LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England The titles below are available from booksellers, or, in case of difficulty, from Cambridge University Press. 34 36 39 46 50 59 66 69 74 76 77 79 83
86 87 88 89 90 92 93 94 95 96 97 98 99 100 101 103 104 105
106 107 108 109 110 113 114 115 116 117 118 119 121 122 125 126 127 128 129 130 131 132 133 134 135
136 137 138 139
Representation theory of Lie groups, M.F. ATIYAH et al Homological group theory, C.T.C. WALL (ed) Affine sets and affine groups, D.G. NORTHCOTT p-adic analysis: a short course on recent work, N. KOBLITZ Commutator calculus and groups of homotopy classes, H.J. BAUES Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Several complex variables and complex manifolds II, M.J. FIELD Representation theory, I.M. GELFAND et al Symmetric designs: an algebraic approach, E.S. LANDER Spectral theory of linear differential operators and comparison algebras, H.O. CORDES Isolated singular points on complete intersections, E.J.N. LOOIJENGA Probability, statistics and analysis, J.F.C. KINGMAN & G.E.H. REUTER (eds) Homogeneous structures on Riemannian manifolds, F. TRICERRI & L. VANHECKE Topological topics, I.M. JAMES (ed) Surveys in set theory, A.R.D. MATHIAS (ed) FPF ring theory, C. FAITH & S. PAGE An F-space sampler, N.J. KALTON, N.T. PECK & J.W. ROBERTS Polytopes and symmetry, S.A. ROBERTSON Representation of rings over skew fields, A.H. SCHOFIELD Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) Representations of general linear groups, G.D. JAMES Low-dimensional topology 1982, R.A. FENN (ed) Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN Localization in Noetherian rings, A.V. JATEGAONKAR Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Groups and geometry, ROGER C. LYNDON Surveys in combinatorics 1985, I. ANDERSON (ed) Elliptic structures on 3-manifolds, C.B. THOMAS A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG Syzygies, E.G. EVANS & P. GRIFFITH Compactification of Siegel moduli schemes, C-L. CHAI Some topics in graph theory, H.P. YAP Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) An introduction to surreal numbers, H. GONSHOR Lectures on the asymptotic theory of ideals, D. REES Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG An introduction to independence for analysts, H.G. DALES & W.H. WOODIN Representations of algebras, P.J. WEBB (ed) Homotopy theory, E. REES & J.D.S. JONES (eds) Skew linear groups, M. SHIRVANI & B. WEHRFRITZ Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) Non-classical continuum mechanics, R.J. KNOPS & A.A. LACEY (eds) Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE Van der Corput's method of exponential sums, S.W. GRAHAM & G. KOLESNIK New directions in dynamical systems, T.J. BEDFORD & J.W. SWIFT (eds) Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK Model theory and modules, M. PREST Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S. PUTCHA Number theory and dynamical systems, M. DODSON & J. VICKERS (eds) Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds) Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (eds) Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds)
140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170
Geometric aspects of Banach spaces, E.M. PEINADOR & A. RODES (eds) Surveys in combinatorics 1989, J. SIEMONS (ed) The geometry of jet bundles, D.J. SAUNDERS The ergodic theory of discrete groups, PETER J. NICHOLLS Introduction to uniform spaces, I.M. JAMES Homological questions in local algebra, JAN R. STROOKER Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO Continuous and discrete modules, S.H. MOHAMED & B.J. MULLER Helices and vector bundles, A.N. RUDAKOV et al Solitons, nonlinear evolution equations and inverse scattering, M. ABLOWITZ & P. CLARKSON Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds) Number theory and cryptography, J. LOXTON (ed) Classification theories of polarized varieties, TAKAO FUJITA Twistors in mathematics and physics, T.N. BAILEY & RJ. BASTON (eds) Analytic pro -p groups, J.D. DIXON, M.P.F. DU SAUTOY, A. MANN & D. SEGAL Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMAYER (eds) Groups St Andrews 1989 volume 1, C.M. CAMPBELL & E.F. ROBERTSON (eds) Groups St Andrews 1989 volume 2, C.M. CAMPBELL & E.F. ROBERTSON (eds) Lectures on block theory, BURY-HARD KULSHAMMER Harmonic analysis and representation theory, A. FIGA-TALAMANCA & C. NEBBIA Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SHRIKANDE & S.S. SANE Groups, combinatorics & geometry, M.W. LIEBECK & J. SAXL (eds) Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) Stochastic analysis, M.T. BARLOW & N.H. BINGHAM (eds) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) Boolean function complexity, M.S. PATERSON (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK
171
Squares, A.R. RAJWADE Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) Lectures on mechanics, J.E. MARSDEN
172 173 174 175 176 177 178 179 180 181 182 183
184 185 186 187 188 189 190 191
192 193 194 195 196 197 198 199
200 201
202 203 204 207
Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE, & A. PITTS (eds) Lower K- and L-theory, A. RANICKI Complex projective geometry, G. ELLINGSRUD et at Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory H, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, O. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D.J.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY Polynomial invariants of finite groups, D.J. BENSON Finite geometry and combinatorics, F. DE CLERCK et at Symplectic geometry, D. SALAMON (ed) Computer algebra and differential equations, E. TOURNIER (ed) Independent random variables and rearrangement invariant spaces, M. BRAVERMAN Arithmetic of blowup algebras, WOLMER VASCONCELOS Microlocal analysis for differential operators, A. GRIGIS & J. SJOSTRAND Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI, W. METZLER & A.J. SIERADSKI (eds) The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN Invariant potential theory in the unit ball of Cn, MANFRED STOLL The Grothendieck theory of dessins denfant, L. SCHNEPS (ed) Singularities, JEAN-PAUL BRASSELET (ed) The technique of pseudodifferential operators, H.O. CORDES Hochschild cohomology, A. SINCLAIR & R. SMITH Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT, J. HOWIE (eds) Groups of Lie type and their geometries, W.M. KANTOR & L. DI MARTINO (eds)
London Mathematical Society Lecture Note Series. 202
The Technique of Pseudodifferential Operators H.O. Cordes
Emeritus, University of California, Berkeley
AMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521378642
© Cambridge University Press 1995
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1995
A catalogue record for this publication is available from the British Library ISBN 978-0-521-37864-2 paperback Transferred to digital printing 2008
To my 6 children, Stefan and Susan Sabine and Art, Eva and Sam
TABLE OF CONTENTS
Chapter 0. Introductory discussions
1
0.0. Some special notations, used in the book 0.1. The Fourier transform; elementary facts 0.2. Fourier analysis for temperate distributions on 0.3. The Paley-Wiener theorem; Fourier transform for
1
3
In
general uE D'
0.4. The Fourier-Laplace method; examples 0.5. Abstract solutions and hypo-ellipticity
9
14
20 30
0.6. Exponentiating a first order linear differential operator
31
0.7. Solving a nonlinear first order partial differential equation
36
0.8. Characteristics and bicharacteristics of a linear PDE
40
0.9. Lie groups and Lie algebras for classical analysts
45
Chapter 1. Calculus of pseudodifferential operators 1.0. Introduction I.I. Definition of pdo's
52 52
52
1.2. Elementary properties of ipdo's
56
1.3. Hoermander symbols; Weyl pdo's; distribution kernels
60
1.4. 1.5. 1.6. 1.7.
The composition formulas of Beals The Leibniz' formulas with integral remainder Calculus of 1pdo's for symbols of Hoermander type Strictly classical symbols; some lemmata for application
Chapter 2. Elliptic operators and parametrices in Mn 2.0. Introduction
64 69 72
78 81 81
2.1. Elliptic and md-elliptic Vdo's 2.2. Formally hypo-elliptic pdo's
82
2.3. Local md-ellipticity and local md-hypo-ellipticity
87
2.4. Formally hypo-elliptic differential expressions 2.5. The wave front set and its invariance under yxlo's
91
84
93
Contents
viii 2.6. Systems of ,do's
Chapter 3. L2-Sobolev theory and applications 3.0. Introduction
3.1. L2-boundedness of zero-order do's 3.2. L2-boundedness for the case of 6>0 3.3. Weighted Sobolev spaces; K-parametrix and Green inverse 3.4. Existence of a Green inverse 3.5. Hs-compactness for ftpdo's of negative order
97
99 99 99 103
106 113
117
Chapter 4. Pseudodifferential operators on manifolds with conical ends 4.0. Introduction
118 118
4.1. Distributions and temperate distributions on manifolds
119
4.2. Distributions on S-manifolds; manifolds with conical ends 4.3. Coordinate invariance of pseudodifferential operators 4.4. Pseudodifferential operators on S-manifolds 4.5. Order classes and Green inverses on S-manifolds Chapter 5. Elliptic and parabolic problems 5.0. Introduction 5.1. Elliptic problems in free space; a summary 5.2. The elliptic boundary problem
123
129 134 139 144 144
147 149
5.3. Conversion to an &n-problem of Riemann-Hilbert type
154
5.4. Boundary hypo-ellipticity; asymptotic expansion mod av
157
5.5. A system of fide's for the Vj of problem 3.4
162
5.6. Lopatinskij-Shapiro conditions; normal solvability of (2.2).
169
5.7. Hypo-ellipticity, and the classical parabolic problem
174
5.8. Spectral and semi-group theory for ado's
179
5.9. Self-adjointness for boundary problems 5.10. C*-algebras of tpdo's; comparison algebras
186
Chapter 6. Hyperbolic first order systems 6.0. Introduction 6.1. First order symmetric hyperbolic systems of PDE
189
196 196 196
6.2. First order symmetric hyperbolic systems of fide's on
n .
6.3. The evolution operator and its properties
200
206
Contents
ix
6.4. N-th order strictly hyperbolic systems and 210
symmetrizers.
6.5. The particle flow of a single hyperbolic pde
215
6.6. The action of the particle flow on symbols 6.7. Propagation of maximal ideals and propagation of singularities
219
Chapter 7. Hyperbolic differential equations 7.0. Introduction 7.1. Algebra of hyperbolic polynomials 7.2. Hyperbolic polynomials and characteristic surfaces 7.3. The hyperbolic Cauchy problem for variable coefficients 7.4. The cone h for a strictly hyperbolic expression of type e°
226 226 227 230
235 238
7.5. Regions of dependence and influence; finite propagation speed 7.6. The local Cauchy problem; hyperbolic problems on manifolds Chapter 8. Pseudodifferential operators as smooth operators of L(H) 8.0. Introduction 8.1.
223
241
244 247 247
,do's as smooth operators of L(H0)
248
8.2. The 11DO-theorem
251
8.3. The other half of the gbbO-theorem
257
8.4. Smooth operators; the V -algebra property; 'Wdo-calculus
8.5. The operator classes 'S and 'IVL
261 ,
and their
symbols 8.6
8.7
265
The Frechet algebras y"x0, and the Weinstein-
Zelditch class
271
Polynomials in x and ax with coefficients in 'TX
275
Characterization of qtX by the Lie algebra Chapter 9. Particle flow and invariant algebra of a semistrictly hyperbolic system; coordinate invariance 8.8
279
of OpWxm.
282
9.0. Introduction
282
9.1. Flow invariance of V10
283
9.2. Invariance of Vsm under particle flows
286
9.3. Conjugation of Optpx with eiKt , KE Opy)ce
289
9.4. Coordinate and gauge invariance; extension to S-manifolds 9.5. Conjugation with eiKt for a matrix-valued K=k(x,D)
293
296
Contents
x
9.6. A technical discussion of commutator equations
301
9.7. Completion of the proof of theorem 5.4 Chapter 10. The invariant algebra of the Dirac equation 10.0. Introduction
305
10.1. A refinement of the concept of observable 10.2. The invariant algebra and the Foldy-Wouthuysen transform
314
10.3. The geometrical optics approach for the Dirac algebra P
310 310
319
324
10.4. Some identities for the Dirac matrices 10.5. The first correction z0 for standard observables 10.6. Proof of the Foldy-Wouthuysen theorem
329
10.7. Nonscalar symbols in diagonal coordinates of
350
10.8. The full symmetrized first correction symbol zS 10.9. Some final remarks
356
334 343
367
References
370
Index
380
P R E F A C E
It is generally well known that the Fourier-Laplace transform converts a linear constant coefficient PDE P(D)u=f on Rn to
an equation P()u ()=f"(), for the transforms u , f- of u and f, so that solving P(D)u=f just amounts to division by the polynomial The practical application was suspect, and ill understood, however, until theory of distributions provided a basis for a logically consistent theory. Thereafter it became the Fourier-Laplace method for solving initial-boundary problems for standard PDE. We recall these facts in some detail in sec's 1-4 of ch.0. The technique of pseudodifferential operator extends the Fourier-Laplace method to cover PDE with variable coefficients, and to apply to more general compact and noncompact domains or manifolds with boundary. Concepts remain simple, but, as a rule, integrals are divergent and infinite sums do not converge, forcing lengthy, often endlessly repetitive, discussions of 'finite parts' (a type of divergent oscillatory integral existing as distribution integral) and asymptotic sums (modulo order -oo). Of course, pseudodifferential operators (abbreviated ado's) are (generate) abstract linear operators between Hilbert or Banach spaces, and our results amount to 'well-posedness' (or normal solvability) of certain such abstract linear operators. Accordingly both, the Fourier-Laplace method and theory of ipdo's, must be seen
in the context of modern operator theory. To this author it always was most fascinating that the same type of results (as offered by elliptic theory of ipdo;'s) may be
obtained by studying certain examples of Banach algebras of linear operators. The symbol of a ipdo has its abstract meaning as Gelfand function of the coset modulo compact operators of the abstract operator in the algebra.
On the other hand, hyperbolic theory, generally dealing with a group exp(Kt) (or an evolution operator U(t)) also has its manifestation with respect to such operator algebras: conjugation with
Preface
xii
exp(Kt) amounts to an automorphism of the operator algebra, and of the quotient algebra. It generates a flow in the symbol space essentially the characteristic flow of singularities. In [Ci], [C2] we were going into details discussing this abstract approach. We believe to have demonstrated that ado's are not necessary to understand these fact. But the technique of pdo's, in spite of its endless formalisms (as a rule integrals are always 'distribution integrals', and infinite series are asymptotically convergent, not convergent), still provides a strongly simplifying principle, once the technique is mastered. Thus our present discussion of this technique may be justified. On the other hand, our hyperbolic discussions focus on invariance of ido-algebras under conjugation with evolution operators, and do not touch the type of oscillatory integral and further discussions needed to reveal the structure of such evolution operators as Fourier integral operators. In terms of Quantum mechanics we prefer the Heisenberg representation, not the Schroedinger representation. In particular this leads us into a discussion of the Dirac equation and its invariant algebra, in chapter X. We propose it as algebra of observables. The basis for this volume is (i) a set of notes of lectures given at Berkeley in 1974-80 (chapters I-IV) published as preprint at U. of Bonn, and (ii) a set of notes on a seminar given in 1984 also at Berkeley (chapters VI-IX). The first covers elliptic (and parabolic) theory, the second hyperbolic theory. One might say that we have tried an old fashiened PDE lecture in modern style. In our experience a newcomer will have to reinvent the theory before he can feel at home with it. Accordingly, we did not try to push generality to its limits. Rather, we tend to focus on the simplest nontrivial case, leaving generalizations to the reader. In that respect, perhaps we should mention the problems (partly of research level) in chapters I-IV, pointing to manifolds with conical tips or cylindrical ends, where the 'Fredholm-significant symbol' becomes operator-valued. The material has been with the author for a long time, and was subject of many discussions with students and collaborators. Especially we are indebted to R. McOwen, A.Erkip, H. Sohrab, E. Schrohe, in chronological order. We are grateful to Cambridge University Press for its patience, waiting for the manuscript. Berkeley, November 1993
Heinz 0. Cordes
Chapter 0. INTRODUCTORY DISCUSSIONS.
