Solvable Algebras of Pseudodifferential Operators 303128397X, 9783031283970

Table of contents :
Contents
Introduction
1 Preliminaries
1.1 Pseudodifferential Operators
1.1.1 Amplitudes
1.1.2 Pseudodifferential Operators
1.1.3 The Kernel of a Pseudodifferential Operator
1.1.4 Smoothing Operators
1.1.5 Properly Supported Pseudodifferential Operators
1.1.6 Pseudodifferential Operators in Generalized Function Spaces
1.1.7 Symbols
1.1.8 Asymptotic Expansions in the Classes Sμ(Ω)
1.1.9 The Symbol of Proper Pseudodifferential Operator
1.1.10 Symbolic Calculus of Pseudodifferential Operators
1.1.11 Change of Variables in Pseudodifferential Operators
1.1.12 Classes bμ(Rn)
1.1.13 The Boundedness of DO in L2(Rn)
1.1.14 DO in Sobolev Spaces
1.1.15 Elliptic Pseudodifferential Operators
1.2 Meromorphic Pseudodifferential Operators
1.2.1 Integral Transforms on a Sphere
1.2.2 Canonical Meromorphic Pseudodifferential Operators
1.2.3 The Kernel of a Canonical Pseudodifferential Operator
1.2.4 Operations on Canonical Meromorphic Pseudodifferential Operators
1.2.5 General Meromorphic Pseudodifferential Operators
1.2.6 Change of Variables in Meromorphic Pseudodifferential Operators
1.3 C*-algebras
1.3.1 C*-algebras and Their Morphisms
1.3.2 Representations of C*-algebras
1.3.3 Spectrum of C*-algebra
1.3.4 Criteria for an Element of an Algebra to Be Invertible or to Be Fredholm
1.3.5 Continuous Field of C*-algebras
1.3.6 A Sufficient Triviality Condition for the Fields of Elementary Algebras
1.3.7 Solvable Algebras
1.3.8 Maximal Radical Series
1.3.9 The Localization Principle
2 C*-algebras of Pseudodifferential Operators on Smooth Manifolds with Discontinuities in Symbols Along a Submanifold
2.1 Algebras Generated by Pseudodifferential Operators with Smooth Symbols
2.2 Algebras of Pseudodifferential Operators with Isolated Singularities in Symbols
2.2.1 Algebras A and S
2.2.2 Proof of the Inclusion C0(R)K L2(Sn-1)S
2.2.3 Auxiliary Results
2.2.4 The Spectrum of Algebra S
2.2.5 The Spectrum of Algebra A
2.3 Algebras of Pseudodifferential Operators with Discontinuities in Symbols Along a Submanifold
2.3.1 The Statement of Basic Theorems
2.3.2 Algebras L(θ): Irreducibility
2.3.3 Localization in the Algebra L (θ)
2.3.4 The Spectrum of Algebra L(θ)
2.3.5 The Spectrum of the Algebra of Pseudodifferential Operators with Symbols Discontinuous Along a Submanifold
3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols on a Smooth Manifold
3.1 Algebra A and Its Irreducible Representations
3.1.1 Stratification of Manifold M. Algebra A
3.1.2 The Irreducible Representations of the Algebra A (Formulation of a Theorem)
3.1.3 Proof of Theorem 3.1.4
3.2 The Spectral Topology of Algebra A
3.2.1 Description of the Jacobson Topology (Formulation of the Theorem)
3.2.2 Proof of Theorem 3.2.1
3.3 Solving Series
3.3.1 Construction of a Solving Series. Formulation of the Theorem
3.3.2 Proof of Theorem 3.3.1
Verification of Formulas (3.3.23)
4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges
4.1 Pseudodifferential Operators in Rmn
4.1.1 Amplitudes
4.1.2 Pseudodifferential Operators
4.1.3 Symbols
4.1.4 Composition of do: Adjoint Operator
4.1.5 Conditions for do to Belong to Classes -∞0 and -∞
4.1.6 Elliptic do
4.2 Operators on Manifolds with Wedges
4.2.1 Admissible Diffeomorphisms of Subsets of Rmn
4.2.2 Change of Variables in do
4.2.3 Pseudodifferential Operators on a Wedge
4.2.4 W-manifolds
4.2.5 do on w-manifold
4.3 Pseudodifferential Operators in Weighted Spaces
4.3.1 Boundedness of Proper do of Non-positive Order in the Spaces L2, τ
4.3.2 Pseudodifferential Operators in the Spaces Hsτ
4.3.3 Pseudodifferential Operators on Spaces with Weighted Norms on w-Manifolds
5 C*-Algebra of Pseudodifferential Operators on Manifold with Edges
5.1 Classes Ψμ
5.2 C*-Algebra Generated by Proper do. Local Algebras
5.3 Algebras L (θ, Rn) and L (0, Rn)
5.3.1 A Special Representation of Generators of L (0, Rn )
5.3.2 Coincidence of Algebras L(θ, Rn) and L0(θ)
5.4 Localization in L (θ, K)
5.5 Localization in L(0, K)
5.6 Invariant Description of Local Algebras
5.7 The Spectrum of C*-Algebra of Pseudodifferential Operators on Manifold with Edges
Bibliographical Sketch
References

Citation preview

Pseudo-Differential Operators Volume 15

Associate Editors Guozhen Lu, University of Connecticut, Storrs, CT, USA Alberto Parmeggiani, Università di Bologna, Bologna, Italy Luigi G. Rodino, Università di Torino, Torino, Italy Bert-Wolfgang Schulze, Universität Potsdam, Potsdam, Germany Johannes Sjöstrand, Université de Bourgogne, Dijon, France Sundaram Thangavelu, Indian Institute of Science at Bangalore, Bangalore, India Maciej Zworski, University of California at Berkeley, Berkeley, CA, USA Managing Editor M. W. Wong, York University, Toronto, Canada

Pseudo-Differential Operators: Theory and Applications is a series of moderately priced graduate-level textbooks and monographs appealing to students and experts alike. Pseudodifferential operators are understood in a very broad sense and include such topics as harmonic analysis, PDE, geometry, mathematical physics, microlocal analysis, timefrequency analysis, imaging and computations. Modern trends and novel applications in mathematics, natural sciences, medicine, scientific computing, and engineering are highlighted.

Boris Plamenevskii • Oleg Sarafanov

Solvable Algebras of Pseudodifferential Operators

Boris Plamenevskii Department of Higher Mathematics and Mathematical Physics St. Petersburg State University St. Petersburg, Russia

Oleg Sarafanov Department of Higher Mathematics and Mathematical Physics St. Petersburg State University St. Petersburg, Russia

ISSN 2297-0355 ISSN 2297-0363 (electronic) Pseudo-Differential Operators ISBN 978-3-031-28397-0 ISBN 978-3-031-28398-7 (eBook) https://doi.org/10.1007/978-3-031-28398-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Pseudodifferential Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 The Kernel of a Pseudodifferential Operator. . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Smoothing Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Properly Supported Pseudodifferential Operators . . . . . . . . . . . . . . . . . . 1.1.6 Pseudodifferential Operators in Generalized Function Spaces . . . . . 1.1.7 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.8 Asymptotic Expansions in the Classes S μ () . . . . . . . . . . . . . . . . . . . . . . 1.1.9 The Symbol of Proper Pseudodifferential Operator . . . . . . . . . . . . . . . . 1.1.10 Symbolic Calculus of Pseudodifferential Operators . . . . . . . . . . . . . . . . 1.1.11 Change of Variables in Pseudodifferential Operators. . . . . . . . . . . . . . . μ 1.1.12 Classes b (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.13 The Boundedness of DO in L2 (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.14 DO in Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.15 Elliptic Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Meromorphic Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Integral Transforms on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Canonical Meromorphic Pseudodifferential Operators . . . . . . . . . . . . . 1.2.3 The Kernel of a Canonical Pseudodifferential Operator . . . . . . . . . . . . 1.2.4 Operations on Canonical Meromorphic Pseudodifferential Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 General Meromorphic Pseudodifferential Operators . . . . . . . . . . . . . . . 1.2.6 Change of Variables in Meromorphic Pseudodifferential Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 C ∗ -algebras and Their Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Representations of C ∗ -algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Spectrum of C ∗ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 4 9 12 13 17 18 19 21 26 29 32 35 37 38 40 40 43 46 49 51 52 53 53 54 55 v

vi

Contents

1.3.4 1.3.5 1.3.6 1.3.7 1.3.8 1.3.9

Criteria for an Element of an Algebra to Be Invertible or to Be Fredholm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Field of C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Sufficient Triviality Condition for the Fields of Elementary Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solvable Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximal Radical Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Localization Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 58 60 61 63 64

C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with Discontinuities in Symbols Along a Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.1 Algebras Generated by Pseudodifferential Operators with Smooth Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.2 Algebras of Pseudodifferential Operators with Isolated Singularities in Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Algebras A and S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.2.1 2.2.2 Proof of the Inclusion C0 (R) ⊗ KL2 (S n−1 ) ⊂ S . . . . . . . . . . . . . . . . . . 73 2.2.3 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.2.4 The Spectrum of Algebra S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.2.5 The Spectrum of Algebra A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.3 Algebras of Pseudodifferential Operators with Discontinuities in Symbols Along a Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.3.1 The Statement of Basic Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.3.2 Algebras L(θ): Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.3.3 Localization in the Algebra L(θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.3.4 The Spectrum of Algebra L(θ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2.3.5 The Spectrum of the Algebra of Pseudodifferential Operators with Symbols Discontinuous Along a Submanifold . . . . 116

2

.

3

Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols on a Smooth Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Algebra A and Its Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Stratification of Manifold M. Algebra A . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 The Irreducible Representations of the Algebra A (Formulation of a Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Proof of Theorem 3.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Spectral Topology of Algebra A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Description of the Jacobson Topology (Formulation of the Theorem). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Proof of Theorem 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 123 125 129 139 139 142

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vii

3.3 Solving Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 3.3.1 Construction of a Solving Series. Formulation of the Theorem. . . . 158 3.3.2 Proof of Theorem 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4

5

Pseudodifferential Operators on Manifolds with Smooth Closed Edges . . . . . 4.1 Pseudodifferential Operators in Rm n ........................................... 4.1.1 Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Pseudodifferential Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Composition of do: Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Conditions for do to Belong to Classes 0−∞ and  −∞ . . . . . . . . . 4.1.6 Elliptic do . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Operators on Manifolds with Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Admissible Diffeomorphisms of Subsets of Rm n .................... 4.2.2 Change of Variables in do . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Pseudodifferential Operators on a Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 W-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 do on w-manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Pseudodifferential Operators in Weighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Boundedness of Proper do of Non-positive Order in the Spaces L2,τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Pseudodifferential Operators in the Spaces Hτs . . . . . . . . . . . . . . . . . . . . . 4.3.3 Pseudodifferential Operators on Spaces with Weighted Norms on w-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 168 168 169 172 177 180 183 185 185 187 192 195 197 200

C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges . . . . . . . 5.1 Classes  μ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 C ∗ -Algebra Generated by Proper do. Local Algebras . . . . . . . . . . . . . . . . . . . . . 5.3 Algebras L(θ, Rn ) and L(0, Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 A Special Representation of Generators of L(0, Rn ) . . . . . . . . . . . . . . . 5.3.2 Coincidence of Algebras L(θ, Rn ) and L0 (θ ) . . . . . . . . . . . . . . . . . . . . . . 5.4 Localization in L(θ, K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Localization in L(0, K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Invariant Description of Local Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 The Spectrum of C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209 209 213 216 217 222 226 228 229

.

200 202 206

234

Bibliographical Sketch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Introduction

Pseudodifferential operators (.DO) are widely used in the theory of partial differential equations, mathematical physics, functional analysis, and topology. Several monographs [9, 15, 18, 39, 42, 43] are devoted to such operators in the smooth situation (i.e., on smooth manifolds and with smooth symbols). For a long time, “one-dimensional” singular integral operators with discontinuous coefficients on composite contours have been playing an important role in studying the boundary value problems on plain domains with non-smooth boundary. These operators have been discussed from different points of view in [10,12,20], etc. At present, there are also several monographs (for example, [17, 21, 35, 36]) dealing with the .DO on manifolds of dimension .n ≥ 2 with singularities. This monograph has a few intersections with [21]; it has no essential intersections with other books. To read this monograph, it is desirable to have some familiarity with the smooth theory of .DO and with .C ∗ -algebras. However, in the first introductory chapter, some necessary preliminaries are given. Now let us explain the subject of the book. In what follows, algebra and morphism mean ∗ .C -algebra and .∗-morphism, .BH denotes the algebra of all bounded operators in a Hilbert  of algebra .A space H , and .KH is the ideal of all compact operators in H . The spectrum .A is the set of all (equivalence classes of) irreducible representations of this algebra endowed with a natural topology (the so-called Jacobson topology). Let .M be a smooth compact manifold without boundary and let .A be the algebra generated in .L2 (M) by scalar .DO with smooth symbols. We assume that the principal symbols of such .DO are defined on the bundle of non-zero cotangent vectors, are homogeneous functions of degree zero on every fiber, and belong to the class .C ∞ (S ∗ (M)), where .S ∗ (M) is the cospherical bundle. As is known, the algebra .A contains the ideal .KL2 (M) and the quotient algebra ∧ (i.e., the space of maximal .A/KL2 (M) is commutative. The spectrum .(A/KL2 (M)) ∗ ideals) can be identified with the bundle .S (M). If .P ∈ A and . is the principal symbol of the operator P , then the map (A/KL2 (M))∧ π → P (π ) := π[P ]

.

is implemented as the function .π → (π ) ∈ C; here, .[P ] is the equivalence class of the operator P in the algebra .A/KL2 (M). The operator P is Fredholm if and only if ix

x

Introduction

π [P ] = 0 for all .π ∈ (A/KL2 (M))∧ . (An operator .A ∈ BH is called Fredholm if its range .R(A) is closed and the spaces .kerA and .cokerA are finite-dimensional.) If the manifold .M or the symbols of .DO have singularities, then the quotient algebra .A/KL2 (M) is non-commutative. Among its irreducible representations, there are infinitedimensional ones (as .dim M > 1). For .A ∈ A, the map .

(A/KL2 (M))∧ π → A(π ) := π [A]

.

is an analogue of the principal symbol. In contrast to the commutative case, not only can scalars serve as .π[A] but also non-trivial operators in infinite-dimensional Hilbert spaces. The “principal symbol” still gives a criterion for an operator to be Fredholm: .A ∈ A is Fredholm if and only if any operator .A(π ) = π[A] is invertible as .π ∈ (A/KL2 (M))∧ . Thus, to apply this criterion, one must list all (equivalence classes of) irreducible representations of the quotient algebra .A/KL2 (M) and to find implementations of these representations. One of this book’s purposes is to describe the spectra of algebras generated by .DO with discontinuous symbols (coefficients) on piecewise smooth manifolds. For the algebra generated by .DO with smooth symbols on a smooth manifold, the following relations hold: {0} ⊂ KL2 (M) ⊂ A,

.

A/KL2 (M) C(S ∗ (M)).

In the general situation, one can try to simplify the study of the (non-commutative) quotient algebra .A/KL2 (M) by taking the additional quotient modulo some ideal .J ⊃ KL2 (M). The resulting loss of information is not large if the algebra .J /KL2 (M) is comparatively simple. Definition An algebra .L is called solvable if there is a composition series .{0} = I−1 ⊂ I0 ⊂ · · · ⊂ IN = L of ideals .Ij , such that the successive quotients .Ij /Ij −1 consist of continuous operator-functions with compact values; more precisely, if there is an isomorphism Ij /Ij −1 C0 (Xj ) ⊗ KHj ,

.

j = 0, . . . , N < ∞,

where .Hj is a Hilbert space, .Xj is a locally compact Hausdorff space, and .C0 (Xj ) is the algebra of continuous functions on .Xj tending to zero at infinity. A composition series possessing such property is called solving, and the number N is called the length of the solving composition series. Thus, in the smooth situation, the series .{0} ⊂ KL2 (M) ⊂ A is solving, and its length is 1. Let, for example, the algebra .A be generated by .DO with smooth symbols on a smooth manifold and by operators of multiplication by functions a continuous everywhere except at a fixed point .x0 . At that point, the coefficients may have discontinuities “of the

Introduction

xi

first kind” (there exists .lim a(x) as .x → x0 that depends on the direction in which x approaches .x0 ). The ideal .com A spanned by the commutators of the elements in .A is larger than .KL2 (M). The algebra .A is solvable, and the (shortest) solving series is {0} ⊂ KL2 (M) ⊂ com A ⊂ A,

.

while .

com A/KL2 (M) C0 (R) ⊗ KL2 (S m−1 ), A/com A C(S ∗ (C)),

where .C is obtained by gluing the boundary (i.e., a sphere) to the manifold .M \ x0 . A solving composition series allows us to determine the collection of symbols .σ1 (A), . . . , .σN (A) for an operator A in a solvable algebra .L. The symbol .σj (A) is an operator-valued function on the space .Xj . The invertibility of .σj (A), . . . , σN (A) at each point is necessary and sufficient for the invertibility of A up to a summand in the ideal .Ij −1 . Composition series can be useful for studying the groups .K∗ (L) of the operator K-theory related to the algebra .L. In this book, we present solving series for considered algebras; other questions mentioned in this paragraph are not discussed. In the first part of the book, we study the algebras generated by pseudodifferential operators on a smooth manifold with continuous symbols and by operators of multiplication by discontinuous coefficients. Thus, the initial objects, i.e. the generators of the algebras, do not require special definitions in this situation. The second part is devoted to operators on manifolds with (smooth non-intersecting) “edges” of arbitrary dimensions (conical points are edges of dimension 0). On such manifolds, we must begin with the definition of pseudodifferential operators. Here, we introduce .DO of arbitrary order and study their properties in detail, then we describe the spectrum of .C ∗ -algebras generated by the operators of zero order. In [29, 30], this theory is developed for a wider class of “stratified” piecewise smooth manifolds (informally, manifolds with intersecting edges). The results of these papers are not included in the book. Let us briefly describe the content of the chapters. Chapter 1 is divided into three sections. Section 1.1 contains a standard introduction into the smooth theory of “classical” .DO. In Sect. 1.2, the necessary facts about special meromorphic .DO are listed. Here, we restrict ourselves by formulations, the proofs can be found in the monograph [21]. The meromorphic .DO are used everywhere in the book; they participate in the implementation of irreducible representations of the considered algebras. Section 1.3 contains some facts about .C ∗ -algebras given without proofs (with references to the monograph [3]) and several results presented earlier only in papers; these results are accompanied by detailed proofs.

xii

Introduction

In Chap. 2, we consider the algebra .A generated in the space .L2 (M) on a smooth compact n-dimensional manifold .M by the operators of two classes. One of the classes comprises zero order pseudodifferential operators with smooth symbols. The other class consists of the operators of multiplication by functions (“coefficients”) that may have discontinuities along a given submanifold .N . All the equivalence classes of irreducible representations of .A are listed. If .0 < dim L < n − 1, then the algebra .A has, in addition to the one-dimensional representations, two series of infinite-dimensional irreducible representations; if .dim L = n − 1, then the representations in one of these series become  is described, and the shortest solving two-dimensional. The topology on the spectrum .A series is constructed; it coincides with the series of maximal radicals. This chapter is the core of the book: although, in the next chapters, we consider more complicated algebras, which can possess many distinct series of infinite-dimensional irreducible representations, but any such series is in a sense analogous to one of the two mentioned series of representations of the algebra .A. In Chap. 3, the study of algebras of .DO with discontinuous symbols on a smooth manifold is continued. As in Chap. 2, the algebra .A is spanned by .DO with smooth symbols and by operators of multiplication on “coefficients” which now may have discontinuities along a given set of submanifolds (with boundary) of different dimensions; the submanifolds are allowed to have non-empty intersections. All equivalence classes of irreducible representations are listed, the topology on the spectrum is described, and a solving composition series is constructed. Chapter 3 is not used in Chaps. 4 and 5. We start to study .DO on manifolds with smooth closed edges in Chap. 4. Here, we have to start with the definition of pseudodifferential operators. A class of manifolds is described, on which .DO of arbitrary order are introduced and general properties of these operators are discussed. Shortly, this chapter generalizes the theory from Sect. 1.1 for manifolds with edges. In Chap. 5, the spectrum of .C ∗ -algebras generated by zero order .DO is studied. Thus, in Chaps. 4 and 5, the results of Chap. 2 are generalized for manifolds with smooth closed edges. In the body of the book, we restrict ourselves to the technical references. The detailed references are collected in the Bibliographical sketch at the end of the book.

1

Preliminaries

This chapter presents preliminary information that is of use in the book. Section 1.1 contains an elementary introduction (with proofs) to the theory of .DO  s with smooth symbols. This section is addressed to the reader having no knowledge of pseudodifferential operators. Section 1.2 is devoted to a special class of “meromorphic .DO  s” depending on complex parameter. Such operators are involved in the implementation of .C ∗ -algebra irreducible representations in the remaining chapters (except Chap. 4). Here, we restrict ourselves to statements. The proofs can be found in [21]. In Sect. 1.3, a summary of some facts in the general theory of .C ∗ -algebras has been given; the summary is accompanied by references to the monograph [3]. Moreover, Sects. 1.3.6–1.3.9 contain results (with proofs) before only published in papers.

1.1

Pseudodifferential Operators

For .x ∈ Rn , we introduce .x = (1 + |x|2 )1/2 and denote by .Z+ the set of nonnegative integers and by .Zn+ the set of multi-indices .α = (α1 , . . . , αn ), where .αj ∈ Z+ . As it usually is, .∂ = ∂x = (∂/∂x1 , . . . , ∂/∂xn ) and .D = Dx = −i∂x = (−i∂/∂x1 , . . . , −i∂/∂xn ).

1.1.1

Amplitudes

Let . be an open set in .Rn , which we will call a domain for brevity, and let .μ ∈ R.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Plamenevskii, O. Sarafanov, Solvable Algebras of Pseudodifferential Operators, Pseudo-Differential Operators 15, https://doi.org/10.1007/978-3-031-28398-7_1

1

2

1 Preliminaries

Definition 1.1.1 A function .a ∈ C ∞ ( ×  × Rn ) is called an amplitude of order .μ (in n . × ), if for any compact .K ⊂  ×  and for all .α, β, γ ∈ Z+ there exists a constant .C = C(α, β, γ , K) such that |∂xα ∂yβ ∂ξ a(x, y, ξ )| ≤ Cξ μ−|γ | for (x, y, ξ ) ∈ K × Rn . γ

.

(1.1.1)

The collection of all .μ order amplitudes is denoted by .S μ (, ), or more simply, by .S μ .  We also set .S −∞ (, ) = μ S μ (, ) (the amplitudes of order .−∞) and .S(, ) =  μ μ S (, ). The following amplitude properties are evident. 1. For each .μ, the set .S μ is a complex linear space. 2. From .μ1 ≤ μ2 it follows that .S μ1 ⊂ S μ2 , so the amplitude order is not uniquely determined. 3. Given .a ∈ S μ1 and .b ∈ S μ2 , .ab ∈ S μ1 +μ2 holds. β γ 4. Given .a ∈ S μ , .∂xα ∂y ∂ξ a ∈ S μ−|γ | holds.  Example 1 Assume that . aα ∈ C ∞ ( × ). Then, .a(x, y, ξ ) = |α|≤μ aα (x, y)ξ α is an amplitude of order .μ. In particular, for .μ = 0, the function .(x, y, ξ ) → a(x, y) with ∞ .a ∈ C ( × ) is an amplitude of order 0. Example 2 Let a function .a ∈ C ∞ ( ×  × Rn ) be homogeneous of degree .μ ∈ C with respect to .ξ for large .|ξ |, that is, .a(x, y, tξ ) = t μ a(x, y, ξ ), where .t ≥ 1 and .|ξ | ≥ 1. Then .a ∈ S Re μ . Example 3 For . μ ∈ R, the function .(x, y, ξ ) → ξ μ is the amplitude of order .μ. Every space .S μ (, ) can be endowed with a locally compact topology. To this end, for .a ∈ S μ with .μ > −∞, we set pαβγ K (a) = sup{|∂xα ∂yβ ∂ξ a(x, y, ξ )|ξ |γ |−μ ; (x, y, ξ ) ∈ K × Rn },

.

γ

(μ)

(1.1.2)

where .α, .β, and .γ are any multi-indices and .K ⊂  ×  is an arbitrary compact. It is clear (μ) that .pαβγ K (a) is the minimal constant C satisfying (1.1.1). Formula (1.1.2) defines on .S μ (μ)

the function .pαβγ K having all seminorm properties. The family of all such seminorms makes .S μ into a Frechet space, that is, into a complete metrizable locally compact space. The space .S −∞ is provided with the projective limit topology so that aj → a in S −∞ ⇔ aj → a in S μ ∀ μ > −∞.

.

In what follows, we make use of the next simple assertions.

1.1 Pseudodifferential Operators

3

(i) For any .μ1 , μ2 ∈ [−∞, +∞) there is a continuous bilinear mapping S μ1 × S μ2  (a, b) → ab ∈ S μ1 +μ2 .

.

(ii) For any .α, .β, and .γ ∈ Zn+ and for .μ ∈ [−∞, +∞) there is a continuous mapping S μ  a → ∂xα ∂yβ ∂ξ a ∈ S μ−|γ | . γ

.

(iii) Given .μ1 < μ2 , the mapping .S μ1  a → a ∈ S μ2 is continuous. (iv) Let .χ ∈ C ∞ (Rn ) for each .γ ∈ Zn+ satisfy .|∂ γ χ (ξ )| ≤ Cγ ξ −|γ | , that is, .χ is an amplitude of order 0, independent of x and y. For every .ε > 0, we set .χε (ξ ) = χ (εξ ). Assume also that .a ∈ S μ . Then, χε a →ε→0 χ (0)a in S μ+1 .

.

We prove (iv). In view of (i), it suffices to verify that .χε → χ (0) in .S 1 , which means .

   sup∂ γ χε (ξ ) − χ (0) ξ |γ |−1 →ε→0 0

∀γ ∈ Zn+ .

(1.1.3)

  Assume that .γ = 0. Then, (1.1.3) follows from .χε (ξ ) − χ (0) ≤ C|ε ξ | ≤ Cεξ , where .C = sup |∇χ |. Suppose now that .γ = 0. Since .ε → 0, we can consider .ε ∈ (0, 1). Then  γ  ∂ χε (ξ ) − χ (0) ξ |γ |−1 = |(∂ γ χ )(ε ξ )|ε|γ | ξ |γ |−1 ≤

.

≤ Cγ ε ξ −|γ | ξ |γ |−1 ε|γ | ≤ Cγ ε; we have used the relations ε ξ −|γ | ξ |γ |−1 ≤ ε ξ 1−|γ | ξ |γ |−1 =

.

= ε1−|γ |

(1 + |ξ |2 )(|γ |−1)/2 = (1 + |ε ξ |2 )(|γ |−1)/2

 1 + |ξ |2 (|γ |−1)/2 ≤ ε1−|γ | . ε−2 + |ξ |2

Note that property (iv) remains valid when .S μ+1 changes for .S μ+δ with any positive μ can be approximated in the .δ. According to the property, every amplitude .a ∈ S μ+δ by amplitudes whose supports are compact with respect to .ξ . topology of .S

4

1 Preliminaries

1.1.2

Pseudodifferential Operators

Denote by .Cc∞ () the subspace in .C ∞ () consisting of functions with compact supports. By definition, .uj → u in .Cc∞ () if and only if .{supp uj } ⊂ K for a certain compact .K ⊂  and .∂ α uj ⇒ ∂ α u on K for all .α. Recall also that .uj → u in .C ∞ () .⇔ .∂ α uj ⇒ ∂ α u on each compact K for any multi-index .α. Proposition 1.1.2 Assume that .a ∈ S(, ) and .u ∈ Cc∞ (). Then, the formula

v(x) =

.

ei(x−y)ξ a(x, y, ξ )u(y) dydξ

(1.1.4)

defines a function v in .C ∞ () (by the integral is meant an iterated integral). Proof Let .K1 ⊂  be an arbitrary compact and let .K2 = supp u. We also set .Dx 2 = 1 − = 1 + (Dx21 + . . . + Dx2n ). Taking account of the equality ei(x−y)ξ = Dy 2N ei(x−y)ξ ξ −2N ,

.

N ∈ N,

we obtain

v(x) =

.

Rn ×K2

ei(x−y)ξ Dy 2N (a(x, y, ξ )u(y))ξ −2N dydξ.

(1.1.5)

Since .a ∈ S(, ), we have .a ∈ S μ (, ) with a certain .μ. From the amplitude properties it follows that |Dy 2N (a(x, y, ξ )u(y))ξ −2N | ≤ Cξ μ−2N ,

.

(x, y, ξ ) ∈ K1 × K2 × Rn .

For large N such that .μ − 2N < −n, there holds the equality





.

Rn ×K2

ξ μ−2N dydξ = mes K2

ξ μ−2N dξ < +∞.

Hence, for .x ∈ K1 the integrand in (1.1.5) has a summable majorant independent of x. Therefore, integral (1.1.5) uniformly with respect to .x ∈ K1 converges and the function

K1  x →

.

Rn ×K2

ei(x−y)ξ Dy 2N (a(x, y, ξ )u(y))ξ −2N dydξ



≡

ei(x−y)ξ a(x, y, ξ )u(y) dydξ

1.1 Pseudodifferential Operators

5

is continuous. The compact .K1 has arbitrarily been chosen, so formula (1.1.4) defines v ∈ C(). Now let .α be any multi-index. From (1.1.5), it formally follows

.



∂ v(x) =

.

α

Rn ×K2

ei(x−y)ξ Dy 2N (b(x, y, ξ )u(y))ξ −2N dydξ,

(1.1.6)

where  α (iξ )β ∂xα−β a(x, y, ξ ) .b(x, y, ξ ) = β β≤α

is an amplitude of order .μ + |α|. According to the first part of the proof, for .N > (μ + |α| + n)/2, integral (1.1.6) converges uniformly with respect to .x ∈ K1 , while .K1 ⊂  is   any compact. This implies formula (1.1.6) and the inclusion .∂ α v ∈ C(). Integrating (“in the opposite direction”) the right-hand side of (1.1.6) by parts, we obtain

α .∂ v(x) = ∂xα (ei(x−y)ξ a(x, y, ξ )u(y)) dydξ. Therefore, when calculating the derivatives of function (1.1.4), one can commute the differentiation and integration. Assume that .a ∈ S(, ). According to Proposition 1.1.2, the formula Au(x) = (2π )−n



.

ei(x−y)ξ a(x, y, ξ )u(y) dydξ

(1.1.7)

defines a linear operator .A : Cc∞ () → C ∞ (), which will sometimes be denoted by .Op a. Definition 1.1.3 An operator of the form (1.1.7) is called a pseudodifferential operator (.DO) in the domain .. In the case of .a ∈ S μ (, ) and .μ ∈ [−∞, +∞), the number .μ is called the order of .DO A. The collection of all .DO of order .μ is denoted by   μ −∞ () = μ μ . (). We also set . μ  () and .() = μ  (). (Because of the μ μ inclusion . 1 () ⊂  2 () for .μ1 ≤ μ2 , the .DO order is not uniquely determined.) Proposition 1.1.4 The bilinear mapping S μ (, ) × Cc∞ ()  (a, u) → (Op a)u ∈ C ∞ ()

.

is continuous for any .μ ∈ [−∞, +∞).

6

1 Preliminaries

Proof Suppose that .aj → a in .S μ and .uj → u in .Cc∞ (). We have to show that ∞ .(Opaj )uj → (Opa)u in .C (). Because of (Opaj )uj − (Opa)u = (Op(aj − a))uj + (Opa)(uj − u),

.

we can assume that at least one of the sequences .{aj } or .{uj } tends to zero. Then, we have to verify .(Opaj )uj → 0 in .C ∞ (). We set .vj = (Opaj )uj . Let .α be any multi-index and choose a number .N ∈ N such that .μ + |α| − 2N < −n. When proving Proposition 1.1.2, we established that ∂ α vj (x) = (2π )−n



ei(x−y)ξ σj (x, y, ξ ) dydξ,

.

where σj (x, y, ξ ) = Dy 2N (bj (x, y, ξ )uj (y))ξ −2N ,  α (iξ )β ∂xα−β aj (x, y, ξ ). bj (x, y, ξ ) = β

.

β≤α

Each function .uj can be taken as a zero order amplitude independent of .(x, ξ ), while the convergence .uj → u in .Cc∞ () implies that in .S 0 (, ). This and assertions (i)–(iii) in 1.1.1 lead to σj →j →∞ 0 in S μ+|α|−2N .

(1.1.8)

.

Let .K1 ⊂  be any compact. Denote by .K2 (⊂ ) a compact containing each of the sets supp uj . By virtue of (1.1.8),

.

|σj (x, y, ξ )| ≤ Cj ξ μ+|α|−2N ,

.

(x, y, ξ ) ∈ K1 × K2 × Rn ,

where .Cj → 0 as .j → ∞. Hence

|∂ vj (x)| ≤ Cj mesK2

.

α

ξ μ+|α|−2N dξ,

x ∈ K1 .

Since .μ + |α| − 2N < −n, the integral is finite. Therefore .∂ α v ⇒ 0 on .K1 . Because .α and .K1 have arbitrarily been chosen, we obtain .vj → 0 in .C ∞ ().   Corollary 1.1.5 For each fixed .a ∈ S(, ), there is the continuous mapping Cc∞ ()  u → (Op a)u ∈ C ∞ ();

.

1.1 Pseudodifferential Operators

7

for each fixed .u ∈ Cc∞ (), there is a continuous mapping S μ (, )  a → (Op a)u ∈ C ∞ ().

.

Corollary 1.1.6 Assume that .χ ∈ Cc∞ (Rn ) and .χ (0) = 1. For any .ε > 0 we set .χε (ξ ) = χ (εξ ). Then   Op (aχε ) u →ε→0 (Op a)u in C ∞ ()

.

for .a ∈ S(, ) and .u ∈ Cc∞ (). Proof Let .μ be the order of the amplitude a. By virtue (iv) in 1.1.1, .aχε → a in .S μ+1 . It   remains to apply Corollary 1.1.5. Now, we show that .S(, )  a → Op a ∈ () is not a one-to-one mapping. Proposition 1.1.7 For any .DO A there exist infinitely many amplitudes a such that A = Op a.

.

Proof Let an amplitude a and a polynomial P be such that the function (x, y, ξ ) → a(x, y, ξ )/P (x − y)

.

(1.1.9)

is an amplitude as well (for instance, this is the case if .P (x) = 0 . ∀x). By Corollary 1.1.6, setting .A = Op a, we have



 a(x, y, ξ )  χε (ξ )u(y) dydξ = P (Dξ )ei(x−y)ξ ε→0 P (x − y)

 a(x, y, ξ ) −n χε (ξ ) u(y) dydξ. = lim (2π ) ei(x−y)ξ P (−Dξ ) ε→0 P (x − y)

Au(x) = lim (2π )−n

.

Let .μ be the order of amplitude (1.1.9). Then, P (−Dξ )

.

 a(x, y, ξ )

a(x, y, ξ ) χε (ξ ) → P (−Dξ ) P (x − y) P (x − y)

in .S μ+1 , so Au(x) = (2π )−n

.



ei(x−y)ξ P (−Dξ )

a(x, y, ξ ) u(y) dydξ. P (x − y)

8

1 Preliminaries

  Therefore, .A = Op b with .b(x, y, ξ ) = P (−Dξ ) a(x, y, ξ )/P (x − y) . Varying P , we arrive at the needed conclusion.   For a .DO .A = Op a, we set t

.

a ∗ (x, y, ξ ) = a(x, y, ξ )

a(x, y, ξ ) = a(y, x, −ξ ),

and introduce the transposed operator .tA = Op t a and the “adjoint” operator .A∗ = Op a ∗ . The operators .tA and .A∗ belong to the same class . μ () as does A. There hold the equalities Au, v = u, tAv,

.

(Au, v) = (u, A∗ v) ,

u, v ∈ Cc∞ (),

where the pairings .·, · and .(·, ·) are defined by



u, v =

(u, v) =

u(x)v(x) dx,

.

u(x)v(x) dx.

Examples (1) A differential operator 

P (x, D) =

.

pα (x)D α (qα (x) ·),

pα , qα ∈ C ∞ (),

|α|≤μ

is a .DO of order .μ: P (x, D)u(x) = (2π )−n

.



ei(x−y)ξ



pα (x)ξ α qα (y) u(y) dydξ.

|α|≤μ

In the case of .qα = 1 for all .α, the formula takes the form P (x, D)u(x) = (2π )

.

−n



= (2π )−n/2

ei(x−y)ξ P (x, ξ )u(y) dydξ

ˆ ) dξ, eixξ P (x, ξ )u(ξ

where .uˆ denotes the Fourier transform of u, u(ξ ˆ ) = (2π )

.

−n/2



e−iyξ u(y) dy.

1.1 Pseudodifferential Operators

9

(2) An integral operator with a smooth kernel

Au(x) =

.

G(x, y)u(y) dy,

G ∈ C ∞ ( × ),

is a .DO of order .−∞. To prove this, we set a(x, y, ξ ) = e−i(x−y)ξ G(x, y)χ (ξ ),

.

where .χ ∈ S(Rn ) is any function that satisfies .(2π )−n take .χ ∈ Cc∞ (Rn )). Then, .a ∈ S −∞ and (Op a)u(x) = (2π )−n

ei(x−y)ξ a(x, y, ξ )u(y) dydξ



= (2π )

1.1.3

χ (ξ ) dξ = 1 (one can even



.

−n



G(x, y)χ (ξ )u(y) dydξ =

G(x, y)u(y) dy = Au(x).

The Kernel of a Pseudodifferential Operator

Suppose that .a ∈ S μ (, ) and .A = Op a. Assuming .μ < −n − k and .k ∈ Z+ , we change the integration order in Au(x) = (2π )−n



ei(x−y)ξ a(x, y, ξ )u(y) dydξ,

.

u ∈ Cc∞ (),

and obtain

Au(x) =

.

GA (x, y)u(y) dy,

(1.1.10)

where GA (x, y) = (2π )−n

.

ei(x−y)ξ a(x, y, ξ ) dξ.

(1.1.11)

The function .GA ∈ C k ( × ) is called the kernel of A. It can be identified with an element in .D ( × ) acting on the functions .w ∈ Cc∞ ( × ) by the formula

GA , w =

.

GA (x, y)w(x, y) dxdy,

(1.1.12)

10

1 Preliminaries

where .D ( × ) stands for the general function space adjoint to .Cc∞ ( × ). In more detail, GA , w = (2π )

.

−n



ei(x−y)ξ a(x, y, ξ )w(x, y) dxdydξ ;

(1.1.13)

because of .aw ∈ L1 ( ×  × Rn ), the integration order plays no role. For u, .v ∈ Cc∞ () we set .(v ⊗ u)(x, y) = v(x)u(y) and in (1.1.12) choose .w = v ⊗ u. Taking into account (1.1.10), we arrive at GA , v ⊗ u = Au, v.

.

(1.1.14)

For arbitrary .μ, the kernel of .DO A is, by definition, the distribution   .GA given by . . . dxdy. To (1.1.13); however, this time the integral means the iterated integral . dξ prove that .GA is a distribution, we have to verify the existence of such an integral and its continuous dependence on w. It can be done by means of the formula GA , w = (2π )−n



.

ei(x−y)ξ Dy 2N (a(x, y, ξ )w(x, y))ξ −2N dxdydξ

(compare with the proofs of Propositions 1.1.2 and 1.1.4). In fact, for any .μ ∈ R, the integral (1.1.13) is a continuous function of .(a, w) ∈ S μ (, ) × Cc∞ ( × ). Assume that .χ ∈ Cc∞ (Rn ), .χ (0) = 1, and .χε (ξ ) := χ (εξ ) for .ε > 0. As .ε → 0, we have .χε a → a in .S μ+1 , .μ being the amplitude order of a. Hence, GA , w = lim (2π )

.

−n

ε→0

= lim (2π )

−n



ei(x−y)ξ a(x, y, ξ )χε (ξ )w(x, y) dxdydξ





w(x, y) dxdy

ε→0

ei(x−y)ξ a(x, y, ξ )χε (ξ ) dξ. (1.1.15)

Therefore, GA (x, y) = lim (2π )−n



.

ε→0

ei(x−y)ξ a(x, y, ξ )χε (ξ ) dξ

in the sense of the convergence in .D ( × ). This formula has usually been written in the form (1.1.11). We substitute .w = v ⊗ u into (1.1.15) and obtain the formula GA , v ⊗ u = lim (2π )

.

ε→0

−n





v(x) dx

ei(x−y)ξ a(x, y, ξ )χε (ξ )u(y) dydξ

=

v(x) Au(x) dx = Au , v,

1.1 Pseudodifferential Operators

11

which coincides with (1.1.14). Now, it is easily seen that (1) Distinct operators have distinct kernels. (2) For .H ∈ D ( × ) such that .H, v ⊗ u = Au, v with any .u, v ∈ Cc∞ (), there holds the equality .H = GA . In particular, .GA is independent of the choice of the amplitude a in (1.1.15); it goes without saying that the equality .A = Op a must be fulfilled. (Recall that the linear combinations of functions of the form .v ⊗ u are dense in .Cc∞ ( × )). We also note that the kernel of the transposed operator .DO .tA is defined by the equality GtA , w = GA , t w, where

.

t

w(x, y) = w(y, x),

and the kernel of the adjoint .DO .A∗ satisfies GA∗ , w = GA , w ∗ , where w ∗ (x, y) = w(y, x).

.

In the case that the kernel belongs to .L1loc ( × ), equality (1.1.14) can be written in the form



. v(x) dx GA (x, y)u(y) dy = v(x)Au(x) dx, which implies (1.1.10). Let .1 and .2 be domains in .Rn and .G ∈ D (1 × 2 ). For a fixed .u ∈ Cc∞ (2 ), the mapping Cc∞ (1 )  v → G(v ⊗ u)

.

(1.1.16)

is a linear continuous functional on .Cc∞ (1 ); in other words, it belongs to .D (1 ). Therefore, we have defined an operator .Cc∞ (2 ) → D (1 ) that takes .u ∈ Cc∞ (2 ) to functional (1.1.16). It is easy to verify that the operator is continuous. We denote this operator by A and have Au, v = G, v ⊗ u.

.

(1.1.17)

Conversely, if .A : Cc∞ (2 ) → D (1 ) is a linear continuous operator, there exists a unique distribution .G ∈ D (1 × 2 ) that satisfies (1.1.17). The distribution is called the Schwartz kernel of the operator A, and the above statement is the Schwartz kernel theorem. Let A be a .DO in a domain . ⊂ Rn . Then, A implements a continuous mapping ∞ ∞ ∞  .Cc () → C (). The canonical embedding .C () → D () is continuous, so A

12

1 Preliminaries

can be considered as continuous operator .Cc∞ () → D (). For any .u, v ∈ Cc∞ (), the Schwartz kernel of such an operator satisfies (1.1.17) and consequently coincides with its kernel in the sense of the definitions given in the present section.

1.1.4

Smoothing Operators

Definition A pseudodifferential operator A is called smoothing if its kernel belongs to C ∞ ( × ). Proposition 1.1.8 The following assertions are equivalent: (1) A is a smoothing DO. (2) A ∈  −∞ (). (3) A ∈  −∞ (). Proof (1)⇒(2). For G ∈ C ∞ ( × ), the operator

Au(x) =

GA (x, y)u(y) dy

.

is integral with a smooth kernel, so A ∈  −∞ () (see Example 2 in Sect. 1.1.2). (2)⇒(3). It is evident. (3)⇒(1). For any k ∈ N, there exists an amplitude ak ∈ S −n−k such that A = Op ak . Therefore, GA (x, y) = (2π )

.

−n

ei(x−y)ξ ak (x, y, ξ ) dξ

is a function of the class C k−1 ( × ). Because k is arbitrary, we have GA ∈ C ∞ ( × ).   Proposition 1.1.9 Let an amplitude a vanish on V × Rn , where V ⊂  ×  is a neighborhood of the diagonal := {(x, y) ∈  ×  : x = y}. Then, Op a ∈  −∞ (). Proof From the proposition assumption, it follows that, for any N ∈ N, the function (x, y, ξ ) → a(x, y, ξ )/|x − y|2N is an amplitude (of same order μ as a). As shown in Sect. 1.1.2, Op a = Op bN with bN = (− ξ )N a(x, y, ξ )/|x − y|2N . Moreover, bN ∈   S μ−2N and N has arbitrarily been chosen, so Op a ∈  −∞ (). Proposition 1.1.10 The kernel GA of each DO A is smooth outside the diagonal .

1.1 Pseudodifferential Operators

13

Proof Assume that a ∈ S(, ), ϕ ∈ C ∞ ( × ), and set A = Op a and B = Op(ϕa); then GB = ϕGA . Indeed, for any w ∈ Cc∞ ( × ), we have GB , w = (2π )

.

−n



ei(x−y)ξ a(x, y, ξ )ϕ(x, y)w(x, y) dxdydξ = = GA , ϕw = ϕGA , w.

Let θ ∈ C ∞ (R) satisfy θ (t) = 0 for t ≤ 1 and θ (t) = 1 for t ≥ 2. We take an arbitrary δ > 0 and set ϕδ (x, y) = θ (|x − y|/δ). It is obvious that ϕδ ∈ C ∞ (Rn × Rn ), ϕδ = 0 in the δ-neighborhood Vδ of the diagonal , and ϕδ = 1 outside V2δ . Assume that A = Op a is any DO and Aδ = Op (aϕδ ). Since aϕδ = 0 in a neighborhood of the diagonal, we have Aδ ∈  −∞ () (Proposition 1.1.9). Therefore, GAδ ∈ C ∞ ( × ) (Proposition 1.1.8). According to the remark at the beginning of the proof, GAδ = ϕδ GA . Hence, GAδ = GA on ( × ) \ V2δ , so GA is a smooth function on  ( × ) \ V2δ . Because δ > 0 has arbitrarily been chosen, which completes the proof. 

1.1.5

Properly Supported Pseudodifferential Operators

A mapping .f : X → Y of topological spaces is called proper if .f −1 (K) is compact for any compact .K ⊂ Y . A distribution .G ∈ D ( × ) is said to have proper support if both projections π1 , π2 : supp G → 

.

are proper mappings. Definition 1.1.11 A pseudodifferential operator is called properly supported (or simply proper) if its kernel has proper support. Any differential operator is proper because the support of its kernel belongs to the diagonal. In the case that A is a proper .DO, the operators .tA and .A∗ are proper as well. For an amplitude a, we denote by .suppx,y a the closure of the projection of .supp a into . ×  . An amplitude a is said to have proper support if both projections π1 , π2 : suppx,y a → 

.

are proper mappings. Lemma 1.1.12 For a .DO A (that need not be proper), the following assertions are valid:

14

1 Preliminaries

(a) In the case that .χ ∈ C ∞ ( × ) and .χ = 1 in a neighborhood of the set .supp GA , there holds the equality .A = Op (χ a). (b) .supp GA ⊂ suppx,y a. Proof (a) Setting .B = Op (χ a), we have .GB = χ GA = GA , so .A = B. (b) From GA , w = (2π )−n



.

ei(x−y)ξ a(x, y, ξ )w(x, y) dxdydξ, w ∈ Cc∞ ( × ),

it follows that .GA , w = 0 provided .suppw ∩ suppx,y a = ∅. Therefore, .GA = 0 on .( × ) \ suppx,y a, so .supp GA ⊂ suppx,y a.   Proposition 1.1.13 The following assertions are equivalent: (1) A is a proper .DO. (2) There exists an amplitude a with proper support such that .A = Op a. Proof (1).⇒(2). Assume that .A = Op a and .χ ∈ Cc∞ ( × ) with proper support equal to 1 in a neighborhood of the set .supp GA . Then, .χ a is an amplitude with proper support and (by Lemma) .A = Op (χ a). (2).⇒ (1). The amplitude a has proper support and (by Lemma) .supp GA ⊂ suppx,y a,   therefore, .GA has proper support as well. Proposition 1.1.14 The following assertions are equivalent: (1) A is a proper .DO. (2) For any compact .K2 ⊂ , there exists a compact .K1 ⊂  such that u ∈ Cc∞ (K2 ) ⇒ Au, tAu ∈ Cc∞ (K1 ).

(1.1.18)

.

Proof (1). ⇒ (2). Assume that .A = Op a, where a is an amplitude with proper support. We choose any compact .K2 ⊂  and set .K1 = π1 (π2−1 (K2 )), where .π1 and .π2 are the projections .suppx,y a → . In the case of .x ∈ K1 , the function .(y, ξ ) → a(x, y, ξ ) vanishes on the set .K2 × Rn . Consequently, for .u ∈ Cc∞ (K2 ), we have Au(x) = (2π )−n

.



ei(x−y)ξ a(x, y, ξ )u(y) dydξ = 0

for x ∈ K1 ,

1.1 Pseudodifferential Operators

15

which means that .Au ∈ Cc∞ (K1 ). The inclusion .tAu ∈ Cc∞ (K1 ) can for a certain compact    .K ⊂  be established in a similar way. It remains to set .K1 = K ∪ K . 1 1 2 (2). ⇒ (1). Denote by G the kernel of .DO A, and by .π1 and .π2 denote the projections .supp G → . We have to verify that the mappings .π1 and .π2 are proper. Let .E ⊂  be an arbitrary compact and .V ⊃ E a domain with compact closure .K2 := V ⊂ . By assumption, there exists a compact .K1 ⊂  such that there holds (1.1.18). Then, G, v ⊗ u = Au, v = 0 for all u ∈ Cc∞ (V ), v ∈ Cc∞ ( \ K1 ).

.

For such u and v, the linear combinations of functions of the form .v ⊗ u are dense in ∞ .Cc (( \ K1 ) × V ), so G, w = 0 for w ∈ Cc∞ (( \ K1 ) × V ).

.

Hence, .supp G ∩ (( \ K1 ) × V ) = ∅ and π2−1 (V ) ∩ (( \ K1 ) × V ) = ∅ ⇒ π2−1 (E) ∩ (( \ K1 ) × E) = ∅

.

⇒ π2−1 (E) ⊂ K1 × E . Being a closed subset of the compact .K1 × E, the set .π2−1 (E) is compact. This proves that mapping .π2 is proper. Such a fact for the mapping .π2 : supp t G → , where . t G is the kernel of .tA , can be verified in a similar way. Let s stand for the homeomorphism supp G → supp t G : (x, y) → (y, x).

.

Then, .π1 = π2 ◦ s, and hence, the mapping . π1 is proper.

 

Proposition 1.1.15 Every proper .DO A implements a continuous mapping .Cc∞ () → Cc∞ (). ∞ ∞ Proof Assume that .{uj }∞ j =1 ⊂ Cc () and .uj → 0 in .Cc (). There exists a compact .K2 ⊂  such that .supp uj ⊂ K2 for all j . We employ Proposition 1.1.14 and obtain

supp Auj ⊂ K1 (∀ j ),

.

with a certain compact .K1 ⊂ . Moreover, Auj → 0 in C ∞ ().

.

It follows that .Auj → 0 in .Cc∞ ().

 

16

1 Preliminaries

Proposition 1.1.16 Let A be a proper .DO. Then, A extends to the continuous operator A : C ∞ () → C ∞ ().

.

Proof Let a be a proper amplitude such that .A = Op a. We set v(x) = (2π )−n

.



ei(x−y)ξ a(x, y, ξ )u(y) dydξ.

It was shown in Sect. 1.1.2 that from .u ∈ Cc∞ () there follows .v ∈ C ∞ (); to prove this, we integrated by parts. This was possible because the support of the function .y → a(x, y, ξ )u(y) was compact owing to the compact support of u. Being proper, the amplitude a provides a compact support for the product au and the support of u need not be compact. Therefore, the inclusion .v ∈ C ∞ () can be verified in the same way as in Proposition 1.1.2. Thus, the formula Au(x) = (2π )−n

.



ei(x−y)ξ a(x, y, ξ )u(y) dydξ,

u ∈ C ∞ (),

defines a linear operator .A : C ∞ () → C ∞ (). We show that the operator is continuous. Assume that .K1 ⊂  is any compact and .K2 = π2 (π1−1 (K1 )) , .π1 , π2 : suppx,y a →  being the projections. Choose a function .ψ ∈ Cc∞ () to satisfy .ψ|K2 = 1. We also suppose that .{uj } ⊂ C ∞ () and .uj → 0 in .C ∞ () as .j → ∞. From the equality .a(x, y, ξ ) = a(x, y, ξ )ψ(y) for .x ∈ K1 , it follows that Auj = (Op a)ψuj = A(ψuj ) on K1 .

.

Since .ψuj → 0 in .Cc∞ (), we have .A(ψuj ) → 0 in .C ∞ (). In particular, .A(ψuj ) ⇒ 0 on .K1 together with all derivatives. The compact .K1 ⊂  has arbitrarily been chosen, so ∞ .Auj → 0 in .C ()   The space .Cc∞ () is dense in .C ∞ (). (Indeed, let .{Kj } be an exhaustive sequence of compact subsets in . and let the sequence .{ψj } ⊂ Cc∞ () satisfy .ψj |Kj = 1. Then .uψj → u in .C ∞ () for .u ∈ C ∞ ().) Therefore, the extension .A of A given by Proposition 1.1.16 is unique. In what follows we write A instead of .A. Note that any .DO A admits the representation .A = A1 + A2 , where .A1 is a proper .DO and .A2 is a smoothing .DO. To see that, it suffices to take a properly supported function .χ ∈ C ∞ ( × ) equal to 1 in a neighborhood of the diagonal and to set .A1 = Op (χ a), .A2 = Op ((1 − χ )a).

1.1 Pseudodifferential Operators

1.1.6

17

Pseudodifferential Operators in Generalized Function Spaces

Denote by .D () and .E  () the general function spaces adjoint to .Cc∞ () and .C ∞ (), respectively. The space .E  () consists of the elements in .D () with compact supports in  ∞ .. We assume that .A ∈ (), fix .u ∈ E () and define a functional .fu on .Cc () by the equality fu , v = u, tAv.

.

(1.1.19)

Clearly, the functional is continuous, so .fu ∈ D (). We set Au = fu .

(1.1.20)

Au, v = u, tAv.

(1.1.21)

.

Then, (1.1.19) takes the form .

If the function u fixed above turns out to be in .Cc∞ (), the equality .Au, v = u, tAv leads to .Au = Au. Therefore, formula (1.1.20) defines an extension of .DO .A : Cc∞ () → C ∞ () to the operator .A : E  () → D (). This operator is continuous. Indeed, if .uj → 0 in .E  (), we obtain .uj , tAv → 0 for .v ∈ Cc∞ () and then, taking account of (1.1.21), arrive at .Auj , v → 0. In the sense of functional analysis, the operator .A is adjoint to the operator .tA: A

D () ←− E  ()

.

tA

Cc∞ () −→ C ∞ (). In what follows, we write A instead of .A. Now suppose that A is a proper .DO. Such an operator implements continuous mappings A : Cc∞ () → Cc∞ ()

.

A : C ∞ () → C ∞ (). Repeating the preceding arguments, we conclude that A extends to the continuous operators A : E  () → E  (),

.

A : D () → D ().

18

1 Preliminaries

1.1.7

Symbols

Definition 1.1.17 A function a ∈ C ∞ ( × Rn ) is called a symbol of order μ ∈ R if, for any α, β ∈ Zn+ and for each compact K ⊂ , there exists a constant C = C(α, β, K) such that β

|∂xα ∂ξ a(x, ξ )| ≤ Cξ μ−|β| for (x, ξ ) ∈ K × Rn .

.

Denote by S μ () the set of all symbols of order μ. We also introduce S −∞ () =  μ μ μ S (), S() = μ S (). Every symbol can be considered as an amplitude independent of y, so S μ () ⊂ μ S (, ). The topology of S μ (, ) induces that of S μ (). For μ > −∞, such a topology can be given by the seminorm family 

pαβK (a) = sup{|∂xα ∂ξ a(x, ξ )|ξ |β|−μ : (x, ξ ) ∈ K × Rn }.

.

β

(μ)

For S μ () endowed with the above topology, assertions (i)–(iii) in 1.1.1 are valid. For any symbol a, we introduce the DO A = Op a: Au(x) = (2π )

.

−n



e

i(x−y)ξ

−n/2

a(x, ξ )u(y) dydξ = (2π )

ˆ ) dξ, eixξ a(x, ξ )u(ξ

where u(ξ ˆ ) = (2π )−n/2



.

e−iyξ u(y) dy.

In what follows, we use the notation a(x, D) along with Op a. Examples  α ∞ (1) P (x, ξ ) = |α|≤μ aα (x)ξ with aα ∈ C () is a symbol of order μ. The corresponding DO is the differential operator P (x, D) =

.

 |α|≤μ

aα (x)D α .

(2) Let, for μ ∈ C, the function a ∈ C ∞ ( × Rn ) be degree μ homogeneous for large |ξ |, that is, a(x, tξ ) = t μ a(x, ξ ), where t ≥ 1 and |ξ | ≥ 1. Then, a ∈ S Re μ (). (3) The function (x, ξ ) → ξ μ is a symbol of order μ. (4) Assume that a˜ ∈ S μ (, ) and set a(x, ξ ) = a(x, ˜ x, ξ ). Then, a ∈ S μ ().

1.1 Pseudodifferential Operators

1.1.8

19

Asymptotic Expansions in the Classes S μ ()

Let .aj ∈ S μj (), .j = 0, 1, . . . , .μj → −∞ for .j → ∞ and let .a ∈ C ∞ ( × Rn ). The relation a(x, ξ ) ∼

∞ 

aj (x, ξ )

(1.1.22)

aj (x, ξ ) ∈ S μk (),

(1.1.23)

.

j =0

means that for any .k ∈ N a(x, ξ ) −

k−1 

.

j =0

where .μk = maxj ≥k μj . In particular, from (1.1.23) it follows that .a ∈ S μ0 . It is easy to verify the following assertions: (1) Relation (1.1.22) defines a up to a term in .S −∞ (). (2) From (1.1.22), it follows that β

∂xα ∂ξ a(x, ξ ) ∼

.

∞ 

β

∂xα ∂ξ aj (x, ξ )

(∀ α , β)

j =0

(the asymptotic expansions can be differentiated). (3) Any permutation of the series terms does not violate (1.1.22). By permutation, we can always provide the monotonically decreasing sequence .μj of the symbol orders and obtain .μk = μk .  (4) The relations (1.1.22) and .aj (x, ξ ) ∼ ∞ k=0 aj k (x, ξ ) with .j = 0, 1, . . . imply a(x, ξ ) ∼



.

aj k (x, ξ ).

j,k

Theorem 1.1.18 If .aj ∈ S μj (), .j = 0, 1, . . ., and .μj → −∞ as .j → ∞, there exists a function .a ∈ S μ0 () that satisfies (1.1.22). Proof We can assume that .μj ≥ μj +1 for all j . Choose .χ ∈ C ∞ (Rn ) such that .χ (ξ ) = 0 for .|ξ | ≤ 1 and .χ (ξ ) = 1 for .|ξ | ≥ 2. We also assume .{Kj } to be an exhaustive sequence of compact subsets in .. Since .χ (εξ ) →ε→+0 0 in .S 1 (), for every j there is a number .εj > 0 such that |∂xα ∂ξ (χ (εj ξ )aj (x, ξ ))| ≤ 2−j ξ μj −|β|+1

.

β

(1.1.24)

20

1 Preliminaries

for .(x, ξ ) ∈ Kj × Rn with .α and .β subject to .|α| + |β| ≤ j . We can consider that .εj → 0 as .j → 0. Let us define a function a by the equality a(x, ξ ) =

∞ 

.

(1.1.25)

χ (εj ξ )aj (x, ξ ).

j =0

The sum on the right-hand side is locally finite, that is, for any compact .K ⊂ Rn , there are only finitely many terms different from zero on . × K, so the definition of a is correct. It is evident that .a ∈ C ∞ ( × Rn ) and β

∂xα ∂ξ a(x, ξ ) =

.

∞ 

 β ∂xα ∂ξ χ (εj ξ )aj (x, ξ ) .

j =0

Any remainder of series (1.1.25) has similar properties. We show that, for every .k ≥ 0, there holds the inclusion ∞  .

χ (εj ξ )aj (x, ξ ) ∈ S μk ().

(1.1.26)

j =k

Let .K ⊂  be an arbitrary compact and .α and .β be any multi-indices. We have to verify the inequality ∞    β ∂xα ∂ξ χ (εj ξ )aj (x, ξ )  ≤ Cξ μk −|β| , 

.

(x, ξ ) ∈ K × Rn .

j =k

There exists a number l such that .K ⊂ Kl , .|α + β| ≤ l and .μl + 1 ≤ μk . Because ∞  .

l−1 ∞    β ∂xα ∂ξ χ (εj ξ )aj (x, ξ ) = ... + ...,

j =k

j =k

j =l

it suffices to check that both of the sums on the right are majorized by .ξ μk −|β| . For the first sum, it is evident because the sum belongs to .S μk −|β| (). For the second one, the needed estimate follows from (1.1.24): ∞ ∞ ∞      β ∂xα ∂ξ χ (εj ξ )aj (x, ξ )  ≤ 2−j ξ μj −|β|+1 ≤ ξ μl −|β|+1 2−j 

.

j =l

j =l

≤ 2−l+1 ξ μk −|β| .

j =l

1.1 Pseudodifferential Operators

21

The inclusion (1.1.26) has been established. Now, for any .k ≥ 1, we have a(x, ξ ) −

k−1 

.

aj (x, ξ ) =

j =0

k−1 ∞   (χ (εj ξ ) − 1)aj (x, ξ ) + χ (εj ξ )aj (x, ξ ) ∈ S μk (); j =k

j =0

the first sum on the right vanishes for large .|ξ |, so it belongs to .S −∞ (), while the second   sum is in .S μk by virtue of (1.1.26).

1.1.9

The Symbol of Proper Pseudodifferential Operator

Let A be a proper .DO. We set σA (x, ξ ) = e−ξ Aeξ (x),

.

where .eξ (x) = eixξ . The function .σA is called the symbol of A. Examples (1) Let .P (x, D) be a differential operator in .. Taking into account that .P (x, D)eξ (x) = P (x, ξ )eξ (x), we obtain .σP (x, ξ ) = P (x, ξ ). (2) Let

A : u →

G(x, y)u(y) dy,

.

u ∈ C ∞ (),

be an integral operator with a properly supported smooth kernel. Then σA (x, ξ ) = e

.

−ixξ



ˇ G(x, y)eiyξ dy = (2π )n/2 e−ixξ G(x, ξ ),

ˇ is the inverse Fourier transform of G with respect to the second argument. where .G   In the case that A is a proper .DO and .A = Op a with properly supported amplitude a, we have e−ξ Aeξ (x) = e−ξ (x) (2π )

.

= (2π )−n



−n



ei(x−y)η a(x, y, η)eξ (y) dydη

ei(x−y)η a(x, y, η)ei(y−x)ξ dydη = (2π )−n



ei(x−y)(η−ξ ) a(x, y, η) dydη .

22

1 Preliminaries

Therefore, σA (x, ξ ) = (2π )−n



.

ei(x−y)θ a(x, y, ξ + θ ) dydθ.

Theorem 1.1.19 Let .a ∈ S μ (, ) be a proper amplitude and .A = Op a. Then, (1) .σA ∈ S μ ().   1 α α  (2) .σA (x, ξ ) ∼ α! Dy ∂ξ a(x, y, ξ ) y=x . (3) .A = Op σA .

α

 Note that .Dyα ∂ξα a(x, y, ξ ) is an amplitude of order .μ−|α|. Hence, . Dyα ∂ξα a(x, y, ξ )y=x is a symbol of the same order. Therefore, the terms of series (2) are symbols of decreasing orders. Before proving Theorem 1.1.19, we verify the following lemma. Lemma 1.1.20 Assume that .a ∈ S μ (, ) is a proper amplitude, .h ∈ C[0, 1], and .α ∈ Zn+ . We set



σ (x, ξ ) =

.

1

ei(x−y)θ a(x, y, ξ + tθ )θ α h(t) dtdydθ.

(1.1.27)

0

Then, .σ ∈ S μ (). Proof Using the equality .ei(x−y)θ = Dy 2N ei(x−y)θ θ −2N , we rewrite (1.1.27) in the form



1

σ (x, ξ ) =

.

ei(x−y)θ Dy 2N a(x, y, ξ + tθ )θ α θ −2N h(t) dtdydθ.

(1.1.28)

0

Let .K1 ⊂  be any compact and .K2 = π2 (π1−1 (K1 )), where .π1 , π2 : suppx,y a →  are the projections. For .x ∈ K1 , we have Dy 2N a(x, y, ξ + tθ ) = 0 for y ∈ K2 ,

.

|Dy 2N a(x, y, ξ + tθ )| ≤ C1 ξ + tθ μ ≤ C1 2|μ|/2 ξ μ tθ |μ| ≤ C1 2|μ|/2 ξ μ θ |μ| for y ∈ K2 (C1 = C1 (K1 )). These relations lead to |Dy 2N a(x, y, ξ + tθ )θ α θ −2N h(t)| ≤ C2 χ (y)ξ μ θ |α|+|μ|−2N ,

.

x ∈ K1 , (1.1.29)

1.1 Pseudodifferential Operators

23

where .χ is the characteristic function of the set .K2 . For the case that .|α| + |μ| − 2N < −n, we obtain



1

.

χ (y)θ |α|+|μ|−2N dtdydθ < +∞.

(1.1.30)

0

Introduce the notation .BR = {ξ ∈ Rn : |ξ | ≤ R} with .R > 0. From (1.1.29) and (1.1.30) it follows that for .(x, ξ ) ∈ K1 × BR the integrand in (1.1.28) has the summable majorant .C2 χ (y) maxξ ∈BR ξ μ θ |α|+|μ|−2N independent of .(x, ξ ). Hence, integral (1.1.28) converges uniformly with respect to .(x, ξ ) ∈ K1 × BR , and formula (1.1.28) defines a continuous function .σ on the set .K1 × BR . Since the compact .K1 and number .R > 0 have arbitrarily been chosen, there holds the inclusion .σ ∈ C( × Rn ). Moreover, (1.1.28) and (1.1.29) provide the inequality |σ (x, ξ )| ≤ C(K1 )ξ μ ,

.

x ∈ K1 .

(1.1.31)

Assume now that .β and .γ are arbitrary multi-indices. Differentiating (1.1.28), we obtain γ

∂xβ ∂ξ σ (x, ξ ) =

.

 β i |β−ν| σν (x, ξ ) ν

(1.1.32)

ν≤β

with



σν (x, ξ ) =

.

0

1

ei(x−y)θ Dy N ∂xν ∂ξ a(x, y, ξ + tθ )θ α+β−ν θ −2N h(t) dtdydθ. γ

(1.1.33) Every function (1.1.33) is of the form (1.1.28) (with various a and .α). We choose N to satisfy .|α + β| + |μ| − 2N < −n. Then, from the first part of the proof, it follows that for all .ν integral (1.1.33) uniformly converges on any set of the form .K1 × BR . Hence, γ the differentiation in (1.1.28) was admissible and .σν ∈ C( × Rn ). Because .∂xν ∂ξ a ∈ S μ−|γ | (, ), the analogue of (1.1.31) for .σν is of the form |σν (x, ξ )| ≤ C(ν, K1 )ξ μ−|γ | ,

.

x ∈ K1 .

Therefore (see (1.1.32)), |∂xβ ∂ξ σ (x, ξ )| ≤ Cξ μ−|γ | ,

.

γ

x ∈ K1 .

In view of freedom in choosing the compact .K1 , we arrive at .σ ∈ S μ ().

 

24

1 Preliminaries

Proof of Theorem 1.1.19 Taking into account the equality σA (x, ξ ) = (2π )−n



ei(x−y)θ a(x, y, ξ + θ ) dydθ,

.

and the Taylor expansion a(x, . y, ξ +θ ) =

 |α|≤N −1

 N 1 α ∂ξ a(x, y, ξ )θ α+ α! α! |α|=N

0

1

(1−t)N −1 ∂ξα a(x, y, ξ+tθ )θ α dt,

we obtain 

σA (x, ξ ) =

.

|α|≤N −1

1 (α) σ (x, ξ ) + rN (x, ξ ), α!

where σ (α) (x, ξ ) = (2π )−n



.

rN (x, ξ ) = (2π )

.

−n

 ei(x−y)θ ∂ξα a(x, y, ξ )θ α dydθ = Dyα ∂ξα a(x, y, ξ )y=x ,

 N

1 ei(x−y)θ ∂ξα a(x, y, ξ + tθ )θ α (1 − t)N −1 dtdydθ. α! 0

|α|=N

By Lemma 1.1.20, the terms in the last sum, being considered as functions of .(x, ξ ), belong to .S μ−N (). Therefore, for any N σA (x, ξ ) = (2π )−n



.

|α|≤N −1

 1 α α Dy ∂ξ a(x, y, ξ )y=x + rN (x, ξ ), α!

rN ∈ S μ−N ().

This proves assertion 2 of the theorem, which implies assertion 1. It remains to verify that .A = Op σA . The representation of .σA can be written in the form

−n .σA (x, ξ ) = (2π ) ei(x−y)(η−ξ ) a(x, y, η) dydη. Therefore, (2π )−n

.



ˆ ) dξ = (2π )−2n eixξ σA (x, ξ )u(ξ

eixξ



ei(x−y)(η−ξ ) a(x, y, η) dydη u(ξ ˆ ) dξ,

that is, (Op σA )u(x) = (2π )

.

−2n



ˆ ) dydηdξ. eixη e−iy(η−ξ ) a(x, y, η)u(ξ

(1.1.34)

1.1 Pseudodifferential Operators

25

Now, we show that for fixed x the function

(ξ, η) →

.

eixη e−iy(η−ξ ) a(x, y, η)u(ξ ˆ ) dy = eixη u(ξ ˆ )



e−iy(η−ξ ) a(x, y, η) dy (1.1.35)

decreases rapidly as .|ξ | + |η| → ∞. Because of .uˆ ∈ S(Rn ), it suffices to obtain the estimate 

   e−iy(η−ξ ) a(x, y, η) dy  ≤ CN ξ 2N ημ−2N . for any .N ∈ N. In view of .e−y(η−ξ ) = Dy 2N e−iy(η−ξ ) η − ξ −2N , 



  e−iy(η−ξ ) a(x, y, η) dy  ≤ |Dy 2N a(x, y, η)|dy η − ξ −2N 

.

≤ CN ημ η−2N ξ 2N = CN ξ 2N ημ−2N , as we wanted to obtain. Due to the rapid decrease of function (1.1.35), we can in (1.1.34) permute the integrals with respect to .η and .ξ : (Op σA )u(x) = (2π )

.

−2n



ˆ ) dydξ dη. ei(x−y)η eiyξ a(x, y, η)u(ξ

By the integral in brackets is meant a double one (for fixed x and .η, the function .(y, ξ ) → a(x, y, η)u(ξ ˆ ) vanishes for large y and rapidly decreases as .ξ → ∞). Integrating over .ξ , we arrive at

−n .(Op σA )u(x) = (2π ) ei(x−y)η a(x, y, η)u(y) dydη = Au(x).   The asymptotic expansion of the symbol of a proper .DO A remains valid even in the case that the amplitude a satisfying the condition .A = Op a is not proper. Indeed, choose a properly supported function .χ ∈ C ∞ ( × ) equal to 1 in a neighborhood of the set . ∪ supp GA . Then, .A = Op (χ a) (see Lemma 1.1.12). Since the amplitude .χ a is proper, from Theorem 1.1.19, it follows that σA (x, ξ ) ∼

.

 1  1   Dyα ∂ξα (χ a)(x, y, ξ )y=x = Dyα ∂ξα a(x, y, ξ )y=x . α! α! α α

26

1 Preliminaries

The equalities .σA (x, ξ ) = eξ (x)Aeξ (x) and .A = Op σA establish a bijection between the proper .DO and their symbols. If A is an arbitrary .DO, it is usual to introduce −∞ . Such a symbol is .σA = σ A1 , where .A1 is a proper .DO and .A − A1 ∈  not uniquely defined; however, any two symbols differ by a function in .S −∞ . More precisely, the correspondence .A ↔ σA generates the isomorphism . μ ()/ −∞ () ∼ = S μ ()/S −∞ () for every .μ.

1.1.10 Symbolic Calculus of Pseudodifferential Operators 1◦ . Symbol expansions of transposed and adjoint operators. Theorem 1.1.21 Let A be a proper DO and σA its symbol. Then, σ tA (x, ξ ) ∼

.

 1 Dxα ∂ξα σA (x, −ξ ), α! α

(1.1.36)

 1 Dxα ∂ξα σA (x, ξ ). α! α

(1.1.37)

σA∗ (x, ξ ) ∼

.

Proof Because A = OpσA , we have Au(x) = (2π )−n

t

.



ei(x−y)ξ σA (y, −ξ )u(y) dydξ,

u ∈ Cc∞ ().

By Theorem 1.1.19, σ tA (x, ξ ) ∼

.

 1  Dyα ∂ξα σA (y, −ξ )y=x , α! α

which coincides with (1.1.36). The relation (1.1.37) can in a similar way be derived from the equality ∗

A u(x) = (2π )

.

−n



ei(x−y)ξ σA (y, ξ )u(y) dydξ.  

The function σ˜ A (x, ξ ) := σ tA (x, −ξ ) is sometimes called the dual symbol of A. From (1.1.36), it follows that σ˜ A (x, ξ ) ∼

.

 1 Dxα (−∂ξ )α σA (x, ξ ). α! α

1.1 Pseudodifferential Operators

27

2◦ . Symbol expansion of operator composition. Let A and B be arbitrary DO. In the case that u ∈ Cc∞ (), we have Bu ∈ C ∞ () but generally Bu ∈ Cc∞ (). Therefore, in the general case, the composition AB makes no sense. However, the composition AB is defined if at least one of the operators, A or B, is proper. Theorem 1.1.22 Let A and B be proper DO of order μ1 and μ2 , respectively. Then, 1. AB is a proper DO of order μ1 + μ2 .  1 α 2. σAB ∼ α α! ∂ξ σA (x, ξ )Dxα σB (x, ξ ). Proof Since B = t (t B) and t B = Op σ tB , we have Bu(x) = (2π )

.

−n

= (2π )−n



ei(x−y)ξ σ tB (y, −ξ )u(y) dydξ



eixξ dξ

e−iyξ σ˜ B (y, ξ )u(y) dy.

Therefore, ∧



(Bu) (ξ ) =

.

e−iyξ σ˜ B (y, ξ )u(y) dy.

Setting v = Bu in the equality Av(x) = (2π )−n

.

ˆ ) dξ, eixξ σA (x, ξ )v(ξ

we obtain −n

ABu(x) = (2π )

.



ei(x−y)ξ σA (x, ξ )σ˜ B (y, ξ )u(y) dydξ.

(1.1.38)

The function (x, y, ξ ) → σA (x, ξ )σ˜ B (y, ξ ) is an amplitude of order μ1 + μ2 , so AB ∈  μ1 +μ2 (). Now, we verify that DO AB is proper. According to Proposition 1.1.14, for any compact K2 ⊂ , there exists a compact K  ⊂  such that u ∈ Cc∞ (K2 ) ⇒ Bu ∈ Cc∞ (K  ).

.

In turn, for K  there exists a compact K1 ⊂  such that v ∈ Cc∞ (K  ) ⇒ Av ∈ Cc∞ (K1 ).

.

28

1 Preliminaries

Consequently, u ∈ Cc∞ (K2 ) ⇒ ABu ∈ Cc∞ (K1 ).

.

The operator t (AB) = t B tA has a similar property. Again employing Proposition 1.1.14, we conclude that AB is a proper DO. The asymptotic expansion of σAB remains to be obtained. Making use of Theorem 1.1.19, we derive from (1.1.38) that σAB (x, ξ ) ∼

.

.

=

 1  Dyα ∂ξα [σA (x, ξ )σ˜ B (y, ξ )]y=x α! α

 1  1  α! β γ ∂ξα [σA (x, ξ )Dxα σ˜ B (x, ξ )] = ∂ξ σA · ∂ξ Dxα σ˜ B α! α! β!γ ! α α β+γ =α

 1 β γ β+γ ∂ σA · ∂ξ Dx σ˜ B . β!γ ! ξ

=

β,γ

Furthermore, σ˜ B ∼

.

 1  (−1)|δ| γ +δ β+γ +δ γ β+γ Dxδ (−∂ξ )δ σB ⇒ ∂ξ Dx σ˜ B ∼ ∂ξ D x σB . δ! δ! δ

δ

Hence, σAB ∼

.

 (−1)|δ| β  1 β  (−1)|δ| γ +δ β+γ +δ ∂ξ σA ·∂ξ Dx ∂ξ σA ·∂ξν Dxβ+ν σB . σB = β!γ !δ! β! γ !δ!

β,γ ,δ

β,ν

γ +δ=ν

In the equality (x + y)ν =



.

γ +δ=ν

ν! γ δ x y γ !δ!

we set x = −y = (1, 1, . . . , 1) and obtain .

 (−1)|δ| =0 γ !δ!

γ +δ=ν

for ν = 0. Therefore, σAB ∼

.

 1 β ∂ σA Dxβ σB . β! ξ β

 

1.1 Pseudodifferential Operators

29

Corollary 1.1.23 Let A ∈  μ1 () and B ∈  μ2 () be proper DO. Then, [A, B] ≡ AB − BA ∈  μ1 +μ2 −1 (). Let A ∈  μ1 () and B ∈  μ2 (), while the operator B is proper. Then, as was mentioned before Theorem 1.1.22, both compositions AB and BA are defined. We show that they belong to  μ1 +μ2 (). To this end, we represent A in the form A1 + A2 , where A1 is a proper DO, and A2 is a smoothing DO. We have AB = A1 B + A2 B ,

.

BA = BA1 + BA2 .

(1.1.39)

By Theorem 1.1.22, the operators A1 B and BA1 are proper DO of order μ1 +μ2 . Denote by G the kernel of A2 . It is easy to verify that A2 B and BA2 have the smooth kernels t B G(x, y) and B G(x, y), respectively. Therefore, these operators are smoothing, that y x is, they belong to  −∞ (). Taking into account (1.1.39), we obtain the needed result.

1.1.11 Change of Variables in Pseudodifferential Operators Let . and .1 be domains in .Rn and let .f :  → 1 be a .C ∞ -diffeomorphism. Then, the mappings C ∞ (1 )  u → u ◦ f ∈ C ∞ (),

.

Cc∞ (1 )  u → u ◦ f ∈ Cc∞ () are isomorphisms of the linear topological spaces. A pseudodifferential operator A in . is in accordance with a linear continuous operator A1 : Cc∞ (1 ) → C ∞ (1 )

.

defined by A1 u = [A(u ◦ f )] ◦ f −1 .

.

(1.1.40)

Theorem 1.1.24 (1) The mapping  μ ()  A → A1

.

(1.1.41)

defined by (1.1.40), for any .μ, is a linear bijection of . μ () onto . μ (1 ) sending proper operators to proper ones.

30

1 Preliminaries

(2) There holds the asymptotic expansion  (α) .σA (f (x), η) ∼ σA (x, tf  (x)η)ϕα (x, η), 1 α

where .σA (x, ξ ) = ∂ξα σA (x, ξ ), . f  (x) = (∂fi /∂xj ) is the Jacobi matrix of the mapping f , and .ϕα ∈ C ∞ ( × Rn ) are polynomials of .η with degree .≤ |α|/2 being dependent on f but independent of A. (α)

Let us describe the basic stages of the proof. 1. Clearly, the mapping (1.1.41) is linear. It is a bijection because there exists the inverse mapping given by the formula .Av = [A1 (v ◦ f −1 )] ◦ f . In what follows, we write g instead of .f −1 . 2. Let .A = Op a with .a ∈ S μ (, ) and let .χ ∈ C ∞ ( × ) be a properly supported function equal to 1 in a neighborhood of the diagonal. We write an operator A in the form .B + C, where .B = Op (χ a) is a proper operator and .C = Op ((1 − χ )a) is a smoothing operator. Under the mapping (1.1.41), the operator C is in accordance with a −∞ ( ), so to prove .A ∈ ( ) it suffices to verify that .B ∈  μ (). .DO .C1 ∈  1 1 1 1 3. There exists a neighborhood V of the diagonal . 1 := {(x, y) ∈ 1 × 1 : x = y} such that, for all pairs .(x, y) ∈ V , the line segment joining x and y belongs to .1 . In the neighborhood V , the equality g(x) − g(y) = h(x, y)(x − y)

.

holds, where h is a matrix-valued function defined by the integral

h(x, y) =

.

