Advances In Pseudo Differential Operators
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| Author | : Michael Ruzhansky |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 712 |
| Release | : 2009-12-29 |
| Genre | : Mathematics |
| ISBN | : 3764385146 |
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.
| Author | : Xavier Saint Raymond |
| Publisher | : Routledge |
| Total Pages | : 120 |
| Release | : 2018-02-06 |
| Genre | : Mathematics |
| ISBN | : 1351452932 |
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 | : Ryuichi Ashino |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 252 |
| Release | : 2004-08-20 |
| Genre | : Mathematics |
| ISBN | : 9783764371401 |
This volume consists of the plenary lectures and invited talks in the special session on pseudo-differential operators given at the Fourth Congress of the International Society for Analysis, Applications and Computation (ISAAC) held at York University in Toronto, August 11-16, 2003. The theme is to look at pseudo-differential operators in a very general sense and to report recent advances in a broad spectrum of topics, such as pde, quantization, filters and localization operators, modulation spaces, and numerical experiments in wavelet transforms and orthonormal wavelet bases.
| Author | : Ryuichi Ashino |
| Publisher | : Birkhäuser |
| Total Pages | : 236 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3034878400 |
This volume consists of the plenary lectures and invited talks in the special session on pseudo-differential operators given at the Fourth Congress of the International Society for Analysis, Applications and Computation (ISAAC) held at York University in Toronto, August 11-16, 2003. The theme is to look at pseudo-differential operators in a very general sense and to report recent advances in a broad spectrum of topics, such as pde, quantization, filters and localization operators, modulation spaces, and numerical experiments in wavelet transforms and orthonormal wavelet bases.
| Author | : Nicolas Lerner |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 408 |
| Release | : 2011-01-30 |
| Genre | : Mathematics |
| ISBN | : 3764385103 |
This book is devoted to the study of pseudo-di?erential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We have tried here to expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators. The?rstchapter,Basic Notions of Phase Space Analysis,isintroductoryand gives a presentation of very classical classes of pseudo-di?erential operators, along with some basic properties. As an illustration of the power of these methods, we give a proof of propagation of singularities for real-principal type operators (using aprioriestimates,andnotFourierintegraloperators),andweintroducethereader to local solvability problems. That chapter should be useful for a reader, say at the graduate level in analysis, eager to learn some basics on pseudo-di?erential operators. The second chapter, Metrics on the Phase Space begins with a review of symplectic algebra, Wigner functions, quantization formulas, metaplectic group and is intended to set the basic study of the phase space. We move forward to the more general setting of metrics on the phase space, following essentially the basic assumptions of L. H ̈ ormander (Chapter 18 in the book [73]) on this topic.
| Author | : Bert-Wolfgang Schulze |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 294 |
| Release | : 2010-03-01 |
| Genre | : Mathematics |
| ISBN | : 3034601980 |
Consists of the expository paper based on the 6-hour minicourse given by Professor Bert-Wolfgang Schulze, and sixteen papers based on lectures given at the workshop and on invitations.
| Author | : Serge Alinhac |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 178 |
| Release | : 2007 |
| Genre | : Mathematics |
| ISBN | : 0821834541 |
This book presents two essential and apparently unrelated subjects. The first, microlocal analysis and the theory of pseudo-differential operators, is a basic tool in the study of partial differential equations and in analysis on manifolds. The second, the Nash-Moser theorem, continues to be fundamentally important in geometry, dynamical systems and nonlinear PDE. Each of the subjects, which are of interest in their own right as well as for applications, can be learned separately. But the book shows the deep connections between the two themes, particularly in the middle part, which is devoted to Littlewood-Paley theory, dyadic analysis, and the paradifferential calculus and its application to interpolation inequalities. An important feature is the elementary and self-contained character of the text, to which many exercises and an introductory Chapter $0$ with basic material have been added. This makes the book readable by graduate students or researchers from one subject who are interested in becoming familiar with the other. It can also be used as a textbook for a graduate course on nonlinear PDE or geometry.
| Author | : M.A. Shubin |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 296 |
| Release | : 2011-06-28 |
| Genre | : Mathematics |
| ISBN | : 3642565794 |
I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudo-differential operator" but I was unable to check how much the results ofthese two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews.
| Author | : Man-wah Wong |
| Publisher | : World Scientific Publishing Company |
| Total Pages | : 195 |
| Release | : 2014-03-11 |
| Genre | : Mathematics |
| ISBN | : 9814583103 |
The aim of this third edition is to give an accessible and essentially self-contained account of pseudo-differential operators based on the previous edition. New chapters notwithstanding, the elementary and detailed style of earlier editions is maintained in order to appeal to the largest possible group of readers. The focus of this book is on the global theory of elliptic pseudo-differential operators on Lp(Rn).The main prerequisite for a complete understanding of the book is a basic course in functional analysis up to the level of compact operators. It is an ideal introduction for graduate students in mathematics and mathematicians who aspire to do research in pseudo-differential operators and related topics.
| Author | : Hitoshi Kumanogō |
| Publisher | : MIT Press (MA) |
| Total Pages | : 455 |
| Release | : 1982-01 |
| Genre | : Mathematics |
| ISBN | : 9780262110808 |
This self-contained and formal exposition of the theory and applications of pseudo-differential operators is addressed not only to specialists and graduate students but to advanced undergraduates as well. The only prerequisite is a solid background in calculus, with all further preparation for the study of the subject provided by the book's first chapter. This chapter introduces the fundamental concepts of spaces of functions and Fourier transforms, and covers such topics as linear operators, linear functionals, dual spaces, Hilbert spaces, distributions, and oscillatory integrals. The second chapter develops the theory of pseudo-differential operators themselves on the basis of elementary calculus and concepts presented in the opening chapter, while the third chapter extends the theory of Sobolev spaces. The major applications of the theory, most of them the result of work done since 1965, are in the study and solution of linear partial differential equations, which are found in many branches of pure and applied mathematics and are ubiquitous throughout the sciences and technology. The final seven chapters of Pseudo-Differential Operators take up a range of applications, and deal with such problems as hypoellipticity, local solvability, local uniqueness, index theory, elliptic boundary values, complex powers, initial values, well-posedness, the fixed point theorem of Atiyah-Bott-Lefschetz, Fourier integral operators, and propagation of singularities. For this English edition, the last chapter has been greatly extended and appendixes added in order to present the latest developments of the subject. Multiphase Fourier integral operators are applied to initial-value problems, the micro-local theory is developed from the notion of the "wave front set," and the Nirenberg-Treves existence theorem for the solutions of partial differential equations is discussed. The systematic use of the "multiple symbols" introduced by K. O. Friedrichs provides elegant proofs of otherwise lengthy developments. Hitoshi Kumano-Go teaches in the Mathematics Department at Osaka University.
