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?erentialoperatorsonthetorusandmoregeneralLiegroupsinChapters 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