In the present introductory chapter we give comprehensive discussions of a variety of nonrelated topics. All of these bear on the concept of pseudo-differential operator, at least in the author's mind. Some are only there to make studying Ado's appear a natural thing, reflecting the author's inhibitions to think along these lines. In sec.! we discuss the elementary facts of the Fourier transform, in sec.'s 2 and 3 we develop Fourier-Laplace transforms of temperate and nontemperace distributions. In sec.4 we discuss the Fourier-Laplace method of solving initial-value problems and free space problems of constant coefficient partial differential equations. Sec.5 discusses another problem in PDE, showing how the solving of an abstract operator equation together with results on hypo-ellipticity and "boundary-hypo-ellipticity" can lead to existence proofs for classical solutions of initialfor boundary problems. Sec.6 is concerned with the operator eLt ,
a first order differential expression L Sec.'s 7 and 8 deal with the concept of characteristics of a linear differential expression .
and learning how to solve a first order PDE. Sec.9 gives a miniintroduction to Lie groups, focusing on the mutual relationship between Lie groups and Lie algebras. (Note the relation to i,do's
discussed in ch.8).
We should expect the reader to glance over ch.0 and use it to have certain prerequisites handy, or to get oriented in the serious reading of later chapters. 0. Some special notations. The following notations, abbreviations, and conventions will be used throughout this book. (a)
xn
(b)
(x) = (l+Ix12)!/2
(2n)-n/2
,
ox = xndx1dx2...dxn = xndx
(1+1
,
!
12)!,2
,
.
etc.
0. Introductory discussions
2
(c) Derivatives are written in various ways, at convenience: For u=u(x)=u(x1,...,xn) we write u(a)=aXU=axlax2...u = a a =aatlax",...a n/ax nu. Or, ulx =ax u, ulx to denote the n-vector j
J
with components ulx , V"u for the k-dimensional array with compoJ
nents
ulxi xi
z
'
For a function of (x,i;)=(xl
..
'.'fin)
"
it is often convenient to write u(a)=aaXU. (d) A multi-index is an n-tuple of integers a=(a1,...an) We write laI=lall+...+lanl al=all...an! ((3)=(R )...( ), a a xa= xl 1...xn n etc., IIn={all multi-indices} ,
,
,
.
(e) Some standard spaces: &n = n-dimensional Euclidean space Bn=directional compactification of Ien (one infinite point «x added
in every direction (of a unit vector x). (f) Spaces of continuous or differentiable complex-valued functions over a domain or differentiable manifold X (or sometimes only X=1en): C(X) = continuous functions on X ; CB(X)= bounded continuous functions on X; CO(X)= continuous functions on X vanishing at a ; CS(X) = continuous functions with directional limits; CO(X) = continuous functions with compact support; Ck(X)= functions with derivatives in C, to order k, (incl. k=oo). CB00(X)="all derivatives exist and are bounded". The Laurent-Schwartz notations D(X)=CO(X), E(X)=C°(X) are used. Also S= S(Jk)= "rapidly decreasing functions" (All derivatives decay stronger as any power of x). Also, distribution spaces D', E', SO. (g) LP-spaces: For a measure space X with measure dµ we write Lp(X)=Lp(X,dµ)={measurable functions u(x) with luIp integrable} for lsp
Y
.
.
L(X) (i) Classes of linear operators (X= Banach space) (K(%))= continuous (compact) operators; GL(X) (U(H)) = invertible (unitary) operators of L(X) (of L(H), H=Hilbert space); Un U(Mn). :
For operators X- Y, again, L(X,Y), etc. (j) The convolution product: For u,v E LI(len) we write w(x)
=(u*v)(x)=xnfdyu(x-y)v(y) (Note the factor xn
(2n)-n/2).
(k) Special notation: " X CC Y " means that X is contained in a compact subset of Y .
0.1. The Fourier transform
3
(1) For technical reason we may write limE-Oa(E)
= alE..O
(m) Abbreviations used: ODE (PDE) = ordinary (partial) differential equation (or "expression"). FOLPDE (or folpde)= first order linear partial differential equation (or "expression"); 'pdo=
pseudodifferential operator. (n) Integrals need not be existing (proper or improper) Riemann or Lebesgue integrals, unless explicitly stated, but may be distribution integrals By this term we mean that either (i) the integral may be interpreted as value of a distribution at a testing function-the integrand may be a distribution, or (ii) the limit of Riemann sums exists in the sense of weak convergence of a sequence of (temperate) distributions, or (iii) the limit defining an improper Riemann integral exists in the sense of weak convergence, as above, or (iv) the integral may be a 'finite part' (cf. 1,4). (o) Adjoints: For a linear operator A we use 'distribution adjoint' A and 'Hilbert space adjoint' A*, corresponding to trans-
pose AT and adjoint vely. For a symbols
AT=A*,
in case of a matrix A=((a.k)), respectia* (or a+) may denote the symbol of
as specified in each section. (p) supp u (sing supp u (or s.s.u)) denotes the (singular) support of the distribution u. the adjoint 'tpdo of a(x,D)
,
1. The Fourier transform; elementary facts. Let u E L1(&n) be a complex-valued integrable function. Then we define the Fourier transform u^= Fu of u by the integral (1.1)
with x =x. =
(1.2) Note that u^
(1.3)
x E &n ,
u^(x) = J
an existing Lebesgue integral. Clearly,
u^ (x) I s
IIu11L1
=
n9lxlu(x)l
is uniformly continuous over In : We get
u^ (x)-u^ (y) I s 2f¢t s NIX-111 llullL1 +
I
R1
where the right hand side is <E if N is chosen
for
<E/4,
0. Introductory discussions
4
) Moreover, we get u^E CO(&n), i.e.,
and then Ix-yi< a/(2NIIuiI L
limlxI.00u^(x)=0, a fact, known as the Riemann-Lebesgue lemma.
To prove the latter, we reduce it to the case of u E CO(&n): 00 The space CO is known to be dense in LI By (1.1) we get .
(1.4)
Iu^ (x)-v^ (x) I s IIu-vilL1 < e/2
Hence lim
IXI
OOv^ (x)=0 implies
I u^ I
x is chosen according to Iv^I < e/2
,
as v E CO ,
IIu-vuIL1<E/2.
I u^ -v^ I+ I v^ I< e whenever
s .
But for v E CO the Fourier integral extends over a ball ICI s N only, since v=0 outside. We may integrate by parts for (1.5)
=_f
IxI2u^(x) =-J0
=-(Av)^(x)
with the Laplace differential operator e _=1a 2
Clearly we
have tv E CO C LI as well, whence (1.1) applies to AV , for
(1.6)
Iv^ (x) I s
11 AV ll
L
1/1X12
0
as IxI
completing the proof. The above partial integration describes a general method to be applied frequently in the sequel. (1.6) may be derived under the weaker assumptions that vE C2, and that all derivatives v(a), Ials2
,
are in LI (cf. pbm.5). On the other hand, there are simple
examples of u(4 LI such that u^ does not decay as rapidly as (1.6) indicates. In particular, uE L' exists with u^(4 L'
(cf.pbm.4).
This matter becomes important if we think of inverting the linear operator F:LI - CO defined by (1.1), because formally an inverse seems to be given by almost the same integral. Indeed, define the (complex) conjugate Fourier transform F:LI -> CO by Fu = (Fu)
(1.7)
,
or, u"= Fu
,
u" (x) =
where
f
u E LI (len)
.
Then, in essence, it will be seen that F is the inverse of the operator F. More precisely we will have to restrict F to a (dense) subspace of LI, for this result. Or else, the definition of the operator F must be extended to certain non-integrable functions, for which existence of the Lebesgue integral (1.7) cannot be expected. Both things will be done, eventually.
0.1. The Fourier transform
5
It turns out that F induces a unitary operator of the Hilbert space L2(&n): We have Parseval's relation: (1.8)
JRn4IxIu^(x)12 = fRnoxlu(x)12
,
for all u E_= L'
(,n) ,2 (e)
Formula (1.8) is easier to prove as the Fourier inversion formula, asserting u^'=u`^=u for certain u: We may write
(1.9)
Q
_1
glxu^ (x)v^ (x) = N
N
e
ix 7. ( .-1. ) j -"j qIx.
-N
assuming that u,vE LI(&n) , with the 'cube' QN {lx.IsN,j=1,..,N} some integer N>0. Indeed, the interchange of integrals leading to (1.9) is legal, since the integrand is L'(QNxinxlen)
Note that
fNeistdt = 2sis sN
s340
,
= 2N
,
s=0, allowing
-N
evaluation of the inner integrals at right of (1.9). With f A A _ f Af 911 , and it = - IN , A = N n,, , (1.9) assumes the form
fQN 41xu^ (x)v^ (x) =
(1.10)
where cp(t) = (2/n) 1/2 sin t
,
j t340, continuously extended into t=0.
For vE C(Rn), as N-- , the function will converge independent of Thus one expects the inner integral at
to
.
right of (1.10) to converge to
j=
since
P sin t dt/t = n/2
(1.11)
0
Legalization of this argument will confirm Parseval's relation, since the right hand converges to the right hand side of
(1.8), as N-. With uE L' and vE Co (setting cen(t)=ikp(t.)) write (1.12)
f Rng1xuv
To show that the inner integral at left goes to 0 as N->oo
it is
more skilful to use the integration variable O=1;/N, dt=NndO. For n=1
,
fsin NO
f101sb + fjolab
= I0 + I. .
Here we get (with w(O) =
JI01 sOIIv'III N((w(O)cos(NO)le=&b + The latter gives 100 s
(11vIILW+ 11v' 11L00)
,
f1OjZbcos(NO)wl0(O)dO).
with a constant c, only
depending on the volume of supp v, i.e., it is fixed after fixing v . The estimates imply the inner integral to go to 0, uniformly
6
0. Introductory discussions
as xE lgn. For uE L'
the Lebesgue theorem then implies the left
hand side of (1.12) to tend to 0, as N-- , for each fixed vE CO For general n the proof is a bit less transparent, but remains the same: Split the inner integral into a sum of integrals over a small neighbourhood of 0 and its complement. In the first term use differentiability of v; in the second an integration by parts. We now have a 'polarized' Parseval relation, in the form
(1.13)
RngIxu^ V.
ngxuv
for u E L1 , v E
,
COO
For u E LIf1L2 pick a sequence ujECO with IIu-ujIILl - 0,
11u-uill L2 0,
as is possible. Then, since uj-ulE CO C L2 (1.13) with u=v=uj-vj implies Iluj^-u1^IIL2=IIuj-u11IL2 0, j,l - . In other words, uj and ,
uj^ both converge in L2 Clearly, uj u^. Indeed, initially we showed uniform convergence over &n, while the V -limit z=lim u .
satisfies (u^
f (u^ -z)cpdx=0 for
,(p)=f z pdx for all cpE Cc". This yields
all such cp, hence u '=z (almost everywhere), since CO is dense in
V. Substituting u=v=uj in (1.13), letting j-3, it follows that (1.8) is valid for all u ELI(1L2 confirming Parseval's relation. Clearly (1.13) also holds for all u,vE L'(lL2. We use it to prove the Fourier inversion Let n=1. For vE L'("IP, u=X[ 0,x0 ]' some xo>0 apply (1.13). Confirm by calculation of the integral that ,
(1.14)
(2n)1/2u^(x) =
(e-ixxo_
1)/(-ix) = hxo(x)
x # 0
,
,
hence xo
(1.15)
v(x)dx = gIxv^ (x)hxo (x)dx
0
The Fourier inversion formula is a matter of differentiating (1.15) for xo under the integral sign, assuming that this is legal Consider the difference quotient:
xo+b (1.16)
JgIxv^(x)eixxo sin 6x
v(x)dx =
(20)-I
xo -b
Assuming only that v (1.17) limn-0(26)-nfQ
v^ both are in LI
,
xo ,s
v(x)dx =f9lxv^
(Actually, our proof works for n=1 be extended to all xo
,
,
and general n
,
it follows indeed that (x)eixxo=
xo .
> 0 only
(v^ )I (xo ), xoE 2n. ,
but can easily
One must replace the deri-
vative d/dxo by a mixed derivative an/(axol...axo n).
)
Indeed,
0.1. The Fourier transform
7
letting 6-0 in (1.17) we obtain (1.15), using that sin(6x) /(Sx) - 1 uniformly on compact sets, and boundedly on & , as 6-0 If v is continuous at x0 then clearly the left hand side of (1.17) equals v(xo) giving the Fourier inversion formula, as it is well known. For n=1, if v has a jump at xo then the left hand side of (1.17) equals the mean value (v(xo+0)+v(xo-0))/2 exists, if only Again for n=1 a limit of (1.16), as .
,
(1.18)
limaloo
v^ (x)pix
,
-a the principal value, exists (cf. pbm.6), without requiring VA E-= L'
We summarize our results thus far: Proposition 1.1.
The Fourier transform u^ of (1.1) and its com-
plex conjugate u" =(u^)
have u^
,
u" E CO(1Qn)
relation (1.8)
.
.
are defined for all u E L1(&n)
,
and we
For u E L1(In)1L2(In) we have Parseval's
If both u E L1(In) and u^ E L1(1en) hold, then we
have u^" (x) = u"^ (x) = u(x) for almost all x E In
It is known that the Banach space L1(&n) is a commutative Banach algebra under the convolution product (1.19)
u*v = w , w(x) = fOyu(x-Y)w(Y) = f41Yv(x-Y)u(Y)
Indeed, (1.20)
IIwIIL1=JIw(x)Idx s xnfdxfdyIu(x-Y)IIv(Y)I = KnIIuIIL1IIvIIL1
Prop.1.2, below, clarifies the role of the Fourier transform F for this Banach-algebra: F provides the Gelfand homomorphism.
Proposition 1.2.
For u,v E L'
(1.21)
w^ (
let w = u*v
.
Then we have
E &n
) = u^
Proof. We have fpiye
fgixe
The substitution x-y=z , dy=dz thus confirms (1.21), q.e.d. The importance of the Fourier transform for PDE's hinges on Proposition 1.3.
(1.22)
If u(P) E L1 for all (3s a then
u(a)^
h
E &n
.
8
0. Introductory discussions
Proof.Partial integration gives
fdxe
(with vanishing boundary integrals), implying (1.21), q.e.d. Given a linear partial differential equation (1.23)
P(D)u = f
,
where fE L1(&n), Dx =-iax j
P(D) =
ja sN
ax
a Da
one might attempt to find solutions by
,
J
taking the Fourier transform. With proper assumptions (1.21) gives (1.24)
Assuming that e = (P(X))' exists, (1.24) will assume the form
(1.25)
u"
e,
which by prop.1.2 (and Fourier inversion) is equivalent to (1.26)
u(x) = fgIye(x-y)f(y)
.
