Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics

This monograph develops a global quantization theory of pseudo-differential operators on compact Lie groups. Traditionally, the theory of pseudo-differential operators was introduced in the Euclidean setting with the aim of tackling a number of important problems in analysis and in the theory of partial differential equations. This also yields a local theory of pseudo-differential operators on manifolds. The present book takes a different approach by using global symmetries of the space which are often available. First, a particular attention is paid to the theory of periodic operators, which are realized in the form of pseudo-differential and Fourier integral operators on the torus. Then, the cases of the unitary group SU(2) and the 3-sphere are analyzed in extensive detail. Finally, the monograph also develops elements of the theory of pseudo-differential operators on general compact Lie groups and homogeneous spaces. The exposition of the book is self-contained and provides the reader with the background material surrounding the theory and needed for working with pseudo-differential operators in different settings. The background section of the book may be used for independent learning of different aspects of analysis and is complemented by numerous examples and exercises.