1

g  (y + t (x − y)) dt.

0

For .x = y, we have .h(x, x) = g  (x). Since .|g  (x)| = 0, by continuity .|h(x, y)| = 0 for all .(x, y) in the vicinity of . 1 . Diminishing the neighborhood V , we can obtain .|h(x, y)| = 0 for .(x, y) ∈ V . 4. We choose .χ (see stage 2) so that the support of the function .χ1 defined by .χ1 (x, y) := χ (g(x), g(y)) belongs to V . We have B1 u(x) = [B(u ◦ f )] ◦ g(x)

.

= (2π )−n = (2π )

−n

−n

= (2π )









ei(g(x)−y)ξ a(g(x), y, ξ )χ (g(x), y)(u ◦ f )(y) dydξ ei(g(x)−g(y))ξ a(g(x), g(y), ξ )χ1 (x, y)|g  (y)|u(y) dydξ ei(x−y) h(x,y)ξ a(g(x), g(y), ξ )χ1 (x, y)|g  (y)|u(y) dydξ. t

1.1 Pseudodifferential Operators

31

Thus, B1 u(x) = (2π )−n



.

ei(x−y)η a(g(x), g(y), t h(x, y)−1 η)χ1 (x, y)×

× |g  (y)||h(x, y)|−1 u(y) dydη .

(1.1.42)

It can be verified that the function .(x, y, η) → a(g(x), g(y), t h(x, y)−1 η) × χ1 (x, y) is in .S μ (1 , 1 ). This proves the inclusion .B1 ∈  μ (1 ) and, consequently, .A1 ∈  μ (1 ). If A is a proper .DO, the equality .A = B + C implies that C is proper. Since .A1 = B1 + C1 and both operators on right are proper, A is proper as well. Thus, mapping (1.1.41) sends a proper .DO to a proper .DO. 5. If A is a proper .DO, from the very beginning, we can take .σA for the role of a. Then, (1.1.11) takes the form B1 u(x) = (2π )−n



ei(x−y)η a1 (x, y, η)χ1 (x, y)u(y) dydη,

.

where a1 (x, y, η) = σA (g(x), t h(x, y)−1 η) |g  (y)| |h(x, y)|−1 .

.

(1.1.43)

We apply Theorem 1.1.19 and obtain (taking into account .χ1 = 1 in a neighborhood of the diagonal . 1 ) σA1 (x, η) ∼ σB1 (x, η) ∼

.

 1  Dyβ ∂ηβ a1 (x, y, η)y=x ; β!

(1.1.44)

β

the first relation meaning that .σA1 −σB1 ∈ S −∞ (1 ) follows from . A1 = B1 +C1 , where −∞ ()). From (1.1.43), it follows that the terms of (1.1.44) are sums of terms of .C1 ∈  (β+γ ) (g(x), t g  (x)−1 η) × η δ , where .c ∈ C ∞ (1 ) and .|γ | = |δ| ≤ |β|. the form .c(x)σA We substitute these sums into (1.1.44), collect similar terms with the same .β + γ in one group, and arrive at  (α) .σA (x, η) ∼ ψα (x, η)σA (g(x), t g  (x)−1 η), 1 α

where .ψα (x, η) are polynomials in .η of degree .≤ |α|/2. Changing x for .f (x), we finally obtain  (α) .σA (f (x), η) ∼ ϕα (x, η)σA (x, tf  (x)η), 1 α

where .ϕα (x, η) = ψα (f (x), η) (we took into account .g  (f (x))−1 = f  (x)). .

32

1 Preliminaries

Remark We set . fx (y) = f (y) − f (x) − f  (x)(y − x). It can be shown that ϕα (x, η) =

.

1 α ifx (y)η  . D e y=x α! y

μ

1.1.12 Classes b (Rn ) This section is preparatory for proving the continuity of .DO in .L2 (Rn ). 1◦ . Amplitudes, symbols, and .DO. For any .μ ∈ R, we denote by .Sb (Rn , Rn ) the set of all amplitudes .a ∈ S μ (Rn , Rn ) satisfying μ

.

|∂xα ∂yβ ∂ξ a(x, y, ξ )| ≤ Cαβγ ξ μ−|γ | , γ

.

x, y, ξ ∈ Rn ,

μ

for all .α, β, γ ∈ Zn+ . We also let .Sb (Rn ) denote the set of all symbols .σ ∈ S μ (Rn ) such that β

|∂xα ∂ξ σ (x, ξ )| ≤ Cαβ ξ μ−|β| ,

.

x, ξ ∈ Rn . μ

The symbols can be considered as amplitudes independent of y so that .Sb (Rn ) ⊂  μ  μ  μ μ Sb (Rn , Rn ). We set .Sb−∞ = μ Sb , .Sb = μ Sb and .b (Rn ) = μ b (Rn ), where μ

μ

b (Rn ) = {Op a : a ∈ Sb (Rn , Rn )},

.

μ

μ ∈ [−∞, ∞). μ

Since . Sb (Rn , Rn ) is a linear subspace in .S μ (Rn , Rn ), the set .b (Rn ) is a linear subspace in . μ (Rn ). ◦ .2 . Symbol of .DO Proposition 1.1.25 For any .DO .A ∈ bm (Rn ) there exists the only symbol .σ ∈ Sbm (Rn ) such that .A = Op σ . Proof The uniqueness is evident because

.

ˆ ) dξ = 0 for ∀ u ∈ Cc∞ (Rn ) ⇒ σ = 0 eixξ σ (x, ξ )u(ξ μ

μ

(here, the conditions .A ∈ b (Rn ) and .σ ∈ Sb (Rn ) are not used; it suffices to assume that μ n μ n .A ∈  (R ) and .σ ∈ S (R )).

1.1 Pseudodifferential Operators

33 μ

Let us prove the existence. For .∀ s ≥ 0, there is an amplitude .a ∈ Sb (Rn , Rn ) such that .A = Op a and |∂xα ∂yβ ∂ξ a(x, y, ξ )| ≤ Cαβγ x − y−s ξ μ−|γ | γ

.

(∀ α, β, γ ).

(1.1.45)

μ

Indeed, if .A = Op a, ˜ .a˜ ∈ Sb (Rn , Rn ), one can take .a(x, y, ξ ) = Dξ 2N a(x, ˜ y, ξ )x − −2N , .N > s/2 (see Sect. 1.1.2). y μ Let .A = Op a, where the amplitude .a ∈ Sb (Rn , Rn ) satisfies (1.1.45) and .s > n. We set

−n .σ (x, ξ ) = (2π ) (1.1.46) ei(x−y)θ a(x, y, ξ + θ ) dydθ (cp. Sect. 1.1.9). Integrating by parts, we obtain σ (x, ξ ) = (2π )−n



.

ei(x−y)θ Dy 2N a(x, y, ξ + θ ) θ −2N dydθ.

(1.1.47)

If . N is sufficiently large, the integrand in (1.1.47) is summable (see estimate (1.1.48) below) and the integral (1.1.46) exists as an iterated one. We have   Dy 2N a(x, y, ξ + θ ) θ −2N  ≤ C x − y−s ξ + θ μ θ −2N

.

≤ 2|μ|/2 C x − y−s ξ μ θ |μ|−2N .

(1.1.48)

Therefore, |σ (x, ξ )| ≤ 2|μ|/2 C ξ μ

.



x − y−s θ |μ|−2N dydθ ≤ C  ξ μ .

β

The estimates .|∂xα ∂ξ σ (x, ξ )| ≤ Cξ μ−|β| can be established in a similar way. Thus, .σ ∈ μ   Sb (Rn ). The equality .Op σ = A is verified in the same way as in Theorem 1.1.19. μ

μ

We now assume that . A ∈ b (Rn ). A function .σ ∈ Sb (Rn ) satisfying .A = Op σ will be called a symbol of A and denoted by .σA . According to Proposition 1.1.25, any operator n n .A ∈ b (R ) (which does not need to be proper) has the only symbol. If .A ∈ b (R ) is a proper .DO, the symbol .σA just defined coincides with .σA in Sect. 1.1.9. Repeating the proof of Theorem 1.1.19 with evident modifications, we obtain the asymptotic expansion σA (x, ξ ) ∼

.

 1  Dyα ∂ξα a(x, y, ξ )y=x ; α! α

(1.1.49)

34

1 Preliminaries

here, a is any amplitude that defines A, and (1.1.49) means that .σA − μ−k Sb (Rn )



|α|≤k−1

... ∈

for all .k ∈ N.

3◦ . Symbols of transposed .DO and adjoint .DO. For .a ∈ Sb we have .t a , a ∗ ∈ Sb . μ μ Therefore, .A ∈ b implies .t A and .A∗ ∈ b . The expansions μ

.

σ (x, ξ ) ∼

 1 Dyα ∂ξα σA (x, −ξ ), α! α

σA∗ (x, ξ ) ∼

 1 Dyα ∂ξα σA (x, ξ ) α! α

. t A

μ

(1.1.50)

follow from (1.1.49) (see Sect 1.1.10). μ 4◦ . Composition of .DO. We assume .A ∈ b (Rn ) and note that in the integral

.

Au(x) = (2π )−n/2

.

ˆ ) dξ eixξ σA (x, ξ )u(ξ

(1.1.51)

one can change .u ∈ Cc∞ (Rn ) for any function in the Schwartz class .S(Rn ). Proposition 1.1.26 The operator .A : S(Rn ) → S(Rn ) is continuous. Proof Since .uˆ ∈ S, the integral (1.1.51) is convergent. Integrating by parts, we obtain Au(x) = (2π )

.

−n/2



ˆ )) dξ · x−2N . eixξ Dξ 2N (σA (x, ξ )u(ξ

We now note that .Dξ 2N (σA (x, ξ )u(ξ ˆ )) is a linear combination of the functions γ δ ˆ ) with .|γ + δ| ≤ 2N and every of these functions is majorized .∂ σA (x, ξ ) .× ∂ u(ξ ξ ξ by const.ξ s for any .s ∈ R. Therefore, taking .s < −n, we obtain

|Au(x)| ≤ C

.

ξ s dξ · x−2N = C  x−2N .

Because the number N can be chosen arbitrarily, the function Au rapidly decays as .|x| → ∞. A similar assertion for the derivatives .∂ α (Au) follows from the fact that .∂ α (Au) is a linear combination of integrals of the form (1.1.51): ∂ α (Au)(x) = (2π )−n



.

β≤α

i |β|



α ˆ ) dξ. eixξ ∂xα−β σA (x, ξ )ξ β u(ξ β

1.1 Pseudodifferential Operators

35

Verification of the continuity of the mapping .A : S → S is left to the reader. (The convergence .uj → 0 in .S means that, for any .α ∈ Zn+ and .s ∈ R, there exists a number   sequence .Cj → 0 such that .|∂ α uj (x)| ≤ Cj xs ). μ

In what follows, we assume that all operators in .b are defined on .S(Rn ). It is convenient because every such operator sends .S(Rn ) to .S(Rn ), and the composition makes sense for any A and B in .b (Rn ), which do not need to be proper. Let us show for μ1 μ2 μ1 +μ2 n n .A ∈  (Rn ). We use b (R ) and .B ∈ b (R ) that .AB ∈ b ABu(x) = (2π )−n



.

ei(x−y)ξ σA (x, ξ )σ˜ B (y, ξ )u(y) dydξ, μ

μ

where .σ˜ B (y, ξ ) = σtB (y, −ξ ) (see Sect. 1.1.10). Since .σA ∈ Sb 1 (Rn ) and .σ˜ B ∈ Sb 2 (Rn ), μ +μ the function .(x, y, ξ ) → σA (x, ξ )σ˜ B (y, ξ ) belongs to .Sb 1 2 (Rn , Rn ). Therefore, .AB ∈ μ1 +μ2 (Rn ). The expansion b σAB (x, ξ ) ∼

.

 1 ∂ξα σA (x, ξ )Dxα σB (x, ξ ) α! α

(1.1.52)

can be verified in the same way as in Sect. 1.1.10.

1.1.13 The Boundedness of DO in L2 (Rn ) Let .(u, v) and .$u$ be the inner product and norm in .L2 (Rn ) and let .BL2 (Rn ) be the set of all bounded operators in .L2 (Rn ). An operator .A ∈ (Rn ) is said to be bounded and n n .A ∈ BL2 (R ) if A extends to a bounded operator in .L2 (R ). Every following condition is equivalent to .A ∈ BL2 (Rn ): (i) .$Au$ ≤ C$u$ ∀ u ∈ Cc∞ (Rn ). (ii) .|(Au, v)| ≤ C$u$$v$ ∀ u, v ∈ Cc∞ (Rn ). For .A ∈ b (Rn ), we have .(Au, v) = (u, A∗ v) with any .u, v ∈ Cc∞ (Rn ). By continuity, the equality extends to .u, v ∈ S(Rn ). Since .$Au$2 = (Au, Au) = (A∗ Au, u) ≤ $A∗ Au$$u$, the inclusion .A∗ A ∈ BL2 (Rn ) ∈ BL2 (Rn ) implies .A ∈ BL2 (Rn ). Moreover, .A∗ A ∈ BL2 (Rn ) follows from .(A∗ A)∗ (A∗ A) = (A∗ A)2 ∈ BL2 (Rn ) and k so on. Therefore, if the operator .(A∗ A)2 is bounded for a certain .k ∈ Z+ , we obtain n .A ∈ BL2 (R ).

36

1 Preliminaries

Lemma 1.1.27 Let .G ∈ C(R2n ) and let A be an integral operator given for .u ∈ Cc∞ (Rn ) by

Au(x) =

G(x, y)u(y) dy.

.

Under the conditions

.C1 := sup |G(x, y)| dx < +∞ and

C2 := sup

y

x

the operator A belongs to .BL2 (Rn ) and .$A$ ≤

√

|G(x, y)| dy < +∞,

C1 C2 .

Proof Since .Cc∞ (Rn ) is dense in .L2 (Rn ), it suffices to show that .|(Au, v)| ≤ √ C1 C2 $u$$v$ for all .u, v ∈ Cc∞ (Rn ). We have |(Au, v)| ≤ 2

.



|G(x, y)||u(y)||v(x)| dxdy



2



≤

|G(x, y)||u(y)|2 dxdy

|G(x, y)||v(x)|2 dxdy



≤ C1

|u(y)|2 dy · C2

|v(x)|2 dx = C1 C2 $u$2 $v$2 .  

Theorem 1.1.28 Every operator .A ∈ b0 (Rn ) is bounded in .L2 (Rn ). Proof We consider the following three cases. μ

μ

(1) .A ∈ b , .μ < −n. Let .a ∈ Sb (Rn , Rn ) be an amplitude satisfying .A = Op a and |a(x, y, ξ )| ≤ C x − y−(n+1) ξ μ

.

(1.1.53)

(such an amplitude exists, see the proof of Proposition 1.1.25). Since .μ < −n, the function .ξ → a(x, y, ξ ) is summable for fixed x and y. Therefore,

Au(x) =

.

G(x, y)u(y) dy,

u ∈ Cc∞ (Rn ),

where G(x, y) = (2π )

.

−n

ei(x−y)ξ a(x, y, ξ ) dξ.

1.1 Pseudodifferential Operators

37

From (1.1.53), it follows that .|G(x, y)| ≤ C  x − y−(n+1) . Then,

C1 := sup

|G(x, y)| dx ≤ C

.

y

C2 := sup x





z−(n+1) dz < +∞

|G(x, y)| dy < +∞.

By Lemma 1.1.27, the operator A extends to a continuous operator in .L2 (Rn ). k μ (2) .A ∈ b , .μ < 0. For .k ∈ Z+ the order of .DO .(A∗ A)2 is equal to .μ2k+1 . If k is k chosen to satisfy .μ2k+1 < −n, the operator .(A∗ A)2 is bounded, and consequently, A is bounded as well. (3) .A ∈ b0 . We set .b = (M 2 − |σA |2 )1/2 with a constant M such that .sup |σA | < M. It is easy to verify .b ∈ Sb0 . Let .B = Op b. From the expansions (1.1.50) and (1.1.52), it follows that σA∗ A = |σA |2 + σ1 ,

.

σ1 ∈ Sb−1 ,

σB ∗ B = M 2 − |σA |2 + σ2 ,

σ2 ∈ Sb−1 .

Hence, .σA∗ A + σB ∗ B = M 2 + σ , where .σ ∈ Sb−1 . Therefore, A∗ A + B ∗ B = M 2 I + C ,

.

C ∈ b−1 ;

here, .I : u → u is the identity operator. Since .M 2 I + C ∈ BL2 (Rn ), we obtain ∗ ∗ n .A A + B B ∈ BL2 (R ). Then,   $Au$2 ≤ $Au$2 + $Bu$2 = (A∗ A + B ∗ B)u, u

.

≤ $A∗ A + B ∗ B$$u$2 . Thus, .A ∈ BL2 (Rn ).  

1.1.14 DO in Sobolev Spaces For .s ∈ R, the Sobolev space .H s (Rn ) is the completion of the set .S in the norm

 1/2 −n .$u$s := (2π ) < +∞. |u(ξ ˆ )|2 ξ 2s dξ

38

1 Preliminaries

For .s ∈ Z+ , the formula  .$u$s

=

 

|∂ α u|2 dx

1/2

|α|≤s

defines an equivalent norm in .H s (Rn ). We set .Dt = F −1 ξ t F for .t ∈ R. It is easy to see that the operator . Dt : H s (Rn ) → H s−t (Rn ) is unitary for all s. μ

Theorem 1.1.29 Every .DO .A ∈ b (Rn ) extends to a bounded operator .As : H s (Rn ) → H s−μ (Rn ) (∀ s). Proof Let us write A in the form .A = Dμ−s (Ds−μ AD−s )Ds . The operators s : H s (Rn ) → H 0 (Rn ) and .Dμ−s : H 0 (Rn ) → H s−μ (Rn ) are unitary, and .D the operator in parentheses belongs to .b0 (Rn ) and consequently extends to a bounded   operator .H 0 (Rn ) → H 0 (Rn ).

1.1.15 Elliptic Pseudodifferential Operators Definition 1.1.30 A symbol a ∈ S μ () is called an elliptic symbol of order μ if, for any compact K ⊂ , there exist positive constants c and t such that |a(x, ξ )| ≥ c ξ μ

.

for x ∈ K , |ξ | ≥ t.

(1.1.54)

We denote by ES μ () the set of all elliptic symbols of order μ. Unlike S μ (), the classes ES μ () are pairwise non-intersecting. If a ∈ ES μ (), the function (x, ξ ) → a(x, ξ )−1 is an elliptic symbol for large |ξ |. More precisely, let K ⊂  be an arbitrary compact and let c and t be constants corresponding to K (see (1.1.54)). Then,  β |∂xα ∂ξ a(x, ξ )−1 | ≤ Cξ −μ−|β|

.

for x ∈ K , |ξ | ≥ t

 β (∀ α, β; C = C(α, β, K)). Indeed, it is easy to verify by induction that ∂xα ∂ξ a(x, ξ )−1 is a linear combination of functions of the form a −k b, 1 ≤ k ≤ |α + β| + 1, b ∈ S (k−1)μ−|β| (). From the last inclusion, it follows that |b(x, ξ )| ≤ C ξ (k−1)μ−|β| ,

.

(x, ξ ) ∈ K × Rn ;

moreover, |a(x, ξ )−1 | ≤ c−1 ξ −μ for x ∈ K, |ξ | ≥ t. Therefore, |a(x, ξ )−k b(x, ξ )| ≤ Cc−k ξ −μ−|β|  β ⇒ |∂xα ∂ξ a(x, ξ )−1 | ≤ C  ξ −μ−|β| , x ∈ K, |ξ | ≥ t, .

what was needed. The ellipticity of a(x, ξ )−1 is evident.

1.1 Pseudodifferential Operators

39

We denote by E μ () the class of all proper DO A ∈  μ () satisfying σA ∈ ES μ (). Definition 1.1.31 An operator A ∈  μ () is called elliptic if A = A1 + A2 ,

.

A1 ∈ E μ (), A2 ∈  −∞ ().

Definition 1.1.32 A proper DO B is called a parametrix of a DO A if BA = I + R1 ,

.

AB = I + R2 ,

R1 , R2 ∈  −∞ ().

A parametrix, if it exists, is uniquely defined modulo  −∞ (). More precisely, let B be a left parametrix and B  a right parametrix of A, that is, BA = I + R1 , AB  = I + R2 , and R1 , R2 ∈  −∞ ()). Then, B − B  ∈  −∞ (). Indeed, BAB  = (I + R1 )B  = B  + R1 B  ,

.

BAB  = B(I + R2 ) = B + BR2 .

Therefore, B − B  = R1 B  − BR2 ∈  −∞ (). Theorem 1.1.33 For an elliptic DO A ∈  μ (), there exists a parametrix B ∈ E −μ (). Lemma 1.1.34 (1) For any symbol σ ∈ S μ (), there exists a proper DO B ∈  μ () such that σB −σ ∈ S −∞ ().  μj (2) For any asymptotic series ∞ j ∈ S (), there exists a proper DO B ∈ j =0 σj , σ  μ (), μ = max μj , such that σB ∼ ∞ j =0 σj . Proof of Lemma (1) B = Op (χ a), where χ ∈ C ∞ ( × ) is a function with proper support equal to 1 in a neighborhood of the diagonal.  (2) There exists a symbol σ ∈ S μ () satisfying σ ∼ j σj . It remains to use assertion (1).   Proof of Theorem We can assume that A ∈ E m (). Let σA ∈ ES m () be a symbol ∞ of A, {ψj }∞ j =1 ⊂ Cc () a partition of unity in , and cj , tj constants corresponding to the compact supp ψj (see (1.1.54)). We choose χ ∈ C ∞ (Rn ) such that χ (ξ ) = 0 for  |ξ | ≤ 1 and χ (ξ ) = 1 for |ξ | ≥ 2 and set σ˜ (x, ξ ) = j ψj (x)χ (ξ/tj )σA (x, ξ )−1 . Then, σ˜ ∈ S −m ().

40

1 Preliminaries

Let B0 ∈  −m () be a proper DO such that σB0 − σ˜ ∈ S −∞ (). Since σ˜ σA − 1 ∈ S −∞ (), we have σB0 σA − 1 ∈ S −∞ (). Taking into account σB0 A − σB0 σA ∈ S −1 () and σAB0 − σA σB0 ∈ S −1 (), we obtain B0 A = I + R0 ,

.

AB0 = I + R0 ,

R0 , R0 ∈  −1 ().

We denote by σj the symbol of R0 , j ≥ 0. Because R0 ∈  −j (), we conclude that  σj ∈ S −j () and the series j (−1)j σj is asymptotic. Now, we consider a proper DO  C of order zero such that σC ∼ j (−1)j σj . Then, CB0 A = I + R1 and R1 ∈  −∞ (). Indeed, for any N ∈ N there hold the equalities j

C = Op

N−1 

.

j

N−1  j (−1)j σj + Op σ (N ) = (−1)j R0 + Op σ (N ) ,

j =0

j =0

where σ (N ) ∈ S −N (). This implies CB0 A =

N−1 

.

j (−1)j R0 + Op σ (N ) (I + R0 ) = I − (−1)N−1 R0N + Op σ (N ) · (I + R0 ).

j =0

Therefore, CB0 A − I ∈  −N () for all N. Thus, the operator B := CB0 is a left  parametrix of A. A right parametrix is defined by B  = B0 C  , where σC  ∼ j (−1)j σj and σj is a symbol of (R0 )j . Since B −B  ∈  −∞ (), the operator B is a right parametrix as well.  

1.2

Meromorphic Pseudodifferential Operators

1.2.1

Integral Transforms on a Sphere

Let .ϕ = (ϕ1 , . . . , ϕn ) and .ω = (ω1 , . . . , ωn ) be unit vectors in .Rn , let .ϕω = ϕ1 ω1 + · · · + ϕn ωn , and let .S n−1 be the unit .(n − 1)-dimensional sphere with center at the coordinate origin. For .Re μ > −1 and .u ∈ C ∞ (S n−1 ), we introduce the operators ± .(Jμ u)(ϕ) μ

=

S n−1 μ

(±ϕω + i0)μ u(ω) dω,

(1.2.1) μ

μ

where .(ϕω + i0)μ = (ϕω)+ + eiμπ (ϕω)− , .(−ϕω + i0)μ = eiμπ (ϕω)+ + (ϕω)− , and, for example, .(ϕω)+ = ϕω for .ϕω ≥ 0 and .(ϕω)+ = 0 for .ϕω < 0.

1.2 Meromorphic Pseudodifferential Operators

41

Proposition 1.2.1 ([21], Prop. 1.1.2) The maps .Jμ± : C ∞ (S n−1 ) → C ∞ (S n−1 ) are continuous. The operator-valued functions .μ → Jμ± admit analytic extension to the complex .μ-plain. For all complex .λ, with the exception of .λ = i(k + n/2), where .k = 0, 1, . . . , and for ∞ n−1 ), we set .u ∈ C (S π

(E(λ)u)(ϕ) = (2π )−n/2 ei 2 (iλ+n/2) (iλ + n/2)

(−ϕω + i0)−iλ−n/2 u(ω) dω. ×

.

(1.2.2)

S n−1

Let us recall that the function .μ → (μ) is meromorphic throughout the .μ-plane. The poles of .-function are located at the points .μ = 0, −1, . . . , are simple, and res(μ)|μ=−k = (−1)k /k!.

.

(1.2.3)

According to Proposition 1.2.1, the operator-valued function .λ → E(λ) : C ∞ (S n−1 ) → C ∞ (S n−1 ) is analytic throughout except for the simple poles indicated above. The residue at .λ = i(k + n/2) is a finite-dimensional operator; taking into account (1.2.3), we obtain resE(λ)u|λ=i(k+n/2)

.

(−i)k+1  1 γ x = y γ u(y) dy. γ! (2π )n/2 S n−1

(1.2.4)

|γ |=k

We introduce the Mellin transform for the functions in .Cc∞ (Rn \ 0) by (Mr→λ u)(λ, ϕ) ≡ u(λ, ˜ ϕ) = (2π )

.

−1/2



+∞

r −iλ−1 u(r, ϕ) dr, λ ∈ C,

(1.2.5)

0

where .r = |x| and .ϕ = x/|x| with .x ∈ Rn \ 0. The inversion formula u(r, ϕ) =

.

−1 (Mλ→r u)(r, ˜ ϕ)

−1/2



= (2π )

r iλ u(λ, ˜ ϕ) dλ

(1.2.6)

Im λ=τ

and the Parseval equality



|u(λ, ˜ ϕ)|2 dλ =

.

Im λ=τ

+∞

r 2β |u(r, ϕ)|2 dr, τ = β + 1/2

(1.2.7)

0

hold. (These formulas can be obtained from the corresponding properties of the onedimensional Fourier transform by the change of variable .r = et .)

42

1 Preliminaries

Proposition 1.2.2 ([21], Prop. 1.2.1) The equality −1 Fx→ξ u = M(in/2−λ)→ρ Eϕ→ψ (λ)Mr→(λ+in/2) u

.

holds for .u ∈ Cc∞ (Rn \0), where .Im λ = n/2, .r = |x|, .ρ = |ξ |, .ϕ = x/|x|, and .ψ = ξ/|ξ |; the Fourier transform is written in the form (F u)(ξ ) = (2π )

.

−n/2



e−iξ x u(x) dx.

For all .λ ∈ C, except for .λ = −i(k + n/2) with .k = 0, 1, . . . , we introduce the operator π

(E(λ)−1 v)(ϕ) = (2π )−n/2 ei 2 (n/2−iλ) (n/2 − iλ)

(ϕω + i0)iλ−n/2 v(ω) dω, ×

(1.2.8)

.

S n−1

assuming .v ∈ C ∞ (S n−1 ). According to Proposition 1.2.1, the function .λ → E(λ)−1 : C ∞ (S n−1 ) → C ∞ (S n−1 ) is analytic throughout, with the exception of the mentioned points, which are simple poles. The residue at .λ = −i(k + n/2) is a finite-dimensional operator resE(λ)−1 v|λ=−i(k+n/2) =

.

i k+1  1 γ y γ v(y) dy. x n−1 γ! (2π )n/2 S

(1.2.9)

|γ |=k

We denote by .E(λ)∗ the operator adjoint to .E(λ) with respect to the inner product in n−1 ). .L2 (S Proposition 1.2.3 ([21], Props. 1.4.1 and 1.4.4) The following assertions are valid with .k = 0, 1, . . . : (1) .E(λ) and .E(λ)−1 are inverse to each other for .λ = ±i(k + n/2). ¯ −1 for .λ = i(k + n/2). (2) .E(λ)∗ = E(λ) We will consider the operators .E(λ)±1 in the spaces .H s (λ, S n−1 ) of (generalized) functions on the sphere with norm depending on .λ ∈ C. We first introduce the spaces. Let .M be a smooth compact manifold without boundary and let .{U, χ } be an atlas on n .M, that is, .{U } is a finite open covering of .M, and .χ : U → R are coordinate maps. We denote by .{ζ } a partition of unity subject to the covering. The space .H s (λ, M) is the completion of .C ∞ (M) with respect to the norm  $u; H (λ, M)$ =

.

s



U

1/2 |F (ζχ u)(ξ )| (1 + |ξ | + |λ| ) dξ 2

2

2 s

,

(1.2.10)

1.2 Meromorphic Pseudodifferential Operators

43

where .ζχ u = ζ u ◦ χ −1 on .χ (U ) and .ζχ u = 0 outside .χ (U ). For any fixed number .λ norm (1.2.10) is equivalent to the norm in the Sobolev space .H s (M). Another partition of unity and another (equivalent) atlas lead to an equivalent norm in .H s (λ, M). Proposition 1.2.4 ([21], Prop. 1.5.5 and Cor. 1.5.6) If .λ = i(k + n/2) (or .λ = −i(k + n/2)), where .k = 0, 1, . . . , the map .E(λ) : H s (λ, S n−1 ) → H s+Im λ (λ, S n−1 ) (resp. −1 : H s (λ, S n−1 ) → H s−Im λ (λ, S n−1 )) is continuous. On every closed set .F lying .E(λ) in the strip .{λ ∈ C : |Im λ| < h} and not containing the points .λ = i(k + n/2) (resp. .λ = −i(k + n/2)) the estimate $E(λ); H s (λ, S n−1 ) → H s+Im λ (λ, S n−1 )$ ≤ c(F)

(1.2.11)

$E(λ)−1 ; H s (λ, S n−1 ) → H s−Imλ (λ, S n−1 )$ ≤ c(F))

(1.2.12)

.

(resp. .

holds. The operator .E(λ) : L2 (S n−1 ) → L2 (S n−1 ) is unitary for .Im λ = 0.

1.2.2

Canonical Meromorphic Pseudodifferential Operators

To motivate the definition of a meromorphic .DO, we consider the convolution operator −1 .A = F ξ →x (ξ )Fy→ξ taking . as a homogeneous function of degree a, .Re a > −n/2, smooth on .S n−1 . In view of Proposition 1.2.2,

a −1/2 .ρ (ω)(F u)(ρ, ω) = (2π ) ρ i(in/2−λ) (ω) (1.2.13) Im λ=Re a

×E(λ − ia)u(λ ˜ − ia + in/2, ·) dλ. The integrand is holomorphic in the strip between the lines .Im λ = 0 and .Im λ = Re a. ˜ + in/2, ·) rapidly decreases We suppose that .u ∈ Cc∞ (Rn \ 0), then the function .λ → u(λ as .λ → ∞ in any strip .|Im λ| < h. Together with (1.2.11), this allows us to change the integration line in (1.2.13) for .Imλ = 0. Now, applying the inverse Fourier transform .F −1 and taking into account Propositions 1.2.2 and 1.2.3, we obtain

+∞ −1/2 .Au(r, ϕ) = (2π ) r i(in/2+λ) Eω→ϕ (λ)−1 (ω)Eψ→ω (λ − ia) (1.2.14) −∞

= (2π )−1/2

×u(λ ˜ − ia + in/2, ψ) dλ



r i(in/2+λ+ia) Aψ→ϕ (λ)u(λ ˜ + in/2, ψ) dλ, Im λ=−Re a

where .Aψ→ϕ (λ) = Eω→ϕ (λ + ia)−1 (ω)Eψ→ω (λ).

44

1 Preliminaries

Definition 1.2.5 Let .a ∈ C and . ∈ C ∞ (S n−1 × S n−1 ). The operator Aψ→ϕ (λ) = Eω→ϕ (λ + ia)−1 (ϕ, ω)Eψ→ω (λ),

.

(1.2.15)

where .λ = i(k + n/2) and .λ = −i(k + a + n/2), .k = 0, 1, . . . , is called a canonical meromorphic .DO of order a. Proposition 1.2.6 ([21], Prop. 3.1.1) Let .A be an operator of the form (1.2.15). Then, the estimate $A(λ); H s (λ, S n−1 ) → H s−Re a (λ, S n−1 )$ ≤ c(s, F)

.

holds on every closed set .F located in the strip .{λ ∈ C : |Im λ| < h} and not containing poles of the meromorphic operator-function .λ → A(λ). If the function . in (1.2.15) is independent of .ϕ, this assertion follows from Proposition 1.2.4. In the general case, we expand . in a series (ϕ, ω) =

.



amk (ϕ)Ymk (ω),

where .Ymk is a spherical function of order m. Since the coefficients .amk rapidly decrease as .m → ∞, the series  .

amk Eω→ϕ (λ + ia)−1 Ymk (ω)Eψ→ω (λ)

converges in the operator norm .$· ; H s (λ, S n−1 ) → H s−Re a (λ, S n−1 )$ uniformly on the set .F. Proposition 1.2.7 Let .A be an operator of the form (1.2.15), where .(ϕ, ω) = 0 in a neighborhood of the manifold .{(ϕ, ω) : ϕω = 0}. Then, for any .q ∈ R and every fixed .λ ∈ C such that .λ = i(k + n/2), .λ = −i(k + a + n/2) for .k = 0, 1, . . . the map A(λ) : H s (S n−1 ) → H s+q (S n−1 )

.

is continuous. To verify this assertion, recall (1.2.11) and take into account that the kernel of the integral operator .T (λ)v(ϕ) := Eω→ϕ (λ + ia)−1 (ϕ, ω)v(ω) belongs to .C ∞ (S n−1 × S n−1 ). We now describe the asymptotic behavior of .e−iμg(ϕ) Aψ→ϕ (λ)eiμg(ψ) u(ψ) as .|λ|2 + μ2 → ∞, where the operator .A is defined by (1.2.15), u and g are in .C ∞ (S n−1 ), and .|Im λ| ≤ h, where .μ ∈ R and g is a real function. In particular, the following

1.2 Meromorphic Pseudodifferential Operators

45

theorem justifies the name “pseudodifferential” for operators of the form (1.2.15) (see the “invariant” definition of .DO on a manifold given by Hörmander in [14]). We assume for the function . in (1.2.15) that the maps .ϕ → (ϕ, ω) and .ω → (ϕ, ω) are extended to n .R \ 0 as homogeneous functions of degree a, and u and g are homogeneous functions of degree zero. Theorem 1.2.8 ([21], Prop. 3.5.1) For any number .P ≥ 0, there exist non-negative numbers N and Q such that   1  −iμg(ϕ) (α) (ϕ, μ∇g(ϕ) + σ ϕ) Aψ→ϕ (λ)eiμg(ψ) u(ψ) − e α! |α|≤N     × Dyα [u(ψ)ρ i(λ+in/2) exp{iμ(g(ψ) − g(ϕ)) − i(y − ϕ, μ∇g(ϕ) + σ ϕ)}] ρ=1, ϕ=ψ  .

≤ c(μ2 + σ 2 )−P /2 $u; C Q (S n−1 )$, (1.2.16) where .σ = Re λ, .ρ = |y|, .ψ = y/|y|, and .(α) (ϕ, ω) = Dωα (ϕ, ω). According to the mentioned invariant definition of .DO in [14], to prove that .A(λ) is a .DO, it suffices to verify that .

exp (−iμg(ϕ))Aψ→ϕ (λ) exp (iμg(ψ))u(ψ)

expands in an asymptotic series as .μ → ∞ for fixed .λ. Formula (1.2.16) provides such an expansion. Let us show, in particular, how to obtain from (1.2.16) the principal symbol for the operator .A(λ) of order zero. The principal term of the asymptotic expansion is of the form .(ϕ, μg(ϕ) + σ ϕ)u(ϕ). Here .σ = Reλ, the functions g and . are extended as order zero homogeneous to .Rn \ 0, and .g(ϕ) is calculated in Cartesian coordinates. The vector .g(ϕ) is tangent to the sphere at .ϕ; hence, .ϕ and .g(ϕ) are orthogonal. A value of the sought principal symbol for the covariant vector tangent to the sphere at .ϕ and directed along .g(ϕ) is equal to .

lim (ϕ, μg(ϕ) + σ ϕ) = (ϕ, g(ϕ)).

μ→+∞

Choosing various functions g, we calculate the principal symbol for all pairs .(ϕ, ω), where .ϕ is a point of the sphere, and .ω is a cotangent vector at .ϕ. Thus, A(λ) = Eω→ϕ (λ)−1 (ϕ, ω)Eψ→ω (λ)

.

46

1 Preliminaries

is a .DO of order zero on the sphere .S n−1 and its principal symbol is the function .V  (ϕ, ω) → (ϕ, ω) on the set V of pairs .(ϕ, ω) of orthonormal vectors. If .(ϕ, ω) = 0 throughout V , the operator .A(λ) is elliptic and, consequently, Fredholm in .L2 (S n−1 ). In the representation theory of .C ∗ -algebras of pseudodifferential operators, operatorvalued functions of the form R  λ → A(λ) = Eω→ϕ (λ)−1 (ϕ, ω)Eψ→ω (λ) : L2 (S n−1 ) → L2 (S n−1 )

.

provide a series of irreducible representations (see Chap. 2). There arise questions about the existence of the inverse operator .A(λ)−1 for all .λ ∈ R and about the inequality $A(λ)−1 ; L2 (S n−1 ) → L2 (S n−1 )$ ≤ C

.

with constant C independent of .λ. In this connection, the condition (ϕ, ω) = 0 ∀ (ϕ, ω) ∈ V

.

is not sufficient, and the requirement (ϕ, ω) = 0 ∀ (ϕ, ω) ∈ S n−1 × S n−1 ,

.

is necessary, which can be interpreted as “ellipticity with regard to parameter.”

1.2.3

The Kernel of a Canonical Pseudodifferential Operator

Let us consider operator (1.2.15). We first assume that .−n/2 < Re a < 0, set .f = E(ia + in/2)−1 , and introduce the homogeneous function G of order .−n − a, G(x) = r −n−a f (ϕ), r = |x|, ϕ = x/|x|.

.

For .u ∈ Cc∞ (Rn \) we define the operator

(G  u) =

.

G(x − y)u(y) dy.

Since (F G)(ξ ) = |ξ |a Eϕ→θ (ia + in/2)f (ϕ) = |ξ |a (θ )

.

1.2 Meromorphic Pseudodifferential Operators

47

with .θ = ξ/|ξ |, we have .G  u = F −1 |ξ |a (θ )F u. In view of (1.2.14),

−1/2

(G  u)(x) = (2π )

.

r i(in/2+λ+ia) Aψ→ϕ (λ)u(λ ˜ + in/2, ψ) dλ, Im λ=−Re a

(1.2.17) where .Aψ→ϕ (λ) = Eω→ϕ (λ + ia)−1 (ω)Eψ→ω (λ). We extend f as an order zero homogeneous function to .Rn \ 0 and apply to .G  u the Mellin transform .Mr→λ+ia+in/2 with .Im λ = 0. Setting .x = rϕ, .y = ρψ, we have .

(2π )



−1/2

0

= (2π )−1/2



r −i(λ+ia+in/2)−1 (G  u)(x) dr

+∞

r −i(λ+ia+in/2)−1 dr

S n−1

+∞

dψ S n−1

dψ

0



=

+∞

+∞

G(rϕ − ρψ)u(ρψ)ρ n−1 dρ

0

t −i(λ+ia+in/2)−1 G(tϕ − ψ)u(λ ˜ + in/2, ψ) dt.

(1.2.18)

0

From (1.2.17) and (1.2.18), it follows that

Aψ→ϕ (λ)u(λ ˜ + in/2, ψ) =

G(ϕ, ψ; λ)u(λ ˜ + in/2, ψ) dψ,

.

where

G(ϕ, ψ; λ) =

.

+∞

r −i(λ+ia+in/2)−1 G(tϕ − ψ) dt.

(1.2.19)

0

Thus,

Aψ→ϕ (λ)v(ψ) =

.

G(ϕ, ψ; λ)v(ψ) dψ

(1.2.20)

for .v ∈ C ∞ (S n−1 ). We obtained representation (1.2.20) for the real .λ under the condition .−n/2 < Re a < 0. Let us define the representation for all complex a and .λ by analytic extension throughout with the exception of the poles. The operator .E(ia + in/2)−1 is not defined for .a = −n − l, .l = 0, 1, . . . and is not an isomorphism for .a = 0, 1, . . . (it annihilates a finitedimensional space, see [21], Prop. 1.4.3). Proposition 1.2.9 Let .a = 0, 1, . . . and .a = −n − l, where .l = 0, 1, . . . . Let also λ = i(n/2 + k), .λ = −i(n/2 + k + a), .k = 0, 1, . . . . Then, formula (1.2.20) holds for the operator .Aψ→ϕ (λ) = Eω→ϕ (λ + ia)−1 (ω)Eψ→ω (λ). The kernel .G(ϕ, ψ; λ) is

.

48

1 Preliminaries

defined by (1.2.19), where the integral is understood in the sense of analytic extension in λ. (Explicit formulas for the analytic extension are presented in [21], § 3.1.)

.

We now consider the operator .A(λ) for the values of a excluded in Proposition 1.2.9. Let a be a non-negative integer. The function . can be represented as a sum .0 + 1 , where .0 is subject to the conditions

.

S n−1

0 (θ )θ γ dθ = 0

for all multi-indices .γ such that .|γ | = a, and .1 is of the form 1 (θ ) =

[a/2] 

.

ha−2j (θ ),

j =0

hj being a harmonic polynomial of degree j . For .λ = i(n/2 + k), .λ = −i(n/2 + k + a), we have

.

A(λ) = E(λ + ia)−1 0 (θ )E(λ) + E(λ + ia)−1 1 (θ )E(λ).

.

(1.2.21)

For the first term on the right in (1.2.21), a formula of the form (1.2.20) holds with .G(x) = r −n−a Eω→ϕ (ia+in/2)−1 (ω) (note that .E(ia+in/2)−1  = E(ia+in/2)−1 0 ) and the second term is a differential operator of order a on the sphere .S n−1 in which the parameter .λ enters as in a polynomial of degree a. What has been said in this section on the operator .Aψ→ϕ (λ) = Eω→ϕ (λ + ia)−1 (ω) .×Eψ→ω (λ) can be generalized for operator (1.2.15). In particular, for .a = −n − a, the function G is defined by .G(ϕ, x) = r −n−a Eω→ϕ (ia + in/2)−1 (ϕ, ω), and the kernel .G(ϕ, ψ; λ) is defined by

+∞

G(ϕ, ψ; λ) =

.

r −i(λ+ia+in/2)−1 G(ϕ, tϕ − ψ) dt.

0

Using a representation of the form (1.2.20), one can verify Proposition 1.2.10 ([21], Prop. 3.1.6) Let .η, ζ ∈ C ∞ (S n−1 ) and .supp η ∩ supp ζ = ∅. Then, for operator (1.2.15) the estimate $ζ A(λ)η; H s (λ, S n−1 ) → H s+p (λ, S n−1 )$ ≤ c(F, p, s)

.

(1.2.22)

holds, where .F is an arbitrary closed set located in a strip of the form .{λ ∈ C : |Im λ| < h} and not containing poles of the function .λ → A(λ); p is any real number.

1.2 Meromorphic Pseudodifferential Operators

1.2.4

49

Operations on Canonical Meromorphic Pseudodifferential Operators

Here, we describe composition and the operations of taking the adjoint, differentiation, and shift with respect to the parameter for canonical meromorphic pseudodifferential operators. In what follows, we consider meromorphic operator-functions that have in every strip .{λ ∈ C : |Im λ| < h < ∞} at most finitely many poles. Any closed set located in a strip .{λ ∈ C : |Im λ| < h} and not containing poles of the operators considered will be called admissible. Let .σ ∈ C ∞ (S n−1 ) and . ∈ C ∞ (S n−1 × S n−1 ). We extend .σ to .Rn \ 0 as a homogeneous function of degree .ν and extend the function .ω → (ϕ, ω) as a homogeneous function of degree a; the numbers .ν and a are taken arbitrary complex. Proposition 1.2.11 ([21], Prop. 3.2.1) For an operator .A of the form (1.2.15), the formula A(λ)σ =

.

N  1 γ ∂ σ (ϕ)E(λ + iν + i(a − |γ |))−1 Dωγ (ϕ, ω)E(λ + iν) + RN (λ) γ!

|γ |=0

(1.2.23) holds with any .N ∈ Z+ ; the operator .RN is subject to the inequality $RN (λ); H s (λ; S n−1 ) → H s+N +1−Re a (λ; S n−1 )$ ≤ c(F, N)

.

(1.2.24)

on every admissible .F. (Here and in what follows, the letter .R denotes various operators that are remainders in “asymptotic” formulas.) Setting .ν = iμ, .σ (x) = |x|iμ , and .ϕ = x/|x| in (1.2.23) and then changing .λ − μ for .λ, we obtain the following assertion. Proposition 1.2.12 Let .μ ∈ C and let .A be an operator of the form (1.2.15). Then, A(λ + μ) − A(λ) =

.

N  1 γ iμ  (∂x |x| ) |x|=1 Eω→ϕ (λ + i(a − |γ |))−1 γ!

|γ |=1

×Dωγ (ϕ, ω)Eψ→ω (λ) + RN (λ, μ)

(1.2.25)

and $RN (λ, μ); H s (λ; S n−1 ) → H s+N +1−Re a (λ; S n−1 )$ ≤ c(F, N)|μ|

.

(1.2.26)

50

1 Preliminaries

on every admissible set .F. If .|μ| < δ, with .δ an arbitrary small number, then a set .F that γ is admissible for the operators .E(λ + i(a − |γ |))−1 Dω (ϕ, ω)E(λ), .0 ≤ |γ | ≤ N, is admissible also for all operators .λ → RN (λ, μ), and the constant .c(F, N) in (1.2.26) can be chosen to be independent of .μ. It follows from (1.2.25) that if .A is an operator of order a, then the difference .A(λ + μ) − A(λ) is an operator of order .a − 1. Taking into account (1.2.25) and (1.2.26), we obtain Proposition 1.2.13 For .N = 1, 2, . . . the derivative of operator (1.2.15) admits the representation N   1 ργ (x)|x|=1 Eω→ϕ (λ + i(a − |γ |))−1 γ!

∂λ A(λ) =

.

|γ |=1

×Dωγ (ϕ, ω)Eψ→ω (λ) + RN (λ),

(1.2.27)

where .ργ (x) = limμ→0 μ−1 ∂x |x|iμ , while .RN is subject to estimate (1.2.24). γ

We now consider the composition of canonical meromorphic pseudodifferential operators. Let A(λ) = E(λ + ia)−1 (ϕ, ω)E(λ),

.

B(λ) = E(λ + ib)−1 (ϕ, ω)E(λ).

.

We will assume that the function . is extended to .Rn \ 0 in each argument .ϕ and .ω as a homogeneous function of degree a, and . as a homogeneous function of degree b. Proposition 1.2.14 The formula A(λ)B(λ) =

.

N  1 E(λ + i(a + b − |γ |))−1 Dωγ (ϕ, ω)∂ϕγ (ϕ, ω)E(λ) + RN (λ) γ!

|γ |=0

(1.2.28) holds for any .N ∈ Z+ . The estimate $RN (λ); H s (λ; S n−1 ) → H s+N +1−Re (a+b) (λ; S n−1 )$ ≤ c(F, N)

.

is valid on every admissible set .F.

1.2 Meromorphic Pseudodifferential Operators

51

 Proof We expand . in a series of spherical harmonics, .(ϕ, ω) = amk (ϕ)Ymk (ω). The coefficients .amk are taken to be homogeneous functions of degree b. The series B(λ) =

.



amk (ϕ)E(λ + ib)−1 Ymk (ω)E(λ)

(1.2.29)

converges in the norm of operators from .H s (λ, S n−1 ) to .H s−Re b (λ, S n−1 ). According to (1.2.23), A(λ)amk =

.

N  1 γ ∂ amk (ϕ)E(λ + i(a + b − |γ |))−1 Dωγ (ϕ, ω)E(λ + ib) + RN (λ). γ! ϕ

|γ |=0

It remains to apply (1.2.29).

 

Proposition 1.2.15 ([21], Prop. 3.2.5) Let .A(λ) = E(λ + ia)−1 (ϕ, ω)E(λ) and let ∗ n−1 ). Then, .A(λ) be the operator adjoint to .A(λ) with respect to the duality in .L2 (S A(λ)∗ =

.

N  1 E(λ + i(a − |γ |))−1 ∂ϕγ Dωγ (ϕ, ω)E(λ) + RN (λ), γ!

|γ |=0

where .N = 0, 1, . . . , and the operator .RN is subject to inequality (1.2.24). The function . is taken to be extended to .Rn \ 0 with respect to .ϕ and with respect to .ω as a homogeneous function of degree a.

1.2.5

General Meromorphic Pseudodifferential Operators

The composition of canonical meromorphic pseudodifferential operators is not, in general, a canonical .DO. The operations of taking the adjoint and differentiation with respect to parameter also lead out the class of canonical operators. We define a larger class of meromorphic pseudodifferential operators, which is invariant already under the mentioned operations. Let .a0 , a1 , . . . be a sequence of complex numbers, .Re aj ≥ Re aj +1 , and .Re aj → ∞ n−1 × S n−1 ) assuming that −∞. We denote by .{}∞ j =0 a sequence of functions in .C (S the functions .ϕ → j (ϕ, θ ) and .θ → j (ϕ, θ ) are extended to .Rn \ 0 as homogeneous functions of degree .aj . Definition 1.2.16 An operator-valued function .λ → A(λ) that is meromorphic in the complex plane is called a meromorphic pseudodifferential operator of order .a0 if every

52

1 Preliminaries

strip of the form .{λ ∈ C : |Imλ| < h} contains at most finitely many poles of .A and the inequality $A(λ) −

N 

.

Aj (λ); H s (λ, S n−1 ) → H s−Re aN+1 (λ, S n−1 )$ ≤ c(F, N, s)

(1.2.30)

j =0

holds for any .N ∈ Z+ on every set .F that is admissible for the operators .A and .Aj , −1  (ϕ, θ )E .j = 0, 1, . . . , N, where .Aj (λ) = Eθ→ϕ (λ + iaj ) j ψ→θ (λ).  The formal series . j is called the complete symbol of .A, and the function .0 will  be called the principal symbol. A series . ∞ j =0 Aj consisting of canonical meromorphic pseudodifferential operators is called an asymptotic series for a meromorphic pseudodifferential operator .A if inequalities of the form (1.2.30) hold for all N and .F. In the sequel,   the notation .A ∼ Aj means that . Aj is an asymptotic series for .A. We note, for example, that by Proposition 1.2.12 the function .λ → A(λ + μ) with .A defined by (1.2.15) is a meromorphic .DO of order a with symbol

.

∞  1 −iμ γ iμ γ |x| ∂x |x| Dθ (x, θ ) γ!

|γ |=0

(the function . is homogeneous of degree a with respect to x and with respect to .θ ). Theorem 1.2.17 ([21], Thm. 3.3.3) There exists a meromorphic .DO with any given  symbol . j .

1.2.6

Change of Variables in Meromorphic Pseudodifferential Operators

Let .g : S n−1 → S n−1 be a diffeomorphism, .g(σ ) = (g(σ )1 , . . . , g(σ )n ), .σ ∈ Rn , n .|σ | = 1, and .|g(σ )| = 1. We assume that the functions .gj are defined on .R \ 0 and are n   homogeneous of degree one, that .g (σ ) = $∂gj /∂σk $j,k=1 and that .|g (σ )| is a modulus of the determinant .det g  (σ ). Theorem 1.2.18 ([21], Thm. 3.8.1) Let .A be an operator of the form (1.2.15). Then, ˆ τ →σ (λ) |g  (τ )| u(τ Aψ→ϕ (λ)u(ψ) = |g  (σ )−1 | A ˆ ),

.

1.3 C ∗ -algebras

53

ˆ τ →σ is a meromorphic .DO with where .ϕ = g(σ ), .ψ = g(τ ), .u(τ ˆ ) = u(g(τ )), and .A symbol

.

∞  1 γ ih(ω,σ,τ )  D γ (g(σ ), (g  (σ )−1 )∗ ω), ∂ e τ =σ ω γ! ω

|γ |=0

where, moreover, .h(ω, σ, τ ) = (ω, τ − g  (σ )−1 g(τ )) = O(|σ − τ |2 ).

1.3

C ∗ -algebras

1.3.1

C ∗ -algebras and Their Morphisms

Let .A be an algebra over the field .C. We say that a map .x → x ∗ of .A into itself is an involution if the following conditions hold: .(x ∗ )∗ = x, .(x + y)∗ = x ∗ + y ∗ , .(λx)∗ = λx ∗ , and .(xy)∗ = y ∗ x ∗ . An algebra endowed with involution is called involutive. An involutive normed algebra is a normed algebra .A with involution such that .$x ∗ $ = $x$ for all .x ∈ A. A complete involutive normed algebra is called an involutive Banach algebra. If .$x ∗ $2 = $x ∗ x$ for every element .x ∈ A, the involutive Banach algebra is called a ∗ .C -algebra. Let .A be an involutive algebra. A subalgebra of .A that is mapped into itself under involution is, by definition, an involutive subalgebra. Any closed involutive subalgebra of a .C ∗ -algebra is a .C ∗ -algebra. Let us denote by .BH the algebra of all bounded linear operators on a Hilbert space H . Involution in .BH means transition to the adjoint operator. Every closed involutive subalgebra of .BH is a .C ∗ –algebra. Let .A and .B be involutive algebras. A morphism .f : A → B is a linear map such that ∗ ∗ .f (xy) = f (x)f (y) and .f (x ) = f (x) for all .x, y ∈ A. A bijective morphism is called an isomorphism. It turns out that any morphism of .C ∗ -algebras .A and .B (simply regarded as a morphism of involutive algebras) is continuous, and its norm is no greater than one. If .f : A → B is an injective morphism, then .$f (x)$ = $x$ for all .$x$ ∈ A. Every .C ∗ algebra is isomorphic to a closed involutive subalgebra of an algebra .BH . Let .A be a .C ∗ -algebra and J a closed two-sided ideal in .A. Then, J is self-adjoint (i. e., is preserved as a set under involution) and the quotient algebra .A/J , endowed with the quotient norm and corresponding involution, is a .C ∗ -algebra. Let .A and .B be .C ∗ -algebras, .ϕ : A → B a morphism, and I a kernel of .ϕ. Then, I is a closed ideal in .A and the image .ϕ(A) is closed in .B. Indeed, the map .ϕ is continuous because any morphism of .C ∗ -algebras is continuous. Hence, I is a closed ideal, and .A/I is a .C ∗ -algebra. The morphism .A/I → B obtained by factoring through .ϕ is injective; hence, it is an isometry. Thus, the .ϕ(A) is closed and complete in .B.

54

1.3.2

1 Preliminaries

Representations of C ∗ -algebras

In the sequel, an algebra and morphism will always mean .C ∗ -algebra and .∗-morphism. A representation of an algebra .A in a Hilbert space H is a morphism .π : A → BH . The space H is called the representation space of .π , and the (Hilbert) dimension of H is called the dimension of the representation. Representations .π and .π  of an algebra .A in spaces H and .H  are equivalent, by definition, if there is a Hilbert space isomorphism .U : H → H  such that .U π(x) = π  (x)U for all .x ∈ A. We say that a vector .ξ ∈ H is cyclic (or totalizing) for a representation .π of an algebra .A in H if the closure of the set .π(A)ξ coincides with H . A representation .π of an algebra .A ∈ H is called irreducible if .π satisfies either one of the following (equivalent) conditions: (1) The only closed subspaces in H that are invariant under .π(A) are 0 and H . (2) The commutant of .π(A) in .BH consists of scalars. (3) Either every nonzero vector .ξ ∈ H is totalizing for .π or .π is the null representation of dimension 1. A subalgebra .A ⊂ BH is called irreducible if its identity representation .A → BH : a → a is irreducible. The following assertion shows that, for every algebra .A, there are “sufficiently many” irreducible representations. Proposition 1.3.1 ([3], 2.7.3) There exists a family .{πi } of irreducible representations of an algebra .A such that .$a$ = supi $πi (a)$ for all .a ∈ A. Proposition 1.3.2 Let .π be an irreducible representation of an algebra .A in a Hilbert space H . Then: (1) If I is a two-sided ideal in .A and .π(I ) = 0, then the restriction .π |I of .π to I is irreducible. (2) If .I1 and .I2 are two-sided ideals in .A and .π(I1 ) = 0 and .π(I2 ) = 0, then .π(I1 I2 ) = 0. Proof (1) We set .E = {x ∈ H : π(I )x = 0}. Since .π(A)E ⊂ E and by requirement .E = H , we have .E = 0. Hence, .π(I )ξ = 0 regardless the nonzero vector .ξ ∈ H . The subspace .π(I )ξ is invariant under .π(A). Since .π is irreducible, we find .π(I )ξ = H . Thus, any nonzero vector .ξ ∈ H is cyclic for .π |I . (2) As was proven in (1), .π(I2 )H = H and .π(I1 )π(I2 )H = H . Hence, .π(I1 I2 ) = 0.  

1.3 C ∗ -algebras

55

The kernel of an irreducible representation of an algebra .A is called a two-sided primitive ideal in .A. Every closed two-sided ideal in .A is the intersection of the primitive ideals containing it ([3], 2.9.7). For example, the ideal .comA generated by commutators .[a, b] = ab − ba, where .a, b ∈ A, coincides with the intersection of the kernels of all one-dimensional representations of .A. Proposition 1.3.3 ([3], 2.11.4) Let .I1 and .I2 be two-sided ideals in an algebra .A and let I be a primitive ideal. If .I1 I2 ⊂ I (in, particular, if .I1 ∩ I2 ⊂ I ), then either .I1 ⊂ I or .I2 ⊂ I . Proof Assume the contrary, i. e., .I1 ⊂ I and .I2 ⊂ I . Applying the second assertion of Proposition 1.3.2 to an irreducible representation .π of .A with kernel I , we find .π(I1 I2 ) =   0, whence .I1 I2 ⊂ I . Proposition 1.3.4 ([3], 4.1.5) Let .A = KH be the algebra of compact operators in a Hilbert space H . Then, every nonnull irreducible representation of .A is equivalent to the identity representation. We say that .A is an algebra of type I if the set .π(A) contains .KH for every irreducible representation .π of .A in H . Proposition 1.3.5 ([3], 4.1.10) Let .π be an irreducible representation of an algebra .A in H . If .π(A) ∩ KH = 0, then .KH ⊂ π(A) and every irreducible representation of .A with the same kernel as .π is equivalent to .π . Proposition 1.3.6 ([3], 2.10.2) (extension of a representation) Let .B be a subalgebra of A and .ρ a representation of .B in a Hilbert space G. Then, there exist a Hilbert space H containing G as a subspace and a representation .π of .A in H such that .ρ(x) = π(x)|G for all .x ∈ B. If .ρ is irreducible, .π can be chosen to be irreducible.

.

1.3.3

Spectrum of C ∗ -algebra

Let Prim.A be the set of two-sided primitive ideals of an algebra .A and .T ⊂ PrimA. We denote by .I (T ) the intersection of all ideals in T . The set .I (T ) is a two-sided ideal in .A. Moreover, let .T¯ be the set of primitive ideals containing .I (T ). It turns out that there exists a unique topology on Prim.A such that .T¯ is the closure of T for all .T ⊂ PrimA in this topology, which is called the Jacobson topology on Prim.A.  of equivalence classes of nonnull irreducible representations We introduce the set .A  → PrimA. The of an algebra .A. The map .π → kerπ defines a canonical surjection .A  spectrum of .A is the set .A endowed with the topology that is a preimage of the Jacobson  → PrimA. topology under the canonical map .A

56

1 Preliminaries

 Let We also provide another description of the Jacobson topology on the spectrum .A. : .{xi } be a family of elements of .A that is everywhere dense in .A and let .Zi = {π ∈ A  $π(xi )$ > 1$. Then, the set of .Zi is a base of the topology on .A. Thus, if the algebra .A is  admits a countable base. separable, then the topology on .A We say that a topological space is a .T0 -space if, given two points of this space, there is a neighborhood of one point not containing the other point. The following three conditions are equivalent ([3], 3.1.6):  is a .T0 -space. (1) .A (2) Two irreducible representations of .A with the same kernel are equivalent.  → PrimA is a homeomorphism. (3) The canonical map .A Proposition 1.3.7 ([3], 3.2.1 and 3.2.2) (1) Let I be a closed two-sided ideal of an algebra .A and let .p : A → A/I be the I = {π ∈ A  : π |I = 0}. Then, the maps canonical morphism. We set .A I , (A/I )  π → π ◦ p ∈ A

.

\ A I  π → π |I ∈ I A

are homeomorphisms. (2) The map .I → I is a bijection of the set of the closed two-sided ideals of .A onto the  Moreover, .I1 ⊂ I2 ⇔ I1 ⊂ I2 . set of open parts of the spectrum .A. As before, let .KH be the algebra of compact operators in a Hilbert space H and X a locally compact space. We consider the algebra .A = C0 (X) ⊗ KH of the continuous functions taking values in .KH and tending, in the operator norm, to zero at infinity. The norm in .A is defined by $f $ = sup $f (x) ; BH $.

.

x∈X

We set .π(x)f = f (x) for .x ∈ X. It is evident that .π(x) is an irreducible representation of A in H . The following proposition is an instance of assertion 10.4.4 in [3].

.

Proposition 1.3.8 The map .x → π(x) is a homeomorphism of X onto .(C0 (X) ⊗ KH )∧ . A subalgebra .B of an algebra .A is called rich if the following conditions hold: (1) For every irreducible representation .π of .A, the representation .π |B is irreducible. (2) If .π1 and .π2 are nonequivalent irreducible representations of .A, the representations .π1 |B and .π2 |B are nonequivalent.

1.3 C ∗ -algebras

57

Proposition 1.3.9 ([3], 11.1.4) If .B is a rich subalgebra of the algebra .C0 (X)⊗KH , then B = C0 (X) ⊗ KH .

.

1.3.4

Criteria for an Element of an Algebra to Be Invertible or to Be Fredholm

Proposition 1.3.10 Let A be an algebra with identity. An element a ∈ A is invertible if and only if the operator π(a) is invertible for every irreducible representation π of A (in the space Hπ of the representation). Proof The necessity of the condition for an element a ∈ A to be invertible is evident. Let us verify the sufficiency. Let us assume that a does not have a right (or left) inverse. Then, the element b = b∗ = aa ∗ (respectively, a ∗ a) cannot be inverted. We introduce the commutative algebra B generated by b and the identity of A. It is well-known ([3], 1.5) that B is isomorphic to the algebra of continuous functions on the spectrum Spb. Since b is noninvertible, 0 belongs to the spectrum Spb, and there is a one-dimensional representation  extending ρ ρ of B annihilating b. By Proposition 1.3.6, there is a representation π ∈ A in the sense that Hρ is a one-dimensional subspace of Hπ and π(x)|Hρ = ρ(x) for all   x ∈ B. Since π(b)|Hρ = 0, the operator π(b) (hence also π(a)) is noninvertible. Proposition 1.3.11 ([3], 1.3.10) Let B be a subalgebra of an algebra A that has an identity that, moreover, belongs to B. Then, for any x ∈ B we have SpA x = SpB x, where SpC y denotes the spectrum of an element y in the algebra C. Thus, if an element x ∈ B is invertible in A, then it is also invertible in B. A continuous linear operator A : E1 → E2 , where E1 and E2 are Banach spaces, is called Fredholm if the range ImA is closed and the kernel and cokernel are finitedimensional, dim kerA < ∞ and dim cokerA < ∞. Proposition 1.3.12 Let A be a subalgebra of BH and let A contain both the ideal KH and the identity operator. Then, an operator A ∈ A is Fredholm if and only if every operator π(A) is invertible, where π ∈ (A/KH ). Proof According to the Atkinson theorem (e.g., see [19], Theorem 1.4.16), an operator A is Fredholm if and only if the class [A] ∈ BH /KH is invertible. Taking into account Proposition 1.3.11, we obtain that A is Fredholm if and only if the class [A] is invertible   in A/KH . It remains to apply Proposition 1.3.10.

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1 Preliminaries

1.3.5

Continuous Field of C ∗ -algebras

We denote by .B(E, E  ) the space of linear operators acting from a Banach space E into a Banach space .E  ; for .E = E  , we write .B(E). Let .C(T , L) be the space of continuous functions on T with values in L, where T and L are topological spaces. We assume that for every point .t ∈ T , there corresponds a Banach space .Et . Every map x:T →



.

Et ,

t∈T

subject to the condition .x(t) ∈ Et for all .t ∈ T , is called a vector field on T . Sometimes it  is convenient to identify a vector field x with its image .{x(t)} ⊂ Et . A continuous field .E of Banach spaces on T is a family of Banach spaces .{Et }t∈T endowed with a set . of vector fields such that (i) (ii) (iii) (iv)

 is a linear space. The set .{x(t) : x ∈ } is dense in .Et for every .t ∈ T . The function .t → $x(t)$ is continuous for every .x ∈ .  If .y ∈ t∈T Et is a vector field, and for any .t0 ∈ T and .ε > 0, there exists a field .x ∈  for which .$y(t) − x(t)$ < ε in a neighborhood of .t0 , then .y ∈ . .

The elements .x ∈  are called continuous vector fields in .E. The space . is a module over the ring .C(T ) ([3], 10.1.9). An isomorphism of continuous vector fields .E = ({Et }t∈T , ) and .E  = ({Et }t∈T ,   ) is a family .ϕ = {ϕ(t)}t∈T of linear operators .ϕ(t) : Et → Et such that (1) For every t, the operator .ϕ(t) is an isometric isomorphism .Et onto .Et . (2) .ϕ() ⊂   (this requirement is equivalent to the condition .ϕ() =   ). Example We consider a product .T × E, where E is a Banach space, and assume that Et = {t} × E, .q : T × E → E is a projection, and .qt = q|Et . The map q allows us to endow every fiber .Et with the structure of a Banach space so that the operators .qt : Et → E become isometric isomorphisms. Let us choose as a linear space . the set of all vector fields of the form .t → (t, f (t)), where .f ∈ C(T , E). Then .({Et }, ) is a continuous field of Banach spaces; it is called a constant field. We denote this field (not quite correctly) by .T × E and will write .(T × E) instead of .. A field that is isomorphic to a constant field is called trivial. .

Let .E = ({Et }t∈T , ) be a continuous field of Banach spaces on T and . ⊂ T an open set. We denote by . the set of vector fields on . which are the limits of elements in .| with respect to local uniform convergence. Then .({Et }t∈ ,  ) is a continuous field of Banach spaces on ., which is denoted by .E| and is called a field induced by .E on ..

1.3 C ∗ -algebras

59

A field .E is called locally trivial if, for any point t, there is a neighborhood . such that the field .E| is trivial. Let .E be a locally trivial field of Banach spaces on T and let .{i }i∈I be an open covering of T such that all fields .E|i are trivial and .ϕi : i × Ei → E|i is an isomorphism of trivial fields. A family of cards .{(ϕi , i )}i∈I is called a trivializing atlas (or simply an atlas) for .E. Let .E = ({Et }t∈T , ) be a continuous field of Banach spaces. We say that a set . ⊂  is total if for every .t ∈ T the set .{x(t) : x ∈ } is total in .Et , i. e., the closure of its linear hull coincides with .Et . A field .E is called separable if . contains a countable total subset. Lemma 1.3.13 ([3], 10.2.7) Let T be a separable metrizable space and .E = ({Et }, } a locally trivial continuous field of Banach spaces on T . If every .Et is separable, then also .E is separable. Lemma 1.3.14 ([3], 10.8.7) Let T be a finitely dimensional (in the sense of topological dimension) paracompact space and .E = ({Et }t∈T , ) a separable continuous field of Hilbert spaces; the dimension of every space .Et is .ℵ0 . Then, the field .E is trivial. We now discuss an isomorphism of trivial fields in more detail and assume .ϕ : T ×E → T × E  to be such an isomorphism. Then .ϕ(t, e) = (t, At e) for every .t ∈ T , where .At is a linear isometry .E → E  and .At = qt ϕ(t)qt−1 . Lemma 1.3.15 Let .ϕ : T × E → T × E  be an isomorphism of trivial fields and .At = qt ϕ(t)qt−1 for .t ∈ T . Then, the function .t → At ∈ B(E, E  ) is continuous with respect to strong operator topology. Proof We fix .e ∈ E. Since the vector field .t → (t, e) belongs to .(T × E), its image  .t → (t, At e) under the isomorphism .ϕ belongs to .(T ×E ). Hence, the function .t → At e is continuous.   A continuous field of .C ∗ -algebras on T is, by definition, a continuous field .({At }, ) of Banach spaces, where every space .At is endowed with multiplication and involution that turn it into a .C ∗ -algebra, and . is invariant under multiplication and involution. An isomorphism of fields .({At }, ) and .({At },  ) is defined by a family .{ϕ(t)}t∈T , where   .ϕ(t) : At → At is an isomorphism for every t and .ϕ() ⊂  . The triviality and local triviality of the field of .C ∗ -algebras are introduced with regard to that definition. Let .({At }, ) be a continuous field of .C ∗ -algebras and every .At elementary .C ∗ -algebra (i. e., every algebra .At is isomorphic to the ideal of compact operators in a certain Hilbert space). Then, .({At }, ) is called a field of elementary .C ∗ -algebras. Let us assume that T is a locally compact space and .L = ({At }, ) is a continuous field of .C ∗ -algebras on T . We denote by .A the set of vector fields .x ∈  such that .$x(t)$ tends to zero as .t → ∞. Then, .A is an involutive subalgebra of .. For .x ∈ A, we set

60

1 Preliminaries

$x$ = sup{$x(t)$ ; t ∈ T }. Being endowed with such a norm, .A is a .C ∗ -algebra; it is called a .C ∗ -algebra defined by the field .L. If .L is a trivial field, then .A ∼ = C0 (T , B) ∼ = C0 (T ) ⊗ B, where .B is a certain .C ∗ -algebra. In the specific case of a trivial field .L of elementary algebras, we have .A ∼ = C0 (T ) ⊗ KH .

.

Proposition 1.3.16 ([3], 10.4.3) Let T be a locally compact space, let .L = ({At }, ) be a continuous field of algebras on T , and let .A be the algebra defined by the field .L. Moreover, let .ρπ denote the irreducible representation .x → π(x(t)) of .A for .t ∈ T and t onto .A.  t . Then, .π → ρπ is a bijection of the union of the sets .A .π ∈ A  was described in [9]. We Under the conditions of Proposition 1.3.16, the topology on .A present the corresponding result for the case of a trivial field .L. Proposition 1.3.17 ([3], 10.10.2) Let T be a locally compact space, let .B be a .C ∗ algebra, and let .A be a .C ∗ -algebra of continuous maps T into .B tending to zero at infinity.  is canonically identified with the space .T × B.  Then, the space .A

1.3.6

A Sufficient Triviality Condition for the Fields of Elementary Algebras

Let T be a finite-dimensional paracompact space and H a Hilbert space of .dim H = ℵ0 . Considering the trivial fields . × H and . × KH , where . is an open subset of T , we denote “fibers” .{t} × H and .{t} × KH by .Ht and .(KH )t . Let .E = ({Et }t∈T , ) be a separable continuous Hilbert space field, .dim Et = ℵ0 , and ∗ .L = ({KEt }t∈T , ) a continuous field of elementary.C -algebras (for notation uniformity in what follows, we write .(L) instead of .). According to Lemma 1.3.14, there exists an isomorphism .ϕ : T ×H → E. If for any .K(·) ∈ (L) the vector field .t → ϕ(t)−1 K(t)ϕ(t) belongs to .(T × KH ), the field .L is trivial. In particular cases, this simple triviality condition of the field .L is not very useful because of the necessity to construct the isomorphism .ϕ. Usually, in applications, there arise local trivializations of the field .E. The following proposition suggests a sufficient triviality condition of the field .L in terms of such trivializations. Proposition 1.3.18 Let T , H , .E, and .L be the same as above and let .{(ϕi , i )}i∈I be an atlas for .E. Assume that, for all .i ∈ I and .K ∈ (L), the vector field .t → ϕi (t)−1 K(t)ϕi (t) on .i belongs to .(i × KH ). Then the field .L is trivial, hence the ∗ .C -algebra defined by this field is isomorphic to .C0 (T ) ⊗ KH . Before proving this proposition, we present the next lemma, where .U (H ) denotes the unitary group of a Hilbert space H .

1.3 C ∗ -algebras

61

Lemma 1.3.19 Assume that .t → R(t) ∈ KH and .t → u(t) ∈ U (H ) are defined on an open set . ⊂ T , R is continuous in the uniform operator topology, and u is continuous in the strong topology. Then, the function .t → u(t)R(t)u(t)−1 is continuous in the uniform topology. Proof of the Lemma It is evident that the function uR is continuous in the uniform   topology. Then, .uRu−1 = (u(uR)∗ )∗ is also continuous in the same topology. Proof of the Proposition Let .ϕ = {ϕ(t)}t∈T be an isomorphism .T × H → E. For every vector field .K(·) ∈ (L), we define the vector field T  t → (t, qt ϕ(t)−1 K(t)ϕ(t)qt−1 ) ∈ (KH )t ,

.

where .qt is the restriction of the projection .q : T × H → H to the “fiber” .Ht . The proposition will be proven if we verify the continuity of the function t → Q(t) := qt ϕ(t)−1 K(t)ϕ(t)qt−1

.

in the uniform topology. To this end, we note that for any .i ∈ I the equality (Q|i )(t) = ui (t)Ri (t)ui (t)−1

.

(1.3.1)

holds, where .Ri (t) = qt ϕi (t)−1 K(t)ϕi (t)qt−1 and .ui (t) = qt (ϕ −1 ◦ ϕi )qt−1 . From the assumption of the proposition it follows that the function .Ri belongs to the space .C(i , KH ). Moreover, the function .ui is continuous in the strong operator topology; this follows from Lemma 1.3.15, since .ϕ −1 ◦ ϕi is an isomorphism of the fields .i × H and .(T × H )|i . Taking into account Lemma 1.3.19 and equality (1.3.1), we find that the function .Q|i is continuous in the uniform topology for any .i ∈ I ; hence, .Q ∈   C(T , KH ).

1.3.7

Solvable Algebras

Definition 1.3.20 An algebra A is called solvable if there exists a composition series {0} = J−1 ⊂ J0 ⊂ · · · ⊂ Jl = A of ideals such that there is an isomorphism Jk /Jk−1 ' C0 (Xk ) ⊗ KHk

.

(1.3.2)

for k = 1, . . . , l, where Hk is a Hilbert space, Xk is a locally compact Hausdorff space, and C0 (Xk ) is the algebra of continuous functions tending to zero at infinity. A composition series with this property is said to be solving, and the number l is called the length of the series. The least one of the solving series lengths is called the length of the algebra A.

62

1 Preliminaries

Algebras L of the form C0 (X)⊗KH ' C0 (X, KH ) play the role of “elementary” objects. Every nonzero irreducible representation of such an algebra L is the map L  f →  coincides with f (x) ∈ KH for a certain x ∈ X. The Jacobson topology on the spectrum L  the topology of X. Therefore, L can be identified with the space X (Proposition 1.3.8). Thus, a solvable algebra consists of elementary blocks. In view of (1.1.39), (Jj /Jj −1 )∧ = j Xj and hence Jj = Gj , where Gj = i=0 Xi . Being the spectrum of an ideal, Gj is an  of the algebra A. open subset of the spectrum A Let us discuss the solvable algebra definition in more detail. Let A be an arbitrary C ∗ algebra. To every element a ∈ A there corresponds the function aˆ given on the spectrum  by a(π A ˆ ) = π(a) (∈ BHπ ). The functions aˆ form an algebra isomorphic to A since } = $a$ $a$ ˆ = sup{ $a(π ˆ ) ; BHπ $ ; π ∈ A

.

  (Proposition 1.3.1). In general, for every set  ⊂  BHπ of vector fields on A π ∈A ∗ containing {aˆ : a ∈ A}, the C -algebra field ({π(A)}π ∈A, ) is not continuous. The  endowed with the Jacobson topology is not Hausdorff cause is that, as a rule, the space A and therefore the functions π → $a(π ˆ ); BHπ $ are not continuous (one can guarantee semicontinuity from below only [3, 3.3.2]). Let  ∅ = G−1 ⊂ G0 ⊂, . . . ⊂ Gl = A

.

 such that the space Xk := Gk \ Gk−1 is Hausdorff for every k ∈ be open subsets of A {0, . . . , l} (we suppose that such Gk exist). We connect with each set Gk the ideal  \ Gk }. Jk = {a ∈ A : a(π ˆ ) = 0 for π ∈ A

.

The spectrum of this ideal can be identified with Gk and the spectrum of the quotient algebra Jk /Jk−1 is identified with Xk . Any element a in Jk defines the vector field aˆ on Xk depending on the residue class a + Jk−1 only. Let k be the local closure of the set of such fields. Then, ({π(Jk )}π ∈Xk , k ) is a C ∗ -algebra continuous field (the continuity follows from the separability of Xk [3, 3.3.9]). The solvability of an algebra A means that the sets Gk subject to the above conditions can be chosen to satisfy, for all k, the requirements: (i) π(Jk ) = KHπ for π ∈ Xk . (ii) The algebra field Lk := ({KHπ }π ∈Xk , k ) is trivial. In this case, the algebra Ak defined by the field Lk is isomorphic to C0 (Tk ) ⊗ KH (see Sect. 1.3.5). Since Ak ∼ = Jk /Jk−1 , k = 1, . . . , l [3, 10.5.4], there is isomorphism (1.3.2) for every k.

1.3 C ∗ -algebras

63

The solvability of an algebra in the sense of Definition 1.3.20 will be temporarily called the solvability in the narrow sense. To define the solvability in the wide sense, we change condition (ii) for the condition (ii) The fields Lk are locally trivial. Being solvable in the narrow sense, an algebra is also solvable in the wide sense; the converse is generally false. The algebras of pseudodifferential operators in Chaps. 2–4 turn out to be solvable in the narrow sense. A solving series allows us to determine the collection of symbols σ1 (A), . . . , σN (A) for an operator A in an arbitrary solvable algebra A (see [6]). The symbol σj (A) is an operator-valued function on the space Xj . In terms of such symbols, the notions of a j parametrix and a j -symbol are introduced for A. Solving series plays an important role in studying the groups K∗ (A) of operator K-theory related to an algebra A [34]. These questions are not discussed in the book.

1.3.8

Maximal Radical Series

Definition 1.3.21 The maximal radical m(A) of an algebra A is the intersection of all two-sided maximal ideals of A. The composition series · · · ⊂ m(m(A)) ⊂ m(A) ⊂ A is called the maximal radical series. In what follows, we describe sufficient conditions providing the coincidence of the shortest solving series with the maximal radical series. Definition 1.3.22 A composition series {0} = J−1 ⊂ J0 ⊂ . . . Jn in an algebra A is said to be stratified if π(Jk ) = π(Jk+1 ) = . . . π(Jn+1 )

.

for every irreducible representation π of A satisfying π(Jk ) = 0. (In other words, the requirement π(Jk ) = π(Jk+1 ) has to be fulfilled for any irreducible representation π of the ideal Jk+1 , 0 ≤ k ≤ n − 1.) Proposition 1.3.23 If a solving composition series is stratified, it coincides with the maximal radical series. Proof Assume that Jk /Jk−1 ' C0 (Xk ) ⊗ KHk . Since m(C0 (Xk ) ⊗ KHk ) = m(C0 (Xk )) ⊗ KHk = 0,

.

we have m(Jk /Jk−1 ) = 0. This means that m(Jk ) ⊆ Jk−1 . Indeed, let p be the projection Jk → Jk /Jk−1 . The preimage p−1 (μ) of an arbitrary maximal ideal μ ⊂ Jk /Jk−1 is a

64

1 Preliminaries

maximal ideal in Jk . Taking into account that   .