Presently, (1.26) can only have a formal meaning, since noror u(4 LL, in practical applications. However, as to be discussed in the sections below, the Fourier transform may be extended to more general classes of functions and to generalized functions. Then (1.26) yields a powerful tool for solving problems in constant coefficient PDE's (cf. sec.4).
mally (1/P)(4 L', or f (4 L'
,
Problems. 1) For n=1 obtain the Fourier transforms of the functions a) (a2+x2)-1, a>O; b) (sin2ax)/x2, a>O; c) 1/cosh x a_ ax 2
2) For general n obtain the Fourier transform of
,
a>0
3) Obtain the Fourier transform of f(x) = (1+IxI2)-v, where v>n/2 (Hint: A knowledge of Bessel functions is required for this problem). 4) Construct a function f(x) E L1(&n) such that f^(4 L1
5)The Riemann-Lebesgue lemma states that f^E CO whenever f E L1 Is it true that even f (x) = O((x)-E) for each f E L1 with some 8>O ? 6) Combining some facts, derived above, show that, for n=1, every piecewise smooth function f(x) E L1(&) has a Fourier transform satisfying f(x) = O(1/x) as Jxj is large, and satisfying ,
(1.27)
(f(x+0)+f(x-0))/2
=limfa91yelxyf^(y)
,
x E It
Here 'piecewise smooth' means, that & may be divided into finitely many closed subintervals in each of which f is C1 changing its value at boundary points.
,
possibly after
0.2. Fourier analysis for temperate distributions
9
2. Fourier analysis for temperate distributions on &n We assume that the reader is familiar with the concept of distribution, as a continuous linear functional on the space D(In) = CO(&n) . A linear functional f:D - C is said to be continuous if (f,gj)-.0 whenever (pj-0 in D. The latter means that (i) fj K independent of j, (iii) E D, j=1,2,..., (ii) supp cpjE KCC &n ,
sup{I cp(a)(x)I: xE &n} - 0, as j->oo, for every a. The space of dist-
ributions on In is called D'=D'(&n). The space Llloc(&n) of locally integrable functions is naturally imbedded in DI by defining
(2.1)
(f,(p) = ff(x)g(x)dx
for fE L11oc
,
The derivatives f(a)=aaf of a distribution f E D' are defined by (f(a),(P)
( 2 . 2 )
= ( - 1 )
I a l ( f
,
( , ( a ) )
,
q E D
the product of a distribution f E D' and a C"O(&n) function g by (2.3)
(gf,lp) = (f,g(q)
pED.
,
Thus Lf is defined for any distribution
fE D,(Rn) and linear dif-
ferential operator L=Yaaact with coefficients aa() E C,(&n) While the value f(x) of a distribution at a point x is a meaningless concept, one may talk about the restriction fl!a of fE D.(&n) to an open subset S2 and its properties: First of all, the space DI(S2) of distributions over a consists of the continuous lin,
ear functionals on D(c)=CO(SZ), with continuity defined as for &n. For fE D,(&n), the restriction fID(St) defines a distribution of D'(a), denoted by f192. Thus, for example, it is meaningful to say
that fE DI(&n) is a function (a Ck(f)-function, etc.) in an open set 0C &n - it means that fISz has this property. For a distribution fE DI (a) on an open set the derivatives and product with gE COO(S2) is defined as in (2.2)
,
(2.3)
The support supp f (singu-
.
lar support sing supp f) of fE D' is defined as the smallest closed set E (intersection of all closed sets E) such that f=0 (such C00 that f is ) in the complement of E .
The concept of Fourier transform can be generalized to distributions on &n, with multiple benefit: Some non-L'-functions will get distributions as Fourier transforms. Certain distributions will get functions as Fourier transforms. The Fourier inversion formula and many assumptions (limit interchanges) will simplify.
0. Introductory discussions
10
We used the Fourier integral of (1.1) only for uE L'(&n). It is practical to introduce a growth restriction for uE D'(&n) if we want u^ to be a distribution again. Later on (sec.3) we also , but it no longer will be a distdefine u^ for general uE D'(&n) ribution in D'(&n). We follow [Schwl] here, but [GS] in sec.3. The growth restriction is imposed by requesting that uE D' allows an extension to a larger space of testing functions called S. Here S - the space of rapidly decreasing functions- consists of all cpE C"O(&n) such that for all multi-indices a and k=1,2,...,
(2.4)
sP(a)(x) = 0((X)-k) (x)-k
- the derivatives of y decay faster than every power Note that, equivalently, we could have prescribed that for every a one (and the same) of the following conditions be satisfied: (x)ku(a)(x) (for every k=1,2,..), or xOu(a)(x) (for every (3),
(2.5)
or
(AM) (a) (for every 0), is 0(1)
CB , or CO
,
or L2
,
or is o(1)
,
or is
or LP (for some l sps )
,
Indeed, for a given a one of these conditions may be weaker or stronger than the other. However for all a simultaneously all conditions are equally strong. One must use Leibniz' formula to handle interchanges of as and multiplications (cf. lemma 2.8). The above at once gives the following: Theorem 2.1. We have SC L1(&n), so that u^ of (1.1) (and u") are defined on S. Moreover, for uE S, we have u^, u"E S, and
(2.6)
(u^ )" (x) = (u" )^ (x) = u(x) , x E In
.
The Fourier transform and its conjugate therefore define bijective linear maps S -S , inverting each other. Proof. Using repeated partial integration and get
fdxe
=iIaI+ISIa
(2.7)
hence
(xRu(a))A
In fact, we get xsu(a) E L'
,
for every a,(3
,
by the equivalence
0.2. Fourier analysis for temperate distributions (2.5)
,
for u E S
.
11
Therefore the right hand side is in CO
,
so that u^E S
,
for
again by the equivalence (2.5). Thus we get u^ E S for all u E S Similarly for """ . Also, the Fourier inversion formula holds for u E S and the left hand side of every a,(3
,
.
,
(1.17) equals v(x). This implies (2.5), also by taking complex conjugates. The bijectivity then follows at once, q.e.d. Following Schwartz we introduce distributions with controlled growth at infinity - so called temperate distributions - over sz=&n
as continuous linear functionals over S. The space of all temperaso that a functe distributions is denoted by S'. Clearly, S DD tional u over S induces a functional over D - its restriction uID. ,
Definition 2.2. A sequence of functions cpj ES is said to converge to 0 (in S) if for every multi-index a and k = 0,1,2,... the sequence (x)kcpj(a)(x) converges to zero uniformly for all x E In Definition 2.3. A linear functional u over S is said to be conti-
nuous if cpjE S
,
(p,-.0 in S implies (u,gj) - 0
Temperate distributions are distributions. More precisely speaking: For uE S' the restriction uID determines u uniquely, and uIDE D' (1n). To confirm this we must prove: Lemma 2.4. a) If cpjE D, (Pj-> 0 in D, then we also have cpj- 0 in S. b)
For cpE S there exists a sequence cpjE D such that q)-pj->0 in S
From lemma 2.4 it follows that for WE S' the restriction v= uID is continuous over D : If cpj-0 in D , then c.->0 in S (by (a)),
hence (v, p) =(u, (pj) -0. Hence vE D' . Furthermore, if u, wE S' have uID=wID=vE D', then for gE S let (p. be a sequence of (b) above.
Get u-wE S' , (u-w, c-cpj) -0. Hence 0=(u-w, (pj) _(v-v, cpj) -+( u-w, (p) , implying that (u, cp) =(v, (p) for all y e S, or u=v, so that indeed uE S' is uniquely determined by its restriction v=uIDE D'
.
Proof of lemma 2.4. (a): IfkgjE D, cpj-0 in D then supp Tea) C K= In, while the functions (x) are bounded in K. Thus the uniform convergence (x)kcp.(a)(x)-,0 in pn follows from the uniform conver7 gence j(a)(x)-0 in I and weCoO(&n) have y.- 0 in S, proving (a). To prove (b), let X(x)EE satisfy x(x)=l near 0. For a qE S define cpj(x)=T(x)x(x/j), j=1,2,... so that qjE D. Setting w.(x)=1-x(x/j), get p.=g-c.=Tw.=0 in Ixlsl for large j. Note, ,
,
x)
is a linear
combination of ORY ' j=(x) kcp(') wj (Y) ,
(3+y =a
where sup{IOPY'j(x)I:x E &n} s sup at right goes to 0 as 1-- (i.e.,as j-co).
Also, sup{Iwj(Y)I}=j-IYIsup{w(x):xE 2n}s c. Thus Vj->0 in S, q.e.d.
0. Introductory discussions
12
Note that polynomials, and delta functions S(a)(x-a) are examples of temperate distribution. However, ex (4 S(le) (pbms.2,3).
To generalize F we still require the following. Corollary 2.5. The transforms F and F both have the property that
cjE S
,
Tj- 0 in S implies Fcpj- 0 FT j- 0 in S
It is sufficient to prove this for F. Again we need an equiin S' valence like (2.2), now for the property :
j=1..... Then 'ypj- 0 in S Proposition 2.6. Let q E S equivalent to each of the following conditions: .
,
(x)kgpj(a) (2.8)
0
,
or Apj(a) a 0
or (xs(pj)(a)
,
'
is
0
or k=0,1,2,..., in one (and
for all multi-indices a ,
the same) of the norms of CB(1n) or Lp(&n)
,
1spso .
For the proof cf. lemma 2.8. Using prop.2.6, lemma 2.5 is a matter of (1.2), and (2.7). hence Indeed, if pj- 0 in S we have IlxRcpj(a)IILI-> 0 j ,
,
(xaq)j^ ) (P) IICB_ 0 , implying (pj^ - 0 , For a given u E S'
(2.9)
,
q.e.d.
observe that u^
qES
(u^ ,p) =
, defined by ,
defines a functional in S', since q.- 0 in S implies p^- 0 in S (by cor.2.5) hence (u,cp^) - 0 If U E LI(2n) then it follows that u E S' (cf. pbm.3). In that case we have ,
(2.10)
.
ix
(u,cP^) =
(p E S
=
by Fubini's theorem, since the integrand is L'(12n). Thus, for uE L', (2.10) implies that the functional (2.9) coincides with that of the Fourier transform u^ of (1.1). Accordingly, for a general u E S' we define the Fourier transform u^ as the functional of (2.9) and the conjugate Fourier transform uv by
(2.11)
(u" ,(P) = (u,cp°)
,
(P E S
.
It is clear at once that we have Theorem 2.7. The (conjugate) Fourier transform coincides with the
0.2. Fourier analysis for temperate distributions
13
(conjugate) Fourier transform previously defined for L1-functions (cf.(1.1), and (1.7))
(2.12)
. We have the Fourier inversion formula
(u^ )` = (u` )^
= u , for all u E S'.
E S' Also, for u E S' we have and (2.7) holds as well. Prop.2.6 and (2.2) follow from the (evident) lemma, below. Lemma 2.8. a) We have (using Leibniz' formula and its adjoint) ,
(2.13)
(xau) (R) = IC
with finite sums and constants caRY b) We have (2.14)
Ixals(x)lal
,
and (x)ksc
xau(1) = Ed
,
a(3Yxa-Yu(R-Y)
,
(xa-Y u) (R-Y)
,
aRY
daRY
k laIsk
(x011 with a constant c k
c) We have
(2.15)
(lull
Lp
s II(x)-kll
Lp
II(x)kull
Loo
,
Ispn/p
d) We have
(2.16)
Ilull
lscll (1+Ixl )n+1u^ II
scIIU^ II L
=C
L
E
L
lalsn+l
II (U
(a)
"o s c
)^ II
L
c
Ilxau^ II
7,
lalsn+l
I
lalsn+l
IN
L
(a) II
L
Problems: 1) Show that the following functionals define distributions in D'(&n): a) (f,(p)=q(a)(x°), for given multiindex a and x°
E &n; b) (f,(p)=
J
(p(x)dS, dS=surface measure
;
c) (p.v.X,cp)=
Ixl=1
Ixarl
rp(x)dxX (for n=1). 2) Obtain the first partials of the
distributions of pbm.1. 3) Show that distributions f.E DI(I) are defined by (f+,T)=lime-0,s>0 -J cp()xE. Relate f+ with p.v.X of pbm.1. 4) The distribution derivative satisfies Leibniz' formula and its adjoint (cf. [C,],I,(1.23)). 5) Show that a distribution f E D'(R) with f(a)E C(st), lalsk is a function in Ck(S). 6) Let L1po1
be the class of all uE L'1oc(Rn) with (x) -kuE L' (i") for some k= k(u). Show that Lpo1C S'. 7) Show that p(x)=
aax' E Lpo1C S'.
asm
Also that CB(&n)C Lpol, and LP(&n)C Lpol, lspso. 8) Show that eax
0. Introductory discussions
14
E D'(2), but eax
O. 9) Let Tpol be the class of all
S', as Re a
k aE C00(2n) with a(a)(x)=O((x) a), for some kaE Z, for every a. Show
that differentiation and multiplication by aE Tpol leaves S' invariant. That is, for uE S' , aE Tpol, aE 2+ we have auE S',
u(a)E
S'. 10) Obtain the Fourier transform of the following distributions (If necessary, show, they are in S'): a) xa, a E 2+; b) S
xo
(P),
E Zn +
;
c) eiax , aE 2n. 11) Obtain (p.v.1)^, for the dis3F
1
tribution of pbm.1. 12) Define a distribution x E S', using the same kind of 'principal-value integral' as in pbm 1. 13) Obtain the Fourier transform of a Calculate (p.v.sln-h X)^ 2n-periodic C00(2)- function a(x). Hint: Use that a(x) has a uni2n
formly convergent Fourier series a(x)=y00 ameimx, am
ae-iz°xdx 0
14) Let f(x)=sin xJ
. Show that fE S' and evaluate f^.
3. The Paley-Wiener theorem; Fourier transform of a general uE D'. The support of a distribution uE D' was defined as smallest closed set Q with u=O in 1\Q. We now consider u with supp u0 a. A simple but important remark is that a compactly supported distribution uE D'(fz), as linear functional over D(fz), admits a natural extension to the larger space E=C00(c). (The notation was introduced by Schwartz again.) Indeed, for a given X(x)E CD(f) with x(x)=1 near supp u, define the extension of (u,.) to E by
(3.1)
(u,(p) = (u,x(p)
,
for all q) E E(a) = C,(fz)
.
This defines an extension: if yE D(fz) , then (1-x)TE D(fz), and supp (1-x)TC supp (1-x) is disjoint from supp u, hence (u,(1-x)(0= 0, or, (u,(p)=(u,x(p). The extension is independent of the choice
of X. If OE D(Q) has the property of x then r-x=0 near supp u,
(3.2)
(u,Ocp) = (u,xq))
,
for all q> E E(c)
.
The class of all distributions uE D'(fz) with compact support is commonly denoted by E'(fz). We have seen that E'(fz) is naturally
identified with a class of linear functionals on the space E(fz). Proposition 3.1. The set El(fl) of all (above extensions of) compactly supported uE D'(fz) coincides with the set of continuous lin-
0.3. The Paley-Weiner theorem
15
ear functionals over E(n) (i.e., the functionals u over E(at) such
that jE E, q .-'O in E implies (u, (pj) ->0) . Here ypj-0 in E means that Tj (a)(x) - 0 uniformly on compact sets of f, for all a .
