Ebook: 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?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.

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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.

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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.

Content:
Front Matter....Pages i-xiv
Introduction....Pages 1-6
Front Matter....Pages 7-7
Sets, Topology and Metrics....Pages 9-78
Elementary Functional Analysis....Pages 79-113
Measure Theory and Integration....Pages 115-189
Algebras....Pages 191-218
Front Matter....Pages 219-220
Fourier Analysis on ? n ....Pages 221-258
Pseudo-differential Operators on ? n ....Pages 259-296
Periodic and Discrete Analysis....Pages 297-331
Pseudo-differential Operators on $ mathbb{T}^n $ ....Pages 333-412
Commutator Characterisation of Pseudo-differential Operators....Pages 413-426
Front Matter....Pages 427-427
Groups....Pages 429-444
Topological Groups....Pages 445-489
Linear Lie Groups....Pages 491-513
Hopf Algebras....Pages 515-526
Front Matter....Pages 527-528
Pseudo-differential Operators on Compact Lie Groups....Pages 529-593
Fourier Analysis on SU(2)....Pages 595-630
Pseudo-differential Operators on SU(2)....Pages 631-665
Pseudo-differential Operators on Homogeneous Spaces....Pages 667-681
Back Matter....Pages 683-709

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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.

Content:
Front Matter....Pages i-xiv
Introduction....Pages 1-6
Front Matter....Pages 7-7
Sets, Topology and Metrics....Pages 9-78
Elementary Functional Analysis....Pages 79-113
Measure Theory and Integration....Pages 115-189
Algebras....Pages 191-218
Front Matter....Pages 219-220
Fourier Analysis on ? n ....Pages 221-258
Pseudo-differential Operators on ? n ....Pages 259-296
Periodic and Discrete Analysis....Pages 297-331
Pseudo-differential Operators on $ mathbb{T}^n $ ....Pages 333-412
Commutator Characterisation of Pseudo-differential Operators....Pages 413-426
Front Matter....Pages 427-427
Groups....Pages 429-444
Topological Groups....Pages 445-489
Linear Lie Groups....Pages 491-513
Hopf Algebras....Pages 515-526
Front Matter....Pages 527-528
Pseudo-differential Operators on Compact Lie Groups....Pages 529-593
Fourier Analysis on SU(2)....Pages 595-630
Pseudo-differential Operators on SU(2)....Pages 631-665
Pseudo-differential Operators on Homogeneous Spaces....Pages 667-681
Back Matter....Pages 683-709
....