μ

 p

−1

(μ) /Jk−1 =

  (p−1 (μ)/Jk−1 ) = μ = m(Jk /Jk−1 ) = 0, μ

we obtain Jk−1 = ∩p−1 (μ) ⊇ m(Jk ). Now, we show that m(Jk ) ⊇ Jk−1 for k = 0, . . . , n. We have to verify that every maximal ideal M ⊂ Jk contains Jk−1 . Since M is primitive, there exists an irreducible representation π of the ideal Jk such that ker π = M. By our assumption, π(Jk−1 ) = π(Jk ). On the other hand, π(Jk ) ' Jk /M and the quotient algebra is simple, i. e., it   contains no nontrivial ideals. Therefore, π(Jk−1 ) = 0 and M ⊇ Jk−1 . Here is an example of an algebra whose shortest solving series does not coincide with the maximal radical series. Let X be a compact space and let A = KH ⊕ C(X). Then, the series 0 ⊂ KH ⊂ KH ⊕ C(X) is solving, and the length of A is equal to 1. However, m(A) = 0.

1.3.9

The Localization Principle

Let .A be a .C ∗ -algebra, let .C be a commutative subalgebra of .A, and let .C be the maximal  we set .Ix = {c ∈ C : c(x)  is ˆ = 0}, where .cˆ ∈ C0 (C) ideal space of .C. For every .x ∈ C, the Gelfand transformation of .c ∈ C. Let .Jx be the ideal of .A spanned by the elements in .Ix and let .Ax = A/Jx . The transition from .A to .Ax is called the localization at x; the algebras .Ax are called the local algebras. Proposition 1.3.24 Let J be an ideal in .A. Assume that  (i) .π |C = 0 for all .π ∈ A. (ii) For any .x1 , x2 ∈ C such that .x1 = x2 , there exist .c1 , c2 ∈ C satisfying .cˆj (xj ) = 0, .j = 1, 2, and .c1 Ac2 ⊂ J .  (iii) For any .π ∈ J and .x ∈ Cthere exist .a ∈ J and .c ∈ Ix satisfying .π(ac) = 0. Then,  = ∪ A   A x∈C x ∪ J .

.

(1.3.3)

Remark 1.3.25 The requirement (i) is automatically fulfilled if the algebra .A contains a unity .e ∈ C.

1.3 C ∗ -algebras

65

 = (A/J )∪ J, it suffices to verify that Proof Since .A x . (A/J )= ∪x∈CA

.

(1.3.4)

Let .π be an irreducible representation of .A/J ; then .π can be considered as a representation of .A that vanishes on J . Let us consider the representation .π |C of the algebra .C. There ˆ = 0}. The set E is not empty, exists a subset .E ⊂ C such that .ker(π |C) = {c ∈ C : c|E otherwise, the equality .ker(π |C) = C would be fulfilled, which contradicts (i). We shall prove that E is a singleton. Let us assume, to the contrary, that .x1 , x2 ∈ E and .x1 = x2 . We denote by .c1 and .c2 elements in .C satisfying condition (ii) and by .J (cj ), .j = 1, 2, the ideal in .A generated by .cj . From (ii), it follows that .J (c1 )J (c2 ) ⊂ ker π and, because .ker π is a primitive ideal, either .J (c1 ) ⊂ ker π or .J (c2 ) ⊂ ker π . Therefore, in view of .c1 ∈ J (c1 ) ∩ C and .c2 ∈ J (c2 ) ∩ C, we have either .c1 ∈ ker (π |C) or .c2 ∈ ker (π |C). However, neither of the two last inclusions is possible since .cˆi |E = 0, .j = 1, 2.  Then, .ker (π |C) = Ix , .ker π ⊃ Thus, .E = {x}, where x is a certain point of the set .C.  Jx , and .π ∈ (A/Jx )≡ Ax . As a result, we find x . (A/J )⊂ ∪x∈CA

.

x and consider .π as a Let us verify the inverse inclusion. We assume that .π ∈ A representation of .A such that .ker π ⊃ Jx . We have .π |J = 0. Indeed, otherwise, we would obtain .(π |J ) ∈ J. By virtue of (iii), there exist .a ∈ J and .c ∈ Ix such that .π(ac) = 0. However, .π(ac) = π(a)π(c) = 0 because .π |Ix = 0. This contradiction   shows that .π |J = 0. Therefore, .π ∈ (A/J )ˆand we arrive at (1.3.3). We also have to use another version of the localization principle in the situation where an algebra being exposed to localization does not contain a commutative subalgebra needed for such a procedure. Let .B and .C be subalgebras of an algebra .L. We suppose that .C is commutative, .Ix is defined as at the beginning of this section, and .Jx is the ideal in .L generated by the set .Ix . Let .Bx denote the image of .B under the canonical map .px : L → L/Jx . The algebras .Bx are called the local algebras. The choice of the algebra .L does not play any role; changing .L to a subalgebra of .L containing .B and .C, we would obtain the local algebras isomorphic to the old ones. Let H be a Hilbert space, .dim H = ∞, and let . denote the canonical map .BH → BH /KH . Proposition 1.3.26 Let subalgebras .B, C ⊂ BH satisfy the following conditions: (1) .KH ⊂ B; (2) the algebra .C is commutative and contains the identity operator; (3) if .C1 , C2 ∈ C and .C1 C2 ∈ KH , then .C1 BC2 ∈ KH for every .B ∈ B. Moreover, let .C contain a nonscalar operator and let the map . : C → BH /KH be an isometry. Then,  = x ∪ [I d]. B .B (1.3.5) x∈C

C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with Discontinuities in Symbols Along a Submanifold

.

This chapter deals with .C ∗ -algebras generated by pseudodifferential operators (.DOs) of zero order in the space .L2 (M) on a closed smooth manifold .M. We discuss the three types of algebras: (1) the algebras spanned by the .DOs with smooth symbols; (2) the algebras generated by .DOs with smooth symbols and the operators of multiplication by functions (coefficients) that may have discontinuities “of the first kind” at finitely many points; (3) the algebras spanned by smooth .DOs and the coefficients with discontinuities along a submanifold .N of the positive dimension. These algebras contain the ideal .KL2 (M) of compact operators. In the first case, the corresponding algebra .A becomes commutative after taking the quotient by the ideal .KL2 (M). In the second case, the quotient algebra .A/KL2 (M) is not commutative; besides one-dimensional representations, the spectrum .(A/KL2 (M)) includes infinitedimensional representations. Namely, to every point .x 0 ∈ M where the coefficient discontinuities are admitted, there corresponds a series of irreducible representations 0 .π(x ; λ) ∈ (A/KL2 (M)) , .λ ∈ R, in the space .L2 (S(M)x 0 ); here .S(M)x 0 denotes the unit sphere of the tangent space at .x 0 . Finally, in the third case, the spectrum .(A/KL2 (M)) , in addition to one-dimensional representations and those of the type .π(x; λ) for .x ∈ N , contains a series of representations parametrized by the points of the cospheric bundle .S ∗ (N ) over the manifold .N . The last series consists of two-dimensional representations if .codim N = 1 and infinite-dimensional ones for .codim N > 1. For the mentioned algebras, we indicate all equivalence classes of irreducible representations, describe the spectral topology, and present solving composition series that turn out to be maximal radical series. The results of this chapter are essentially used in Chaps. 3 and 5; they are not needed for Chap. 4.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Plamenevskii, O. Sarafanov, Solvable Algebras of Pseudodifferential Operators, Pseudo-Differential Operators 15, https://doi.org/10.1007/978-3-031-28398-7_2

67

2

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

68

2.1

Algebras Generated by Pseudodifferential Operators with Smooth Symbols

Let .M be a smooth n-dimensional compact manifold without boundary and let .{Vj , vj } be an atlas on .M, where .{Vj } is an open covering of .M and .vj is a coordinate diffeomorphism. The covering is chosen to be fine so that the union .Vi ∪ Vj for any two indices i and j can serve as a coordinate neighborhood, possibly not connected. We denote by .vij a diffeomorphism of .Vi ∪ Vj onto an open subset . (≡ ij ) in .Rn . Definition 2.1.1 An operator .A : C ∞ (M) → C ∞ (M) is called a .DO of order m on −1 ∗ ∗ m .M if .(v ij ) χi Aχj vij ∈  () for any i, j . The set of all .DOs of order m is denoted m by . (M). Let .νj be the measure on .Vj induced by the Lebesgue measure on .vj (Vj ). Glueing the local measures by a partition of unity, we define the measure .ν on .M and set .L2 (M) := L2 (M, ν). The operators in . 0 (M) are bounded on .L2 (M). Proposition 2.1.2 Let .P be the algebra generated by the operators in . 0 (M) on .L2 (M). Then: (1) the algebra .P is irreducible; (2) .KL2 (M) ⊂ P; (3) the quotient algebra .P/KL2 (M) is commutative. Proof (1) The algebra .P contains every operator of multiplication by a smooth function. Therefore, any subspace invariant for .P is contained in a subspace of the form .χE L2 (M), where .χE is the characteristic function of the set E with .0 < |E| < |M|; for example, .|E| denotes the measure of E. Indeed, let .G be an invariant subspace and .P : L2 (M) → G an orthogonal projection. Then, P commutes with all operators in .P, in particular, with the operators of multiplication by functions in .C ∞ (M). Consequently, .P (f ) = f P (1), i.e., P acts as the operator of multiplication by the function .P (1). Moreover, .P 2 (f ) = P (f ) = f P (1); hence, .P (1)2 = P (1). Thus, the function .P (1) can take the values 0 and 1 only, so the function is characteristic for a certain set E. For .φ, ψ ∈ C ∞ (M), we set  Bφ,ψ u(x) = φ(x)

.

ψ(y)u(y) dν(y).

Let us choose functions .u ∈ G and .ψ so that the integral would not be zero. Then, .supp (Bφ,ψ u) = supp φ. Since .Bφ,ψ ∈  −∞ (M), we have .Bφ,ψ ∈ P and

2.1 Algebras Generated by Pseudodifferential Operators with Smooth. . .

69

supp(Bφ,ψ u) ⊂ E. The last inclusion contradicts the possibility of choosing .φ arbitrarily in .C ∞ (M).

.

(2) The operator .Bφ,ψ is compact in .L2 (M), so .KL2 (M) ∩ P = 0. It remains to apply Proposition 1.3.5. (3) If .A1 and .A2 in . 0 (M), the commutator .[A1 , A2 ] = A1 A2 − A2 A1 belongs to −1 (M) (see Corollary 1.1.23). Therefore, the map .[A , A ] : L (M) → H 1 (M) . 1 2 2 is continuous. Since the embedding .H 1 (M) ⊂ L2 (M) is compact, we obtain .[A1 , A2 ] ∈ KL2 (M).  The algebra .C(M) of continuous functions on M is a subalgebra in .P. For .x ∈ M, we set Ix := {f ∈ C(M) : f (x) = 0} and denote by .Jx the ideal in .P generated by the set .Ix .

.

Proposition 2.1.3 The equality .

 = ∪x∈M (P/Jx )∪ [I d] P

holds, where .[I d] is the equivalence class of the identity representation. Proof We are going to apply Proposition 1.3.24 for .A = P, .J = KL2 (M), and .C = C(M). Let us verify that the assumptions of the proposition are fulfilled. It suffices to show that, for .a ∈ C(M) and .A ∈ P, the commutator .[a, A] = aA − Aa is compact. We can consider that the inclusions .a ∈ C ∞ (M) and .A ∈  0 (M) are valid. Then, −1 (M) (Corollary 1.1.23) and .[a, A] ∈ KL (M). It remains to take into .[a, A] ∈  2 account that .C(M)= M.  ¯ 0 (M) be the set of operators in . 0 (M) subject to the following Definition 2.1.4 Let . ¯ 0 (M) and .σ is the symbol of an operator of the form additional conditions. If .A ∈  −1 ∗ ∗ .(v ij ) χi Aχj vij , then: (i) There exists the limit .σ 0 (x, ξ ) := limt→+∞ σ (x, tξ ). (ii) The function .(x, ξ ) → σ (x, ξ ) − ζ (ξ )σ 0 (x, ξ ) belongs to the class .S −1 (), where ∞ n .ζ ∈ C (R ), .ζ (ξ ) = 0 for .|ξ | ≤ 1/2, and .ζ (ξ ) = 1 for .|ξ | ≥ 1. ¯ 0 (M) in a local coordinate system V . We assume that .σV is a symbol of .A ∈  0 ¯ (M), there exists the limit .σ 0 (x, ξ ) = According to the definition of the class . V limt→+∞ σV (x, tξ ). For a fixed x, the function .ξ → σV0 (x, ξ ) is homogeneous of zero degree. Thus, we obtain the function .σ 0 defined on the cospheric bundle .S ∗ (M) over .M. We call .σ 0 the principal symbol of A.

70

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

¯ 0 (M) on .L2 (M). Let .A denote the algebra generated by the operators of the class . Propositions 2.1.2 and 2.1.3 remain valid (as well as their proof) with the algebra .P replaced by .A. The next assertion describes the local algebras .A/Jx . ¯ 0 (M) and let .σ 0 be the principal symbol of A. For .z ∈ Proposition 2.1.5 Let .A ∈  M, the map .A → σ 0 extends to an isomorphism .A/Jz ∼ = C(S ∗ (M)z ), where as usual ∗ ∗ .S (M)z is the fiber of .S (M) over z. Proof The map .A → σ 0 is compatible with algebraic operations, i.e., it is linear, multiplicative, and commutes with involution. Therefore, it suffices to show that .

inf{A + J ; J ∈ Jz } = σ 0 |Sz∗ ; C(Sz∗ )

¯ 0 (M). for every .A ∈  We first verify the inequality .

inf{A + J ; J ∈ Jz } ≥ σ 0 |Sz∗ ; C(Sz∗ ).

(2.1.1)

Let .χ ∈ Cc∞ (V ), where V is a coordinate neighborhood of z, .|χ | ≤ 1, and .χ (z) = 1. Since the principal symbols of the operators A and .χ Aχ coincide on .Sz∗ and .A + J  ≥ χ Aχ + χ J χ , we can consider that .V = Rn and .z = 0. Now, the role of .A is played by ¯ 0 (Rn ), while .Jz = J0 is the ideal the algebra spanned by the operators .χ Aχ , where .A ∈  in .A generated by the operators of multiplication by the functions .ζ ∈ Cc∞ (Rn ) equal to 0 at the coordinate origin. Let us introduce the operator .Ut : u(·) → t n/2 u(t·) with .t > 0; the operator is unitary on .L2 (Rn ). For .A = Opσ , we have −1 .(Ut AUt u)(x)

−n/2

= (2π )



eixξ σ (tx, ξ/t)u(ξ ˆ ) dξ, u ∈ Cc∞ (Rn ).

We set .A0 := F −1 σ 0 (0, ξ )F and, for any .J ∈ J0 , obtain .Ut (A + J )Ut−1 → A0 in the strong operator topology as .t → 0. This implies the relation A + J  ≥ A0  = sup{|σ 0 (0, ξ )|; ξ ∈ S n−1 },

.

which leads to (2.1.1). To verify the converse inequality, we set 0 Bu(x) = Fξ−1 →x σ (0, ξ )(1 − χ (ξ ))Fy→ξ u(y),

.

2.1 Algebras Generated by Pseudodifferential Operators with Smooth. . .

71

where .χ ∈ Cc∞ (Rn ), .0 ≤ χ ≤ 1, and .χ (0) = 1. Moreover, let .b1 (x, ξ ) = σ (x, ξ ) − σ (0, ξ ) and let .b2 (x, ξ ) = σ (0, ξ ) − σ 0 (0, ξ )(1 − χ (ξ )). Then χ (A − B)χ = χ Opb1 χ + χOpb2 χ .

.

(2.1.2)

The operator .χ Opb1 χ belongs to the ideal .J0 . To prove the inclusion .χOpb2 χ ∈ J0 , we note that .(1 − χ )K ∈ J0 for any compact operator K; hence, .KL2 (Rn ) ∩ J0 = 0. Being a nonzero ideal of irreducible algebra, .J0 is irreducible (Proposition 1.3.2, (2)); therefore, n .KL2 (R ) ⊂ J0 (Proposition 1.3.5). From the equality .limξ →∞ b2 (x, ξ ) = 0, it follows that the operator .χ Opb2 χ is compact and, consequently, belongs to .J0 . Now, in view of (2.1.2), we obtain .χ (A − B)χ ∈ J0 . Finally, .

inf χ Aχ + J  ≤ χ Bχ  ≤ sup |σ 0 (0, ξ )(1 − χ (ξ ))| ≤ sup |σ 0 (0, ξ )|. 

We set .ϒ = S ∗ (M)∪e, where e is a point, and introduce a topology on .ϒ. We assume that e is an open set and that the fundamental system of neighborhoods of a point .ξ ∈ S ∗ (M) consists of the sets .V ∪ e, where .V is a neighborhood of .ξ in .S ∗ (M). Theorem 2.1.6 Let .A be the algebra of operators on .L2 (M) generated by the .DOs in ¯ 0 (M). Then: ¯ 0 (M) and let .σ 0 be the principal symbol of .A ∈   A

.

(1) The map .A → σA0 extends to an isomorphism .j : A/KL2 (M) → C(S ∗ (M)). (2) Any irreducible representation of .A is either one-dimensional or equivalent to the identity representation I d. A one-dimensional representation can be implemented as 0 0 .π(ξ ) : A → σ (ξ ), where .σ A A = j ([A]), .[A] is the equivalence class of A in ∗ .A/KL2 (M), and .ξ ∈ S (M).  onto .ϒ. (3) The relation .π(ξ ) → ξ , .I d → e determines a bijection from the spectrum .A The topology on .ϒ coincides with the Jacobson topology. (4) An operator .A ∈ A is Fredholm if and only if the symbol .j ([A]) is distinct from zero throughout .S ∗ (M). (5) The maximal radical .m(A) of the algebra .A, i.e., the intersection of all twosided maximal ideals in .A, coincides with .KL2 (M). The composition series .0 ⊂ KL2 (M) ⊂ A is solving and coincides with the maximal radical series. The length of .A is equal to 1. To verify this theorem, one should compare Propositions 2.1.3, 2.1.5, and 1.3.12; details are left to the reader.

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

72

2.2

Algebras of Pseudodifferential Operators with Isolated Singularities in Symbols

2.2.1

Algebras A and S

We fix a point .x 0 ∈ M and introduce the set .M of continuous functions on .M \ x 0 such that, for .f ∈ M, there exists a finite limit .limr→0 f (r, ϕ) = f 0 (ϕ) uniform with respect to .ϕ, where .(r, ϕ) are local spheric coordinates with the origin at .x 0 . In this section, .A ¯ 0 (M) and the operators .f · denotes the algebra generated on .L2 (M) by the operators in . of multiplication by functions .f ∈ M. We can use the same localizing subalgebra .C(M) as in Sect. 2.1. Just as in the proof of Proposition 2.1.5, it is easy to see that the local algebra .Ax 0 := A/Jx 0 is generated on .L2 (Rn ) by operators of the form Au(x) = (2π )−n/2

.

 eixξ σ 0 (ξ )u(ξ ˆ ) dξ

(2.2.1)

and f 0 · u(x) = f 0 (x/|x|)u(x),

.

(2.2.2)

where .σ 0 is a zero degree homogeneous function, while .f 0 and .σ 0 belong to .C ∞ (S n−1 ). In what follows, we write . and a instead of .σ 0 and .f 0 . Recollecting Sect. 1.2.2, in particular (1.2.14)), we write (2.2.1) in the form −1/2

Au(r, ϕ) = (2π )

.



+∞

−∞

r i(n/2+λ) Aψ→ϕ (λ)u(λ ˜ + in/2, ψ) dλ,

(2.2.3)

where Aψ→ϕ (λ) = Eω→ϕ (λ)−1 (ω)Eψ→ω (λ).

.

(2.2.4)

We apply the Parseval equality to the Mellin transform and obtain that the algebra .Ax 0 is isomorphic to the algebra .S spanned by functions of the form R  λ → B(λ) = a(ϕ)Eω→ϕ (λ)−1 (ω)Eψ→ω (λ),

.

(2.2.5)

where a and . are in .C ∞ (S n−1 ), with pointwise operations and the norm B; S = sup {B(λ); BL2 (S n−1 ), λ ∈ R}.

.

We intend to use Proposition 1.3.24 for the algebra .S taking the ideal .C0 (R)⊗KL2 (S n−1 ) as J . Therefore, we first verify the inclusion .C0 (R) ⊗ KL2 (S n−1 ) ⊂ S. Before describing the local algebras, we deal with some technical preparations; they mainly relate to the

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

73

behavior of operator (2.2.4) under local straightening of the sphere .S n−1 . All irreducible representations (to within equivalence) of the algebra .S are given by Theorem 2.2.11.  and describe the spectrum .A.  Then, we investigate the Jacobson topology on the set .S

Proof of the Inclusion C0 (R) ⊗ KL2 (S n−1 ) ⊂ S

2.2.2

Let .S(λ) denote the algebra spanned by operators of the form (2.2.5) on .L2 (S n−1 ), where .λ is a fixed real number. Proposition 2.2.1 The following assertions are valid: (1) The algebra .S(λ) is irreducible for every .λ ∈ R. (2) .KL2 (S n−1 ) ⊂ S(λ), and the quotient algebra .S(λ)/KL2 (S n−1 ) is commutative. Proof (1) The algebra .S(λ) contains every operator of multiplication by a smooth function. Therefore, any invariant subspace of this algebra is contained in a subspace of the form .χ L2 (S n−1 ), where .χ is the characteristic function of the set . ⊂ S n−1 with n−1 | (see the proof of Proposition 2.1.2). Let us suppose that there exists .0 < || < |S a nontrivial invariant subspace and let u be an element of such a subspace. We can assume that the support of u is in an open semisphere and choose a point .ω0 so that the set .{ψ ∈ S n−1 : |ψω0 | < ε} is disjoint from .supp u for sufficiently small .ε. Denote by .{Gm } a sequence of smooth averaging kernels such that  .

S n−1

Gm (θ, ψ)u(ψ) dψ → u(θ ) in L2 (S n−1 ).

Then,  .

S n−1

Gm (θ, ψ)(−ω0 ψ + i0)−iλ−n/2 u(ψ) dψ → (−ω0 θ + i0)−iλ−n/2 u(θ ) (2.2.6)

for almost all .θ. We fix a point .θ = θ0 such that the limit (2.2.6) exists and is nonzero. Then, for sufficiently large m,  .

S n−1

Gm (θ0 , ψ)(−ω0 ψ + i0)−iλ−n/2 u(ψ) dψ = 0.

(2.2.7)

Obviously, the function .ψ → ν(ψ) = Gm (θ0 , ψ)u(ψ) belongs to the invariant subspace.

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

74

Denote by .{k } a .δ-shaped sequence of smooth functions such that the sets .supp k shrink to the point .ω0 . Further, let a be a smooth function equal to 0 in a small neighborhood of the equator .{ϕ : ϕω0 = 0} and to 1 outside another small neighborhood of this equator. Setting .Ak (λ) = a(ϕ)E(λ)−1 k (ω)E(λ), we have  Ak (λ)ν → C(λ)a(ϕ)(ϕω0 + i0)iλ−n/2

.

S n−1

(−ω0 ψ + i0)−iλ−n/2 ν(ψ) dψ, (2.2.8)

where .C(λ) = (2π )−n exp (iπ n/2)(−iλ + n/2)(iλ + n/2). From (2.2.7) and the equality −iλ−n/2

(−ϕω0 + i0)−iλ−n/2 = e−i(iλ+n/2)π (ϕω0 )+

.

−iλ−n/2

+ (ϕω0 )−

,

it follows that the right-hand side of (2.2.8) cannot vanish almost everywhere in .S n−1 \ . We have obtained a contradiction. (2) Let .a ∈ C ∞ (S n−1 ) and let .A(λ) be an operator of the form (2.2.4). From Proposition 1.2.11, it follows that the commutator .[a, A(λ)] is compact on .L2 (S n−1 ). Thus, n−1 ) = 0. In view of Proposition 1.3.5, .KL (S n−1 ) ⊂ S(λ). .S(λ) ∩ KL2 (S 2  Proposition 2.2.2 Let .λ1 and .λ2 be distinct real numbers. Then, the representations n−1 ) are nonequivalent. .π(λj ) : B(·) → B(λj ), .j = 1, 2, of the algebra .S on .L2 (S Proof Let . ∈ C ∞ (S n−1 × S n−1 ) and .A(λ) = Eω→ϕ (λ)−1 (ϕ, ω)Eψ→ω (λ). It is clear that .A ∈ S. By virtue of Proposition 1.2.11, A(λ)ψj = ϕj A(λ + i) + E −1 (λ)Dωj (ϕ, ω)E(λ + i),

.

(2.2.9)

where .ϕ = (ϕ1 , . . . , ϕn ) ∈ Rn , .|ϕ| = 1, and so on. Let us replace .(ϕ, ω) by .ϕω(ϕ, ω) in (2.2.9). Taking into account the definition of the operator .E(λ)−1 (see (1.2.8)) and the relation .(ϕω + i0)iμ ϕω = (ϕω + i0)i(μ−i) , we obtain A(λ + i) = E(λ + i)−1 ϕω(ϕ, ω)E(λ + i) = (λ + in/2)E(λ)−1 (ϕ, ω)E(λ + i).

.

Now, equality (2.2.9) takes the form .

Eω→ϕ (λ)−1 ϕω(ϕ, ω)Eψ→ω (λ)ψj = ϕj (in/2 + λ)Eω→ϕ (λ)−1 (ϕ, ω)Eψ→ω (λ + i) +Eω→ϕ (λ)−1 Dωj (ϕω(ϕ, ω))Eψ→ω (λ + i).

(2.2.10)

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

75

The definition (1.2.2) of .E(λ) and the equality .(−ωψ +i0)−iμ ωψ = −(−ωψ +i0)−i(μ+i) imply that E(λ)−1 (ϕ, ω)E(λ)ωψ = −(iλ − 1 + n/2)E(λ)−1 (ϕ, ω)E(λ + i). (2.2.11)

.

We multiply (2.2.11) by .ϕj and subtract the result from (2.2.10). Then, E(λ)−1 ϕω(ϕ, ω)E(λ)ψj − ϕj E(λ)−1 (ϕ, ω)E(λ)ωψ

.

(2.2.12)

= E(λ)−1 (ϕ, ω)E(λ + i), where .(ϕ, ω) = i(n − 1)ϕj (ϕ, ω) + Dωj (ϕω(ϕ, ω)). It is obvious that the lefthand sides of (2.2.11) and (2.2.12) belong to the algebra .S. Let us replace in (2.2.11) the function . by .. The operator-functions λ → M(λ) := (iλ − 1 + n/2)E(λ)−1 (ϕ, ω)E(λ + i),

.

λ → N(λ) := E(λ)−1 (ϕ, ω)E(λ + i)

.

also belong to .S. Suppose that, for some distinct .λ1 and .λ2 , the representations .π(λ1 ) and .π(λ2 ) are equivalent. Then, the norms of both the operators .M(λ1 ) and .M(λ2 ) and those of .N(λ1 ) and .N(λ2 ) must coincide. If .λ1 = −λ2 , this is impossible since .|iλ1 − 1 − n/2| = |iλ2 − 1 − n/2|. In case of .λ1 = −λ2 , the values of the function .M(λ) − (iλ1 − 1 + n/2)N(λ) = i(λ − λ1 )N(λ) have distinct norms at .λ1 and .λ2 .  Proposition 2.2.3 The algebra .S contains the ideal .C0 (R) ⊗ KL2 (S n−1 ). Proof The intersection .S ∩ (C0 (R) ⊗ KL2 (S n−1 )) is a closed two-sided ideal in .S. The restriction of the representations .π(λ1 ) and .π(λ2 ) to this ideal is irreducible and nonequivalent (Proposition 1.3.7). It follows that .S ∩ (C0 (R) ⊗ KL2 (S n−1 )) is a rich subalgebra of the algebra .C0 (R) ⊗ KL2 (S n−1 ); therefore, .S ∩ (C0 (R) ⊗ KL2 (S n−1 )) = C0 (R) ⊗ KL2 (S n−1 ) (Proposition 1.3.9). 

2.2.3

Auxiliary Results

This section is mainly devoted to describing the behavior of operator (2.2.4) under local straightening the sphere .S n−1 . Keeping in mind further applications, we consider here a somewhat more general situation than it is necessary for studying the algebra .S introduced in (2.2.5).  Let L be a subalgebra of the algebra .L∞ (S n−1 ) and .C(S n−1 ) ⊂ L. We denote by .S the algebra generated by operator-functions of the form (2.2.5), where .a ∈ L and . ∈

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

76

 is defined by the equality C ∞ (S n−1 ). The norm in .S  = sup{B(λ); BL2 (S n−1 ), λ ∈ R}. B; S

.

Let .N = (0, . . . , 0, 1) be the northern pole of the sphere .S n−1 and let .J (N ) be the ideal  that is spanned by the operators .ζ I d, where .ζ ∈ C(S n−1 ) and .ζ (N ) = 0. of the algebra .S Proposition 2.2.4 Assume that .{χq } is a sequence of functions in .C(S n−1 ) such that .χq (N ) = 1, .|χq | ≤ 1, and the sets supp .χq shrink to the point .N . Then the relation  → 0 is equivalent to the inclusion .B ∈ J (N ) for an element .B ∈ S.  .χq B; S  = B − (1 − χq )B; S  → 0. Since the operatorProof We suppose that .χq B; S function .λ → (1 − χq )B(λ) belongs to the ideal .J (N ), it follows that .B ∈ J (N ).  → 0. It suffices Now, we assume that .B ∈ J (N ) and prove the relation .χq B; S   to consider that .B = j k ζj k Aj k ηj k , where the operators .Aj k are of the form (2.2.5), n−1 ) (j and k take finitely many values), and at least one of the factors .ζ .ζj k , ηj k ∈ C(S jk  or .ηj k in every product . k ζj k Aj k ηj k vanishes at .N . Let us choose a sequence .{κq } of functions in .C(S n−1 ) that satisfy the same conditions as .{χq } and, in addition, are subject to the requirement .κq χq = χq . For example, if .ηj l (N ) = 0, then the j -th term in the sum .χq B(λ) can be written in the form .

χq



k ζj k Aj k (λ)ηj k

(2.2.13)

= κq K(λ) + ζj 1 Aj 1 (λ)ηj 1 . . . ζj l Aj l (λ)ηj l χq . . . ζj M Aj M (λ)ηj M , where .K(λ) ∈ KL2 (S n−1 ) and .K(λ) → 0 as .λ → ∞. Since .ηj l χq → 0 in .C(S n−1 ), the  second term in (2.2.13) tends to 0 with respect to the norm in .S.  It remains to be shown that .κq K; S → 0. We choose a large number T so that n−1 ) < ε for .|λ| > T and a given .ε > 0. For any .λ ∈ [−T , T ], there .κq K(λ); BL2 (S exists a finite .ε-net for the set .Q(λ) = {v : v = K(λ)u, u; L2 (S n−1 ) = 1}. This and the continuity of the function .λ → K(λ) imply the existence of a finite .ε-net .{w} for the union .∪Q(λ), .|λ| ≤ T . For every .u ∈ L2 (S n−1 ) with unit norm and any .λ ∈ [−T , T ], we have κq K(λ)u; L2 (S n−1 ) ≤ κq (K(λ)u − w); L2 (S n−1 ) + κq w; L2 (S n−1 ),

.

where w is such an element of the net that the first term on the right is less than .ε. The inequality .κq w; L2 (S n−1 ) < ε is fulfilled for all elements of the net for sufficiently  ≤ 2ε. large q. In summary, we arrive at .κq K; S   + denote the subalgebra of the algebra .S  generated by the operator-functions Let .S n−1 .λ → ζ A(λ)η, where .A is a function of the form (2.2.5) and .ζ, η ∈ Cc (S+ ); from now

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

77

n−1 on, the inclusion .χ ∈ Cc (S+ ) means that .χ ∈ C(S n−1 ) and supp.χ belongs to the open n−1  semisphere .S+ = {x = (x , xn ) ∈ Rn : |x| = 1, xn > 0} with pole .N = (0 , 1). We set  + ∩ J (N ). .J+ (N ) = S

 induces an isomorphism + → S Proposition 2.2.5 The embedding .S  + /J+ (N )  S/J  (N ). j :S

.

 + /J+ (N ) → S/J  (N ) follows from the Proof The continuity of the map .j : S norm definition in a quotient algebra. Therefore, it suffices to verify the inequality  (N ) ≥ [A]; S  + /J+ (N ). Let .{χq } be the sequence in Proposition 2.2.4. .j ([A]); S/J For a certain .Kε ∈ J (N ), we have  (N ) ≥ A + Kε ; S  − ε ≥ χq A; S  − χq Kε ; S  −ε j ([A]); S/J

.

 + /J+ (N ) − χq Kε ; S  − ε. ≥ [A]; S  → 0 as .q → ∞. It remains to note that, by virtue of Proposition 2.2.4, .χq Kε ; S



The next proposition describes the transformation of .ζ A(λ)η under local straightening the sphere in a neighborhood of the point .N ; here, .A is an operator of the form (2.2.4), and n−1 .ζ and .η are in .Cc (S+ ). Let .x = (x  , xn ) ∈ Rn , .r(x  ) = (1 + |x  |2 )1 /2, and .(x  ) =    (x /r(x ), 1/r(x )). Then, the volume element .dψ of .S n−1 is .r(x  )−2(n−1) dx  , where .ψ = n−1 (x  ). For .v ∈ L2 (S+ ), we set (U (λ)v)(x  ) = v((x  ))r(x  )iλ−(n−1) .

.

n−1 The values of the function .R → U (λ) : L2 (S+ ) → L2 (Rn−1 ) are unitary operators. −n ∞ n−1 Let .G(x) = f (x/|x|)|x| with .f ∈ C (S ) and let the function f be orthogonal to 1 in .L2 (S n−1 ). Then, the operator (2.2.4) with .(ξ ) = Fx→ξ G(x) admits the representation

 (A(λ)v)(ϕ) =



∞

v(ψ) dψ

.

S n−1

t −i(λ+in/2)−1 G(tϕ − ψ) dt

(2.2.14)

0

(see Sect. 1.2.3). n−1 Proposition 2.2.6 Let .u ∈ L2 (Rn−1 ) and let .η and .ζ be in .Cc (S+ ). Then, .

(U (λ)ζ A(λ)ηU (λ)−1 u)(x  ) = ζ ((x  ))r(x  )1−n/2  ∞  × u(y  )η((y  ))r(y  )1−n/2 dy  t −i(λ+in/2)−1 G(tx  − y  , t − 1) dt. Rn−1

0

(2.2.15)

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

78

Proof According to (2.2.14), (U (λ)ζ A(λ)ηU (λ)−1 u)(x  ) = ζ ((x  ))r(x  )iλ−n+1



u( −1 (ψ))

.

×η(ψ)r( −1 (ψ))−iλ+n−1 dψ = ζ ((x  ))r(x  )iλ−n+1  ×



∞

S n−1

t −i(λ+in/2)−1 G(t(x  ) − ψ) dt

0



Rn−1

∞

u(y  )η((y  ))r(y  )−iλ−n+1 dy 

t −i(λ+in/2)−1 G(t(x  ) − (y  )) dt.

(2.2.16)

0

Taking into account the definition of . and the homogeneity of the function G, we write the inner integral in the form 

+∞

.

t −i(λ+in/2)−1 G(t(x  ) − (y  )) dt

0



= r(y  )n

+∞

t −i(λ+in/2)−1 G(tr(y  )x  /r(x  ) − y  , tr(y  )/r(x  ) − 1) dt

0  n





= r(y ) (r(x )/r(y ))

−i(λ+in/2)



+∞

t −i(λ+in/2)−1 G(tx  − y  , t − 1) dt.

0



It remains to substitute this expression into (2.2.16). We set (B(λ)u)(x  ) = r(x  )1−n/2



.



(D(λ)u)(x ) =





Rn−1

Rn−1

u(y ) dy



u(y  )r(y  )1−n/2 dy  



+∞

t−i(λ+in/2)−1 G(tx  − y  , t − 1) dt,

0 +∞

−∞

e−iλs G(x  − y  , s) ds.

Proposition 2.2.7 Let .{χq } be a sequence of functions in .Cc (Rn−1 ) such that .χq (0) = 1 and .|χq | ≤ 1. Moreover, let the supports of .χq shrink to the coordinate origin 0. Then, the relation .

sup{χq (B(λ) − D(λ))σ ; BL2 (Rn−1 ); λ ∈ R} → 0

holds for any function .σ ∈ Cc (Rn−1 ) as .q → ∞. Proof We represent .K1 (λ) := χq (B(λ) − D(λ))σ in the form K1 (λ) = χq (K2 (λ) + K3 (λ) + K4 (λ))σ,

.

(2.2.17)

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

79

where (K2 (λ)u)(x  ) =



.

Rn−1



u(y  ) dy 

∞

(t −i(λ+in/2)−1 − e−iλ(t−1) )G(tx  − y  , t − 1) dt,

0

(2.2.18) (K3 (λ)u)(x  ) =



.

Rn−1

u(y  ) dy 



∞

−∞

  e−iλs G((1 + s)x  − y  , s) − G(x  − y  , s) ds, (2.2.19)





(K4 (λ)u)(x ) = −

.



Rn−1

u(y ) dy





−1 −∞

e−iλs G((1 + s)x  − y  , s) ds.

(2.2.20)

We will prove that .

sup{χq Kj (λ)σ ; BL2 (Rn−1 ); λ ∈ R} → 0

(2.2.21)

for .q → ∞, .j = 2, 3, 4: (A) We consider operator (2.2.18). Assume that .ζ ∈ C ∞ (R), .ζ (t) = 1 for .|t − 1| < 1/2, and .ζ (t) = 0 for .|t − 1| > 3/4. Integrating by parts, we write the inner integral in (2.2.18) in the form 

∞

.

0

(1 − ζ (t))(t −i(λ+in/2)−1 − e−iλ(t−1) )G(tx  − y  , t − 1) dt



+

∞

ζ  (t)h(t, λ)G(tx  − y  , t − 1) dt +

0



∞

ζ (t)h(t, λ) 0

∂ G(tx  − y  , t − 1) dt, ∂t (2.2.22)

where h(t, λ) = (t −i(λ+in/2) − 1)/(iλ − n/2) − (e−iλ(t−1) − 1)/ iλ.

.

According to (2.2.22), .K2 (λ) can be written as the sum .K2 (λ) = K21 (λ) + K22 (λ) + K23 (λ). In the two first integrals in (2.2.22), the factor at .G(tx  − y  , t − 1) vanishes if .|t − 1| < 1/2. Therefore, σ1 (K21 (λ) + K22 (λ))σ2 ∈ KL2 (Rn−1 )

.

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

80

for all .λ ∈ R and .σj ∈ Cc (Rn−1 ). Moreover, .lim σ1 K2j (λ)σ2  = 0 as .λ → ∞, .j = 1, 2. (It is obvious for .K22 ; for .K21 , it follows from the fact that the first term in (2.2.22) reduces to the form 

∞

.

0



e−iλ(t−1) − eiλ t −iλ+n/2 − iλ − n/2 iλ

((1 − ζ (t))G(tx  − y  , t − 1))t dt

by means of integration by parts.) We show that .K23 has the same properties. Let us first verify the inequality .|(t − 1)−2 ζ (t)h(t, λ)| ≤ c|ζ (t)|. To this end, we consider the function .g : (α, β) → (e−iαβ − 1)/(iα) for .α ∈ C, .β ∈ R, and .Im (αβ) ≤ M. For any pair of points .(αj , βj ), .j = 1, 2, |g(α1 , β1 ) − g(α2 , β2 )| ≤ C(M)(|β1 − β2 | + β22 |α1 − α2 |).

.

(2.2.23)

Indeed, .|∂g/∂β| = eIm (αβ) ; hence, |g(α1 , β1 ) − g(α1 , β2 )| ≤ C(M)|β1 − β2 |.

.

Moreover, |g(α1 , β2 ) − g(α2 , β2 )| = |β2 ||(e−iα1 β2 − 1)/(iα1 β2 ) − (e−iα2 β2 − 1)/(iα2 β2 )|.

.

Since the derivative of the function .z → (e−iz − 1)/(iz) is .O(eImz ), we have |g(α1 , β2 ) − g(α2 , β2 )| ≤ C(M)β22 |α1 − α2 |,

.

which leads to (2.2.23). We set in (2.2.23) .α1 = λ + in/2, .α2 = λ, .β1 = ln t, and β2 = t − 1 and obtain

.

|ζ (t)h(t, λ)| ≤ C(| ln t − (t − 1)| + (t − 1)2 n/2)|ζ (t)| ≤ C1 (t − 1)2 |ζ (t)|.

.

Thus, .|(t − 1)−2 ζ (t)h(t, λ)| ≤ const|ζ (t)|. Now, using this inequality, we prove the estimate   k(λ, x  , y  ) := 

∞

ζ (t)h(t, λ)

.

0

  ∂ G(tx  − y  , t − 1) dt  ≤ C|x  − y  |2−n ∂t

(2.2.24)

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

81

under the condition .n ≥ 3 and .|x  | + |y  | ≤ const. For .n = 2, the estimate is valid with right-hand side .C ln |x  − y  |−1 + C1 . For .x = (x  , 1) and .y = (y  , 1), k(λ, x  , y  ) ≤ C



.

 ≤C

∞

(t − 1)2 |ζ (t)

0 ∞

(t − 1)2 |ζ (t)||tx − y|−1−n dt ≤ C

0



∞

∂ G(tx − y)| dt ∂t

|ζ (t)||tx − y|1−n dt

0

because .|tx − y|2 ≥ (t − 1)2 . Furthermore, |tx − y| = |x|((t − a)2 + b2 )1/2 ,

.

where .a = xy/|x|2 and .b2 = (|x|2 |y|2 − (xy)2 )/|x|4 . Therefore, k(λ, x  , y  ) ≤ C|x|1−n



∞

.

|ζ (t)|((t − a)2 + b2 )(1−n)/2 dt.

0

Changing the integration variable, we have k(λ, x  , y  ) ≤ C|x|1−n b2−n



+∞

.

−∞

|ζ (bt + a)|(t 2 + 1)(1−n)/2 dt.

(2.2.25)

If .n ≥ 3, the integral on the right in (2.2.25) is bounded uniformly with respect to a and b. Since b2 = ((1 + |x  |2 )(1 + |y  |2 ) − (1 + x  y  )2 )/|x|4

.

≥ (|x  |2 − 2x  y  + |y  |2 )/|x|4 = |x  − y  |2 /|x|4 , from (2.2.25), we obtain k(λ, x  , y  ) ≤ C|x|n−3 |x  − y  |2−n

.

and estimate (2.2.24). Suppose now that .n = 2. Then, (2.2.25) takes the form 





−1

k(λ, x , y ) ≤ C|x|

.

+∞

−∞

−1/2

|ζ (bt + a)|(t + 1) 2

 dt ≤ C

(β−a)/b

(t 2 + 1)−1/2 dt,

(α−a)/b

where .[α, β] is a finite interval containing .supp ζ . The points .x  and .y  do not quit a bounded set, so M = max{sup |α − a|, sup |β − a|} < ∞.

.

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

82

Therefore, 





k(λ, x , y ) ≤ C

.

M/b

0

(t 2 +1)−1/2 dt ≤ C(1+ln+ M/b) ≤ C(1+ln+ (M|x|2 |x  −y  |−1 ).

Increasing M, if needed, and noticing that .|x| ≥ 1, we obtain inequality (2.2.24) with right-hand side replaced by .C ln |x  − y  |−1 + C1 . From (2.2.24), it follows that .σ1 K23 (λ)σ2 ∈ KL2 (Rn−1 ) for all .λ ∈ R and .σj ∈ Cc (Rn−1 ). We show that .lim σ1 K23 (λ)σ2  = 0 as .λ → ∞. Let .κj ∈ C(R) with .j = 1, 2, .0 ≤ κj ≤ 1, and .κ1 + κ2 = 1, while .κ1 (t) = 1 for .|t| < ε and .κ1 (t) = 0 for .|t| > 2ε. We represent .K23 (λ) as the sum .Q1 (λ) + Q2 (λ), where .Qj (λ) is the operator with kernel κj (|x  − y  |)



∞

h(t, λ)ζ (t)

.

0

∂ G(tx  − y  , t − 1) dt. ∂t

The kernel of .σ1 Q2 (λ)σ2 has compact support on .Rn−1 × Rn−1 and does not exceed −1 , so .σ Q (λ)σ  → 0 as .λ → ∞. Moreover, it can be deduced from (2.2.24) .C|λ| 1 2 2 that .σ1 Q1 (λ)σ2  is .O(ε) in the case .n ≥ 3 and .O(ε ln ε−1 ) in the case .n = 2; hence, .lim σ1 K23 (λ)σ2  = 0. Thus, .σ1 K2l (·)σ2 ∈ C0 (R) ⊗ KL2 (Rn−1 ) for .l = 1, 2, 3; therefore, .σ1 K2 (·)σ2 ∈ C0 (R) ⊗ KL2 (Rn−1 ). The last inclusion implies (2.2.21) for .j = 2; this can be established as in the proof of Proposition 2.2.4 (with .KL2 (S n−1 ) changed for .KL2 (Rn−1 )). (B) Let us turn to operator (2.2.19). We have 



(K3 (λ)u)(x ) =

.

Rn−1



u(ξ ˆ ) dξ

√  = 2π





+∞

−∞

ˆ  , s)(ei(1+s)ξ  x  − eiξ  x  )e−iλs ds G(ξ  

Rn−1

u(ξ ˆ  )((ξ  , λ − ξ  x  ) − (ξ  , λ))eix ξ dξ  ,

where ˆ  , s) = Fx  →ξ  G(x  , s). (ξ ) = Fx→ξ G(x), u(x ˆ  ) = Fx  →ξ  u(x  ), G(ξ

.

(The same letter F denotes the Fourier transform in .Rn and that in .Rn−1 .) Let .χ ∈ Cc∞ (Rn−1 ), .|χ | ≤ 1, .χ (0) = 1, and .χq (x  ) := χ (qx  ) for .q = 1, 2, . . . . (By virtue of Proposition 2.2.4, we can consider that the same sequence participates in the statement of Proposition 2.2.7.) Let ζ (x  , ξ  , λ) =

.

√ 2π ((ξ  , λ − ξ  x  ) − (ξ  , λ))

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

83

and ηq (z , ξ  , λ) = Fx  →z (χq (x  )ζ (x  , ξ  , λ)).

.

Then, (χq K3 (λ)u)∧ (z ) =



.

Rn−1

ηq (z − ξ  , ξ  , λ)u(ξ ˆ  ) dξ  .

To prove (2.2.21), it suffices to verify that, for any .u, v ∈ L2 (Rn−1 ), the estimate |(χq K3 (λ)u)∧ , v)| ≤ c(q)uv

.

(2.2.26)

holds, where .(·, ·) is the inner product in .L2 (Rn−1 ), .c(q) is independent of u and v, and .c(q) → 0 as .q → ∞. We have 

∧

|(χq K3 (λ)u) , v)| ≤

.

 ×

|u(ξ ˆ  )|2 |ηq (z − ξ  , ξ  , λ)| dz dξ 

(2.2.27)

|v(z )|2 |ηq (z − ξ  , ξ  , λ)| dz dξ  .

It turns out that for any .s ≥ 0 |ηq (z , ξ  , λ)| ≤ Cq 2s−n (1 + |z |2 )−s

.

(2.2.28)

(we will check the inequality below). Employing (2.2.28) for .s ∈](n − 1)/2, n/2[, we find   |ηq (z − ξ  , ξ  , λ)| dz ≤ Cq 2s−n (1 + |z − ξ  |2 )−s dz , .  .

|ηq (z − ξ  , ξ  , λ)| dξ  ≤ Cq 2s−n



(1 + |z − ξ  |2 )−s dξ  .

The right-hand sides of the two last inequalities tend to 0 as .q → ∞; therefore, (2.2.27) implies (2.2.26). We now prove estimate (2.2.28). First, assume that s is a non-negative integer. Since s −iz x  = (1 + |z |2 )s e−iz x  , we have .(1 − x  ) e ηq (z , ξ  , λ) = (2π )(1−n)/2 (1 + |z |2 )−s

.



 

(1 − x  )s (χq (x  )ζ (x  , ξ  , λ))e−iz x dx  .

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

84

It suffices to verify     β  γ   −iz x    . dx  ≤ Cq 2s−n  ∂x  χq (x )∂x  ζ (x , ξ , λ))e

(2.2.29)

for all multi-indices .β and .γ such that .|β + γ | ≤ 2s. We have .|∂x  ζ (x  , ξ  , λ)| ≤ C|x  |τ (γ ) , where .τ (γ ) = 0 for .|γ | ≥ 1 and .τ (γ ) = 1 for .γ = 0. Therefore, γ

|∂x  χq (x  )∂x  ζ (x  , ξ  , λ))| ≤ C|x  |τ (γ ) q |β| |(∂ β χ )(qx  )|.

.

γ

β

It follows that the left-hand side of (2.2.29) is no greater than Cq |β|



|x  |τ (γ ) |(∂ β χ )(qx  )| dx  ≤ Cq |β|−n+1−τ (γ ) ≤ Cq 2s−n .

.

Thus, inequality (2.2.28) has been established for integers .s ≥ 0. Let s be any non-negative number. We choose integers .s1 , .s2 and non-negative .μ1 , .μ2 so that .0 ≤ s1 ≤ s ≤ s2 , .μ1 + μ2 = 1, and .μ1 s1 + μ2 s2 = s. We write (2.2.28) for .s = sj , raise it to the power .μj with .j = 1, 2, and multiply the obtained inequalities. The result is (2.2.28) with given .s ≥ 0. (C) Finally, consider operator (2.2.20). Integrating by parts in the inner integral, we obtain n−1 ) for .σ ∈ C (Rn−1 ) with .j = 1, 2. This, as in part .σ1 K4 (·)σ2 ∈ C0 (R) ⊗ KL2 (R j c B) of the proof, leads to (2.2.21) (for .j = 4).  Now we list for later use properties of the operators −1 n−1 Fη→x ) → L2 (Rn−1 ),  (η, λ)Fy  →η : L2 (R

.

(2.2.30)

where F is the Fourier transform in .Rn−1 , . is a zero degree homogeneous function, n−1 ∈ C ∞ (S n−1 ), and .λ ∈ R. .|S Proposition 2.2.8 Let .D(λ) be an operator of the form (2.2.30). Then, the following assertions are valid: (1) The function .R \ 0  λ → D(λ) is continuous with respect to the operator norm. (2) .D(λ) → D(0) in the strong operator topology as .λ → ∞; .D(λ) ≤ max D(±1) for all .λ ∈ R. (3) If .χ ∈ Cc∞ (Rn−1 ) and .λ ∈ R, the commutator .[χ , D(λ)] belongs to .KL2 (Rn−1 ). (4) .[χ , D(λ)] → 0 with respect to the operator norm as .λ → ∞.

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

85

Proof (1) The equality .(η, λ) − (η, μ) = n (η, λ, μ)(λ − μ) holds, where 

1

n (η, λ, μ) =

∂n (η, μ+t (λ−μ)) dt, ∂n (η1 , . . . , ηn ) = (∂/∂ηn )(η1 , . . . , ηn ).

.

0

The function .Rn−1  η → n (η, λ, μ) is bounded uniformly with respect to .μ in a neighborhood of the point .λ = 0. (2) For .t > 0,    −(n−1)/2 eix η (η, tλ)u(η) . D(tλ)u(x ) = (2π ) ˆ dη −(n−1)/2

= (2π )





eitx η (η, tλ)t n−1 u(tη) ˆ dη = Ut D(λ)Ut−1 u(x  ),

where .Ut u(x  ) = t (n−1)/2 u(tx  ). The operator .Ut is unitary on .L2 (Rn−1 ); hence, .D(λ) = D(tλ). If .t → 0, then .D(tλ) → D(0) in the strong operator topology; hence, .D(0) ≤ max D(±1). (3) Let .ζ ∈ Cc∞ (Rn−1 ) and .ζ (η) = 1 in a neighborhood of the coordinate origin. We represent .D(λ) in the form D(λ) = F −1 ζ (η)(η, λ)F + F −1 (1 − ζ (η))(η, λ)F =: D1 (λ) + D2 (λ).

.

The commutator .[χ , D2 (λ)] is a .DO of order .−1. For a ball B with finite radius in Rn−1 , the embedding .H 1 (B) ⊂ L2 (B) is compact; therefore, .[χ , D2 (λ)] belongs to n−1 ). The operator .D (λ) is a convolution with a smooth function. Therefore, .KL2 (R 1 the inclusion .[χ , D1 (λ)] ∈ KL2 (Rn−1 ) follows from the just mentioned embedding theorem and the compactness of .supp χ. (4) The kernel of the integral operator .[χ , D(λ)] can be written in the form  +∞ . eiλs (χ (x  ) − χ (y  ))G(x  − y  , s) ds .

−∞

=

i λ



+∞

−∞

n

eiλs

χi (x  , y  )(xi − yi )∂s G(x  − y  , s) ds =: k(x  , y  , λ)/λ,

i=1

where .G(x) = Fξ−1 →x (ξ ) and χi (x  , y  ) = −



1

.

∂i χ (x  + t (y  − x  )) dt.

0

The operator .D (λ) with kernel .k(x  , y  , λ) is bounded uniformly with respect to .λ. 

86

2.2.4

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

The Spectrum of Algebra S

The study of the irreducible representations of the algebra .A in essence reduces to that of the algebra .S introduced in Sect. 2.2.1. We apply to .S the localization principle in Proposition 1.3.24. The role of the ideal J is played by .C0 (R)⊗K(L2 (S n−1 )). This allows to find all equivalence classes of irreducible representations of the algebra .S. Then, we describe the spectral topology for .S and prove that .S is a solvable algebra of length 1. The next assertion justifies the use of the localization principle for the algebra .S. Proposition 2.2.9 Let J denote the ideal .C0 (R) ⊗ KL2 (S n−1 ) in .S and .C = C(S n−1 ). Then, .C is a commutative subalgebra in .S and the assumptions of Proposition 1.3.24 are fulfilled. Proof It is obvious that the operator of multiplication by a function .α ∈ C belongs to .S, i.e., .C ⊂ S. The algebra .C contains unity, so assumption (i) of Proposition 1.3.24 is fulfilled. The sphere .S n−1 coincides with the maximal ideal space .Cˆ of .C. Let .ϕ1 , ϕ2 ∈ S n−1 and .ϕ1 = ϕ2 . We choose elements .c1 and .c2 in .C so that .cj (ϕj ) = 1 for .j = 1, 2 and .supp c1 ∩ supp c2 = ∅. Let us show that .c1 Sc2 ⊂ J . It suffices to verify that, for any generator .A of the algebra .S and function .α ∈ C, the commutator .[A, α] belongs to J . Therefore, as .A, we can take an operator-function of the form (2.2.5). According to Proposition 1.2.11, the commutator .[A, α] belongs to .C0 (R) ⊗ KL2 (S n−1 ); hence, assumption (ii) of Proposition 1.3.24 is also fulfilled. Finally, assumption (iii) follows from the fact that every irreducible representation of the ideal J is equivalent to a representation .π(λ) : A → A(λ).  Let .LN be a subalgebra of the algebra .L∞ (S n−1 ), let .C(S n−1 ) ⊂ LN , and let every n−2 element .a ∈ LN satisfy the following condition. Denote by .SN the unit sphere in n−1 the tangent space for .S at the point .N (the northern pole .(0 , 1) of .S n−1 ). Almost n−2 everywhere on .SN , there exists a limit .

lim a(ϕ) = a(N ; θ ),

ϕ→N

which is uniform with respect to the approach direction .θ of .ϕ to the pole .N . We introduce the algebra .SN generated by the operator-functions of the form (2.2.5), where .a ∈ LN and . ∈ C ∞ (S n−1 ). The algebra .SN is endowed with the norm A; SN  = sup {A(λ); BL2 (S n−1 ); λ ∈ R}.

.

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

87

Let .J (N ) be the ideal in .SN spanned by the functions .ζ Id, where .ζ ∈ C(S n−1 ) and .ζ (N ) = 0. We denote by .L the algebra generated by the functions of the form −1 n−1 R  λ → a(N ; ·)Fη→x ) → L2 (Rn−1 ),  (η, λ)Fy  →η : L2 (R

.

where .a(N ; ·) is the function .Rn−1 \ 0  x  → a(N ; x  /|x  |) and D; L = sup {D(λ); BL2 (Rn−1 ); λ ∈ R}

.

for .D ∈ L. When describing local algebras, we will use the following proposition many times in later sections; for studying the algebra .S, we need only the special case .LN = C(S n−1 ). Proposition 2.2.10 Let .A be an operator-function of the form (2.2.5). The map −1 A(λ) → a(N ; ·)Fη→x  (η, λ)Fy  →η

.

extends to an isomorphism .SN /J (N )  L. Proof We set .

Aj k (λ) = Eω→ϕ (λ)−1 j k (ω)Eψ→ω (λ), −1 Dj k (λ) = Fη→x  j k (η, λ)Fy  →η

with .j k ∈ C ∞ (S n−1 ) and .aj k ∈ LN . It suffices to show that







aj k Aj k ; SN /J (N ) = aj k (N ; ·)Dj k ; L ,

.

j

k

j

(2.2.31)

k

where j and k run over finitely many values. We first verify that







aj k Aj k ; SN /J (N ) ≤ aj k (N ; ·)Dj k ; L .

.

j

k

j

(2.2.32)

k

In view of Proposition 2.2.5, this inequality is equivalent to







ζj k aj k Aj k ηj k ; S+ /J+ (N ) ≤ aj k (N ; ·)Dj k ; L

.

j

k

j

k

(2.2.33)

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

88

n−1 with functions .ζj k and .ηj k that belong to .Cc∞ (S+ ) and equal 1 in a neighborhood of the pole .N . Without loss of generality, we suppose that the functions .j k are orthogonal to 1 in .L2 (S n−1 ) and introduce .Gj k (x) = Fξ−1 →x j k (ξ ). Let

(Bj k (λ)u)(x  ) = r(x  )1−n/2



.



+∞

×

Rn−1

r(y  )1−n/2 u(y  ) dy 

(2.2.34)

t −i(λ+in/2)−1 Gj k (tx  − y  , t − 1) dt.

0

Setting .P(λ) =

  j

k

χq ζj k aj k Aj k ηj k , we have



ζj k aj k Aj k ηj k ; S+ /J+ (N ) ≤ P; SN .

.

j

(2.2.35)

k

According to Proposition 2.2.6,



P(λ); BL2 (S n−1 ) = (χq ζj k aj k ) ◦ κBj k ηj k ◦ κ; BL2 (Rn−1 ) .

.

j

k

To simplify notation, in what follows, we write .ηj k instead of .ηj k ◦ κ and so on. By virtue of Propositions 2.2.7 and 2.2.8,



P(λ); BL2 (S n−1 ) = (χq ζj k aj k Dj k (λ)ηj k + χq Qj k (λ)); BL2 (Rn−1 ) ,

.

j

k

where .χq Qj k ; SN  → 0 as .q → ∞. For sufficiently large q, we obtain the equality χq ζj k = χq and the estimate

.



P(λ); BL2 (S n−1 ) ≤ χq aj k (N ; ·)Dj k (λ)ηj k ; BL2 (Rn−1 ) + ε

.

j

(2.2.36)

k

for a given .ε > 0 and all .λ ∈ R. We can assume in these formulas that k runs over the same set of values for every j (otherwise, we could add several factors .ζj k and .ηj k that coincide with the unity in the quotient algebra .SN /J (N )). Thus, by Proposition 2.2.8,  .

j

k

χq aj k (N ; ·)Dj k (λ)ηj k =

 k

χq

  j

aj k (N ; ·)Dj k (λ) + χq Q(λ),

k

(2.2.37) where .Q(λ) ∈ KL2 (Rn−1 ), the function .R \ 0  λ → Q(λ) is continuous in the operator norm, and .Q(λ); BL2 (Rn−1 ) → 0 as .λ → ∞.

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

89

We would like to make the function .χq Q(λ) small for all .λ by choosing .χq . However, there is an obstacle caused by the singularity of .Dj k at .λ = 0. To overcome this difficulty, we consider the restrictions of the functions in the algebra .SN (and in the other algebras which we deal with) to the set .{λ ∈ R : |λ| > δ} with a positive .δ and define the norms n−1 ); |λ| > δ} and so on. The norms .·; S /J (N ) .A; SN δ = sup {A(λ); BL2 (S N and .·; SN  in (2.2.35) can be replaced by the norms .·; SN /J (N )δ and .·; SN δ . We show that .

sup {χq Q(λ); BL2 (Rn−1 ); |λ| > δ} < ε

(2.2.38)

for sufficiently large q and any fixed positive .δ and .ε. We choose a large T so that n−1 ) < ε for .|λ| > T . For every .λ ∈ [−T , −δ] ∪ [δ, T ] and any .ε > 0, .χq Q(λ); BL2 (R there exists a finite .ε-net for the set .H (λ) = {v : v = Q(λ)u; u; L2 (Rn−1 ) = 1}. In view of the continuity of the function .λ → Q(λ), this implies the existence of a finite .ε/2net .{w} for the union .∪H (λ), δ ≤ |λ| ≤ T . For every .λ and an arbitrary .u ∈ L2 (Rn−1 ) with unit norm χq Q(λ)u; L2 (Rn−1 ) ≤ χq (Q(λ)u − w); L2 (Rn−1 ) + χq w; L2 (Rn−1 ),

.

where the element w of the net is chosen so that the first term on the right is less than ε/2. The inequality .χq w; L2 (Rn−1 ) < ε/2 is fulfilled for all elements of the net for sufficiently large q. Thus, we arrive at estimate (2.2.38). Coming back to inequality (2.2.35) (with the mentioned change of the norms) and taking into account (2.2.36)–(2.2.38), we obtain

.



   



ζj k aj k Aj k ηj k ; S+ /J+ (N ) ≤ χq aj k (N ; ·)Dj k + χq Q; L

.

j

δ

k

k

j

k

δ

   

≤ χq aj k (N ; ·)Dj k ; L + χq Q; Lδ k

j

k

δ



≤ aj k (N ; ·)Dj k ; L + ε. j

k

This leads to inequalities (2.2.33) and (2.2.32) because the numbers .δ and .ε are arbitrary and the functions .Aj k are continuous. Now, we prove the inequality that is converse for (2.2.32) or, equivalently, for (2.2.33).   Setting .A(λ) = j k ζj k aj k Aj k ηj k and again employing Propositions 2.2.6 and 2.2.7, we obtain



n−1 .A(λ); BL2 (S ) = ζj k aj k Dj k (λ)ηj k + Q(λ); BL2 (Rn−1 ) , (2.2.39) j

k

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

90

where .sup {χq Q(λ); BL2 (Rn−1 ), λ ∈ R} → 0 as .q → 0. The operator .Ut defined for (n−1)/2 u(tx) is unitary on .L (Rn−1 ). The equalities .t > 0 by the equality .(Ut u)(x) = t 2 .D(λ)Ut = Ut D(λ/t) and .a(·)Ut = Ut a(·/t) hold for an operator .D(λ) of the form (2.2.30) and a function a on .Rn−1 . Therefore, the right-hand side of (2.2.39) coincides with

 

−1 

. Ut ζj k aj k Dj k (λ)ηj k + Q(λ) Ut j

k



(ζj k aj k )(·/t)Dj k (λ/t)ηj k (·/t) + Ut−1 Q(λ)Ut . = j

(2.2.40)

k

Given .ε > 0, we choose a large q so that .χq Q(λ); BL2 (Rn−1 ) < ε for all .λ ∈ R. From (2.2.39) and (2.2.40), it follows that



A(λ); BL2 (S n−1 ) ≥ (ζj k aj k )(·/t)Dj k (λ/t)ηj k (·/t)

.

j

(2.2.41)

k

+(1 − χq (·/t))Ut−1 Q(λ)Ut ; BL2 (Rn−1 ) − ε. Let us change .λ for .λt in (2.2.41). The operators .(1 − χq (·/t))Ut−1 Q(λt)Ut weakly converge to the zero operator as .t → +∞ for an arbitrary .λ ∈ R. Moreover, the operators     . j k (ζj k aj k )(·/t)Dj k (λ)ηj k (·/t) converge to . j k aj k (N ; ·)Dj k (λ) in the strong topology. Therefore,



lim t→+∞ A(λt); BL2 (S n−1 ) ≥ aj k (N ; ·)Dj k (λ); BL2 (Rn−1 )

.

j

(2.2.42)

k

for .λ ∈ R. If an element .B ∈ S+ belongs to .J+ (N ), then .χq B; SN  → 0 as .q → ∞ (Proposition 2.2.4); hence, replacing .A by .A + B with some .B ∈ J+ (N ), we do not change the right-hand side of (2.2.42). Thus, .

sup {A(λ) + B(λ); BL(S n−1 ); λ ∈ R} ≥ lim t→+∞ A(λt) + B(λt); BL2 (S n−1 )



aj k (N ; ·)Dj k (λ); BL2 (Rn−1 ) , ≥ j

k

which leads to the inequality







ζj k aj k Aj k ηj k ; S+ /J+ (N ) ≥ aj k (N ; ·)Dj k ; L .

.

j

k

j

k

Taking into account estimate (2.2.33), we complete the proof.



2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

91

Theorem 2.2.11 Let .A ∈ S and .Aψ→ϕ (λ) = Eω→ϕ (λ)−1 (ϕ, ω)Eψ→ω (λ), where . ∈ C ∞ (S n−1 × S n−1 ) and .λ ∈ R. Then, the maps .

π(λ) : A → A(λ) : L2 (S n−1 ) → L2 (S n−1 ), π(ϕ, ω) : A → (ϕ, ω),

λ ∈ R, .

(ϕ, ω) ∈ S n−1 × S n−1

(2.2.43) (2.2.44)

extend to irreducible representations of the algebra .S. These representations are pairwise nonequivalent. Every irreducible representation of .S is equivalent to one of the representations (2.2.43) or (2.2.44). Proof According to Proposition 2.2.9, we can apply the localization principle (Proposition 1.3.24) to the algebra .S setting .J = C0 (R) ⊗ KL2 (S n−1 ) and .C = C(S n−1 ). Then, .

 = ∪ϕ∈S n−1 S  ϕ ∪ J. S

(2.2.45)

Every irreducible representation of the ideal J is a map .π(λ) : K → K(λ) with .K ∈ J (Proposition 1.3.7). The map uniquely extends to a representation of .S in .L2 (S n−1 ), and we obtain .π(λ) in (2.2.43).  ϕ of a local algebra, we employ Proposition 2.2.10. The role To find the spectrum .S N of .L is played by the algebra .C(S n−1 ). At a fixed point .ϕ ∈ S n−1 , the algebra .Sϕ is isomorphic to the commutative algebra .L of the operator-functions −1 R  λ → D(λ) = Fη→x (ϕ, η, λ)Fy→η : L2 (Rn−1 ) → L2 (Rn−1 )

.

with .D; L = sup {D(λ); BL2 (Rn−1 ), λ ∈ R}. The function .ξ → (ϕ, ξ ) is homogeneous of zero degree and belongs to .C ∞ (Rn \ 0). Therefore, the algebra .Sϕ is isomorphic to .C(S n−1 ), and the map (2.2.44) defines all (to within equivalence) irreducible representations of the algebra .Sϕ .  According to Proposition 2.2.1, .KL2 (S n−1 ) ⊂ S(λ), and the algebra .S(λ)/KL2 (S n−1 ) is commutative. Clearly, every representation of .S(λ) is also a representation of .S. In particular, the following theorem shows that the algebras .S(λ)/KL2 (S n−1 ) are isomorphic for all .λ ∈ R and differ from .S/(C0 (R) ⊗ KL2 (S n−1 )). Theorem 2.2.12 For .λ ∈ R, the algebra .S(λ)/KL2 (S n−1 ) is isomorphic to the algebra .C(V (n, 2)) of the continuous functions on the manifold .V (n, 2) of the pairs .(ϕ, ω) of orthogonal unit vectors in .Rn . The maps π(ϕ, ω) : A(λ) → (ϕ, ω), (ϕ, ω) ∈ V (n, 2),

.

(2.2.46)

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

92

extend to irreducible representations of the algebra .S(λ)/KL2 (S n−1 ) (the notations are the same as in Theorem 2.2.11). The representations are pairwise nonequivalent. Any irreducible representation of .S(λ) is equivalent either to one of the representations (2.2.46) or to the identity representation. Proof can be found in [21], 5.3.  of the algebra We are going to describe the Jacobson topology on the spectrum .S n−1 n−1 .S. Denote by . the union of the product .S ×S and the line .l = R. From Theorem 2.2.11, it follows that the correspondence .π(ϕ, ω) → (ϕ, ω), .π(λ) → λ  onto .. We introduce a topology on .. If .(ϕ, ω) ∈ defines a bijection of the spectrum .S n−1 n−1 S ×S and the scalar product .ϕω of unit vectors .ϕ and .ω in .Rn satisfies the inequality .ϕω > 0 (< 0), then a fundamental neighborhood system of the point .(ϕ, ω) comprises the sets .V(ϕ, ω) ∪ {λ ∈ l : λ > N (< N)}, where .V(ϕ, ω) is a neighborhood of .(ϕ, ω) in the product .S n−1 × S n−1 and .N ∈ R. In the case .ϕω = 0, the fundamental system is formed by the sets .V(ϕ, ω) ∪ R. A neighborhood of a point .λ ∈ l ⊂  is an interval in l containing .λ.  induced by that of . coincides with the Jacobson Theorem 2.2.13 The topology of .S topology.  is formed by the sets of the Proof Recall that the base of the Jacobson topology on .S classes of equivalent irreducible representations .π such that .πA > 1, where .A runs over a dense set in .S (see Sect. 1.3.3)). Let .λ0 be an arbitrary point in the line l and let .A be an element in .S such that .A(λ0 ) > 1. The set .{π : π A > 1} is a neighborhood of .λ0 in the Jacobson topology. Since the function .λ → A(λ) is continuous in the norm, the set is a neighborhood of .λ0 in the usual topology on the real axis. On the other hand, if f and g are elements in .L2 (S n−1 ) and c is an arbitrary continuous function vanishing at infinity, the operatorfunction .λ → c(λ)(·, f )g belongs to .S. Therefore, any open interval on the real axis l is open in the Jacobson topology. Now, we consider neighborhoods of a point .(ϕ0 , ω0 ) ∈ S n−1 × S n−1 assuming that −1 (ϕ, ω)E(λ). Since .π(ϕ, ω)A = .ϕ0 ω0 > 0. Let .(ϕ0 , ω0 ) > 1 and .A(λ) = E(λ) (ϕ, ω), the inequality .π(ϕ, ω)A > 1 holds for all .(ϕ, ω) in a neighborhood of .(ϕ0 , ω0 ) on .S n−1 × S n−1 . Moreover, .π(λ)A > 1 for all .λ ∈ (N, +∞), where N is a sufficiently large positive number. To show that, we choose a function v in .C ∞ (S n−1 ) with support in a small neighborhood of .ϕ0 such that .v; L2 (S n−1 ) = 1 and denote by g a smooth function on the sphere extended to .Rn \ 0 as a zero degree homogeneous function. In addition, assume that the gradient .∇g does not vanish on the support of v. From Theorem 1.2.8, it follows that e−iμg(ϕ) Aψ→ϕ (λ)eiμg(ψ) v(ψ) − (ϕ, μ∇g(ϕ) + σ ϕ)v(ϕ) = O((|μ| + |σ |)−1 ) (2.2.47)

.

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

93

as .|μ| + |σ | → ∞, where .μ ∈ R and .σ = λ. Note that .ϕ∇g(ϕ) = 0 and .ϕ0 ω0 = 0. Therefore, given large .σ > 0, we can choose .μ and g so that the vectors .μ∇g(ϕ0 ) + σ ϕ0 and .ω0 are not parallel. Then, in view of (2.2.47), .π(λ)A > 1 for all .λ > N. Thus, a neighborhood of .(ϕ0 , ω0 ) in the Jacobson topology is also a neighborhood in the topology on .. The following fact (Lemma 5.4.8 in [21]) will be used below. Assume that .A(λ) = E −1 (λ)(ϕ, ω)E(λ), where .(ϕ, ω) = 0 for .ϕω ≤ 0 (.ϕω ≥ 0). Then, .A(λ) = O(e−π|λ| ), as .λ → −∞ (resp., .λ → +∞). Let .V(ϕ0 , ω0 ) ∪ {λ ∈ l : λ > N } be an arbitrary neighborhood of .(ϕ0 , ω0 ) in the space .. We verify that it is also a neighborhood of the point in the Jacobson topology. Let us choose a function . ∈ C ∞ (S n−1 × S n−1 ) so that .(ϕ0 , ω0 ) > 1 and . = 0 outside the set .V(ϕ0 , ω0 ) ∩ {(ϕ, ω) : ϕω > δ} with small positive .δ. Then, for .A(λ) = E −1 (λ)E(λ), we obtain .{(ϕ, ω) : π(ϕ, ω)A > 1} ⊂ V(ϕ0 , ω0 ). From the fact formulated above, it follows that .π(λ)A → 0 for .λ → −∞. Moreover, according to Proposition 1.2.7, the operator .A(λ) is compact for every .λ. In view of Proposition 2.2.3, the function .λ → c(λ)A(λ), where .c ∈ C0∞ (R), belongs to .S. We choose the function c so that .0 ≤ c ≤ 1 and .c(λ) = 1 for .λ ∈ [−M, M], where M is a large number. Changing, if needed, .A for .A − cA, we obtain an operator-function subject to the relation .{π : πA > 1} ⊂ V(ϕ0 , ω0 ) ∪ {λ ∈ l : λ > N }. We have shown that the neighborhood systems of a point .(ϕ0 , ω0 ) with .ϕ0 ω0 > 0 in the topology . and in the Jacobson topology coincide. Such a coincidence for the points .(ϕ0 , ω0 ) with .ϕ0 ω0 < 0 can be verified in a similar way. Assume now that .ϕ0 ω0 = 0, .(ϕ0 , ω0 ) > 1, and .A(λ) = E(λ)−1 E(λ). We choose the function g in (2.2.47) so that .ω0 = ∇g(ϕ0 ). Letting .μ to infinity, we obtain the inequality .π(λ)A > 1 for all .λ ∈ l. Therefore, a neighborhood of .(ϕ0 , ω0 ) in the Jacobson topology is a neighborhood in the space . as well. The converse is evident.   is not separable; .S  is a .T0 -space (one Thus, the topology on the spectrum of the algebra .S of the two any points of the space has a neighborhood not containing the other point). The reason why the topology is not Hausdorff is already evident for .n = 1. In this case, in the operators .E(λ)± (see (1.2.2) and (1.2.8)), the “integral” over a 0-dimensional sphere is the sum of the integrand values at the points .±1. Therefore, .

(E(λ)u(1)) (E(λ)u)(−1)

(1/2 + iλ) = √ 2π



e−i(π/2)(1/2+iλ) ei(π/2)(1/2+iλ) (2.2.48) ei(π/2)(1/2+iλ) e−i(π/2)(1/2+iλ)

u(1) × , u(−1)

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

94



(E(λ)−1 v(1)) (E(λ)−1 v)(−1)



(1/2 − iλ) = √ 2π



ei(π/2)(1/2−iλ) e−i(π/2)(1/2−iλ) (2.2.49) e−i(π/2)(1/2−iλ) ei(π/2)(1/2−iλ)

v(1) × . v(−1)

Setting .c(ϕ) = ((ϕ, 1)+(ϕ, −1))/2 and .d(ϕ) = ((ϕ, 1)−(ϕ, −1))/2 for .ϕ = ±1, we obtain

c(1) + d(1) tanh π λ id(1)/ cosh π λ −1 .A(λ) = E(λ) (ϕ, ω)E(λ) = . −id(−1)/ cosh π λ c(−1) − d(−1) tanh π λ (2.2.50) The maps (2.2.43) and (2.2.44) take the form π(λ) : A → A(λ) : C2 → C2 ,

.

π(±1, ω) : A → c(±1) + d(±1)sgn ω, ω = ±1. It is obvious that the representations .π(λ) are irreducible for all real .λ. Comparing the traces, we find that the representations are pairwise nonequivalent. Moreover, .

lim A(λ) = diag(c(1) + d(1), c(−1) − d(−1)),

λ→+∞

lim A(λ) = diag(c(1) − d(1), c(−1) + d(−1)).

λ→−∞

In other words, at the points .±∞, the irreducible representation .π(λ) has, as a “limit,” the sum of two irreducible one-dimensional representations. This explains the non-separability at the points .π(±, ω). To prove the solvability of the algebra .S, we need: Proposition 2.2.14 Let .A(λ) = E −1 (λ)(ϕ, ω)E(λ), where . ∈ C ∞ (S n−1 × S n−1 ). The map .p : A →  extends to an isomorphism p : S/(C0 (R) ⊗ KL2 (S n−1 ))  C(S n−1 × S n−1 ).

.

(2.2.51)

Proof Let .J = C0 (R) ⊗ KL2 (S n−1 ), .Aj k (λ) = E −1 (λ)j k (ϕ, ω)E(λ), and .j k ∈ C ∞ (S n−1 × S n−1 ), where j and k run over finite sets. According to Proposition 1.2.14,  .

j

k

Aj k (λ) = E(λ)−1

 j

k

j k (ϕω)E(λ) + T (λ),

2.2 Algebras of Pseudodifferential Operators with Isolated Singularities. . .

95

where .T ∈ J . Representations (2.2.44) annihilate the ideal J ; they can be considered as representations of the quotient algebra .S/J . Every representation .S/J is equivalent to one of representations (2.2.44) (Theorem 2.2.11). By virtue of Proposition 1.3.1, 



.

j

Aj k ; S/J  = sup{π(ϕ, ω)

k

 j

Aj k ; (ϕ, ω) ∈ S n−1 × S n−1 }

k

=

 j

j k ; C(S n−1 × S n−1 ).

k

Therefore, the map p extends to an isomorphism (2.2.51).



The next result immediately follows from Propositions 2.2.14 and 1.3.23. Theorem 2.2.15 The composition series .0 ⊂ C0 (R) ⊗ KL2 (S n−1 ) ⊂ S is solving and n−1 ))  C(S n−1 × S n−1 ). This series coincides with the maximal .S/(C0 (R) ⊗ KL2 (S radical series. The length of the algebra .S is equal to 1. Remark 2.2.16 Denote by .com S the ideal of .S generated by the commutators .[A, B] := AB − BA, where .A, B ∈ S. Then, .com S = C0 (R) ⊗ KL2 (S n−1 ). Therefore, the composition series in Theorem 2.2.15 can be written in the form .0 ⊂ com S ⊂ S. Indeed, .com S ⊂ J := C0 (R) ⊗ KL2 (S n−1 ) because the algebra .S/J is commutative. For any .λ ∈ R, the algebra .S(λ) is noncommutative, and the restriction .π(λ)|comS is nonzero. According to Proposition 1.3.7, the maps .π(λ1 )|comS and .π(λ2 )|comS are irreducible and nonequivalent representations of the ideal .com S for .λ1 = λ2 , i.e., .com S is a rich subalgebra in J . Thus, .comS = J (Proposition 1.3.9).

2.2.5

The Spectrum of Algebra A

¯ 0 (M) Recall that the algebra .A is generated on .L2 (M) by the operators of the class . and the operators of multiplication .f · by the functions f in .M (see the beginning of ¯ 0 (M) is a smooth function on Sect. 2.2.1). The principal symbol .σ 0 of an operator .A ∈  ∗ 0 the cospheric bundle .S (M). For .f ∈ M, we set .f (ϕ) := limr→0 f (r, ϕ), where .(r, ϕ) are local spheric coordinates centered at the point .x 0 . For such generators f and A of the algebra .A, we introduce the maps π(x, ω) : A → σ 0 (ω), π(x, ω) : f → f (x),

.