Clearly the extension (3.1) to E of uE D' with supp u Cc n is a continuous linear functional over E, in the above sense: If ypjE E, (pj- 0 in E , then xroj* 0 in D, as a consequence of Leibniz'
formula. Vice versa, for a continuous linear functional u over E the restriction v=uID is a distribution in D', since cpjE D, Tj-0 in D trivially implies Tj-. 0 in E. Prop.3.1 follows if we can show that supp v CC st. Suppose not, then a sequence of balls Bj may be constructed such that um0 in Bj, while every set KCC st is disjoint from all but finitely many of the B. Construct T.E D ,
supp W C Bj with (u,(pj)=1
.
Observe that cpj-> 0 in E while ?u,(Pj)
=1 does not tend to zero, a contradiction.
Q.E.D.
For a compactly supported distribution on &n we always have a Fourier transform in the sense of sec.2, i.e.,we get E'(len)C S':
Theorem 3.2. All compactly supported distributions over mn are temperate. Moreover, for uEE'C S', u^ is a Co function given by
(3.3)
u^ (x) =
fne
(u,ex)
,
ex(
) = e ix
with a distribution integral, given by the third expression (3.3). In fact, the function u^(x) is entire analytic, in the n complex variables xj, in the sense that v(z)=(u,ez),
ez(x)=a-izx,
is meaningful for all zE Cn, (not only &n), and defines an extension of u^ of (3.3) to Cn having continuous partial derivatives in the complex sense with respect to each of the variables zl,...,zn. Note that formula (3.3) is meaningful only by virtue of our extension (3.1) of u E=-El to all of E .
Proof. For uE D'(En), supp u CC &n, the natural extension to E may be restricted to S again to provide a continuous linear functional on S, since "(pj- 0 in S " implies "q)j->0 in E". Hence uE S'. The function v(z) indeed is meaningful for all zE Cn. Existence of av/azj is a matter of the continuity of the functional u over E: For a fixed z h E Cn, form the difference quotient ,
(3.4)
we = (v(z+eh)-v(z))/E = (u,(ez+Eh ez)/e)
For the directional derivative Vhez
(3.5)
,
E > 0
of ez at z , we get
V. = (ez+eh ez)/E - Ohez
0 in E
Indeed, this only means that aXVE, 0 uniformly on KCC 2n, as rea-
0. Introductory discussions
16
dily verified for
Continuity of u then implies
limE-0,Ex0wE _ (u,Vhez)
(3.6)
,
confirming that v(z) is analytic for all z. Formally we then get
(3.7)
(u^ ,q)) =
rP(U)) =
(u,e
f91
with v(x) as defined, where the interchange of limit leading to the second equality remains to be confirmed. Clearly (3.7) implies u^=v, i.e., (3.3) and thm.3.2 follows. For the interchange of limit show existence of the improper Riemann integral f g in the sense of convergence in E: For KCC in we must show that
in E, as k-. Here Sk is any sequence of Riemann sums, with maximum partition diameter tending to 0 as k-'o. Also, that f
as K runs through a sequence Kj with LX.=&n,
R n \K
again, with convergence in E. Again, convergence in E just means local uniform convergence with all derivatives. One confirms easily the local uniform convergence in the parameter x since the function e,(x) = e-'x is continuous. Similarly for the x-derivatives, again continuous in x and This, and the fact that the x-derivatives of the Riemann sums are Riemann sums again, indeed allows to confirm the desired convergences. Q.E.D. As examples for Fourier transforms of compactly supported ,
.
distributions we mention those of the delta-function and its derivatives. As seen in 2,pbm.5 we get 60(a)^= i1alxnxa In fact, this is an immediate consequence of (3.3), above. We observe that the entire analytic function u^(z) of (3.3), as a function of complex arguments z has a growth property which characterizes the Fourier transforms of compactly supported distributions. The result is called the Paley-Wiener theorem. .
,
Tn
Theorem 3.3. An entire analytic function v(z) over is the Fourier transform of a compactly supported distribution uE D'(&n) if and only if there exists an integer k > 0 and a real i>O such that
(3.8)
v(z) = O((z)ke'11Im z1)
for all z ETn ,
(z)=(I+Ilzj12)1/2
Moreover, the constant I may be chosen as the radius of the smallest ball lxlsr containing supp u Furthermore, uE D(&n) if and only if (3.8) holds for all k with rl=max{lxi: xE supp u} .
0.3. The Paley-Weiner theorem
17
Proof. For u E E' we must have (3.9)
I(u,(P) I s c sup{ IyP (a) (x) I: x E K
Ialsk }
for some c, k, and some compact K J supp u and all TEE E. Otherwise
for every c=k=j and Ixlsj there exists T=q E E with (u,T.)=1 ">" holds in (3.9). Or, I(pja)(x)Is for all lalsj 2,... implying uniform convergence Tja)(x)- 0 ,j-*oo ,
Ix1sj ,
,
,
and j=1,
a contra-
diction, since 1=(u, c).) does not tend to 0. We get u^(z)=(u,Xzez) x(t)=1
xz=X(IzI(IxI-i)) where xEC00(R)
, x decreasing. It follows that supp X. C } so that (XZez)(a)(x)= O(e'1IImzI+1(z)k). Combining this {IxI511+ with (3.9) we get (3.8) with the proper constant i Next assume uE C0(f). We trivially get (3.9) with k=0 and K= ,
t1
supp u CC n, since uE L1. Similarly for u(a). Accordingly, for all a we get
1zau^(z)I=I(u,e(a))I=O(e1IIm
z1),
hence (3.8) for all k.
Vice versa, (3.8) for all k implies xavllnE L'C S'. Then v" is given by the conjugate Fourier integral. We have u=(vl&n)`E CO, and even u(a)E CO, i.e., uE Coo(&n). To show that supp uCC in write (3.10)
u(x) = J
If 0 E In is given arbitrary then we also may write (3.10) as u(x) =
(3.11)
Indeed, this is a matter of Cauchy's integral theorem, applied for a rectangle in the complex i;j-plane with sides Re tj=±A, In =0 or Oj. In such a rectangle the integrand elx'v(i;) is holomorphic
as a function of ti for constant other variables, so that the complex integral over the boundary vanishes. For A-- the integrals over Re tj=±A, O
for example. The integration pathes have length 0i and the integrand is O((A)-2e(1-x)IOI). The integral (3.10) may be written as n-fold iterated integral over It. The above proceedure allows the transfer of the integration from R to the line { j+i0 Next let us estimate (3.11): (3.12)
u(x) =
xjE I}
O(e1101-xO)
setting k=n+1 (it holds for every k), and using that (1+i0) z M. The 'Q(.)- constant' is independent of 0. Hence we can set 0=tx, The exponent is <0 as 1xI>1, and t>0, for u(x)=O(etlxl(11-Ixl)),
0. Introductory discussions
18
u(x)=0 follows. Thus supp uC {IxIsi}, uE D, if (3.8) for all k. Finally, if (3.8) holds for some k, let X(x)E=- D, supp x C
{jxIs1}, x(x)2:0, fx(x)¢x=1. For e>0 let XE(x)=nx(E). Note that
x
(
Moreover, for any cpE S get
)=x^ (ei;)-+x^ (0)=1, as
qe^-w+0
in S. Since supp XEC {IxjsE} we have (3.8) for xE^(t) with rf rep-
laced by a for all k. Hence the product v(z)xE^(z) satisfies (3.8) with ti replaced by T1+e again for all k. It follows that vxe^=ue^
with uEE D, supp uEC {jxjsri+E}. Also (uE,(P}=(uE^ ,Cpl}=(v,xe^V") (v, q)-) _(u, q) , for all qE S. The latter implies that supp uC xI2M+e}, all a>O. It follows that uE E, supp uC {IxIsr1}, q.e.d.
Let Z denote the space of all entire analytic functions v(z) =v(z1,...,zn) in n complex variables such that for k=0,1,2,..., and some rjz0 we have (3.8) satisfied. We shall say that a sequence
vj E Z tends to 0 in Z if (i) estimates (3.8) hold with constants Nn}-0,
independent of j, and (ii) mj=Max{Ivj(x)I: x E
as
Corollary 3.4. The Fourier transform F: u -, u^ establishes a linear bijection D H Z which is continuous in either direction, in
the sense that u
0 in D holds if and only if uj^-> 0 in Z
.
Proof. After thm.3.3 we focus on continuity only. If uj-> 0 in D
then supp uj C {jxIsa} for a independent of j
.
This yields (3.8)
with r)=a independent of j, by thm.3.3. Inspecting the first part
of thm.3.3's proof we also find the 0(.) constant independent of J. Vice versa, if in Z, then (3.8) with r) independent of j implies supp uj° C But (3.8), for real z=x, implies vj= O((x)-k), uniformly in x and j, for every k. Thus conclude from cdn.(ii) that jlxav1I,
1-> 0, as j-. -. For the inverse Fourier trans-
0, so that indeed
form uj=vj^ we get 1Iuj(a)ii
in D.
Q.E.D.
L 00 _>
Following [GS] we now define a Fourier transform of a general distribution f E D'(111) regardless of growth at infinity, as a continuous linear functional f^:Z-'Q. Here of course "f^ contin-
uous" means that "(f^ ,(pj) -'0, whenever qj->0 in Z". We define f^ by (f^ ,(p) = (f,(p^)
(3.13)
,
for all p E Z
,
taking into account that p^
E D for T E Z This definition is compatible with the earlier ones. Indeed, we have Z C S in the sense that for uE Z the restriction ulin determines u uniquely and is contained in S . Moreover, Z is dense ,
in S, since Z=D^
, and D is dense in S while F and F
are continu-
19
0.3. The Paley-Weiner theorem
ous maps S->S. Also T j-0 in Z implies cp.-> 0 in S. For uE S' the restriction u1Z determines u and we have ulZE V. Hence get a natural imbedding S' 4Z' . For uE S' C Z' we earlier defined (u^ , (P) _(u, V ) for TE S. The restriction u^IZ gives our present functional,q.e.d. for uE D' in general is not a distribution, Notice that u^ as defined in sec.2. It is a linear functional on Z , not on D . Recall that for a function f E LIloc(R) with f=0 in x
,
f-(r;)= f glxe lxtf(x)
(3.14)
,
Im t < -c
0
where the integral exists and defines a holomorphic function in the complex half-plane Im i; <-c (we have modified the standard
definition, by a factor i). The inverse transform then is given by
f(x) =
(3.15)
-oo+iy
+«'+iy plteixif
(i;)
with a complex curve integral along the parallel Im = y <-c We now will identify f- with the Fourier transform f^E Z' of the distribution fE D'. For cpE D , supp T C {jxjsi} , we know that q is entire analytic, satisfying (3.8). For y<-c we have
(3.16)
Im =y
f- (t) q), (t) dt = P91xf (x)
f eixtg. (t) dt _ (f,g)
Imt=y
The integral dxdIl;I exists absolutely, hence the interchange, by
Fubini's theorem. Also, we get
= fR
f
,
at right, by the pro-
perties of the (analytic) integrand. Then (3.16) follows from Fourier inversion for functions in D. Or, f^E Z' may be written as
(3.17)
(f^
t)) =
f f- (t)cp(z)dt
,
cpE Z
Im?=y where we must choose y < -c with c of (3.14) (for f) Thus for a function fE LIloc(R) of exponential growth and =0 in x<0 the Fourier transform f^ is given as the complex integral (3.17) involving the Laplace transform f- of f .
Problems. 1) Obtain the Laplace transforms of the following functions (Each is extended zero for x<0). a) xk k=0,1,..., b) eax; c) cos bx ; d) eaxsin bx ; e) sin In each case, discuss the .
Fourier transform - i.e., the linear functional on Z. 2) Obtain the inverse Laplace transform of a) ; b) log(1+Z ). (In each case specify a branch of the (multi-valued) function well defined in a half-plane Im z < y .) 3) For uE D'(ien) with supp uC {xix0}=
0. Introductory discussions
20 In
and ecx'uE S', for some c, show that u^ may be defined by a complex integral like (3.17), with u replaced by ut with respect to (x2,...,xn). 4) The convolution product w=u*v, so far defined for u,vE L'(&n), by (1.19), may be defined for general distributions u,vE D'(&n) under a support restriction -for example (i) if supp U= 1n, supp v general, or (ii) if supp vC {xlzo}, supp v C {jxjscx,}
One then defines (w,(p>=ffdxdyu(x)v(y)T(x+y),
.
with a distribution integral (for precise definition cf.[Schw1], or, Show that (1.21) is valid for this convolution product as well, assuming in case (ii) the cdns. of pbm.3 for u,v. 5) Let TA be the space of all entire functions x(z) satisfying (3.8) for some k. Show that XPE Z, for pE Z , XE TA . Moreover, setting show that fE Z' allows definition of a product XfE Z' (xf,(p)=(f,xcp), (pE Z All polynomials p(x) belong to TA . 6) Show ,
.
that (1.22) is valid for general distributions
u E D'(&n)
4. The Fourier-Laplace method; examples. We now will discuss the 'Fourier-Laplace method' for 'free space'-problems of the following constant coefficient operators: (4.1)
A ==lax 2
(the Laplace operator)
(4.2)
A + k2
(the Helmholtz operator)
H = axo - A = at - A
(4.3)
(4.4)
= axo2 - A = at A
(4.5)
+m2
(the heat operator)
,
,
(the wave operator)
(the Klein-Gordon operator)
.
The last 3 operators act on the n+1 variables x0=t, (x1,...,xn)=x. The first two act on x only, distinguishing xo from the others. The discussion around (1.23)-(1.26) was a formal attempt to solve constant coefficient PDE in free space (in all In). We found e= ()' for a P(D), of special interest. Now we are prepared to implicate this technique, called the Fourier-Laplace method. Certain initial-boundary problems may be converted into free space problems: (a) An initial-value problem for (4.3),(4.4), or ,
0.4. The Fourier-Laplace method (4.5)
21
seeks to find solutions u of P(D)u=f in some half-space,
say, t=xo>O, where f is given in ta0, together with initial data of u at t=0. Such problem may be written as a free space problem by extending u=0 and f=0 into t0, u=0 as x1=0, then consider the odd extensions of u and f to &n: v(xl,...,xn)=u(x), x1>0, =-u(-xl,x2,...,xn), x,<0, similarly g extending f. It follows that Av=g in Rn, again converting the half-space Dirichlet problem of A to a free space problem over I
.
