Pseudo-differential operators and symmetries

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This monograph focuses on the development of the theory of pseudo-differential operators on spaces with symmetries, including Euclidean space, the n-torus, compact Lie groups, and compact homogeneous spaces. It is structured in several parts, aiming to present new results while drawing parallels between various approaches to pseudo-differential operators across different spaces. The material is designed to be self-contained, making it accessible for newcomers. However, the diverse spaces necessitate different backgrounds. For instance, operators on Euclidean space in Chapter 2 rely on established Euclidean Fourier analysis, while those on the torus and general Lie groups in Chapters 4 and 10 require knowledge of discrete analysis and representation theory, which are introduced in Chapter 3 and Part III, respectively. Readers interested in pseudo-differential operators on Lie groups will benefit from a solid understanding of representation theory, which is covered in Part III to minimize the need for external resources. Additionally, the foundational material for the theory of pseudo-differential operators on S and SU(2) in Chapter 12 is provided in a self-contained manner in Chapter 11, ensuring immediate applicability.