(2.2.52)

where .ω ∈ S ∗ (M)x and .x ∈ M \ x 0 ; π(ϕ, ω) : A → σ 0 (ω), π(ϕ, ω) : f → f 0 (ϕ),

.

(2.2.53)

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

96

where .(ϕ, ω) ∈ S(M)x 0 × S ∗ (M)x 0 ; π(λ) : A → Eω→ϕ (λ)−1 σ 0 (ω)Eψ→ω (λ) : L2 (S(M)x 0 ) → L2 (S(M)x 0 ),

.

(2.2.54)

π(λ) : f → f 0 · : L2 (S(M)x 0 ) → L2 (S(M)x 0 ), where .f 0 · denotes the operator of multiplication by a function .f 0 on the space .L2 (S(M)x 0 ). Taking into account Sect. 2.2.1, Propositions 1.3.26 and 2.1.5, and Theorem 2.2.11, we obtain the following result. Theorem 2.2.17 Maps (2.2.52)–(2.2.54) extend to representations of the quotient algebra A/KL2 (M). These representations are irreducible and pairwise nonequivalent. Every irreducible representation of the algebra .A is equivalent to one of the representations (2.2.52)–(2.2.54) or to the identity representation. The algebra .A is an algebra of type I .

.

Let us discuss the spectral topology. We denote by . the disjoint union of the sets S ∗ (M)|(M \ x 0 ), .S(M)x 0 × S ∗ (M)x 0 , and .R. By Theorem 2.2.17, the correspondence ∧ .π(x, ω) → ω, .π(ϕ, ω) → (ϕ, ω), and .π(λ) → λ defines a bijection of .(A/KL2 (M)) onto .. Therefore, it suffices to introduce a topology on .. A basis of neighborhoods for a point .λ ∈ R is formed by the open intervals in .R containing .λ. Neighborhoods of a point in .S ∗ (M)|(M \ x 0 ) are also defined in a usual way. To describe neighborhoods of a point .(ϕ, ω) ∈ S(M)x 0 × S ∗ (M)x 0 , we consider the union .S ∗ (M)|(M \ x 0 ) ∪ (S(M)x 0 × S ∗ (M)x 0 ) as a compact manifold X with boundary. If .ϕω > 0, a neighborhood of a point .(ϕ, ω) is introduced as the union of a neighborhood .W(ϕ, ω) of this point in X and a set of the form .{λ ∈ R : λ > N} with some N. In the case .ϕω < 0, a neighborhood of .(ϕ, ω) is the union .W(ϕ, ω) ∪ {λ ∈ R : λ < N }. Finally, for .ϕω = 0, a basis of neighborhoods is formed by the sets .W(ϕ, ω) ∪ R. Using Theorem 2.2.13, the reader can verify the following assertion. .

Theorem 2.2.18 The topology carried over from the space . to .(A/KL2 (M))∧ coincides with the Jacobson topology. Let us consider the composition series 0 ⊂ KL2 (M) ⊂ com A ⊂ A,

.

(2.2.55)

where .com A denotes the ideal in .A generated by the commutators of the elements in .A. It is obvious that all one-dimensional representations vanish on the ideal .com A. From Theorem 2.2.17, it follows that .com A/KL2 (M)  C0 (R) ⊗ KL2 (S n−1 ). Finally, .A/comA  C(X), where X is a compact manifold with a boundary obtained by gluing

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

97

S(M)x 0 × S ∗ (M)x 0 to .S ∗ (M)|(M \ x 0 ). Thus, composition series (2.2.55) is solving, and the length of the algebra .A is equal to 2. Series (2.2.55) is stratified and, consequently, coincides with the maximal radical series (see Sect. 1.3.7). Indeed, by Theorem 2.2.17, all irreducible representations but the identity one annihilate .KL2 (M). Among the nonzero irreducible representations of the ideal .comA, in addition to the identity one, there are representations of the form .π(λ), n−1 ), while the algebra .π(λ)A contains some .λ ∈ R. However, .π(λ)(comA) = KL2 (S noncompact operators (for example, the identity operator on .L2 (S n−1 )). Summing up these facts, we obtain the following result. .

Theorem 2.2.19 Composition series (2.2.55) is solving and comA/KL2 (M)  C0 (R) ⊗ KL2 (S n−1 ),

.

A/comA  C(X), where X is a compact manifold with a boundary obtained by gluing .S(M)x 0 × S ∗ (M)x 0 to .S ∗ (M)|(M \ x 0 ). The length of the algebra .A is equal to 2. Series (2.2.55) coincides with the maximal radical series.

2.3

Algebras of Pseudodifferential Operators with Discontinuities in Symbols Along a Submanifold

2.3.1

The Statement of Basic Theorems

Let .M be a smooth m-dimensional compact Riemannian manifold without boundary and let .N be a submanifold of codimension n, where .1 ≤ n ≤ m − 1. We denote by .MN the set of smooth functions f on .M \ N that have a limit .f 0 (z, ϕ) = limx→z f (x) smoothly depending on .z ∈ N and on .ϕ ∈ ν(N )z , where .ν(N ) is the bundle of the unit vectors normal to .N ; thus, the functions f are continuous on the compact obtained by gluing the boundary of a tubular neighborhood of .N to .M \ N . We consider the algebra .A ¯ 0 (M) (see Definition 2.1.4) and the generated on .L2 (M) by the operators of the class . multiplication operators .f · for .f ∈ MN . This algebra is irreducible and contains the ideal .KL2 (M). ¯ 0 (M) and let .σ 0 be the principal symbol of the operator A. We introduce Let .A ∈  the maps: π(x, ω) : A → σ 0 (ω), π(x, ω) : f → f (x),

.

(2.3.1)

where .ω ∈ S ∗ (M)x and .x ∈ M \ N ; π(z, ϕ, ω) : A → σ 0 (ω), π(z, ϕ, ω) : f → f 0 (z, ϕ),

.

(2.3.2)

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

98

where .z ∈ N , .ϕ ∈ ν(N )z , and .ω ∈ S ∗ (M)z ; −1 π(z, θ ) : A → Fη→x σ 0 (η, θ )Fy→η ∈ BL2 (Rn ),

.

(2.3.3)

π(z, θ ) : f → f 0 (z, ·)· ∈ BL2 (Rn ), where F is the Fourier transform on .Rn = T (N )⊥ z ⊂ T (M)z , T is the tangent bundle, ∗ 0 n .θ ∈ S (N )z , and .f (z, ·)· denotes the operator of multiplication by the function .R \ 0  x → f 0 (z, x/|x|); π(z, λ) : A → A(z, λ) := Eω→ϕ (λ)−1 σ 0 (ω, 0)Eψ→ω (λ) ∈ BL2 (z ),

.

(2.3.4)

π(z, λ) : f → f 0 (z, ·)· ∈ BL2 (z ), where .z = ν(N )z is the unit sphere in the space .T (N )⊥ z , .ω ∈ z . Theorem 2.3.1 The maps (2.3.1)–(2.3.4) extend to irreducible pairwise nonequivalent representations of the algebra .A/KL2 (M). Every irreducible representation of the algebra .A is equivalent either to the identity representation or to one of those in (2.3.1)– (2.3.4). The algebra .A is a type I -algebra. If .codim N = 1, representations (2.3.4) turn out to be two-dimensional, see (2.2.48)– (2.2.50). As a rule, in what follows, we do not emphasize this fact by special remarks. We now proceed to describing the spectral topology. Let .MN be the compact manifold with the boundary obtained by gluing .ν(N ) to .M \ N and .p : MN → M the projection. We denote by .p∗ (S ∗ (M)) the induced bundle over .MN and by . the disjoint union of the sets .p∗ (S ∗ (M)), .S ∗ (N ), and .N ×R. According to Theorem 2.3.1, the correspondence π(x, ω) → (x, ω) ∈ p ∗ (S ∗ (M)), x ∈ M \ N ,

.

∗

(2.3.5)

∗

π(z, ϕ, ω) → (z, ϕ, ω) ∈ p (S (M)), (z, ϕ) ∈ ν(N )z ⊂ MN , π(z, θ ) → (z, θ ) ∈ S ∗ (N )z , z ∈ N , π(z, λ) → (z, λ) ∈ N × R is a bijection of the set .(A/KL2 (M))∧ onto .. We introduce a topology on . by indicating typical neighborhoods of the points, i.e., neighborhoods forming a fundamental system. A neighborhood in . of a point .ω ∈ p ∗ (S ∗ (M))z , .z ∈ N , is a usual neighborhood ∗ ∗ .V(ω) of this point in .S (M), .V(ω) ∩ (S (M)|N ) = ∅. A neighborhood in . of a point ∗ ∗ .θ ∈ S (N )z , .z ∈ N , is its neighborhood in the bundle .S (N ). A neighborhood of a point .(z, λ) ∈ N × R is defined as the union of an open set .U ⊂ N × R containing .(z, λ) and the part of .S ∗ (N ) over the projection of .U into .N .

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

99

Let .z0 ∈ N , .ϕ0 ∈ ν(N )z0 , and .ω0 ∈ S ∗ (M)z0 . We can assume that, near .z0 , the manifold .M coincides with .Rm = {x = (x (1) , x (2) ) : x (1) = (x1 , . . . , xn ), x (2) = (xn+1 , . . . , xm )} and .N = Rm−n = {x ∈ Rm : x = (x (1) , x (2) ), x (1) = 0}. Setting (1) (2) (2) .ω0 = (ω 0 , ω0 ), we first consider the case .ω0 = 0. A typical neighborhood of a point 0 .(z , ϕ0 , ω0 ) in . is the union U (z0 , ϕ0 , ω0 ) ∪ (V(z0 ) × W(θ0 )),

.

where .U(z0 , ϕ0 , ω0 ) is a usual neighborhood of .(z0 , ϕ0 , ω0 ) in the bundle .p∗ (S ∗ (M)) not containing the points .(z, ϕ, ω) with .ω(2) = 0; .V(z0 ) is a neighborhood of .z0 in .N ; (2) (2) m−n−1 . .θ0 = ω 0 /|ω0 | and .W(θ0 ) is a neighborhood of .θ0 in .S (1) (1) For .ω0 = (ω0 , 0), .ϕ0 ω0 = 0, a typical neighborhood of .(z0 , ϕ0 , ω0 ) is of the form U (z0 , ϕ0 , ω0 ) ∪ (V(z0 ) × S m−n−1 ) ∪ (V(z0 ) × R),

.

where .U(z0 , ϕ0 , ω0 ) is an arbitrary neighborhood of .(z0 , ϕ0 , ω0 ) in .p∗ (S ∗ (M)). If (1) .ϕ0 ω 0 ≷ 0, then the line .R must be changed for the set .{λ ∈ R : λ ≷ Q} with any (1) real number Q. In case of .codim N = 1, the points subject to the conditions .ϕ0 ω0 = 0 are absent, and any point .(z0 , ϕ0 , ω0 ) has a neighborhood containing only a part of the line .R. Theorem 2.3.2 The topology carried over from the space . to .(A/KL2 (M))∧ by the bijection (2.3.5) coincides with the Jacobson topology. The spectrum .(A/KL2 (M))∧ is a .T0 -space. Theorem 2.3.3 Let I denote the ideal in the algebra .A equal to the intersection of the kernels of all one-dimensional representations and all representations .π(z, λ) for .(z, λ) ∈ N × R. Then, the composition series .0 ⊂ KL2 (M) ⊂ I ⊂ com A ⊂ A is solving and I /KL2 (M)  C(S ∗ (N )) ⊗ KL2 (Rn ),

.

com A/I  C0 (N × R) ⊗ KL2 (S n−1 ), A/com A  C(p∗ (S ∗ (M))). The length of .A is equal to 3. The composition series is the maximal radical series.

2.3.2

Algebras L(θ): Irreducibility

Let us introduce Cartesian coordinates in the tangent plane to .M at a point .z0 ∈ N choosing axes .x1 , . . . , xn to be orthogonal to the submanifold .N . A point x in the tangent

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

100

plane .Rm will be written in the form .x = (x (1) , x (2) ), where .x (1) = (x1 , . . . , xn ) and (2) = (x .x n+1 , . . . , xm ). We also introduce the operators A = Fξ−1 →x (ϕ, ξ )Fy→ξ ,

.

(2.3.6)

where F is the Fourier transform in .Rm , .(ϕ, ξ ) = f 0 (ϕ)σ 0 (ξ ), .ϕ = x (1) /|x (1) |, and .σ 0 is a degree zero homogeneous function, while .f 0 ∈ C ∞ (S n−1 ) and .σ 0 ∈ C ∞ (S m−1 ). Denote by .L the algebra spanned by the operator-functions −1 S m−n−1  θ → A(θ ) = Fη→x (ϕ, η, θ )Fy→η ∈ BL2 (Rn ),

.

(2.3.7)

with .A(·); L = sup{A(θ ); BL2 (Rn ), θ ∈ S m−n−1 } (here, F denotes the Fourier transform in .Rn ). Let .L(θ ) with .θ ∈ S m−n−1 be the algebra generated by the operators n .A(θ ) on .L2 (R ). To apply the localization principle (Proposition 1.3.24) to the algebra .A, we choose .C(M) as a localizing subalgebra and .J = KL2 (M). Let .Az0 be a local algebra obtained by localizing .A at a point .z0 . Proposition 2.3.4 (1) The algebra .Az0 for .z0 ∈ N is generated on .L2 (Rm ) by the operators of the form (2.3.6). (2) The algebras .Az0 and .L are isomorphic. The equalities  = ∪θ∈S m−n−1 L(θ  ) L

.

(2.3.8)

 and .L(θ  ) are the spectra of the algebras .L and .L(θ ). hold, where .L Proof (1) Let V be a coordinate neighborhood of a point .z0 in .M and let . be a coordinate diffeomorphism projecting V into the tangent space to .M at .z0 so that .(z0 ) = 0,  0 (1) , x (2) ). Moreover, let B denote a product . (z ) = I , and .V  z → (z) = (x of finitely many generators of the algebra .A and let .B denote the operator B written in local coordinates. For .t > 0, we introduce the unitary operator .Ut = t m/2 u(t·) on −1 m ∞ .L2 (R ). Finally, we choose .χ ∈ Cc (V ) and consider .Ut (χ Bχ ) Ut , which is a product of finitely many operators of the form  v →

.

eixξ f (tx)σ (tx, ξ/t)v(ξ ˆ ) dξ,

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

101

¯ 0 (M) and f is a function in .MN written in where .σ is a symbol of operator in . local coordinates. Therefore, in .L2 (Rm ), there exists a strong limit Q0 := lim Ut (χ Bχ ) Ut−1 .

.

t→0

The operator .Q0 is a product of finitely many operators of the form (2.3.6) and Q0 ≤ |χ (0)|2 Q,

.

where .Q = (ψBψ) , .ψ ∈ Cc∞ (V ), and .ψχ = χ. Let .Q be the algebra spanned in .L2 (Rm ) by operators of the form .(χ Bχ ) and let .J be the ideal in .Q generated by the operators of multiplication by the functions .ζ ∈ Cc∞ (Rm ) such that .ζ (0) = 0. From the inequality .Q0 ≤ |χ (0)|2 Q, it follows that the map .Q → Q0 extends to an epimorphism .q : Q/J → Q0 , where .Q0 is the algebra spanned by operators of the form (2.3.6). We show that q is an isomorphism. Let .φ ∈ Cc∞ (Rm ) and .φ(0) = 1. Then, .φQ0 φ ∈ Q and .q : [φQ0 φ] → Q0 , where .[φQ0 φ] is the residue class of the operator .φQ0 φ in the algebra .Q/J. The fact that q is a monomorphism follows from the relations .Q−φQ0 φ ∈ J and .φQ0 φ ≤ |φ|2 Q0 . Let .Jz0 be the ideal in .A generated by the operators of multiplication by the functions 0 .η ∈ C(M) with .η(z ) = 0. An isomorphism .Az0  Q0 is established by the map Az0 = A/Jz0  [A] → [( −1 )∗ (χ Aχ )()∗ ] ∈ Q/J → Q0 ;

.

the latter arrow indicates the map q, .χ ∈ Cc∞ (V ), and .χ (z) = 1 near .z0 . (2) We denote by .u(y ˆ (1) , ξ (2) ) = Fy (2) →ξ (2) u(y (1) , y (2) ) the partial Fourier transform of a function u. Then, from (2.3.6), it follows that (1) (2) (Au)ˆ(x (1) , ξ (2) ) = Fξ−1 , ξ )Fy (1) →ξ (1) u(y ˆ (1) , ξ (2) ). (1) →x (1) (ϕ, ξ

.

(2.3.9)

Setting .X = x (1) |ξ (2) | and .Y = y (1) |ξ (2) |, we rewrite (2.3.9) as (Au)ˆ(X|ξ (2) |−1 , ξ (2) ) = (2π )−n



.

Rn

 eiXη (ϕ, η, θ ) dη e−iY η u(Y ˆ |ξ (2) |−1 , ξ (2) ) dY Rn

(2.3.10) because . is a homogeneous function. Let us write x, y (.∈ Rn ) instead of X, Y and .v(y) and .A(θ )v(x) instead of .u(Y ˆ |ξ (2) |−1 , ξ (2) ) and .(Au)ˆ(X|ξ (2) |−1 , ξ (2) ). Equality (2.3.10) now coincides with the formula −1 A(θ )v(x) = Fη→x (ϕ, η, θ )Fy→η v(y), θ ∈ S m−n−1 .

.

(2.3.11)

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

102

Since the transform .Fy (2) →ξ (2) is unitary on .L2 (Rm−n ), we find that .Az0 and .L are isomorphic. Formula (2.3.8) follows from Proposition 1.3.16.  Proposition 2.3.5 The algebra .L(θ ) is irreducible, i.e., every subspace in .L2 (Rn ) invariant with respect to .L(θ ) is either 0 or .L2 (Rn ). Proof We first obtain a new representation for operators (2.3.11). According to Proposition 1.2.2, the Fourier transform is a composition of the Mellin transform, .E(λ), and the inverse Mellin transform. We substitute the corresponding expressions for .F ±1 in (2.3.11) and, after the change of variable .ρ = et , arrive at 1 A(θ )u(x) = √ 2π



∞

.

r iλ−n/2 Eω→ϕ (λ)−1 μ→λ (ϕ, ω, θ )Eψ→ω (μ)

(2.3.12)

0

×u(μ ˜ + in/2 , ψ) dμ, where .u˜ denotes the Mellin transform of u and μ→λ (ϕ, ω, θ )g(μ) =

.

1 2π



+∞ −∞

eiλt (ϕ, ω, e−t θ ) dt



+∞ −∞

e−itμ g(μ) dμ. (2.3.13)

Let .H ⊂ L2 (Rn ) be a subspace invariant with respect to .L(θ ). We show that it is also invariant with respect to all operators Bu(x) = (2π )−1/2



.

∞

r iλ−n/2 Eω→ϕ (λ)−1 (ϕ, ω)Eψ→ω (λ)u(λ ˜ + in/2, ψ) dλ,

0

(2.3.14) where . ∈ C ∞ (S n−1 × S n−1 ). It suffices to verify that every operator (2.3.14) is the limit of a sequence of operators in .L(θ ) in the strong topology. We first consider the case where the function . depends only on .ω ∈ S n−1 . Let .χ ∈ C ∞ (R), let .χ (t) = 1 for .t > 0, and let .χ (t) = 0 for .t < −1. We smoothly extend the function .ω → (ω) inside the sphere .S n−1 and introduce .k (ξ ) = 2 m (ξ 1 /|ξ |)χ√ k (|ξ |/|ξ |), where .k ∈ N, .ξ ∈ R \ 0, and the function .χk is defined −t by .χk (e / 1 + e−2t ) = χ (t + k). Denote√by .Ak (θ ) operator (2.3.12) assuming that −t θ ) ≡  (ω, e−t θ ) = (ω)χ (e−t / 1 + e−2t ). .(ω, e k k The sequence of the operators .h → k h on .L2 (R, L2 (S n−1 )) strongly converges to the operator .h → h as .k → ∞. Therefore, the sequence .{Ak (θ )} strongly converges to the operator B because the Mellin transform .M : L2 (Rn ) → L2 (R, L2 (S n−1 )) and .E(λ) : L2 (S n−1 ) → L2 (S n−1 ) are unitary as well as the Fourier transform on .L2 (R, L2 (S n−1 ))

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

103

sending a function h to the map 1 .(t, ω) → √ 2π



+∞ −∞

e−itλ h(λ, ω) dλ.

We have assumed that the function . does not depend on .ϕ ∈ S n−1 . To eliminate this restriction, it suffices to observe that the operator (2.3.14) with an arbitrary function ∞ n−1 × S n−1 ) can be approximated in the norm by operators in which the role . ∈ C (S  of .(ϕ, ω) is played by the finite sums . aj (ϕ)bj (ω). The orthogonal complement .H⊥ of the subspace .H is also an invariant subspace for the algebra .L(θ ). It follows from the foregoing that .H⊥ is invariant under all the operators (2.3.14). The subspace .H is thereby an invariant subspace for the algebra n .L(0) generated by operators of the form (2.3.14) on .L2 (R ). The algebra .L(0) is isomorphic to the algebra .S spanned by the operator-valued functions .λ → A(λ) = E(λ)−1 (ϕ, ω)E(λ) ∈ BL2 (S n−1 ), see Sect. 2.2.1. According to Proposition 2.2.3, we have .C0 (R) ⊗ KL2 (S n−1 ) ⊂ S. This means that the algebra .L(0) contains all operators of the form N   L2 (Rn )  f → M −1 h(λ) (Mf )(λ + in/2 , ·), ek ek ,

.

(2.3.15)

k=1

where . , and .{ek } are the inner product and a basis in .L2 (S n−1 ), and .h ∈ C0 (R). It is clear that the sequence of operators (2.3.15) converges strongly as .N → ∞ to the operator −1 (h(λ)(Mf )(λ + in/2 , ·)). Hence, this operator carries the subspace .H, which .f → M is invariant under .L(θ ), into itself. Let f and g be elements of .H and .H⊥ , respectively. It follows from the above that 

+∞

.

−∞

h(λ)f˜(λ + n/2 , ·) , g(λ ˜ + in/2 , ·) dλ = 0.

Since .h ∈ C0 (R) is arbitrary, we obtain f˜(λ + in/2 , ·) , g(λ ˜ + in/2 , ·) = 0

.

for almost all .λ ∈ R. Every operator in .L(0) can be written in the form .M −1 A(λ)M, where .A ∈ S. Therefore, .f˜(λ + in/2 , ·) can be replaced by .A(λ)f˜(λ + in/2 , ·). Thus, for almost all .λ ∈ R ˜ + in/2 , ·) , g(λ A(λ)(λ ˜ + in/2 , ·) = 0.

.

(2.3.16)

The algebra .S(λ) generated on .L2 (S n−1 ) by the operators .A(λ) is irreducible for each n−1 ) is thereby .λ ∈ R (Proposition 2.2.1). Every nonzero vector .f˜(λ + in/2 , ·) ∈ L2 (S

104

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

totalizing for .S(λ), i.e., the set .{A(λ)f˜(λ+in/2 , ·) : A(λ) ∈ S(λ)} is dense in .L2 (S n−1 ). Consequently, if .f ∈ H, .g ∈ H ⊥ , and .λ ∈ R are such that (2.3.16) holds and .g(λ ˜ + in/2 , ·) = 0, then .f˜(λ + in/2 , ·) = 0. Assume that there is a nonzero element g in .H⊥ . Then, any function .λ → f˜(λ+in/2 , ·) with .f ∈ H vanishes almost everywhere on the set .{λ ∈ R : g(λ ˜ + in/2 , ·) = 0} of positive measure. We show that this is possible only if .H = 0. This concludes the proof of the proposition. Choose a sequence .{χj } ⊂ C0∞ (R) converging to the function .t → δ(t − t0 ). We define the functions .j ∈ C ∞ (S m−1 ) by .j (ω(1 + e−2t )−1/2 , e−t θ (1 + e−2t )−1/2 ) = χj (t), where .ω ∈ S n−1 , .θ ∈ S m−n−1 , and .t ∈ R, and extend .j to .Rm \ 0 as a homogeneous function of zero degree. Let .Aj (θ ) denote the right-hand side of (2.3.12) with . = j . The equality .vj = Aj (θ )f can be rewritten in the form 1 .E(λ)vj (λ + in/2 , ·) = 2π





+∞

−∞

e

iλt

χj (t) dt

+∞ −∞

e−itμ E(μ) × f˜(μ + in/2 , ·) dμ. (2.3.17)

Assume that .H = 0 and .f ∈ H is a nonzero element. Taking into account that the operator of multiplication by a function .h ∈ C0 (R) does not lead out of the subspace .MH, we can take .f ∈ H such that the function .λ → f˜(λ + in/2 , ·) has compact support and  +∞ . e−it0 μ E(μ)f˜(μ + in/2 , ·) dμ = 0. −∞

Clearly, the operator .v(λ ˜ + in/2 , ·) → E(λ)v(λ + in/2 , ·) does not change the support of a function in .L2 (R, L2 (S n−1 )). Therefore, choosing the number j in (2.3.17) sufficiently large, we obtain that the function .λ → v˜j (λ + in/2, ·) has no zero values on any given finite interval. Take this interval so that the measure of its intersection with the set .{λ : g(λ ˜ + in/2 , ·) = 0} is positive. Then, the function .λ → g(λ ˜ + in/2 , ·) does not vanish on the indicated intersection, which contradicts the inclusion .vj ∈ H. 

2.3.3

Localization in the Algebra L(θ)

n−1 ¯ n be the compact set obtained by adjoining the .(n − 1)-dimensional sphere .S∞ Let .R n to the space .R at infinity. We are going to apply Proposition 1.3.26 for localization in ¯ n ) as a localizing algebra .C. We first prove some technical the algebra .L(θ ) choosing .C(R results for verification of the conditions of this proposition.

Lemma 2.3.6 Let .Rm \ 0  ξ → (ξ ) = |ξ |a f (ξ/|ξ |) be a homogeneous function of a complex degree a with .Re a ≤ 0, .f ∈ C ∞ (S m−1 ), and −1 B(θ ) = Fη→x (η, θ )Fy→η ,

.

(2.3.18)

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

105

where .η ∈ Rn and .θ ∈ S m−n−1 . Then, .(B(θ )v)(λ + ia + in/2, ϕ) is equal to  =

Eω→ϕ (λ + ia)−1 μ→λ (ω, θ )Eψ→ω (μ)v(μ ˜ + in/2, ψ)  dψ H˜ (ϕ, ψ; λ, λ − μ, θ )v(μ ˜ + in/2, ψ) dμ;

.

S n−1

(2.3.19)

Imμ=0

here, .Im λ = 0, the operator .μ→λ is defined by (2.3.13) (with .(ϕ, ω, e−t θ ) replaced by −t θ )), .v ∈ C ∞ (Rn \ 0), and .(ω, e 0 H˜ (ϕ, ψ; λ, ν, θ ) = (2π )−1/2



∞

.

ρ −iν−1 dρ



0

∞

t −i(λ+ia+in/2)−1 Hˆ (tϕ − ψ, ρθ ) dt,

0

Hˆ (x, θ ) = (2π )(n−m)/2

(2.3.20)

 Rm−n

e−iθz H (x, z) dz,

where .H = F −1  (in the last equality, F is the Fourier transform on .Rm , and in (2.3.18), it denotes the Fourier transform on .Rn ). In (2.3.20) .ϕ = ψ, the Mellin transform with respect to .ρ is understood in the sense of the theory of generalized functions, and the inside integral (with respect to the variable t) is defined by means of analytic extension with respect to the parameter a (an explicit formula is written below for this extension). The relation (2.3.19) holds everywhere in the half-plane .Rea ≤ 0 except at the poles .a = −n/2 − k + iλ, where .k = 0, 1, . . . Proof Let us first assume that .0 ≥ Re a > −n/2. We write operator (2.3.18) in the form  B(θ )v(x) =

.

Rn

Hˆ (x − y, θ )v(y) dy.

Applying the Mellin transform to this equality, we obtain (B(θ )v)(λ + ia + in/2, ϕ)

.

= (2π )−1/2 = (2π )−1/2



 dψ

S n−1



∞

r −i(λ+ia+in/2)−1 dr

0 ∞



∞

v(ρ, ψ)ρ n−1 dρ

0



∞

× 0

Rm

Hˆ (x − y, θ )v(y) dy

Hˆ (rϕ − ρψ, θ )r −i(λ+ia+in/2)−1 dr

0

= (2π )−1/2

(2.3.21)





 dψ

S n−1

∞

v(ρ, ψ)ρ −i(λ+in/2)−1 dρ

0

ρ i(λ−in/2) r −i(λ+ia+in/2)−1 Hˆ (rϕ − ρψ, θ ) dr.

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

106

The quantity .Hˆ (x, ρθ ) tends to zero faster than any power of .ρ as .ρ → ∞ for .x = 0 and .0 ≥ Re a > −n/2. Since H is a homogeneous function of degree .−m − a, it follows that ˆ (tx, θ ) = Hˆ (x, tθ )t −a−n for any .t > 0. Therefore, .H 

∞

.

ρ i(λ−in/2) r −i(λ+ia+in/2)−1 Hˆ (rϕ − ρψ, θ ) dr

0

 =

∞

(2.3.22)

t −(λ+ia+in/2)−1 Hˆ (tϕ − ψ, ρθ ) dt.

0

We take account of the fact that  ∞  −i(λ+in/2)−1 . ρ v(ρ)w(ρ) dρ =

w(λ ˜ − μ)υ(μ ˜ + in/2) dμ.

Imμ=0

0

From this, (2.3.21), and (2.3.22), we obtain

=

(B(θ )v)(λ + ia + in/2, ϕ)

.





H˜ (ϕ, ψ; λ, λ − μ, θ )v(μ ˜ + in/2, ψ) dμ.

dψ S n−1

(2.3.23)

Imμ=0

However, for .0 ≥ Re a > −n/2, (B(θ )v)(λ + ia + in/2) = Eω→ϕ (λ + ia)−1 μ→λ (ω, θ )Eψ→ω (μ)v(μ ˜ + in/2, ψ)

.

(cf. the derivation of (2.3.12)). Thus, (2.3.23) can be rewritten in the form (2.3.19). The function .λ → E(λ)−1 is meromorphic on the whole plane (the poles are located at the points .λ = −i(n/2 + k), .k = 0, 1, . . . ). Therefore, the left-hand side of (2.3.19) can be extended analytically to the half-plane .Re a ≥ 0 except for the points .a = −n/2 − k + iλ. Obviously, the right-hand side of (2.3.19) has the same property. The analytic extension of the inside integral in (2.3.20) on the right is implemented in the strip .0 ≥ Re a > −n/2−p by the formula 

∞

.

(−1)p = (−is) . . . (−is + p − 1)

 0

t −is−1 χ0 (t)Hˆ (tϕ − ψ, ρθ ) dt (2.3.24)

0 ∞

t −is+p−1

dp (χ0 (t)Hˆ (tϕ − ψ, pθ )) dt, dt p

where .s = λ + ia + in/2, .χ ∈ C ∞ (R¯ + , .χ (t) = 1 for .t < 1/4, and .χ (t) = 0 for .t > 1/2.  Lemma 2.3.7 Let .A(θ ) be the operator defined by (2.3.11) with a function . independent of .ϕ and homogeneous of degree 0. Furthermore, suppose that .G(x) = Fξ−1 →x (ξ ), a zero

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

107

degree homogeneous function .σ belongs to .C ∞ (Rn \ 0), and let .ζ1 and .ζ2 be arbitrary functions in .C ∞ (S n−1 ) such that the union .suppζ1 ∪ suppζ2 of their supports lies in a semisphere. Then, for all .λ ∈ R except perhaps for .λ = 0, N (−1)|γ | (ζ1 ∂ϕγ σ )(ϕ)Eω→ϕ (λ − i|γ |)−1 γ!

(ζ1 A(θ )ζ2 σ u)(λ + in/2, ϕ) =

.

(2.3.25)

|γ |=0

(γ ) ˜ + in/2, ψ) + (ζ1 RN ζ2 u)(λ ˜ + in/2, ϕ), ×μ→λ (ω, θ )Eψ→ω (μ)ζ2 (ψ)u(μ

where .∂ϕ = ∂ |γ | /∂ϕ1 1 . . . ∂ϕnn , the operator .μ→λ is defined by (2.3.13) with −t θ ) replaced by .(−1)|γ | ∂ γ (ω, e−t θ ), and .(ϕ, ω, e ω γ

γ

γ

(γ )

 (ζ1 RN ζ2 u)(λ ˜ + in/2, ϕ) =

 dψ

.

S n−1

∞

˜ N (ϕ, ψ; λ, λ − μ, θ ) ζ1 (ϕ)G

0



∞



1

×ζ2 (ψ)u(μ ˜ + in/2, ψ) dμ,  ∞ ρ −iν−1 dρ t −i(λ−i(N+1)+in/2)−1

1 ζ1 (ϕ)GN (ϕ, ψ ; λ, ν, θ )ζ2 (ψ) = √ 2π 0 0 ˆ × G(tϕ − ψ, ρθ )(tϕ − ψ)γ γ (ϕ, ψ, t) dt, |γ |=N+1

(N

γ (ϕ, ψ, t) =

+ 1)(−1)N+1 γ!

ζ1 (ϕ)ζ2 (ψ)

(1 − s)N (∂ γ σ )(ϕ + s(t −1 ψ − ϕ)) ds.

0

(The inside integral with respect to t is understood in the sense of analytic extension with respect to .λ; the extension is implemented by a formula of the form (2.3.24).) Proof By Taylor’s formula, ζ1 (ϕ)σ (ψ)ζ2 (ψ) =

.

N (−1)|γ | ζ1 (ϕ)ζ2 (ψ)(∂ γ σ )(ϕ)(ϕ − t −1 ψ)γ γ!

|γ |=0



+

(ϕ − t −1 ψ)γ γ (ϕ, ψ, t).

|γ |=N+1

According to Lemma 2.3.6, (ζ1 A(θ )ζ2 σ u)(λ + in/2, ϕ) =

.

 ×

 dψ S n−1

N (−1)|γ | (ζ1 ∂ϕγ σ )(ϕ) γ!

|γ |=0

0∞ ζ2 (ψ)v(μ ˜ + in/2, ψ)

˜ γ (ϕ, ψ; λ, λ − μ, θ ) dμ + (ζ1 RN ζ2 v)(λ ×G ˜ + in/2, ϕ),

(2.3.26)

108

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

and .



∞

˜ γ (ϕ, ψ; λ, ν, θ ) = √1 G 2π



∞

ρ −iν−1 dρ

(2.3.27)

0

ˆ − ψ, ρθ )(tϕ − ψ)γ dt. t −i(λ−i|γ |+in/2) G(tϕ

0

Using (2.3.19) for the terms on the right-hand side of (2.3.26) and taking into account (2.3.27), we conclude the proof.  Lemma 2.3.8 Let T be the operator on .L2 (R, L2 (S n−1 )) defined by  (T v)(λ) =

+∞

.

−∞

g(λ, μ)v(μ) dμ.

(2.3.28)

Assume that the kernel g is continuous on .R × R with respect to the norm of operators on L2 (S n−1 ), its values belong to .KL2 (S n−1 ), and

.

 .

R×R

g(λ, μ)2 dλdμ < ∞.

Then, T is a compact operator. The proof is left to the reader. Proposition 2.3.9 Suppose that . ∈ C ∞ (S n−1 × (Rm \ 0)), .ξ → (ϕ, ξ ) is a homogeneous function of degree zero, and .(ϕ, ω, 0) ≡ f (ϕ), where .ω ∈ S n−1 and f is an arbitrary element of .C ∞ (S n−1 ). Moreover, let .A(θ ) be the operator defined by (2.3.11). Then, for .σ ∈ C ∞ (S n−1 ), the commutator .[A(θ ), σ ] = A(θ )σ − σ A(θ ) is compact on n .L2 (R ). Proof Without loss of generality, it can be assumed that .(ϕ, ω, 0) ≡ 0 (if not, then (ϕ, ω, θ ) is replaced by the difference .(ϕ, ω, θ ) − f (ϕ)). The compactness of the commutator .[A(θ ) , σ ] on .L2 (Rn ) is equivalent to the compactness of the commutator −1 , σ ] on .L (R, L (S n−1 )). .[MA(θ )M 2 2 We first assume that . is independent of .ϕ. Denote by .{ηj } a partition of unity on n−1 such that the union .suppη ∪ suppη of any two intersecting supports lies in some .S j  k hemisphere. It is clear that .A(θ ) = j,k ηj A(θ )ηk . Let j and k be such that .suppηj ∩ suppηk = ∅. We use (2.3.25) setting .ζ1 = ηj and .ζ2 = ηk . The operator .

Tγ ≡ Eω→ϕ (λ − i|γ |)−1 μ→λ (ω, θ )Eψ→ω (μ)

.

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

109

is compact on .L2 (R, L2 (S n−1 )) for .|γ | ≥ 1. Indeed, we write .Tγ in the form (2.3.28), γ −1 where .g(λ, μ) = E(λ − i|γ |)−1 h(λ − μ)E(μ) and .h(ν) = Ft→ν Dω (ω, e−t θ ). For each ∞ n−1 ) and .h(ν) decreases rapidly as .ν → ∞. The operator .ν ∈ R, we have .h(ν) ∈ C (S −1 is compact on .L (S n−1 ) and .E(λ − i|γ |)−1  ≤ .E(μ) is unitary, while .E(λ − i|γ |) 2 −|γ | c(1 + |λ|) , . λ, .μ ∈ R (see Sect. 1.2.1). The compactness of .Tγ , and hence that of .ηj Tγ ηk , now follows from Lemma 2.3.8. The compactness of the operator .u ˜ → ηj RN ηk u˜ on .L2 (R, L2 (S n−1 )) is ensured by the smoothness of its kernel for sufficiently large N(see Lemma 2.3.7) and the same Lemma 2.3.8. Using (2.3.25), we obtain that the commutator −1 η , σ ] is compact on the indicated space. The arguments only simplify in .[ηj MA(θ )M k the case .suppηj ∩ suppηk = ∅.  If . depends on .ϕ, then we expand . in a series .(ϕ, ξ ) = j,k Yj(n) k (ϕ)aj k (ξ ), where (n)

Yj k are spherical harmonics, and the coefficients .aj k satisfy the condition .aj k (ω, 0) ≡ 0.  (n) According to this, .A(θ ) can be represented as a series . Yj k (ϕ)Aj k (θ ), where .Aj k (θ ) is the operator (2.3.11) with . = aj k . It remains to use the first part of the proof in connection with the operators .Aj k (θ ). 

.

¯ n denotes the compact obtained by adjoining the .(n − 1)-dimensional Recall that .R n−1 sphere .S∞ to the space .Rn at infinity. The next assertion follows from [2], Theorem C. ¯ n ). Then, the Proposition 2.3.10 Let .A(θ ) be an operator of the form (2.3.7) and .a ∈ C(R n commutator .[A(θ ), a] is compact on .L2 (R ). Now, we are ready to describe localization in the algebra .L(θ ) directly. The assumptions of Proposition 1.3.26 are fulfilled for .L(θ ) and the localizing algebra .C = C(R¯ n ). Indeed, if the operator .A(θ ) is subject to the requirements of Proposition 2.3.9 and .σ ∈ C ∞ (S n−1 ), then .[A(θ ), σ ] ∈ KL2 (Rn ). Since the algebra .L(θ ) is irreducible, it contains the ideal n .KL2 (R ). Therefore, the first condition of Proposition 1.3.26 is fulfilled. The second condition is obvious, and the third one is ensured by Proposition 2.3.10. Let .L(θ )z denote the local algebra of .L(θ ) at a point .z ∈ R¯ n . −1 (ϕ, η, θ )F For the operators .A(θ ) = Fη→x y→η in (2.3.7), we introduce the maps: (i) .p(z) : A(θ ) → (ϕ, ·, 0) ∈ C(S n−1 ), z ∈ Rn \ 0, and ϕ = z/|z|. (ii) .p(0) : A(θ ) → A, where .A is the function R  λ → A(λ) = Eω→ϕ (λ)−1 (ϕ, ω, 0)Eψ→ω (λ) ∈ BL2 (S n−1 ).

.

(iii) .p(ϕ, θ ) : A(θ ) → (ϕ, ·, θ ) ∈ C(R¯ n ). Proposition 2.3.11 The maps (i)–(iii) extend to isomorphisms n−1 ) for .z ∈ Rn \ 0; .p(z) : L(θ )z  C(S

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

110

.p(0) : L(θ )0  S, where as in Sect. 2.2.1, .S is the algebra generated by the operatorfunctions .A; n−1 and the vector .ϕ is directed to the point z. ¯ n ), where .z ∈ S∞ .p(ϕ, θ ) : L(θ )z  C(R

¯ n ). We denote by .Jz the ideal in Proof Let .L(θ ) be the algebra spanned by .L(θ ) and .C(R n ¯ ) that vanish at .z ∈ R ¯ n ). We have .KL2 (Rn ) ⊂ .L(θ ) generated by the functions .c ∈ C(R Jz for all .z ∈ R¯ n . The assertions about .p(z) and .p(0) can be verified as in the proof of Proposition 2.1.5. Let us consider the map (iii). We set −1 Aj k (θ ) = aj k (ϕ)Fη→x j k (η, θ )Fy→η ,

.

¯ n ), where .ϕ = x/|x|, .aj k ∈ C ∞ (S n−1 ), and .j k ∈ C ∞ (S m−1 ), and choose .χj k ∈ C(R n−1 and let .ϕ 0 ∈ S n−1 where the indices j and k run over finite sets. Moreover, let .z0 ∈ S∞ 0 n be the vector directed to .z . On .L2 (R ), we introduce the unitary operator .Ut u(x) = −1 (η, θ )F u(x + tϕ 0 ) for .t ∈ R. Since the operators .A(θ ) = Fη→x y→η and .Ut commute, the equality ⎞

⎛



Ut ⎝

.

j

⎛



χj k aj k Aj k (θ )⎠ Ut−1 u = ⎝

k

j

⎞ Ut (χj k aj k )Aj k (θ )⎠ u

(2.3.29)

k

holds. It is clear that Ut (χj k (x)aj k (x)) = Ut (χj k (x))Ut (aj k (x)) = χj k (x+tϕ 0 )aj k (ϕ+tϕ 0 )−→χj k (z0 )aj k (ϕ 0 )

.

as .t → +∞. Therefore, there exists a strong limit  .

j

Ut (χj k aj k )Aj k (θ ) −→

k

 j

χj k (z0 )aj k (ϕ 0 ))Aj k (θ ).

k

   χ a A (θ ) + J Ut−1 strongly This and (2.3.29) imply that the operator .Ut j k j k j k j k tends to the same limit as .t → +∞, J being any element of the ideal .Jz0 . Taking into account the unitarity of .Ut and the property of strong limit “the norm of strong limit does not exceed the low limit of the norms,” we obtain the inequality 



.

j

≤ inf{ J

 j

k

χj k (z0 )aj k (ϕ 0 ))Aj k (θ ); BL2 (Rn ) ≤

k

χj k aj k Aj k (θ ) + J ; BL2 (Rn ); J ∈ Jz0 }.

(2.3.30)

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

111

Obviously, the operators .aj k (ϕ 0 ))Aj k (θ ) and .aj k (ϕ))Aj k (θ ) are in the same class in the quotient algebra .L(θ )/Jz0 . Therefore, the inequality converse for (2.3.30) holds. Thus, .

inf{ J

 j

χj k aj k Aj k (θ ) + J ; BL2 (Rn ); J ∈ Jz0 } =

(2.3.31)

k

=

 j

=

χj k (z0 )aj k (ϕ 0 ))Aj k (θ ); BL2 (Rn ) =

k

 j

χj k (z0 )aj k (ϕ 0 ))j k (θ ); C(Rn ).

k



2.3.4

The Spectrum of Algebra L(θ)

10 . Representations. The following theorem contains a list of all equivalence classes of irreducible representations of the algebra L(θ ) generated on L2 (Rn ) by operators −1 (ϕ, η, θ )F of the form A(θ ) = Fη→x y→η , where (ϕ, η, θ ) = f (ϕ)σ (η, θ ), ϕ = x/|x|, and f ∈ C ∞ (S n−1 ); the function (Rm \ 0)  ξ → σ (ξ ) is homogeneous of zero degree, σ ∈ C ∞ (S m−1 ), and θ ∈ S m−n−1 . Introduce the maps: (1) π(ϕ, η, θ ) : A(θ ) → (ϕ, η, θ ) for ϕ ∈ S n−1 and η ∈ Rn . (2) π(ϕ, ω) : A(θ) → (ϕ, ω, 0) for ϕ, ω ∈ S n−1 . (3) π(λ) : A(θ ) → A(λ), where A(λ) = Eω→ϕ (λ)−1 (ϕ, ω, 0)Eψ→ω (λ) ∈ BL2 (S n−1 ) and λ ∈ R. Theorem 2.3.12 The algebra L(θ ) is irreducible and contains the ideal KL2 (Rn ). The maps (1)–(3) extend to representations of the quotient algebra L(θ )/KL2 (Rn ) that are irreducible and nonequivalent. Any irreducible representation of L(θ ) is equivalent either to one of those listed or to the identity representations ι(θ ). Proof The irreducibility of the algebra L(θ ) is established by Proposition 2.3.5. Together with Proposition 2.3.9, this provides the inclusion KL2 (Rn ) ⊂ L(θ ). It remains to apply the localization principle (Proposition 1.3.26) and to recall the description of the local algebras (Proposition 2.3.11) and the list of the irreducible representations of the algebra S given in Theorem 2.2.11.  We will also write representations (1) and (2) in a somewhat different form. Let Rn+1 (θ ) be the subspace in Rm spanned by the plane {ξ ∈ Rm : ξ =

112

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

(η1 , . . . , ηn , 0, . . . , 0)} and the vector θ = (0, . . . , 0, θ1 , . . . , θm−n ), |θ | = 1. Denote n (θ ) the open hemisphere with pole θ in Rn+1 (θ ). Since Rm \ 0 → (ϕ, ξ ) by S+ is a homogeneous function of zero degree, we have (ϕ, η, θ ) = (ϕ, ξ ), where ξ = (η/(|η|2 + 1)1/2 , θ/(|η|2 + 1)1/2 ). Therefore, the representations in (1) can be n (θ ) and those in (2) as written as π(ϕ, ξ ) : A(θ ) → (ϕ, ξ ) for (ϕ, ξ ) ∈ S n−1 × S+ n n−1 π(ϕ, ξ ) for (ϕ, ξ ) ∈ S × ∂S+ (θ ). 20 . The topology on the spectrum. Denote by #(θ ) the disjoint union of the sets S n−1 × n (θ ), R, and the point ι. We introduce a topology on #(θ ). As before, neighborhoods S+ making up a fundamental system will be called typical. The point ι is an open set in the space #(θ ). The closure of ι coincides with #(θ ). A typical neighborhood in #(θ ) n (θ ) is taken to be the union of an ordinary neighborhood of a point (ϕ, ξ ) ∈ S n−1 × S+ n n−1 of it in S × S+ (θ ) with the point ι. n (θ ) and let ϕξ = 0 (here ξ is regarded as a vector in R n , since Let (ϕ, ξ ) ∈ S n−1 × ∂S+ its last m − n coordinates are equal to zero). A neighborhood of (ϕ, ξ ) is the union of an n (θ ) with the point ι and the line {λ : λ ∈ R}. In ordinary neighborhood of it in S n−1 × S+ the case ϕξ ≷ 0, the line is replaced by a set {λ ∈ R : λ ≷ N}, where N is an arbitrary real number. Finally, a neighborhood of a point λ ∈ R in #(θ ) is the union of an interval on R containing λ with the point ι. n (θ ), Theorem 2.3.13 The correspondence ι(θ ) → ι, π(ϕ, ξ ) → (ϕ, ξ ) ∈ S n−1 × S+  ) onto the space #(θ ). The π(λ) → λ ∈ R determines a bijection from the spectrum L(θ  ) coincides with the topology carried over by means of this bijection from #(θ ) to L(θ Jacobson topology.

Proof The correspondence indicated in the theorem is denoted by h. The fact that h is a bijection follows from Theorem 2.3.12 and the remark after its proof. It remains to see that  ) → #(θ ) and h−1 : #(θ ) → L(θ  ) are continuous (we mean that the the maps h : L(θ  ) is endowed with the Jacobson topology). space L(θ Denote by J (θ ) the intersection of the kernels of all representations (2) and (3) in n (θ ) and Theorem 2.3.12. Thus, the representations π(ϕ, ξ ) for (ϕ, ξ ) ∈ S n−1 × ∂S+ n π(λ) for λ ∈ R vanish on J (θ ). The ideal J (θ ) is irreducible, KL2 (R ) ⊂ J (θ ), and by Theorem 2.3.12, the spectrum Jˆ(θ) consists of the equivalence classes of the n (θ ); such representations π(ϕ, ξ ) representations ι(θ ) and π(ϕ, ξ ) for (ϕ, ξ ) ∈ S n−1 × S+  It constitute the spectrum (J (θ )/KL2 (Rn ))∧ . Recall that B = sup{π B, π ∈ B}. n n n−1 follows that the algebras J (θ )/KL2 (R ) and C0 (S × S+ (θ )) are isomorphic, where n (θ )) denotes the algebra of the continuous functions on S n−1 × S n (θ ) C0 (S n−1 × S+ + n (θ ). The spectrum (L(θ )/J (θ ))∧ consists of the equivalence that vanish on S n−1 × ∂S+ classes of the representations π(λ) for λ ∈ R and the representations π(ϕ, ξ ) for n (θ ). This and Theorem 2.2.11 imply that the algebras L(θ )/J (θ ) (ϕ, ξ ) ∈ S n−1 × ∂S+ and S are isomorphic.

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

113

Let I be a closed two-sided ideal of the algebra B. By virtue of Proposition 1.3.7, we  = I ∪ (B/I )∧ , and the set I is open in B,  while (B/I )∧ is closed. Therefore, have B  ) = K  ∪ (J (θ )/K)∧ ∪ (L(θ )/J (θ ))∧ , where K := KL2 (Rn ). The sets K  and K ∪ L(θ ∧   (J (θ )/K) = J (θ ) are open in L(θ ) (being the spectra of ideals).  = ι(θ ). Thus, the points ι(θ ) and ι = h(ι(θ )) are open in L(θ  ) and #(θ ), Recall that K −1 respectively. The maps h and h are thereby continuous at these points. The spectrum n (θ ) ⊂ #(θ ), and a homeomorphism (J (θ )/K)∧ is homeomorphic to the set S n−1 × S+  ) with is implemented by the map h. Since any neighborhood of a point π(ϕ, ξ ) ∈ L(θ n n−1 (ϕ, ξ ) ∈ S × S+ (θ ) contains ι(θ ), while the neighborhoods of (ϕ, ξ ) contain ι, we find n (θ )) ∪ ι is a homeomorphism. that h : J(θ ) → (S n−1 × S+ n (θ ). Let l = R if ϕ ξ = 0, and We now consider a point (ϕ0 , ξ0 ) ∈ S n−1 × ∂S+ 0 0 let l = {λ ∈ R : λ ≷ N}, where N is a number if ϕ0 ξ0 ≷ 0. Let  be some open n (θ ) containing the point (ϕ , ξ ). Then,  ∪ l is a neighborhood of subset of S n−1 × S+ 0 0 (ϕ0 , ξ0 ) in #(θ ). From Theorem 2.2.13, it follows that the algebra S contains an element   A = j k Aj k (j and k run through finite sets of values), where n Aj k (λ) = E(λ)−1 j k (ϕ, ω)E(λ), j k ∈ C ∞ (S n−1 × ∂S+ (θ )),

.

having the properties: (1) {λ : A(λ) 1} ⊂ l. >  n−1 × ∂S n (θ )).   (2) {(ϕ, ω) : + j k j k (ϕ, ω) > 1} ⊂  ∩ (S (A union of sets of the form (1) and (2) is a typical neighborhood of (ϕ0 , ω0 ) in the spec Let j k be functions in C ∞ (S n−1 × S m−1 ) such that j k (ϕ, ω, 0) = j k (ϕ, ω) trum S.) n (θ ) and (S n−1 × S n (θ )) ∩ supp  on S n−1 × ∂S+ j k ⊂ . We consider the operator +  −1  (ϕ, η, θ )F A(θ ) = Aj k (θ ) in the algebra L(θ ), where Aj k (θ ) = Fη→x jk y→η and the functions ξ → j k (ϕ, ξ ) are extended as homogeneous of zero degree to Rm \ 0. It is clear that  ) : π(A(θ )) > 1}) ⊂  ∪ l ∪ ι. h({π ∈ L(θ

.

Thus, the map h is continuous at π(ϕ0 , ξ0 ). We verify that the inverse map h−1 is continuous at (ϕ0 , ξ0 ). Let −1 Aj k (θ ) = Fη→x j k (ϕ, η, θ )Fy→η

.

be arbitrary elements in L(θ ) and A(θ ) =



Aj k (θ ). We assume that

   π(ϕ0 , ξ0 )A(θ ) =  j k (ϕ0 , ω0 , 0) > 1

.

114

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

and denote by A the element of S  L(θ )/J (θ ) corresponding to the residue class  ) : π(A(θ )) > 1} is a neighborhood [A(θ )] ∈ L(θ )/J (θ ). The set V := {π ∈ L(θ  ). Clearly, this set contains the identity representation ι(θ ) and all the of π(ϕ0 , ξ0 ) in L(θ n (θ ) sufficiently close to (ϕ , ξ ). Again, representations π(ϕ, ξ ) with (ϕ, ξ ) ∈ S n−1 × S+ 0 0 using Theorem 2.2.13, we obtain that if ϕ0 ω0 = 0, then A(λ) > 1 for all λ ∈ R, while if ϕ0 ω0 ≷ 0, then A(λ) > 1 for λ ≷ N , where N is some number. It follows from the foregoing that the set h(V ) contains a neighborhood of (ϕ0 , ξ0 ) in #(θ ). This shows that n (θ ).  ) is continuous at (ϕ0 , ω0 ) ∈ S n−1 × ∂S+ the map h−1 : (θ ) → L(θ −1 It remains to be proven that h and h are continuous at the respective points π(λ) and  λ. By Theorem 2.2.13, there exists an element A = Aj k ∈ S subject to the following conditions: (1) Aj k (λ) = E(λ)−1 j k (ϕ, ω)E(λ). (2) {λ ∈ R : A(λ) > 1} ⊂ (λ0 − ε , λ + ε).  n (θ ). (3) max j k (ϕ, ω) < 1, (ϕ, ω) ∈ S n−1 × ∂S+ Suppose that j k ∈ C∞ (S n−1 × S m−1 ), j k (ϕ, ω, 0) = j k (ϕ, ω) for (ϕ, ω) ∈ n (θ ), and max    < 1 on S n−1 × S m−1 . As earlier, we introduce × ∂S+ jk   ) : the operators Aj k (θ ) ∈ L(θ ) and A(θ ) = Aj k (θ ). It is clear that h({π ∈ L(θ π(A(θ )) > 1} ⊂ (λ0 − ε , λ0 + ε). This gives the continuity of h at π(λ0 ). Conversely, if A(θ ) ∈ L(θ ) and π(λ0 )A(θ ) = A(λ0 ) > 1, then A(λ) > 1 at all points λ close to λ0 . Thus, h−1 is continuous at λ0 . 

S n−1

30 . Solvability and the length of the algebra L(θ ). Theorem 2.3.14 (1) KL2 (Rn ) ⊂ comL(θ ). (2) The composition series 0 ⊂ KL2 (Rn ) ⊂ com L(θ ) ⊂ L(θ ) is solving and com L(θ )/KL2 (Rn )  C0 (R) ⊗ KL2 (S n−1 ),

.

n (θ )). L(θ )/com L(θ )  C(S n−1 × S+

(3) The length of L(θ ) is equal to 2. (4) The above composition series is the maximal radical series. Proof (1) Since the algebra L(θ ) is irreducible, the same is true for the ideal com L(θ ). If A1 (θ ) and A2 (θ ) are elements of the ideal J (θ ), then [A1 (θ ), A2 (θ )] ∈ KL2 (Rn ). Therefore, KL2 (Rn ) ⊂ com L(θ ).

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

115

(2) We show that the algebras com L(θ )/KL2 (Rn ) and C0 (R) ⊗ KL2 (S n−1 ) are isomorphic. If A ∈ com L(θ ) and p : L(θ ) → L(θ )/J (θ )  S is the projection, then p(A) ∈ com S. Moreover, com S  C0 (R) ⊗ KL2 (S n−1 ) (see Remark 2.2.16). The map q : comL(θ )/KL2 (Rn ) → C0 (R) ⊗ KL2 (S n−1 ) is defined by the equality q([A]) = p(A), where [A] is the residue class of the element A in com L(θ )/KL2 (Rn ) (this definition is unambiguous in view of the inclusion KL2 (Rn ) ⊂ J (θ )). Let us verify that q is an isomorphism. Since the spectrum of the algebra C0 (R)⊗KL2 (S n−1 ) coincides with R, we have q([A]) = sup{A(λ) : BL2 (S n−1 ); λ ∈ R},

.

(2.3.32)

where A = q([A]). The spectrum of com L(θ )/KL2 (Rn ) consists of the representations of L(θ )/KL2 (Rn ) that do not annihilate com L(θ )/KL2 (Rn ). Taking into account Theorem 2.3.12, we see that the spectrum com L(θ )/KL2 (Rn )∧ consists of representations of the form π(λ). Hence,



[A] = sup π(λ)A = sup A(λ).

.

λ∈R

λ∈R

Together with (2.3.32), this gives us that q is a monomorphism. It is obviously an epimorphism. Thus, com L(θ )/KL2 (Rn )  C0 (R) ⊗ KL2 (S n−1 ).

.

The spectrum of L(θ)/com L(θ ) consists of all the representations of L(θ ) vanishing on com L(θ ), i.e., of all the one-dimensional representations. By Theorem 2.3.12, n (θ ). (L(θ )/comL(θ ))∧ = S n−1 × S+

.

n (θ )). In other words, L(θ )/com L(θ )  C(S n−1 × S+ (3) It follows from (2) that the length of L(θ ) is no greater than 2. Since the algebra L(θ ) has irreducible representations of different dimensions (see Theorem 2.3.12), L(θ ) cannot be isomorphic to an algebra of the form C0 (X) ⊗ K(H ). Hence, the length of L(θ ) ≥ 1. Assume that the length is equal to 1. Then, there exists an ideal I = 0 such that I  C0 (X1 ) ⊗ K(H1 ) and L(θ )/I  C0 (X2 ) ⊗ K(H2 ), where X1  ). Every and X2 are locally compact spaces. The spectrum I is an open subset of L(θ  ) contains the point ι(θ ) corresponding to the identity open subset of the spectrum L(θ representation (Theorem 2.3.13). Since the point ι(θ ) is not closed, the spectrum I would not be a Hausdorff space in the case ι(θ ) = I, and that is impossible because I is homeomorphic to X1 . Hence, I = ι(θ ) and I = KL2 (Rn ). By Theorem 2.3.12, the quotient algebra L(θ )/KL2 (Rn ) has representations of different dimensions and thus

2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

116

cannot be isomorphic to an algebra C0 (X2 ) ⊗ KH2 . Thus, the length of L(θ ) is equal to 2. (4) This assertion is immediately verified by means of Theorem 2.3.12. 

2.3.5

The Spectrum of the Algebra of Pseudodifferential Operators with Symbols Discontinuous Along a Submanifold

Here, we prove the theorems stated in Sect. 2.3.1 for the algebra .A (we keep the same notation). Let us first describe the spectrum of the local algebra .Az0 at a point .z0 ∈ N . This algebra is generated on .L2 (Rm ) by the operators (2.3.6) whose symbols may have discontinuities on an .(m−n)-dimensional plane. According to Proposition 2.3.4, .Az0  L, where .L is the algebra spanned by the functions −1 S m−n−1  θ → A(θ ) = Fη→x (ϕ, η, θ )Fy→η ∈ BL2 (Rn )

.

(2.3.33)

with .ϕ = x (1) /|x (1) | and . ∈ C ∞ (S n−1 ×(Rm \0)), while .ξ → (ϕ, ξ ) is a homogeneous function of degree zero. 10 . The spectrum of the algebra .L. Let .A(·) be an algebra of the form (2.3.33). Introduce the notations:

.

(1) .π(θ ) : A → A(θ ) ∈ BL2 (Rn ) for .θ ∈ S m−n−1 . (2) .π(λ) : A(·) → A(λ) ∈ BL2 (S n−1 ), where .A(λ) = Eω→ϕ (λ)−1 (ϕ, ω, 0)Eψ→ω (λ) and .λ ∈ R. (3) .π(ϕ, ξ ) : A(·) → (ϕ, ξ ) for .(ϕ, ξ ) ∈ S n−1 × S m−1 . Theorem 2.3.15 The mappings (1)–(3) extend to irreducible pairwise nonequivalent representations of .L. Every irreducible representation of .L is equivalent to one of those listed. Proof It suffices to take into account Proposition 2.3.4, Theorem 2.3.12, and the remark after its proof.  Let us discuss a spectral topology. Denote by .# the disjoint union of the sets .S m−n−1 , n−1 × S m−1 , and .R. We introduce a topology on .#. The typical neighborhoods in .# of a .S point .θ ∈ S m−n−1 coincide with its neighborhoods in .S m−n−1 . If .ξ = (ξ (1) , ξ (2) ), .ξ (1) = (ξ1 , . . . , ξn ), .ξ (2) = (ξn+1 , . . . , ξm ), and .ξ (2) = 0, then a neighborhood in .# of the point n−1 × S m−1 is defined as a union .(U (ϕ) × V(ξ )) ∪ W(θ ), where .θ = ξ (2) /|ξ (2) |, .(ϕ, ξ ) ∈ S

2.3 Algebras of Pseudodifferential Operators with Discontinuities in. . .

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while .U(ϕ), .V(ξ ), and .W(θ ) are neighborhoods of the points .ϕ, .ξ , and .θ in .S n−1 , .S m−1 , and .S m−n−1 , respectively, and .V(ξ ) ∩ {ξ : ξ (2) = 0} = ∅. For .ξ = (ξ (1) , 0), .ϕξ (1) = 0, a typical neighborhood of a point .(ϕ, ξ ) is of the form .S m−n−1 ∪ (U(ϕ) × V(ξ )) ∪ R, and for 1 .ϕξ ≷ 0, the line .R is replaced by the set .{λ ∈ R : λ ≷ N}, where N is an arbitrary real number. A neighborhood of a point .λ ∈ R is defined as a union .(λ − ε, λ + ε) ∪ S m−n−1 , .ε > 0. With its topology, .# is a .T0 -space. Theorem 2.3.16 The correspondence π(θ ) → θ ∈ S m−n−1 , π(λ) → λ ∈ R, π(ϕ, ξ ) ∈ S n−1 × S m−1

.

 onto the set .#. The topology carried over via this bijection is a bijection of the spectrum .L  coincides with the Jacobson topology. from .# to .L Proof Denote by J the closed two-sided ideal in .L spanned by the functions .θ → A(θ ) ∈ J (θ ), where .J (θ ) is the ideal in .L(θ ) introduced in the proof of Theorem 2.3.13. We show that any continuous function .S m−n−1  θ → A(θ ) ∈ BL2 (Rn ) for .A(θ ) ∈ J (θ ) belongs to J . Let .J0 be a closed two-sided ideal in .L(θ ) generated by operators of the form (2.3.33), where the function .η → (ϕ, η, θ ) vanishes for .|η| > R and all .ϕ ∈ S n−1 , while R is a sufficient large number. From Theorem 2.3.12, it follows that .J0 (θ ) is a rich subalgebra in .J (θ ) and, therefore, .J0 (θ ) = J (θ ). This property of J means that the algebra J is generated by the continuous field .{J (θ ), J } of the algebras .J (θ ). Therefore, a topology on the spectrum .J = ∪θ J(θ ) can be described with the help of Proposition 1.3.17. We now verify that the algebras .L/J and .S are isomorphic. Let a map .r : L → S be the composition of the morphisms L → L(θ ) → L(θ )/J (θ ) ∼ = S,

.

where .θ is a fixed point, the first arrow denotes the calculation of a value at .θ, .L  A → A(θ ) ∈ L(θ ), the second arrow is a projection, and .∼ = stands for the isomorphism established in the proof of Theorem 2.3.13. It is clear that r vanishes on J ; therefore, .r : L/J → S is a well-defined epimorphism. Moreover, from Theorems 2.3.12 and 2.3.15, it follows that .[A]; L/J  = r(A); S for all .A ∈ L, i.e., .r : L/J → S is a monomorphism. Thus, .L/J  S.  Therefore, the typical neighborhoods of the points The set .J is open in the spectrum .L. (1) (2) (2) .(ϕ, ξ ) for .ξ = (ξ , ξ ) with .ξ = 0 in the space .J can be taken as neighborhoods of  Neighborhoods in .L  of the points .(ϕ, ξ ) for .ξ = (ξ (1) , 0) and those these points also in .L.  can be obtained by evident modifications in the proof of the points .λ (i.e., the points in .S)  of Theorem 2.3.13.

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Theorem 2.3.17 Let .I ⊂ L be the ideal in .L equal to the intersections of the kernels of all representations of the form .π(λ) and .π(ϕ, ξ ) (defined in Theorem 2.3.15). Then, the composition series .{0} ⊂ I ⊂ comL ⊂ L is solving, and I∼ = C(S m−n−1 ) ⊗ KL2 (Rn ),

.

comL/I ∼ = C0 (R) ⊗ KL2 (S n−1 ),

.

L/comL ∼ = C(S n−1 × S m−1 ).

.

The length of .L is equal to 2. The composition series is the maximal radical series. Proof We saw earlier that .L is isomorphic to a certain algebra of operator-valued functions S m−n−1  θ → A(θ ) ∈ L(θ ) that contains all continuous functions .S m−n−1  θ → A(θ ) ∈ J (θ ) (see the proof of Theorem 2.3.16). In particular, .C(S m−n−1 ) ⊗ KL2 (Rn ) is a subalgebra of .L. It follows from the definition of the representations .π(ϕ, ξ ) and .π(λ) that each of them is a representation of .L(θ ) for some .θ. The intersections of the kernels of such representations of .L(θ ) coincide with the ideal .KL2 (Rn ) ⊂ L(θ ). Therefore, m−n−1 ) ⊗ KL (Rn ) is the intersection of the kernels of .π(ϕ, ξ ) and .π(λ), regarded as .C(S 2 representations of .L, i.e., .I ∼ = C(S m−n−1 ) ⊗ KL2 (Rn ). ∼ As in Theorem 2.3.16, let r be the composition of the morphisms .L → L/J → S. It is clear that .r(com L) ⊂ comS. Define a morphism .q : com L/I → com S by the equality .q([A]) = r(A), where .[A] is the residue class of an element A in .com L/I ; this definition is unambiguous by virtue of the inclusion .I ⊂ J . Recall that .com S ∼ = C0 (R) ⊗ KL2 (S n−1 ). As in the proof of Theorem 2.3.14, we see that q is an isomorphism. The relation .L/com L ∼ = C(S n−1 × S m−1 ) follows from Theorem 2.3.15. Thus, the composition series indicated in the theorem is solving. We show that the length of .L equals 2. Assume that there exists a solving composition series of the form .{0} ⊂ Q ⊂ L, i.e., it is shorter than that in the formulation of the theorem. It follows from Theorem 2.3.15 that the spectrum of the ideal Q must contain either all infinite-dimensional representations or all one-dimensional representations (the spectrum .Qˆ must consist of representations of a single dimension). The first case is ˆ is then not a Hausdorff space, and the composition series is not solving. impossible since .Q The second case cannot be because the set of all one-dimensional representations is not open (Theorem 2.3.16), while the spectrum of an ideal is always an open part of the spectrum of an algebra.  .

Proof of Theorem 2.3.1 It suffices to compare the localization principle (Proposition 1.3.26), Theorem 2.3.15, and Proposition 2.1.5.  Proof of Theorem 2.3.2 We identify the sets .(A/KL2 (M))∧ and . with the help of bijection (2.3.5). It must be verified that the same subsets of . play the role of the

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typical neighborhoods of points both in the .-topology (defined before the statement of Theorem 2.3.2) and in the Jacobson topology. This is clear for the points .(x, ω) ∈ p∗ (S ∗ (M)) with .x ∈ M \ N . Let .z0 ∈ N and .θ 0 ∈ S ∗ (N )z0 . We choose an element .A ∈ comA that is annihilated by all representations of the form .π(z, λ) (and, of course, by all one-dimensional representations of .A). Moreover, we assume that the set .{θ ∈ S ∗ (N )z0 : π(z0 , θ )A > 1} coincides with a given neighborhood of .θ 0 in .S ∗ (N )z0 . (The needed operator A can be related to a neighborhood of .z0 in .M and constructed in local coordinates.) Multiplication by functions in .C ∞ (M) does not remove from the algebra .A. Therefore, it is possible to choose such a function .χ ∈ C ∞ (M) that the set .{π ∈ (A/KL2 (M))∧ : π(χ A) > 1} is a sufficiently small neighborhood of .(z0 , θ 0 ) in .. On the other hand, the same set is a neighborhood of .(z0 , θ 0 ) in the Jacobson topology as well. Thus, the same sets are neighborhoods of .(z0 , θ 0 ) in both topologies. Let .(z0 , λ0 ) ∈ N × R and let .A0 ∈ Az0 be such that .{π ∈ Aˆ z0 : π(A0 ) > 1} is a union of an interval in .R and the sphere .S ∗ (N )z0 (the existence of .A0 follows from Theorem 2.3.16). Choosing .χ ∈ C ∞ (M) and .A ∈ comA so that .pz0 (A) = A0 , where .pz A → Az is the canonical localizing map, one can argue as in the preceding case. Let us now consider a neighborhood of a point .(z, ϕ0 , ω0 ) ∈ p ∗ (S ∗ (M)), where 1 1 .(z, ϕ0 ) ∈ ν(N )z ⊂ MN . Assume, for example, that .ω0 = (ω , 0) and .ϕ0 ω = 0. Let A be 0 0 an operator in .A whose symbol . satisfies .|(z, ϕ, ω)| < 1 outside a small neighborhood V of .(z, ϕ0 , ω0 ) on .p∗ (S ∗ (M)) (see notation before (2.3.5)) and .|(z, ϕ, ω)| > 1 in another neighborhood U of this point, .U ⊂ V . Furthermore, let .U be the projection of U in .N . Theorem 2.3.16 gives us the inequalities .π(z, λ)A > 1 for .(z, λ) ∈ U × R and .π(z, θ )A > 1 for .θ ∈ S ∗ (N )z , .z ∈ U. From this, it is easy to deduce that every neighborhood of .(z, ϕ0 , ω0 ) in the .- topology (the Jacobson topology) contains a neighborhood of this point in the Jacobson topology (the .-topology), which is what was needed. The cases .ϕ0 ω01 = 0 and .ω0 = (ω01 , ω02 ) with .ω02 = 0 are handled similarly.  Proof of Theorem 2.3.3 Let .pz : A → Az be the canonical localizing map; as usual, .Az is the local algebra at a point .z ∈ M, and .C(M) is a localizing algebra. We consider the algebra .D generated by functions of the form M  z → α(z) := pz (A) ∈ Az ,

.

Where .a ∈ A. The operations in .D are pointwise, and the norm is defined by α; D = sup{α(z); Az , z ∈ M}.

.

Taking into account Theorem 2.3.1, one can see that there is an isomorphism .D  A/KL2 (M).

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2 .C ∗ -algebras of Pseudodifferential Operators on Smooth Manifolds with. . .

Let I be the ideal in the statement of Theorem 2.3.3 and let .J := I /KL2 (M). We first show that the composition series .0 ⊂ J ⊂ comD ⊂ D is solving. The elements of .D can be multiplied by functions in .C ∞ (M). This implies that the algebra .comD|N of restrictions to .N of elements in .comD is defined by the continuous field of algebras .{comDz , comD|N }, while .J |N is defined by the continuous field .{Jz , J |N }, where, for example, .Jz is the algebra of values at z of the functions in J (see [3], 10.4.2). By Theorem 2.3.17, for .z ∈ N , we have .Jz  C(S ∗ (N )z ) ⊗ KL2 (Rnz ) and .comDz /Jz  C0 (R) ⊗ KL2 (z ); here .Rnz = T (N )⊥ z is the subspace of the tangent space .T (M)z orthogonal to the submanifold .N , and .z is the unit sphere in .Rnz . This means that .J |N is defined by the continuous field on .S ∗ (N ) of the elementary algebras .KL2 (Rnz ), while .comD|N is defined by the continuous field on .N × R of the elementary algebras .KL2 (z ). The fields .J |N and .comD|N are locally trivial; the local trivializations of the fields come from the local trivializations of the normal bundle over the submanifold .N . According to Proposition 1.3.18, these fields are trivial. Therefore, .J |N  C(S ∗ (N )) ⊗ KL2 (Rn ) and n−1 ). The elements of .comD are identically .(comD|N )/(J |N )  C0 (N × R) ⊗ KL2 (S equal to zero outside .N . Hence, in the last two relations, we can replace .comD|N and .J |N by .comD and J , respectively. Finally, the quotient algebra .D/comD is commutative, and its spectrum coincides with .p∗ (S ∗ (M)) (see (2.3.5)), i.e., .D/comD  C(p∗ (S ∗ (M))). Thus, the composition series .0 ⊂ J ⊂ comD ⊂ D is solving. Let us now consider the composition series 0 ⊂ KL2 (M) ⊂ I ⊂ comA ⊂ A.

.

We have the isomorphisms .I /KL2 (M)  J , .comA/KL2 (M)  comD, and A/KL2 (M)  D. Therefore,

.

I /KL2 (M)  C(S ∗ (N )) ⊗ KL2 (Rn ),

.

comA/I  C0 (N × R) ⊗ KL2 (S n−1 ), A/comA  C(p∗ (S ∗ (M))), i.e., the composition series is solving. We show that the length of .A is equal to 3. It follows from the last paragraph that the length is at most 3. It must be shown that if a composition series .0 ⊂ I0 ⊂ I1 ⊂  · · · ⊂ IN = A is solving, then .N ≥ 3. According to Theorem 2.3.2, the spectrum .A is homeomorphic to the space obtained by adjoining to . a single point .ι corresponding to the identity representation; this point is an open set, and its closure coincides with the whole space. The spectra .Iˆj of the ideals .Ij form an increasing sequence of open subsets  Any open subset contains the point .ι; in particular, .ι ∈ Iˆ0 . In the case .ι = Iˆ0 , of .A. the spectrum .Iˆ0 would not be a Hausdorff space; the isomorphism .I0  C0 (X) ⊗ KH would be impossible. Hence, .Iˆ0 = ι, .I0 = KL2 (M). Assume that .N = 1. Then, .I1 /I0 = A/KL2 (M). However, the spectrum .(A/KL2 (M))∧ contains representations of different

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dimensions (Theorem 2.3.1), and therefore, the isomorphism .A/KL2 (M)  C0 (X)⊗KH is impossible. Consequently, .N > 1. Finally, assume that .N = 2. Each open subset of .Aˆ intersects .S ∗ (N ) (Theorem 2.3.2). In particular, .S ∗ (N ) ∩ Iˆ1 = ∅. In the case .Iˆ1 ⊂ S ∗ (N ), the spectrum .(I1 /I0 )∧ is not a Hausdorff space, and a relation of the form .I1 /I0  C0 (X)⊗ KH does not hold. However, if .Iˆ1 ⊂ S ∗ (N ), then the space .(A/I1 )∧ turns out not to be Hausdorff, and again the isomorphism .A/I1  C0 (X) ⊗ KH cannot be. Therefore, the smallest possible number N in a solving series is equal to 3. 

Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols on a Smooth Manifold

On a smooth compact m-dimensional manifold .M without boundary, we consider the .C ∗ algebra .A generated on .L2 (M) by the operators of two classes. One of the classes consists of zero order pseudodifferential operators with smooth symbols. The other class comprises the operators of multiplication by functions (“coefficients”) that may have discontinuities along a given collection of submanifolds (with boundary) of various dimensions; the intersections of the submanifolds under nonzero angles are admitted. The situation is formally described by a stratification of the manifold .M. All the equivalence classes of irreducible representations of .A are listed; the topology on the spectrum is described; solving composition series are presented.

3.1

Algebra A and Its Irreducible Representations

3.1.1

Stratification of Manifold M. Algebra A p

Let .T = {sα } be a finite partition of manifold .M into subsets (strata). In what follows, the discontinuities in coefficients will be supported by the strata of positive codimension. A p stratum .sα is a connected p-dimensional submanifold of .M (possibly, nonclosed). The partition consisting of a single element (the manifold .M itself) is not ruled out. The p p boundary .s¯α \ sα of a stratum is the union of strata of smaller dimension (or the empty p p p set). The collection of all strata whose boundary contains .sα is called the star .st(sα ) of .sα . Let us subject a stratification .T of .M to some additional requirements. Namely, p we suppose that for any point .x ∈ sα there exist a neighborhood U in .M and a diffeomorphism . of U onto a neighborhood of the origin in .Rm that locally rectifies p q p p .sα and all .s β ∈ st(sα ); this means that .(sα ∩ U ) is a neighborhood of the origin in q p m : y = ··· = y .R = {y = (y1 , . . . , ym ) ∈ R 1 m−p = 0} and .(sβ ∩ U ) coincides near

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Plamenevskii, O. Sarafanov, Solvable Algebras of Pseudodifferential Operators, Pseudo-Differential Operators 15, https://doi.org/10.1007/978-3-031-28398-7_3

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3

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . . p

(sα ∩ U ) with the product .Rp × K q−p , where .K q−p is a .(q − p)-dimensional cone in the subspace .Rm−p orthogonal to .Rp . It will be assumed further that .(x) = 0 and .  (x) = 1. Let .S m−p−1 be the unit sphere in this subspace .Rm−p . It is assumed that the projections q q m−p−1 , where .s q ∈ st(s p ), in turn, form a partition of .pr(s ) of the sets .(s ∩ U ) onto .S α β β β p m−p−1 .S satisfying the above requirements; denote the partition by .T (sα ) (the point x is not indicated in the notation for simplicity). Under the given conditions, the partition .T of .M is said to be admissible. p q We give to the partition .T (sα ) and the stratum .pr(sβ ), .q ≤ m − 1, the roles of .T p p q and .sα . This determines a partition of the sphere .S m−q−1 ; denote it by .T (sα , sβ ). In general, let .s0 , . . . , sk be a chain of strata such that .dim sp = dp , .dk ≤ m − 1, .si ∈ st(sj ) for .i > j (simplifying the notation, we write strata with a single subscript). Continuing the procedure, we obtain partitions .T (s0 ), .T (s0 , s1 ), . . . , T (s0 , . . . , sk ). If .T (s0 , . . . , sk ) consists only of a single element, i.e., the sphere .S m−dk −1 (for .dk < m − 1), or of two points, i.e., the sphere .S 0 (for .dk = m−1), then the chain .s0 , . . . , sk is said to be complete. .