Similarly for the Neumann problem, using even extensions. Technique (a) works as well for a more general initial surface t=0(x) x(-= &n. Both techniqes may be combined to reduce cer,
tain initial-boundary problems to free space problems. The above will emphasize the power of the Fourier Laplace method. (4.1)-(4.5) are a crossection of popular PDE's. We control parabolic and hyperbolic initial value problems, elliptic boundary problems and initial-boundary problems in half spaces, etc., with Green's (Riemann) functions, using results on special functions. From now on interprete the equation P(D)v=g , xE Rn, as a PDE involving distributions v,gE D'(1n). The Fourier transform exists without restrictions: Using 3,pbm.6, we get P(x)v^=g^ , where v^, g^E Z'. If eE D'(&n) solves P(D)e=(2n)n/2S we get P(x)e^=1. In the cases corresponding to (4.1)-(4.5) (4.6)
P(x) =-Ix12
,
=k2-Ix12
,
=it+x12
,
we get, respectively, =Ixl2-t2
,
=m2+Ix12-t2,
where t=x0 again. Generally, P(x) (4 Liloc' except for (4.3) and (4.1), n2t3, due to zeros of P. Some pbm's of sec's 2,3 (and, more generally, [C,],II) discuss distributions p.v.a associated to a
Llloc' P(x)z=1 may have many solutions zE D' (or E Z'). For (4.3)-
(4.5) we will be interested in z=e^, P(x)z=1, with supp eC {to}, because then supp a*gC {t2-0} whenever supp gC {tao}, so that u= (e*(g+h))I{t2-0} will solve the initial-value problem for P(D)u=f
ta0, we started with in (a) above. Indeed, a proper z exists: In pbm's I and 3 of sec.2 we defined p.v.1 , and f+ all 3 distinct, xf=1. Only f has its inverse Fourier transform =0 for x<0. ,
22
0. Introductory discussions
D,(Rn+1) solving For (4.3)-(4.5) we will construct such eE supp e C {t2-0}, using the setup of sec.3, pbm.3: P(D)e--%r2_xn+I6 ,
Such e, if ecxeE S', will have a Fourier transform in x=(xi,..) and a Laplace transform in xo=t. Accordingly we must seek an inverse Laplace transform of an inverse Fourier transform of a suitable solution z of Pz=1, or vice versa, in appropriate variables. Our proofs will be sketchy, in part, due to overflow of details. The lemma below is convenient, due to spherical symmetry of P. Lemma 4.1. Given a spherically symmetric function f(x)=co(Ixi) LI()e+,rn-1dr). where w(r) E Then the transforms f^ and f, are spherically symmetric as well: f^(x)=f"(x)=X(Ixl), where X(r) and co(r) are related by the Hankel transform A , v=2 1. In detail we have ,
r(n-1)/2X(r) = Hn/2-1(r(n-1)/2(o(r)) (4.7)
r(n-1)/2w(r) = Hn/2-1(r(n-1)/2X(r))
.
where (4.8)
H (X(r))(p) =
fV pr Jv(rp) ?(r) dr
, Re v > -2
0
with the Bessel function J v(z) . The second formula (4.7) is valid L1(&+,rn-1dr) if in addition X E .
T paxe-ix
Proof. For an orthogonal nxn-matrix 0 get Jxe-i(O
T
X)
T
Thus f^ has the same
w(IOTxI) = folye
symmetry:
with some X(p). We may set
RIXe-ipx w(r)=Kn f rn-1w(r) f e-irpz, dS 0 where the inner integral I is over the unit sphere Izl=1. Evaluate
(4.9)
f^
Il;I
, 0 , ... , 0)=f
this inner integral by converting it to an integral on the n-1-dimensional ball IXI2 sl, setting z=(z1,?,). We know that dS= With a contribution from the upper and lower hemisphere where z,= 1-72
, writing dX=an-2dodl, o=IXI, etc., we get 1
I=2
on-2 1662dEcos(rp)=tan-1 f 6n-2cos(rp) 0
A substitution a= sin 0 of integration variable yields x /2
(4.10)
1 = 2an-1
d9 sinn-20 cos(rpcos 0) 0
23
0.4. The Fourier-Laplace method (n-1)/2 ,
2 with an-1 ='((n-1)/2)
the area of the n-l-dimensional unit sphere. Using Poisson's formula ([MOS],p.79) we get fn /2
(4.11)
cos(rpcosO) sine-28 d9 =2n/2-2V.(n21)Jn/2-1(rp)
0
Substituting into (4.10) and I into (4.9) confirm (4.7). No change a-ix in (4.9) is replaced by elx . Thus lemma 4.1 follows. if Recall also (1.26), now under the aspect of 3,pbm.4. In details, the convolution product v*w of two distributions v,wE D'(&n) may be defined by setting (with a distribution integral)
(4.12)
(v*w,(P) = fv(x)w(y)g(x+y)dxdy = (v(ga,V) , V(x,y)=T(x+y)
for all q)E D(&n)=Cp(&n)
,
if v and w satisfy the support condition
(4.13) Ka (supp v x supp w)fl{jx+yIsa} is compact for every a>0
.
Here supp v and supp w are regarded as subsets of in of the variables x and y, respectively (cf. [Schwl] or [C1], 1,8). The distribution v=e*g is defined for e as constructed above whenever g E E' for (4.1)-(4.3) and gE D'(1en+1) , g=o as t<0 for (4.4) and (4.5) since condition (4.13) holds under these assumptions. Moreover, P(D)v=g follows, leading to a solution of the free space problem, and the related initial-boundary problems. ,
,
Now let us attempt a detailed construction of the proper e. Ia) Consider the operator A of (4.1), i.e., the potential
equation Au=f. For n2:3 the function -IX is Lpol, hence a distribution in S'. This is a homogeneous distribution of degree -2 Hence e is homogeneous of degree 2-n . It is also spherically symmetric. Conclusion: , with a constant cn. Clearly e E Lpol The constant cn may be evaluated by looking at e(x)=cnIxI2-n
.
cnfAW(x)1x12-ndx =cnlims-0
(2n)n/2g(0)=(e,Ag)= =cnan(2-n)T(0) (4.14)
.
dSicpdrr2-n
f
r=e
It follows that
en(x) =(n2)w 1xI2-n n
n- n 2
n
For n=1 we first define a distribution (4.15)
e^=-p.f.XZ
=
!I (p.v.1
involving the distribution derivative and p.v.X of pbm.l,sec.2.
We confirm that e^ solves -xz e^ =l . Using a=e^ (4.16)
e(x) _jxI .
one finds that
0. Introductory discussions
24
For n=2 we can define (cf.[C1],II,(2.11), for 1=2) (4.17)
a"(x)=-p.f.IX :=-Lj=1( x Zxjloglxl)Ix.7=-2A((log Ix2
again with distribution derivatives. However, it is easier to con-
firm directly that e(x) = x1loglxl
(4.18)
is a spherically symmetric L11oc-function satisfying Ae=b (just evaluate the integral (Ae, q) _(e, A(p) , using partial integration). Ib) Consider (4.2), i.e, the Helmholtz equation (D+X)u=f also known as the time-independent wave equation if X=k2>0, and as the resolvent equation of the Laplace operator A if XE C, X#k2
.
In the latter case get a^(x)=(X-1xI2)-1E C°n Lpol. An evaluation We get of e(x) is possible, using lemma 4.1, as long as ns3 .
(4.19)
IxI1-n/2
e(x) =
Pn/2X
J
Jn/2-1(Plxl)
For larger n this integral ceases to exist. However, it still will exist as improper integral in the sense of distributions A that is, as limAO where the limit exists in weak convergence ,
0
of D' (i.e., (lim.,(p) = lim(.,(p)
). For odd n the Bessel function Jn/2-1 may be expressed by trigononmetric functions. For example,
in case n=3 we get J1/2(z)
Or,
e(x) = VnII J-- 24 sinpixl
(4.20)
0
We may write X = x2 sinpr = 2
00
,
picking the root x with In x>O eirpxdP
KPp sinpr =41 -00
e(x) =-7
(4.21)
.
= 2 ixlxl
Then Hence
e'xlxl/lxl
(4.21) may be confirmed, noting that e=eiicr/r solves (A+X)e=2n6
For x=k real is not Ltioc (4.22)
P(x)=k2-lx,2
vanishes at the set Ixl=k, and
Then look at p.f.(P(1x)). Or else, observe that
lim£.,0,s>0 ek+is(x) = e(x)
,
ex(x) as e(x) in (4.21)
in the sense of distributions. This implies that (4.23)
a+(x) _ -Ve±iklx1/IxI
0.4. The Fourier-Laplace method
both will solve (A+k2)e=(2n)3/26
25
The proper sign may be chosen
.
by imposing a 'radiation condition' at - . For general n2:2 we still may evaluate the integral (4.19).
Using a formula by Sonine and Gegenbauer (cf. [MOS], p.105) we get (4.24)
e(x)=-(
1XI
)n/2-1Kn/2_1(KIxI)
x > 0
,
with the modified Hankel function Kv(z). Again get (4.24) more directly, observing that e(x)=y(Ixl) solves (A+X)e=0, hence y(r) solvn-1, we obtain es the ODE y"+n-1y'-x2y=0. Substituting y=rvb, v
the modified Bessel equation S"+rS'-(1+j.)5=0, showing that the only spherically symmetric solutions of (A+X)u=0 in S' are the multiples of (4.24). A partial integration shows that fe(A+X)tpdx= g(x) for all TEED , fixing the remaining multiplicative constant. II) In the case (4.3) of the heat equation Hu=ut Au=f we may use (4.24): Applying FX1 to P for the second and third polynomial (4.6) gives the same result, if we set ?=k2=-x2=it. That is, x= e-in14Jt,
Re x >0, in (4.24) will define F-'(.) , and we then must obtain the inverse Laplace transform. It is more practical, however, to first obtain Ft Note
has inverse Laplace transform 04(t)-
8(t)=(it+a)-1
ta0, =0, t<0
,
2ne-at,
f e-ate-ittdt. For e'=F- 1(P) get
calculating
0
(4.25)
(2n)1/2e-t1x1 2
e' (t) =
Recall that (e -
as ta0
,
,
e' (t)=0
as t<0
,
.
x 2/2)"=e- xjz/2 (in n dimensions), by a complex
integration. Also for the function ga(x) = g(ax) we get
(4.26)
g0" (x) = a ng" (x/a)
,
a E &+
,
as shown by an integral substitution. Choosing a--V_2t we thus get (4.27)
e(t,x) _()
_I X12
ta0
11
=0
t<0
This is the well known fundamental solution of the heat opera tor. It is not L1(Rn+1). Use it to solve the initial value problem (4.28)
Hu = ax u-Au = f
,
x0a0
,
u(x0,x) = q(x)
,
0
where f,p are given
Setting u and f zero in t<0 to
26
0. Introductory discussions
obtain functions v and g we get Hv = g + 6(t)(9p(x) = h
(4.29)
Thus v=e*h , or, (4.30)
u(t,x)=xn+lftstgltgTye(t-t,x-y)f(y) + xn+1r
J9Tye(t,x-y)w(y).
III) Now we look at (4.4), or, the wave equation u = (at - O)u = f
(4.31)
We apply the Fourier-Laplace method as for (II): The function
P =
(4.32)
-t'
IxI
21x7
{t+Ixi - tIxI}
has inverse Laplace transform (in t) given by
(4.33)
2i Ix
(e-
itjxl_ itlxl
)=
sin( x t)
V2n
1xI
,
ta0
(and zero for t<0). Looking for F;1 of the function (4.33) we cannot apply a Fourier integral, since the function (4.33) is not L1.
First set n=2. Writing FXlw=w° for a moment, we have
(4.34)
(e,cp) =
(cos2rt (Plt) = (f,mlt)
using that (cos Ixlt)lt = -Ixl sin IxIt
, roE D(Rn)
,
r=IxI
Now the inverse Fourier
.
integral of f= cos2rt may be calculated as improper Riemann
integral limAjl xIsA , although still that function is not LI Using (4.7), (4.8) - where v=2
j,(,)=V2
z , for n=3 - we get
,
00
f°=w(IXI)
w(r)=(V_2n)(N(_2
)
f '(r) sin r13 cos2pt dp
.
Or,
0
w(r) = r
= r{
(4.35)
Psin rp cos pt
sgn(t-r) - 2}
.
p
=r fco(sin p(t-r) - sin p(t+r)) P
Conclusion:
e(t,x) =li atH(t-IxI) , as t>0
,
=0
,
as t<0
,
n=3
with the distribution derivative at and the Heaviside function H(t)=1, tZ0 , H(t)=0 t
,
0.4. The Fourier-Laplace method (4.36)
27
, t>0 , u=p , ult=V at t=0
ultt-Au=p
into a full space problem we get vltt Av = S'(t)Clp(x) + 6(t)W(x) + g = h
(4.37)
where v and g are u and f extended zero for t<0 e*h we get the solution of (4.36) in the form ,
u(t,x) =4iJ (4.38)
.
Evaluating
yf(t-lx-yl,y)
lx-ylst
+ 4n at(t,Ilzl=1T(x+tz)dSZ) +
4n Jlz1=1V(x+tz)dSz
(4.38) is known as the Kirchhoff formula.
For general n2t2 let j=[n21] . As in (4.35) write (e,y)) _
+t or fj=tcPs,1t (with p=lxl) for V-(f j,((at)j(p)where fj=±sPn° even (odd) J. The inverse Fourier integral of fj exist at least as
limAJ lxlsA' (4.39)
we apply lemma 4.1 as above. For odd n we get
fj°(t,x)=wj(t,lxl)
,
wj(t,r)=*r v P7V(rp)ls'n cos pt}
for even n this will be of the form (4.40)
fj°(t,x)=wj(t,lxl)
v
wj(t.r)=tr
J JV(rp){cos pt} dp 0
with a convergent improper Riemann integral. More precisely, in (4.39) and (4.40) we have "(-1)1cos pt " as j is odd, and "(-1)lsin pt " as j is even, with 1=[Z] Our distribution e then is given by (4.41) below, with a distribution derivative at .
(4.41)
e(t,x) = (-1)JV-2natwj(t,lxl)
Now we must consult different formulas on Bessel function integrals where as a general reference we quote [MOS] (or [MO]). Case (a): n even, j odd. Then j=[n21]-2-1=v, 1=, n=0(mod 4). We use a Weber-Schafheitlin integral ([MOS] p.99, last formula). Since v=j is odd, we have for t>r the formula (4.42)
I = f Jv(rp)cos pt dp= (rz-t2)-1/2cos(vsin-1(r))=4Pj_1(F)' 0
with a certain polynomial Pj-1(t) of degree j-1. Note that the integral enters (4.41) under the derivative at so that a=0 for ,
0. Introductory discussions
28
tr-
For r
t>lxl
by the same
r-j(t+ tx-rx)-j
formula. Summarizing: For n=0(mod 4) we get
e(t,x) _-r 821+1(Z1(t,r))
,
with
(4.43)
Z1(t,r) = r 21Q214P, t>r
t
,
-rL)21+1
(t+
=0(-i)1 m(22m1)T2m(1_i2)1-m
where Q21(2) =
Case (b): n even, j even. We still have j=v=?-1, 1=2, n=2(mod 4) Use formula 3 on p. 51 of [MO]. The result is as in (4.43) with
Z1 (t,r) = r n/2Q21-14P (4.43)b Q 21
-1 (t) =
,
t>r , =-p (t+ t -
1(_1)l-m( 21 )t 2m+1(i_T2 )1-m-1
_-0
t
,
) 21
,
n!2>0,
1
4
2m+1
while for n=2 (i.e.,[nn-1] 1=0) we n2 set Q2i_1=0 =j, n=2j+1, 1='-', j=21+1, n=3(mod 4). Case (c): n,j odd, .
