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Pseudo-Differential Operators and Symmetries

Background Analysis and Advanced Topics

  • Textbook
  • © 2010

Overview

Authors:
  1. Michael Ruzhansky

    1. Department of Mathematics, Imperial College London, London, UK

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    You can also search for this author in PubMed   Google Scholar

  2. Ville Turunen

    1. Institute of Mathematics, Helsinki University of Technology, Finland

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    You can also search for this author in PubMed   Google Scholar

  • Provides an easy-to-read introduction to the theory of pseudo-differential operators and its more advanced aspects
  • Contains a collection of necessary analytic tools which makes it excellent as a textbook on the subject
  • Uniquely combines different parts of analysis and representation theory in a simple accessible form. Usually these tools are scattered over different sources
  • Includes supplementary material: sn.pub/extras

Part of the book series: Pseudo-Differential Operators (PDO, volume 2)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Introduction

    1. Introduction

      • Michael Ruzhansky, Ville Turunen
      Pages 1-6

  3. Foundations of Analysis

    1. Front Matter

      Pages 7-7
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    2. Sets, Topology and Metrics

      • Michael Ruzhansky, Ville Turunen
      Pages 9-78

    3. Elementary Functional Analysis

      • Michael Ruzhansky, Ville Turunen
      Pages 79-113

    4. Measure Theory and Integration

      • Michael Ruzhansky, Ville Turunen
      Pages 115-189

    5. Algebras

      • Michael Ruzhansky, Ville Turunen
      Pages 191-218

  4. Commutative Symmetries

    1. Front Matter

      Pages 219-220
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    2. Fourier Analysis on ℝn

      • Michael Ruzhansky, Ville Turunen
      Pages 221-258

    3. Pseudo-differential Operators on ℝn

      • Michael Ruzhansky, Ville Turunen
      Pages 259-296

    4. Periodic and Discrete Analysis

      • Michael Ruzhansky, Ville Turunen
      Pages 297-331

    5. Pseudo-differential Operators on \( \mathbb{T}^n \)

      • Michael Ruzhansky, Ville Turunen
      Pages 333-412

    6. Commutator Characterisation of Pseudo-differential Operators

      • Michael Ruzhansky, Ville Turunen
      Pages 413-426

  5. Representation Theory of Compact Groups

    1. Front Matter

      Pages 427-427
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    2. Groups

      • Michael Ruzhansky, Ville Turunen
      Pages 429-444

    3. Topological Groups

      • Michael Ruzhansky, Ville Turunen
      Pages 445-489

    4. Linear Lie Groups

      • Michael Ruzhansky, Ville Turunen
      Pages 491-513

    5. Hopf Algebras

      • Michael Ruzhansky, Ville Turunen
      Pages 515-526

  6. Non-commutative Symmetries

    1. Front Matter

      Pages 527-528
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    2. Pseudo-differential Operators on Compact Lie Groups

      • Michael Ruzhansky, Ville Turunen
      Pages 529-593
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Keywords

About this book

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.

Reviews

From the reviews:

“This book presents recent results on the global theory of pseudo-differential operators on compact Lie groups along with a bulk of background material. … this very welcomed book is a profitable read for specialists and—due to the included background material, which makes it self-contained—also for graduate students.” (R. Steinbauer, Monatshefte für Mathematik, 2013)

“The present book is devoted to the theory of pseudodifferential operators on spaces with symmetries … . Summing up, this welcome book is nicely written and looks very appealing for researchers in the fields of Euclidean and abstract harmonic analysis and pseudodifferential operators, as well as for any Ph.D. student interested in those topics.” (Fabio Nicola, Mathematical Reviews, Issue 2011 b)

“The book is devoted to a global theory of pseudo-differential operators on compact Lie groups. … In the whole, including background material and new results about operators on Lie groups, the book offers a pleasant and profitable reading to graduate students and specialists.” (Luigi Rodino, Zentralblatt MATH, Vol. 1193, 2010)

Authors and Affiliations

  • Department of Mathematics, Imperial College London, London, UK

    Michael Ruzhansky

  • Institute of Mathematics, Helsinki University of Technology, Finland

    Ville Turunen

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