Example 3.1.1 A partition .T of .S 2 is formed by a zero-dimensional stratum consisting of the point .P+ = (0, 0, 1)) and by the two-dimensional stratum .s = S 2 \ P+ . It is clear that 1 .T (P+ ) = S . A complete chain is formed by .P+ . Example 3.1.2 A partition .T of .S 2 is formed by zero-dimensional strata: the points .P± = (0, 0, ±1), a one-dimensional stratum .μ = {x = (x1 , x2 , x3 ) : |x| = 1, x1 > ¯ A locally 0, x2 = 0} (the meridian), and the two-dimensional stratum .s = S 2 \ μ. rectifying diffeomorphism in a neighborhood of the points .P± or a point .x ∈ μ acts as the orthogonal projection onto the tangent plane at the corresponding point. There are three complete chains at all: .P+ , .μ; .P− , .μ; and .μ. The chain .P+ , μ generates a partition .T (P+ ) consisting of two strata: a zero-dimensional one .p(μ) and a one-dimensional .S 1 \ p(μ), and a partition .T (P+ , μ) = S 0 . The partition .T (μ) = S 0 corresponds to the complete chain .μ. Example 3.1.3 A partition .T of the sphere .S m−1 consists of an arbitrary k-dimensional m−1 \γ . Then, .T (γ ) = S m−k−2 . .(k < m−1) smooth closed submanifold .γ and a stratum .S To determine the generators of .A, we first describe the coefficients. On a manifold .M endowed with an admissible partition .T , we introduce a class .M(T ) ≡ M(T , M) of functions. The operators aI with .a ∈ M(T ) form one type of the generators of .A. We define the classes .M(T , M) using induction on .dim M. For .dim M = 0, such a class is the set of all functions on .M (by .M is meant a finite collection of points with discrete topology). Assume that the classes have been defined for .k = dim M ≤ m − 1 and turn to the case .dim M = m. We suppose that .T is an admissible partition of .M and .|T | is the union of all strata (as subsets of the manifold .M) of dimension no greater than .m − 1. By definition, a smooth function f given on .M \ |T | belongs to the class .M(T , M) if a

3.1 Algebra A and Its Irreducible Representations

125

representation of the form (f ·  −1 )(y, z) = (y, z) + 1 (y, z)

.

(3.1.1)

holds in a neighborhood U of any point of every stratum .sj , .dj ≤ m − 1; here, .(y, z) ∈ (U ) ⊂ Rm , .y = (y1 , . . . , ym−dj ), .z = (z1 , . . . , zdj ), .(y, z) = (y/|y|, z) for .y = 0, and .(·, z) ∈ M(T (sj ), S m−dj −1 ) for all z; the terms in (3.1.1) depend continuously on z in the norm of .L∞ and .1 (y, z) = o(1) for .y → 0. The function .(·, z) is called the limit value of f at the point .z ∈ sj (instead of . −1 (0, z), we write simply z). The other type of the generators of the algebra .A is formed by pseudodifferential ¯ 0 (M) (with smooth symbols) introduced by Definition 2.1.4. operators of the class . Thus, we assume that an admissible partition .T is given on a smooth compact .M without boundary and let .A stand for the algebra generated on .L2 (M) by the operators aI ¯ 0 (M). with a in .M(T , M) and by .ψDOs of the class .

3.1.2

The Irreducible Representations of the Algebra A (Formulation of a Theorem)

We suppose that the manifold .M is endowed with the Riemannian metric. Let .s0 , . . . , sk be a chain of strata of the partition .T , i.e., .si ∈ st(sj ) for .i > j , .dim sj = dj , and .dk ≤ m − 1, where .m = dim M. For the generators of .A, we define a localization procedure along the chain of strata. Let us start with the coefficients .a ∈ M(T , M). Let .a(z0 ; ·) ∈ M(T (s0 ), S m−d0 −1 ) be the limit value of a at a point .z0 ∈ s0 . The .(d1 − d0 − 1)-dimensional stratum .pr(s1 ) of the partition .T (s0 ) corresponds to the stratum .s1 . Denote by .a(z0 , z1 ; ·) the limit value of .a(z0 ; ·) at a point .z1 ∈ pr(s1 ). Continuing the process, we arrive at the collection 0 0 k 0 j m−dj −1 ). We .a(z ; ·), . . . , a(z , . . . , z ; ·), where .a(z , . . . , z ; ·) ∈ M(T (s0 , . . . , sj ), S say that this collection is obtained by localizing a along the chain .s0 , . . . , sk . If the chain is complete, then .a(z0 , . . . , zk ; ·) ∈ C(S m−dk −1 ). We extend the localization procedure to generators of the other type, i.e., to the .ψDOs in ¯ 0 (M). Let .A ∈  ¯ 0 (M) and let .(z0 ; ·) be the principal symbol of A at a point .z0 ∈ s0 . . The function .ξ → (z0 ; ξ ) is homogeneous of zero degree and smooth on any fiber ∗ .T (M)z0 \ 0 of nonzero cotangent vectors. In the tangent space .T (M)z0 , we introduce orthogonal coordinates .(x1 , . . . , xm ), where the axes .xm−d0 +1 , . . . , xm are parallel to the stratum .s0 . Assume that the symbol .(z0 ; ·) is written in that coordinate system. We introduce operators “of the first generation” −1 A(z0 ; θ 0 ) = Fη→x (z0 ; η, θ 0 )Fy→η ∈ BL2 (Rm−d0 ),

.

(3.1.2)

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

where .η = (η1 , . . . , ηm−d0 ), .θ 0 = (θ1 , . . . , θd0 ) ∈ Rd0 , .|θ 0 | = 1, and F is the Fourier transform in .Rm−d0 ; A(z0 ; λ) = Eω→ϕ (λ)−1 (z0 ; ω, 0)Eψ→ω (λ) ∈ BL2 (S m−d0 −1 ),

.

(3.1.3)

where .ϕ, .ψ, .ω ∈ Rm−d0 , .|ϕ| = |ψ| = |ω| = 1, .λ ∈ R, and the operators .E(λ)±1 are defined by (1.2.2) and (1.2.8) for .n = m − d0 . The operators (3.1.2) are not defined for .d0 = 0. We now choose in .T (M)z0 new Cartesian coordinates .x˜1 , . . . , x˜m with the same origin. The axes .x˜m−d0 +1 , . . . , x˜m coincide with the old .xm−d0 +1 , . . . , xm . The axis .x ˜m−d0 is directed toward the point .z1 ∈ pr(s1 ), .dim pr(s1 ) = d1 − d0 − 1; the axes .x ˜m−d1 , . . . , x˜m−d0 −1 are parallel to the .(d1 − d0 − 1)-dimensional space tangent to the stratum .pr(s1 ) at .z1 . The rest axes .x˜1 , . . . , x˜m−d1 −1 are parallel to the space tangent to n−1 and orthogonal to .pr(s ) at .z1 . The coordinate transformation (with block-diagonal .S 1 ˜ In what follows, the old matrix) will be written in the form .x = diag(J (z1 ), Im−d0 )x. coordinates do not appear, and the new ones are denoted by x. We introduce operators “of the second generation” −1 A(z0 , z1 ; θ 0 , θ 1 ) = Fη→x (z0 ; J (z1 )(η, θ 1 ), θ 0 )Fy→η ∈ BL2 (Rm−d1 ),

.

(3.1.4)

where .θ 0 ∈ Rd0 , .θ 1 ∈ Rd1 −d0 , .|θ 0 |2 + |θ 1 |2 = 1, and F is the Fourier transform in .Rm−d1 ; A(z0 , z1 ; λ) = Eω→ϕ (λ)−1 (z0 ; J (z1 )(ω, 0), 0)Eψ→ω (λ) ∈ BL2 (S m−d1 −1 ), (3.1.5)

.

where .λ ∈ R and the operators .E(λ)±1 are defined by (1.2.2) and (1.2.8) for .n = m − d1 . If .d0 = 0, then in (3.1.4), the operator A(z0 , z1 ; θ 1 ) = Fη→x (z0 ; J (z1 )(η, θ 1 ))Fy→η

.

is defined instead of .A(z0 , z1 ; θ 0 , θ 1 ), and the operator A(z0 , z1 ; λ) = Eω→ϕ (λ)−1 (z0 ; J (z1 )(ω, 0))Eψ→ω (λ)

.

is defined in (3.1.5). Continuing the process, we arrive at the collection of the operators A(z0 , . . . , zj ; θ 0 , . . . , θ j ), .A(z0 , . . . , zj ; λj ), where .0 ≤ j ≤ k, .λj ∈ R, .θ 0 ∈ Rd0 , j ∈ Rdj −dj −1 for .j ≥ 1, and .|θ 0 |2 + · · · + |θ j |2 = 1. We say that such a collection is .θ ¯ 0 (M) along the chain .s0 , . . . , sk . obtained by localizing .A ∈  Although the operators .A(z0 , . . . , zj ; θ 0 , . . . , θ j ), .A(z0 , . . . , zj ; λj ) are constructed for special coordinate systems, they are independent of the arbitrariness in their definition. To explain this, suppose that there is chosen a chain of strata .s0 , . . . , sj . Let .

3.1 Algebra A and Its Irreducible Representations

127

us glue the strata .sj −1 , . . . , s0 to .sj . The collection .(z0 , . . . , zj ) determines a .dj dimensional subspace .T (z0 , . . . , zj ) of .T (M)z0 tangent to .sj . The point .(θ 0 , . . . , θ j ) runs over the unit sphere .S ∗ (z0 , . . . , zj ) in the cotangent space .T ∗ (z0 , . . . , zj ). Denote by .Rm−dj (z0 , . . . , zj ) the orthogonal complement of .T ∗ (z0 , . . . , zj ) in .T (M)z0 and by m−dj −1 (z0 , . . . , zj ) the unit sphere in the mentioned orthogonal complement. Recall .S ¯ 0 (M) is defined on the bundle of nonzero that the principal symbol . of .A ∈  0 cotangent vectors. The operator .A(z , . . . , zj ; θ 0 , . . . , θ j ) acts in .L2 (Rm−dj (z0 , . . . , zj )) and .A(z0 , . . . , zj ; λj ) does in .L2 (S m−dj −1 (z0 , . . . , zj )). This implies that the operators are independent of arbitrariness in their definition. The functions a in .M(T , M) are smooth on .M \ |T |. The set .M \ |T | is embedded in a compact .C, and the functions a extend to .C by continuity; we now describe .C. We first glue the boundary of a tubular neighborhood (in .M) of every .(m − 1)-dimensional stratum to .M \ |T | (distinct points have to be glued to distinct ones). Denote the obtained set by .Cm−1 . The boundary of a tubular neighborhood in .M of each .(m − 2)-dimensional stratum s is cut by the .(m − 1)-dimensional strata of the star .st s. We glue the boundary to .Cm−1 so that it be identified along the cut with the boundary of an already glued tubular neighborhood of .(m − 1)-dimensional stratum (distinct points have to be identified with distinct ones). Denote the new set by .Cm−2 . Descending to strata of smaller dimensions, we finally obtain the compact .C := C0 . Let .p : C → M be the natural projection and let .S ∗ (C) := p∗ S ∗ (M), where .S ∗ (M) is the cospherical bundle over .M and .p∗ S ∗ (M) the induced bundle. In what follows, the one-dimensional representations of the algebra .A will be parametrized by the points of ∗ .S (C). We now tour to listing the irreducible representations of .A. As before, let .s0 , . . . , sk be a chain of strata of the partition .T , .dim sj = dj . We introduce the following series of ¯ 0 (M) and .a ∈ M(T , M) of .A. mappings for the generators .A ∈  m−d 0 j Mapping into .BL2 (R (z , . . . , zj )): π(z0 , . . . , zj ; θ 0 , . . . , θ j ) : A →A(z0 , . . . , zj ; θ 0 , . . . , θ j ) ∈ BL2 (Rm−dj (z0 , . . . , zj )),

.

(3.1.6) π(z0 , . . . , zj ; θ 0 , . . . , θ j ) : a → a(z0 , . . . , zj ; ·)I ∈ BL2 (Rm−dj (z0 , . . . , zj ))I, where .j = 0, . . . , k and the function .a(z0 , . . . , zj ; ·) is zero degree homogeneous on m−dj \0. For .d = 0, the mappings .π(z0 ; θ 0 ) are absent; given a chain of strata, the map.R 0 pings .π(z0 , . . . , zj ; θ 0 , . . . , θ j ) in (3.1.6) are changed for .π(z0 , z1 , . . . , zj ; θ 1 , . . . , θ j ), where in particular .π(z0 , . . . , zj ; θ 1 , . . . , θ j )A = A(z0 , . . . , zj ; θ 1 , . . . , θ j ). In what follows, we, as a rule, make no remarks related to the specificity of the case .d0 = 0.

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

Mapping into .BL2 (S m−dj −1 (z0 , . . . , zj )): π(z0 , . . . , zj ; λj ) : A → A(z0 , . . . , zj ; λj ) ∈ BL2 (S m−dj −1 (z0 , . . . , zj )), (3.1.7)

.

π(z0 , . . . , zj ; λj ) : a → a(z0 , . . . , zj ; ·)I ∈ BL2 (S m−dj −1 (z0 , . . . , zj )), where .j = 0, . . . , k and .λj ∈ R. Mapping into .C: π(x, ξ ) : A → (p(x), ξ ) ∈ C,

.

(3.1.8)

π(x, ξ ) : a → a(x), where .(x, ξ ) ∈ S ∗ (C)x , while .S ∗ (C)x is the fiber over .x ∈ C of the cospherical bundle ∗ .S (C), .p : C → M is the projection, and . is the principal symbol of A. From Sect. 2.1, it follows that the algebra .A is irreducible and contains the ideal .KL2 (M). The next theorem describes all (up to equivalence) irreducible representations of the algebra .A. Theorem 3.1.4 The mappings (3.1.6)–(3.1.8) extend to representations of the quotient algebra .A/KL2 (M). The representations (3.1.6) and (3.1.7) corresponding to all complete chains of strata of the partition .T are irreducible and nonequivalent. Any irreducible representation .π of .A with .dim π > 1 is equivalent either to one of the representations (3.1.6), (3.1.7) or to the identity representation. Any one-dimensional representation of .A coincides with one of those in (3.1.8). The algebras .π(z0 , . . . , zj ; θ 0 , . . . , θ j )A contain the subalgebras spanned by the ¯ 0 (M), and by the coefficients operators .π(z0 , . . . , zj ; θ 0 , . . . , θ j )A, where .A ∈  0 j m−d −1 j .a(z , . . . , z ; ·) smooth on .S . Hence, due to Theorem 2.3.12, the inclusion KL2 (Rm−dj ) ⊂ π(z0 , . . . , zj ; θ 0 , . . . , θ j )A

.

holds. Moreover, the algebras .π(z0 , . . . , zj ; λj )A contain the subalgebras generated ¯ 0 (M), and by the coefficients by the operators .π(z0 , . . . , zj ; λj )A, where .A ∈  0 j m−d −1 j .a(z , . . . , z ; ·) smooth on .S . Proposition 2.2.1 leads to the inclusion KL2 (S m−dj −1 ) ⊂ π(z0 , . . . , zj ; λj )A.

.

Thus, Theorem 3.1.4 implies the following assertion. Corollary 3.1.5 .A is an algebra of type I.

3.1 Algebra A and Its Irreducible Representations

3.1.3

129

Proof of Theorem 3.1.4

10 . Plan of the proof. In what follows, if no misunderstanding is possible, we write Rm−dj and S m−dj −1 instead of Rm−dj (z0 , . . . , zj ) and S m−dj −1 (z0 , . . . , zj ). We apply the localization principle (Proposition 1.3.24) to the algebra A, taking C(M) as a localizing algebra and J = KL2 (M). It is clear that the local algebra Ax at a point x ∈ M \ |T | is isomorphic to C(S ∗ (M)x ) (Proposition 2.1.6). If z0 is a point of a d0 -dimensional stratum s0 and d0 > 0, then the same argument as in the proof of Proposition 2.3.4 shows that the local algebra Az0 is isomorphic to the algebra of functions on S d0 −1 ranging in BL2 (Rm−d0 ) spanned by the functions θ 0 → A(z0 ; θ 0 ) = π(z0 ; θ 0 )A and ¯ 0 (M) and a ∈ by the operators of multiplication by a(z0 ; ·)I = π(z0 ; θ 0 )a; here A ∈  0 0 0 M(T , M), while A(z ; θ ) and a(z ; ·) are defined as in 3.1.2. Localization of A(z0 ; θ 0 ) leads in particular to the algebra S(z0 ) generated by the functions R  λ → A(z0 , λ) (see (3.1.3)) and by the operators of multiplication by the coefficients a(z0 ; ·)|S n−1 , where n = m − d0 . The operations in S(z0 ) are pointwise, and a norm is given by A(·) = sup{A(λ); BL2 (S n−1 ); λ ∈ R}.

.

If d0 = 0, then the algebra A(z0 ; θ 0 ) does not appear, and instead, S(z0 ) arises even at the first step of localization. The algebras A(z0 ; θ 0 ) and S(z0 ) are said to be algebras of the first generation. Localizing such algebras, we obtain algebras of the second generation (of the same type), and so on. The last generation consists of commutative algebras. In essence, description of the dynasty of local algebras is tantamount to listing all the equivalence classes of irreducible representations of the algebra A. We proceed to implement this plan. 20 . Localization of the algebra A(z0 , θ 0 ). To apply the localization principle from ¯ n ) as a localizing algebra, where n = m − d0 and R ¯n Proposition 1.3.26, we take C(R n n−1 is a compact obtained by adding to R the sphere S∞ at infinity. The conditions of the Proposition 1.3.26 are fulfilled; this can be proved in the same way as in Sect. 2.3.3 (before Proposition 2.3.11). Remind that A(z0 , θ 0 ) is generated in L2 (Rn ) by the operators of the form −1 A(z0 ; θ 0 ) = Fη→x (z0 ; η, θ 0 )Fy→η ∈ BL2 (Rn )

.

(see (3.1.2) for notations) and the operators of multiplication by the coefficients Rn \ 0  x → a(z0 ; x/|x|), a(z0 ; ·)|S n−1 ∈ M(T (s0 ), S n−1 ).

.

On such generators, we define some mappings that, as shown below, arise under ¯ n ; in items (i)–(v), the mapping l(z) is acting on the algebra localization at a point z ∈ R lz .

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

(i) For z ∈ Rn \ 0, z/|z| ∈ S n−1 \ |T (s0 )|, we set lz = C(S n−1 ) and introduce the mapping l(z) : a(z0 ; ·) → a(z0 ; z/|z|),

.

(3.1.9)

l(z) : A(z0 ; θ 0 ) → (z0 ; ·, 0). (ii) For z = 0, we denote by lz the algebra generated on L2 (Rn ) by operators of the form −1 A(z0 ; 0) = Fη→x (z0 ; η, 0)Fy→η

.

and the operators of multiplication by the functions Rn \ 0  x → a(z0 ; x/|x|). Introduce the mapping l(z) : a(z0 ; ·) → a(z0 ; ·)I,

.

(3.1.10)

l(z) : A(z0 ; θ 0 ) → A(z0 ; 0). n−1 and ϕ ∈ S n−1 \ |T (s )|, where ϕ is the vector directed toward z, we (iii) For z ∈ S∞ z 0 z ¯ n ) and introduce the mapping set lz = C(R

l(z) : a(z0 ; ·) → a(z0 ; ϕz ),

.

(3.1.11)

l(z) : A(z0 ; θ 0 ) → (z0 ; ·, θ 0 ). ¯ n \ 0 for ϕz ∈ |T (s0 )|. We pass on to the mappings corresponding to the points z ∈ R Let s1 ∈ st(s0 ), dim s1 = d1 , so that pr(s1 ) is a (d1 − d0 − 1)-dimensional stratum of the partition T (s0 ) of the sphere S n−1 . We choose new coordinates x˜1 , . . . , x˜n in Rn with the same origin. The axis x˜n is directed toward ϕz ∈ pr(s1 ); the axes x˜n−d1 +d0 , . . . , x˜n−1 are parallel to the space tangent to the stratum pr(s1 ) at ϕz ; the remaining axes are orthogonal to the stratum pr(s1 ) at the point ϕz and parallel to the plane tangent to the sphere S n−1 at that point. We write the coordinate transformation in the form x = J (z)x. ˜ Further, the old coordinates do not appear, and the new ones are denoted by x. A point x = (x1 , . . . , xn ) will be written in the form x = (x 1 , x 2 ), where x 1 = (x1 , . . . , xn−(d1 −d0 )−1 ) and x 2 = (xn−(d1 −d0 ) , . . . , xn ). (iv) For z ∈ Rn \ 0 and z/|z| ∈ pr(s1 ), we denote by lz the algebra generated on L2 (Rn ) by operators of the form 0 B(z0 , z; 0) = Fξ−1 →x (z ; J (z)ξ, 0)Fy→ξ

.

and the operators of multiplication by the functions Rn \ {x = (x 1 , x 2 ) : x 1 = 0}  x → a(z0 , z; x) = a(z0 , z; x 1 /|x 1 |).

.

3.1 Algebra A and Its Irreducible Representations

131

(Remind that the function Rn \0  x → a(z0 ; x) is zero degree homogeneous; hence, a(z0 , z; x 1 /|x 1 |) = a(z0 , z/|z|; x 1 /|x 1 |).) Introduce the mapping l(z) : a(z0 ; ·) → a(z0 , z; ·)I,

.

(3.1.12)

l(z) : A(z0 ; θ 0 ) → B(z0 , z; 0). n−1 and ϕ ∈ pr(s ), by l , we denote the algebra generated on L (Rn ) by (v) For z ∈ S∞ z 1 z 2 operators of the form 0 0 B(z0 , z; θ 0 ) = Fξ−1 →x (z ; J (z)ξ, θ )Fy→ξ

.

and the operators of multiplication by the functions x →= a(z0 , z; x 1 /|x 1 |); here a(z0 , z; x 1 /|x 1 |) = a(z0 , ϕz ; x 1 /|x 1 |).

.

Introduce the mapping l(z) : a(z0 ; ·) → a(z0 , z; ·)I,

.

(3.1.13)

l(z) : A(z0 ; θ 0 ) → B(z0 , z; θ 0 ). ¯ n obtained by the Proposition 3.1.6 Let A(z0 , θ 0 )z be the local algebra at z ∈ R 0 0 n ¯ ). The mapping l(z) defined localization of A(z , θ ) with commutative algebra C(R 0 0 on the generators of A(z ; θ ) by (3.1.9)–(3.1.13) extends to an isomorphism l(z) : ¯ n. A(z0 ; θ 0 )z → lz for every z ∈ R Proof Items (i)–(iii) can be treated in the same way as in Proposition 2.3.11. Let us turn to the mapping (iv). Introduce the unitary operator (Ut u)(x) = t m/2 u(z + t (x − z)) on L2 (Rn ) for t > 0 and consider the composition Ut QUt−1 , where Q is a product of finitely ¯ n ). In view of the properties of the many generators of A(z0 ; θ 0 ) and functions in C(R coefficients, there exists a strong limit Q0 = lim Ut QUt−1 .

.

t→0

(3.1.14)

Since .

lim (Ut a(z0 ; ·)Ut−1 u)(x) = a(z0 , z; x 1 /|x 1 |)u(x),

t→0

(3.1.15)

−1 0 0 −1 0 lim (Ut Fξ−1 →x (z ; J (z)ξ, θ )Fy→ξ Ut u)(x) = (Fξ →x (z ; J (z)ξ, 0)Fy→ξ u)(x),

t→0

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

the operators Q0 generate on L2 (Rn ) the algebra lz defined in (iv) before Proposition 3.1.6. ¯ n ). From (3.1.13) and (3.1.14), it Let Q be the algebra spanned by A(z0 ; θ 0 ) and C(R follows that the mapping Q → Q0 extends to an epimorphism q : Q/J → lz , where J ¯ n ), χ (z) = 0. We show that q is an is the ideal generated in Q by the functions χ in C(R ∞ n isomorphism. If the support of χ ∈ Cc (R ) belongs to a sufficiently small neighborhood of z, then χ Q0 is in Q, and if χ (z) = 1, then the operators Q and χ Q0 , where Q0 = limt→0 Ut QUt−1 , represent the same residue class [Q] in Q/J. Hence, [Q] ≤ χ Q0  ≤ sup |χ |Q0 , which means that q is a monomorphism. Thus, q is an isomorphism. Let us finally turn to the mapping (v). We can argue as in the part of the proof of Proposition 2.3.11 relating to the case (iii). Namely, as Aj k (θ ), we take operators of the form 0 0 Bj k (z0 , z; θ 0 ) = Fξ−1 →x j k (z ; J (z)ξ, θ )Fy→ξ ,

.

and as coefficients, we choose aj k (z0 ; ·), which are zero degree homogeneous functions satisfying aj k (z0 ; ·)|S n−1 ∈ M(T (s0 ), S n−1 ). Then, there holds an equality similar to (2.3.29), Ut (



.

j

={

(3.1.16)

k

 j

χj k aj k (z0 ; ·)Bj k (z0 , z; θ 0 ))Ut−1 u

(Ut (χj k aj k (z0 ; ·))Bj k (z0 , z; θ 0 )}u,

k

where (Ut u)(x) = u(x + tϕz ). We have .

lim (Ut aj k (z0 ; ·))(x) = aj k (z0 , z; ·).

t→+∞

Therefore, instead of (2.3.31), we obtain .

inf{

 j

χj k aj k (z0 ; ·)Bj k (z0 , z; θ 0 ) + K; BL2 (Rn ); K ∈ Jz }

k

=

 j

χj k (z)aj k (z0 , z; ·)Bj k (z0 , z; θ 0 ); BL2 (Rn ).

k

  30 . Localization of the algebra S(z0 ). If a point z0 belongs to a zero-dimensional stratum s0 of the partition T of M, then under localization of the initial algebra A there arises 0 the algebra spanned by operators of the form a(z0 ; x)Fξ−1 →x (z ; ξ )Fy→ξ ; here, F is the

3.1 Algebra A and Its Irreducible Representations

133

Fourier transform on Rm . Such an algebra is isomorphic to the algebra S(z0 ) generated by the operator-valued functions R  λ → a(z0 ; ϕ)Eω→ϕ (λ)−1 (z0 ; ω)Eψ→ω (λ) ∈ BL2 (S m−1 );

.

the norm in S(z0 ) is defined by B(·); S(z0 ) = sup{B(λ); BL2 (S m−1 ); λ ∈ R} (see 2.2.1). If z0 ∈ s0 and dim s0 = d0 > 0, then S(z0 ) arises under localization of the algebra A(z0 , θ 0 ) at z = 0. In more detail, first comes the algebra l0 of the operators a(z0 ; x)A(z0 ; 0) (see item (ii) before Proposition 3.1.6), which is isomorphic to the algebra S(z0 ) spanned by the functions R  λ → a(z0 ; ϕ)Eω→ϕ (λ)−1 (z0 ; ω, 0)Eψ→ω (λ) ∈ BL2 (S n−1 ),

.

where n = m − d0 (compared with Sect. 2.2.1 and Proposition 2.3.11). To apply the localization principle (Proposition 1.3.24) to S(z0 ), we take C0 (R) ⊗ KL2 (S n−1 ) as the ideal J and C n−1 (S n−1 ) as the localizing algebra. The fulfillment of the hypotheses of Proposition 1.3.24 is in fact verified in the proof of Proposition 2.2.9. Indeed, in contrast with the algebra S(z0 ), the role of coefficients in S from 2.2.1 is played by functions in C ∞ (S n−1 ). Hence, J = C0 (R)⊗KL2 (S n−1 ) ⊂ S ⊂ S(z0 ) (the inclusion J ⊂ S is proved in 2.2.2). This easily implies that the proof of Proposition 2.2.9 verifies the applicability of Proposition 1.3.24 to the algebra S(z0 ), too. Therefore, the equality  0 ) = ∪ϕ∈S n−1 S(z  0 )ϕ ∪ J S(z

.

(3.1.17)

holds; here, S(z0 )ϕ = S(z0 )/Jϕ is the local algebra at a point ϕ, and Jϕ is the ideal in S(z0 ) spanned by the functions χ ∈ C(S n−1 ) vanishing at ϕ. We then describe the local algebras S(z0 )/Jϕ . Assume that z0 ∈ s0 , dim s0 = d0 ≥ 0, and T (M)z0 = T (m) (M)z0 is the tangent space of M at z0 . Denote by T (n) (M)z0 , where n = m − d0 , the subspace of T (m) (M)z0 , orthogonal to the stratum s0 . We choose arbitrarily a unit vector z1 in T (n) (M)z0 and introduce in T (n) (M)z0 new orthogonal coordinates x˜1 , . . . , x˜n (in the same way as in a similar situation before). Namely, the axis x˜n , n = m − d0 , is directed toward z1 ; if s1 ∈ st(s0 ), dim s1 = s0 , and z1 ∈ pr(s1 ), then the axes x˜m−d1 , . . . , x˜m−d0 −1 are parallel to the (d1 − d0 − 1)-dimensional space tangent to the stratum pr(s1 ) at z1 . The remaining axes x˜1 , . . . , x˜m−d1 −1 are orthogonal to pr(s1 ) at z1 . We write the coordinate transformation in the form x = J (z1 )x˜ with orthogonal matrix J (z1 ). In what follows, we use the new coordinates, which will be given the notation x1 , . . . , xn . The algebra S(z0 ) is spanned by the operators of multiplication a(z0 ; ·)I ∈ BL2 (S n−1 ) and the functions R  λ → A(z0 ; λ) ∈ BL2 (S n−1 ); here A(z0 ; λ) = Eω→ϕ (λ)−1 (z0 ; J (z1 )ω, 0)Eψ→ω (λ),

.

while 0 is the row (0, . . . , 0) containing d0 elements (which is absent for d0 = 0).

(3.1.18)

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

Denote by L(z0 , z1 ) the algebra generated on L2 (Rn−1 ) by the operators a(z0 , z1 ; ·)I and functions of the form −1 0 1 n−1 R  λ → D(z0 , z1 ; λ) = Fη→x ); (3.1.19)  (z ; J (z )(η, λ), 0)Fy  →η ∈ BL2 (R

.

as usual, a(z0 , z1 ; ·) denotes a zero degree homogeneous function Rn−1 \ 0  x  → a(z0 , z1 ; x  ) and a norm in L(z0 , z1 ) is introduced by D(·); L = sup{D(λ); BL2 (Rn−1 ); λ ∈ R}.

.

The next assertion follows from Proposition 2.2.10. Proposition 3.1.7 Let A and D be the same as in (3.1.18) and (3.1.19). Then, the mapping a(z0 ; ·) → a(z0 , z1 ; ·), A(z0 ; ·) → D(z0 , z1 ; ·),

.

defined for generators of the algebra S(z0 ) extends to an isomorphism S(z0 )z1  L(z0 , z1 ),

.

where S(z0 )z1 is the local algebra S(z0 )/Jz1 at z1 ∈ S n−1 . 40 . The dynasty of local algebras. In this section, with every (complete) chain of strata s0 , . . . , sk of the partition T of M, we associate the dynasty of k + 1 generations of some local algebras. These algebras arise under iterations of the localization procedure. In the next section, any irreducible representation π of A (up to equivalence) with dim π > 1 will be implemented in terms of such local algebras. The successive generations consist of algebras of the same type, however, increasingly simple ones: the operators of the algebras act in the spaces L2 (Rp ) or L2 (S q ) with certain p and q; as a generation is succeeded, the p and q decrease. The last generation of a dynasty is degenerate: localization of the algebras of the last generation provides only commutative algebras. We apply the localization principle (Proposition 1.3.24) to A with the localizing algebra C(M) and J = KL2 (M). The following three types of local algebras Az0 can arise: 1) if z0 ∈ M \ |T |, then Az0 is commutative; 2) if z0 ∈ s0 , where s0 is a stratum of T with dim s0 = d0 > 0, then the algebra Az0 is generated by the functions S d0 −1  θ 0 → a(z0 ; ·)A(z0 ; θ 0 ) ∈ BL2 (Rm−d0 )

.

(the notations are explained in 10 ); 3) if z0 ∈ s0 and d0 = dim s0 = 0, then Az0 is spanned by the functions R  λ → a(z0 ; ·)A(z0 ; λ) ∈ BL2 (S m−1 ).

.

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135

First, we consider case 2). As in 10 , by A(z0 ; θ 0 ), we denote the algebra generated on L2 (Rm−d0 ) by the operators of the form a(z0 ; ·)A(z0 ; θ 0 ) (for a fixed θ 0 ). The algebra A(z0 ; θ 0 ) is irreducible. The equality z0 = A



.

 0; θ 0) A(z

(3.1.20)

θ 0 ∈S d0 −1

is valid (compared with Proposition 2.3.4). Now, we apply Proposition 3.1.6. The local algebras A(z0 ; θ 0 )z in (i) and (iii) are commutative (see items (i)–(v) before Proposition 3.1.6); we postpone for a while their discussion and turn to items (ii), (iv), and (v). The algebra A(z0 ; θ 0 )z for z = 0 (item (ii)) is isomorphic to the algebra S(z0 ) of functions that range in BL2 (S n−1 ), where n = m − d0 ; see 10 and 30 . In (iv), the local algebra A(z0 ; θ 0 )z is isomorphic to the algebra of the functions S d1 −d0 −1  θ 1 → a(z0 , z; ·)A(z0 , z; 0, θ 1 ),

.

(3.1.21)

where −1 0 1 m−d1 A(z0 , z; 0, θ 1 ) = Fη→x ). (3.1.22) (1) (z ; J (z)(η, θ ), 0)Fy (1) →η ∈ BL2 (R

.

We pass on to item (v). Denote by u(y ˆ (1) , ξ (2) ) = Fy (2) →ξ (2) u(y (1) , y (2) ) the partial Fourier transform of u. Then, 0 (1) (2) (B(z0 , z; θ 0 )u)ˆ(x (1) , ξ (2) ) = Fξ−1 , ξ ), θ 0 )Fy (1) →ξ (1) u(y ˆ (1) , ξ (2) ). (1) →x (1) (z ; J (z)(ξ

.

We set p = (|ξ (2) |2 + 1)1/2 , X = x (1) p, Y = y (1) p and obtain

= (2π )−(m−d0 )

(B(z0 , z; θ 0 )u)ˆ(Xp−1 , ξ (2) ) (3.1.23)

.

 Rm−d1

exp(iXp−1 ξ (1) )(z0 ; J (z)(ξ (1) , ξ (2) ), θ 0 ) dξ (1)

×

 Rm−d1

−1 (2) −1 exp (−iξ (1) Yp−1 )u(Yp ˆ , ξ )p dY.

We write x, y (∈ Rm−d1 ) instead of X, Y , and v(y) and w(x) instead of u(Yp−1 , ξ (2) ) and (B(z0 , z; θ 0 )u)ˆ(Xp−1 , ξ (2) ). Then, (3.1.23) takes the form −1 w(x) = Fη→x (z0 ; J (z)(η, ξ (2) /p), θ 0 /p)Fy→η v(y).

.

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

Thus, the algebra A(z0 , θ 0 )z is isomorphic to the algebra of the functions Rd1 −d0  ξ (2) → a(z0 , z; x/|x|) ×

.

(3.1.24)

−1 ×Fη→x (z0 ; J (z)(η, ξ (2) /p), θ 0 /p)Fy→η v(y) ∈ BL2 (Rm−d1 ).

Denote by A(z0 , z; θ 0 , θ 1 ) the algebra generated on L2 (Rm−d1 ) by operators of the form −1 a(z0 , z; x/|x|)Fη→x (z0 ; J (z)(η, θ 1 ), θ 0 )Fy→η

.

for fixed θ 1 ∈ Rd1 −d0 and θ 0 ∈ Rd0 such that θ 0 = 0 and |θ 0 |2 + |θ 1 |2 = 1. From (3.1.24), it follows that  0 ; θ0 )z = A(z



.

 0 , z; θ 0 /(1 + |θ 1 |2 )1/2 , θ 1 /(1 + |θ 1 |2 )1/2 ) A(z

(3.1.25)

θ 1 ∈Rd1 −d0

(compared with Proposition 2.3.4). We turn to case 3). Let z0 be a point of a zero-dimensional stratum s0 . As the local algebra Az0 , there arises the algebra S(z0 ) for n = m − 1, see 10 and 30 ; remind that such an algebra had already arisen in 40 with d0 = dim s0 > 0 as the local algebra A(z0 ; θ 0 )z for z = 0. We assume that A(z0 ; θ 0 ) and S(z0 ) (for d0 ≥ 0) form the first generation of local algebras. (The commutative algebras will be considered in their own right.) We are going to make up the second generation of local algebras. In the generation, we first include the algebras A(z0 , z1 ; θ 0 , θ 1 ) introduced after formula (3.1.24) (we changed the notation z for z1 ). To clarify the contribution of the algebras S(z0 ) in the second generation, we turn to Proposition 3.1.7. If z1 ∈ S n−1 \ |T (s0 )|, n = m − d0 , then the algebra S(z0 )z1 is isomorphic to the algebra C(S n−1 ) (of the functions S n−1  ω → (z0 ; J (z1 )ω, 0), see (3.1.19)). If z1 belongs to a stratum pr(s1 ) of the partition |T (s0 )|, then the algebra S(z0 )z1 is isomorphic to the algebra of operator-valued functions defined by (3.1.21) and (3.1.22) with z = z1 . Therefore, to the second generation, S(z0 ) gives the algebras A(z0 , z1 ; 0, θ 1 ). In addition, under localization of A(z0 , z1 ; θ 0 , θ 1 ), the algebras S(z0 , z1 ) appear (as local ones at the origin) spanned by functions of the form R  λ → a(z0 , z1 ; ϕ)Eω→ϕ (λ)−1 (z0 ; J (z1 )(ω, 0), 0)Eψ→ω (λ) ∈ BL2 (S m−d1 ).

.

The algebras S(z0 , z1 ) also have to be included in the second generation. Before listing the rest of the generations, we will simplify the notation. Let s0 , . . . , sk be a chain of strata. Let us write the stratum p(s1 ) of the partition T (s0 ) in the form s1 (s0 ); by s2 (s0 , s1 ), we denote the stratum of T (s0 , s1 ) generated by s2 ; the notations s3 (s0 , . . . , s2 ), . . . , sk (s0 , . . . , sk−1 ) have a similar meaning. Applying the above localization procedure to the second and succeeding generations, with a chain s0 , . . . , sk , we associate the dynasty spanning k + 1 generations, where the j -

3.1 Algebra A and Its Irreducible Representations

137

th generation consists of the algebras A(z0 , . . . , zj −1 ; θ 0 , . . . , θ j −1 ) and S(z0 , . . . , zj −1 ) with j ≤ k + 1, A(z0 , . . . , zj −1 ; θ 0 , . . . , θ j −1 ) ⊂ L2 (Rm−dj −1 ), z0 ∈s0 , . . . , zj −1 ∈sj −1 (s0 , . . . , sj −2 ),

.

θ j ∈ Rqj , |θ 0 |2 + · · · + |θ j −1 |2 = 1, q0 + · · · + qj −1 = dj −1 = dim sj −1 , S(z0 , . . . , zj −1 ) ⊃ C0 (R) ⊗ KL2 (S m−dj −1 −1 ). The algebra A(z0 , . . . , zj −1 ; θ 0 , . . . , θ j −1 ) is generated by the operators of multiplication a(z0 , . . . , zj −1 ; ·) and the operators A(z0 , . . . , zj −1 ; θ 0 , . . . , θ j −1 ) (similar types of operators appeared in (3.1.2) and (3.1.4)). The algebra S(z0 , . . . , zj −1 ) is spanned by the functions R  λ → a(z0 , . . . , zj −1 ; ·)A(z0 , . . . , zj −1 ; λ) ∈ BL2 (S m−dj −1 −1 )

.

(see (3.1.5)). If the chain s0 , . . . , sk is complete (see 3.1.1), then localization of the algebras in the (k + 1)-th generation provides only commutative algebras. 50 . A list of irreducible representations π of the algebra A with dim π > 1. We will show that any irreducible representation π of the algebra A such that dim π > 1 is equivalent to either one of the representations (3.1.6), (3.1.7) or the identity representation. Recall that the algebra A is irreducible and contains the ideal KL2 (M). Thus, the identity representation of A is irreducible. We start searching for other representations by localization of A (with Proposition 1.3.24; see 10 ). This implies, in particular, that all irreducible representations that are not equivalent to the identity one turn out to be representations of the quotient algebra A/KL2 (M). The local algebras at the points z ∈ M\|T | are commutative; their irreducible representations are one-dimensional. Therefore,  with dim π > 1 can be found only under localization along the representations π ∈ A chains of strata. Let s0 , . . . , sk be a complete chain of strata, dim sj = dj . If d0 > 0 and z0 ∈ s0 , then formula (3.1.20) is valid. The algebras A(z0 ; θ 0 ) are irreducible. Hence, mappings of the form (3.1.6) for j = 0 extend to irreducible representations π(z0 ; θ 0 ) of the algebra A/KL2 (M). Such representations for distinct pairs (z0 , θ 0 ) are nonequivalent (they have distinct kernels). If d0 = 0, then the local algebra Az0 is isomorphic to the algebra S(z0 ) (see 30 ) and the equality (3.1.17) holds. The mappings (3.1.7) for j = 0 extend to the representations π(z0 ; λ0 ) that arise in (3.1.17) as representations of the ideal J . These representations are irreducible and pairwise nonequivalent. If the partition T contains no zero-dimensional strata, then S(z0 ) appears all the same under localization of A(z0 ; θ 0 ), so there appear the representations (3.1.7) as well for j = 0. From the description of localization for the algebras A(z0 ; θ 0 ) and S(z0 ), it follows that new irreducible representations of dimension > 1 can occur only in the spectrum of local algebras of succeeding generations. The same argument as applied to the algebras A(z0 , z1 ; θ 0 , θ 1 ) and S(z0 , z1 ) provides the representations π(z0 , z1 ; θ 0 , θ 1 )

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

and π(z0 , z1 ; λ1 ). Going from a generation to the succeeding generation in a dynasty of local algebras, we find all representations (3.1.6), (3.1.7). According to the localization principle, such a list contains all (up to equivalence) irreducible representations of the algebra A of dimension >1. 60 . The one-dimensional representations of the algebra A. We list all one-dimensional representations arising at various stages of the localization procedure. It is necessary to take their origin into account when describing the topology on the spectrum of A. Assume that x ∈ M \ |T |. The local algebra Ax is commutative; any representation of Ax is included in (3.1.8); in that case p(x) = x and the coefficients a are continuous at the point x. We now suppose that s0 is a stratum of the partition T , d0 = dim s0 > 0, and z0 = s0 . Let T (z0 , s0 ) (T ∗ (z0 , s0 )) be the tangent (cotangent) space to M that is orthogonal to s0 . Let also z ∈ T (z0 , s0 ), z = 0, and ϕ = z/|z| ∈ |T (s0 )|. Finally, we denote by ω a unit vector in T ∗ (z0 , s0 ). According to Proposition 3.1.6 and formulas (3.1.9), the mapping π(z0 ; ϕ, ω, 0) : a(z0 ; ·) → a(z0 ; ϕ),

.

π(z0 ; ϕ, ω, 0) : A(z0 ; θ 0 ) → (z0 ; ω, 0)

(3.1.26)

defines a one-dimensional representation of the algebra A(z0 ; θ 0 ). Suppose that z is a point in the infinitely distant sphere of T (z0 , s0 ), ϕ is the unit vector directed toward z, and ϕ ∈ |T (s0 )|. Then, by virtue of (3.1.11), the mapping a(z0 ; ·) → a(z0 ; ϕ),

.

A(z0 ; θ 0 ) → (z0 ; η, θ 0 )

(3.1.27)

defines a one-dimensional representation of A(z0 ; θ 0 ) for any η ∈ T ∗ (z0 , s0 ). Considering that the function ξ → (z0 ; ξ ) is homogeneous, we can combine (3.1.26) and (3.1.27). As a result, we obtain that the mapping π(z0 ; ϕ, ξ ) : a(z0 ; ·) → a(z0 ; ϕ),

.

π(z0 ; ϕ, ξ ) : A(z0 ; θ 0 ) → (z0 ; ξ )

(3.1.28)

defines a one-dimensional representation of A(z0 ; θ 0 ) for any ξ ∈ S ∗ (M)z0 and every unit vector ϕ ∈ T (z0 , s0 ). Representations of the form (3.1.28) are contained among those in (3.1.8). Note that π(z0 ; ϕ, ω, 0) in (3.1.26) are representations of the algebra S(z0 ) as well; see Theorem 2.2.11 and formulas (3.1.18). In particular, π(z0 ; ϕ, ω, 0) for ϕω = 0 are representations of the algebra Sλ (z0 ) spanned by multiplications by a(z0 ; ·)I and by operators (3.1.3) with a fixed λ ∈ R. In the case d0 = dim s0 = 0, the representations (3.1.27) are absent, and formulas (3.1.26) and (3.1.28) coincide and define representations of S(z0 ).

3.2 The Spectral Topology of Algebra A

139

In essence, listing the one-dimensional representations under further localization along chains of strata reduces to the argument just given above. As a result, all representations in (3.1.8) appear as representations of A(z0 , . . . , zj ; θ 0 , . . . , θ j ). From the localization principle, it follows that there are no other one-dimensional representations of the algebra A.  

3.2

The Spectral Topology of Algebra A

3.2.1

Description of the Jacobson Topology (Formulation of the Theorem)

10 . Parametrizing the spectrum. We first introduce a set parametrizing the spectrum of A. Assume that s0 , . . . , sk is a complete chain of strata of the partition T of M, z0 , . . . , zk are variables related to the chain: z0 runs over Z0 = s0 , and zj runs over the stratum Zj of the partition T (s0 , . . . , sj −1 ), 1 ≤ j ≤ k. The stratum s0 is a submanifold of the manifold M, while the stratum sj , 1 ≤ j ≤ k, is a submanifold of the corresponding sphere. We set Zj = Z0 × · · · × Zj for j = 0, . . . , k. A point (z0 , . . . , zj ) in Zj indicates the direction in M of approach to the point z0 ∈ s0 and, by doing so, defines the dj -dimensional subspace T (sj ; z0 , . . . , zj ) of T (M)z0 tangent to the stratum sj . As before (Theorem 3.1.4), denote by S ∗ (sj ; z0 , . . . , zj ) the unit sphere in the cotangent space T ∗ (s; z0 , . . . , zj ). Let S ∗ (Zj ) stand for the induced bundle with base Zj and fiber S ∗ (sj ; z0 , . . . , zj ) over a point (z0 , . . . , zj ). We introduce the disjoint union  of the sets S ∗ (C),

k 

.

(T ) j =0

S ∗ (Zj ),

k 

Zj × R,

(T ) j =0

where ∪(T ) is taken over all complete chains of strata in T . From Theorem 3.1.4, it follows that the spectrum of the quotient algebra A/KL2 (M) can be parametrized by the points of . Assume that point e corresponds to the identity representation of A. Then the spectrum of A is parametrized by the points of the set e ∪ . In what follows, we compare representations obtained by localization along distinct q chains of strata. Let us explain how to do that. Assume that sl 0 , . . . , s l is a chain consisting of some strata of an original chain s0 , . . . , sj , 0 ≤ l 0 < · · · < l q = j . We say that the new chain is a subchain of the original one. We now describe which of the one-type representations π(z0 , . . . , zj ; θ 0 , . . . , θ j ) and π(ζ 0 , . . . , ζ q ; τ 0 , . . . , τ q ) or π(z0 , . . . , zj ; λ) and π(ζ 0 , . . . , ζ q ; μ) must be considered as close ones; here, π(z0 , . . . , zj ; ·) and π(ζ 0 , . . . , ζ q ; ·) are obtained by localization along the chain and subchain, respectively. Let a point ζ 0 ∈ sl 0 belong to a sufficiently small tubular neighborhood of the stratum s0 . Then, the “cylindrical coordinates” (z0 , r 0 , α 0 ) are defined for the point, ¯ + , and α 0 ∈ S m−d0 −1 . For the points α 0 in a tubular where z0 ∈ s0 = Z0 , r 0 ∈ R neighborhood of the stratum Z1 , the coordinates (z1 , r 1 , α 1 ) are defined and so on.

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

For a couple (ζ 0 , ζ 1 ), we obtain a row (z0 , r 0 , . . . , zp , r p , ζ 1 ), where p = l 0 − 1 and ζ 1 ∈ S m−dp −1 . Continuing the procedure, we rewrite the row (ζ 0 , . . . , ζ q ) in the form (z0 , r 0 , . . . , zj −1 , r j −1 , r j ), where j = l q , zj = ζ q ∈ S m−dj −1 −1 . Collections j (z00 , . . . , z0 ) and ζ 0 , . . . , ζ q ) ≡ (z0 , r 0 , . . . , zj −1 , r j −1 , zj ) are said to be close if the nonnegative numbers r 0 , . . . , r j −1 are sufficiently small, while z0ν and zν are close as points of the same stratum Zν (ν = 0, . . . j ). Furthermore, points (θ 0 , . . . , θ j ) and (τ 0 , . . . , τ q ) with |θ 0 |2 + · · · + |θ j |2 = 1 and |τ 0 |2 + · · · + |τ q |2 = 1 belong to dj -dimensional spheres; j for close (z00 , . . . , z0 ) and (ζ 0 , . . . , ζ q ), the spheres can be identified in a natural way with a “standard” sphere (local trivialization). Representations π(z0 , . . . , zj ; θ 0 , . . . , θ j ) j and π(ζ 0 , . . . , ζ q ; τ 0 , . . . , τ q ) are said to be close if the collections (z00 , . . . , z0 ) and 0 q 0 (ζ , . . . , ζ ) are close (in the sense mentioned above), while the points (θ , . . . , θ j ) and (τ 0 , . . . , τ q ) differ little in the standard sphere S dj . The proximity of representations π(z0 , . . . , zj ; λ) and π(ζ 0 , . . . , ζ q ; μ) can be described in a similar way. 20 . Fundamental system of neighborhoods. We introduce a topology on the set e ∪  by indicating typical neighborhoods of the points that form a fundamental system. (Recall that the set e ∪  parametrizes the spectrum of the algebra A.) By U (x; T ), we denote an arbitrary neighborhood of a point x in a topological space T . Neighborhoods of the points in S ∗ (Zj ). Let (z0 , . . . , zj ; θ 0 , . . . , θ j ) ∈ S ∗ (Zj ), 0 ≤ j ≤ k, where θ j ∈ Rdj −dj −1 , d−1 = 0, and |θ 0 |2 + · · · + |θ j |2 = 1. If θ 0 = 0, then a typical neighborhood U ((z0 , . . . , zj ; θ 0 , . . . , θ j ); ) consists of the sets U ((z0 , . . . , zq ); Zq ) × U (q ; S dq −1 ),

.

(3.2.1)

for q = 0, . . . , j , where q = (θ 0 , . . . , θ q )/(|θ 0 |2 + · · · + |θ q |2 )1/2 ∈ S dq −1 , and the neighborhood contains all sufficiently close representations of the same types obtained by localization along subchains. (We write the part of S ∗ (Zj ) over a “sufficiently small” neighborhood U ((z0 , . . . , zq ); Zq ) as the product (3.2.1).) Suppose now that θ 0 = 0, . . . , θ i = 0, θ i+1 = 0. A neighborhood U ((z0 , . . . , zj ; 0 θ , . . . , θ j ); ) consists of the sets (3.2.1) for q = i + 1, . . . , j , the products U ((z0 , . . . , zp ); Zp ) × S dp −1 , p = 0, . . . , i, .

(3.2.2)

U ((z0 , . . . , zr ); Zr ) × R, r = 0, . . . , i − 1,

(3.2.3)

.

and the product U ((z0 , . . . , zi ); Zi ) × {λ ∈ R : λ ≶ N }

.

in the case zi+1 θ i+1 ≶ 0 or the product U ((z0 , . . . , zi ); Zi ) × R

.

3.2 The Spectral Topology of Algebra A

141

for zi+1 θ i+1 = 0; here, N is an arbitrary real and zi+1 θ i+1 is the projection of θ i+1 onto the unit vector zi+1 . (The vector zi+1 ∈ Rm−di is directed from the coordinate origin to a point in Zi+1 , while θ i+1 lies in the space spanned by zi+1 and by the space tangent to the stratum Zi+1 at the point zi+1 .) Neighborhoods of the points in Zj × R. A typical neighborhood U ((z0 , . . . , zj ; λ); ) is the union of sets (3.2.2) (3.2.3) for i = j , the product U ((z0 , . . . , zj ); Zj )×U (λ; R), and close representations obtained by localization along subchains. Neighborhoods of the points in S ∗ (C). Typical neighborhoods in  of the inner points of the manifold S ∗ (C) are the same as in the cospherical bundle S ∗ (M \ |T |). Consider the points (z0 , . . . , zj ; ϕ j , ξ ) ∈ Zj × (S m−dj −1 \ |T (s0 , . . . , sj )|) × S m−1 in the boundary of S ∗ (C). Let us write such a point in the form (z0 , . . . , zj ; ϕ j , J (z0 )(ξ 0, θ 0 )), where ξ i = J (z0 , . . . , zi+1 )(ξ i+1 , θ i+1 ), i = 0, . . . , j − 1; as before, θ i ∈ Rdi −di−1 , ξ j ∈ Rm−dj ; the Jacobi matrices J (z0 , . . . , zi+1 ) are defined by induction (see Sect. 3.1.2; there we have taken J (z0 ) = 1 and written J (z1 ) instead of J (z0 , z1 )). In what follows, the points (z0 , . . . , zj ; ϕ j , ξ ) are denoted by (z0 , . . . , zj ; ϕ j , ξ j , θ 0 , . . . , θ j ). From now on, we tacitly assume that a neighborhood, apart from explicitly listed representations, contains those of the same type obtained by localization along subchains and close to the listed representations. If θ 0 = 0, then a neighborhood in  of a point (z0 , . . . , zj ; ϕ j , ξ j , θ 0 , . . . , θ j ) is the union of the sets (3.2.1) for q = 0, . . . , j and the product U ((z0 , . . . , zi ); Zi ) × U ((ϕ j , ξ ); (S m−dj −1 \ |T (s0 , . . . , sj )|) × S m−1 )

.

(in such a case, the second factor contains no points with θ 0 = 0). Assume now that θ 0 , . . . , θ i = 0, θ i+1 = 0, 0 ≤ i ≤ 1. Then, a typical neighborhood consists of the set (3.2.2), the set (3.2.1) for q = i + 1, . . . , j , the set (3.2.3), and the product U ((z0 , . . . , zi ); Zi ) × {λ ∈ R : λ ≶ N }

.

in the case zi+1 θ i+1 ≶ 0 or U ((z0 , . . . , zi ); Zi ) × R

.

in the case zi+1 θ i+1 = 0. Finally, suppose that θ 0 = 0, . . . , θ j = 0. A neighborhood consists of the union of the sets (3.2.2) and (3.2.3) for i = j , the product U ((z0 , . . . , zi ); Zi ) × {λ ∈ R : λ ≶ N} in the case ϕ j ξ j ≶ 0, or U ((z0 , . . . , zi ); Zi ) × R in the case ϕ j ξ j = 0. The topology defined on  by the above neighborhoods is inseparable;  is a T0 -space (at least one of any two points has a neighborhood not containing the other of the points). Recall that the sets  and (A/KL2 (M))are connected by the bijection stated owing to Theorem 3.1.4).

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

Theorem 3.2.1 The topology on  coincides with the Jacobson topology on the spectrum (A/KL2 (M))∧ . The topology on the spectrum of A can be easily obtained from Theorem 3.2.1. Introduce a topology on  ∪ e. Point e will be taken as an open set. A neighborhood in  ∪ e of any point π ∈  is, by definition, a neighborhood π in  supplemented with e. Corollary 3.2.2 The topology on  ∪ e coincides with the Jacobson topology on the  of the algebra A. spectrum A

3.2.2

Proof of Theorem 3.2.1

Neighborhoods defining the topology on . were listed before Theorem 3.2.1. Moving along this list, we establish the coincidence of the topology on . (.-topology) with the Jacobson topology (J -topology) on the spectrum of .(A/KL2 (M))∧ . (The spectrum is parametrized by points of ..) 0 .1 . Connections between representations of the algebra .A. To verify Theorem 3.2.1, we need some relations between representations of .A. They can be seen from the localization procedure. For ease of reference, these relations are described in the following two lemmas. We use the notation of Theorem 3.1.4. Let .θ 0 ∈ Rd0 , .θ i ∈ Rdi −di−1 for .i ≥ 1. If .(θ 0 , . . . , θ j ) = 0, then we let .j = (θ 0 , . . . , θ j )/(|θ 1 |2 + · · · + |θ j |2 )1/2 . Lemma 3.2.3 (1) Suppose that .(θ 0 , . . . , θ j −1 ) = 0. Then for all .A ∈ A, the following inequality holds: π(z0 , . . . , zj ; j )A; BL2 (Rm−dj ) ≤ π(z0 , . . . , zj −1 ; j −1 )A; BL2 (Rm−dj −1 ).

.

(2) Suppose that .θ j ∈ Rdj −dj −1 , .j ≥ 1, and .|θ j | = 1. Then, for all .A ∈ A, π(z0 , . . . , zj ; 0, . . . , θ j )A; BL2 (Rm−dj )

.

≤ sup{π(z0 , . . . , zj −1 ; λ)A; BL2 (S m−dj −1 −1 ), λ ∈ R}. (3) Let .(z0 , . . . , zj ; ξ j , θ 0 , . . . , θ j ) = (z0 ; J (z0 , z1 )(ξ 1 , θ 1 ), θ 0 ), where ξ i = J (z0 , . . . , zi+1 )(ξ i+1 , θ i+1 )

.

for .i ≤ j − 1 (see explanation of notation before Theorem 3.2.1). Denote by I (z0 , . . . , zj ; .θ 0 , . . . , θ j ) the ideal generated in the algebra .π(z0 , . . . , zj ; θ 0 , . . . , θ j )

.

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143

A by the elements 1 j j 0 j Fξ−1 j →x (z , . . . , z ; ξ , θ , . . . , θ )Fy→ξ j ,

.

for which .(z0 , . . . , zj ; ξ j ; 0, . . . , 0) ≡ 0. Then, for all .A ∈ A and .λ ∈ R, π(z0 , . . . zj ; λ)A; BL2 (S m−dj −1 ) ≤ [π(z0 , . . . , zj ; θ 0 , . . . , θ j )A],

.

where the right-hand side is the norm of a residue class .[·] in the quotient algebra (π(z0 , . . . , zj ; θ 0 , . . . , θ j )A)/I (z0 , . . . , zj ; θ 0 , . . . , θ j ).

.

Proof The representations .π(z0 , . . . , zj ; j ) of .A arise in the localization procedure in the algebra .π(z0 , . . . , zj −1 ; j −1 )A. Any morphism of .C ∗ -algebras (considered as a morphism of involutive algebras) is continuous, and its norm does not exceed 1. This implies the first assertion of Lemma. While localizing in .S(z0 , . . . , zj −1 ), the representations .π(z0 , . . . , zj ; 0, . . . , θ j ) appear (see the description of the dynasty of local algebras in Sect. 3.1.3). Hence, the second assertion of Lemma is valid. For verification of the third assertion, one can reason in the same way as in the proof of Proposition 3.1.6 in case (iv).   As in the list of neighborhoods before Theorem 3.2.1, the one-dimensional representations .π(z0 , . . . , zj ; ϕ j , ξ ) will also be denoted by .π(z0 , . . . , zj ; ϕ j , ξ j , θ 0 , . . . , θ j )). Lemma 3.2.4 For all .A ∈ A, the following inequalities hold: |π(z0 , . . . , zj ; ϕ j , ξ j , j )A| ≤ π(z0 , . . . , zj ; j )A; BL2 (Rm−dj ),

.

(3.2.4)

where, as before, .j = (θ 0 , . . . , θ j ), .|j | = 1; |π(z0 , . . . , zj ; ϕ j , ξ j , 0, . . . , 0)A| ≤ sup{π(z0 , . . . , zj ; λ)A; BL2 (S m−dj −1 ), λ ∈ R}. (3.2.5)

.

Moreover, |π(z0 , . . . , zj ; ϕ j , ξ j , 0, . . . , 0)A| ≤ π(z0 , . . . , zj ; λ)A; BL2 (S m−dj −1 ) (3.2.6)

.

for any .λ ∈ R provided .ϕ j ξ j = 0.

144

3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

Proof The representations .π(z0 , . . . , zj ; ϕ j , ξ j , j ) of .A appear in the localization procedure as representations of the algebra .π(z0 , . . . , zj ; ϕ j , ξ j , j )A (see the listing of one-dimensional representations of .A in the proof of Theorem 3.1.4); this implies inequalities (3.2.4). The representations .π(z0 , . . . , zj ; ϕ j , ξ j , 0, . . . , 0) appear as representations of .S(z0 , . . . , zj ); thus, the formulas (3.2.5) are valid. Finally, we notice that if .ϕ j ξ j = 0, then the representations .π(z0 , . . . , zj ; ϕ j , ξ j , 0, . . . , λ) are also representations of 0 j 0 j .π(z , . . . , z ; λ)S(z , .. . . , z ) for any .λ ∈ R. Indeed, from Theorem 2.2.12, it follows 0 j that .π(z , . . . , z ; λ) is a representation of the ideal in .π(z0 , . . . , zj ; λ)S(z0 , . . . , zj ) spanned by the coefficients .a(z0 , .. . . , zj ; ·) vanishing on the set of all singularities of all the remaining coefficients .a(z0 , . . . , zj ; ·). Each representation of an ideal of any .C ∗ algebra extends uniquely to an irreducible representation of the whole algebra. Therefore,   inequality (3.2.6) holds as well. 20 . Neighborhoods of points in .S ∗ (Zj ). Assume that .s0 , . . . , sk is a chain of strata of the partition .T , i.e., .si ∈ st(sj ) for .i > j , .dim sj = dj , and .dk ≤ m − 1, where .m = dim M. .

Lemma 3.2.5 Let .(z00 , . . . , z0 ; ψ0 ) be an arbitrary point in the product .Zj × (S m−dj −1 \ |T (s0 , . . . , sj )|); here, as before, .Zj = Z0 × · · · × Zj , .0 ≤ j ≤ k. Then, there exists a function .a ∈ M(T ) subject to the following conditions: j

j

j

j

1. .a(z00 , . . . , z0 ; ψ0 ) = 1, .a(z0 , . . . , zj ; ψ j ) = 0 outside a small neighborhood of the j j point .(z00 , . . . , z0 ; ψ0 ) on .Zj × (S m−dj −1 \ |T (s0 , . . . , sj )|). q q+1 2. .a(z00 , . . . , z0 ; ·) = 0 outside a neighborhood of .z0 on .S m−dq −1 for .0 ≤ q ≤ j − 1, 0 and .a(·) = 0 outside a neighborhood of .z0 on .M. 3. .a(z0 , . . . , zj ; ·) ∈ C(S m−dj −1 ), and .a(z0 , . . . , zp ; ·) ≡ 0 for .j + 1 ≤ p ≤ k. Proof Let .χ ∈ C(S m−dj −1 ) be such that .χ (ψ0 ) = 1 and .χ = 0 outside a neighborhood j of .ψ0 . We regard .(zj , x) ∈ Zj × Rm−dj −1 as cylindrical coordinates in a tubular neighborhood of the stratum .Zj on .S m−dj −1 −1 , and we introduce the function .(zj , x) → j fj (zj , x) = χ (x/|x|). Let .χj ∈ C(S m−dj −1 −1 ), .χj (z0 ) = 1, and .χj = 0 outside a j neighborhood of .z0 on .S m−dj −1 −1 . Extending .χj fj by zero outside the support of .χj , we obtain a function denned on .S m−dj −1 −1 \ |T (s0 , . . . , sj −1 )|. We give the roles of .χ and .S m−dj −1 −1 to the function .χj fj and the sphere m−d m−dj −2 −1 \ j −2 −1 . Repeating the argument, we define the function .χ .S j −1 fj −1 on .S |T (s0 , . . . , sj −2 )|. Continuing this process, we arrive at an element .a = χ0 f0 ∈ M(T ). It follows from the construction that the function a satisfies conditions 1 and 2. Condition 3 is an obvious consequence of 1.   j

j

j

Fix an arbitrary point .(z00 , . . . , z0 ; θ00 , . . . , θ0 ), .0 ≤ j ≤ k. We consider several cases separately.

3.2 The Spectral Topology of Algebra A

145

(a) The case .θ00 = 0. We show that every neighborhood .U (π0 ; ) of the point .π0 = j j π(z00 , . . . , z0 ; θ00 , . . . , θ0 ) in hte .-topology contains some neighborhood .U (π0 ; J ) of this  : π A > point in the J -topology. For an arbitrary element .A ∈ A, the set .UA = {π ∈ A 1} is open in the J -topology. Our goal is to determine an A such that .π0 ∈ UA ⊂ U (π0 ; ). As in Lemma 3.2.3, let (z0 , . . . , zj ; ξ j , θ 0 , . . . , θ j ) = (z0 ; J (z0 , z1 )(ξ 1 , θ 1 ), θ 0 ),

(3.2.7)

.

where .ξ i = J (z0 , . . . , zi+1 )(ξ i+1 , θ i+1 ) for .i ≤ j − 1. We choose a zero degree homogeneous function .(z0 ; ·) on the fiber .T ∗ (M)z0 \ 0 of nonzero cotangent vectors j j j such that .(z00 , . . . , z0 ; ξ0 , θ00 , . . . , θ0 ) > 1 and . = 0 outside a small neighborhood j j of the point .ζ0 /|ζ0 | on .S ∗ (M)z0 ; here .ζ 0 = (ξ01 , θ00 , . . . , θ0 ) and .ξ0 is an arbitrary point in .S m−dj −1 ⊂ S ∗ (M)z0 . We assume that this neighborhood does not intersect the set of ¯ 0 (M) with principal points such that .θ 0 = 0. Furthermore, let P be a .ψdo in the algebra . 0 symbol coinciding with the previously chosen function .(z ; ·) on the fiber .T ∗ (M)z0 \ 0. Finally, suppose that .a ∈ M(T , M) is the function in Lemma 3.2.5, and let .A = aP . Then, for some .ξ0 ∈ S ∗ (M)z0 , the relation j

j

j

j

j

j

j

π(z00 , . . . , z0 ; ψ 0 , ξ0 )A = a(z00 , . . . , z0 ; ψ0 )(z00 , . . . , z0 ; ξ0 , θ0 , . . . , θ0 ) > 1

.

j

j

j

j

holds. We write .π(z00 , . . . , z0 ; ψ 0 , ξ0 ) as .π(z00 , . . . , z0 ; ψ 0 , ξ0 , θ00 , . . . , θ0 ) and use the inequality (3.2.4), obtaining the inclusion .π0 ∈ UA . The representation .π(z0 , . . . , zq ; q ) annihilates the operator A if .(z0 , . . . , zq ; q ) does not belong to small neighborhood q q p 0 0 p .U ((z , . . . , z ;  ); ), .q = 0, . . . , j (see (3.2.1). The representations .π(z , . . . , z ;  ) 0 0 0 0 with .p ≥ j + 1 do not fall in .UA because of condition 3, which is satisfied by function a in Lemma 3.2.5. The set .UA does not contain representations of the type .π(z0 , . . . , zq ; λ), .1 ≤ q ≤ k. Indeed, there are no points in the support of . for which .θ 0 = 0. This means that all the representations of the indicated type annihilate the operator A and hence cannot belong to .UA . Thus, only those representations being considered also fall in .UA that are in .U (π0 ; ). The infinite-dimensional representations in .UA obtained by localization along subchains are contained also in the neighborhood .U (π0 ; ); as is easy to see, this is ensured by the choice of the operator A (more precisely, by the choice of function a). It remains to consider the one-dimensional representations. Assume that there exists an  : π B > 1} does not contain oneelement .B ∈ A such that the set .UB = {π ∈ A dimensional representations and .π0 ∈ UB . Then, .UA ∩ UB is a neighborhood of .π0 in the J -topology and .Ua ∩ UB ⊂ U (π0 ; ). We determine the necessary element B. Suppose that a is the function in Lemma 3.2.5, the function . is defined in (3.2.7), and A(λ) = Eω→ϕ (λ)(z0 , . . . , zj ; ω, 0 . . . , 0)Eψ→ω (λ) : L2 (S m−dj −1 ) → L2 (S m−dj −1 ),

.

146

3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . . j

where .ω = ξj /ξ j . This time, we choose . so that the commutator .[a(z00 , . . . , z0 ; ·), A(λ)] does not vanish identically on the axis .−∞ < λ < +∞. (This possibility is ¯ 0 (M) easily seen, for example, from Proposition 1.2.11.) Assume that P is a .ψdo in . with principal symbol coinciding on the fiber .T ∗ (M)z0 \ 0 with the chosen function j 0 0 .(z ; ·). Let .B = [a, P ] ∈ A. Then, .π(z , . . . , z ; λ)B = 0 for some .λ ∈ R; it can 0 0 j be assumed that .π(z00 , . . . , z0 ; λ)B > 1. According to assertion 3 in Lemma 3.2.3), we j have .π0 B ≥ π(z00 , . . . , z0 ; B > 1 and .π0 ∈ UB . Obviously, each one-dimensional representation annihilates the operator B and thus is not contained in .UB . Therefore, an arbitrary neighborhood .U (π0 ; ) of .π0 contains some neighborhood of this point in the J -topology. We now prove that any neighborhood .U (π0 ; J ) also contains some neighborhood of 0  : π A > 1}, where .A = .π in the .-topology. Let .U (π0 ; J ) = UA = {π ∈ A   l k Alk , and .Alk are generators of the algebra .A. Iterating the inequalities in part 1 of q Lemma 3.2.3, we arrive at the estimates .π(z00 , . . . , z0 ; q )A > 1, where .q = 0, . . . , j − 1. The functions .(z0 , . . . , zq ; q ) → π(z0 , . . . , zq ; q )A are continuous. Therefore, the indicated inequalities remain valid for all representations .π in a small neighborhood .U (π0 ; ), i.e., .U (π0 ; ) ⊂ U (π0 ; J ). (b) The case .θ00 = 0, .θ01 = 0. We show first that every neighborhood .U (π0 ; J ) contains the half-line .{λ ∈ R : λ ≷ N}, if .z01 θ01 ≷ 0, or the whole line, if .z01 θ01 = 0. For that, we need several lemmas. Suppose that .T (S n−1 ) is an admissible partition of the sphere .S n−1, .aj k ∈ M(T , S n−1 ), the functions .j k are zero degree homogeneous on .Rn , .j k ∈ C ∞ (S n−1 ), and −1 Aj k (λ) = Eω→ϕ (λ)j k (ω)Eψ→ω (λ), Dj k (λ) = Fη→x j k (η, λ)Fy→η .

.

We will assume that the point .z01 of stratum .s 1 ∈ T (S n−1 ) coincides with the north pole  .(0 , 1). Lemma 3.2.6 Assume that j and k run through finite sets. Then, lim λ→±∞ 



.

j

aj k Aj k (λ); BL2 (S n−1 )≥

k

 j

aj k (z01 ; ·)Dj k (±1); BL2 (Rn−1 ).

k

(3.2.8) n−1 Proof Suppose that the functions .ζj k and .ηj k belong to the class .C ∞ (S+ ) and are equal to 1 in a neighborhood of the pole .(0 , 1). Furthermore, let .A(λ) =   j k ζj k aj k Aj k (λ)ηj k . In the proof of Proposition 2.2.10, inequality (2.2.42) was obtained:

lim t→+∞ A(λt); BL2 (S n−1 ) ≥ 



.

j

k

aj k (z01 ; ·)Dj k (λ); BL2 (Rn−1 )

(3.2.9)

3.2 The Spectral Topology of Algebra A

147

with .λ ∈ R. We set .λ = ±1 in (3.2.9) and arrive at lim λ→±∞ A(λ); BL2 (S n−1 ) ≥ 



.

j

aj k (z01 ; ·)Dj k (±1); BL2 (Rn−1 ).

k

(3.2.10) It remains to get rid of the cut-off functions .ζj k , .ηj k appearing in .A(λ). According to Proposition 1.2.11, the commutators .[Aj k , σ ] belong to the algebra .C0 (R) ⊗ KL2 (s n−1 ) for .σ ∈ S n−1 . Therefore, A(λ)χ =

 

.

j

k

aj k Aj k (λ)



ζj k ηj k χ + T (λ)χ = B(λ)χ + T (λ)χ ,

k

(3.2.11)   n−1 where .T ∈ C0 (R) ⊗ KL2 (s n−1 ), .B(λ) = j k aj k Aj k (λ), .χ ∈ Cc (S+ ), while  .0 ≤ χ ≤ 1, .χ = 1 near the pole .(0 , 1), .χ ζj k = χ , .χ ηj k = χ . The operator .B(λ)χ differs from .A(λ) only in the form of the cut-off functions. Consequently, inequality (3.2.10) holds with .A(λ) replaced by .A(λ)χ . By (3.2.11), we have .A(λ)χ  ≤ B(λ) + T (λ) and .T (λ) → 0 as .λ → ∞, which implies (3.2.8).   Let .(x = (x (1) , x (2) ) ∈ Rk , .x (1) = (x1 , . . . , xl ) ∈ Rl , .x (2) = (xl+1 , . . . , xk ), and .l < k. We consider the algebras .D± generated on .L2 (Rk ) by the convolution −1 (η, ±1)F operators .Fη→x y→η and the operators of multiplication by the functions .x → (1) (1) a(x /|x |); here, . are zero degree homogeneous functions on .Rk+1 \ 0 and a are elements of some algebra of functions on .S l−1 (for example, of type .M(T , S l−1 )). We also k−l  θ → A(θ ) : L2 (Rl ) → introduce the algebras .D± of operator-valued functions .S± L2 (Rl ), where .A(θ ) = Fζ−1 →t (ζ, θ )Fs→ζ or .A(θ ) ≡ a(·)I (the operator of multiplication l = {θ = (s, σ ) : s ∈ Rl , σ ∈ R, |s|2 + σ 2 = 1, σ ≷ 0}. by the function .s → a(s/|s|)), .S± l }. The algebra .D± is equipped with the norm .A; D = sup{A(θ ); BL2 (Rl ), θ ∈ S± Lemma 3.2.7 The algebra .D± is isomorphic to the algebra .D± . Proof Applying the Fourier transform .Fx (2) →ζ to the elements .A ∈ D± , we obtain that .D± is isomorphic to the algebra of operator-valued functions Rk−l  ζ → A± (ζ ) : L2 (Rl ) → L2 (Rl ),

.

148

3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

−1 (η, ζ, ±1)F where .A± (ζ ) = Fη→x or .A± (ζ ) ≡ a(·)I , with norm .sup{A± (ζ );  y→η l k−l 2 BL2 (R ), .ζ ∈ R }. Let .t = |ζ | + 1 and define a unitary operator .(Ut u)(x) = t l/2 u(tx) in .L2 (Rl ). We have   l/2 .A± (ζ )Ut u = t (2π )−l eixη (η, ζ, ±1) dη e−iηy u(ty)dy =

= t l/2 (2π )−l



 eixξ t (ξ t, θ± ) dξ

e−iξ z u(z) dz = Ut A(θ± )u,

 l . Moreover, the operator .U commutes with where .θ± = (ζ, ±1)/ |ζ |2 + 1 ∈ S± t operators of multiplication by functions .Rl \ 0  x → a(x/|x|). All this leads to an   isomorphism .D±  D± . j

Proposition 3.2.8 Each neighborhood .U (π0 ; J ) of the point .π0 = π(z00 , z01 , . . . , z0 ; 0, θ01 , j 1 1 .. . . , θ ) in the J -topology (Jacobson topology) with .z θ ≷ 0 contains a set of the form 0 0 0 0 0 0 .{π(z ; λ)}, where .(z , λ) runs through a set of the form .U (z ; Z0 ) × {λ ∈ R : λ ≷ N }, 0 0 whereas .U (z0 ; Z0 ) is a small neighborhood, and N is a sufficiently large number (see formulas (3.2.2), (3.2.3) and the accompanying text).   Proof It can be assumed that .U (π0 ; J ) = UA = {π : π A > 1}, where .A = l k Alk , ¯ 0 (M) with principal symbol .lk (z0 ; ·). .Alk = alk Plk , and .Plk denotes a .ψdo in . 0 According to part 1 of Lemma 3.2.3, the inequality .π0 A > 1 implies the estimate π(z00 , z01 ; 0, θ01 /|θ01 |)A > 1.

.

Recalling the definition of .π(z00 , z01 ; 0, θ01 /|θ01 |) (see (3.1.6) and (3.1.4)), we now deduce from Lemma 3.2.7 that  . alk (z00 , z01 ; ·)Dlk (±1); BL2 (Rm−d0 −1 ) > 1 l

k

in the case .z01 θ01 ≷ 0. [We assume that .z01 coincides with the north pole. A point .θ 1 with positive (negative) last coordinate appears as .θ+ (.θ− ) in Lemma 3.2.7. The symbols .lk of the operators .Dlk (λ) are determined by the equalities .lk (η, λ) = lk (z00 ; η, λ, θ 0 )|θ 0 =0 .]   The proof is concluded by using Lemma 3.2.6. Let us proceed to the case .z01 θ01 = 0. Instead of Lemma 3.2.6, we use: Lemma 3.2.9 In the notation of Lemma 3.2.6, the inequality   . alk Alk (λ); BL2 (S n−1 ) ≥  alk (z01 ; ·)Dlk (0); BL2 (Rn−1 ) l

k

holds for all .λ ∈ R.

l

k

3.2 The Spectral Topology of Algebra A

149

Proof Passing in (2.2.41) to the limit as .t → +∞, we obtain A(λ); BL2 (S n−1 ) ≥ 



.

l

alk (z01 ; ·)Dlk (0); BL2 (Rn−1 ) − ε.

(3.2.12)

k

The purpose now is to eliminate the cut-off functions .ζlk and .ηlk in the operator .A(λ). n−1 ), .0 ≤ χq ≤ 1, equal to 1 near We choose a sequence of functions .χq ∈ Cc∞ (S+   the pole .(0 , 1) and with supports shrinking to .(0 , 1). The inequality (3.2.12) holds for each operator .A(λ)χq . Suppose that .{uq } ⊂ L2 (S n−1 ), .uq  = 1, and .A(λ)χq uq  ≥ A(λ)χq ; BL2 (S n−1  − ε. This and (3.2.11) imply that B(λ)χq uq  ≥ A(λ)χq uq  − T (λ)χq uq  ≥

.

≥ A(λ)χq ; BL2 S

n−1

(3.2.13)

) − T (λ)χq uq  − ε.

It is clear that the sequence .{χq uq } converges weakly to zero. Since .T (λ) is a compact operator on .L2 (S n−1 ), we have .T (λ)χq uq  → 0 as .q → ∞. Therefore, by (3.2.12) and (3.2.13), B(λ); BL2 (S n−1 ) ≥ B(λ)χq uq  ≥ 

.



alk (z01 ; ·)Dlk (0); BL2 (Rn−1 ) − 2ε.  

Proposition 3.2.10 Let .z01 θ01 = 0. Then, every neighborhood .U (π0 ; J ) of .π0 = j j π(z00 , z01 , . . . , .z0 ; 0, θ01 , . . . , θ0 ) in the J -topology contains some set .{π(z0 , λ)}, where 0 0 .z runs through a small enough neighborhood .U (z ; Z0 ) and .λ ∈ R. 0 Proof As in the proof of Proposition 3.2.8, suppose that .U (π0 ; J ) = Ua and A has the previous meaning. Applying Theorem 3.1.4 to the algebra generated on .L2 (Rm−d0 −1 ) by −1 (η, 0)F the operators of the form .a(z00 , z01 ; ·)D(0), where .D(0) = Fη→x y→η , we get that .π0 is a representation of this algebra. (Formally speaking, we use not Theorem 3.1.4 itself, but a result obtained in its proof.) Therefore, the inequality .π0 A > 1 implies 



.

l

alk (z00 , z01 ; ·)Dlk (0); BL2 (Rm−d0 −1 ) > 1.

k

This and Lemma 3.2.9 imply that .π(z00 ; λ)A; BL2 (S m−d0 −1 ) > 1.

 

Thus, in Propositions 3.2.8 and 3.2.10, we have determined which representations j j π(z0 ; λ) fall in the neighborhood .U (π0 ; J ) of .π0 = π(z00 , z01 , . . . , z0 ; 0, θ01 , . . . , θ0 ) with 1 0 0 .θ = 0. We now turn to the representations .π(z , θ ). 0 .

150

3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

Proposition 3.2.11 Each neighborhood .U (π0 ; J ) contains some set .{π(z0 ; θ 0 )}, where 0 0 0 .z varies in a small enough neighborhood .U (z ; Z0 ) and .θ does in the whole sphere 0 ∗ .S (Z0 )z0 . Proof As before, assume that .U (π0 ; J ) = UA , where A is a finite combination of generators of .A. If .π0 A > 1, then there exist .λ ∈ R such that .π(z00 ; λ)A > 1 (Propositions 3.2.8 and 3.2.10). According to part 3 of Lemma 3.2.3, the inequality 0 0 0 0 ∈ S ∗ (Z ) . It remains .π(z ; λ)A > 1 implies that .π(z ; θ )A > 1 for each .θ 0 z0 0 0 to use the continuity of the function .S ∗ (Z0 )  (z0 ; θ 0 ) → π(z00 ; θ 0 )A.