2
00
Let I= f Jv(rp)cospt
=v 2 f Jj-112(rp)J-112(tp)dp. Use formula 2,
p.50 of [MO] with (a,b)=(r,t), (v,n)=(1+?,1). Get v-n=2, I=O, t>r,
(4.44)
I=N"" 2r
1.3.5... (21-1)xF,
246
2.1
2
2
r
t
2r
where the hypergeometric function (hence Q21(z)) 1 (11 2Fi(1+-1;2;z)=1-1(21+1)z+li =](21+1)(21+3)2r + x
(4.45)
...
is a polynomial in z of degree 1=n43. Combining (4.39) and (4.44), (4.46)
e(t,x)
=(_i)1+1xr 21-1x21+1 t IQ21(r)H(r ,)} t
Case (d): n odd, j even, j=n41, n=2j+1, 1=
_ r-lxl
__n
1--n4
,
, n=i(mod 4)
.
Now we
need I= f Jv(rp)sinpt r = n2 f Jj_i,,2(rp)Ji/2(tp)dp . Again use 0 Get the formula of case (c), now with
(4.47)
I=V7(t)3.5..
2r r 2.4
21+1
2t-2 2F, (1+1,1-1;3;tx 2 2 r )_`/
21-1(t, r
t
and 1=0 for t>r , with a polynomial Q21-1(z) of degree 21-1. Hence (4.48)
e(t,x) _ (-1)lnr 21at1{Q21-1(r)H(r-t)}
n41
,
r=IxI
IV) Finally, look at the Klein-Gordon equation (4.5), n=3 only
29
0.4. The Fourier-Laplace method
(4.49)
utt Au + mz u = f
(p+mz ) u =
Equation (4.32) now assumes the form
p = Xz ltz ={t+a. - tip}
a(x)- m2+ x
FesintVx m x VTP m For n=3, v=2, j=n21-1 use (4.7),(4.8) on x(IxI) of (4.51), for 00
)p3/2/Vg -+W dp
Vrw(r) = nVt f J1/2(rp)J1/2(t
(4.52)
0
This integral diverges, just as for m=0. Applying a formula of Sonine (1880) and Gegenbauer (1884) (first on p.104 of [MOS]), for s>O , we get (µ,v)=(2,2 E) (a,b)=(t,r) m=z ,
,
,
foo
(4.53)
+p )
J1/2(rp)J1/2+E (t
0
p"I
pJ m
',
1-E
limn
Js-1 (MV-t2
IF
with existing weak limit in D'(13)
.
Or,
e(t,x) = nm limE->O,E>O{H(t-IxI)JE-l(m t2 rz)/ t-E
(4.54)
Recall, for v=s-1 we have Z/2) E-1
(4.55)
Jv(z) = r(s)
-
z 2 0 r( +m+E
6>0
m+
The second term, at right, goes to -J1(z), as E-0, uniformly on K and weakly in D'. For a cE D write (e, (p) =T, +Tz , by the decomposition (4.55). Then Tz - -nm(Jl (m tz -rz )/ t2-rz ,(p) , Ti M &n
,
2n lime-Of
rd(t)
(tt x )1-E
, where r(E)=E(1+c,E+ ...). Hence
Ixlst
T1=2n lime.,O{f (4.56)
dxdt t+rx (tEr)E =2n lim
=-2n1imE-Of1xlstdxdtat{r+t x }(t-r)E =-2nfH(t-r){ In other words, TI=n(-:-.a tH(t-r),g)
(4.57)
fdtdr{(t-r)E}t
IxIst
.
e(t,x) = r tH(t-Ixl) -
}tdtdx
Result:
J1(m tz-xz)
30
0. Introductory discussions
5. Abstract solutions and hypo-ellipticity. In sections 1,2,3,4 we have deployed the Fourier-Laplace method of solving certain problems involving constant coefficient PDE's under sufficiently simple initial boundary conditions. The focus was on the Fourier transform. In a distribution setting it
reduces the problem of solving P(D)v=g to the division problem speaP(x)v^=g^, providing an 'inverse' for P(D) in the form king roughly. In later chapters, this type of inverse construction will be extended to variable coefficients - and more general boundaries, but the central role of the Fourier transform is maintained, and the inverses obtained have similar features. On the other hand, the theory of constructing an inverse of an abstract operator between two linear spaces is well studied, in present times. It will be ever present in the background of our discussions. The role of certain Hilbert spaces- L2-Sobolev spaces - in theory of 1pdo's will have to be studied. Theory of (bounded and unbounded) Fredholm operators in Hilbert spaces will be crucial for elliptic i,do's, as well as compactness properties.
Often existence of a generalized solution to a PDE (a boundary problem) can be derived by purely abstract arguments. But then theory of pdo's may have to be used to derive properties of such solution -even differentiability, and that we have a solution in the classical sense. Let us discuss an example. For an open domain czC &n, consider the Hilbert space H=L2(n) with norm and inner product (5.1)
(u,v) =
The Laplace operator A=
1lull2 = (u,u)
, u,v E H
ax 2 defines a linear map CD(cZ)-C'(S)C H 7
which may be interpreted as an unbounded operator H0 of H with domain dom H0=Co(st), using that C0(a) is a dense subspace of H
In fact, H0 is hermitian and negative, i.e., we have (5.2)
(u,H0u) real and s0
,
for every u E dom H0
.
This implies that H0 has self-adjoint extensions. A distinguished such extension H , called the Friedrichs extension, may be constructed using a rather simple form-closing principle (cf.
[Ka1],[RN],[Wm1],[Yo1], or [C2],I, thm.2.7 ). To express the above in different terms: H is also a linear
0.6. Exponentiating a fast order operator
31
H We have (5.2) for H instead of HO, but H is maximal with respect to this property: No proper extension of H still satisfies (5.2). This maximality has important consequences: Any operator H satisfying (5.2) must be hermitian. A maximal such operator must be self-adjoint- it possesses a spectral measure, as we will not discuss here. Moreover, it follows that the unbounded operator K= 1-H is invertible in the following sense: There exists a continuous operator R E L(H), called K 1 = (1-H)-1, such that R:H-> dom H C H is a bi-jection inverting the linear map K:dom H -> H . Accordingly, due to existence of the inverse R, it follows that the linear equation (5.3)
(1-H)u = f
,
f E H ,
admits one and only one solution u E dom H C H , for every f E H After this abstract discussion we now ask: Does u of (5.3),
a function of H whose existence is shown by a chain of abstract arguments, solve the differential equation u-Au=f ? Also, does u satisfy any kind of boundary condition? The corresponding question is meaningful not only for A but for any smooth differential expression a(x,D) defined in sa such that the corresponding operator L0: C0 ->C0C H satisfies (5.2).
For general a(x,D), with the Friedrichs extension L of LO
,
the unique uE dom. LC H solving (1-L)u=f, for given fE HC L'loc(n) C Du(st) is a distribution solution of u-a(x,D)u =f, since
(u,c)-a(x,D)"cp) = (u,g)-((Lu) that (Lu,T)=(u,a(x,D)p), gE D(st)=CO(st)
q,ED(R), using
Question: (a) If, in addition, we have fE C"(c), will u be smooth - and a classical solution of u-a(x,D)u=f? (b) If, in addi-
tion, st has a smooth boundary r, and fE COO(clLE') , will also u be C0O(szlf), and will it satisfy boundary conditions?
For a(x,D)=A both questions have a positive answer. In particular, u must satisfy the Dirichlet condition u=0 on r. The same remains true if a(x,D) is elliptic and of second order (cf. ch.5). Generally, if a(x,D) is hypoelliptic the answer to (a) will be
positive. An answer to (b) is given in V,4. We speak of a concept called boundary hypoellipticity, in that respect. 6. Exponentiating a first order partial differential operator.
In this section we discuss the formal linear operator e, G
32
0. Introductory discussions
with a first order differential expression (a 'folpde') (6.1)
G
=1 b3a+ j
=
p
, j= aa/a x
J
with real Co coefficients bj(x) j=1,...,n, and complex-valued p()E C, all defined in some domain S2 C &n. By definition eG will ,
be the solution operator of the initial-value problem
(6.2)
atu=au/at =Gu , xES2 , to 0 , u=u0 , t = 0
,
C00
i.e., u(x,t)=eGtuo(x), uoE (12), is the unique solution of (6.2). The single first order PDE atu=Gu becomes an ODE along certain curves, called characteristic curves, defined as solutions of a system of ODE's, the system of characteristic equations, (6.3)
t'
= 1
xI = -bj(x)
,
,
j = 1,...,n
,
" " = d/ds
for (t(s),x(s))=(t(s),x1(s),...,xn(s)) with real-valued functions t,xj of a parameter s. The first equation (6.3) gives t-s=c=const. Thus set t=s, reducing (6.3) to the autonomous system (for xj(t)) (6.4)
-bj(xl,...,xn)
,
j = 1,...,n
d/dt .
,
Since the functions bj(x) are real-valued and Coo it is clear that the orbits (i.e. the curves in Sz, given by the parametric repre-
sentation x=x(t), for x(t) solving (6.4)), provide a family of nonintersecting smooth curves covering the entire domain R . A solution x(t) with x(0)=x0 will exist in a maximal interval Ost
where either t0=oo or x (t) E St\K for tE [ tK, t0 ) for every compact set KC Sz with some tK
.
In particular t0=co whenever for every
t1>0 a compact set Kt C Sz can be found such that the 'apriori
estimate' x(t)E Kt x(t) solves (6.4)
,
as 0stst,, can be verified from the fact that
and x(0)=x0. An orbit may degenete to a point,
or, may be closed -i.e., x(t) may be periodic. The relation between (6.2) and (6.3) is this: A function uE
C1, defined for (t,x) near (t° ,x° ), x°E S2, solves the PDE (6.2) if and only if along every solution curve (t,x(t)) of (6.3) the composite function u(t,x(t))=y(t)
solves the ODE
(6.5)
Y(t)=p(x(t))
dt = Y(t)q ,
Indeed, atj(t) _ -b.(x(t)) implies that
EbJulx )(t,x(t))
(6.6)
J
j
Hence if u solves (6.2), we will get (6.5) along every curve x(t)
0.6. Exponentiating a first order operator
33
for which the composition u(t,x(t)) is defined. Vice versa, if u(t,x) is a function defined and C1 near (t°,x°) such that p(t) satisfies (6.5) for every curve x(t) solving (6.4) in some neighbourhood of x°
Then u(t,x) solves (6.2) near (t°,x°).
.
Let u(O,x)=uo(x) be given for all xE St. Then a unique solution u(t,x) of (6.2) may be constructed as follows: At each x°E $1
construct the solution x(t) of (6.4) through x°, defined for a maximal t-interval (t-,t+), tt=t(x°). Define y(t)=p(x(t), and p(t)= u0(x°)exp{ f y(r)dt}, tE (t-,t+). Then set u(t,x(t))
(t) along
0
the curve x=x(t), tE (t-,t+), and do this for all such curves.
Assume that we have t+=tao for every x° E 12. Then the map x° x(t°,x°) defined by following the orbit through x° from t=O to t=
t° defines a diffeomorphism v,to :S2
n. Indeed, this map S2-S2 is 1-1
-
by construction, and it may be inverted by solving the reverse initial-value problem (follow x(t) through x° for a t-interval of length t° in negative t-direction). The map vt is Coo by standard results of dependence of solutions of an ODE on initial values. We thus obtain a 1-parameter family F = {vt : tE &} of diffeomorphisms vt St - St having the group property :
(6.7)
,
V
to nt
= nt+t
,
t' t E 1e
,
since following the curves for t+t units has the same effect than following first for t then for t units.- Such a group of diffeomor-
phisms is commonly called a flow. The flow defined by (6.4) is called the characteristic flow of the PDE (6.2) In this terminology we have proven: Theorem 6.1. Assume that all solutions of the characteristic sys-
tem (6.4) extend for tE
Then the
problem (6.2) has a uni-
que solution u(t,x)E Coo(&xn), for uoE C00(SZ), where u is given by (6.8)
ft(pov-t)(x)dt}
u(t,x) =
0
Indeed our above description of u(t,x) translates to (6.8). In particular the diffeomorphism vt is inverted by v-t, hence u is defined for all x E 12 and all t and (6.8) follows. ,
Note that (6.8) establishes an abelian group {etG
linear maps etG:C"O(S2)-C00(St) (6.9)
, defined by
etGuO(x) = u(t,x)
We also have
,
u0 E C'(c)
,
t e & .
:
tE le} of
0. Introductory discussions
34
etG
(6.10)
:
CO(Sc) - CO(SZ) ,
tE&
The relation between equations (6.2) and (6.4) remains invariant under a coordinate transform of 11, i.e., the characteristic
equations (6.4) go into those of the transformed equation (6.2). Accordingly theorem 6.1 also holds for a folpde G defined on a
differentiable manifold a with local representation (6.2) and assuming that characteristic equations locally given by (6.4) our condition remains satisfied - that all solutions of x'=-b(x), ,
regardless of the choice of initial value x°E n , extend into an infinite time interval, passing of x(t) between charts of different coordinates being permitted. Now we want to address the problem of finding a more concrete condition for the assumption of thm.6.1 that all solutions of the
characteristic system (6.3) extend indefinitely. Assume that a carries a Riemmannian metric ds2=EhJkdxJ,dxk, under which it is complete. Let d(x,x0) denote the distance from x to x0 on n . Proposition 6.2. Assume that the principal part tensor (b3) of the expression G of (6.1) satisfies the estimate (6.11)
Ib(x)I = {(Ehjkb3bk)(x)}1/2 = 0(1+d(x,x0)) , x E 11
,
with some fixed x0E n. Then all x(t) of (6.4) extend to -oc
.
Proof. By well known results on continuation of solutions of ODE'S it suffices to get an apriori estimate. Let x(t) solving (6.4) be
defined for tE [t0,t1] ,
x(to )=x°
.
y(t) is Lipschitz continuous in t (6.12)
Idy/dtl
s
Let y (t)=1+d(x° ,x(t)). We know and that
{Ehjkxj'xk'}1/2 = Ib(x(t))I = O(y(t))
,
noting that (6.11) trivially holds for each fixed x0 E n if it is valid for only one x0. Now (6.12) implies log y(t) =O(t-t0), or (6.13)
x(t) E {d(x,x0) s
ec(t-to)
-1 }
.
Since n is complete, the sphere (6.13) is compact for t=t1, and we have shown that x(t) stays in it. This proves the proposition. Now we look at G and eG in a Hilbert space H=L2(n,dµ), with (6.14)
(u,v) = fnuvdµ
,
IIu!I=
as inner product and norm, dµ denoting a positive Co measure on Q, C00 . We still assume that the chalocally of the form dµ=Kdx, 0<x(=-
0.6. Exponentiating a fast order operator
35
racteristic flow F is defined: All solutions of (6.4) extend indefinitely. Clearly the restriction GO=GICO(st) may be interpreted as unbounded operator G0 of H with domain dom G0 = CO(n).
Assume G skew-selfadjoint: with real-valued bj , p we have + ip
G = Ebjaj + 1/2 x- I(xbJ)
(6.15)
lx
Ij then G0 with domain CO(n) clearly is skew-hermitian: We have 00 (u,GOu) + (G0u,u) = 0
(6.16)
If u(t,x) solves (6.2)
,
, u E dom G0 = CO(ft)
for some u0E CO(sz), (6.10) implies that C00(1),
u(t,.)E D(st), for each fixed t. The y(t)=Ilu(t,.)III E (6.17)
and
dy/dt = (u,ult) + (ult,u) = (u,GOu) + (GOu,u) = 0 , tE t
Accordingly,
(6.18)
IletGuOII
= IIu(t,.)II = I1uOII = const., for all u0E Co(st).
