In the 19th century, the Fourier transformation was introduced to study various problems of partial differential equations. Since 1960, this old tool has been developed into a well-organized theory called microlocal analysis that is based on the concept of the pseudo-differential operator. This book provides the fundamental knowledge non-specialists need in order to use microlocal analysis. It is strictly mathematical in the sense that it contains precise definitions, statements of theorems and complete proofs, and follows the usual method of pure mathematics. The book explains the origin of the theory (i.e., Fourier transformation), presents an elementary construcion of distribution theory, and features a careful exposition of standard pseudodifferential theory. Exercises, historical notes, and bibliographical references are included to round out this essential book for mathematics students; engineers, physicists, and mathematicians who use partial differential equations; and advanced mathematics instructors.
Author(s): Xavier Saint Raymond
Series: Studies in Advanced Mathematics
Publisher: CRC
Year: 1991
Language: English
Pages: 118
Front Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 8
Introduction......Page 10
1.1 Functions in R^n......Page 11
1.2 Fourier transformation and distributions in R^n......Page 18
1.3 Sobolev spaces......Page 26
Exercises......Page 32
Notes on Chapter I......Page 36
Introduction to Chapters 2 and 3......Page 37
2.1 Definition and approximation of symbols......Page 38
2.2 Oscillatory integrals......Page 41
2.3 Operations on symbols......Page 46
Exercises......Page 52
3.1 Action in S and S'......Page 56
3.2 Action in Sobolev spaces......Page 61
3.3 Invariance under a change of variables......Page 67
Exercises......Page 70
Notes on Chapters 2 and 3......Page 76
Introduction......Page 78
4.1 Local solvability of linear differential operators......Page 79
4.2 Wave front sets of solutions of partial differential equations......Page 85
4.3 The Cauchy problem for the wave equation......Page 92
Exercises......Page 98
Notes on Chapter 4......Page 103
Bibliography......Page 106
Index of Notation......Page 112
Index......Page 116