0

 

We turn to a comparison of the J - and .-topologies. Proposition 3.2.12 For each neighborhood .U (π0 ; ), there exists a neighborhood U (π0 ; J ) belonging to it.

.

Proof Suppose that the functions ., . and the operator A are the same as in part a) (i.e., in the case .θ00 = 0), but now the support of . does not intersect the set of q q points for which .θ 1 = 0. Then, .π(z0 , . . . , z0 ; 0 )A > 1 for .q = 0, . . . , j − 1. As before, the neighborhood .UA = {π : π A > 1} does not contain representations q 0 q .π(z0 , . . . , z ; θ , . . . , θ ), .q ≥ 1, that are not in .U (π0 ; ). It is also clear that .UA does not contain representations of the type .π(z0 , . . . , zq ; λ), .1 ≤ q ≤ k; these representations annihilate A, since . = 0 for .θ 1 = 0. The one-dimensional representations are excluded for the same reason as in part a) (i.e., .UA is replaced by the intersection .U = UA ∩ UB , which now does not contain one-dimensional representations). According to Proposition 3.2.11, the neighborhood U contains the set .{π(z0 ; θ 0 ) : z0 ∈ U (z00 , Z0 ), θ 0 ∈ S ∗ (Z0 )z0 }; the same set also appears in .U (π0 ; ). If .z01 θ01 = 0, then 0 0 0 .{π(z ; λ) : z ∈ U (z ; Z0 ), λ ∈ R} is a subset of both .U (π0 ; ) and U . 0 Suppose finally that .z01 θ01 ≷ 0 and .U (π0 ; ) ⊃ {π(z0 , λ) : z0 ∈ U (z00 ; Z0 ), λ ≷ N }.  : π P  > 1} and Below, we determine an operator .P ∈ A such that .π0 ∈ UP = {π ∈ A UP ∩ {π(z0 ; λ) : z0 ∈ U (z00 ; Z0 ), λ ∈ R} ⊂ {π(z0 , λ) : z0 ∈ U (z00 ; Z0 ), λ ≷ N}.

.

The set .U ∩ UP is open in the J -topology and contains only those .π(z0 , λ) belonging to .U (π0 ; ). By the choice of A, .U ∩ UP does not contain representations not in .U (π0 ; ). Thus, .π0 ∈ U ∩ UP ⊂ U (π0 ; ). It remains to produce the operator P . Let . and let . be defined just as in part 3 of Lemma 3.2.3 (these are new functions, not j connected with those used above for the operator A). Choose an arbitrary .ξ0 and denote j j j by .ξ0 ∈ S m−1 a point for which .(z00 ; ξ0 ) = (z00 , z01 , . . . , z0 ; ξ0 , 0, θ01 , . . . , θ0 ). We assume that .(z00 ; ξ0 ) > 1 and .(z0 ; ξ ) = 0 for .ξ outside a small neighborhood of .ξ0 on .S m−1 or for .z0 outside a small neighborhood of .z00 on .M. For .m > n, the inclusion m−d0 −1 ⊂ S m−1 is valid, where .S m−d0 −1 = {η ∈ Rm : |η| = 1, η .ξ0 ∈ S m−d0 +1 = · · · =

3.2 The Spectral Topology of Algebra A

151

ηm = 0}. Furthermore, let .a ∈ C(S m−d0 −1 ) be such that .a(z01 ) ≥ 1 and .a = 0 outside a small neighborhood of .z01 on .S m−d0 −1 . We introduce the operator .P = aQ, where Q ¯ 0 (M) with principal symbol .. Let us ensure that P has the necessary is a .ψdo in . properties. Since j

j

j

j

j

j

j

π(z00 , z01 , . . . , z0 ; ϕ0 , ξ0 , θ01 , . . . , θ0 )P = a(z01 )(z00 , z01 , . . . , z0 ; ξ0 , 0, θ01 , . . . , θ0 ) > 1,

.

the inequality .π0 P  > 1 follows from (3.2.4), i.e., .π0 ∈ UP . By the inclusion .S m−d0 −1 ⊂ S m−1 , we can regard .z01 as a point in .Rm . Recalling the definition of .ξ0 , we obtain the equality .z01 ξ0 = z01 θ01 . Suppose for definiteness that.z01 θ01 > 0. Since the supports of a and 0 . are small, the function .(ϕ, ω) → a(ϕ)(z ; ω, 0) is equal to zero on the set 0 {(ϕ, ω) ∈ S n−1 × S n−1 : ϕω ≤ 0}.

.

Therefore, Lemma 5.4.8 in [21] leads to .

lim π(z00 ; λ)P ; BL2 (S n−1 ) = 0.

λ→−∞

(3.2.14)

(An analogous relation was used in the proof of Theorem 2.2.13, where in particular, the mentioned lemma from [21] was formulated.) If the inclusion .{λ : π(z00 ; λ)P  > 1} ⊂ {λ ∈ R : λ > N} holds, then P possesses all the needed properties. However, if the latter inclusion does not hold, then P must be replaced by the new operator .P  described below. Since the function .(ϕ, ω) → a(ϕ)(z00 ; ω, 0) is equal to zero for .ϕω = 0, it follows that .π(z00 ; λ)P ∈ KL2 (S n−1 ) for all .λ ∈ R (Theorem 2.2.12). Let .χ ∈ C(R) satisfy the conditions .0 ≤ χ ≤ 1, .χ (λ) = 1 for .λ < N, and .χ (λ) = 0 for 0 n−1 ). .λ > N + 1. Then, it follows from (3.2.14) that .χ (·)π(z ; ·)P ∈ C0 (R) ⊗ KL2 (S 0 Assume that .Lcont is an algebra generated on .L2 (Rm ) by the operators of the form −1 (1) (1) (1) (2) (1) = (x , . . . , x ), .a(ϕ)F 1 n ξ →x (ξ )Fy →ξ , where .ϕ = x /|x |, .x = (x , x ), .x (2) = (x .x , . . . , x ), and . n = m − d − 1; the functions a and .  are continuous n+1 m 0 on spheres .S n−1 and .S m−1 , respectively. Let also .Scont be an algebra spanned by the operator-valued functions .R  λ → B(λ) = a(ϕ)Eω→ϕ (λ)−1 (ω)Eψ→ω (λ) with the norm .sup{B(λ); BL2 (S n−1 ), λ ∈ R}. We introduce a mapping p on the generators of .Lcont by p(aF −1 (ξ )F ) = a(ϕ)Eω→ϕ (·)−1 (ω, 0)Eψ→ω (·) ∈ Scont ,

.

where .ω = ξ (1) /|ξ (1) |, .ξ (1) = (ξ1 , . . . , ξn ). It can be shown that p extends to an epimorphism .p : Lcont → Scont (see [25], Lemma 2.2, and Proposition 2.3). Therefore, since .P ∈ Lcont , there exists an element .P  ∈ Lcont satisfying .p(P  ) = (1 − χ (·)π(·)P . We verify that .P  is the desired operator. The inclusion .{λ : π(λ)P   > 1} ⊂ {λ ∈ R : λ > N} obviously follows from the definition of .P  . The restriction of the

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . . j

j

j

j

representation .π(z00 , . . . , z0 ; ϕ0 , ξ0 , 0, θ01 , . . . , θ0 ) to .Lcont can be written in the form 1 n−1 × S n−1 . This one-dimensional representation of .L .π(ϕ0 , ξ0 ), .ϕ0 = z , .(ϕ0 , ξ0 ) ∈ S cont 0 is also a representation of the quotient algebra .Scont /C0 (R) ⊗ KL2 (S ( n − 1)). Therefore, π(z00 , . . . , z0 ; ϕ0 , ξ0 , 0, θ01 , . . . , θ0 )P  = π(ϕ0 , ξ0 )P  = π(ϕ0 , ξ0 )P j

.

j

j

j

j

j

j

j

= π(z00 , . . . , z0 ; ϕ0 , ξ0 , 0, θ01 , . . . , θ0 )P , i.e., the operator .P  inherits the necessary properties of P .

 

Proposition 3.2.13 Every neighborhood .U (π0 ; J ) contains some neighborhood .U (π0 ; ). Proof According to Propositions 3.2.8 and 3.2.10, every neighborhood .U (π0 ; J ) contains some set of the form .U (z00 ; Z0 ) × {λ ∈ R : λ ≷ N } if .z01 θ01 > 0 or of the form 0 1 1 .U (z ; Z0 ) × R if .z θ 0 0 0 = 0. Proposition 3.2.11 ensures that .U (π0 ; J ) is a subset of 0 0 0 0 0 ∗ .{(z ; θ ) : z ∈ U (z ; Z0 ), θ ∈ S (Z0 )z0 }. The sets of the form (3.2.1) fall in .U (π0 ; J ) 0 in view of assertion 1 in Lemma 3.2.3. It remains to recall the definition of .U (π0 ; ) (see   the list of neighborhoods before Theorem 3.2.1). (c) The case .θ00 = 0, . . . , θ0i = 0, .θ i+1 = 0, .i ≥ 1. Here, we confine ourselves to a verification that every neighborhood .U (π0 ; J ) contains the sets of all those types that occur in .U (π0 ; ) (according to the list of neighborhoods). The formal proofs of the analogues of Propositions 3.2.12 and 3.2.13 are left to the reader. j j Now, let .π0 = π(z00 , z01 , . . . , z0 ; 0, . . . , 0, θ i+1 , . . . , θ0 ), .θ i+1 = 0. It can be assumed    : π A > 1}, where .A = that .U (π0 ; J ) = UA = {π ∈ A l k Alk and .Alk are generators of .A. Proposition 3.2.14 The set .UA contains products of the form (3.2.1), where q q (z0 , . . . , zq ) = (z00 , .. . . , z0 ), .q = 0 , .q = i + 1, . . . , j , and neighborhoods q q 0 d −1 .U ((z , . . . , z ); Zq ) and .U ( ; .S q ) (the factors in these products) are sufficiently 0 0 0 small. .

Proof Assertion 1 of Lemma 3.2.3 implies the inequalities j −1

π0 A ≤ π(z00 , . . . , z0

.

j −1

; 0

)A ≤ · · · ≤ π(z00 , . . . , z0i+1 ; i+1 0 )A.

(3.2.15)

Therefore, the estimate .π0 A > 1 and the continuity of the functions .Zq × S dq −1  (z0 , . . . , zq ; q ) → π(z0 , . . . , zq ; q )A lead to the relations .π(z0 , . . . , zq ; q )A > 1 for all (z0 , . . . , zq ; q ) ∈ U (z00 , . . . , z0 ; Zq ) × U (0 ; S dq −1 )

.

with .q = i + 1, . . . , j .

q

q

 

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153

Proposition 3.2.15 The set .UA contains products of the form .U (z00 , . . . , z0 ; Zp ) × S dp −1 , p where .p = 1, . . . , i and .U (z00 , . . . , z0 ; Zp ) is a sufficiently small neighborhood. p

0 i+1 ; 0, . . . , 0, θ i+1 /|θ i+1 |), it follows Proof Since .π(z00 , . . . , z0i+1 ; i+1 0 ) = π(z0 , . . . , z 0 0 from (3.2.15) and part 2 of Lemma 3.2.3 that .π(z0 , . . . , z0i ; λ∗i )A > 1 for some .λ∗i ∈ R. Then in view of part 3 of the same lemma

π(z00 , . . . , z0i ; θ 0 , . . . , θ i )A > 1

.

(3.2.16)

for all .θ 0 , . . . , θ i , .|θ 0 |2 + · · · + |θ i |2 = 1. This means that .UA contains a set of the form .U (z00 , . . . , z0i ; Zi ) × S di −1 . Formula (3.2.16) gives, in particular, the estimate i 0 i i .π(z , . . . , z ; .0, . . . , 0, θ )A > 1 for .|θ | = 1 and thereby enables us to repeat the 0 0 argument just used with i replaced by .i − 1. This procedure continues to the end of the proof.   Proposition 3.2.16 (1) .UA contains the products .U (z00 , . . . , z0r ; Zr ) × R (i.e., the representations 0 r 0 r 0 r .π(z , . . . , z ; λr ), where .(z , . . . , z ) ∈ U (z , . . . , z ; Zr ) and .λr ∈ R), whereas 0 0 .r = 1, . . . , i − 1. (2) .UA contains the set .U (z00 , . . . , z0i ; Zi ) × {λi ∈ R : λi ≷ N} if .z0i+1 θ0i+1 ≷ 0 or the set i+1 i+1 i 0 .U (z , . . . , z ; Zi ) × R} if .z = 0. 0 0 0 θ0 Proof (1) We consider the algebra .S(z00 , . . . , z0i−1 ) by the operator-valued functions .R  λi−1 → π(z00 , . . . , z0i−1 ; λi−1 )B : L2 (S m−di−1 −1 ) → L2 (S m−di−1 −1 ), where .B ∈ A. The norm in .S(z00 , . . . , z0i−1 ) is defined by π(z00 , . . . , z0i−1 ; ·)B = sup{π(z00 , . . . , z0i−1 ; λ)B, λ ∈ R}.

.

This algebra is generated by the functions of the form λi−1 → a(z00 , . . . , z0i−1 ; ·)E(λi−1 )−1 (ω)E(λi−1 )

.

with . ∈ C(S m−di−1 −1 ). We assume that the point .z0i coincides with the north pole of the sphere .S m−di−1 −1 . Lemma 3.2.9 gives the inequality 



.

l

≥

alk (z00 , . . . , z0i−1 ; ·)Alk (λi−1 ); BL2 (S m−di−1 −1 ) ≥

k

 l

k

alk (z00 , . . . , z0i−1 , z0i ; ·)Dlk (0); BL2 (Rm−di−1 −1 ),

(3.2.17)

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

for all .λi−1 ∈ R, where this time .Alk (λ) = E(λ)−1 lk (ω)E(λ), .Dlk (0) = −1  (η, 0)F m−di−1 −1 \0. Fη→x lk y→η , and .lk is a homogeneous function of zero degree on .R The algebra generated by the operators of the form a(z00 , . . . , z0i ; ·)F −1 (η, 0)F : L2 (Rm−di−1−1 ) → L2 (Rm−di−1−1 ),

.

is isomorphic to the algebra .S(z00 , . . . , z0i ) of operator-valued functions .R  λi → π(z00 , . . . , z0i ; .λi )B. This and (3.2.17) give us that π(z00 , . . . , z0i−1 ; λi−1 )A ≥ sup{π(z00 , . . . , z0i ; λi )A, λi ∈ R}.

.

(3.2.18)

In verifying Proposition 3.2.15, from the estimate .π0 A > 1, we derived the inequality .π(z00 , . . . , z0i ; λ∗i )A > 1 for some .λ∗i ∈ R. Using (3.2.18), we obtain i−1 0 .π(z , . . . , z 0 0 ; λi−1 )A > 1 for all .λi−1 ∈ R. This means that .UA contains the set i−1 0 .U (z , . . . , z 0 0 ; Zi−1 ) × R. The argument can be repeated with i replaced by .i − 1 and with an arbitrary real number chosen as .λ∗i−1 . As a result, we obtain the inclusion i−2 0 .U (z , . . . , z 0 0 ; Zi−2 ) × R ⊂ UA and so on. (2) In the first part of the proof, Lemma 3.2.9 was applied to the elements of the algebra .S(z00 , . . . , z0i−1 ) to derive (3.2.17). We replace .S(z00 , . . . , z0i−1 ) by the algebra i−1 i 0 .S(z , . . . , z 0 , .z0 ) and use Lemma 3.2.6 instead of Lemma 3.2.9. Then, we obtain 0 lim λi →±∞ 



.

l

≥

 l

alk (z00 , . . . , z0i ; ·)Alk (λi ); BL2 (S m−di −1 ) ≥

k

alk (z00 , . . . , z0i , z0i+1 ; ·)Dlk (±1); BL2 (Rm−di −1 ).

k

Applying Lemma 3.2.9 to the elements of .S(z00 , . . . , z0i−1 , z0i ), we arrive at the inequality 



.

l

≥

alk (z00 , . . . , z0i−1 , z0i ; ·)Alk (λi ); BL2 (S m−di −1 ) ≥

k

 l

alk (z00 , . . . , z0i , z0i+1 ; ·)Dlk (0); BL2 (Rm−di −1 )

k

for all .λi ∈ R. Now it remains to repeat the proof of Proposition 3.2.8 with the obvious changes in the case .z0i+1 θ0i+1 ≷ 0 and to repeat the proof of Proposition 3.2.10 in the case .z0i+1 θ0i+1 = 0.   We remark that together with the representations considered in Propositions 3.2.14– 3.2.16 the neighborhood .UA includes close representations of the corresponding types obtained by freezing along subchains of strata.

3.2 The Spectral Topology of Algebra A

155

30 . Neighborhoods of points in .Zj × R. We proceed to a discussion of neighborhoods   j of points in .π0 = π(z00 , . . . z0 ; λ0j ). Let .A = l k Alk , .Alk being generators of .

j

the algebra .A, and suppose that .π(z00 , . . . z0 ; λ0j )A > 1. Using inequality (3.2.18) successively for .i = j, j −1, . . . , 0, we obtain the inclusions .U(z00 , . . . , z0i ; Zi )×R ⊂ UA for .i = 0, . . . , j − 1 and sufficiently small neighborhoods .U(z00 , . . . , z0i ; Zi ); here, as  : π A > 1}. According to part 3 of Lemma 3.2.3, the before, .UA = {π ∈ A j estimate .π(z00 , . . . , z0 ; θ 0 , . . . , θ j )A > 1 holds for all .θ 0 , . . . , θ j such that .|θ 0 |2 + · · · + |θ j |2 = 1. 1. This, together with part 1 of the same lemma and the continuity of the functions .Zi × S di −1  (z0 , . . . , zi ; i ) → π(z0 , . . . , zi ; i )A, gives us that i 0 d −1 ⊂ U , .i = 0, . . . , j . Continuity of the function .U (z , . . . , z ; Zi ) × S i A 0 0 Zj × R  (z0 , . . . , zj ; λj ) → π(z0 , . . . , zj ; λj )A

.

j

supplies the inclusion .U (z00 , . . . , z0 ; Zj ) × U(λ0j ; R) ⊂ UA . The whole of the foregoing amounts to: j

Proposition 3.2.17 Every neighborhood .U (π0 ; J ) with .π0 = π(z00 , . . . , z0 ; λ0j ) contains some neighborhood .U (π0 ; ). This part of the proof concludes with a verification of the following assertion. j

Proposition 3.2.18 Every neighborhood .U (π0 ; ) of the point .π0 = π(z00 , . . . , z0 ; λ0j ) contains some neighborhood .U (π0 ; J ). Proof The product .C0 (R) ⊗ KL2 (S m−dj −1 ) is contained in the algebra of operator-valued j functions .R  λj  π(z00 , . . . , z0 ; λj )B, where .B ∈ com A. We choose an element B satisfying the conditions π(z00 , . . . , z0 ; ·)B ∈ C0 (R) ⊗ KL2 (S m−dj −1 ), π(z00 , . . . , z0 ; λ0j )B > 1 j

.

j

j

and .π(z00 , . . . , z0 ; λj )B = 0 for .λj ∈ [λ0j − ε, λ0j + ε], where .ε is a given positive number. Let .a ∈ M(T ) be a function in Lemma 3.2.5 such that j

j

j

a(z00 , . . . , z0 ; ·)π(z00 , . . . , z0 ; λ0j )B = π(z00 , . . . , z0 ; λ0j )aB > 1.

.

 : π A > 1}. We check that .UA ⊂ U(π0 ; ). Let .A = aB and .U(π0 ; J ) = UA = {π ∈ A The representations obtained by “deeper” localization (i.e., the representations of the j j +1 j +1 j +2 form .π(z00 , . . . , z0 , z0 ; ·), .π(z00 , . . . , z0 , z0 ; ·), etc.) annihilate the operator A to property 3 of the function a in Lemma 3.2.5. All the one-dimensional representations also vanish on A, since .A ∈ com A. Every representation .π(z0 , . . . , zi ; ·), .0 ≤ i ≤ j , for points .(z0 , . . . , zi ) not belonging to a small neighborhood .U(z00 , . . . , z0i ; Zi ) is equal to

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

zero on A in view of part 1 in Lemma 3.2.5. Hence, all these representations do not fall in UA . It remains to recall the definition of .U (π0 ; ). 0 ∗ .4 . Neighborhoods of points in .S (C). Finally, to conclude the proof of Theorem 3.2.1, we must still compare neighborhoods of the one-dimensional representations of .A in the J - and .-topologies. Typical neighborhoods in the space . of inner points of the manifold .S ∗ (C) are the same as in the cospherical bundle .S ∗ (M \ |T |). Let us consider points .(z0 , . . . , zj ; ϕ j , ξ ) ∈ Zj × (S m−dj −1 \ |T (s0 , . . . , sj )|) × S m−1 in the j j boundary of the manifold .S ∗ (C). We write the point .(z00 , . . . , z0 ; ϕ0 , ξ0 ) in the form j j j j 0 0 .π0 = π(z , . . . , z ; ϕ , ξ , θ , . . . , θ ) (as in Sect. 3.2.1).   0 0 0 0 0 0 .

Proposition 3.2.19 For every neighborhood .U (π0 ; J ), there exists a neighborhood U (π0 ; ) such that .U (π0 ; ) ⊂ U (π0 ; J ).

.

 : π A > 1}. Proof Suppose that .A ∈ A, .π0 A > 1, .U (π0 ; J ) = UA = {π ∈ A j

j

(a) .θ00 = 0. By (3.2.4), .π(z00 , . . . , z0 ; 0 )A > 1. Then, part 1 in Lemma 3.2.3 leads to the inequalities .π(z00 , . . . , z0i ; i0 )A > 1, .i = 0, . . . , j . Since the functions 0 i i 0 i i .(z , . . . , z ;  ) → π(z , . . . , z ;  )A are continuous, we have .U (π0 ; ) ⊂ UA for “small” neighborhoods .U (π0 ; ). (b) .θ00 = 0, .θ01 = 0. Again, using (3.2.4) and part 1 of Lemma 3.2.3, we obtain the inequalities .π(z00 , . . . , z0i ; i0 )A > 1 for .i = 1, . . . , j (that is, .UA contains the sets .U (z00 , z01 , . . . , z0i ; Zi ) × U (i0 ; S di −1 )). By part 2 of Lemma 3.2.3, the estimate 0 1 0 1 0 ∗ 1 1 1 .π(z , z ;  )A .= π(z , z ; 0, θ /|θ |)A > 1 implies that .π(z ; λ )A > 1 0 0 0 0 0 0 0 0 ∗ 0 for some .λ ∈ R. Then, part 3 of the same lemma means that .π(z ; θ 0 )A > 1 for j j all .{(z0 , θ 0 ) : z0 ∈ U (z00 ; Z0 ), θ 0 ∈ S ∗ (Z0 )z0 }. Since .π(z00 , . . . , z0 ; 0 )A > 1, it follows from Proposition 3.2.8 (in the case .z01 θ01 ≷ 0) that .U (z00 ; Z0 ) × {λ ∈ R : λ ≷ N} ⊂ UA , and we obtain from Proposition 3.2.10 (in the case .z01 θ01 = 0) the inclusion 0 .U (z ; Z0 ) × R ⊂ UA . Therefore, the small neighborhood .U (π0 ; ) belongs to .UA . 0 j j (c) .θ00 , . . . , θ0i = 0, .θ0i+1 . As in the preceding case, .π(z00 , . . . , z0 ; 0 )A > 1. Using now Propositions 3.2.14–3.2.16 instead of Propositions 3.2.8 and 3.2.10, we again obtain the inclusion .U (π0 ; ) ⊂ UA . j (d) .θ00 = 0, . . . , θ0 = 0. It follows from the inequality .π0 A > 1 and (3.2.5) that j 0 ∗ ∗ ∈ R. Then, part 3 of Lemma 3.2.3 gives .π(z , . . . , z ; λ )A > 1 for some .λ j j 0 0 j

us that .π(z00 , . . . , z0 ; j )A > 1 for all .j = (θ, . . . , θ j ) with .|j | = 1. Setting .θ0 = 0, . . . , θ j −1 = 0 and using Propositions 3.2.14–3.2.16, we see that .UA p contains the sets (3.2.2) and (3.2.3) for .i = j , .(z0 , . . . , zp ) = (z00 , . . . , z0 ), and small p 0 .U (z , . . . , z ; Zp ), .p = 0, . . . , j . 0 0 j j

We proceed to the representations .π(z0 , . . . , zj ; λj ). If .ϕ0 ξ0 = 0, then (3.2.6) ensures j j j the inclusion .π(z00 , . . . , z0 ; λj ) ⊂ UA for all .λj ∈ R. Suppose that .ϕ0 ξ0 ≷ 0 and let .χ ∈

3.2 The Spectral Topology of Algebra A

157

C(S m−dj −1 ) be a function such that .0 ≤ χ ≤ 1, .χ (ϕ0 ) = 1, and .suppχ ∩|T (s0 , . . . , sj )| = j ∅, i.e., it vanishes on the discontinuities of the coefficients .a(z00 , . . . , z0 ; ·), .a ∈ M(T ). It   can be assumed that .A = l k Alk , where .Alk are generators of the algebra .A. We write j the operators .π(z00 , . . . , z0 ; λj )Alk in the form j

π(z00 , . . . , z0 ; λj )Alk = Alk (λj ) = Eω→ϕ (λj )−1 lk (ϕ, ω)Eψ→ω (λj ) ∈ BL2 (S m−dj −1 ). j

.

Let Q be a sufficiently large natural number. We have 



.

l



Alk (λj ) ≥ χ Q

k

l

Alk (λj ) ≥ 

k



χ Q−Ql

l



χAlk (λj ) + K(λj ),

k

(3.2.19) where . ·  = ·; BL2 (S m−dj −1 ) and .K(·) ∈ C0 (R) ⊗ KL2 (S m−dj −1 ) (we used Proposition 1.2.11). The functions .(ϕ, ω) → χ (ϕ)lk (ϕ, ω) are continuous on .S m−dj −1 ×   j j S m−dj −1 . Since .π0 A = l k lk (ϕ0 , ξ0 ) and .|π0 A| > 1, it follows that 



.

l

χ Q−Ql



χ Alk (λj ) > 1

k

for .λj ≷ N with some .N ∈ R (see Theorem 2.2.13). This and (3.2.19) give us that j π(z00 , . . . , z0 ; λj )A > 1, if .λj ≷ N  . Hence,

.

U (z00 , . . . , z0 ; Zj ) × {λj ∈ R : λj ≷ N  } ⊂ UA .

.

j

Summarizing all the foregoing, we arrive at the relation .U(π0 ; ) ⊂ UA some neighborhood .U(π0 ; ).   j

j

j

Proposition 3.2.20 For every neighborhood .U (π0 ; ) with .π0 = π(z00 , . . . , z0 ; ϕ0 , ξ0 , j θ00 , . . . , .θ0 ), there exists a neighborhood .U (π0 ; J ) such that .U(π0 ; J ) ⊂ U(π0 ; ). The proof will only be sketched. We look for an element .A ∈ A such that .UA = {π ∈  : π A > 1} can be taken as .U (π0 ; J ). In cases a) and b) (see Proposition 3.2.19), A the difficulty lies in the fact that .UA need not contain “large” subsets of the spheres .S di −1 appearing in the products .U (z00 , z01 , . . . , z0i ; Zi ) × U (i0 ; S di −1 ). We show, for example, how this difficulty is overcome for .j = 0, i.e., for the points .π(z00 ; ϕ00 , ξ00 , θ00 ), where 0 0 m−d0 −1 \ |T (s )|. Let . ∈ C(Rm \ 0) be a homogeneous function of .θ = 0 and .ϕ ∈ S 0 0 0 zero degree such that .(ξ1 , . . . , ξd0 , 0) ≡ 0. For every .χ ∈ C(S d0 −1 ), the homogeneous function . of zero degree defined for .|θ 0 | = |ξ 0 | = 1 by the equality .(ξ 0 , θ 0 ) = χ (θ 0 )(ξ 0 , θ 0 ) belongs to the class .C(Rm \ 0). Therefore, there exist an .a ∈ M(T ) and a .χ satisfying .a(z0 ; ϕ 0 )(ξ 0 , θ 0 ) = 0 outside a small neighborhood of the point

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

(z00 ; ϕ00 , ξ00 , θ00 ) and .a(z00 ; ϕ00 )(ξ00 , θ00 ) > 1. It is not hard to verify that the operator in .A defined in a coordinate neighborhood of .z00 ∈ s0 ⊂ M by .A = a(x)Fη→x (η)Fy→η : L2 (Rm ) → L2 (Rm ), is the desired one. In cases (c) or (d), no new difficulties arise in general under the condition j j i+1 i+1 .z = 0 or .ϕ0 ξ0 = 0 because .U (π0 ; ) contains the set .U (z00 , . . . , z0i ; Zi ) × R 0 θ0 j or .U (z00 , . . . , z0 ; Zj ) × R, respectively. If, for example, .z0i+1 θ0i+1 > 0 in case c), then we have to produce a neighborhood .U (π0 ; J ) containing the set .U (z00 , . . . , z0i ; Zi ) × {λ ∈ R : λ > N} only for sufficiently large N . Such a neighborhood can be constructed by using the considerations connected with the operator P in the proof of Proposition 3.2.12. .

3.3

Solving Series

3.3.1

Construction of a Solving Series. Formulation of the Theorem

Let .T be an admissible partition of manifold .M. The union of all strata with dimension no greater than j is called the j -dimensional skeleton of .T and denoted by .Tj . Generally, the sequence .∅ = T−1 , T0 , . . . , Tm−1 can contain coinciding elements. We form a subsequence .T−1 , Tm1 , . . . , Tmk , where .−1 ≤ mk ≤ m − 1 and .Tmp = Tmq only for .p = q. To this goal, moving from left to right, we include to the subsequence every element not encountered before. In what follows, we, instead of .Tmj , will write simply .Tj , where .dim Tj = mj . Introduce all kinds of skeleton collections (Tj1 , . . . , Tjq ), 1 ≤ j1 < · · · < jq ,

.

of the partition .T . To simplify notation once again, we write .(Tj1 , . . . , Tjq ) instead of .(j1 , . . . , jq ). Thus, in a row .(j1 , . . . , jq ) the element .jk stands for the skeleton of dimension .mjk . We put in order the set of skeleton collections according to the rule: .(j1 , . . . , jq ) ≺ (l1 , . . . , lp ) if and only if either .j1 = l1 , . . . , jh = lh and .jh+1 > lh+1 or .p > q and .j1 = l1 , . . . , jq = lq . It is clear that one of any two skeleton collections is subject to the other. Introduce the empty collection .T∅ and assume that .T∅ is subject to the minimal collection .Tmin = (k). Collections T and .T  are called neighboring if .T ≺ T  , and there is no such a collection .T  that .T ≺ T  ≺ T  . Thus, all collections form an increasing sequence of neighbors T∅ ≺ T1 ≺ · · · ≺ T Q ,

.

(3.3.1)

where .T1 = Tmin = (k) and .TQ = Tmax = (1, . . . k). The passage from .T = (j1 , . . . , jq ) to the neighboring .T   T can be performed by one of the following operations: (1) If .jq < k, then .T  = (j1 , . . . , jq , k) (the number of components increases).

3.3 Solving Series

159

(2) If .jq = k > 1 and either .jq−1 < k − 1 or .q = 1, then .T  = (j1 , . . . , jq−1 , k − 1) (the number of components does not vary). (3) If .jq = k, .jq−1 = k − 1, . . . , jq−p = k − p > 1 and either .jq−p−1 < k − p − 1 or .q − p = 1, then .T  = (j1 , . . . , jq−p−1 , jq−p − 1) (the number of components decreases). If none of the above operations is applicable, then T coincides with the maximal collection .(1, . . . , k). Sequence (3.3.1) can be written explicitly: T∅ ≺ (k) ≺ (k − 1) ≺ (k − 1, k) ≺ (k − 2) ≺ (k − 2, k) ≺ (k − 2, k − 1) ≺

.

(3.3.2)

≺ (k − 2, k − 1, k) ≺ (k − 3) ≺ · · · ≺ (1, 2, . . . , k). With every collection .Tν in (3.3.1), we associate an ideal .A(Tν ) in .A. To do that, we need some notation. Let I be the intersection of the kernels of all one-dimensional representations of the algebra .A, I=



.

ker π1 (x, ξ ),

(3.3.3)

where .(x, ξ ) runs over the cospherical bundle .S ∗ (C), .x ∈ C, and .ξ ∈ S ∗ (C)x , see (3.1.8). Denote the intersection of the kernels of all representations of the form (3.1.6) by .(T∅ ), (T∅ ) =



.

ker π(z0 , . . . , zj ; θ 0 , . . . , θ j ).

(3.3.4)

\T∅

Moreover, let .(T∅ ) stand for the intersection of the kernels of all representations of the form (3.1.7), (T∅ ) =



.

ker π(z0 , . . . , zj ; λ).

(3.3.5)

\T∅

We set A(T∅ ) = I

.



(T∅ )



(T∅ ).

(3.3.6)

Thus, .A(T∅ ) is the intersection of the kernels of all irreducible representations of .A, except for the identity representation. We introduce the ideal .(T1 ) for .mk > 0 as the intersection of the kernels of all representations (3.1.6) except for representations of the form .π(z0 ; θ 0 ), where a point .z0 runs over all the strata s of dimension .mk , while .(z0 ; θ 0 ) runs over the cospherical bundle

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

S ∗ (s) over each stratum s. Let us write .(T1 ) in the form

.

(T1 ) =



ker π(z0 , . . . , zj ; θ 0 , . . . , θ j )

.

(3.3.7)

\T1

(the .\T1 indicates the representations that are absent in the intersection considered). The representations .π(z0 ; θ 0 ), where .z0 ∈ s, are not defined for the zero-dimensional strata s (however, generally, the representations .π(z0 , z1 ; θ 1 ) and similar ones make sense). Therefore, in the case of .mk = 0, we obtain (T1 ) = (T∅ ).

.

(3.3.8)

In a similar manner, we set (T1 ) =



.

ker π(z0 , . . . , zj ; λ),

(3.3.9)

\T1

assuming that the right-hand side of (3.3.9) is the intersection of the kernels of all representations (3.1.7) except for the representations .π(z0 ; λ), where .z0 runs over all .mk dimensional strata s and .λ runs the real axis. Suppose that the ideals .(T ), .(T ) have already been defined for all T in the sequence (3.3.1) such that .T ≺ Tν , while .(T ) and .(T ) are given as the intersections of some primitive ideals, i.e., the kernels of irreducible representations. We introduce ideals .(Tν ), .(Tν ). Let .Tν = (j1 , . . . , jq ) and consider all chains of strata .s1 , . . . , sq satisfying .sα ∈ st (sβ ) for .α > β and .dim sα = mjα , .α = 1, . . . , q. If there is no such a chain, then we set (Tν ) = (Tν−1 ), (Tν ) = (Tν−1 ).

.

(3.3.10)

Assume now that some chains with the mentioned property exist and .π(z1 , . . . , zq ; θ 1 , . . . , θ q ) are the representations obtained by localization along such a chain. From (3.3.2), it follows that the kernels of these representations participate in the intersection, which defines the ideal .(Tν−1 ). Removing the kernels .ker π(z1 , . . . , zq ; θ 1 , . . . , θ q ) of all such representations from the mentioned intersection, we obtain the ideal .(Tν ), (Tν ) =



.

ker π(z1 , . . . , zq ; θ 1 , . . . , θ q ).

(3.3.11)

\Tν

Analogously, removing the .ker π(z0 , . . . , zj ; λ) from the intersection of ideals equal to .(Tν−1 ), we obtain the ideal (Tν ) =



.

\Tν

ker π(z0 , . . . , zj ; λ).

(3.3.12)

3.3 Solving Series

161

Furthermore, if .Tν+1 = (j1 ) provided .mj1 = 0 (which means that the collection .Tν+1 is represented by the zero-dimensional skeleton), then the same reason as in the case of (3.3.8) leads to (Tν+1 ) = (Tν ).

.

(3.3.13)

Continuing the process, we define the ideals .(T ) and .(T ) for all collections T in the sequence (3.3.1). In particular, .(TQ ) = A and .(TQ ) = A. Introduce the ideals A(T ) := I ∩ (T ) ∩ (T ),

.

(3.3.14)

where I is the same ideal as in (3.3.3). It is evident that if .T ≺ T  , then .(T ) ⊂ (T  ),   .(T ) ⊂)(T ) and therefore .A(T ) ⊂ A(T ). Among .A(T ), identical ideals can occur; see (3.3.10). Let .T0 := T∅ , .T1 := Ti1 , . . . , Tp−1 := Tip−1 , .Tp := TQ be a subsequence of sequence (3.3.1) such that the composition series A(T0 ) ⊂ A(T1 ) ⊂ · · · ⊂ A(Tp ) ⊂ A

.

(3.3.15)

consists of all different ideals of the form .A(T ) (and of only those ones); note that always .i1 = 1, i.e., .T1 = T1 = Tmin = (k), see (3.3.1). Generally, the series (3.3.15) is not solving. To obtain a solving composition series, we introduce intermediate ideals .J (Tj ). For this purpose, we consider neighboring ideals .A(Tj ) and .A(Tj +1 ) in (3.3.15). According to (3.3.14), we have .A(Tj ) = I ∩ (Tj ) ∩ (Tj ). Set J (Tj ) = I ∩ (Tj +1 ) ∩ (Tj )

.

(3.3.16)

for .j = 0, . . . , p − 1. If the zero-dimensional skeleton .T0 of the partition .T of .M is not empty, then for a certain .ν, the equality J (Tν ) = A(Tν )

.

(3.3.17)

holds (see (3.3.13)), while for .j = ν, we have the strict inclusions A(Tj ) ⊂ J (Tj ) ⊂ A(Tj +1 ).

.

(3.3.18)

Thus, if the skeleton .T0 is empty, then (3.3.18) holds for all .j = 0, . . . , p − 1, and we obtain the composition series 0 ⊂ A(T0 ) ⊂ J (T0 ) ⊂ A(T1 ) ⊂ J (T1 ) ⊂ · · · ⊂ A(Tp ) ⊂ A.

.

(3.3.19)

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3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

If .T0 = ∅, then, for the number .ν in (3.3.17), the series (3.3.19) contains the link .A(Tν ) ⊂ A(Tν+1 ) instead of .A(Tν ) ⊂ J (Tν ) ⊂ A(Tν+1 ). Let .Z0 × · · · × Zq be the submanifold defined for a chain of strata .s0 , . . . , sq as at the beginning of Sect. 3.2.1. We say that a chain .s0 , . . . , sq is connected with a skeleton collection .Tν = (j0 , . . . , jq ) if .dim sα = mjα , .α = 0, . . . , q (compared with the description of formulas (3.3.10) and (3.3.11)). Denote by .ϒ ν the union of manifolds of the form .Z0 × · · · × Zq corresponding to all chains connected with the collection .Tν . Finally, we set .δν = dim mjq . Theorem 3.3.1 The composition series (3.3.19) is solving. Moreover, A(T0 )  KL2 (M), .

.

∗

A/A(Tp )  C(S (C)),

(3.3.20) (3.3.21)

where .C(S ∗ (C)) is the cospherical bundle over the compact .C (see the description of .C after (3.1.5)) and .A(Tp ) = comA. If the zero-dimensional skeleton .T0 of the partition .T of .M is empty, then J (Tj )/A(Tj )  C0 (S ∗ (ϒ j )) ⊗ KL2 (Rm−δj ),

.

(3.3.22)

where .j = 0, . . . , p − 1, and A(Tj )/J (Tj −1 )  C0 (ϒ j × R) ⊗ KL2 (S m−δj −1 ),

.

(3.3.23)

where .j = 1, . . . , p. If .T0 = ∅, then for the number .ν in (3.3.13), instead of the link .A(Tν ) ⊂ J (Tν ) ⊂ A(Tν+1 ), the series (3.3.19) contains .A(Tν ) ⊂ A(Tν+1 ) and the two formulas (3.3.22) for .j = ν and (3.3.23) for .j = ν + 1 are replaced by the only relation A(Tν+1 )/A(Tν )  C(|T0 | × R) ⊗ KL2 (S m−1 ).

.

3.3.2

(3.3.24)

Proof of Theorem 3.3.1

10 . Verification of formulas (3.3.20) and (3.3.21). We first consider (3.3.20). From Theorem 3.1.4, it follows that all irreducible representations of the algebra A, except for the identity one, vanish on the ideal A(T0 ). Since the algebra A is irreducible, the ideal A(T0 ) is irreducible as well. Moreover, the ideal contains some compact operators, for example, the commutators [a, P ], where a ∈ C ∞ (M), a|Tm−1 = 0, and P is a ψDO in ¯ 0 (M). Therefore, KL2 (M) ⊂ A(T0 ). The identity representation is the only irreducible  representation of A(T0 ), so A(T0 )  KL2 (M).

3.3 Solving Series

163

Let us turn to (3.3.21). It was noticed before (3.3.14) that (Tp ) = (Tp ) = A. Recall that Tp = TQ and therefore, by virtue of (3.3.14), A(Tp ) = I , where I is the intersection of the kernels of all one-dimensional representations of A. Hence, A(Tp ) = comA. It remains to take into account that, according to Theorem 3.1.4, the list (3.1.8) contains any one-dimensional representation of A. 20 . Verification of formulas (3.3.22). We first consider the case j = 0 and the equality J (T0 )/A(T0 )  C0 (S ∗ (ϒ 0 )) ⊗ KL2 (Rm−δ0 );

.

here, J (T0 ) = I ∩ (T1 ) ∩ (T0 ), the ideal (T1 ) = (T1 ) is defined by (3.3.7) (for mk > 0, which is supposed to be fulfilled for the time being), and (T0 ) is given by (3.3.5). Thus, the spectrum J (T0 )∧ consists of representations of the form π(z0 ; θ 0 ), where z0 runs over the union ϒ 0 of all strata of (maximal) dimension mk and (z0 ; θ 0 ) runs over the cospherical bundle S ∗ (ϒ 0 ). Let us consider the algebra π(z0 ; θ 0 )A for a fixed point (z0 ; θ 0 ). Note that the representations π(z0 ; λ) of A arose in the localization procedure as representations of the algebra π(z0 ; θ 0 )A. Therefore, inclusion A ∈ ker π(z0 ; λ) implies π(z0 ; λ)π(z0 ; θ 0 )A = 0. It is clear that all one-dimensional representations of π(z0 ; θ 0 )A are also representations of A. Hence, all irreducible representations of π(z0 ; θ 0 )A are described by Theorem 2.3.12. k ) for all operators A in It now follows from this Theorem that π(z0 ; θ 0 )A ∈ KL2 (Rm−m z0 k and its lifting to the fiber S ∗ (ϒ 0 )z0 ; J (T0 ). (We use the same notation for the space Rm−m z0

k −1 the same is true for Szm−m .) 0 We introduce a continuous field of elementary algebras on S ∗ (ϒ 0 ). To this goal with k ). Any element A of every point (z0 ; θ 0 ) ∈ S ∗ (ϒ 0 ), we associate the algebra KL2 (Rm−m z0 the ideal J (T0 ) gives rise to the vector field k FA : (z0 ; θ 0 ) → π(z0 ; θ 0 )A ∈ KL2 (Rm−m ). z0

.

(3.3.25)

All representations of π(z0 ; θ 0 ) vanish on the ideal A(T0 ), so the field FA depends only on the residue class [A] of an operator A in J (T0 )/A(T0 ). On the set of vector fields of the form (3.3.25), we introduce the norm k FA ; F0  = sup{π(z0 ; θ 0 )A; KL2 (Rm−m ); (z0 ; θ 0 ) ∈ S ∗ (ϒ 0 )}; z0

.

in what follows, F0 stands for the algebra of fields (3.3.25) with that norm. Assume in addition that 0 is the set of vector fields that are limits of the fields in F0 with respect to the local uniform convergence. Taking the elements in 0 as continuous vector fields, we k )}, 0 ) of elementary algebras on S ∗ (ϒ 0 ). introduce the continuous field ({KL2 (Rm−m z0

164

3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols. . .

According to Theorem 3.1.4, the spectrum of J (T0 )/A(T0 ) can be parametrized by the points of the set S ∗ (ϒ 0 ). Therefore, for the residue class [A] ∈ J (T0 )/A(T0 ) of the operator A ∈ J (T0 ), there holds the equality k [A] = sup{π(z0 ; θ 0 )A; KL2 (Rm−m ); (z0 ; θ 0 ) ∈ S ∗ (ϒ 0 )}. z0

.

Hence, the mapping [A] → FA is an isomorphism of C ∗ -algebras J (T0 )/A(T0 )  F0 .

.

(3.3.26)

From Theorem 3.2.1, it follows that the Jacobson topology on the spectrum (J (T0 )/A(T0 ))∧ coincides with the topology of cospherical bundle S ∗ (ϒ 0 ), i.e., the space (J (T0 )/A(T0 ))∧ is finite-dimensional, Hausdorff, and locally compact. The set {π ∈ (J (T0 )/A(T0 ))∧ : π A ≥ α} is compact for any α > 0 and every operator A ∈ J (T0 ) ([3], 3.3.7). This means that every vector field FA , S ∗ (ϒ 0 )  π → π(A), k )}, 0 ) tends to zero at infinity. The algebra defined by the continuous field ({KL2 (Rm−m z0 is isomorphic to F0 ([3], 10.5.4). k )}, 0 ) is locally trivial; as local trivializations, one can use The field ({KL2 (Rm−m z0 those of the normal bundle over ϒ 0 . (In connection with the normal bundle over ϒ j , see the definition of the fiber Rm−dq (z0 , . . . , zq ) over a point (z0 , . . . , zq ) ∈ Zq = Z0 × · · · × Zq given before Theorem 3.1.4.) Proposition 1.3.18 provides the triviality of this field. Therefore, F0  C0 (S ∗ (ϒ 0 )) ⊗ KL2 (Rm−mk ). Taking into account (3.3.26), we obtain (3.3.22) for j = 0. Assume now that mk = 0; then, k = 1, and the series (3.3.19) is of the form 0 ⊂ A(T0 ) ⊂ A(T1 ) ⊂ A. Thus, instead of (3.3.22) for j = 0 and (3.3.23) for j = 1, we must prove the isomorphism A(T1 )/A(T0 )  C0 (|T0 | × R) ⊗ KL2 (S m−1 );

.

(3.3.27)

here, A(T1 ) = I = com A, A(T0 ) = I ∩ (T0 ), and (T0 ) = (z,λ) ker π(z; λ), while λ runs over the real axis and z runs over the finite set |T0 |. According to (3.3.20), A(T0 ) = KL2 (M). We consider functions of the form R  λ → π(z; λ)A ∈ BL2 (S m−1 ), where A ∈ A(T1 ) and z is a fixed point in |T0 |. The representations π(z; λ) vanish on the ideal A(T0 ), so the function π(z; ·)A depends only on the residue class [A] in A(T1 )/A(T0 ). Such a function is an element of the algebra S(z). From the definition of the ideal A(T1 ), it follows that the ideal spanned by the functions in S(z) belongs to the kernel of every onedimensional representation of S(z) and, consequently, coincides with the ideal com S  C0 (R) ⊗ KL2 (S m−1 ). This leads to (3.3.27). We now turn to (3.3.22) for j > 0. Assume that Tj = (j1 , . . . , jq ), the ideals (Tj ) and (Tj ) are defined in (3.3.11) and (3.3.12), A(Tj ) = I ∩ (Tj ) ∩ (Tj ), and J (Tj ) = I ∩ (Tj +1 ) ∩ (Tj ). From Theorem 3.1.4 and the definition of the

3.3 Solving Series

165

ideals A(Tj ) and J (Tj ), it follows that the spectrum (J (Tj )/A(Tj ))∧ consists of the representations π(z1 , . . . , zq ; θ 1 , . . . , θ q ), where z := (z1 , . . . , zq ) ∈ ϒ j and θ := (θ 1 , . . . , θ q ) ∈ S ∗ (ϒ j )z . For a fixed point (z, θ ), we consider the algebra π(z; θ )J (Tj ). From Theorem 3.1.4 (and the localization procedure in its proof), it follows that the element π(z; θ )A for A ∈ J (Tj ) depends only on the residue class [A] ∈ J (Tj )/A(Tj ) and the only irreducible representation (up to equivalence) of the quotient algebra J (Tj )/A(Tj ) is the identity one. Moreover, as in the case j = 0, there hold the inclusion KL2 (Rm−δj ) ⊂ π(z; θ )J (Tj ) and the equality π(z; θ )(J (Tj )/A(Tj )) = KL2 (Rm−δj ). To complete the verification of (3.3.22) for j > 0, it remains to repeat with evident modifications the proof of the formula for j = 0.

Verification of Formulas (3.3.23) The spectrum .(A(Tj )/J (Tj −1 ))∧ can be parametrized by the points .(z; λ), where .z = (z1 , . . . , zq ) ∈ ϒ j and .λ ∈ R. The operator .π(z; λ)A with .A ∈ A(Tj ) depends only on the residue class .[A] ∈ A(Tj )/J (Tj −1 ). For a fixed z, the function .λ → π(z; λ)A is an element of the algebra .S(z). The ideal in .S(z) spanned by such functions with .A ∈ A(Tj ) m−δ −1 coincides with the ideal .com S(z)  C0 (R) ⊗ KL2 (Sz j ). Define a field of elementary algebras on .ϒ j × R. With every point .(z; , λ) ∈ ϒ j × R, m−δ −1 we associate the algebra .KL2 (Sz j ). Denote by .Gj the algebra of vector fields of the form m−δj −1

GA : (z; λ) → π(z; λ)A ∈ KL2 (Sz

.

(3.3.28)

)

endowed with the norm m−δj −1

Ga ; Gj  = sup{π(z; λ)A; BL2 (Sz

.

), (z; λ) ∈ ϒ j × R}.

Let .j stand for the set of limits of the fields in .Gj with respect to the local uniform converm−δ −1

gence. We introduce the continuous field of elementary algebras .({KL2 (Sz j )}, j ), where .j plays the role of a set of continuous vector fields. The mapping .[A] → GA performs an isomorphism .A(Tj )/J (Tj −1 )  Gj . The algebra defined by the field m−δ −1

({KL2 (Sz j )}, .j ) is isomorphic to .Gj . The triviality of the field is provided by Proposition 1.3.18, as by verification of formulas (3.3.22). This leads to the relation   (3.3.23).

.

4

Pseudodifferential Operators on Manifolds with Smooth Closed Edges

In this chapter, we define pseudodifferential operators of arbitrary order on “manifolds with edges” and discuss the general properties of these operators. The presented results will be used in Chap. 5 devoted to .C ∗ -algebras generated by pseudodifferential operators of zero order. In Chaps. 4 and 5, we limit ourselves to manifolds with smooth closed nonintersecting edges. To clarify the definition of the operators, let us consider the subspace .x 1 = 0 in .Rm = {x = (x 1 , x 2 ) : x 1 = (x1 , . . . , xn ), x 2 = (xn+1 , . . . , xm )} as a “wedge.” Our operator class representative is, for example, a .μ-order .do of the form −m/2

(2π )

.

 eixξ a(x, |x 1 |ξ )u(ξ ˆ ) dξ,

(4.0.1)

where a is a function subjected to the estimates |x 1 ||α| |∂xα ∂ηβ a(x, η)| ≤ C(α, β)ημ−|β|

.

for all multi-indices .α, β; as usual, .uˆ is the Fourier transform of u and .η = (1 + |η|2 )1/2 .  Note that a differential operator of the form .a(x, |x 1 |Dx ) = |α|≤μ aα (x)(|x 1 |Dx )α is a natural and traditional object in the theory of boundary value problems in domains with edges. Manifolds with edges that we will consider can be locally represented as a surface of the form .K × Rm−n , where K is an n-dimensional conical surface smooth outside its origin. Rectifying a neighborhood of the base of the cone K, we obtain the set .K × Rm−n , where n .K is an open cone in .R . Therefore, studying .do on a manifold with edges begins with 1 2 m 1 consideration of operators of the form (4.0.1) in .Rm n := {x = (x , x ) ∈ R : x = 0}.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Plamenevskii, O. Sarafanov, Solvable Algebras of Pseudodifferential Operators, Pseudo-Differential Operators 15, https://doi.org/10.1007/978-3-031-28398-7_4

167

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4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

4.1

Pseudodifferential Operators in Rm n

4.1.1

Amplitudes

Let m and n be integers, .1 ≤ n ≤ m. A point .x ∈ Rm will be written in the form .(x 1 , x 2 ), 1 2 1 where .x 1 = (x1 , . . . , xn ) and .x 2 = (xn+1 , . . . , xm ). We set .Rm n = {x = (x , x ) : x = 0}. m m Definition 4.1.1 A function .a˜ ∈ C ∞ (Rm n × Rn × R ) is a pre-amplitude of order .μ .(μ ∈ R), if for any multi-indices .α, β, γ there exists a constant .Cαβγ such that

|∂xα ∂yβ ∂ξ a(x, ˜ y, ξ )| ≤ Cαβγ |x 1 |−|α| |y 1 |−|β| ξ μ−|γ | γ

.

m m for all .(x, y, ξ ) ∈ Rm ˜ we assign an amplitude of n × Rn × R . For each pre-amplitude .a, order .μ,

a(x, y, ξ ) = a(x, ˜ y, |x 1 | ξ ).

.

(4.1.1)

˜ μ (Rm The class of all pre-amplitudes (resp., amplitudes) of order .μ will be denoted by .

n) μ m (. (Rn )). Definition 4.1.2 A pre-amplitude .a˜ and the corresponding amplitude a are called proper, ˜ y, ξ ) = 0 for .|x 1 |/|y 1 | ∈ if there exist numbers .δ ∈ (0, 1) and .ε > 0 such that .a(x, −1 2 2 1 (δ, δ ) or .|x − y | > ε(1 + |x |). The class of proper pre-amplitudes (resp., amplitudes) μ m ˜ μ (Rm of order .μ will be denoted by .

n ) (. 0 (Rn )). 0 The following assertions are obvious: ˜ μ and . μ are complex vector spaces. 1. For any .μ ∈ R, the classes .

μ ν ˜ ⊂

˜ and . μ ⊂ ν . 2. If .μ ≤ ν, then .

˜ μ

˜ν ⊂

˜ μ+ν , . μ ν ⊂ μ+ν for all .μ, ν ∈ R. 3. .

μ ˜ 4. Let .a˜ ∈ and .α, .β, .γ ∈ Zm + . Then, the functions (x, y, ξ ) → |x 1 ||α| |y 1 ||β| ∂xα ∂yβ ∂ξ a(x, ˜ y, ξ ),

.

γ

(x, y, ξ ) → |x 1 ||α|−|γ | |y 1 ||β| ∂xα ∂yβ ∂ξ a(x, y, ξ ) γ

˜ μ−|γ | and . μ−|γ | , respectively. belong to the classes .

For singular pre-amplitudes and amplitudes, Properties 1–4 are also valid, and Property 4 can be strengthened.

4.1 Pseudodifferential Operators in Rm n

169

˜ , 5. Let p, .q ∈ R and let the multi-indices .α, .β be such that .p + q = |α| + |β|. If .a˜ ∈

0 then the functions μ

γ

(x, y, ξ ) → |x 1 |p |y 1 |q ∂xα ∂yβ ∂ξ a(x, ˜ y, ξ ),

.

(x, y, ξ ) → |x 1 |p−|γ | |y 1 |q ∂xα ∂yβ ∂ξ a(x, y, ξ ) γ

μ−|γ |

μ−|γ |

˜ belong to the classes .

and . 0 , respectively. 0 μ μ 6. Let .a ∈ 0 and .b(x, y, ξ ) = a(y, x, ξ ). Then, .b ∈ 0 . (If .a ∈ μ , then the analogous assertion is in general false.) In what follows, we use several times the proper amplitude constructed in the next example. Example 4.1.3 Let the numbers .δ, .δ1 , and .ε satisfy .0 < δ < δ1 < 1, .ε > 0, and let the m functions .χ1 , .χ2 ∈ C ∞ (Rm n × Rn ) possess the following properties: (1) .χ1 is independent of .x 2 , .y 2 and homogeneous of zero degree in the variables .x 1 , .y 1 . (2) .χ1 (x, y) = 1 if . |x 1 |/|y 1 | ∈ [δ1 , δ1−1 ] and .χ1 (x, y) = 0 if .|x 1 |/|y 1 | ∈ [δ, δ −1 ]. (3) .χ2 (x, y) = η(|x 2 − y 2 |/(1 + |x 1 |)), where .η ∈ C ∞ (R+ ), .η(t) = 1 for .t < ε/2, and .η(t) = 0 for .t ≥ ε. We set .χ (x, y, ξ ) = χ1 (x, y)χ2 (x, y). Then, .χ ∈ 00 .

4.1.2

Pseudodifferential Operators

For each amplitude a and a function u in .Cc∞ (Rm n ), we introduce the expression  −m .Au(x) = (2π ) ei(x−y)ξ a(x, y, ξ )u(y) dydξ.

(4.1.2)

The integral on the right-hand side of (4.1.2) exists as an iterated integral since, for a fixed x ∈ Rm n , the function  .ξ → ei(x−y)ξ a(x, y, ξ )u(y) dy

.

is in the Schwartz class .S(Rm ). (This follows from the relation 

 .

e

i(x−y)ξ

a(x, y, ξ )u(y) dy =

ei(x−y)ξ Dy 2N (a(x, y, ξ )u(y)) dy ξ −2N , (4.1.3)

which holds for all .N ∈ Z+ .)

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4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

Definition 4.1.4 The operator A defined on function u in .Cc∞ (Rm n ) by (4.1.2), where .a ∈ . The class of all such . do is denoted by . μ . An

μ , is called .do of order .μ in .Rm n operator A with amplitude a will sometimes be written in the form .Op a. Let us represent A as an integral operator and estimate its kernel. If the order .μ of a satisfies .μ < −m, then (4.1.2) can be rewritten in the form  Au(x) =

G(x, y)u(y) dy,

.

where G(x, y) = (2π )

.

= (2π )−m |x 1 |−m

−m



 ei(x−y)ξ a(x, y, ξ ) dξ

˜ y, ξ ) dξ. ei(x−y)ξ/|x | a(x, 1

Lemma 4.1.5 Let .A ∈  μ and .α, .β be some multi-indices. If .μ + |α + β| < −m, then, for any .N ∈ Z+ , the kernel G of A admits the estimate |∂xα ∂yβ G( x, y)| ≤ CαβN |x 1 |N −m−|α+β| (1+|x 1 |/|y 1 |)|β| (|x 1 |2 +|x −y|2 )−N/2 .

.

(4.1.4)

Proof Integrating by parts, we obtain |G(x, y)| ≤ (2π )−m |x 1 |−m (x − y)/|x 1 |−2N

.

 ˜ y, ξ )| dξ |Dξ 2N a(x,

for all .N ∈ Z+ . It follows from the properties of pre-amplitudes that the integral on the right-hand side of this last inequality is uniformly bounded with respect to .(x, y) ∈ m Rm n × Rn ; therefore, for such .(x, y), |G(x, y)| ≤ CN |x 1 |2N −m (|x 1 |2 + |x − y|2 )−N .

.

β

If .μ + |α + β| < −m, then the derivative .∂xα ∂y G is a linear combination of functions .Gα1 β1 of the form 1 −|α+β1 |

Gα1 β1 (x, y) = |x |

.

1 −|β2 |

|y |

 ei(x−y)ξ aα1 β1 (x, y, ξ ) dξ,

4.1 Pseudodifferential Operators in Rm n

171

where .aα1 β1 (x, y, ξ ) = (|x 1 |ξ )α1 +β1 |x 1 ||α2 | |y 1 ||β2 | ∂xα2 ∂y 2 a(x, y, ξ ), while .α1 ≤ α, .β1 ≤ β, .α2 = α − α1 , and .β2 = β − β1 . Since .aα1 β1 ∈ μ+|α1 +β1 | and .μ + |α1 + β1 | < −m, it follows that β

|Gα1 β1 (x, y)| ≤ CN |x 1 |2N −m−|α+β| (|x 1 |/|y 1 |)|β2 | (|x 1 |2 + |x − y|2 )−N .

.

Due to (|x 1 |/|y 1 |)|β2 | < (1 + |x 1 |/|y 1 |)|β| ,

.

we have |Gα1 β1 (x, y)| ≤ CN |x 1 |2N −m−|α+β| (1 + |x 1 |/|y 1 |)|β| (|x 1 |2 + |x − y|2 )−N .

.

Thus, inequality (4.1.4) is proved for even N. Since .|x 1 |(|x 1 |2 +|x −y|2 )−1/2 ≤ 1, formula 

(4.1.4) is valid for any .N ∈ Z+ . Definition 4.1.6 A .do .A = Op a is called proper if a is a proper amplitude. The class  μ μ of all proper .do of order .μ is denoted by .0 . We also set .0 = μ 0 . α Example 4.1.7 Let the function .a ∈ C ∞ (Rm n ) satisfy the estimate .|∂ a(x)| ≤ m 1 −|α| m ∞ m (x ∈ Rn ) for any .α ∈ Z+ . Then, the operator .Cc (Rn )  u → au is Cα |x | in the class .00 since it has the form .Op (χ a), where .χ is the function in Example 4.1.3. ∞ m In particular, the identity operator .I : Cc∞ (Rm n ) → Cc (Rn ) is a proper .do of order 0. 1 |α| α ∞ m Example 4.1.8 For every .α ∈ Zm + , we set .Qα u(x) = |x | D u(x) for .u ∈ Cc (Rn ). |α| Then, .Qα ∈ 0 due to the equality

Qα u(x) = (2π )−m

.

  ei(x−y)ξ χ (x, y)(|x 1 |ξ )α u(y) dydξ,

where .χ is the same as above. Remark 4.1.9 To make inequality (4.1.3) valid, it is sufficient that (together with the condition .a ∈ μ ) the function .y → a(x, y, ξ )u(y) has a compact support for fixed μ m m .x ∈ Rn and .ξ ∈ R . Therefore, if .a ∈ , then the integral (4.1.2) exists as an iterated 0 one for all .u ∈ C ∞ (Rm n ). We also note that the mapping .a → Op a of the set of amplitudes onto the set of .do is not one-to-one. In particular, a proper .do A can be defined by (4.1.2) with a non-proper amplitude a.

172

4.1.3

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

Symbols

m Definition 4.1.10 A function a˜ ∈ C ∞ (Rm n ×R ) is called a presymbol of order μ, μ ∈ R, if for any multi-indices α, γ there exists a constant Cαγ such that

|∂xα ∂ξ a(x, ˜ ξ )| ≤ Cαγ |x 1 |−|α| ξ μ−|γ | ,

.

γ

m (x, ξ ) ∈ Rm n ×R .

For each presymbol a, ˜ we introduce a function a, a(x, ξ ) = a(x, ˜ |x 1 |ξ ),

.

(4.1.5)

which is called a symbol (of order μ). The class of all presymbols (resp., symbols) of order μ is denoted by S˜ μ (resp., S μ ). If a˜ ∈ S˜ μ , then the function (x, y, ξ ) → a(x, ˜ ξ ) is a pre-amplitude of order μ. Thus, presymbols (symbols) can be regarded as pre-amplitudes (amplitudes) independent of y. ˜ μ and μ listed in Sect. 4.1.1 remain valid also Therefore, the properties of the classes

μ μ for the classes S˜ and S . In what follows, we often use the evident fact that the restriction m m of an amplitude of order μ to the set {(x, y, ξ ) ∈ Rm n × Rn × R : x = y} is a symbol of the same order.   μ Definition 4.1.11 Elements of the sets −∞ = μ μ , −∞ = μ 0 , and S −∞ = 0  μ (μ ∈ R) are called amplitudes, proper amplitudes, and symbols of order −∞, μS respectively. An operator A is called a do (respectively, proper do) of order −∞ if −∞ A = Op a, where a ∈ −∞ ( −∞ 0 ). The class of all such do will be denoted by  −∞ (0 ). Some sufficient conditions for do to belong to the classes  −∞ and 0−∞ will be given later. m Definition 4.1.12 Let μ ∈ R, aj ∈ S μ−j , j ∈ Z+ , and a ∈ C ∞ (Rm n × R ). We write

a(x, ξ ) ∼

∞ 

.

aj (x, ξ ),

j =0

if a −

N −1 j =0

aj ∈ S μ−N for any N ∈ N.

The relation (4.1.6) obviously implies a ∈ S μ .

(4.1.6)

4.1 Pseudodifferential Operators in Rm n

173

Theorem 4.1.13 If aj ∈ S μ−j , j ∈ Z+ , then there exists a function a such that (4.1.6) is valid. If another function a  satisfies the same relation, then a − a  ∈ S −∞ . The proof is left to the reader (see the proof of Theorem 1.1.18). Let A be a proper do. We set eξ (x) = eixξ and define a symbol σA of the operator A by σ A (x, ξ ) = e−ξ Aeξ (x).

(4.1.7)

.

If A = Op a with a proper amplitude a, then σ A (x, ξ ) = (2π )−m

 

.

= (2π )−m

ei(x−y)(η−ξ ) a(x, y, η) dydη =

(4.1.8)

  ei(x−y)θ a(x, y, ξ + θ ) dydθ .

(According to Remark 4.1.9, formula (4.1.7) or, equivalently, (4.1.8) makes sense.) Example 4.1.14 The symbol of do u → au in Example 4.1.7 coincides with a. Example 4.1.15 If Q = Qα is the operator in Example 4.1.8, then σQ (x, ξ ) = |x 1 ||α| ξ α . μ

Theorem 4.1.16 Let A ∈ 0 and let σ = σA be the symbol of A. Then: (1) σ ∈ S μ . (2) The formula Au(x) = (2π )−m/2

.



ˆ ) dξ, u ∈ Cc∞ (Rm eixξ σ (x, ξ )u(ξ n ),

(4.1.9)

is valid, where uˆ = F u is the Fourier transform of u. μ (3) If A = Op a, a ∈ 0 , then the following asymptotic expansion holds: σ (x, ξ ) ∼

.

 1  ∂ξα Dyα a(x, y, ξ )y=x . α! α

(4.1.10)

The function (x, y, ξ ) → ∂ξα Dyα a(x, y, ξ ) is an amplitude of order μ−|α|. This implies that the function (x, ξ ) → ∂ξα Dyα a(x, y, ξ )|y=x is in the class S μ−|α| . Thus, the terms of the asymptotic series (4.1.10) are symbols of decreasing orders. Before proving the theorem, we state the following assertion.

174

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges μ

Lemma 4.1.17 Let a ∈ 0 and h ∈ C([0, 1]). For any t ∈ [0, 1], we set σt (x, ξ ) = (2π )−m

 

.

ei(x−y)θ a(x, y, ξ + tθ )u(y) dydθ

and define the function σ by  σ (x, ξ ) =

1

σt (x, ξ )h(t) dt.

.

0

Then, σ ∈ S μ . Proof of the Lemma Let σ˜ t , σ˜ , and a˜ be the functions corresponding to σt , σ , and a due to formulas (4.1.1) and (4.1.5). We have σ˜ t (x, ξ ) = (2π )

.

−m

  ˜ y, ξ + t|x 1 |θ )u(y) dydθ, ei(x−y)θ a(x,  σ˜ (x, ξ ) =

.

1

(4.1.11)

σ˜ t (x, ξ )h(t) dt.

0

It suffices to verify that |∂xα ∂ξ σ˜ t (x, ξ )| ≤ Cαγ |x 1 |−|α| ξ μ−|γ | ,

.

γ

(4.1.12)

where Cαγ are constants independent of t ∈ [0, 1]. In (1.13), we substitute y → |x 1 |y, θ → θ/|x 1 | and integrate by parts. As a result, we obtain σ˜ t (x, ξ ) = (2π )

.

−m



ei(x−y)θ Dθ 2N [Dy 2N bt (x, y, ξ, θ )θ −2N ] × (4.1.13) ×X − y−2N dydθ,

where X = x/|x 1 |, bt (x, y, ξ, θ ) = a(x, ˜ |x 1 |y, ξ + tθ ). Since a˜ is a proper pre-amplitude, there exists a number δ ∈ (0, 1) such that bt (x, y, ξ, θ ) = 0 for |y 1 | ∈ [δ, δ −1 ]. In what follows, we therefore assume that |y 1 | ≥ δ. Since |∂yβ1 ∂θ 2 bt (x, y, ξ, θ )| ≤ Cβ 1 β2 |y 1 |−|β1 | ξ + tθ μ−|β2 | , β1 , β2 ∈ Zm +,

.

β

we have |∂yβ1 ∂θ 2 bt (x, y, ξ, θ )| ≤ Cβ1 β2 ξ μ θ |μ| .

.

β

4.1 Pseudodifferential Operators in Rm n

175

This and (4.1.13), in view of the relation |∂ β (θ −2N )| ≤ CβN θ −2N , imply the estimate  |σ˜ t (x, ξ )| ≤ CN ξ μ

.

X − y−2N θ μ−2N dydθ,

which for N > (m+|μ|)/2 can be rewritten in the form |σ˜ t | ≤ Cξ μ . Thus, the inequality (4.1.12) is proven for α = γ = 0. For arbitrary multi-indices α and γ , this inequality follows from the foregoing γ argument, since |x 1 ||α| ∂xα ∂ξ σ˜ t (x, ξ ) is a sum of integrals of the form (4.1.11) with a˜ replaced by pre-amplitudes of order μ − |γ |. Note that it suffices to consider the case |α + γ | = 1, since one can then reason by induction. 

Proof of Theorem 4.1.16 Using formula (4.1.8) and the Taylor expansion 

a(x, y, ξ ) =

.

+

 N α!

|α|≤N −1



|α|=N

1 0

1 α ∂ a(x, y, ξ )θ α + α! ξ

(1 − t)N −1 ∂ξα a(x, y, ξ + tθ ) dt θ α ,

we obtain 

σ (x, ξ ) =

.

|α|≤N −1

 N 1 (α) σ (x, ξ ) + σ (α) (x, ξ ) , α! α!

(4.1.14)

|α|=N

where σ

.

(α)

(x, ξ ) = (2π )

−m

 

 ei(x−y)θ ∂ξα a(x, y, ξ )θ α dydθ = ∂ξα Dyα a(x, y, ξ )y=x

for |α| ≤ N − 1 and  σ

.

σt(α) (x, ξ ) = (2π )−m for |α| = N. Setting rN (x, ξ ) = as σ (x, ξ ) =

(α)

(x, ξ ) = 0

 

1

(1 − t)N −1 σt(α) (x, ξ ) dt,

ei(x−y)θ ∂ξα Dyα a(x, y, ξ + tθ ) dydθ



|α|=N (N/α!)σ



.

|α|≤N −1

(α) (x, ξ ),

we rewrite equation (4.1.14)

1 (α) σ (x, ξ ) + rN (x, ξ ). α!

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4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

The function (x, y, ξ ) → ∂ξα Dyα a(x, y, ξ ) is an amplitude of order μ − |α|. Lemma 4.1.17 now implies that rN ∈ S μ−N . This proves assertions (3) and (1) of the theorem. It remains to prove formula (4.1.9). Let u ∈ Cc∞ (Rm n ). Formula (4.1.8) implies the following inequality:  .

ˆ ) dξ = (2π )−m ei(x−y)ξ σ (x, ξ )u(ξ

 ˆ ) dξ × eixξ u(ξ

(4.1.15)

  ×

ei(x−y)(η−ξ ) a(x, y, η) dydη.

Let us show that for a fixed x ∈ Rm n the function  (ξ, η) → u(ξ ˆ )

ei(x−y)(η−ξ ) a(x, y, η) dy

.

(4.1.16)

is rapidly decreasing as |ξ | + |η| → ∞. Since uˆ ∈ S(Rm ), it is sufficient to verify the inequality     . ei(x−y)(η−ξ ) a(x, y, η) dy  ≤ CN (x)ημ−2N ξ 2N   β

for any N ∈ Z+ . Using the relations a(x, y, η) = a(x, ˜ y, |x 1 |η) and |∂y a(x, ˜ y, η)| ≤ 1 −|β| μ η , we obtain Cβ |x |      . ei(x−y)(η−ξ ) a(x, y, η) dy  ≤ |Dy 2N a(x, y, η)| dy η − ξ −2N ≤    ≤ CN (x)(1 + |x 1 |−2N )|x 1 |ημ η−2N ξ 2N ≤ CN (x)ημ−2N ξ 2N ,

as needed. Due to the rapid decrease of the function (4.1.16), one can change the order of integration on the right-hand side of (4.1.15):  .

 ×

ˆ ) dξ = (2π )−m ei(x−y)ξ σ (x, ξ )u(ξ

eiyξ u(ξ ˆ ) dξ = (2π )−m/2

  ei(x−y)η a(x, y, η) dydη ×

  ei(x−y)η a(x, y, η)u(y) dydη = (2π )m/2 Au(x). 

Theorem 4.1.18 μ

(1) A proper do A is in the class 0 , μ ∈ [−∞, +∞), if and only if σA ∈ S μ .

4.1 Pseudodifferential Operators in Rm n

177

(2) Formula (4.1.10) for the symbol of a proper do A A = Op a remains valid when the amplitude a is not proper. μ (3)  μ ∩ 0 = 0 (∀μ ∈ [−∞, +∞)).  μ (4) μ 0 = 0−∞ . Proof Assertions (1)–(3) can be proved with the same technique using a function in μ Example 4.1.3. We prove only the first assertion. If A ∈ 0 , then σA ∈ S μ according to Theorem 4.1.16. Conversely, let σA ∈ S μ and A = Op a, where a is some proper m m amplitude. For any function χ ∈ C ∞ (Rm n × Rn ) and any x0 ∈ Rn , we have Op σA (χ (x0 , ·)u)(x) = Op a(χ (x0 , ·)u)(x),

.

x ∈ Rm n.

Setting x0 = x, we obtain Op (σA χ ) = Op (aχ ). Now let χ be a function from Example 4.1.3 such that the equality aχ = a holds. Then, Op (σA χ ) = Op a = A. Since μ μ 

σA χ ∈ 0 , we have A ∈ 0 , as needed. Assertion (4) directly follows from (1). The following theorem shows that any function a ∈ S μ is the symbol of some proper do modulo elements of S −∞ . Theorem 4.1.19 Let μ be an arbitrary real number, and let a ∈ S μ . There exists a do μ A ∈ 0 such that a − σA ∈ S −∞ . Proof Let χ denote a function constructed in Example 4.1.3 (the numbers δ, δ1 , ε in the construction of χ are chosen arbitrarily), and define an operator A by A = Op (χ a). Since μ μ χ a ∈ 0 , we have A ∈ 0 and, by Theorem 4.1.16, σA (x, ξ ) ∼

.

 1  ∂ξα Dyα (χ (x, y)a(x, ξ ))y=x = a(x, ξ ) α! α

(remind that χ = 1 in a neighborhood of the diagonal x = y). According to the definition of an asymptotic expansion, the obtained relation means that a − σA ∈ S μ−N for all N ∈ N. 

By Theorems 4.1.16 and 4.1.19, the mapping  μ  A → σA ∈ S μ (∀μ ∈ R) induces an isomorphism  μ / −∞ ∼ = S μ /S −∞ of linear spaces.

4.1.4

Composition of do: Adjoint Operator

A proper .do maps the set .Cc∞ (Rm n ) into itself. Therefore, the composition of such .do makes sense.

178

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges μ

Theorem 4.1.20 Let .Aj ∈ 0 j , let .σj be the symbol of .Aj , .j = 1, 2, and let .A = A1 A2 . Then: μ +μ

(1) .A ∈ 0 1 2 . (2) The symbol .σ of A admits the asymptotic expansion σ (x, ξ ) ∼

.

 1 ∂ξα σ1 (x, ξ )Dxα σ2 (x, ξ ). α! α

(4.1.17)

Proof μ

μ

(1) Let .aj ∈ 0 j and .Aj = Op aj , .j = 1, 2. We define the transposed .do .tA2 ∈ 0 2 , A2 u(x) = (2π )

t

.

−m

  ei(x−y)ξ a2 (y, x, −ξ )u(y) dydξ.

Let .t σ2 denote the symbol of .tA2 and define the dual symbol .σ2 by .σ2 (x, ξ ) =t σ2 (x, −ξ ). It is easily seen that A2 u(x) = (2π )−m

 

.

ei(x−y)ξ σ2 (y, ξ )u(y) dydξ.

(4.1.18)

Setting .v = A2 u in A1 v(x) = (2π )−m/2

.

 ˆ ) dξ, eixξ σ1 (x, ξ )v(ξ

we obtain Au(x) = (2π )−m

 

.

ei(x−y)ξ σ1 (x, ξ )σ2 (y, ξ )u(y) dydξ

(4.1.19)

(note that the function .(x, y, ξ ) → σ1 (x, ξ )σ2 (y, ξ ) is not an amplitude). Rewrite now the operator .Aj , .j = 1, 2, using the amplitude .aj : Aj u(x) = (2π )

.

−m

  ei(x−y)ξ aj (x, y, ξ )u(y) dydξ.

Simple calculations reveal that Au(x) = (2π )−m

.

  ei(x−y)ξ a(x, y, ξ )u(y) dydξ,

(4.1.20)

4.1 Pseudodifferential Operators in Rm n

179

where a(x, y, ξ ) = (2π )

.

−m

  ei(x−y)θ a1 (x, z, ξ + θ )a2 (z, y, ξ ) dzdθ.

Since .aj , .j = 1, 2, is a proper amplitude, there exist numbers .δj ∈ (0, 1) and .εj > 0 such that .aj (x, y, ξ ) = 0 for .|x 1 |/|y 1 | ∈ (δj , δj−1 ) or .|x 2 − y 2 | > εj (1 + |x 1 |). We set .δ = δ1 δ2 and .ε = ε1 + ε2 . Then, .a(x, y, ξ ) = 0 for .|x 1 |/|y 1 | ∈ [δ, δ −1 ] or .|x 2 − y 2 | ≥ ε(1 + |x 1 |). Let .χ be a function in Example 4.1.3, chosen such that .χ a = a. Reasoning as in the proof of Theorem 4.1.18, we deduce from (4.1.19) and (4.1.20) that Au(x) = (2π )−m

 

.

ei(x−y)ξ σ1 (x, ξ )σ2 (y, ξ )χ (x, y)u(y) dydξ.

(4.1.21)

Since the function .(x, y, ξ ) → σ1 (x, ξ )σ2 (y, ξ )χ (x, y) is a proper amplitude of order .μ1 + μ2 , assertion (1) is now proven. m (2) Since .χ = 1 in a neighborhood of the diagonal .{(x, y) ∈ Rm n × Rn : x = y}, equality (4.1.21) and Theorem 4.1.16 yield the asymptotic expansion σ (x, ξ ) ∼

.

 1 ∂ξα [σ1 (x, ξ )Dxα σ2 (x, ξ )]. α! α

Then, formula (4.1.18) can be written in the form A2 u(x) = (2π )

.

−m

  ei(x−y)ξ (t σ 2 (y, −ξ ))u(y) dydξ.

Since .A2 = t (tA2 ), the operator .tA2 can be similarly expressed via .σ2 : A2 u(x) = (2π )

t

.

−m

  ei(x−y)ξ σ2 (y, −ξ )u(y) dydξ.

Introduce an appropriate cut-off function .χ and obtain t

.

A2 u(x) = (2π )−m

 ei(x−y)ξ σ2 (y, −ξ )χ (x, y)u(y) dydξ.

Again, applying Theorem 4.1.16, we arrive at t

.

σ2 (x, ξ ) ∼

 1 ∂ξα Dxα σ2 (x, −ξ ), α! α

(4.1.22)

180

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

or, equivalently, at σ2 (x, ξ ) ∼

 (−1)|α|

.

α

α!

∂ξα Dxα σ2 (x, ξ ).

(4.1.23)

The asymptotic expansion (4.1.17) can be deduced from (4.1.22) and (4.1.23) with the same combinatorial reasoning as in the usual theory of .do. 