This shows that the operator etG: CO(ft)-+CO(ft) defines an isometry
in the norm of H . Clearly this is an invertible isometry, using the group property of etG. Let us first assume that p of (6.15)
vanishes. Then v(x)=etGu0, for some fixed t, is of the form (6.19)
v(x) = (u0ox)(x)w(x)
,
x(x) = v-t(x)
,
with a real-valued positive Co°-function w(x) independent of u0, by (6.8), with p=(2x)-I(xbj)
.
1
x
From (6.18) and (6.19) we conclude
.
rf
(6.20)
11v112
=
I(u0w)(x(x))12dµ = .IIu0(x)12ds = IIu0II2,u0E Co.
With new integration variables y=x(x) in the first integral we get (6.21)
f11Iu0(x)I2(w2(x)(dµovt)/dµ
- 1) = 0
,
for all u0E Co(st).
But (6.21) implies that (6.22)
w(x(x)) = {(d1iov_t)/dµ}I/2
For p general, etGuO(x) will carry an additional exponential factor. All other functions remain the same. We have proven Proposition 6.3. For a skew-selfadjoint G (6.15) assumes the form Jt
(6.23)
etGuO() = ((u0,v_t){a1Lov_t/a1'}I/2)(x)exp{i
pov_z(x)dT}, 0
defining an invertible isometry COO-COO the sense of the norm of H
0. Introductory discussions
36
00
Then eGt:co->C 0 admits a continuous extension to H, a unitary
operator. We get a group {U(t)=(eGot)closure : tE &} of unitary operators, strongly continuous in t. Its infinitesimal generator (6.24) G1=dU/dt(0)
, dom GI={uE H: lim£-,o
U(£)E-u =G1u exists in H}
is a skew-selfadjoint realization of the folpde G , extending G0. eGot=CO
is dense in H. G1 is This realization is unique, since dom the closure of G0, and iG0 is essentially self-adjoint. Finally consider a dissipative expression G (i.e. GO+G*sO). G must have the form (6.15) with real-valued bj but complex p satisfying In p(x)a0, xE S. Hence (6.16) now assumes the form
(u,GOu)+(GOu,u) _ -Jdµ(Im p)lul2 s 0 , u E C,(a)
(6.25)
Under this assumption the map etGO
,
still of the form (6.23), is
a contraction under the norm of H, for t2t0, i.e., IleGtllsl, t2!0.
This follows by repeating the above argument. 7. Solving a nonlinear first order partial differential equation. In this section we are going to consider the Cauchy problem for a single first order partial differential equation (7.1)
F(x,u,p) = 0 , p = ulx = (ax u, ...
ax u) n
1
,
where F is a given real-valued Co'-function of the 2n+1 real varia-
bles x=(xi,...,xn) , u , p=(pi.... ,pn). For simplicity assume F
defined in
S2xlex1n, with some domain RC ten. Our main application
will be the characteristic equation (8.3), in sec.8, below. There F=aN(x,p) is independent of u and a homogeneous polynomial in p. The Cauchy problem seeks a solution of (7.1) satisfying an initial condition at some n-i-dimensional submanifold rC tt. Let I' be given by x=x(s), s=(si,...,sn-i)E EC &n-i, with ax/6s=((axj/asl)) of rank n-1. Require as initial condition that (7.1')
u(x(s)) = w(s) is given for s E E
.
Then differentiation of (7.1') for s) determines all tangential derivatives of u at r, while (7.1) gives an implicit relation for normal derivative of u. It is natural to assume that the n equations resulting for pj=ulxj locally admit precisely one solution, so that ulx is fully determined along the hypersurface E.
37
0.7. Solving a first order PDE
Thus it is convenient to assume not only u=w(s) but also p= given for sE E. Of course certain conditions on w(s),p(s) must ulx express the fact that p=ulx. In other words, we assume given x = x(s)
(7.2)
,
p = p(s)
, u = w(s)
,
E E
s
,
with ax/as of maximal rank n-1, while x(s), w(s), p(s) satisfy lp -aslx = aslw
(7.3)
,
s E E
,
1 = 1,...,n-1
(called the strip condition) and F(x(s),w(s),p(s)) = 0
(7.4)
,
s E I
.
A (2n+1)-tuple of functions (7.2) satisfying (7.3) is called an ((n-1)-dimensional) strip: The plane in (x,u)-space through the point (x(s),u(s)) with normal vector p(s) is tangential to the n-l-surface (x(s),u(s)) by condition (7.3). If in addition (7.4) holds, then we talk about an integral strip of (7.1). Instead of asking for a function u(x) satisfying (7.1) and (7.1') it is more convenient to ask for a solution of (7.1) extending a given n-i-dimensional integral strip (7.2). That is, (7.5)
u(x(s)) = w(s)
,
uIx(x(s)) = p(s)
,
s E I
.
Thus the Cauchy problem for (7.1) seeks to find u(x) satisfying (7.1) and (7.5) for a given 'initial integral strip' (7.2). We shall see that this reformulation already holds the key for a solution of the problem, while in fact only w(s) may be freely chosen in many cases, perhaps up to a finite choice of p(s). In sec.6 we already solved this problem for a linear equation (6.2). In the present more general setting the system of characteristic equations (6.3) or(6.4) must be replaced by the following: (7.6)
x =Flp(x,u,p), u =PFIP(x,u,P), p =-Flx(x,u,p)-pFlu(x,u,p),
with 'aF'_'''
This is a system of 2n+1 first order ODE's in 2n+1 unknowns x(t), u(t), p(t). The coefficients of (7.6) are Cw in x, u, p, independent of t, hence there exists a unique local solution .
defined for ltI<S for sufficiently small S, satisfying (7.7)
x(0) = x0
,
u(0) = u0
,
p(0) = p0
for arbitrary x0E St, u0E &, p 0 E 2n. The solutions may be extended as long as they stay inside szxIx&n. Their orbits fill SZxXxin as a
nowhere intersecting family of curves. Generally we assume that
0. Introductory discussions
38 (7.8)
Flp(x,u,P) # 0
(x,u,p) E 1 x & x In
,
.
Then the orbits are nondegenerate C° curves. Moreover, x=x(t) defines a nondegenerate Cc'--curve in x-space ien.
We claim that (x(t),u(t),p(t)) also may be interpreted as a 1-dimensional strip in (x,u)-space, along which F is constant. u=u(t) defines a Co curve E in (x,u)-space; at That is, x=x(t) (x(t),u(t))E 'E, the plane through that point with normal p(t), i.e7 ,
(7.9)
u - u(t) = Lj=1p(t)(x-x(t))
= P(t)-(x-x(t))
defines a plane tangent to EE at (x(t),u(t)). Indeed the vector
clearly lies in the plane (7.9). We get
(x (t),u
d/dt(F(x(t),u(t),p(t)) = Flxx + Fluu
+ Flpp
(7.10)
FIxFIP + FIu(pFIP) - FIP(Flx + PFIu) = 0 confirming that F is constant along the strips discussed. A characteristic integral strip is defined to be a 1-dimensio-
nal strip as described above,with the additional property that (7.11)
F(x(t),u(t),p(t)) = 0
along
8 .
For the construction of a solution of the Cauchy problem (7.1) and (7.5) we start from a given n-l-dimensional integral strip (7.2). For each (x(s),w(s),p(s)), sE E, we obtain the unique characteristic integral strip through (x,w,p). That is we construct (7.12)
x(s,t) = x(s1,...,sn-1,t)
w(s,t)
,
,
p(s,t)
s E E
,
solving (7.6) and the initial conditions (7.13)
x(s,0) = x(s)
,
w(s,0) = w(s)
,
p(s,0) = p(s)
The functions (7.12) exist only for (s,t) E no (7.14)
no = {(s,t)
:
s E E
,
Itl
with a suitable function ti(s) > 0
,
,
,
s E E
with
< t1(s)}
defined over E
.
Suppose the function x(s,t) can be inverted near t=O.Since we assume ax/as of maximal rank this means that ax/at=x at r, is ,
linearly independent of the ax/asj. In other words, the projection of the characteristic strip through (x(s),w(s),p(s)) onto x-space must never be tangent to r. Under this condition we call r an
0.7. Solving a fast order PDE
39
admissible initial integral strip. The implicit function theorem then implies that x(s,t) is invertible whenever the initial strip is admissible, and tl(s) is choosen sufficiently small. Let X(x)= (s(x),t(x)) denote the inverse. We claim that a solution of the Cauchy problem (7.1), (7.5) is defined by setting u
(7.15)
=
w,X
,
i.e., u(x) = w(s(x),t(x))
.
so that u(x)=u(x(s))=w(s) From (7.15) we get u(x(s,t)) = w(s,t) Differentiating this for sj and t , using (7.6) and the strip conditions (7.3) , we get Indeed for x e l' we have t = 0
,
.
.
(7.16)
p151
ulxjaslxj = asl = ulxjatxj
pjatxj
= atw =
(s,t) E I
,
i Since the Jacobian ax/a(s,t) is invertible, (7.16) implies that (7.17)
p(s,t) = ulx(x(s,t))
(s,t) E E
.
as xE r hence ulx(x(s)) =p(s). Hence we have (7.5) Also we know that F(x(s,t),w(s,t),p(s,t))= 0. Thus (7.15) and (7.17) imply that the PDE (7.1) is satisfied by u of (7.15) Thus indeed we solved the Cauchy problem. Next let us assume that u(x) satisfies (7.1) and (7.5) for an admissible strip (7.2) . We then may consider the system In particular we again get t=0
,
,
(7.18)
x
= Flp(x,u(x),ulx(x))
,
n'n = d/dt
of n first order ODE's in n unknowns x(t). Let y(s,t) solve (7.18) and the initial cdn's y(s,O)=x(s). Let K(s,t)= u(y(s,t)), q(s,t)= ulx(y(s,t)). We will show that y,K,q solves (7.6) with the same w(s,t) initial conditions as our constructed x(s,t) p(s,t) so that we must have x=y , K=w , p=q , for all sE I ltl
,
,
our previously constructed solution (7.15) of (7.1) and (7.5)
,
so
that we have uniqueness of the solution of this Cauchy problem. Indeed, from (7.18) get y =Flp(y,x,q), assuming u E C2. Also, F(x,u(x),ulx(x))=0 since u solves (7.1). Differentiating we get ,
(7.19)
Flx(x,u,ulx) + Flu(x,u,ulx) + Flp(x,u,ulx)ulxx = 0
.
In (7.18) let x = y(s,t). Then q= ulxx(y)Flp(y,K,q) =-Flx(y,K,q) - Flu(y,x,6)q so that indeed the system (7.6) for y,K,q follows We have proven the result below.
40
0. Introductory discussions
Theorem 7.1. Let r = {x(s) : s E E} be an n-l-dimensional C"-surface in In with local parametric representation, rank as/ax = n-1.
Then for every admissible integral strip (x(s),w(s),p(s)) over r the Cauchy problem (7.1),(7.5) admits a solution uE C°°(SZo), with
some neighbourhood r& of r. The solution is unique within C' (SL ). Remark 7.2. Suppose an n-l-dimensional surface r={xE c:ip(x)=V(x0)}
and F(x,p(x),Vlx(x))=O on r. Then, if x= is given, where VIx#O x(s) represents (some part of) r, we find that w(s)=p(x(s)), p(s)= ,
Vlx(x(s)) defines an n-l-dimensional integral strip of F. Suppose this strip is admissible. Then thm.7.1 yields a solution cp(x), de(x)=yrlx(x) on r. It folfined near r, such that p(x)=p(x), and l
at least locally. lows that r also is given in the form T = c This construction will be useful in sec.8, where we ask whe,
ther a characteristic surface always may be given as a surface of constancy of a solution cp of the characteristic equation (8.3).
However, a closer inspection shows that, for an equation (8.3) the above integral strip is never admissible: We get y=const on r , hence the surface normal is a multiple of iplx(x(s))=p(s) for all s
On the other hand, the projection of the characteristic integral strip at x(s) has direction x'=Flp(x(s),p(s)). Also, since F(x,p) =aN(x,p) of (8.3) is homogeneous of degree N in p , we get p.x' _ p.Flp(x(s),p(s))= NF(x(s),p(s))= 0 along a characteristic integral strip. Accordingly, x' must be in the tangent space of x(s), since it is perpendicular to the surface normal. 8.Characteristics and bicharacteristics of a linear PDE. The conventional approach to the concept of a characteristic (hyper-)surface for a differential expression (8.1)
a(x)Da a
L = a(x,D) = I
aTsK
is the Cauchy-Kowalewska theorem, discussing existence of a unique analytic solution for a PDE with analytic coefficients satisfying analytic initial data. The result directs the attention to certain surfaces along which data cannot be prescribed freely. The concepts also make sense for real-valued solutions of a PDE with real coefficients. However, one has to work with two different sets of assumptions: Either assume real Co coefficients or complex analytic coefficients. The latter case is mostly analogous to the real Co case, and will not be discussed in detail.
41
0.8. Characteristics
Accordingly we assume that L of (8.1) has complex C" coefficients, but that the coefficients of the principal part polynomial (8.2)
a
la =N
0
aa
are real-valued C°°(S), for a domain nC &n, up to a common factor O#yE C00(n) (here assumed 1). The characteristic equation of L is
aN(x,Cplx) = 0
(8.3)
,
with the gradient Tlx of T. Clearly (8.3) is a single first order PDE for the unknown (real-valued) function T(x)EE C00(tt).
In sec.7 we discussed solving such PDE. Since F(x,p,u)=aN(x,p) is independent of u, the characteristic system reduces to x
(8.4)
=
a system of 2n first order ODE for 2n unknowns x(s), F;(s), where we wrote (s) instead of p(s). The equation for u' of (7.6) is p'= NaN(x,(plx)=0, due to (8.3), using Eulers relation for the homogeneous function of degree N in plx). (7.6) splits into this equation and (8.4) to be solved separately. The integral strips are given as solutions of (8.4) together with g(s)=const. A system of the form (8.4) is commonly called a Hamiltonian system. The graph of a solution T of (8.3) is fibered strips x(s) (s) solving (8.4), cp=const.: Through any point (x°,g(x°)) on the graph there is a unique characteristic integral strip
with x(O)=x°
(x(s),c)(x° where
,
(0)=9Plx(x°
solves (8.4). In sec.7 we have seen how solut-
ions of (8.3)
may be composed of such strips. Moreover, since g is constant along such strips, it is clear that they stay on the same level surface of cp Thus also the level surfaces of solut.
ions are fibered by characteristic integral strips of (8.3). A characteristic (hyper-)surface r of the differential expression (8.1) is defined as a surface of constancy of a solution T of the characteristic equation (8.3), with PIx'O. To be precise we require that for every x°E I' there exists a solution T = q) x
(8.3), defined in B={Ix-x°l0 .
Actually one may define somewhat more generally by dropping the assumption that c(x) solves (8.3) near the surface t , and requiring (8.3) on the surface t only. It is clear from the above that a characteristic surface r is fibered by characteristic integral strips: There is precisely
of
42
0. Introductory discussions
one strip through each point x E r . These strips are called bicharacteristic strips of the expression L We speak of a simply characteristic surface r of the exprescplx(x)#0 sion L of (8.1) if r is given by cp(x)=0 with g(x)=0 .
