudo-di?erential- erators on Lie groups will certainly bene?t from a good grasp of certain aspects of representation theory. That is why we present the main elements of this theory in Part III, thus eliminating the necessity for the reader to consult other sources for most of the time. Similarly, the backgrounds for the theory of pseudo-di?erential 3 operators on S and SU(2) developed in Chapter 12 can be found in Chapter 11 presented in a self-contained way suitable for immediate use.

Table of Content

Preface.- Introduction.- Part I Foundations of Analysis.- A Sets, Topology and Metrics.- B Elementary Functional Analysis.- C Measure Theory and Integration.- D Algebras.- Part II Commutative Symmetries.- 1 Fourier Analysis on Rn.- 2 Pseudo-differential Operators on Rn.- 3 Periodic and Discrete Analysis.- 4 Pseudo-differential Operators on Tn.- 5 Commutator Characterisation of Pseudo-differential Operators.- Part III Representation Theory of Compact Groups.- 6 Groups.- 7 Topological Groups.- 8 Linear Lie Groups.- 9 Hopf Algebras.- Part IV Non-commutative Symmetries.- 10 Pseudo-differential Operators on Compact Lie Groups.- 11 Fourier Analysis on SU(2).- 12 Pseudo-differential Operators on SU(2).- 13 Pseudo-differential Operators on Homogeneous Spaces.- Bibliography.- Notation.- Index.