μ

We now turn to the description of an adjoint .do. Let .A = Op a with .a ∈ 0 . The properties of amplitudes listed in Sect. 4.1.1 imply that, for any .τ ∈ R, the function ∗ 1 1 2τ a(y, x, ξ ) is also in the class . μ . Therefore, the operator .τ a : (x, y, ξ ) → (|y |/|x |) 0 ∗ := Op a ∗ is a proper .do of order .μ. The .do . A∗ is called adjoint to A with .τ A τ τ respect to the inner product  .(· , ·)τ = |x 1 |2τ u(x)v(x) dx, since the relation .(Au , v)τ = (u , τ A∗ v)τ u holds for all .v ∈ Cc∞ (Rm n ). The asymptotic expansion for the symbol .τ σ ∗ of .τ A∗ has the form .τ

4.1.5

σ ∗ (x, ξ ) ∼ |x 1 |−2τ

 1   ∂ξα Dyα |y 1 |2τ a(y, x, ξ ) y=x . α! α

(4.1.24)

Conditions for do to Belong to Classes 0−∞ and  −∞

−∞ = Op −∞ . Remind (see Definition 4.1.11) that .0−∞ = Op −∞ 0 , .

Proposition 4.1.21 Let .A = Op a be a proper .do, whereas the amplitude a has a zero −∞ m m of infinite order on the set .{(x, y, ξ ) ∈ Rm n × Rn × R : x = y}. Then, .A ∈ 0 . Proof According to Theorem 4.1.18, it is sufficient to prove that .σA ∈ S −∞ . This inclusion immediately follows from the condition of the Proposition and from formula 

(4.1.10). To expand Proposition 4.1.21 to the case of non-proper .do, we need the next lemma. m Lemma 4.1.22 Let the function .H ∈ C ∞ (Rm n × Rn ) satisfy the following conditions:

(1) There exists a number .ε0 > 0 such that .H (x, y) = 0 for .|x − y| < ε0 |x 1 |. (2) For all .α, .β ∈ Zm + and .N ∈ Z+ |∂xα ∂yβ H (x, y)| ≤ CαβN |x 1 |N −|α| |y 1 |−|β| |x − y|−N .

.

(4.1.25)

4.1 Pseudodifferential Operators in Rm n

181

Let also v be a function in the Schwartz class .S(Rm ). Then, the function b : (x, y, η) → ei(x−y)η H (x, y)v(|x 1 |η)

.

belongs to the class . −∞ . Proof The first condition of the lemma implies that there exists a number .ε1 > 0 such that |x − y| ≥ ε1 (|x 1 | + |y 1 |) for (x, y) ∈ suppH.

.

Indeed, if .|y 1 | ≤ 2|x 1 |, then .|x − y| ≥ (ε0 /3)(|x 1 | + |y 1 |), and if .|y 1 | ≥ 2|x 1 |, then 1 1 1 ˜ y, η) = b(x, y, η/|x 1 |). The inclusion .|x − y| ≥ |y |/2 ≥ (|x | + |y |)/3. We set .b(x, ∼ −∞ −∞ is equivalent to .b˜ ∈ .b ∈

, i.e., to the estimates ˜ y, η)| ≤ C|x 1 |−|α| |y 1 |−|β| η−M |∂xα ∂yβ ∂ηγ b(x,

.

for all .α, .β, .γ ∈ Zm + and .M ∈ Z+ (here and in what follows, we do not indicate dependence of the constant factors on .α, .β, . . . , M and so on). Since the function α β γ ˜ .(x, y, ξ ) → ∂x ∂y ∂η b(x, y, η) is a linear combination of the functions dα1 β1 γ1 (x, y, η) = (∂xα1 ∂yβ1 ∂ηγ1 ei(x−y)η/|x | )(∂xα2 ∂yβ2 H (x, y))(∂ηγ2 v(η)), 1

.

(4.1.26)

where .α1 + α2 = α, .β1 + β2 = β, .γ1 + γ2 = γ , it suffices to prove that |dα1 β1 γ1 (x, y, η)| ≤ C|x 1 |−|α| |y 1 |−|β| η−M

.

for any .α1 ≤ α, .β1 ≤ β, .γ1 ≤ γ . By induction on .|α1 + β1 + γ1 |, one can easily establish that the first factor on the right-hand side of (4.1.26) is estimated by the sum of a finite number of terms of the form ekδ (x, y, η) = |x 1 |−|α1 +β1 |−k |x − y|k |ηδ | ,

k ∈ Z+ , δ ∈ Zm +.

.

Since .v ∈ S(Rm ), it follows that .|ηδ ∂ γ2 v(η)| ≤ Cη−M for all .M ∈ Z+ . Taking into account the estimates (4.1.25), we conclude that   ekδ (x, y, η)∂ α2 ∂ β2 H (x, y) ∂ γ2 v(η) ≤

.

x

y

η

≤ C (|x 1 |−|α1 +β1 |−k |x − y|k )(|x 1 |N −|α2 | |y 1 |−|β2 | |x − y|−N )η−M = = C |x 1 |−|α| |y 1 |−|β| (|x 1 |N −k−|β1 | |y 1 ||β1 | |x − y|k−N )η−M

(4.1.27)

182

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

for any M and .N ∈ Z+ . Let .N ≥ k + |β1 |. Then, |x − y|N −k ≥ (ε1 (|x 1 | + |y 1 |))N −k ≥ ε1N −K |x 1 |N −k−|β1 | |y 1 ||β1 |

.

on .suppH . From here and (4.1.27), we derive   ekδ (x, y, η)∂ α2 ∂ β2 H (x, y)∂ γ2 v(η) ≤ C  |x 1 |−|α| |y 1 |−|β| η−M x y η

.



on the same set.

Proposition 4.1.23 Let a be an amplitude having a zero of infinite order on the set m m {(x, y, ξ ) ∈ Rm n × Rn × R : x = y}. Then, the operator .A = Op a is in the class −∞ . . .

Proof We represent the operator A in the form .A = Op (χ a) + Op (χ1 a), where .χ is a function in Example 4.1.3, .χ1 = 1 − χ . According to Proposition 4.1.21, we have 0 .Op (χ a) ∈ −∞ . It then follows from the construction of the function .χ (x, y) = 0 1 that .|x 1 |/|y 1 | ∈ (δ, δ −1 ) and .|x 2 − y 2 | < ε(1 + |x 1 |) (.δ ∈ (0, 1) and .ε > 0 are some fixed numbers). We put .ε0 = min{1 − δ, ε}. An elementary verification shows that if 1 .|x − y| < ε0 |x |, then .χ (x, y) = 0. Therefore, from the beginning, we can assume that 1 a(x, y, ξ ) = 0 for |x − y| < ε0 |x 1 |.

(4.1.28)

.

For .(x, y, ξ ) ∈ suppa, this implies the relation |x − y| ≥ ε1 (|x 1 | + |y 1 |)

.

(ε1 > 0)

(4.1.29)

(see the proof of Lemma 4.1.22). For each .k ∈ Z+ , we define the function .ak by ak (x, y, ξ ) = (−ξ )k a(x, y, ξ ) |x − y|−2k .

.

Let .a ∈ μ . From (4.1.29), it easily follows that .ak ∈ μ−2k . If .k > (m + μ)/2, then Au(x) = (2π )−m



 ei(x−y)ξ ak (x, y, ξ )u(y) dydξ =

.

G(x, y)u(y) dy,

where G is the kernel of A, G(x, y) = (2π )−m

.

Let .v ∈ S(Rm ), .(2π )−m



 ei(x−y)ξ ak (x, y, ξ ) dξ.

v(η) dη = 1. We set

b(x, y, η) = ei(y−x)η |x 1 |m G(x, y)v(|x 1 |η).

.

(4.1.30)

4.1 Pseudodifferential Operators in Rm n

183

Since (2π )

.

−m



  e

i(x−y)η

b(x, y, η)u(y) dydη =

G(x, y)u(y) dy = Au(x),

we have .A = Op b. Thus, the proposition is proved if the inclusion .b ∈ −∞ is established. To this end, it is sufficient to prove that the function .H : (x, y) → |x 1 |m G(x, y) satisfies the conditions of Lemma 4.1.22. Since condition (1) directly follows from (4.1.28), we need only to obtain the estimates (4.1.25). μ−2k and We fix .α, .β ∈ Zm + and take .k > (m + μ + |α + β|)/2. Since .A = Op ak ∈  .μ − 2k + |α + β| < −m, it follows from Lemma 4.1.5 (with .μ replaced by .μ − 2k and N by .N + |β|) that |∂xα ∂yβ G(x, y)| ≤ C|x 1 |N −m−|α| (1 + |x 1 |/|y 1 |)|β| |x − y|−(N +|β|)

.

(.N ∈ Z+ , .C = C(α, β, N )). Hence, taking into account (4.1.29), we deduce that |∂xα ∂yβ G(x, y)| ≤ C  |x 1 |N −m−|α| |y 1 |−|β| |x − y|−N .

.

(4.1.31)

Since the multi-indices .α and .β are arbitrarily chosen, estimates (4.1.25) follow from (4.1.31) and from the formula α β .∂x ∂y H (x, y)

 α α−γ γ ∂x |x 1 |m · ∂x ∂yβ G(x, y). = γ γ ≤α 

Proposition 4.1.24 Any .do A of arbitrary order .μ admits the representation .A = A0 + μ A1 , where .A0 ∈ 0 and .A1 ∈  −∞ . Proof The needed representation is provided by the equality .Op a = Op χ a+Op (1−χ )a, 

where .χ is a function from Example 4.1.3.

4.1.6

Elliptic do

Definition 4.1.25 A symbol σ ∈ S μ is called elliptic if there exist positive numbers ρ0 and C such that, for |ξ | ≥ ρ0 , the inequality |σ˜ (x, ξ )| ≥ C|ξ |μ ,

.

184

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

holds, where σ˜ denotes the presymbol corresponding to σ in accordance with (4.1.5). A proper do A is called elliptic if σA is an elliptic symbol. μ

If A ∈ 0 and B ∈ 0ν are elliptic, then the relations σAB − σA σB ∈ S μ+ν−1 , σA∗ − μ+ν μ σ A ∈ S μ−1 imply that the operators AB ∈ 0 , A∗ ∈ 0 are elliptic. Definition 4.1.26 A proper do B is called a parametrix of the proper do A if BA = I + R1 , AB = I + R2 ,

.

where R1 , R2 ∈ 0−∞ and I is the identity operator. Theorem 4.1.27 −μ

μ

(1) Any elliptic operator A ∈ 0 possesses a parametrix B ∈ 0 . (2) If B  is another proper do satisfying B  A − I ∈ 0−∞ or AB  − I ∈ 0−∞ , then B  − B ∈ 0−∞ . Proof Since σA is an elliptic symbol of order μ, we have c|ξ |μ ≤ |σ˜ A (x, ξ )| ≤ C|ξ |μ

(C, c > 0, |ξ | ≥ ρ0 ).

.

(4.1.32)

Let σ˜ 0 be a presymbol of order −μ, satisfying σ˜ 0 (x, ξ ) = σ˜ A (x, ξ )−1 for |ξ | ≥ ρ0 , and let σ0 be the corresponding symbol. Adding to σ0 a certain element in S −∞ , we can assume (Theorem 4.1.19) that σ0 is the symbol of some proper do B0 . From (4.1.32), it follows −μ that the operator B0 ∈ 0 is elliptic. Moreover, B0 A = I + R0 , AB0 = I + R0 , where −1  R0 , R0 ∈ 0 . j Denote the symbol of do (−1)j R0 , j ≥ 0, by σ (j ) . By Theorem 4.1.13, one can find an element σ1 ∈ S0 such that σ1 (x, ξ ) ∼



.

σ (j ) (x, ξ ).

j ≥0

Applying Theorem 4.1.19 again, we choose σ1 so that the operator B1 = Op σ1 is proper. We set B = B1 B0 and obtain BA = I + R1 ,

.

0 B ∈ El −μ , R1 ∈ −∞ ,

(4.1.33)

where the operator B is elliptic. In an analogous way, starting from AB0 = I + R0 , we find that AB  = I + R1 ,

.

B  ∈  −μ , R1 ∈ 0−∞ ,

4.2 Operators on Manifolds with Wedges

185

where B  is elliptic. Since BAB  = B(I + R1 ) = (I + R1 )B  , we obtain B − B  ∈ 0−∞ . Therefore, AB = I + R2 ,

.

R2 ∈ 0−∞ .

(4.1.34)

Formulas (4.1.33) and (4.1.34) imply the first assertion of the theorem. The second assertion follows from the fact that, in the proof of the inclusion B  − B ∈ 0−∞ , we 

need only the relations BA − I ∈ 0−∞ and AB  − I ∈ 0−∞ .

4.2

Operators on Manifolds with Wedges

4.2.1

Admissible Diffeomorphisms of Subsets of Rm n

Below, a mapping .f = (f1 , . . . , fm ) : E → Rm = Rn Rm−n (E is an arbitrary set) is often written in the form .f = (f 1 , f 2 ), where .f 1 = (f1 , . . . , fn ), .f 2 = (fn+1 , . . . , fm ). If E is an open subset of .Rm and the mapping f is differentiable, then .det f  denotes the determinant of the Jacobian matrix .f  = (∂fi /∂xj ). Definition 4.2.1 Let W , .W1 ⊂ Rm n be open (not necessarily connected) sets. A diffeomorphism .f : W → W1 (of class .C ∞ ) is called admissible if: (1) coordinate functions of the diffeomorphism f are subjected to the estimates |∂ α fi (x)| ≤ Cα |x 1 |1−|α| ,

.

x ∈ W,

(4.2.1)

for all .α ∈ Zm + , .|α| ≥ 1; (2) there exist constants .C0 , .c0 > 0, such that c0 |x − y|/|x 1 | ≤ |f (x) − f (y)|/|f 1 (x)| ≤ C0 |x − y|/|x 1 |

.

(4.2.2)

for any x, .y ∈ W . Note that condition (2) follows from the relations c0 |x 1 | ≤ |f 1 (x)| ≤ C0 |x 1 | ,

.

c0 |x

− y| ≤ |f (x) − f (y)| ≤

C0 |x

− y|,

x ∈ W, .

(4.2.3)

x, y ∈ W.

(4.2.4)

The converse is also true. To prove this, we rewrite (4.2.2) in the form c0 |f 1 (x)|/|x 1 | ≤ |f (x) − f (y)|/|x − y| ≤ C0 |f 1 (x)|/|x 1 |.

.

(4.2.5)

186

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

Since x and y play equal roles, we have .c0 |f 1 (x)|/|x 1 | ≤ C0 |f 1 (y)|/|y 1 | . Hence, the fraction .|f 1 (x)|/|x 1 | is bounded and separated from zero. Due to (4.2.5), an analogous statement holds for .|f (x) − f (y)|/|x − y|, too. In the following lemma, we gather the properties of the admissible diffeomorphisms that will be used in the proof of Theorem 4.2.7 on the change of variables in .do. Lemma 4.2.2 Let W , .W1 ⊂ Rm n be open sets and .f : W → W1 an admissible diffeomorphism. Then: (1) The diffeomorphism .g : W1 → W inverse to f is also admissible. (2) The Jacobian .det f  is bounded away from zero on the set W . μ (3) If .a ∈ μ (respectively, .a ∈ 0 ) and .suppa ⊂ W × W × Rm , then the function m m .W1 × W1 × R  (x, y, ξ ) → a(g(x), g(y), ξ ) extended to be zero on .(Rm n × Rn × μ Rm ) \ (W1 × W1 × Rm ) belongs to . μ (respectively, . 0 ). Proof In (4.2.2), we change x, .y ∈ W for .g(x), .g(y) (x, .y ∈ W1 ) and obtain that the diffeomorphism g satisfies condition (2) of Definition 4.2.1. Next, the inequality .|g(x) − g(y)| ≤ C|x −y| implies that the derivatives .gpq = ∂gp /∂xq .(1 ≤ p, q ≤ m) are bounded on .W1 . Therefore, the determinant .det g  is also bounded, and hence, the determinant .det f  takes values separated from zero. The estimates .|∂ α gp (x)| ≤ C |x 1 |1−|α| established previously for .|α| = 1 are equivalent to |∂ β gpq (x)| ≤ C|x 1 |−|β| .

.

(4.2.6)

To prove inequality (4.2.6), we note that the entries .gpq ◦ f of the matrix .g  ◦ f are rational functions of the entries .fij of .f  . Since the denominator .det f  of these functions is bounded away from zero, we have |∂zβ (gpq ◦ f )(z)| ≤ C|z1 |−|β| ,

.

β ∈ Zm + , z ∈ W,

β

due to (4.2.1). The derivative .∂x gpq (x) is the sum of functions of the form   ∂zβ1 (gpq ◦ f )(z)z=g(x) (∂ γk gpk (x))lk ,

.

k

  where .|β1 | ≤ |β|, . k lk |γk | = |β|, . k lk = |β1 | (the numbers .pk corresponding to distinct k may coincide; the same is true for the multi-indices .γk ). Reasoning by induction, one can assume that the estimates (4.2.6) are valid with .β replaced by .γ such that .|γ | ≤ |β| − 1. Then, .|∂ γ gp (x)| ≤ C|x 1 |1−|γ | for .|γ | ≤ |β|, and since .|γk | ≤ |β| for all k,

4.2 Operators on Manifolds with Wedges

187

we obtain |∂zβ1 (gpq ◦ f )(g(x))



.

(∂ γk gpk (x))lk | ≤ C|g 1 (x)|−|β1 | |x 1 |l1 (1−|γ1 |)+... =

k

= C|g 1 (x)|−|β1 | |x 1 ||β1 |−|β| ≤ C1 |x 1 |−|β| . 2 2 It remains to prove assertion (3). For short, we further write .(Rm n ) , .W , etc., instead of m m 2 m .Rn × Rn , .W × W , etc. It follows from (4.2.3) and (4.2.4) that if the set .E ⊂ W × R is m 2 m closed in .(Rn ) × R , then its image under the mapping .F : (x, y, ξ ) → (f (x), f (y), ξ ) 2 m is also closed in .(Rm n ) × R . In particular, if the amplitude a satisfies the conditions of Lemma and .b(x, y, ξ ) = a(g(x), g(y), ξ ) (x, .y ∈ W1 ), then the set .W12 × Rm is a 2 m neighborhood of the set .suppb = F (suppa) in .(Rm n ) ×R . Therefore, setting .b(x, y, ξ ) = 2 m 2 m m 0 for .(x, y, ξ ) ∈ ((Rn ) × R ) \ (W1 × R ), we obtain a function of class .C ∞ on m 2 m ˜ y, ξ ) = b(x, y, ξ/|x 1 |). The inclusion .b ∈ μ to be established is .(Rn ) × R . Let .b(x, ∼μ equivalent to .b˜ ∈ , i.e., to the estimates γ ˜ |∂xα ∂yβ ∂ξ b(x, y, ξ )| ≤ Cαβγ |x 1 |−|α| |y 1 |−|β| ξ μ−|γ | .

.

(4.2.7)

˜ y, ξ ) = a(x, If .a˜ is a pre-amplitude corresponding to a by (4.1.1), then .b(x, ˜ y, ξ |g 1 (x)|/ 1 1 1 1 |x |). In accordance with the relations .c0 |x | ≤ |g (x)| ≤ C0 |x |, . |∂ α g(x)| ≤ Cα |x 1 |1−|α| , this implies (4.2.7). μ Finally, let .a ∈ 0 be a proper amplitude. According to Definition 4.1.2, there exist numbers .δ ∈ (0, 1) and .ε > 0 such that .a(x, y, ξ ) = 0 for .|x 1 |/|y 1 | ∈ (δ, δ −1 ) or .|x 2 − y 2 | > ε(1 + |x 1 |). It follows from (4.2.3) and (4.2.4) that the amplitude .b : (x, y, ξ ) → 

a(g(x), g(y), ξ ) possesses the same property.

4.2.2

Change of Variables in do

If W is an open subset of .Rm n , then .ρ(x, ∂W ) denotes the distance from the point .x ∈ W to the boundary .∂W of W . Definition 4.2.3 An open set .W ⊂ Rm n is called a privileged neighborhood of the set 1 .E ⊂ W , if there exists a number .ε > 0 such that .ρ(x, ∂W ) ≥ ε |x | for all .x ∈ E. Lemma 4.2.4 Let W be a privileged neighborhood of E, .f : W → W1 an admissible diffeomorphism, and .E1 = f (E). Then, .W1 is a privileged neighborhood of .E1 . Proof According to Definition 4.2.3, the inequality .ρ(z, ∂W ) ≥ ε|z1 | holds for all .z ∈ E with some .ε > 0. Let .g = (g1 , . . . , gm ) be the diffeomorphism inverse to f . The estimate .ρ(g(x), ∂W ) ≤ Cρ(x, ∂W1 ) follows from the boundedness of the derivatives

188

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

∂gp /∂xq ; here, C is independent of .x ∈ W1 . If .x ∈ E1 , then .g(x) ∈ E and .ρ(x, ∂W1 ) ≥ 

C −1 ρ(g(x), ∂W ) ≥ C −1 ε|g1 (x)| ≥ ε1 |x 1 |.

.

m m If a is a continuous function on the product .Rm n × Rn × R , then .supp1 a (respectively, .supp2 a) denotes the projection of the support of a onto the first (respectively, second) factor of this product. μ Definition 4.2.5 Let .W ⊂ Rm n be an open subset. We denote by . (W ) the class of all μ amplitudes .a ∈ , for which W is a privileged neighborhood of the set .supp1 a ∪ supp2 a. μ μ μ The class . 0 (W ) is defined by . 0 (W ) = 0 ∩ μ (W ). Set also . μ (W ) = Op ( μ (W )), μ μ . (W ) = Op ( (W )). 0 0

Our immediate objective is to prove the theorem on the change of variables in .do of class . μ (W ). To this end, we need the following lemma. Lemma 4.2.6 Let .W1 be a privileged neighborhood of the set .E1 and .g : W1 → W be an admissible diffeomorphism. There exists a number .ε1 > 0 such that the matrix-valued function  h(x, y) =

.

1

g  (y + t (x − y)) dt

(4.2.8)

0

is correctly defined in some neighborhood V of the set D = {(x, y) : x ∈ E1 , |x − y| ≤ ε1 |x 1 |},

.

and, in this neighborhood, its entries .hpq admit the estimates |∂xα ∂yβ hpq (x, y)| ≤ Cαβ |x 1 |−|α| |y 1 |−|β| .

.

(4.2.9)

The number .ε1 can be chosen so that the following inequality holds: .

inf{| det h(x, y)|; (x, y) ∈ V } > 0.

(4.2.10)

Proof According to Definition 4.2.3, we have .ρ(x, ∂W1 ) ≥ ε|x 1 | for all .x ∈ E1 . We may assume that .ε < 2. Set U = {x ∈ W1 : (∃ z ∈ E1 ) |x − z| < ε|z1 |/2 }.

.

Then, U is an open subset of .W1 containing .E1 , and .ρ(x, ∂W1 ) ≥ (ε/2)(1 − ε/2)|x 1 | if .x ∈ U . Let .ε1 = (ε/4)(1 − ε/2), V = {(x, y) : x ∈ U, |x − y| < 2ε1 |x 1 |}.

.

4.2 Operators on Manifolds with Wedges

189

Clearly, V is an open neighborhood of .D. If .(x, y) ∈ V , then the open ball (in .Rm n ) centered at x of radius .2ε|x 1 | contains y. Since .ρ(x, ∂W1 ) ≥ 2ε1 , this ball is a subset of .W1 . Thus, the function h is correctly defined for .(x, y) ∈ V . Since .2ε1 < 1, the condition .(x, y) ∈ V implies .xy ≥ 0. This, in turn, is followed by 1 1 .|tx + (1 − t)y| ≥ max{t|x |, (1 − t)|y |} for any .t ∈ [0, 1]. We now have  |∂xα ∂yβ hpq (x, y)| ≤

1

.

0

|∂xα ∂yβ gpq (y + t (x − y))| dt ≤

≤ Ct |α| (1 − t)|β| |tx + (1 − t)y|−|α+β| ≤ C |x 1 |−|α| |y 1 |−|β| . Note finally that .|h(x, y)| = |g  (x)| ≥ δ0 , where .δ0 > 0 is independent of .x ∈ U . The determinant .det h(x, y) and its entries satisfy (4.2.9). In particular, .|∇y det h(x, y)| ≤ C|y 1 |−1 ≤ C(1 − 2ε)−1 |x 1 | for .(x, y) ∈ V , and therefore, | det h(x, y)| ≥ | det h(x, x)| − | det h(x, x) − det h(x, y)| ≥

.

≥ δ0 − C(1 − 2ε1 )−1 |x 1 |−1 |x − y| ≥ δ0 − 2Cε1 (1 − 2ε1 )−1 . We subject .ε1 to the condition .2Cε1 (1 − 2ε1 )−1 ≤ δ0 /2 and obtain .| det h(x, y)| ≥ δ0 /2 on V .

 Assume now that .A = Op a, .a ∈ μ (W ), and .f : W → W1 , .g : W1 → W are mutually inverse diffeomorphisms. We define an operator B on functions .u ∈ Rm n by Bu = [A(u ◦ f )] ◦ g.

.

(4.2.11)

Note that although the composition .u ◦ f is defined only on W , the expression A(u ◦ f )(x) = (2π )−m

.

  ei(x−y)ξ a(x, y, ξ )(u ◦ f )(y) dydξ

makes sense due to the condition .supp2 a ⊂ W . Since .supp1 a ⊂ W , the function .v = [A(u ◦ f )] ◦ g is in .C ∞ (W1 ) and vanishes near .∂W1 . Setting .v(x) = 0 for .x ∈ Rm n \ W1 , ), which is identified with . Bu. we obtain an element of .C ∞ (Rm n Theorem 4.2.7 Let .f : W → W1 be an admissible diffeomorphism of open subsets of μ Rm n and .g : W1 → W the inverse diffeomorphism. If .a ∈ (W ), .A = Op a, and the μ operator B is defined by (4.2.11), then .B ∈  (W1 ).

.

Proof Fix a number .ε0 ∈ (0, 1). Let .ζ0 ∈ C ∞ (R+ ), .ζ0 (t) = 1 for .t < ε0 /2, and .ζ0 (t) = 0 for .t > ε0 . It is easily seen that .ζ (x, y) = ζ0 (|x − y|/|x 1 |) is in class . 0 . We set .a0 = ζ a, μ 0 .a1 = (1 − ζ )a (a, .a1 ∈ , since .ζ ∈ ) and represent the .do A in the form .A =

190

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

A0 +A1 , where .Aj = Op aj , .j = 0, 1. Let .Bj denote the operator defined by (4.2.11) with A replaced by .Aj . Since .B = B0 + B1 , to prove the inclusion .B ∈  μ (W1 ), it suffices to prove that .Bj ∈  μ (W1 ), .j = 0, 1. First, we consider the operator .B1 . We have   −m ei(g(x)−g(y))ξ b1 (x, y, ξ )u(y) dydξ, u ∈ Cc∞ (Rm .B1 u(x) = (2π ) n ), where .b1 (x, y, ξ ) = a1 (g(x), g(y), ξ )| det g  (y)|, if .(x, y) ∈ W1 , and .b1 (x, y, ξ ) = 0 otherwise. According to Lemma 4.2.2, the diffeomorphism g is admissible. Therefore, the entries .gpq of the determinant .det g  are subject to estimates (4.2.6). This and assertion (3) of the same Lemma imply that .b1 ∈ μ . Moreover, since .|g(x) − g(y)|/|g 1 (x)| ≤ C0 |x − y|/|x 1 |, we have b1 (x, y, ξ ) = 0 for |x − y|/|x 1 | ≤ C0−1 ε0 /2.

.

(4.2.12)

Let .G1 be the kernel of the operator .A1 . In the proof of Proposition 4.1.23, we have obtained the estimate |∂xα ∂yβ G1 (x, y)| ≤ CαβN |x 1 |N−m−|α| |y 1 |−|β| |x − y|−N

.

(4.2.13)

for all .α, .β ∈ Zm + , and .N ∈ Z+ . Denote by .G1 the kernel of .B1 . Then,  −m ei(g(x)−g(y))ξ b1 (x, y, ξ ) dξ = G1 (g(x), g(y)) | det g  (y)|. .G1 (x, y) = (2π ) From the properties of admissible diffeomorphisms and from (4.2.13), it easily follows that  α β 1m  ∂ ∂ (|x | G1 (x, y)) ≤ C 

.

αβN |x

x y

Let .v ∈ S(Rm ) and .(2π )−m



|

1 N−|α|

|y 1 |−|β| |x − y|−N .

(4.2.14)

v(η) dη = 1. We set

b1 (x, y, η) = ei(y−x)η |x 1 |m G1 (x, y)v(|x 1 |η).

.

Due to (4.2.12) and (4.2.14), the function .H : (x, y) → |x 1 |m G1 (x, y) satisfies the conditions of Lemma 4.1.22. Therefore, .b1 ∈ −∞ , and since .B1 = Op b1 , we have −∞ ⊂ μ . Taking into account .supp b ⊂ f (supp a ), .j = 1, 2,, and .B1 ∈  j 1 j 1 Lemma 4.2.4, we conclude that .B1 ∈  μ (W1 ). To prove the inclusion .B0 ∈  μ (W1 ), let us consider the matrix function h defined by (4.2.8). According to Lemma 4.2.6, there exists a number .ε1 > 0 such that the function h is correctly defined in some neighborhood V of the set D = {(x, y) : x ∈ f (supp1 a), |x − y|/|x 1 | ≤ ε1 }

.

4.2 Operators on Manifolds with Wedges

191

and, in this neighborhood, satisfies (4.2.9) and (4.2.10). The support of the function (x, y, ξ ) → a0 (g(x), g(y), ξ ) is a subset of

.

D1 = {(x, y) : x ∈ f (supp1 a) , |g(x) − g(y)|/|g 1 (x)| ≤ ε0 }.

.

In view of .c0 |x − y|/|x 1 | ≤ |g(x) − g(y)|/|g 1 (x)| for .ε0 ≤ c0 ε1 , the inclusion .D1 ⊂ D holds, and therefore, the mentioned support lies in domain V of h. Taking into account that .g(x) − g(y) = h(x, y)(x − y), we have B0 u(x) = (2π )−m

 

.

ei(g(x)−g(y))ξ a0 (g(x), g(y), ξ )| det g  (y)|u(y) dydξ = = (2π )−m

  ei(x−y)η b0 (x, y, η)u(y) dydη,

where .b0 (x, y, η) = a0 (g(x), g(y), t h(x, y)−1 η)| det g  (y)|| det h(x, y)|−1 . Due to Lemma 4.2.2, the function .(x, y, ξ ) → a0 (g(x), g(y), ξ ) belongs to the class . μ . Estimates (4.2.9) now imply that .b0 ∈ μ , and since .W1 is a privileged neighborhood of 

the set .supp1 b0 ∪ supp2 b0 , we have .b0 ∈ μ (W1 ). If in the conditions of Theorem 4.2.7 the amplitude a is proper, then the amplitudes .b0 and .b1 constructed in the proof of the theorem will also be proper (this immediately μ follows from assertion (3) of Lemma 4.2.2). Thus, .B ∈ 0 (W1 ). At the same time, the role of a can be given to the symbol .σA of A. Then, b0 (x, y, η) = ζ (g(x), g(y))σA (g(x), t h(x, y)−1 η)| det g  (y)| | det h(x, y)|−1 ,

.

and therefore, σB (x, η) ∼

.

 1  ∂ηα Dyα (σA (g(x), t h(x, y)−1 η)| det g  (y)| | det h(x, y)|−1 y=x α! α

in view of Theorem 4.1.16 (we have taken into account that the function . (x, y) → ζ (g(x), g(y)) equals 1 in a neighborhood of the set .x = y and that the amplitude .b1 , which vanishes in this neighborhood, does not influence .σB ). Using additional reasoning of combinatorial nature, this expansion can be transformed into σB (x, η) ∼

.

 1 (α)  α if  (z,w)η  σA (z, tf  (z)η)Dw e , w=z α! α

(4.2.15)

where .σA (z, ξ ) = ∂ξα σA (z, ξ ) and .f  (z, w) = f (w) − f (z) − f  (z)(w − z). Therefore, the following theorem is true. (α)

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4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

Theorem 4.2.8 Let the operator A in Theorem 4.2.7 be proper. Then, the .do B defined 

by (4.2.11) is also proper, and its symbol admits the asymptotic expansion (4.2.15).

4.2.3

Pseudodifferential Operators on a Wedge

For any subset D for the unit sphere .S N −1 ⊂ RN , we denote by .K(D) a cone in .RN \ 0 with the base D, K(D) = {z ∈ RN ; z = rϕ, r > 0, ϕ ∈ D}.

.

If D is a compact (without boundary) smooth .(n − 1)-dimensional submanifold of .S N −1 ,

m−n R . then the set .W = K(D) × Rm−n is a smooth m-dimensional surface in .RN Any such a surface will be called a surface of wedge type or just a wedge. If .N = n and n−1 , then .W = Rm . .D = S n Let .U (0) be an open (not necessarily connected) subset of D admitting a diffeomorphic mapping .υ (0) onto some subset of the sphere .S n−1 , .U = K(U (0) ) × Rm−n . We extend

m−n (0) to a mapping .υ = υ : U → Rn R .υ by setting U υ(rϕ, x 2 ) = (rυ (0) (ϕ), x 2 )

.

(r > 0, ϕ ∈ S N −1 , x 2 ∈ Rm−n ).

The pair .(U, υ) is called a local chart on the surface .W. Remark 4.2.9 Replacing, if necessary, the set .U (0) with its (open) subset, we may assume that all the derivatives of the coordinate functions of the mapping .υ (0) (in local coordinates on .S N −1 and .S n−1 ) are bounded. In what follows, we consider only local charts for which this condition is fulfilled. m−n , where K is an open cone Definition 4.2.10 Let .W ⊂ Rm n be a set of the form .K × R n ∞ m in .R . A function .ζ ∈ C (Rn ) is called a homogeneous cut-off function (HCF) serving the set W , if it is independent of .x 2 , homogeneous of degree 0, and .suppζ ⊂ W .

Definition 4.2.11 A pseudodifferential operator (a proper .do) of order .μ on a wedge .W is a linear mapping .A : Cc∞ (W) → C ∞ (W) such that, for any local chart and any .ζ1 , .ζ2 serving the set .W = υ(U ), the operator −1 ∗ ∗ −1 Cc∞ (Rm n )  u → ζ1 (υ ) Aυ (ζ2 u) ≡ ζ1 [A((ζ2 u) ◦ υ)] ◦ υ

.

μ

μ

is in the class . μ (W ) (.0 (W )). The set of all such .do is denoted by . μ (W) ( .0 (W)). To establish the general form of a .do on a wedge-type surface (Theorem 4.2.13), we need the following lemma.

4.2 Operators on Manifolds with Wedges

193

Lemma 4.2.12 Let .(U, υU be .(V , υV ) local charts on the wedge .W, .U ∩ V = ∅, and let .ζ and .η be the HCF serving the sets .υU (U ) and .υV (V ), respectively. We set .W  = υU (U ∩ V ), .W  = υV (U ∩ V ), .p = υV ◦ υU−1 |W  , .θ = η ◦ p. Then:   (1) The function .ζ θ extended to be zero on .Rm n \ W is an HCF serving the set .W .  (2) .W is a privileged neighborhood of the set .supp(ζ θ ). (3) .p : W  → W  is an admissible diffeomorphism.  Proof The function .W   (x 1 , x 2 ) → (ζ θ )(x 1 , x 2 ) extended to be zero on .Rm n \ W 2 is homogeneous of degree zero and independent of .x ; its support coincides with  .υU (supp (ζ ◦ υU )(η ◦ υV )) and is contained in .W . This proves assertion (1). Assertion (2) 

is obvious and (3) directly follows from Remark 4.2.9. (0)

Now consider some finite covering .{Uj }lj =1 of the manifold D such that for all .j, k = (0)

(0)

(0)

(0)

(0)

1, . . . , l the set .Uj k = Uj ∪ Uk is a coordinate neighborhood. Let .υj k : Uj k → S n−1 be a coordinate mapping chosen in accordance with Remark 4.2.9 and .{(Uj k , υj k )} be the corresponding atlas of local charts on .W. Theorem 4.2.13 Let us fix a number .μ ∈ R and for each pair .(j, k) choose a .do .Aj k ∈  μ (Wj k ) arbitrarily. Then, the operator A=



.

∗ υj∗k Aj k (υj−1 k )

(4.2.16)

j,k

is a .do of order .μ on the surface .W. Any .A ∈  μ (W) can be written in the form (4.2.16) with suitable .Aj k . (0)

(0)

Proof Consider a partition of unity .{χj }lj =1 subordinate to the covering .{Uj }lj =1 of D. For .r > 0, ϕ ∈ D, and .x 2 ∈ Rm−n , we set .χj (rϕ, x 2 ) = χj(0) (ϕ) and represent A in the  form .A = j,k χj Aχk . According to Definition 4.2.11, the operator −1 ∗ −1 ∗ ∞ m Aj k : u → (χj ◦ υj−1 k )(υj k ) Aυj k ((χk ◦ υj k )u), u ∈ Cc (Rn ),

.

∗ is in the class . μ (Wj k ) .(j, k = 1, . . . , l). Since .χj Aχk = υj∗k Aj k (υj−1 k ) , the second assertion of the theorem is proved. We now prove the first assertion. Let .aj k ∈ μ (Wj k ), .Aj k = Op aj k , .j, k = 1, . . . , l, and let the operator A be defined by (4.2.16). Since for any .(j, k), the set .Wj k is a privileged neighborhood of .supp1 aj k ∪ supp2 aj k (see Definition 4.2.5), there exists an HCF .ζj k , serving .Wj k such that .ζj k (x)aj k (x, y, ξ )ζj k (y) = aj k (x, y, ξ ). Let us consider

194

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

an arbitrary chart .(V , υV ) . Set .Wj k = υj k (Uj k ∩ V ) and .Wjk = υV (Uj k ∩ V ). If .η1 and ∞ m .η2 are HCFs serving the set .W = υV (V ), then for .u ∈ Cc (Rn ) we have η1 (υV−1 )∗ AυV∗ (η2 u) =



.

[(θj k ζj k Aj k ζj k θj k )(u ◦ pj k )] ◦ pj−1 k, (1)

(2)

(4.2.17)

j,k  where .pj k = υV ◦ υj−1 k |Wj k , .θj k = ηi ◦ pj k , .i = 1, 2, and the prime on the summation sign means that the summation extends only over the pairs .(j, k) for which .Uj k ∩ V = ∅. (i) By Lemma 4.2.12, the function .θj k ζj k , .i = 1, 2, is an HCF serving .Wj k . This implies (i)

that .θj k ζj k Aj k ζj k θj k ∈  μ (Wj k ). Moreover, due to the same lemma, the mapping .pj k : Wj k → Wj k is an admissible diffeomorphism. From Theorem 4.2.7, it now follows that the operator (1)

(2)

(1) (2) −1 Cc∞ (Rm n )  u → [(θj k ζj k Aj k ζj k θj k )(u ◦ pj k )] ◦ pj k

.

is in the class . μ (Wj k ). Taking into account formula (4.2.17) and the inclusion  μ (Wj k ) ⊂  μ (W ), we obtain .η1 (υV−1 )∗ AυV∗ η2 ∈  μ (W ).



.

Remark 4.2.14 If, on the right-hand side of (4.2.16), .Aj k is a proper .do, then the operator A defined by this formula is also proper. From the proof of Theorem 4.2.13, μ it now follows that the converse is also true: any .A ∈ 0 (W) admits a representation of μ the form (4.2.16), where .Aj k ∈ 0 (Wj k ). Corollary 4.2.15 Let .χ1(0) , .χ2(0) ∈ C ∞ (D), where D is the base of the cone K, .supp χ1(0) ∩ (0) (0) supp χ2 = ∅. For .r > 0, .ϕ ∈ D, and .x 2 ∈ Rm−n , we set .χi (rϕ, x2 ) = χi (ϕ), μ .i = 1, 2. Moreover, let .A ∈  (W), where .μ ∈ R is an arbitrary number. Then, .χ1 Aχ2 ∈  −∞ (W). Proof Using formula (4.2.16), we obtain .χ1 Aχ2 =



 ∗ υj∗k Aj k (υj−1 k ) , where .Aj k = (χ1 ◦

−1  μ υj−1 k )Aj k (χ2 ◦ υj k ). From Definition 4.2.11, it follows that .Aj k ∈  (Wj k ). In fact,  −∞ .A (Wj k ) in view of Proposition 4.1.23. 

jk ∈  (0)

(0)

Let .{Uj }lj =1 be a covering of D by coordinate neighborhoods, .{χj }lj =1 a partition of unity subordinate to this covering, and let the elements of this partition be homogeneous (0) cut-off functions. For each .j = 1, . . . , l, we choose a function .ηj ∈ C ∞ (D) satisfying (0)

(0)

⊂ Uj

(0) (0)

(0)

(0)

= χj . As before, we set .χj (rϕ, x 2 ) = χj (ϕ),   (0) 2 .ηj (rϕ, x ) = η ηj Aχj = (1 − ηj )Aχj j (ϕ). Taking into account the equality .A − and Corollary 4.2.15, we obtain the following statement. suppηj

.

and .χj ηj

Corollary 4.2.16 If .A ∈  μ (W), then .A −

 j

ηj Aχj ∈  −∞ (W).

4.2 Operators on Manifolds with Wedges

195

Corollary 4.2.17 Let .μ ∈ R be an arbitrary number and .A ∈  μ (W). Then, A admits μ the representation .A = A0 + A1 , where .A0 ∈ 0 (W), .A1 ∈ −∞ (W). To prove Corollary 4.2.17, one must use formula (4.2.16) and apply Proposition 4.1.24 to operators .Aj k . Concluding the section, we introduce admissible diffeomorphisms of open subsets of a wedge. Definition 4.2.18 Let .D ⊂ S N , .D1 ⊂ S N1 be smooth compact .(n − 1)-dimensional manifolds, and let .W = K(D) × Rm−n , .W1 = K(D1 ) × Rm−n . A diffeomorphism .f : Z → Z1 , where .Z ⊂ W and .Z1 ⊂ W1 are open subsets, is called admissible if for any two local charts .(U, υU ) and .(V , υV ) such that .U ∩ Z = ∅, .V ∩ Z1 = ∅, and .f (U ∩ Z) ∩ V ∩ Z1 = ∅, the mapping υV ◦ f ◦ υU−1 : υU (U ∩ Z ∩ f −1 (V ∩ Z1 )) → υV (f (U ∩ Z) ∩ V ∩ Z1 )

.

is an admissible diffeomorphism in the sense of Definition 4.2.1. Now, by analogy with Definitions 4.2.3 and 4.2.5, we can introduce classes . μ (Z) μ and .0 (Z) for any open .Z ⊂ W. These classes are invariant under admissible diffeomorphisms due to Theorem 4.2.7.

4.2.4

W-manifolds

A compact subset .M of a topological manifold is called a stratified manifold if: (a) .M is a finite union of pairwise disjoint topological manifolds (strata); (b) a closure of any stratum consists of the stratum itself and the union of some (possibly empty) set of strata of lower dimensions. Let .{si }i be a set of strata whose union is a stratified manifold .M. We set .m = max dim si , .ni = m − dim si , .I = {i : dim si < m}, and .M0 = ∪i∈I si . Below, we assume that: (I) .s¯i = si for .i ∈ I , where .s¯i is the closure of .si . (II) For each point of .si , .i ∈ I , there exist a neighborhood .O ⊂ M0 ∪ si of this point, an m-dimensional wedge .W = K × Rm−ni , an open set .V ⊂ W, and a homeomorphism m−ni ). . : O → V such that .(si ∩ O) = V ∩ ({0} × R It follows from conditions (I) and (II) that the (m-dimensional) manifold .M0 is dense in .M, and .M \ M0 is a disjoint union of compact manifolds (edges). The pair .(O, ) in condition (II) is called a special local chart on .M. If .O is a coordinate neighborhood on

196

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

M0 and .  is a homeomorphism of this neighborhood onto an open subset of .Rm , then the chart .(O ,   ) is called standard. An atlas on .M is any finite collection .C = {(Oj , j )} of local charts satisfying the following conditions:

.

 (i) . Oj = M. (ii) If .(Oj , j ), .(Ok , k ) ∈ C are special charts, whereas .Oj ∩ si1 = ∅, .Ok ∩ si2 = ∅, where .i1 , .i2 ∈ I , .i1 = i2 , then .Oj ∩ Ok = ∅. (iii) .Oj ⊂ M0 for any standard chart .(Oj , j ) ∈ C. Definition 4.2.19 An atlas .(Oj , j ) of local charts on a manifold .M with edges defines on .M a structure of w-manifold, if any two charts in this atlas are compatible, i.e., or −1 .Oj ∩ Ok = ∅, or .Oj ∩ Ok = ∅ and the mapping .k ◦  : j (Oj ∩ Ok ∩ M0 ) → j k (Oj ∩ Ok ∩ M0 ) is an admissible diffeomorphism of open subsets of the wedge when .(Oj , j ) and .(Ok , k ) are special charts and a diffeomorphism of smooth manifolds in other cases. The equivalence of two atlases is defined in the usual way. Equivalent atlases define the same w-structure on .M. As follows from Definition 4.2.19, the open dense part .M0 of .M is a smooth manifold. In what follows, when considering local charts on w-manifolds, we always assume that these charts are compatible with the w-structure. Definition 4.2.20 Let . : O → W be a special local chart on .M and let the function ζ ∈ C ∞ (W) possess the following properties: (1) The set .suppζ is bounded, and its closure (in .W) belongs to .V = (O). (2) If .(U, υ) is a local chart .W, then

.

|∂ α (ζ ◦ υ −1 )(x)| ≤ Cα |x 1 |−|α| ,

.

α ∈ Zm +.

Any function .ζ having these properties is called a cut-off function serving the set V . An analogous terminology will be applied to the function .χ = ζ ◦  (extended to be zero on  .M0 \ O) and to the set .O. If .O is a standard coordinate neighborhood, then any function ∞   .χ ∈ C (M0 ) satisfying .suppχ ⊂ O is called a cut-off function serving the set .O . The set of all cut-off functions serving the coordinate neighborhood .O ⊂ M is denoted by .CF(O). For a special coordinate neighborhood, the definition of the set .CF(O) is correct in view of Remark 4.2.9. Let .si , .i ∈ I , be an edge of the w-manifold .M, .{Oj }kj =1 a covering of this edge by special coordinate neighborhoods. Denote by W some neighborhood of .si satisfying k .W ⊂ j =1 Oj . Let .χ0 , χ1 , . . . , χk be a partition of unity on .M0 subordinate to the covering .M0 \ W , .O1 ∩ M0 , . . . , Ok ∩ M0 , while .χj ∈ CF(Oj ) for .j ≥ 1. Let also

4.2 Operators on Manifolds with Wedges

197

j = (j1 , j2 ), .j ≥ 1, be a coordinate mapping of the neighborhood .Oj onto an open subset of the closed surface .Wj = K j × Rm−nj of wedge type. We set

.

i (w) = χ0 (w) +

k 

.

χj (w)|j1 (w)|, w ∈ M0 .

(4.2.18)

j =1

Then, .i is a smooth positive function on .M0 , while .i = 1 in the vicinity of any edge distinct from .si , and .(i ◦  −1 )(x) ∼ r, .x ∈ O ∩ M0 , for any special coordinate system m−ni to the “edge” . : O → W, .O ∩ si = ∅; here, r is the distance from .x ∈ W = K × R m−n −1 i .R , and the relation .(i ◦  )(x) ∼ r means that for any compact set .E ⊂ (O),  , .c there exist constants .c,E ,E > 0 such that   c,E r ≤ (i ◦  −1 )(x) ≤ c,E r , x ∈ E ∩ W.

.

(4.2.19)

According to formula (4.2.18), any edge .si complies with an infinite set of functions depending on the choice of local charts .(Oj , j ) and cut-off functions .χj . Below, .i denotes an arbitrarily fixed function in this set. Introduce a Riemannian metric on the manifold .M0 . Let .C = {(Oj , .j )} be an atlas on .M. If .(Oj , j ) ∈ C is a special chart, then the metric on .Oj ∩ M0 will be defined as the pre-image of the Riemannian metric induced on .j (Oj ∩ M0 ) by the Euclidean metric of the surrounding space. For a standard chart .{(Oj , j )}, a (smooth) metric on .Oj is defined arbitrarily. The metric on the whole manifold .M0 is now pieced together using a partition of unity .{χj }, .χj ∈ CF(Oj ). If we choose another atlas .C or a partition of unity .{χj }, then the metric is changed to an equivalent metric. A smooth measure on .M0 induced by the metric is further denoted by .ν. Note that relations (4.2.19) are equivalent to the estimate     .c dist(·, si ) ≤ i (·) ≤ c dist(·, si ) for .i ; here .c , .c are some constants, and .dist(w, si ) is the distance from .w ∈ M0 to the edge .si defined by the metric.

4.2.5

do on w-manifold

In Sect. 4.2.4, we have defined a .do on a wedge as an operator that in any local coordinates .(U, υ) is an element of some class . μ (W ), .W ⊂ Rm n . To this end, coordinate neighborhoods .U = K(U (0) ) × Rm−n of the same kind were used; for any two points on a wedge, there exists a (possibly not connected) coordinate neighborhood of this kind containing these points. If we define a .do on a w-manifold in an analogous way, then an additional obstacle arises, since there exist neighborhoods of two types: special and standard. Moreover, the special neighborhoods may serve different strata (probably of distinct dimensions). This approach (in a more general situation) is developed in [29, 30]. Here, we choose another way, namely, let us define a .do on a w-manifold .M as a usual .do A on the smooth manifold .M0 , which, in any special coordinate system, is a .do

198

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

on a wedge-type surface. Then, we have to postulate properties of the kernel of operator A outside the diagonal .{(x, y) ∈ M0 × M0 : x = y}. To motivate the following definitions, let us consider any amplitude a in class . μ (Rm n ). ∞ m α 1 −|α| , Assume that functions .χ1 , .χ2 ∈ C (Rn ) satisfy the estimates .|∂ χi (x)| ≤ Cα |x | m m .i = 1, 2 (i.e., .χi ∈ CF(Rn )) and that the closures (in .R ) of their supports are disjoint. Set .b(x, y, ξ ) = χ1 (x)a(x, y, ξ )χ2 (y), .B = Op b. Since the amplitude b and all its derivatives vanish as .x = y, we have .B ∈  −∞ (Rm n ) in view of Proposition 4.1.24. Moreover, .

inf{|x − y|; (x, y) ∈ suppχ1 × suppχ2 } > 0,

and therefore, according to Lemma 4.1.5, for .|x 1 | + |y 1 | ≤ C, the kernel of B satisfies the inequalities |∂xα ∂yβ G(x, y)| ≤ CαβN |x1 |N |y1 |−|β|

.

(α, β ∈ Zm + , N ∈ Z+ ).

Extend these estimates to the case of a w-manifold. Let .{(Oj , j )} be an (finite) atlas  on .M, .{χj }, .χ ∈ CF(Oj ), a partition of unity on .M0 , and . = i∈I i , where .i are functions in (4.2.18). Definition 4.2.21 We will say that the operator  .Bu(z) = G(z, w)u(w) dν(w) , u ∈ Cc∞ (M0 × M0 ),

(4.2.20)

is in the class .ϒ(M) if .G ∈ C ∞ (M × M) and for all .α, .β ∈ Zm + , .N ∈ Z+ , the following inequality holds: sup |∂zα ∂wβ (χj (z)G(z, w)χk (w))| ≤ CαβN (z)N (w)−|β| .

.

(4.2.21)

j,k

The derivatives on the right-hand side of (4.2.21) are calculated in local coordinates on Oj and .Ok . At the same time, the coordinates on any special neighborhood .Oj are defined using maps .υjp ◦ (|O ∩ M0 ) : j−1 (Up ) ∩ Oj → Rm nj , where .{(Ujp , υjp )} is a finite atlas on .Wj . If other atlases .{(Oj , j )}, .{(Ujp , υjp )} and a partition of unity .{χj } are selected, then estimates (4.2.21) remain valid with some other constants .CαβN .

.

Definition 4.2.22 The operator (4.2.20) is called proper .(B ∈ ϒ0 (M)) if there exists a number .δ ∈ (0, 1) such that .G(z, w) = 0 whenever .i (z)/i (w) ∈ (δ, δ −1 ) for some .i ∈ I . The estimate (4.2.21) for the operator kernel .B ∈ ϒ0 (M) takes the form .

sup |∂zα ∂wβ (χj (z)G(z, w)χk (w))| ≤ CαβN (z)N1 (w)N2 . j,k

(4.2.22)

4.2 Operators on Manifolds with Wedges

199

(.N1 , .N2 ∈ Z+ , .C = C(α, β, N1 , N2 )), whereas the adjoint kernel .G∗ (x, y) = G(y, x) is also subject to this estimate. Definition 4.2.23 .do A of order .μ on a smooth manifold .M0 (in the sense of standard definition) is called a .do (proper .do) of order .μ on the w-manifold .M if: (1) For any local chart . : O → V , where V is an open subset of the closed wedge .W, and for any two cut-off functions .ζ1 , .ζ2 serving the set V , the operator1 Cc∞ (W)  u → ζ1 ( −1 )∗ A ∗ (ζ2 u)

.

μ

is in the class . μ (V ∩ W) (.0 (V ∩ W)). (2) .χ Aη ∈ ϒ(M) (.χ Aη ∈ ϒ0 (M)) for any pair of cut-off functions .χ, .η such that .dist (suppχ , suppη) > 0. Let .{(Oj , j )}lj =1 be an atlas on .M. Assume that a chart .(Oj , j ) is special if .j = 1, . . . , k, and standard if .j = k + 1, . . . , l. We set .Zj = j (Oj ). Let .χj , .ηj ∈ CF(Oj ) be functions such that:  (a) . lj =1 χj = 1. (b) .ηj χj = χj , .j = 1, . . . , l. (c) .dist(suppχj , supp(1 − ηj )) > 0, .j = 1, . . . , l. From Definition 4.2.23, it follows that, for any .do .A ∈  μ (M), the equality .A =  ηj Aχj + B holds with .B ∈ ϒ(M). Denoting .ζ1j = ηj ◦ j−1 , .ζ2j = χj ◦ j−1 , we rewrite this equality in the form Au =

.



ζ1j (j−1 )∗ Aj j∗ (ζ2j u) + Bu,

u ∈ Cc∞ (M0 ),

(4.2.23)

where .Aj ∈  μ (Zj ) for .1 ≤ j ≤ k, and for .k + 1 ≤ j ≤ l, the operator .Aj is a “usual” .do of order .μ on .Zj ⊂ Rm . If A is a proper .do, then the operators .Aj , .1 ≤ j ≤ k, and B are also proper. Suppose that the operators .A1 , . . . , Al , B satisfy the following conditions: μ

(1) .Aj ∈  μ (Zj ) (.Aj ∈ 0 (Zj )), .1 ≤ j ≤ k. (2) For .k + 1 ≤ j ≤ l, the operator .Aj is a “usual” .do of order .μ on .Zj ⊂ Rm . (3) .B ∈ ϒ(M) (.B ∈ ϒ0 (M)).

1 For short, we write . and . −1 instead of .|O ∩ M and . −1 |V ∩ W. 0

200

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges μ

Then, the operator A defined by (4.2.23) is in the class . μ (M) (.0 (M)). To prove this assertion, one can repeat (with minor modifications) the proof of Theorem 4.2.13; therefore, we do not do it here. μ To make compositions of proper .do, we have to show that if .A ∈ 0 (M) and 0 .B ∈ ϒ (M), then the operators AB and BA belong to the class .ϒ0 (M). An elementary verification of this fact using the estimates (4.2.22) is given to the reader.

4.3

Pseudodifferential Operators in Weighted Spaces

In this section, we define a scale .Hτs of weighted analogues of Sobolev spaces on a manifold .M with edges and prove the boundedness of .do at this scale.

4.3.1

Boundedness of Proper do of Non-positive Order in the Spaces L2,τ

∞ m For .τ ∈ R, we denote by .L2,τ = L2,τ (Rm n ) the completion of the set .Cc (Rn ) for the norm  1/2 .u; L2,τ  = |x 1 |2τ |u(x)|2 dx . μ

Theorem 4.3.1 Let .A = Op a, where .a ∈ 0 and .μ ≤ 0. Then, for any .τ ∈ R, the operator A defined on .Cc∞ (Rm n ) extends to a continuous mapping .L2,τ → L2,τ . Proof First, we assume that .τ = 0. Consider three cases: (a) .μ < −m. Then,  Au(x) =

.

G(x, y)u(y) dy, u ∈ Cc∞ (Rm n ).

Since the operator A is proper, inequalities (4.1.4) imply that |G(x, y)| ≤ CN |x 1 |N −m /(|x 1 |2 + |x − y|2 )N/2

.

(4.3.1)

for any .N ∈ Z+ . Note that if .N > m/2, then  .

Rm

|x 1 |N −m dy < +∞, (|x 1 |2 + |x − y|2 )N/2

(4.3.2)

4.3 Pseudodifferential Operators in Weighted Spaces

201

whereas the integral is independent of x. Indeed, after the substitution .y → |x 1 |y, this integral takes the form  .

|x 1 |N dy = (|x 1 |2 + |x − |x 1 |y|2 )N/2



dy dy = (1 + ||x 1 |−1 x − y|2 )N/2  dz dz < +∞. = (1 + |z|2 )N/2

Comparing (4.3.1) and (4.3.2), we obtain  C1 := sup

|G(x, y)| dy < +∞.

.

x

(4.3.3)

Since the operator A is proper, along with (4.3.1), the following inequality holds: |G(x, y)| ≤ CN |y 1 |N −m /(|y 1 |2 + |x − y|2 )N/2 ;

.

hence, the estimate  C2 := sup

|G(x, y)| dx < +∞

.

y

(4.3.4)

is also true. For u and v in .Cc∞ (Rm n ), we have |(Au, v)|2 ≤

 

.

|G(x, y)||u(y)||v(x)| dxdy

 ≤

≤

 |G(x, y)||u(y)|2 dxdy 

≤ C2

2

|G(x, y)||v(x)|2 dxdy ≤

 |u(y)| dy · C1 2

|v(x)|2 dx = C1 C2 u2 v2 .

Thus, .A; BL2 (Rm n ) < +∞. (b) .μ < 0. Due to the equality .Au2 = (A∗ Au, u), the boundedness of A follows from the boundedness of .A∗ A, which in an analogous way follows from the boundedness k k of .(A∗ A)2 for some .k ∈ Z+ . Choose k such that the order .μ2k of .(A∗ A)2 is less k than .−m. According to the first part of the proof, for such k, the operator .(A∗ A)2 is bounded. (c) .μ = 0. Denote by .σ and .σ ∗ the symbols of A and its adjoint operator .A∗ = τ A∗ (cf. Sect. 4.1.4). Formulas (4.1.17) and (4.1.24) imply A∗ Au(x) = (2π )−m/2

.



 ˆ ) dξ, eixξ |σ (x, ξ )|2 + σ−1 (x, ξ ) u(ξ

σ−1 ∈ S −1 .

202

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

Let us take .M > sup |σ (x, ξ )| and set .σ0 (x, ξ ) = (M 2 − |σ (x, ξ )|2 )1/2 . Then, for some .A−1 ∈  −1 , we have A∗ Au(x) = (2π )−m/2

.



 ˆ ) dξ + A−1 u(x). eixξ M 2 − σ0 (x, ξ )2 u(ξ

By Theorem 4.1.19, there exists an element .σ−∞ ∈ S −∞ such that the operator .B = Op (σ0 + σ−∞ ) is proper. Comparing B ∗ Bu(x) = (2π )−m/2

.



ˆ ξ ) dξ + A−1 u(x) , A−1 ∈  −1 , eixξ σ0 (x, ξ )2 u(x,

with the formula for .A∗ A, we obtain A∗ A = M 2 − B ∗ B + A−1 ,

.

A−1 ∈  −1 .

(4.3.5)

Since the operator (of multiplication by) .M 2 is a proper .do, it follows from (4.3.5) that the inclusion .A−1 ∈ 0−1 holds. Then, by part (b) of the proof, the operator  : L (Rm ) → L (Rm ) is continuous. Therefore, .A 2 2 n n −1 Au; L2 2 = (A∗ Au, u) = M 2 u2 − Bu2 + (A−1 u, u) ≤

.

≤ (M 2 + A−1 )u2 . This proves the theorem if .τ = 0. For arbitrary .τ , the theorem is now evident since the −τ τ boundedness of a .do A in .L2,τ (Rm n ) is equivalent to the boundedness of .d Ad in m m 1 .L2 (Rn ), where d is the function defined by .Rn  x → d(x) = |x |. 

4.3.2

Pseudodifferential Operators in the Spaces Hτs

This section has an overview character. For each .s ∈ R, we denote by .s some elliptic do of order s and by .Rs a parametrix of .s . Then, .Rs s = I + Ts , .Ts ∈ 0−∞ . Define ∞ m the space .Hτs = Hτs (Rm n ) for .τ, s ∈ R as the completion of .Cc (Rn ) for the norm

.

us,τ = (s u; L2,τ 2 + Ts u; L2,τ 2 )1/2 .

(4.3.6)

.

The function .Cc∞ (Rm n )  u → us,τ is indeed a norm: if .us,τ = 0, then .s u = Ts u = 0 and hence, .u = (Rs s − Ts )u = 0. μ

Theorem 4.3.2 Assume that .τ , s, t, .μ ∈ R, .μ ≤ s − t, and .A ∈ 0 . Then, the operator A extends to a continuous mapping .Hτs → Hτt .

4.3 Pseudodifferential Operators in Weighted Spaces

203

Proof For short, let us write . ·  instead of .·; L2,τ . From .I = Rs s − Ts Rs s + Ts2 , it follows that t Au ≤ t A(Rs − Ts Rs )s u + t ATs2 u.

.

Since .Rs ∈ 0−s , the operator .t A(Rs −Ts Rs ) is in the class .0 , and since .t −s+μ ≤ 0, this operator is continuous in .L2,τ . This is also true for the operator .t ATs . Therefore, t−s+μ

t Au ≤ C1 (s u + Ts u) .

.

(4.3.7)

Analogously, one can prove the inequality Tt Au ≤ C2 (s u + Ts u).

.

Comparing (4.3.7) and (4.3.8), we obtain .Aut,τ ≤ Cus,τ .

(4.3.8) 

Remark 4.3.3 Theorem 4.3.2 is valid for arbitrary elliptic operators .s ∈ 0s , .t ∈ 0t and their parametrices .Rs and .Rt participating in the definition of the norms . · s,τ and . · t,τ . Corollary 4.3.4 Any proper .do of order .−∞ extends to a continuous mapping .Hτs → Hτt for all .τ , s, .t ∈ R. ˜ s are elliptic operators in . s , .Rs , .R˜ s are Assume that .τ , .s ∈ R, .s , and that . 0 ˜ s so that .Rs s = I + Ts , .R˜ s  ˜ s = I + T˜s , whereas .Ts , .T˜s ∈  −∞ . parametrices of .s , . 0 ∞ m In accordance with (4.3.6), we define norms . · s,τ and . · ∼ s,τ on .Cc (Rn ). Theorem 4.3.5 The norms . · s,τ and . · ∼ s,τ are equivalent. Proof Since the norms . · s,τ and . · ∼ s,τ are interchangeable, it is sufficient to prove that ∼ ∞ m ˜ s and .T˜s do not exceed . · s,τ ≤ C · s,τ for .u ∈ Cc (Rn ). The orders of the operators . s; therefore, Theorem 4.3.2 is followed by the inequality ˜ s u20,τ + T˜s u20,τ )1/2 ≤ Cus,τ . (

.

(4.3.9)

According to Remark 4.3.3, we can suppose that .0 = I and .T0 = 0. Then, . · 0,τ = 

·; L2,τ , and inequality (4.3.9) takes the form .u∼ s,τ ≤ Cus,τ .

204

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

Corollary 4.3.6 If .s ∈ Z+ , then the norm . · s,τ is equivalent to the norm us,τ =

s  

.

|x 1 |2(τ +|α|) |∂ α u(x)|2 dx

1/2

.

(4.3.10)

|α|=0

Proof We restrict ourselves to proving this assertion for even s. For any multi-index .α ∈ 1 |α| α Zm + , let us denote by .Qα the differential operator .u → |x | ∂ u. Then, us,τ =

s 

.

Qα u; L2,τ 2

1/2

.

|α|=0 |α|

Since .Qα ∈ 0 , Theorem 4.3.2 implies the estimate .us,τ ≤ C us,τ . To obtain the  converse estimate, we define a norm .·s,τ using the elliptic operator .s = s|α|=0 cα Qα . The inequality .s u; L2,τ  ≤ Cus,τ is then obvious, and we have only to prove  .Ts u; L2,τ  ≤ Cus,τ . However, this is also obvious, since the order of the (proper)  .do .Ts is non-positive and that is why .Ts u ; L2,τ  ≤ Cu; L2,τ  ≤ Cus,τ . 

∞ m Proposition 4.3.7 For .s ≥ t, the identity operator .Cc∞ (Rm n ) → Cc (Rn ) extends to an s t s,t s embedding .Hτ → Hτ , i.e., continuous injective mapping .Iτ : Hτ → Hτt .

Proof Since .I ∈ 00 and .t ≤ s, the operator I extends to a continuous mapping .Iτs,t : Hτs → Hτt (Theorem 4.3.2), and it suffices only to prove that the operator .Iτs,t is injective. ∞ m In other words, we have to show that if a sequence .{uk }∞ k=1 ⊂ Cc (Rn ) is a Cauchy sequence in the norm . · s,τ and tends to zero in the norm . · t,τ , then .uk s,τ → 0. Suppose first that .t = 0 and that s is an even number. According to Corollary 4.3.6, the sequence .{uk } is a Cauchy sequence in the norm . · s,τ . Therefore, for each .α ∈ Zm +, .|α| ≤ s, there exists a function .vα ∈ L2,τ satisfying .

lim |x 1 ||α| ∂ α uk − vα ; L2,τ  = 0.

k→∞

(4.3.11)

Let .w ∈ Cc∞ (Rm n ). From (4.3.11), the inequality   . lim ∂ α uk (x)w(x) dx = vα (x)w(x) dx k→∞

follows. At the same time, since .uk ; L2,τ  = uk 0,τ → 0, we have  .

lim

k→∞

 ∂ α uk (x)w(x) dx = lim

k→∞

uk (x)(−∂)α w(x) dx = 0.

Thus, .vα = 0 almost everywhere, and hence, .|x 1 ||α| ∂ α uk → 0 in .L2,τ for all .α ∈ Zm +, .|α| ≤ s. Then, .uk s,τ → 0 due to Corollary 4.3.6.

4.3 Pseudodifferential Operators in Weighted Spaces

205

Suppose now that s, .t ∈ R, .s − t ∈ Z+ , while .s − t is an even number. From the definitions of the norms . · s,τ and . · t,τ , it now follows that: (a) .{s uk }, .{Ts uk } are Cauchy sequences in the norm of .L2,τ . (b) The sequences .{t uk }, .{Tt uk } tend to zero in this norm. We need to show that then .

lim s uk ; L2,τ  = 0,

lim Ts uk ; L2,τ  = 0

(k → ∞).

(4.3.12)

Since .Ts uk = Ts Rt t uk − Ts Tt uk , the second relation in (4.3.12) follows from (b) and the inclusions .Ts Rt , .Ts ∈ 0−∞ . Prove now the first relation. Set .vk = t uk , .k ∈ N. According to the definition of the norm in .Hτs−t , we have vk − vl s−t,τ = (s−t (vk − vl ); L2,τ 2 + Ts−t (vk − vl ); L2,τ 2 )1/2 .

.

One can assume that .s = s−t t and rewrite the preceding equality in the form vk − vl s−t,τ = (s uk − s ul ); L2,τ 2 + Ts−t (vk − vl ); L2,τ 2 )1/2 .

.

(4.3.13)

From the definition of the sequence .{vk }, taking into account of (b), we deduce that the second summand on the right-hand side of (4.3.13) tends to zero as .k, l → ∞. The first summand also tends to zero in view of (a). Thus, .{vk } is a Cauchy sequence in the norm . · s−t,τ . Moreover, .vk 0,τ ∼ vk ; L2,τ  = t uk ; L2,τ  → 0. Since .s − t ∈ 2Z+ , the conditions of the foregoing case are fulfilled, and we conclude that .lim vk s−t,τ = 0. All the more .

lim s−t vk ; L2,τ  = lim s uk ; L2,τ  = 0,

as needed. Let finally s, .t ∈ R be arbitrary numbers satisfying .t ≤ s. Take .r ∈ (−∞, t] so that the inclusion .s − t ∈ 2Z+ holds. The obvious relation .Iτs,r = Iτt,r ◦ Iτs,t and the injectivity of 

the mapping . Iτs,r imply that .Iτs,t is injective. Theorem 4.3.8 For any .τ , .s ∈ R, the bilinear form ∞ m ∞ m .Cc (Rn ) × Cc (Rn )

  (u, v) →

u(x)v(x) dx

−s extends to a pairing between the spaces .Hτs and .H−τ , i.e., to a continuous bilinear form on −s −s s s .Hτ × H−τ . The spaces .Hτ and .H−τ are mutually conjugate with respect to this pairing.

206

4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges

Theorem 4.3.9 For any .τ , .s ∈ R, the bilinear form ∞ m Cc∞ (Rm n ) × Cc (Rn )  (u, v) →

.

 |x 1 |2τ u(x)v(x) dx

extends to a pairing between the spaces .Hτs and .Hτ−s . These spaces are mutually conjugate with respect to this pairing. We conclude this section with a theorem on continuity of non-proper .do in the scale of .Hτs . Theorem 4.3.10 Let .τ , s, t, .μ ∈ R, .|τ | < n/2, .μ ≤ s − t, and let .A ∈  μ . Then, the operator A extends to a continuous mapping .Hτs → Hτt .

4.3.3

Pseudodifferential Operators on Spaces with Weighted Norms on w-Manifolds

Let .W be an m-dimensional wedge2 and .{(Uj , υj )}lj =1 a partition of unity formed by homogeneous cut-off functions serving the sets .Uj (cf. Definition 4.2.10). For any .τ , .s ∈ R, the space .Hτs (W) can be defined as the completion of .Cc∞ (W) with respect to the norm u; Hτs (W) =

l 

.

(υj−1 )∗ (χj u)2s,τ

1/2

j =1

(the function .χj u is extended on .Rm \ υj (Uj ) as zero). Let .M be an m-dimensional w-manifold and .M0 its open dense part (the union of strata of maximal dimension). Consider an atlas .{(Oj , j )}lj =1 on .M consisting of special local charts .{(Oj , j )}kj =1 and standard charts .{(Oj , j )}lj =k+1 . Remind that for .1 ≤ j ≤ k, the image .υj (Oj ) of the neighborhood .Oj is an open subset of some closed surface .Wj of wedge type. Let .{si }i∈I be the set of strata satisfying .dim si < m. To each index .j = 1, . . . , k, there corresponds an index .i ∈ I equal to the index of the stratum that intersects with the neighborhood .Oj . We denote this correspondence by h and write .i = h(j ). To each stratum .si , .i ∈ I , we assign a real number .τi in an arbitrary manner. We denote the set .{τi }i∈I by .τ .

2 We use notions and notations introduced in Sects. 4.2.3–4.2.5.

4.3 Pseudodifferential Operators in Weighted Spaces

207

For any number .s ∈ R and any set .τ ∈ RI , we introduce the space .Hτs (M) as the completion of .Cc∞ (M0 ) with respect to the norm .

u; Hτs (M) k l (j−1 )∗ (χj u); Hτsi (Wj )2 + = j =1

j =k+1

(j−1 )∗ (χj u); H (s) (Rm )2

1/2

;

here, .i = h(j ), .{χj }lj =1 is a partition of unity on .M0 formed by cut-off functions .χj ∈ CF(Oj ), .H (s) (Rm ) is the Sobolev space in .Rm . Another choice of the atlas .{(Oj , j )} and the functions .{χj } leads to an equivalent norm in .Hτs (M). μ

Theorem 4.3.11 Assume that .τ = {τi }i∈I , s, t, .μ ∈ R, .μ ≤ s − t, and .A ∈ 0 (M) . Then, A extends to a continuous map .Hτs (M) → Hτt (M). If .A ∈  μ (M), then the statement remains valid under the additional condition .τi ∈ (−ni /2, ni /2), .i ∈ I . Proof In view of formula (4.2.23), it is sufficient to apply Theorems 4.3.2 and 4.3.10 and use that if .B ∈ ϒ0 (M), then the mapping .B : Hτs (M) → Hτt (M) is continuous for all s, .t ∈ R, and .τ ∈ RI ; in the case .B ∈ ϒ(M), the mapping .B : Hτs (M) → Hτt (M) is  continuous for .τ = {τi } ∈ RI satisfying .τi ∈ (−ni /2, ni /2) and for arbitrary s, .t ∈ R.

C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

5

.

In this chapter, we consider w-manifolds and pseudodifferential operators subject to some additional “regularity” conditions near edges. The spectrum of the .C ∗ -algebra generated by these operators is studied.

5.1

Classes Ψ μ

m Definition 5.1.1 A function a˜ ∈ C ∞ (Rm n × R ) is called a presymbol of orderμ, μ ∈ C, if

(1) ∂rk ∂x 2 ∂ξ a(r ˜ ·, x 2 , ξ ); C q (S n−1 ) ≤ Ckqβγ ξ Re μ−|γ | for all k, q ∈ Z+ , β ∈ Zm−n + , m γ ∈ Z+ . m (2) There exists the limit a˜ (0) (x, ξ ) = limt→+∞ t −μ a(x, ˜ tξ ), (x, ξ ) ∈ Rm n × (R \ 0). (3) The function (x, ξ ) → a(x, ˜ ξ ) − ζ (ξ )a˜ (0) (x, ξ ), where ζ ∈ C ∞ (Rm ), ζ (ξ ) = 0 for |ξ | < 1/2 and ζ (ξ ) = 1 for |ξ | > 1, satisfies condition 1) with μ replaced by μ − 1. β

γ

Set  = R × S n−1 × Rm−n , + = R+ × S n−1 × Rm−n and denote by q the bijection ∞ m 2 2 m m + × Rm → Rm n × R : (r, ϕ, x , ξ ) → (rϕ, x , ξ ). Assume that C (Rn × R ) = −1 ∞ m {f ◦q ; f ∈ C (+ ×R )}. Condition 1) of Definition 5.1.1 implies that all presymbols ∞ m belong to C (Rm n × R ). Definition 5.1.2 To each presymbol a˜ of arbitrary order μ, we associate a function a(x, ξ ) = a(x, ˜ |x 1 |ξ ), which is called a symbol (of the same order). The class of all μ μ m presymbols (symbols) of order μ is denoted by S˜ (Rm n ) (S (Rn )).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Plamenevskii, O. Sarafanov, Solvable Algebras of Pseudodifferential Operators, Pseudo-Differential Operators 15, https://doi.org/10.1007/978-3-031-28398-7_5

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If a ∈



μ m μ S (Rn ),

then the formula (Au)(x) = (2π )−m/2

.

 eixξ a(x, ξ )u(ξ ˆ ) dξ

∞ m defines a linear operator A = Op a from Cc∞ (Rm n ) to C (Rn ). The kernel of this operator is defined by the oscillatory integral

G(x, y) = (2π )−m

.

 ei(x−y)ξ a(x, ξ ) dξ,

m (x, y) ∈ Rm n × Rn .

Definition 5.1.3 An operator A = Op a with a ∈ Sμ is called a pseudodifferential μ m operator of order μ on Rm n of class Ψ (Rn ). We say that an operator A is proper if for some δ ∈ (0, 1) the support of its kernel is contained in the domain {(x, y) : |x 1 |/|y 1 | ∈ μ (δ, δ −1 ), |x 2 − y 2 | < δ −1 (1 + |x 1 |)}. We denote by Ψ0 (Rm n ) the class of all proper do of order μ. The similarity of the notations μ and Ψ μ for old and new classes of do does not lead to confusion, since only new classes are discussed in this section. It is easy to prove that μ μ μ+ν ν m ∗ m if A ∈ Ψ0 (Rm (Rm n ), B ∈ Ψ0 (Rn ), then A ∈ Ψ0 (Rn ), AB ∈ Ψ0 n ). (The operators Reμ Re(μ+ν) ∗ m m (Rn ), and it remains only to A and AB belong to old classes 0 (Rn ) and 0 prove that the symbols of these operators satisfy the requirements of Definition 5.1.1.) The definitions of the classes Ψ (W) and Ψ0 (W) of pseudodifferential operators on a wedge W are quite similar to the definitions of (W) and 0 (W) in Sect. 4.2.3. For the classes μ Ψ μ (W) and Ψ0 (W), Theorem 4.2.13 remains valid. Assume that W = K × Rm−n , where K is a smooth n-dimensional conical surface (in some space RN ) with base D. We set (W) = R × D × Rm−n and + (W) = R+ × D × Rm−n . Denote by pW the bijection + (W) → W : (r, ϕ, x 2 ) → (rϕ, x 2 ). This bijection can be extended into a continuous mapping pW : + (W) → W. If V is an open subset of W, then its preimage p −1 W (V ) is a manifold with (possibly empty) boundary. Let us consider m-dimensional surfaces W1 = K1 × Rm−n , W2 = K2 × Rm−n , some points (0, x 2 ) ∈ W1 , (0, y 2 ) ∈ W2 , and their neighborhoods V1 ⊂ W1 , V2 ⊂ W2 . Definition 5.1.4 A mapping f : V1 → V2 is called an admissible diffeomorphism if −1 −1 f ◦ (p1 |p −1 1 (V1 )) = p2 ◦ F , where F : p 1 (V1 ) → p 2 (V2 ) is a diffeomorphism (of class ∞ C ) of manifolds with boundary. It is clear that f induces a diffeomorphism of V1 ∩ (0 × Rm−n ) onto V2 ∩ (0 × Rm−n ). Note also that a diffeomorphism admissible in the sense of Definition 5.1.4 may be not admissible in the sense of Definitions 4.2.1 and 4.2.18. However, this is insignificant for what follows: the definition of do on a manifold involves multiplication of the operator by a cut-off function, and, in some neighborhood of the support of this function,

5.1 Classes Ψ μ

211

a diffeomorphism in Definition 5.1.4 satisfies the requirements of the other mentioned definitions. We will use the notations and notions introduced in Sect. 4.2.4. In accordance with the new definition of admissible diffeomorphisms, we change the definition of a w-manifold. Namely, let M be a stratified manifold satisfying conditions (I) and (II) in this section, and let (O1 , 1 ), (O2 , 2 ) be special local charts on M such that O1 ∩ O2 = ∅. These charts are called compatible if the mapping 2 ◦ 1−1 : 1 (O1 ∩ O2 ∩ M0 ) → 2 (O1 ∩ O2 ∩ M0 ) is an admissible diffeomorphism in the sense of Definition 5.1.4. Atlases and a w-structure on M are introduced in the same way as in 4.2.4. Note that a w-structure on M induces a smooth structure on each edge si , i ∈ I . For further needs, we have to take some more narrow classes of cut-off functions. Let ζ −1 be a smooth function on a wedge W. We say that ζ is in the class C ∞ c (W) if ζ = η ◦ pW , ∞ ∞ where η ∈ Cc (+ (W)). By definition, a function ζ in C c (W) is serving an open set V ∈ W if the closure (in W) of its support lies inside V . Let now M be a w-manifold, O a special coordinate neighborhood,  : O → W a coordinate diffeomorphism. A function χ ∈ C ∞ (M0 ) is called a cut-off function serving the set O if χ = ζ ◦ (|M0 ), ζ ∈ C∞ c (W), whereas the function ζ is subject to (O). If O is a standard coordinate neighborhood, then any function in Cc∞ (O) is called a cut-off function serving O. The set of all cut-off functions serving O is denoted (as in Sect. 4.2) by CF(O). Using the new cut-off functions, we can introduce a Riemannian metric on M0 (sf. Sect. 4.2.4) and define functions i by (4.2.18). We denote by ν a smooth measure on M0 induced by the metric. Turn to the description of classes Ψ μ , which repeats the content of Sect. 4.2.5 with minor changes. Consider an arbitrary do in Rm n of the form η1 (Op a)η2 , where a ∈  μ ∞ (Rm ) are such that the closures (in Rm ) of their S is a symbol, and η , η ∈ C 1 2 c n μ supports are disjoint. Let H be the kernel of this operator, H (x, y) = (2π )

.

−m

 ei(x−y)ξ η1 (x)a(x, ξ )η2 (y) dξ.

Then, H ◦ (p, p) ∈ Cc∞ ( × )|(+ × + ); for W = Rm n we write p instead of pW . As simple calculations show (cf. Sect. 4.2.5), |H (x, y)| ≤ CN |x 1 |N for all N ∈ N. Conversely, if a function H satisfies the mentioned properties, then the operator  Bu(x) =

.

H (x, y)u(y) dy

 is a smoothing do, i.e., it belongs to μ∈C Ψ μ (Rm n ). For further needs, we note that the mapping H → H ◦ (p, p) generates an isomorphism m ˆ ∞ m ∼ ∞ C∞ c (Rn )⊗C c (Rn ) = Cc ( × )|(+ × + ).

.

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Definition 5.1.5 We say that an operator  Bu(z) =

.

H (z, w)u(w) dν(w), u ∈ Cc∞ (M0 ),

ˆ ∞ belongs to the class Υ (M) if H ∈ C ∞ c (M0 )⊗C c (M0 ) and the inequality |H (z, w)| ≤ CN (z)N holds for any N ∈ N. The operator B is called proper (B ∈ Υ0 (M)) if there exists a number δ ∈ (0, 1) such that H (z, w) = 0 whenever i (z)/i (w) ∈ (δ, δ −1 ) for at least one i ∈ I . Let U be an open subset of Rm . Denote by Sμ (U), μ ∈ C, the set of all C ∞ -functions a in U × Rm admitting the representation a = a0 + a1 with a0 , a1 ∈ C ∞ (U × Rm ), whereas a0 is a homogeneous function of degree μ for large |ξ |, and a1 satisfies the estimates .

β

sup{|∂xα ∂ξ a1 (x, ξ )|; x ∈ E} ≤ Cα,β,E ξ Reμ−|β|−1

∞ ∞ for all α, β ∈ Zm + and any compact set E. An operator A : Cc (M0 ) → C (M0 ) is called a pseudodifferential operator of order μ on a smooth manifold M0 if, for each (probably non-connected) coordinate neighborhood O ⊂ M0 and each diffeomorphism  : O → U ⊂ Rm , the mapping A : u → [A(u ◦ )] ◦  −1 , u ∈ Cc∞ (U), is a do of order μ in U . The latter means that

A u(x) = (2π )−m/2

.

 eixξ a (x, ξ )u(ξ ˆ ) dξ,

where a is an element in Sμ (U ). (0) (0) Put a (x, ξ ) = limt→+∞ t −μ a (x, tξ ). For a fixed x, the function ξ → a (x, ξ ) is homogeneous of degree μ. It follows from the formula for change of variables in do that a(0) is an expression in local coordinates for some function a (0) well-defined on the complement of zero section of the cotangent bundle over M0 . The function a (0) is called the principal symbol of do A. The class of all do of order μ on a manifold M0 is denoted by Ψ μ (M0 ). According to the standard definition, an operator A ∈ Ψ μ (M0 ) is μ called proper (A ∈ Ψ0 (M0 )) if both projections from the support of its kernel—subset of the product M0 × M0 —onto the factors of this product are proper mappings. The following definition of the classes Ψ (M) is quite similar to Definition 4.2.23. Previously, let us agree that supports of cut-off functions χ1 and χ2 are called disjunctive if the closures of these supports (in M) are disjoint. Definition 5.1.6 The class Ψ μ (M) of pseudodifferential operators of order μ on a wmanifold M consists of all operators A ∈ Ψ μ (M0 ) satisfying the following properties:

5.2 C ∗ -Algebra Generated by Proper do. Local Algebras

213

(1) For any special coordinate system  : O → W and any two cut-off functions χ1 , χ2 ∈ CF(O), the operator u → [(χ1 Aχ2 )(u ◦ 0 )] ◦ 0−1 ,

.

u ∈ Cc∞ (W),

where 0 = |(O ∩ M0 ), belongs to Ψ μ (W); (2) χ1 Aχ2 ∈ Υ (M) for any pair of cut-off functions χ1 , χ2 with disjunctive supports. μ μ We change Ψ μ (M0 ), Ψ μ (W), and Υ (M) for Ψ0 (M0 ), Ψ0 (W), and Υ0 (M) and μ obtain the definition of the class Ψ0 (M) of proper do.

C ∗ -Algebra Generated by Proper do. Local Algebras

5.2

Let, as before, .M be a w-manifold, .M0 its open dense part (i.e., the union of all strata of maximal dimension), .m = dim M0 . Let .A denote the .C ∗ -algebra generated in .L2 (M) ≡ ∞ L2 (M, ν) by proper . do of class .Ψ00 (M). Denote by .C (M0 ) the set of functions f in ∞ .C (M0 ), that near an edge admit the representation .f = h◦(|M0 ), where .h ∈ C∞(W) ∞ and . is a coordinate diffeomorphism. Define also .C(M0 ) as the closure of .C (M0 ) in the algebra .Cb (M0 ) of bounded continuous functions. Multiplication operators .(f ·), ∞ 0 .f ∈ C(M0 ), belong to .A, since .(f ·) ∈ Ψ (M) for .f ∈ C (M0 ). We will consider the 0 algebra .C(M0 ) as a subalgebra of .A. For each .z ∈ M , we put .Iz = {f ∈ C(M) : f (z) = 0} and denote by .Jz the least ideal in .A containing .Iz . Proposition 5.2.1 The following assertions are true: (1) The algebra .A is irreducible. (2) .KL2 (M) ⊂ A. (3) The following equality holds: Aˆ =



.

(A/Jz )∧ ∪ [I d],

z∈M

where .[I d] is the equivalence class containing the identity representation of .A in L2 (M). If .z = x, then the sets .(A/Jz )∧ and .A/Jx )∧ of the spectrum .Aˆ are disjoint.

.

Proof One can prove that .A is irreducible arguing as in the proof of Proposition 2.1.2. If f1 , .f2 ∈ C(M0 ) are functions with disjoint supports, then .f1 Af2 ∈ Υ (M) ⊂ KL2 (M) for any .A ∈ Ψ00 (M). Thus, .KL2 (M) ⊂ A. Note that .C(M) is a subalgebra of both .C(M) and .A. Let us apply Proposition 1.3.24 taking .C(M) and .KL2 (M) as .C and .J . It remains to take into account that .C(M)∧ = M and that the spectrum of .KL2 (M) ∗ ∧ consists of the only element .[I d].   .

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

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We now turn to the description of local algebras .A/Jz . The following statement is in essence similar to Proposition 2.1.5. Proposition 5.2.2 Assume that .A ∈ Ψ00 (M0 ) and that .a (0) is the principal symbol of A. For each .z ∈ M0 , the mapping .A → a (0) |Sz∗ generates an isomorphism .A/Jz ∼ = C(Sz∗ ). Turn to the local algebras .A/Jz for .z ∈ M \ M0 . Let .W = K × Rm−n be a wedge, where K denotes an n-dimensional conical surface in .RN . Let also .L2 (W) be the space of functions square integrable in a Euclidean surface measure and .A(W) be a subalgebra in .BL2 (W) generated by operators of the form .χ Aχ , where .A ∈ Ψ00 (W), .χ ∈ C ∞ c (W). For any .t > 0 we define a unitary operator .Ut : L2 (W) → L2 (W) by .Ut u = t m/2 u(t ·). If 0 .A ∈ Ψ (W) and 0 Aυ u(x) = (2π )−m/2

 eixξ aυ (x, ξ )u(ξ ˆ ) dξ ,

.

u ∈ Cc∞ (υ(U )),

is a representation of A in local coordinates .υ : U → Rm , then (Ut AUt−1 )υ u(x) = (2π )−m/2



.

= (2π )−m/2

eixξ aυ (tx, ξ/t)u(ξ ˆ ) dξ =  eixξ a˜ υ (tx, |x 1 |ξ )u(ξ ˆ ) dξ

(see definition of local coordinates on a wedge in Sect. 4.2.3). This implies that the formula A0 u = lim (Ut AUt−1 )u ,

.

t→0

u ∈ Cc∞ (W),

(5.2.1)

defines an operator .A0 that admits an extension to a bounded operator in .L2 (W), whereas .A0  ≤ A . In local coordinates, (A0 )υ u(x) = (2π )−m/2

.

 eixξ (a˜ υ )0 (ϕ, |x 1 |ξ )u(ξ ˆ ) dξ,

(5.2.2)

where .(a˜ υ )0 (ϕ, ξ ) = lim a˜ υ (tx, ξ ) and .ϕ = x 1 /|x 1 |. t→0

Denote by .A0 (W) the least subalgebra in .BL2 (W) containing all operators .A0 of the form (5.2.2)and by .J0 (W) the ideal in .A(W) generated by operators of multiplication by functions .χ ∈ Cc (W) such that .χ (0) = 0. According to what was said above, the mapping 0 .Ψ (W)  A → A0 extends to an epimorphism .A(W) → A0 (W) whose kernel contains 0 the ideal .J0 (W). Therefore, the epimorphism A(W)/J0 (W)  [A] → A0 ∈ A0 (W)

.

(5.2.3)

5.2 C ∗ -Algebra Generated by Proper do. Local Algebras

215

is defined; actually, it is an isomorphism. To prove this, we note that if .A0 is an operator ∞ corresponding by formula (5.2.1) to a proper . do A, then for any .χ ∈ C c (W) the operator .χ A0 χ is proper (this statement follows from the definition of proper . do and from the equality .t m G(tx, ty) = Gt (x, y), where .t > 0 and G, .Gt denote the kernels of ∞ the operators A, .Ut AUt−1 ). If .χ ∈ C(W) ∩ C c (W) and .χ (0) = 1, then the operator .A0 is the image of the residue class .[χ A0 χ ] under mapping (5.2.3). In view of the relations 2 .A − χ A0 χ ∈ J0 (W) and .χ A0 χ  ≤ (sup |χ | )A0 , which implies that mapping (5.2.3) is injective. Assume now that .M is a w-manifold, .ν is a measure on .M0 introduced in Sect. 5.1, .z ∈ M \ M0 is an arbitrarily fixed point, . : O → W is a local chart on .M, .O  z, and .(z) = 0. The isomorphism .A/Jz ∼ = A(W)/J0 (W) is implemented by the mapping A/Jz  A + Jz → f 1/2 (0−1 )∗ (χ Aχ )0∗ f −1/2 + J0 (W) ∈ A(W)/J0 (W),

.

where .0 = |(O ∩ M0 ), .χ ∈ CF(O), .χ = 1 near z, and f is the density of the measure .(0 )∗ (ν|O ∩ M0 ) with respect to the surface measure on .W. We apply to the residue classes .[(0−1 )∗ (χ Aχ )0∗ ] and .[f ] the mapping (5.2.3) and obtain some element of .A0 (W) denoted again by .A0 and a function .f0 defined by .f0 (x) = limt→+0 f (tx). Thus, there holds 1/2

−1/2

Proposition 5.2.3 For any .z ∈ M\M0 , the mapping .A  A → f0 A0 f0 where .A0 is the operator defined by A0 u = lim Ut (0−1 )∗ (χ Aχ )0∗ Ut−1 u ,

.

t→0

∈ A0 (W),

u ∈ Cc∞ (W),

generates an isomorphism .A/Jz ∼ = A0 (W).

 

The operator .A0 in Proposition 5.2.3 depends on the choice of .. The invariance of elements of the algebra .A/Jz will be determined in Sect. 5.6. Therefore, if .z ∈ M0 , then due to the isomorphism .A/Jz ∼ = C(Sz∗ ) (Proposition 5.2.2) ∧ ∗ the spectrum .(A/Jz ) is identified with .Sz . For .z ∈ M \ M0 , we have an isomorphism .A/Jz ∼ = A0 (W) (Proposition 5.2.3), where .W = K × Rm−n is a wedge, which is a local model for .M in a neighborhood of z. Thus, the problem of description of the spectrum of .A is reduced to an analogous problem for .A0 (W). For further study, it is convenient to replace .A0 (W) with an isomorphic algebra. To this end, we take into account that .A0 in (5.2.1) is invariant under shifts along the edge. Therefore, (A0 u)(·, x 2 ) = Fζ−1 A (ζ )Fy 2 →ζ u(·, y 2 ), →x 2 0

.

(5.2.4)

where F is the Fourier transform along the edge and the function .ζ → A0 (ζ ) is defined on m−n and takes values in the algebra .BL (K) of all bounded operators in the space .L (K) .R 2 2

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

216

on the cone K. Denote by .L(K) the algebra generated by functions .S m−n−1  θ → A0 (θ ), where the function .A0 (·) is related to .A0 ∈ A0 (W) by (5.2.4); put .A0 (·); L(K) = sup{A0 (θ ); BL2 (K); θ ∈ S m−n−1 }. Proposition 5.2.4 The algebras .A0 (W) and .L(K) are isomorphic. Proof Since F −1 A0 (ζ )F u; L2 (W)2 =

 A0 (ζ )u(·, ˆ ζ ); L2 (K)2 dζ,

.

we have A0 ; BL2 (W) = sup{A0 (ζ ); BL2 (K); ζ ∈ Rm−n }.

.

Let .V ⊂ K and let .v : V → v(V ) ⊂ Rn be a coordinate morphism; remind that the mapping v is homogeneous of degree one. In local coordinates, (A0 )(v) (tζ )u(x 1 ) = (2π )−n/2



.

= (2π )−n/2

 eitx

eix 1ξ 1

1ξ 1

(a˜ v )0 (ϕ, |x 1 |ξ 1 , |x 1 |tζ )u(ξ ˆ 1 ) dξ 1

(a˜ v )0 (ϕ, |x 1 |tξ 1 , |x 1 |tζ )t n u(tξ ˆ 1 ) dξ 1 = Ut (A0 )(v) (ζ )Ut−1 u(x 1 ),

(5.2.5)

where .t > 0 and .Ut u(x 1 ) = t n/2 u(tx 1 ). This implies that .A0 (tζ ) = Ut A0 (ζ )Ut−1 and .A0 (ζ ) = A0 (tζ ) for .t > 0 since the operator .Ut is unitary. If .t → 0, then .A0 (tζ ) → A0 (0) in the strong operator topology. Therefore, .A0 (0) ≤ sup{A0 (ζ ); ζ ∈ Rn , ζ = 0}. We arrive at the equality A0 ; BL2 (W) = sup{A0 (θ ); BL2 (K); θ ∈ S m−n−1 },

.

which means that .A0 ; A0 (W) = A0 (·); L(K).

5.3

 

Algebras L(θ, Rn ) and L(0, Rn )

Let .θ ∈ S m−n−1 and .A0 (·) ∈ L(K). Denote by .L(θ, K) the algebra generated by operators .A0 (θ ) in .L2 (K). Let also .L(0, K) be the algebra spanned by operators of the form .A0 (0). In this section, we are preparing for the study of algebras .L(θ, K) as .θ ∈ S m−n−1 and n n .θ = 0; here, we deal with the case .K = R . It turns out that .L(θ, R ) coincides with

5.3 Algebras L(θ, Rn ) and L(0, Rn )

217

the algebra .L(θ ) defined in Sect. 2.3.2, and .L(0, Rn ) is isomorphic to the algebra .L0 (0) generated by operators −1 Fη→x a(ϕ, ω)Fy→η

.

in .L2 (Rn ); here, .a ∈ C ∞ (S n−1 × S n−1 ), .ϕ = x/|x| and .ω = η/|η|. The proof of these statements is the main result of this section. In what follows, the mentioned algebra .L(θ ) and the algebra .S in 2.2.1 will be denoted by .L0 (θ ) and .S0 , respectively. Thus, the algebra .L0 (θ ) is generated by operators of the form −1 A(θ ) = Fη→x (ϕ, η, θ )Fy→η ∈ BL2 (Rn )

.

with a homogeneous function .ξ → (ϕ, ξ ), and .S0 by operator-valued functions R  λ → Eω→ϕ (λ)−1 a(ϕ, ω)Eψ→ω (λ) ∈ BL2 (S n−1 ).

.

A Special Representation of Generators of L(0, Rn )

5.3.1

Taking into account (5.2.5) and simplifying notations, we will temporarily write generators of the algebra .L(0, Rn ) in the form  −n/2 .Au(x) = (2π ) ˜ |x|ξ )u(ξ ˆ ) dξ. (5.3.1) eixξ a(ϕ, The nearest goal is to represent the operator (5.3.1) in terms of the Mellin transform. According to Proposition 1.2.2, the following formula for the Fourier transform is valid: (F u)(ξ ) = (M −1 )in/2−λ→ρ Eψ→ω (λ)M|y|→λ+in/2 u(y)

.

= (M −1 )in/2−λ→ρ Eψ→ω (λ)U (λ, ψ),

(5.3.2)

where .U (λ, ψ) = (Mu)(λ + in/2, ψ), .ψ = y/|y|, .ρ = |ξ |, .ω = ξ/|ξ |, and .λ ∈ R. The inverse transform admits the representation −1 (F −1 v)(x) = Mμ+in/2→r Eω→ϕ (μ)−1 Mρ→in/2−μ v(ρω), μ ∈ R.

.

From (5.3.2) and (5.3.3), it follows that Au(x) = (2π )−3/2



+∞

.



+∞

× 0

 ρ iμ−1 a(ϕ, ˜ rρω) dρ

−∞

+∞

−∞

r iμ−n/2 Eω→ϕ (μ)−1 dμ

ρ −iλ Eψ→ω (λ)U (λ, ψ) dλ.

(5.3.3)

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

218

We put .a 0 (ϕ, η) = limt→∞ a(ϕ, ˜ tη) and .a˜ 1 = a˜ − a 0 ; the function .η → a 0 (ϕ, η) is zero degree homogeneous. We have .A = A(0) + A(1) , where A(0) u(x) = (2π )−1/2



∞

.

−∞

r iλ−n/2 Eω→ϕ (λ)−1 a 0 (ϕ, ω)Eψ→ω (λ)U (λ, ψ) dλ, (5.3.4)

A(1) u(x) = (2π )−3/2



+∞

.



+∞

×

 ρ iμ−1 a˜ 1 (ϕ, rρω) dρ

0

−∞

+∞

−∞

r iμ−n/2 Eω→ϕ (μ)−1 dμ

(5.3.5)

ρ −iλ Eψ→ω (λ)U (λ, ψ) dλ.

Introduce the operator A(0) (λ) = Eω→ϕ (λ)−1 a 0 (ϕ, ω)Eψ→ω (λ)

.

(5.3.6)

and rewrite (5.3.4) in the form (A(0) )u(rϕ) = (2π )−1/2



+∞

.

−∞

r iλ−n/2 A(0) ψ→ϕ (λ)U (λ, ψ) dλ.

(5.3.7)

Let us now obtain a similar representation for .A(1) . To justify the permutation of integrals in (5.3.5), we define the operator .A(1) ˜ 1 (ϕ, rρω) replaced by δ by (5.3.5) with .a .a ˜ δ1 (ϕ, rρω) = ζ (rρδ)ζ (δ/rρ)a˜ 1 (ϕ, rρω); here .δ ∈ (0, 1) and .ζ ∈ C ∞ (R+ ), whereas .ζ (t) = 1 for .t ≤ 1 and .ζ (t) = 0 for .t > 2. We change the variable .ρ → ρ/r and obtain (1) .A δ u(x)





∞

×

dρ 0

+∞

−∞

−3/2



= (2π )

+∞ −∞

Eω→ϕ (μ)−1 dμ

r iλ−n/2 ρ −i(λ−μ)−1 a˜ δ1 (ϕ, ρω)Eψ→ω (λ)U (λ, ψ) dλ.

Since .u ∈ Cc∞ (Rn \ 0), the function .λ → E(λ)U (λ, ·) is rapidly decaying as .λ → ∞. This allows us to switch two inner integrals, so (1)

Aδ u(x) =

.

 ×

+∞ −∞

1 2π



+∞ −∞

Eω→ϕ (μ)−1 dμ

r iλ−n/2 (M a˜ δ1 )(ϕ, λ − μ, ω)Eψ→ω (λ)U (λ, ψ) dλ,

where .(M a˜ δ1 )(ϕ, ν, ω) = Mρ→ν a˜ δ1 (ϕ, ρω). The function (λ, μ) → Eω→ϕ (μ)−1 (M a˜ δ1 )(ϕ, λ − μ, ω)Eψ→ω (λ)U (λ, ψ)

.

(5.3.8)

5.3 Algebras L(θ, Rn ) and L(0, Rn )

219

is holomorphic in the domain .{(λ, μ) : Imλ < n/2, Imμ > −n/2} and decays rapidly if |λ| + |μ| → ∞ provided .|Imλ|, |Imμ| < h. Therefore,

.

−1/2 A(1) δ u(x) = (2π )



+∞

.



−∞

r iλ−n/2 dλ

(5.3.9)

Eω→ϕ (μ)−1 (M a˜ δ1 )(ϕ, λ − μ, ω)Eψ→ω (λ)U (λ, ψ) dμ

× Im μ=−τ

for .τ < n/2. Let .δ → 0 in (5.3.9). We first find the limit of the left-hand side. The equality (1) .A δ u(x)

= (2π )

−n/2

 

ei(x−y)ξ a˜ δ1 (ϕ, |x|ξ )ξ −2N D2N u(y) dy dξ

for .N > n/2 implies that .

lim Aδ u(x) = (2π )−n/2 (1)

 ˆ ) dξ = A(1) u(x). eixξ a˜ 1 (ϕ, |x|ξ )u(ξ

δ→0

(5.3.10)

Consider the right-hand side of (5.3.9). It turns out (cf. Lemma 5.3.2 below) that, for τ ∈ (0, 1) and any .N ≥ 0, the inequality

.

Eω→ϕ (μ)−1 M(a˜ δ1 − a˜ 1 )(ϕ, λ − μ, ω)Eψ→ω (λ)U (λ, ψ); H s+τ (S n−1 )

.

(5.3.11)

≤ CN δ ε (1 + |λ − μ|)−N (1 + |λ|)−N holds, where .Imλ = 0, .Imμ = −τ , .ε ∈ (0, min{τ, 1 − τ }), and s is an arbitrary non-negative number. Therefore, one can pass to the limit as .δ → 0 under the integral sight in the right-hand side of (5.3.9). Taking into account (5.3.10), we obtain the needed representation (A(1) )u(rϕ) = (2π )−1/2



+∞

.

−∞

(1)

r iλ−n/2 Aψ→ϕ (λ)U (λ, ψ) dλ

(5.3.12)

with −1/2

A (λ)v(ϕ) = (2π )

.

(1)

 μ=−τ

Eω→ϕ (μ)−1 (M a˜ 1 )(ϕ, λ − μ, ω)Eψ→ω (λ)v(ψ) dμ, (5.3.13)

where .τ is an arbitrary number in the interval .(0, min(1, n/2)). Together with (5.3.8), this leads to the following statement.

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

220

Proposition 5.3.1 The operator (5.3.1) admits the representation  +∞ −1/2 .(Au)(rϕ) = (2π ) r iλ−n/2 Aψ→ϕ (λ)U (λ, ψ) dλ, −∞

(5.3.14)

where .A(λ) = A(0) (λ) + A(1) (λ), the operator .A(0) is defined by (5.3.6) and the operator (1) by (5.3.13), whereas .u ∈ C ∞ (Rn \ 0), .U (λ, ψ) = M .A |y|→λ+in/2 u(y) c It remains to prove the estimate (5.3.11). This proof is split into three lemmas, which will also be of use in what follows. For . ∈ C ∞ (S n−1 × S n−1 ), introduce the operator Bψ→ϕ (λ, μ) = Eω→ϕ (μ)−1 (ϕ, ω)Eψ→ω (λ).

.

Lemma 5.3.2 Let .F be a closed subset contained in a strip . |Im λ| < h and free of the points . λ = ±i(k + n/2) for .k = 0, 1, . . . . If . (λ, μ) ∈ F × F , .τ = Im (λ − μ) ≥ 0 , then for any . s ≥ 0 , the inequalities B(λ, μ); H s (S n−1 ) → H s (S n−1 ) ≤ C(1 + |λ − μ|)s+h (1 + |λ|)s (1 + |μ|)−τ q , . (5.3.15)

.

B(λ, μ); H s (S n−1 ) → H s+τ (S n−1 ) ≤ C(1 + |λ − μ|)s+h (1 + |λ|)s q (5.3.16) hold, where .q =  ; C q (S n−1 × S n−1 ) and . C , . q depend only on . F and . s . Proof From the inequality .

(1 + |ξ |2 + |μ|2 )t ≤ 2|t| (1 + |λ − μ|2 )|t| (1 + |ξ |2 + |λ|2 )t ,

due to estimates (1.2.11) and (1.2.12), it follows that E(μ)−1 E(λ) ; H s (λ, S n−1 ) → H s+τ (μ, S n−1 ) ≤ C1 (F)(1 + |λ − μ|2 )|s+Im λ|/2 ≤

.

≤ C1 (F)(1 + |λ − μ|)s+h for . (λ, μ) ∈ F × F . Since . B(λ, μ) = E(μ)−1 E(λ)B(λ, λ) and B(λ, λ)w ; H s (λ, S n−1 ) ≤ C2 q w ; H s (λ, S n−1 )

.

for some .q = q(s)), we have B(λ, μ)w ; H s+τ (μ, S n−1 ) ≤ C(F)(1 + |λ − μ|)s+h q w ; H s (λ, S n−1 ) ≤

.

≤ C(F)(1 + |λ − μ|)s+h (1+|λ|)s q w ; H s (S n−1 ) .

5.3 Algebras L(θ, Rn ) and L(0, Rn )

221

The statement of the lemma now follows from the obvious estimates v ; H s+τ (S n−1 ) ≤ v ; H s+τ (μ, S n−1 ),

.

v ; H s (S n−1 ) ≤ (1 + |μ|2 )−τ/2 v ; H s+τ (μ, S n−1 ) applied to . v = B(λ, μ)w .

 

Lemma 5.3.3 Let .a˜ δ1 and .a˜ 1 be the same as in (5.3.11) and let .ε ∈ (0, 1/2). Then, for all non-negative integers N and q, the inequality .

sup{|ν|N M(a˜ δ1 − a˜ 1 )(·, ν, ·); C q (S n−1 × S n−1 ); ε < Im ν < 1 − ε} ≤ cδ ε

holds with a positive constant .c = c(N, q, ε). Moreover, .

sup{|ν|N M a˜ 1 (·, ν, ·); C q (S n−1 × S n−1 ); ε < Im ν < 1 − ε} < ∞.

Proof From definitions, it directly follows that ∂ρk a˜ 1 (ϕ, ρω) = O((1 + ρ)−k−1 )

.

(5.3.17)

for k=0, 1, . . . . Moreover, .∂ρk (ζ (δρ)ζ (δ/ρ)) = O(ρ −k ) uniformly with respect to .δ ∈ (0, 1). We have

= (2π )−1/2 (iν . . . (iν − N + 1))−1

 0

M(a˜ δ1 − a˜ 1 )(ϕ, ν, ω)

.

∞

ρ −iν+N −1 ∂ρN (a˜ δ1 − a˜ 1 )(ϕ, ρω) dρ.

The integrand equals zero as .ρ ∈ (δ, δ −1 ) and is estimated by .ρ Im ν−1 (1+ρ)−1 . Therefore, |M(a˜ δ1 − a˜ 1 )(ϕ, ν, ω)| ≤ c(N)δ ε |ν|−N , Im ν ∈ (ε, 1 − ε).

.

β

The derivatives .∂ϕα ∂ω M(a˜ δ1 (ϕ, ν, ω) − a˜ 1 (ϕ, ν, ω) also admit similar estimates, since β

for all .α, .β the function .∂ϕα ∂ω a˜ 1 (ϕ, ρω) satisfies condition (5.3.17). This implies the first inequality of the lemma. The second inequality follows from the fact that the   corresponding estimate is obvious for the function .a˜ δ1 . Lemma 5.3.4 For any .τ ∈ (0, min(1, n/2)), .s ≥ 0, and .N = 0, 1, . . . , the following estimate is valid: Eω→ϕ (μ)−1 M(a˜ δ1 − a˜ 1 )(ϕ, λ − μ, ω)Eψ→ω (λ)U (λ, ψ); H s+τ (S n−1 ) → L2 (S n−1 )

.

≤ C(N, τ )δ ε (1 + |λ − μ|)−N (1 + |λ|)−N ,

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

222

where .U (λ, ψ) = M|y|→λ+in/2 u(y), .u ∈ Cc∞ (Rn \ 0), .ε is a positive number, .λ is real, .Imμ = −τ . Proof It is sufficient to compare (5.3.16) and the first inequality in Lemma 5.3.3 and to consider that the function .U (λ, ψ) decreases rapidly as .λ → ∞.  

Coincidence of Algebras L(θ, Rn ) and L0 (θ)

5.3.2

We return to Proposition 5.3.1. Formula (5.3.14) can be rewritten in the form .A = M −1 AM, where M is the Mellin transform. Denote by .S the algebra generated by operator-valued functions .R  λ → A(λ) = A(0) (λ) + A(1) (λ) with the norm A; S = sup{A(λ); BL2 (S n−1 ), λ ∈ R}.

.

Since the Mellin transform is a unitary operator from .L2 (Rn ) onto .L2 (R × S n−1 ), the mapping .A → A defined on the set of operators of the form (5.3.1) extends to an isomorphism of the algebra .L(0, Rn ) onto the algebra .S. Remind that .S0 means the algebra spanned by operator-valued functions of the form R  λ → Eω→ϕ (λ)−1 a(ϕ, ω)Eψ→ω (λ),

.

where .a ∈ C ∞ (S n−1 × S n−1 ). (In Sect. 2.2.1, we used the notation .S instead of .S0 .) Proposition 5.3.5 The algebras .S and .S0 coincide. Proof Let .A = A(0) + A(1) be an arbitrary generator of .S. According to Lemma 5.3.2 and to the second inequality in Lemma 5.3.3, the norm of B(λ, μ) = E(μ)−1 (M a˜ 1 )(ϕ , λ − μ , ω)E(λ) ,

.

Im λ = 0 , Im μ = −τ ,

for any .N ∈ N is subject to the estimates .

B(λ, μ) ; L2 (S n−1 ) → H τ (S n−1 ) ≤ CN (1 + |λ − μ|)−N ,  (1 + |λ − μ|)−N (1 + |λ|)−τ . B(λ, μ) ; L2 (S n−1 ) → L2 (S n−1 ) ≤ CN

This implies that the function 1 .R  λ → A (λ) = √ 2π



(1)

B(λ, μ) dμ Im μ=−τ

5.3 Algebras L(θ, Rn ) and L(0, Rn )

223

belongs to the algebra .C0 (R) ⊗ KL2 (S n−1 ). This algebra is contained in .S0 ; therefore, (1) ∈ S. Moreover, .A(0) ∈ S . So, . A = A(0) + A(1) ∈ S and hence .S ⊂ S . .A 0 0 0 We now show that .S0 ⊂ S. Denote by A(0) ( · ) = Eω→ϕ ( · )−1 a(ϕ, ω)Eψ→ω ( · )

.

an arbitrary generator of .S0 , and let .ζ ∈ C ∞ (R), .ζ (t) = 1 for .t < 1/2 and .ζ (t) = 0 for .t > 1. For .δ ∈ (0, 1), we put .aδ (x, ξ ) = (1 − ζ (|x||ξ |/δ))a(ϕ, ω) and consider the . do .Op aδ ∈ L(0, Rn ). Under the isomorphism .L(0, R n ) ∼ = S, to this . do there (1) (0) corresponds an operator-valued function .Aδ = A − Aδ , where for .λ ∈ R, the operator (1) ∞ n−1 ) → C ∞ (S n−1 ) is defined by .A δ (λ) : C (S (1) .A δ (λ)w

1 =√ 2π

 Bδ (λ, μ)w dμ,

(5.3.18)

Im μ=−τ

while Bδ (λ, μ) = E(μ)−1 (M a˜ δ1 )(ϕ, λ − μ, ω)E(λ),

.

a˜ δ1 (ϕ, ξ ) = ζ (|ξ |/δ)a(ϕ, ω). The inclusion .S0 ⊂ S will be proven if we show that .

(1)

lim sup{Aδ (λ); BL2 (S n−1 ); λ ∈ R} = 0.

δ→0

The formula ∂ϕα ∂ωβ (M a˜ δ1 )(ϕ, ν, ω) =

.

= (2π )−1/2 [iν . . . (iν − k + 1)]−1

 0

∞

ρ −iν+k−1 ∂ρk ∂ϕα ∂ωβ a˜ δ1 (ϕ, ρω) dρ,

where .Im ν = −τ , implies, due to the equality .a˜ δ1 (ϕ, ρω) = 0 for .ρ > δ, that |∂ϕα ∂ωβ (M a˜ δ1 )(ϕ , λ , ω)| ≤ Ck |λ − μ|−k δ τ ,

.

k ∈ Z+ .

(5.3.19)

Furthermore, according to Lemma 5.3.2, we have Bδ (λ, μ) ; BL2 (S n−1 )≤C(1+|λ−μ|)τ (1+|μ|)−τ M a˜ δ1 (·, λ−μ, ·); C q (S n−1 ×S n−1 ),

.

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

224

where q is a sufficiently large number. Comparing this formula with (5.3.19), we obtain Bδ (λ, μ) ; BL2 (S n−1 ) ≤ Ck (1 + |λ − μ|)−k δ τ ,

.

k ∈ Z+ .

(1)

From here and (5.3.18), it follows that .Aδ (λ); BL2 (S n−1 ) ≤ Cδ τ .

 

Proposition 5.3.6 The algebra .L(θ, Rn ) coincides with the algebra .L0 (θ ) for .θ = 0 and for all .θ ∈ S m−n−1 . Proof Since .L(0, Rn ) and .L0 (0) are isomorphic to .S and .S0 , respectively, Proposition 5.3.5 implies the coincidence of the algebras .L(0, Rn ) and .L0 (0). Suppose now that m−n−1 . We restrict ourselves to the proof of the inclusion .L (θ ) ⊂ L(θ, Rn ); in .θ ∈ S 0 essence, only this inclusion is used below (for proof of the inverse inclusion see [38]). Let m .Op a be a . do in .Rn with a homogeneous symbol a,  −m/2 eixξ a(ϕ, ξ )v(ξ .(Op a)v(x) = (2π ) ˆ ) dξ, a(ϕ, tξ ) = a(ϕ, ξ ) as .t > 0, .ϕ = x 1 /|x 1 |, and let .ζ ∈ C ∞ (R+ ), .ζ (t) = 1 for .t < 1/2, .ζ (t) = 0 for .t > 1. We put .a ˜ δ (ϕ, ξ ) = ζ (|ξ 1 |/δ)ζ (|ξ 2 |/δ)a(ϕ, ω), .δ > 0 and introduce the operator .

−m/2



(Op (a − aδ ))v(x) = (2π )

.

eixξ (a(ϕ, ξ ) − a˜ δ (ϕ, |x 1 |ξ ))v(ξ ˆ ) dξ.

To the operator .Op aδ there corresponds the function .S m−n−1  θ → (Op aδ )(θ ) ∈ BL2 (Rn ), where (Op aδ )(θ )u(x 1 ) = (2π )−n/2



.

= ζ (|x 1 |/δ)(2π )−n/2

 eix

eix 1ξ 1

1ξ 1

a˜ δ (ϕ, |x 1 |ξ 1 , |x 1 |θ )u(ξ ˆ 1 ) dξ

ζ (|x 1 |ξ 1 /δ)a(ϕ, ξ 1 , θ )u(ξ ˆ 1 ) dξ 1 .

The following equality is valid: Op aδ ; BL2 (Rm ) = sup{(Op aδ )(θ ) ; BL2 (Rn ) ; θ ∈ S m−n−1 }.

.

(5.3.20)

n Since .Op(a − aδ ) ∈ Ψ 0 (Rm n ), to prove the inclusion .L0 (θ ) ⊂ L(θ, R ), it suffices to prove m that .Op aδ ; BL2 (R ) → 0 as .δ → 0. Let us fix .θ ∈ S m−n−1 and let .b˜δ(1) (x 1 , ξ 1 ) = ζ (|ξ 1 |/δ)(a(ϕ, ξ 1 , |x 1 |θ ) − a(ϕ, ξ 1 , 0)), (2) ˜ = ζ (|ξ 1 |/δ)a(ϕ, ξ 1 , 0). Then, .b δ (1)

(2)

(Op aδ )(θ )u(x 1 ) = ζ (|x 1 |/δ)(Op bδ )(θ )u(x 1 ) + ζ (|x 1 |/δ)(Op bδ )u(x 1 )

.

(5.3.21)

5.3 Algebras L(θ, Rn ) and L(0, Rn )

225

for .u ∈ Cc∞ (Rn \ 0). From the proof of Proposition 5.3.5, it follows that (2) n ˜ (1) .Op b δ ; BL2 (R ) → 0 as .δ → 0. Furthermore, the function .bδ satisfies the estimate |∂ρk b˜δ(1) (rϕ, ρω)| ≤ Ck ρ −k r(r 2 + ρ 2 )−1/2 ,

k ∈ Z+ ,

.

(5.3.22)

where . ρ = |ξ 1 | and . ω = ξ 1 /|ξ 1 |. Integrating by parts in 1 (1) (M b˜δ )(rϕ, ν, ω) = √ 2π



∞

.

0

ρ −iν−1 b˜δ (rϕ, ρω) dρ , Im ν > 0, (1)

and assuming that .τ = Im ν, we deduce from (5.3.22) that |(M b˜δ(1) )(rϕ, ν, ω)| ≤ Ck |ν|−k



∞

ρ τ −1 r(r 2 + ρ 2 )−1/2 dρ =

.

=

Ck |ν|−k r τ



∞

0

0

ρ τ −1 (1 + ρ 2 )−1/2 dρ = Ck |ν|−k r τ .

The derivatives .∂ϕα ∂ω (M b˜δ )(rϕ, ν, ω) obey the same inequalities. Due to Proposition 5.3.1 we have  1 (1) .(Op b )u(rϕ) = r iλ−n/2 dλ (5.3.23) √ δ 2π Im λ=0  (1) E(μ)−1 (Mbδ )(rϕ, λ − μ, ω)E(λ)U (λ, ψ) dμ. × β

(1)

Imμ=−τ

The (positive) number .τ in this inequality satisfies .τ < n/2. If .n ≥ 2, then .τ can be taken in (1/2, 1). Then (see proof of Proposition 5.3.5), (1) 2 .ζ (| · |/δ)(Op b δ )u



δ

≤C

r

n−1



r τ −n/2 (1 + |λ|)−τ

dr Imλ=0

0



×U (λ, ·) dλ ≤ C

 0

δ

r 2τ −1 r dr

 Im λ=0

(1 + |λ|)−2τ dλ

2 |λ − μ|−k dμ

Im μ=−τ



U (λ, ·)2 dλ = C  δ 2τ u2 . Im λ=0

In view of (5.3.20) of (5.3.21), it follows from here that .limδ→0 Op aδ ; BL2 (Rm ) = 0. Thus, the inclusion .L0 (θ ) ⊂ L(θ, Rn ) is proven for .n ≥ 2. For .n = 1, the operator-valued

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

226

function .μ → E(μ)−1 has a pole at .μ = −i/2. Therefore, for .τ ∈ (1/2, 1), the right-hand side of (5.3.23) contains the additional summand .

− (2π )−3/2

 r iλ−1/2

 S0

Imλ=0

 (1) (M b˜δ )(rϕ, λ + i/2, ω) dω E(λ)U (λ, ψ) dλ,  

which is estimated as before.

5.4

Localization in L(θ, K)

Remind that the algebra .L(θ, K) is generated by operators .A0 (θ ) in .L2 (K) (cf. formulas (5.2.4) and (5.2.5)). To simplify notations, we write generators .A(θ ) of .L(θ, K) in local coordinates in the form  −n/2 eixξ (a˜ υ )0 (ϕ, |x|ξ, |x|θ )u(ξ .A(υ) (θ )u(x) = (2π ) ˆ ) dξ ; (5.4.1) here, we write x and .ξ instead of .x 1 and .ξ 1 . Proposition 5.4.1 The algebra .L(θ, K) is irreducible, whereas .KL2 (K) ⊂ L(θ, K). Proof Let D be the (smooth) base of the cone K, .D = K ∩ S N−1 , and let .{Vj } be an atlas on K, i.e., .Vj = Vj × R+ , where .{Vj } is an atlas on D. Introduce a partition of unity .{ηj } on K subject to the covering .{Vj } and consisting of homogeneous functions of degree 0. According to Theorem 4.2.13, every element .A(θ ) ∈ L(θ, K) can be written in the form A(θ ) =

.



vij∗ Aij (θ )(vij−1 )∗

(5.4.2)

with .Aij (θ ) = (vij−1 )∗ (ηi A(θ )ηj )vij∗ , while .Aij (θ ) ∈ L(θ, Rn ). The converse is also true: if the last inclusion holds for all .Aij (θ ), then the operator (5.4.2) belongs to .L(θ, K). Owing to Propositions 2.3.5, 2.3.9, and 5.3.6, the algebra .L(θ, Rn ) is irreducible and contains the ideal .KL2 (Rn ). Therefore, any operator of the form .(vij−1 )∗ (ηi Qηj )vij∗ for n .Q ∈ KL2 (K) belongs to the algebra .L(θ, R ). Hence, .Q ∈ L(θ, K) for all .Q ∈ KL2 (K).   Denote by .K¯ the compactification of the cone K obtained by adding the cone vertex and the “infinitely remote copy” .D∞ of the base D (i. e. the set of the “ends” of the cone ¯ of continuous functions on this compact. We generatrices). Introduce the algebra .C(K) ¯ According to apply Proposition 1.3.26 for localization in the algebra .L(θ, K) with .C(K). Proposition 5.4.1, .KL2 (K) ⊂ L(θ, K). From Proposition 2.3.10, we obtain the inclusions

5.4 Localization in L(θ, K)

227

χ A(θ )ζ ∈ KL2 (K) for each operator .A(θ ) ∈ L(θ, K) and for arbitrary functions .χ , ζ ∈ ¯ satisfying .χ ζ = 0. Thus, the requirements of Proposition 1.3.26 are fulfilled. C(K) ¯ we define some algebra .l(θ )z ; below, we will show that .l(θ )z is For any .z ∈ K, isomorphic to the local algebra .L(θ, K)z . Below, .A(θ ) denotes operators obtained from operators of class .Ψ00 (M). Such operators can be taken as generators of the algebra .L(θ, K); they admit a representation of the form (5.4.1). .

(i) Points of the set K. For any .z ∈ K, let us introduce the algebra .l(θ )z := C(S ∗ (K)z ) of continuous functions on a fiber of the bundle .S ∗ (K) of cotangent spheres. Assume that .ϕ is the direction from vertex 0 to point z. (Thus, for .z ∈ K, the algebras .lz (θ ) are independent of .θ . However, it is convenient to keep this parameter to the unity of the notations.) According to Definition 5.1.1, there exists a limit (a 0 )υ (ϕ, ξ, θ ) = lim (a˜ υ )0 (ϕ, |z|tξ, |z|tθ ),

.

t→∞

(5.4.3)

which is a zero degree homogeneous function of the variables .(ξ, θ ). Putting .θ = 0, we obtain the function .ξ → a 0 (ϕ, ξ, 0) on the fiber .S ∗ (K)z and introduce the mapping pz (θ ) : A(θ ) → σ 0 (ϕ, ·, 0) ∈ l(θ )z .

.

(5.4.4)

(ii) Points of the set .D∞ . Let .ϕ be the direction from vertex 0 to .z ∈ D∞ . For .θ = 0, the function .ξ → a 0 (ϕ, ξ, θ ) appearing in (5.4.3) is well-defined and continuous on the fiber .T ∗ (K¯ \ 0)z of the cotangent bundle; remind that the coordinate transforms on K are homogeneous of degree 1. This function extends to be continuous on the compact set .T¯∗ (K¯ \ 0)z obtained by gluing to the fiber the infinitely remote sphere .S n−1 . We put .l(θ )z := C(T¯∗ (K¯ \ 0)z ) and introduce the mapping pz (θ ) : A(θ ) → a 0 (ϕ, ·, θ ) ∈ l(θ )z .

.

(5.4.5)

(iii) The vertex of K. To the point .z = 0, we relate the algebra .l(θ )0 := L(0, K). This algebra is generated by operators .A(0) admitting in local coordinates the representation of the form A(0)(υ) u(x) = (2π )

.

−n/2

 ˆ ) dξ eixξ (a˜ υ )0 (ϕ, |x|ξ, 0)u(ξ

(5.4.6)

(compare with (5.4.1)). By .p0 (θ ), we denote the mapping p0 (θ ) : A(θ ) → A(0) ∈ l(θ )0 .

.

(5.4.7)

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

228

Therefore, if points .z ∈ K are placed on the same generatrix of K, then the mappings pz (θ ) do not depend neither on z nor on .θ ; dependence on .θ is present in .lz (θ ) for ¯ \ 0, the algebras .lz (θ ) are commutative. Actually, the algebra .z ∈ D∞ . For z in .K .l0 (θ ) does not depend on .θ and is commutative. .

To prove the following Proposition 5.4.2, one can use the argument from the proof of Proposition 2.3.11 with obvious changes. Proposition 5.4.2 Let .L(θ, K)z for .z ∈ K¯ be a local algebra obtained by localization in ¯ Then, the localizing mapping .pz (θ ) defined by (5.4.4), .L(θ, K) with the algebra .C(K). (5.4.5), and (5.4.7) extends to an isomorphism .L(θ, K)z  l(θ )z .

5.5

Localization in L(0, K)

Assume, as in Sect. 5.2, that .A stands for the .C ∗ -algebra generated in .L2 (M) by all proper . do in the class .Ψ00 (M) on a w-manifold .M. In Propositions 5.2.2 and 5.2.3, local algebras that arise by the localization in .A are listed. This list and Proposition 5.2.4 show that the study of the local algebras actually reduces to the study of the algebras m−n−1 . After localization in .L(θ, K) (cf. notations at the beginning of Sect. 5.3), .θ ∈ S .L(θ, K) (Proposition 5.4.2), the local algebra .L(0, K) arises; we turn to the study of this algebra. Taking the Mellin transform .Mr→λ along the generatrices of the cone K as an intertwining operator, we pass from .L(0, K) to the isomorphic algebra .S(D); elements of N −1 is the base of .S(D) are some functions .R  λ → A(λ) ∈ BL2 (D), where .D = K ∩ S the cone K. We apply to .S(D) the localization principle formulated in Proposition 1.3.22. The role of J is played by the ideal .C0 (R) ⊗ KL2 (D); here, .C0 (R) is the set of continuous functions vanishing at the infinity, and .KL2 (D) is the ideal of compact operators in .L2 (D). As a commutative algebra .C, we take the algebra .C(D) of continuous functions on D. In this section, we describe the corresponding local algebras. Let .A(0) be a . do in the algebra .L(0, K) (i.e., an operator obtained by localization ∗ vij Aij (0)(vij−1 )∗ similar to (5.4.2), from a . do in .A). We have the equality .A(0) = whereas .Aij (0) = (vij−1 )∗ (ηi A(0)ηj )vij∗ . In view of (5.3.14), for any operator .Aij (0), there corresponds a function .λ → Aij (λ). Denote by .Dij the intersection of the domain of .vij with the sphere .S N −1 . Let .vij0 : Dij → S n−1 be the mapping generating the diffeomorphism .vij . We introduce the operator AD (λ) =

.



(vij0 )∗ Aij (λ)((vij0 )−1 )∗ : L2 (D) → L2 (D).

(5.5.1)

From (5.3.14), it follows that A(0)u = (2π )

.

−1/2



+∞ −∞

r iλ−n/2 AD (λ)U (λ, ·) dλ.

(5.5.2)

5.6 Invariant Description of Local Algebras

229

Due to the Parseval equality for the Mellin transform, we have A(0); BL2 (K)) = sup{AD (λ); BL2 (D); λ ∈ R}.

.

Assume that the algebra .S(D) is generated by functions .λ → AD (λ) defined by (5.5.1) with the norm .AD ; S(D) = sup{AD (λ); BL2 (D); λ ∈ R}. We arrive at the following statement. Proposition 5.5.1 The algebras .L(0, K) and .S(D) are isomorphic. Let us now make sure that the localization principle in Proposition 1.3.22 is applicable to the algebra .S(D). If a function .λ → AD (λ) is defined by (5.5.1), where .uij ∈ C0 (R) ⊗ KL2 (S n−1 ), then .AD ∈ S(D). This implies the inclusion .J := C0 (R) ⊗ KL2 (D)) ⊂ S(D). The localizing algebra .C(D) is a subalgebra of .S(D); the unity of .S(D) belongs to .C(D). For .A ∈ S(D) and .a ∈ C(D), the commutator .[A, a] falls into J . Therefore, conditions (i) and (ii) of Proposition 1.3.22 are fulfilled. Condition (iii) is also fulfilled; this can be easily deduced from the fact that any irreducible representation of the ideal J is equivalent to a representation of the form .π(λ) : A → A(λ). Denote by .Jϕ the ideal generated in the algebra .S(D) by functions in .C(D) equal to zero at .ϕ ∈ D. The local algebra .S(D)/Jϕ is denoted by .S(D)ϕ . Remind that elements .A of .S(D) are related to operators in .L(0, K) by (5.3.14). For .A(0) ∈ L(0, K), the principal symbol .a 0 is defined, .a 0 (ϕ, ·, 0) ∈ C(S ∗ (K)ϕ ), where as usual, .C(S ∗ (K)ϕ ) is the fiber at .ϕ ∈ D ⊂ K of the cospherical bundle .C(S ∗ (K)). The same function .a 0 is also called the principal symbol of the element .A ∈ S connected with .A(0) by (5.3.14). The next assertion follows, in fact, from Proposition 2.2.10. Proposition 5.5.2 Let .A ∈ S(D) and .a 0 be the principal symbol of .A. The mapping S(D)  A → a 0 (ϕ, ·, 0) ∈ C(S ∗ (K)ϕ )

.

generates an isomorphism .S(D)ϕ ∼ = C(S ∗ (K)ϕ ).

5.6

Invariant Description of Local Algebras

Remind that the localization procedure used in Sects. 5.2, 5.4, and 5.5 consisted of several stages. First, the localization was applied to the algebra .A generated in .L2 (M) by operators in .Ψ00 (M). Then, two more stages of localization were carried out in local algebras that appeared in the previous step. As a result, local algebras of three types arise: algebras of continuous (scalar) functions on cotangent spheres and non-commutative algebras .L(K, θ ) and .S(K). The description of .L(K, θ ) and .S(K) depended on the

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

230

choice of local coordinates. We now demonstrate how to make definitions of these algebras “invariant” under the change of variables. Let T be an n-dimensional edge of the manifold .M, .0 ≤ n ≤ m − 1, where .m = dim M. Assume also that .{Oα } is a finite collection of neighborhoods in .M covering T and that .κα : Oα → κα (Oα ) is a homeomorphism onto an open subset of the product ¯ × Rm−n . Since, by definition, any edge T is connected, we can assume that the conical .K manifold K is independent of .α. For neighborhoods .Oα and .Oβ such that .Oα ∩ Oβ = ∅, the mapping .καβ = κβ ◦ κα−1 : Oα ∩ Oβ → Oα ∩ Oβ is an admissible diffeomorphism. Let us choose .z ∈ T ∩ Oα ∩ Oβ and suppose for simplicity that .κα (z) = κβ (z) = 0. Denote by .π (1) and .π (2) the projections .K¯ × Rm−n → K¯ and .K¯ × Rm−n → Rm−n , let (1) (2) (1) ◦ κ (2) ◦ κ , and define .κ αβ and .καβ = π αβ αβ = π kαβ (x 1 ) = lim t −1 καβ (tx 1 , 0), x 1 ∈ K,

.

(1)

(1)

t→+0

kαβ (x 2 ) = lim t −1 καβ (0, tx 2 ), x 2 ∈ Rm−n . (2)

(2)

t→+0

(1) (2) Then, .kαβ is an admissible diffeomorphism .K¯ → K¯ homogeneous of degree 1, and .kαβ is a linear isomorphism. ¯ with base T and fiber .K¯ using the mappings Introduce a bundle .E = (E, T , K) (1) ¯ ¯ .(I dαβ , k αβ ) : (Oα ∩ Oβ ∩ T ) × K → (Oα ∩ Oβ ∩ T ) × K as transition functions. Let

¯ → E|(Oα ∩ T ) hα : (Oα ∩ T ) × K)

.

(5.6.1)

be the natural trivialization, and .χα such a partition of unity on T that .suppχα ⊂ Oα ∩ T . For points on the same generatrix of the cone, addition and multiplication by non-negative numbers are defined. Therefore, the following mapping makes sense: τ : T × K¯ → E : (z, x) →



.

χα (z)hα (z, x).

i

Since .τ is an isomorphism of bundles, we conclude that the bundle .E is trivial. We endow the fiber .E(z) with the measure .μ(z) transferred from .K¯ using the mapping .τ (z) = τ |{z}× ¯ the cone .K¯ is assumed to be endowed with the Euclidean measure. K; Remind that under localization at .z0 ∈ T , we must pass from an operator .A ∈ A to an operator (A0 u)(x) = lim (Ut (κ −1 )∗ (χ Aχ )κ ∗ Ut−1 u)(x)

.

t→0

5.6 Invariant Description of Local Algebras

231

on the wedge .W = K¯ × R m−n , where .χ is a cut-off function, .χ (z0 ) = 1, and .Ut is the unitary operator: .u(·) → t m/2 u(t·) in .L2 (W), while .t > 0 and the point .z0 is taken as the origin. Suppose that .z0 ∈ T ∩ Oα ∩ Oβ and .κα (z0 ) = κβ (z0 ) = 0. Assuming .A(γ ) = ∗ A(β) κ ∗ for .κ −1 −1 (κγ )∗ (χ Aχ )κγ∗ for .γ = α, β, we obtain .A(α) = καβ αβ = κβ ◦ κα . βα = limt→0 Ut A(γ ) Ut−1 . It is clear that

(γ )

Introduce the operator .A0

∗ ∗ A0 = lim (Ut καβ Ut−1 Ut A(β) U −1 Ut κβα Ut−1 ). (α)

.

t→0

Since ∗ ∗ (Ut καβ Ut−1 f )(x) = t m/2 (καβ Ut−1 f )(tx) = t m/2 (Ut−1 f )(καβ (tx)) = f (καβ (tx)/t),

.

we have ∗ ∗ A0 = kαβ A0 kβα for kαβ (x) = lim καβ (tx)/t.

.

(β)

(α)

t→0

Points .x ∈ K¯ × Rm−n will be written in the form .x = (x 1 , x 2 ), where .x 1 ∈ K¯ and 2 m−n . We assume that .((k (1) )∗ u)(x 1 , x 2 ) = u(k (1) x 1 , x 2 ) and .((k (2) )∗ u)(x 1 , x 2 ) = .x ∈ R αβ αβ αβ (2)

u(x 1 , kαβ x 2 ). Introduce the notation .u(·, ˆ ξ 2 ) = Fx 2 →ξ 2 u(·, x 2 ), where F is the Fourier (γ )

transform on .Rq , and represent the operator .A0 , .γ = α, β, in the form  (γ ) (γ ) 2 .(A u)(·, x ) = exp (ix 2 ξ 2 )A0 (ξ 2 )u(·, ˆ ξ 2 ) dξ 2 0

(5.6.2)

(cp. (5.2.4)). Then, ∗ ∗ (kαβ A0 kβα u)(·, x 2 ) (β)

.

 =

(5.6.3)

(2) 2 2 (1) ∗ (1) ∗ (2) ∗ x )ξ }(kαβ ) A0 (ξ 2 )(kβα ) ((kβα ) u)ˆ(·, ξ 2 ) dξ 2 exp {i(kαβ (β)

 =

exp (ix 2 η2 )(kαβ )∗ A0 ((t kαβ )−1 η2 )(kβα )∗ u(·, ˆ η2 ) dη2 . (1)

(β)

(2)

(1)

Using formula (5.6.2) for .A(α) 0 and taking into account (5.6.3), we obtain (1) ∗ (1) ∗ 2 t (2) −1 2 A(α) 0 (η ) = (kαβ ) A0 (( kαβ ) η )(kβα ) . (β)

.

(5.6.4)

Denote by .T(T ) the tangent bundle over T and by .T(T )z the fiber over a point .z ∈ T . Let (2) q 0 −1 .tα : T(T )z0 → R be the derivative of the mapping .κα |T at .z , so that .k αβ = tβ tα . Put ∗ t −1 ∗ A0 (z0 ; ζ ) = (h−1 α ) A0 ( tα ζ )hα

.

(α)

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

232

for .ζ ∈ T∗ (T )z0 . In view of (5.6.4), the operator A0 (z0 ; ζ ) : L2 (E(z0 ); μ(z0 )) → L2 (E(z0 ); μ(z0 ))

.

(5.6.5)

is well-defined for .ζ ∈ T∗ (T )(z0 ). Denote by .A(z0 ; ζ ) the algebra generated by operators (5.6.5) for a fixed .ζ . In the proof of Proposition 5.2.4, it was stated that if .ζ1 and .ζ2 belong to the same ray of the fiber ∗ 0 0 .T (T )z0 , then, for the algebras .A(z ; ζ1 ) and .A(z ; ζ2 ), there exists a unitary intertwining operator. Since we make no distinction between equivalent representations of .A, to each ray .ρ in the fiber .T∗ (T )z0 , we have to relate only one algebra .A(z0 ; ζ ) with an arbitrarily fixed .ζ ∈ ρ. To make the description simpler, let us introduce a Riemannian metric on the edge T and relate to a ray .ρ the algebra .A(z0 ; θ ), where .θ = ρ ∩ S ∗ (T )z0 and .S ∗ (T ) is the cospherical bundle over T . Thus, for an edge T , .0 < dim T < dim M, and for points 0 ∗ 0 .z ∈ T and .θ ∈ S (T )z0 , the local algebra .A(z ; θ ) is the algebra spanned by operators (5.6.5) for .ζ = θ . We turn to the algebra of operator-valued functions .R  λ → A(z0 ; λ). Assume that the bundle .E is equipped with a Riemannian metric and introduce the Mellin transform 1 .(Mu)(λ + in/2, ϕ) := u(λ ˜ + in/2, ϕ) := √ 2π



+∞ −∞

r −i(λ+in/2)−1 u(rϕ) dr,

for a function u on .E(z0 ); here .r = |x| and .ϕ = x/|x| for .x ∈ E(z0 ) \ 0. Define the operator .A(z0 ; ·) = MA0 (z0 ; 0)M −1 , which replaces a function .R  λ → u(λ ˜ + in/2, ·) by .R  λ → A(z0 ; λ)u(λ ˜ + in/2, ·). For .z0 ∈ Oα ∩ Oβ = ∅, we put .A(γ ) (z0 ; ·) = (γ ) M (γ ) A0 (z0 ; 0)(M (γ ) )−1 for .γ = α, β, where 1 (M (γ ) u)(λ + in/2, ϕγ ) := u(λ ˜ + in/2, ϕγ ) := √ 2π



+∞

.

−∞

rγ−i(λ+in/2)−1 u(rγ ϕγ ) drγ ,

rγ = |xγ |, and .ϕγ = xγ /|xγ |. We relate the operators .A(λ) and .A(α) (λ). Using the trivialization .hα from (5.6.1), we obtain

.

∗ (α) ∗ −1 A = MA0 (0)M −1 = M(h−1 α ) A0 (0)hα M

.

∗ (α) −1 (α) ) M A0 (0)(M (α) )−1 M (α) h∗α M −1 = M(h−1 α ) (M (α)

∗ (α) −1 (α) = M(h−1 ) A (M (α) )−1 M (α) h∗α M −1 . α ) (M

(5.6.6)

5.6 Invariant Description of Local Algebras

233

Let us calculate .(M (α) h∗α M −1 u)(λ ˜ + in/2, ϕα ). We have .

(h∗α M −1 u)(x ˜ α ) = (M −1 u(h ˜ α (xα ))  +∞ 1 |hα (xα )|i(λ+in/2) u(λ ˜ + in/2, hα (xα )/|hα (xα )|) dλ. =√ 2π −∞

Therefore, (M (α) h∗α M −1 u)(λ ˜ + in/2, ϕα )

.

=

1 2π



+∞

|xα |−i(λ+in/2)−1 d|xα |

0



+∞

−∞

(5.6.7)

|xα |i(λ+in/2) |hα (xα )/|xα | |i(λ+in/2)

×u(λ ˜ + in/2, hα (xα )/|hα (xα )|) dλ = |hα (ϕα )|i(λ+in/2) u(λ ˜ + in/2, hα (ϕα )/|hα (ϕα )|) (since the mapping .hα is homogeneous of degree 1). From (5.6.7), it follows that ∗ (α) −1 i(λ+in/2) −1 (M(h−1 ) v)(λ ˜ + in/2, ϕ) = |h−1 v(λ ˜ + in/2, h−1 α ) (M α (ϕ)| α (ϕ)/|hα (ϕ)|). (5.6.8)

.

Denote by .((gα−1 )∗ v)(λ+in/2, ˜ ϕ) the right-hand side of (5.6.8) and by .(gα∗ u)(λ+in/2, ˜ ϕα ) the corresponding expression in (5.6.7). Comparing (5.6.6)–(5.6.8), we obtain A(λ) = (gα−1 )∗ A(α) (λ)gα∗ .

.

The operator .A(z0 ; λ) is well-defined for functions on the set .D(z0 ) = {x ∈ E(z0 ) : |x| = 1} and implements a continuous mapping .L2 (D(z0 ); ν(z0 )) → L2 (D(z0 ); ν(z0 )), where .ν(z0 ) is a measure induced on .D(z0 ) by the measure .μ(z0 ) on the cone .E(z0 ). By 0 0 .S(D(z )), we mean the algebra generated by functions of the form .R  λ → A(z ; λ) with the norm A(z0 ; ·); S(D(z0 )) = sup{A(z0 ; λ); BL2 (D(z0 ); ν(z0 )), λ ∈ R}.

.

(5.6.9)

Thus, all local algebras that appeared in Sect. 5.2, 5.4, and 5.5 admit an invariant description.

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

234

5.7

The Spectrum of C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

Introduce a set whose points parametrize one-dimensional representations. Denote by .M the space of maximal ideals of .C(M0 ) (cf. definition of this algebra at the beginning of Sect. 5.2). The set .M0 is homeomorphically embedded into .M. The passage from .M0 to .M can be described as gluing to .M0 the boundary of a tubular neighborhood of each edge. The set .M, in a natural way, is endowed with a structure of a smooth manifold with boundary. Principal symbols of operators in .Ψ00 (M) extend to be continuous functions on the cospherical bundle .S ∗ (M). Let T be a stratum of .M, .0 ≤ dim T = n ≤ m − 1, .dim M = m, and .KT × Rm−n a local model of .M in the vicinity of T ; here, .KT is the cone with the base .DT . As in Sect. 5.6, we introduce locally trivial bundles .E(T ) and .D(T ) with the base space T . A fiber .E(x) of .E over x is a cone diffeomorphic to .KT and generated by rays tangent to .M at x and orthogonal to T . The fiber .D(x) is the base of the cone .E(x). For an operator A in .Ψ00 (M), we introduce mappings π(x, ω) : A → a 0 (x, ω),

.

(5.7.1)

where .(x, ω) ∈ S ∗ (M0 ), .ω ∈ S ∗ (M0 )x , and .a 0 is the principal symbol of A; π(x, ϕ, ω) : A → a 0 (x, ϕ, ω),

.

(5.7.2)

where .x ∈ T , .ω ∈ S ∗ (M)(x,ϕ) (points .(x, ϕ, ω) can be identified with .ω ∈ S ∗ (E(x))ϕ , where .S ∗ (E(x))ϕ is the fiber of .S ∗ (E(x)) over .ϕ ∈ D(x)); π(x, θ ) : A → A(x; θ ) ∈ A(x, θ ) ⊂ BL2 (E(x); μ(x)),

.

(5.7.3)

where .(x, θ ) ∈ S ∗ (T ), .x ∈ T , .θ ∈ S ∗ (T )x , while it is assumed that .dim T > 0; for .dim T = 0 the mappings (5.7.3) are not defined. As .A(x; θ ), we take the operator from formula (5.6.5), and .A(x; θ ) means the algebra defined after this formula, π(x, λ) : A → A(x; λ) ∈ BL2 (D(x); ν(x)),

.

(5.7.4)

where .x ∈ T , .0 ≤ dim T ≤ m − 1, .λ ∈ R and the function .λ → A(x; λ) is defined by (5.6.9). Note that for .dim T < m − 1, the operator .A(x; λ) acts in an infinite-dimensional Hilbert space. If .dim T = m − 1, then the cone .KT is just a finite collection of rays, and all the operators .A(x; λ), for .x ∈ T and .λ ∈ R, act in a space of finite dimension (its dimension is equal to the number of rays in the cone .KT ). Concluding the discussion in Sects. 5.2–5.6, we obtain the following result.

5.7 The Spectrum of C ∗ -Algebra of Pseudodifferential Operators on. . .

235

Theorem 5.7.1 The mappings (5.7.1)–(5.7.4) extend to representations of .A. These representations are irreducible and pairwise non-equivalent. Any irreducible representation of .A is equivalent either to one of the representations (5.7.1)–(5.7.4) or to the identity representation.

taking as a “hint” Theorem 2.3.2 and One can describe the topology on the spectrum .A its proof; this is left to the interested reader. We now present a solving composition series. Denote by . the ideal in .A equal to the overlap of kernels of all representations of the form (5.7.3) for all edges of positive   dimensions, . = ker π(x, θ ). Put . = ker π(x, λ); all representations of the form (5.7.4) for all edges participate in this intersection. As before, .com A denotes the ideal generated by commutators of elements in .A; remind that .com A is the intersection of kernels of all one-dimensional representations. Introduce the following ideals: I0 =  ∩

∩ com A, I1 =

.

∩ com A, I2 = com A.

Theorem 5.7.2 Assume that the set of edges of dimension .≥ 1 is non-empty, and there are no edges of dimension .dim M − 1. Then, the composition series 0 ⊂ I0 ⊂ I1 ⊂ I2 ⊂ A

.

is the shortest solving series; hence, the length of .A is equal to 3. Whereas,

I1 /I0 



∗

(C(S (T ) ⊗ KL2 (KT ))  (

{T : dim T ≥1}

I2 /I1 

 T

I0 = KL2 (M),

.



∗

(5.7.5) .

C(S (T ))) ⊗ KH,

(5.7.6) .

C(T × R)) ⊗ KH,

(5.7.7) .

{T : dim T ≥1}

(C(T × R) ⊗ KL2 (DT ))  (

 T

A/I2  C(S ∗ (M)), (5.7.8) where H is an infinite-dimensional separable Hilbert space. Let us outline the proof. To establish the first (from the left) two isomorphisms (5.7.6) and (5.7.7), one can apply the scheme of the proof of Theorem 2.3.3. The second isomorphism (5.7.6) follows from the fact that, for any T , the space .L2 (KT ) is infinitedimensional and separable. If .dim T < m − 1, then .L2 (DT ) is also infinite-dimensional   and separable. Hence, there holds the second isomorphism in (5.7.7). The restrictions on the dimensions of edges in the theorem are not caused by the merits of the case and needed only to simplify the description. If we refuse these restrictions, then the length of the algebra may change. For example, if the manifold has only zerodimensional edges (conical points), then the length of .A equals 2 (cp. Theorem 2.2.19).

236

5 .C ∗ -Algebra of Pseudodifferential Operators on Manifold with Edges

For an .(m − 1)-dimensional edge T , the cone .KT consists of a finite number of rays and the dimension of the representations .π(x, λ) for .x ∈ T is equal to the number of rays in the cone. Therefore, increasing the number of .(m − 1)-dimensional edges, to which there correspond different numbers of rays, one can increase the length of the algebra.

Bibliographical Sketch

Chapter 1. Section 1.1 is a relatively short and elementary introduction to the theory of pseudodifferential operators. Any book of [15, 39, 42, 43] provides a deeper exposition of this theory. The results of Sect. 1.2 belong to B. A. Plamenevskii; proofs of the propositions formulated in this section can be found in [21]. The notion of solvable algebra was introduced by A. S. Dynin [6]. Examples in [6] are based on papers by L. Boutet de Monvel [1], R. G. Douglas and R. Howe [5], I. Ts. Gohberg and N. Ya. Krupnik [11]. Algebras of Wiener–Hopf operators in “piecewise smooth” cones and algebras of Toeplitz operators on bounded symmetric domains were studied by A. S. Dynin, P. S. Muhly, J. N. Renault, H. Upmeier, V. N. Senichkin, etc. (irreducible representations, spectral topology, solvability); in this regard, see the survey [27]. In particular, H. Upmeier used solving series to obtain formulas for the index of Toeplitz and Wiener–Hopf operators [44, 45]. Papers by C. O. Cordes and his followers (see the monograph [2] and the references therein) are related to this subject. The sufficient triviality condition for the fields of elementary algebras is taken from the paper [28]. The information on maximal radical series is borrowed from [13]. The localization principle in similar formulations was used by R. G. Douglas [4], I. B. Simonenko [40, 41], and A. S. Dynin [7]. Chapter 2. S. G. Mikhlin introduced the notion of symbol for a singular integral operator on .Rn and proved that, if the symbol is nonzero everywhere, then the operator is Fredholm. Irreducible representation of the algebra generated by .DO with smooth symbols on .Rn was described by I. Ts. Gohberg. These results were generalized for manifolds by R. T. Seeley. We present proofs of the mentioned results different from the original ones. The spectrum of the algebra .A (see the description of Chap. 2) and one of the algebra .S generated by meromorphic .DO were studied in [23] (all equivalence classes of irreducible representations are listed, and the spectral topology is described). In this book, another scheme of investigation is used. The major difference from [23] is that the localization principle from Sect. 1.3 is consistently applied. When implementing this

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Plamenevskii, O. Sarafanov, Solvable Algebras of Pseudodifferential Operators, Pseudo-Differential Operators 15, https://doi.org/10.1007/978-3-031-28398-7

237

238

Bibliographical Sketch

scheme, we use some techniques from [26]. Let us note that for one-dimensional singular integral operators on a contour, some infinite-dimensional representations become twodimensional. The operators implementing these representations turn out to be unitary equivalent to the operator symbols introduced in [11]. In this chapter, some results from [25] are presented; we give here new proofs based on the localization principle. Chapter 3. The theorem containing the list of all (equivalence classes of) irreducible representations of the algebra .A considered in Chap. 3 is proved in [22] (earlier this  is theorem was formulated in [26]). The description of the topology on the spectrum .A taken from [26], and the construction of the solving composition series is taken from [22]; the paper [28] contains an outline of such a construction, while the properties of the solving series are not verified there. Another approach to describing algebras of .DO with singularities using the Levi-Civita connections was outlined by A. S. Dynin in [8]. Chapter 4 is based on the paper by V. N. Senichkin [37]. In connection with the theorem on the boundedness of .DO in weighted spaces, we mention another approach discussed in book [21] and papers [24, 31]; in particular, these works contain another approach to definition of bounded non-proper .DO in weighted spaces with weight power outside the Stein interval. Chapter 5. The results of the paper [38] are exposed; proofs are partially reconsidered due to the use of the localization principle. .DO on manifolds with conical singularities were studied in [21]. In [29,30], .DO are defined for a wider class of “stratified” piecewise smooth manifolds than that in Chaps. 4 and 5 (informally, manifolds with intersecting edges of different dimensions). In [29], the approach exposed in Chap. 4 is generalized, and in [30] all (up to equivalence) irreducible representations of the corresponding algebra of .DO are found. The results of [29,30] are not included in the book. Let us mention some studies of .C ∗ -algebras generated by .DO on manifolds with boundary. One of the chapters in [21] is devoted to the study of the spectrum of algebras of .DO on smooth manifolds with boundary; symbols (“coefficients”) allow isolated singularities. The dependence of the spectrum on singularities and on the choice of weighted function spaces is determined. In the paper by A. Yu. Kokotov [16] the spectrum of .C ∗ -algebras generated by pseudodifferential operators in a polyhedron is described. O. V. Sarafanov [32,33] considered pseudodifferential boundary value problems for operators from Chaps. 4 and 5 on manifolds with smooth closed edges on the boundary. In [33], operators composing a boundary value problem are introduced, and in [32], the spectrum of .C ∗ -algebras generated by boundary value problems is studied. Other approaches to .DO with singularities and other references are given in the books [17, 35, 36]. This short sketch is not intended to be complete; its purpose is only to show the origin of the presented results and to point out some related works.

References

1. Boutet de Monvel, L. (1971). Boundary problems for pseudodifferential operators. Acta Math, 126, 11–51. 2. Cordes, H. O. (1987). Spectral theory of linear differential operators and comparison algebras. London Mathematical Society Lecture Note Series (Vol. 76). 3. Dixmier, J. (1977). C*-algebras, Amsterdam: North-Holland Publishing Company. 4. Douglas, R. G. (1972). Banach algebra technics in operator theory. Pure and Applied Mathematics, (Vol. 49). New York, London: Academic Press. 5. Douglas, R. G., & Howe, R. (1971). On the C*-algebras on the quarter plane. Transactions of the American Mathematical Society, 158, 203–217. 6. Dynin, A. S. (1978). Inversion problem for singular integral operators. Proceedings. National Academy of Sciences. United States of America, 75, 4668–4670. 7. Dynin, A. S. (1986). Multivariable Wiener-Hopf operators I. Representation. Integral Equations and Operator Theory, 9, 537–556. 8. Dynin, A. S. (1987). Multivariable Wiener-Hopf operators II. Spectral topology and solvability. Integral Equations and Operator Theory, 10, 554–576. 9. Friedrichs, K. O. (1970). Pseudodifferential operators: An introduction. Lecture Notes, Courant Institute of Mathematical Sciences. 10. Gakhov, F. D. (1966). Boundary value problems. International Series of Monographs on Pure and Applied Mathematics (Vol. 85). Pergamon Press. 11. Gohberg, I. T., & Krupnik, N. Y. (1970). Algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients. Functional Analysis and its Applications, 4(3), 193–201. 12. Gohberg, I. T., & Krupnik, N. Y. (1992). One-dimensional linear singular integral equations. Operator Theory: Advances and Applications. Basel: Birkhäuser Verlag. 13. Handelman, D., & Yin, H.-S. (1988). Toeplitz algebras and rotational automorphisms associated to polydiscs. American Journal of Mathematics, 110(5), 887–920. 14. Hörmander L. (1965). Pseudodifferential operators. Communications on Pure and Applied Mathematics, 18, 501–517. 15. Hörmander, L. (2007). The analysis of linear partial differential operators III. PseudoDifferential operators. Classics in Mathematics. Berlin: Springer-Verlag. 16. Kokotov, A. Yu. (1992). Representation of .C ∗ -algebras generated by pseudodifferential operators in a polyhedron. St. Petersburg Mathematical Journal, 3(3), 541–562. 17. Melrose R. B. (1993). The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics. A K Peters/CRC Press.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 B. Plamenevskii, O. Sarafanov, Solvable Algebras of Pseudodifferential Operators, Pseudo-Differential Operators 15, https://doi.org/10.1007/978-3-031-28398-7

239

240

References

18. Mikhlin, S. G. (1965). Multidimensional singular integrals and integral equations. International Series of Monographs on Pure and Applied Mathematics (Vol. 83), Pergamon Press. 19. Murphy G. (1990). C*-algebras and operator theory. Academic Press. 20. Muskhelishvili, N. I. (1958). Singular Integral Equations: Boundary problems of functions theory and their applications to mathematical physics. Dordrecht: Springer. 21. Plamenevskii, B. A. (1989). Algebras of pseudodifferential operators. Mathematics and its Applications, Soviet Series (Vol. 43). Dordrecht: Kluwer Academic Publishers. 22. Plamenevskii, B. A. (2010). Solvability of the algebra of pseudodifferential operators with piecewise smooth coefficients on a smooth manifold. St. Petersburg Mathematical Journal, 21(2), 317–351. 23. Plamenevskii, B. A., & Senichkin, V. N. (1984). On the spectrum of .C ∗ -algebras generated by pseudodifferential operators with discontinuous symbols. Mathematics of the USSR-Izvestiya, 23(3), 525–544. 24. Plamenevskii, B. A., & Senichkin, V. N. (1987). Representations of .C ∗ -algebras generated by pseudodifferential operators in weighted spaces. Problems in Mathematical Physics, 12, 165– 189. (In Russian). 25. Plamenevskii, B. A., & Senichkin, V. N. (1988). On representations of an algebra of pseudodifferential operators with multidimensional discontinuities in the symbols. Mathematics of the USSR-Izvestiya, 31(1), 143–169. 26. Plamenevskii, B. A., & Senichkin, V. N. (1990). The spectrum of an algebra of pseudodifferential operators with piecewise smooth symbols. Mathematics of the USSR-Izvestiya, 34(1), 147–179. 27. Plamenevskii, B. A., & Senichkin, V. N. (1995). Solvable operator algebras. St. Petersburg Mathematical Journal, 6(5), 895–968. 28. Plamenevskii, B. A., & Senichkin, V. N. (1996). On composition series in algebras of pseudodifferential operators and in algebras of Wiener-Hopf operators. Mathematical Topics (Vol. 11, pp. 373–404). Advances in Partial Differential Equations. Berlin: Akademie Verlag. 29. Plamenevskii, B. A., & Senichkin, V. N. (2000). On a class of pseudodifferential operators in m .R and on stratified manifolds. Sbornik ,191(5), 725–757. 30. Plamenevskii, B. A., & Senichkin, V. N. (2002). Representations of .C ∗ -algebras of pseudodifferential operators on piecewise smooth manifolds. St. Petersburg Mathematical Journal, 13(6), 993–1032. 31. Plamenevskii, B. A., & Tashchiyan, G. M. (1993). A convolution operator in weighted spaces. Journal of Mathematical Sciences, 64, 1363–1381. 32. Sarafanov, O. V. (2004). The spectrum of the .C ∗ -algebra of pseudodifferential boundary value problems on a manifold with smooth edges. Journal of Mathematical Sciences, 120, 1195–1239. 33. Sarafanov, O. V. (2005). Calculus of pseudodifferential boundary value problems on manifolds with smooth edges. American Mathematical Society Translations, Series 2, 214, 183–236. 34. Schochet, C. (1981). Topological methods for C*-algebras I: Spectral sequences. Pacific Journal of Mathematics, 96(1), 193–211. 35. Schulze, B.-W. (1991). Pseudodifferential operators on manifolds with singularities. Studies in Mathematics and its Applications (Vol. 24). Amsterdam: North Holland. 36. Schulze, B.-W. (1994). Pseudo-differential boundary value problems, conical singularities, and asymptotics. Berlin: Akademie Verlag. 37. Senichkin, V. N. (1995). Pseudodifferential operators on manifolds with edges. Journal of Mathematical Sciences, 73, 711–747. 38. Senichkin, V. N. (1997). The spectrum of the algebra of pseudodifferential operators on a manifold with smooth edges. St. Petersburg Mathematical Journal, 8(6), 985–1013. 39. Shubin, M. A. (2001). Pseudodifferential operators and spectral theory. Berlin: Springer-Verlag.

References

241

40. Simonenko, I. B. (1965). A new general method for studying linear operator equations of the type of singular integral equations, I, II. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 29(3,4), 567–586, 757–782. (In Russian). 41. Simonenko, I. B. (1967). Operators of convolution type in cones. Mathematics of the USSRSbornik, 3(2), 279–293. 42. Taylor, M. (1981). Pseudodifferential operators. Princeton, NJ: Princeton University Press. 43. Treves, J.-F. (1980). Introduction to pseudodifferential and Fourier integral operators (Vols. 1– 2). University Series in Mathematics. New York: Springer. 44. Upmeier, H. (1987). Index theory for Toeplitz operators on bounded symmetric domains. Bulletin of the American Mathematical Society, 16, 109–112. 45. Upmeier, H. (1988). Index theory for multivariable Wiener-Hopf operators. Journal fur die Reine und Angewandte Mathematik, 384, 57–79.