,
,
aN(x,cplx(x))=0 on r , as above, and in addition aN1,(x,T,x(x))#0 on r . For a simple characteristic (surface) it is no loss of ecn-
erality to assume that p(x) solves (8.3) near r . Indeed, we may assume that (x°,c)lx(x°))#0, and then solaNI
ve aN(x,T;)=0 for ' . Get a function p, (x,n") with pi (x°
cplx(x° )" )=
gplx.(x°), where p =(p2,...,pn). The plane perpendicular to the integral strip through x°
is noncharacteristic near x°. In that pla-
ne 10 set w(x)=cp(x), p" (x)=cplx(x)- , with our given T(x) solving (8.3) on r near x° . Also set pi =p, (x,p" ). We get an admissible integral strip, and a solution w(x) near x°. We have w(x)=cp(x°) on the strip through x°. Also, w(x)=p(x)=cp(x°) on rn E0. Hence also
w(x)=q(x°)=c on the intersection with r of a small ball around x°. Thus w(x) indeed solves (3.3) and is constant on r near x°. A characteristic surface r along which Fly= aNI (x,Tlx)=0 will be called a multiple characteristic. An example of an expression with multiple characteristics is the heat operator (example b), below). For hyperbolic equations one tends to avoid multiple characteristics, introducing the concept of strictly hyperbolic expressions (cf. VII,2). An expression L having no multiple real characteritics is said to be of principal type (VII,2). As mentioned initially, characteristic surfaces are important for the initial-value problem of (8.1): For a given (hyper-) surface rC &n and given 'data' V. defined on r and f defined near r we ask for existence of u (defined near r such that (8.5)
Lu = f near r, u(a)= aau = 1pa, x(= r ,
for all lal s N-1
The bicharacteristic strips, on the other hand, will be recognized as carriers of singularities of solutions of Lu = f, (cf. ch.6, ch.9). The key result is our version of Egorov's theorem (V1,thm.5.1), accessible only after a study of i,do's. Solving (8.5), it is clear that not all functions 1ya may be prescribed arbitrarily. Assuming r and 1pa smooth, let v(x) be a
smooth transversal vector, defined (locally) near r, and never tangential to r, let us start prescribing arbitrarily the functions u=1p0 , Dvu = 1U1 , , DNu = 1IrN-1 , on r , where
...
(8.6)
43
0.8. Characteristics This will determine all u(a)
lals N-1, on r, as tangential deri-
:
vatives of the Vj: For some local parametric representation x=x(k) 'xn-1) of r the derivatives u(a)(x), xE T, become linear expressions in the derivatives a.yPj. Accordingly, Lu=f, along IT,
translates into a linear equation of the form pDNu
(8.7)
Now if q E C1 with (8.8)
+ Fpaaa j = f yd 0 is constant on F we find that
Tlx
= (TIxj/DvT)Dv + Tj
,
j = 1
,
...
n
,
ax7
with vector fields Ti satisfying T,T = 0 Substituting (8.8) into (8.1) one finds that p in (8.7) is given by .
(8.9)
(Dvq))-N
p=
I
a
-N aa(x)cplxa(x)
If p(x)p60 on t then (8.7) may be solved for DNu in some neigh-
bourhood of T. Moreover, we may apply Dv to the resulting equation for an infinite number of new relations of the form Dmu= linear exv pression in derivatives of lower order, allowing recursive calculation of all Dvu on F. Clearly this allows construction of a unique solution of (8.5) by its Taylor series, assuming that an analytic solution exists. The recursion may be used for estimates proving convergence of the Taylor series and existence of an analytic solution, if the coefficients are analytic. This sketches a proof of the Cauchy-Kowalewska theorem mentioned initially. Moreover the importance of the concept of characteristic surface is clear from the result below, the proof of which is evident, after the above.
Proposition 8.1. The function p(x) of (8.9) vanishes identically on t if and only if r is a characteristic surface. We get p#0 on t if and only if I' is nontangential to any characteristic surface.
To provide more detailed information on characteristic surfaces and bicharacteristics we return to some examples, mainly of sec.4. a) L = t
(8.10)
(of (4.1))
:
The characteristic equation
cpIx2
= Lj=1 'pIx2 = 0
has only constant real-valued solutions, since it requires TIxmo.
Real characteristic surfaces do not exist. However, A has constant (i.e., analytic) coefficients. The Hamiltonian system (8.4) may be regarded as a set of ODE for complex-valued analytic functions. There exist solutions of (8.10) in the n complex variables x1,..., Setting such q constant will give xn, defined in a domain of Tn .
0. Introductory discussions
44
surfaces in Cn, called complex characteristics. Notice that the Helmholtz operator (4.2) has the same principal part, hence the same real or complex characteristics as the operator A . b) L=at-A, in the n+1 variables x0=t, x=(xl,...,xn), (i.e., (4.3)) The characteristic equation is (8.10) again. But we now have one more variable t=x0. Thus (8.10) requires that cp(t,x) is independent of x, but it may depend on t. It follows that the characteri-
stic surfaces all are hyperplanes of the form t=const. Complex characteristics of more general kind may be constructed, of course In sec.4 we solved a modified Cauchy problem, of the form
(8.11)
Lu = f
,
t a t0
,
u = tV at t = t0
for f, WE Co. Only one condition at t=to
,
(instead of 2) is needed
to make the solution of this characteristic Cauchy problem unique. The real characteristics of L all are multiple characteristics: For c(t,x)=t-t0 we have cpjx (1,0), hence a2(glx)=a2101110=0c) L = at - A , (i.e., ex'le (4.4)) , again discussed in n+1 independent variables (t,x) The characteristic equation is .
(8.12)
WIt2 =
=
W1x2
=1 T1.
2
Being only interested in the surfaces T=const., for (WIt,WIx)#0 we may reduce the number of variables: (8.12) implies WIX 0 unless fIt#0.
Hence T=0 may locally be solved for t. Writing t=J(x), get
(8.13)
j=1 JIx,
2
2
= JIx
= 1
This equation is often referred to as the equation of geometrical optics, or the eiconal equation. The bicharacteristic strips (so-
lutions of (8.4), in this case) are straight lines x=at+b with constant a,bE Rn Ia1=1, and constant . The base curves of the bicharacteristic strips are the light rays of geometrical optics. The Cauchy problem (8.5) is well posed for the wave operator along non-characteristic surfaces (cf.sec.4 and ch.7.). Again the Klein-Gordon operator (4.5) has the same principal part and characteristics as the wave operator, and a similar Cauchy problem. d) L is an elliptic operator, with Coo-coefficients. That is, we have O. Again the characteristic equaas xE cz, E n, tion (8.3) implies cplx=0 assuming T real-valued. Real characteristics do not exist. Complex characteristics again may be defined only if the as are analytic in xl,...oxn . ,
e) L = P(D) is a constant coefficient hyperbolic operator (cf. VII,!), with respect to some given real vector h# 0. As a conse-
0.9. Lie groups and Lie algebras
45
quence of VII, prop.1.5. the principal part PN(U) also is a hyperbolic polynomial: We have PN(h)#0, and the algebraic equation for fixed real , has N not necessarily distinct real roots X., j=1,...,n . The characteristic equation is of the form PN((plx)=0. For simplicity let h=(0,...,0,1), as always may be achieved by a linear transformation of independent variables. Then the characteristic equation decomposes into the N equations (8.15)
with
TIxn = tj(roIx1,...,(' Ixn-1)
t
0
the real solutions of
=
for
,
_
In the presence of multiple roots we cannot expect that the
tj depend smoothly on 1If smoothness can be arranged then each of the N equations (8.15) will give its own family of characteristic surfaces and bicharacteristics. Thus one may expect N different types of characteristics for an N-th order expression. The N different types of real characteristics do exist if the N roots
are all distinct, for=(1
0)#0. Then
the t. will be smooth. In this case P(D) is called strictly hyper-
bolic (cf. VII,2). We shall see in ch.7 that the Cauchy problem (8.5) is always well posed, even for variable coefficients, if L is strictly hyperbolic. For constant coefficients the Cauchy problem of a general hyperbolic L is well posed (cf. [Gai], [Hri]). 9. Lie groups and Lie algebras, for classical analysts. Let us discuss the relationship between a Lie group G and A, its Lie algebra, using a form suitable for analysis minded readers.
Generally, G may be represented on some GL(1N) (we will not show this, although the tools are developed). For simplicity let G be a Lie subgroup of GL(1N), as will be true for all our applications. A path ip(t), Itl
the path gV(t) starts at g and (gi(t))'=0'(t) with matrix product. Let y:U- G, or explicitly, y=y(u), y(0)=e, be a charted neighbourhood of e, with an open set UC 1v, v=dim G, OE U. For a gE G the function yg=gy(u), uE U, defines a charted neighbourhood of g. The 'partial derivative' ylu is defined as directional deri7
vative along the coordinate line uk=const, k#j, uj=t (i.e., of) Similarly for ygluj
.
0. Introductory discussions
46
In these coordinates the tangent space at e=y(O) (i.e., the corresponding Lie algebra A) consists of all a=(a3) E &v, representing the matrix Aa = 2aJylu j(0). We get 1-1-correspondences Av= A H {Aa:aE Av} H {a(g):aE &v} with the vector field a(g) def i-
ned by using the same aE my in the local coordinates yg at g :
a(g)=jajyglu (0)
.
- i.e.,
a(g) defines a global folpde La on G
J
a(yg(u))=La(yg(u)), with La, in coordinates y(u), near e, given by
La D Y.A30
( 9 . 1 )
1
(u,O))Y(u)a/aul, where 9(u.v)=Y(Y(u)Y(v)), j
j
similarly for general g with yg . Again A **IL a:aE Av}
,
aE A appears in 4 forms, as a v-vector (a3), a matrix Aa folpde La on G , or again as a(g)=2ajygluj (0)=gAg
so that ,
or a
.
For aE A define a smooth curve V(t) in G as solution of
3 = a(p(t))
(9.2)
,
t near 0
,
V(O) = e
Locally, (9.2) constitutes a system of v ODE's of order 1, with real C"-coefficients. Its Cauchy problem (9.2), even at to instead 0, is uniqely solvable. In fact, (9.2) amounts to y1'=Aa V(0)=e, with the above Aa, and matrix multiplication. This is trivially ,
solved by the exponential function y,(t)=exp(Aat)=F`oAajtj/j!
. By a
continuation argument get yi(t)=exp(Aat)E G , and (9.2) for tE I In particular, {yi(t)} defines a subgroup of G. We summarize:
Proposition 9.1. For each tangent vector aE A=Te(G) there is a iparameter subgroup P(t)=Pa(t), tE 1, defined as unique solution of the Cauchy problem (9.2), with a(g)wLa of (9.1). Corollary 9.2. The connected component GeCO of the unit element e is generated by all the above 1-parameter subgroups, and even by {Va(t): aE A , 0st5s}
,
for any fixed s>0
.
Indeed, any gE Ge may be connected to e by a path r , and F may be replaced by a 'polygon' with sides translates of 'lines' Ostsr's for fixed aE A. This shows that, indeed, g= Va(t),
,
Va 1 (E1)Va (E2) .... Va (Em) 2 M
,
for finite M and sjss.
47
0.9. Lie groups and Lie algebras Corollary 9.3.
(9.3)
We have A t
Va(t) = e a
=
A,,3/j!
,
Aa = Va.(0)_EalY luj(0)
A
The subgroup Ge is generated by all e at, aE A, OO Completing the structure of A, we define a bracket operation: Consider the commutators [Aa,Ab]= AaAb-AbAa and [LaLb]=LaLb LbLa of the matrices and (matrix-valued) first order expressions, resp. Get y(u)y(v)=y(9(u,v)), near u--v=0, with 0(u,v) of (9.1). Differentiating this conclude that [y lu (0),Ylu (O)]= 2jcjlYlu (0) 1
r
r
, with
xjl = 9rlu v (0,0)-9rlu (0,0) . Accordingly, v 7 1 1 j
(9.4)
is 1ajbl)
[AaAb]=Ac , c = (cr)=(
again belongs to the tangent space of G C GL(IN) at g=e. Next, [LaLb] is a first order expression: LetipdI()=11aO v (u,0)Y(u) jl k
The matrices pj(u), ql(u) commute for uE U, hence (9.1) yields
(9.5)
[LaLb]= j71(Pjglluj-gjplluj)au1
At u=O get pj=aJe qj=b3e , 0(O,v)=v , hence 0 1Iv (0,0)=Olk Thus k {q+}lu.(0)={ij (9.6)
}Ylu.(0)+E{bkPllvku,(0,0).
We get
j(pjgllu-gjplluj) _ 7,(ajbl-bjal)Ylu=b1Aa-alAb
with our above Aa, using the symmetry dllu
v k=01lvjuk, at u=v=O.
Accordingly, [La,Lb]Y(0)=I(b1Aa-a1Ab)Ylu1-AaAb AbAa [Aa,Ab]=LcY(0) The argument may be repeated at general gE G , using Yg(u) instead Y(u). One finds that [La,Lb]Yg(0)=Lcyg(O) for all gE G , with c of (9.5) . Accordingly Ac **L , under our 1-1-correspondence. ,
c
This induces a bracket operation in the Lie algebra A, given by
(9.7)
[a,b] H[Aa,Ab]= Ac H[LaLb] = Lc
,
c of (9.5)
.
[.,.] has the usual properties, skew-symmetry, and Jacobi identity. Next we assume given a Lie matrix algebra - i.e., a linear subspace A of some L(IN ) containing all of its (matrix-) commutaProp.9.1 and cor.9.2 tors [A,B]=AB-BA, A,BE A. Assume dim A = v .
0. Introductory discussions
48
then suggest a direction of approach for generation of a corresG should contain ponding Lie group G - a Lie subgroup of GL(in) all matrices eAt, tE &, AE A, and all their finite products. It should be closed under matrix multiplication and inversion. Here :
we are only interested in connected Lie groups (i.e. might get only the component of 1 of a larger group). In fact we will look at the minimal such set - all finite products of eAt, AE A, tE k Observe that Vt = eAteBt , for A, B E A , no longer defines a 1-parameter group, unless A and B commute. For general A, B we will prove the Baker-Campbell-Hausdorff formula: For small Iti ,
Vt = eC(t)
(9.8)
,
where C(t) E A
is a convergent power series in t (cf. lemma 9.5, below). As a first step in this direction: Proposition 9.4. Vt satisfies the differential equation (""'="dtit) (9.9)
Vt'
- (A+eAtBe-At )Vt
,
where the coefficient A+eAtBe-At E A is a convergent power series.
Proof. (9.9) follows trivially by differentiation. Also, the family Bt = eAtBe-At satisfies the differential equation
Bt' = adA Bt
(9.10)
B0 = B
,
,
with the linear operator
(9.11)
adAB = [A,B]
,
adA E L(L(&N) )
Therefore,
(9.12) Bt
eadAtB = L4=Q. (adA) JB = B+[ A, B] t+[ A, [ A, B] ] 22 +...
showing that Bt E A
,
hence A+Bt E A
,
,
q.e.d.
We will prove (9.8) in the following form. Lemma 9.5.
For A , B E A , define C=C(t) = log(eAteBt) by setting
C(t) = X(t)-X(t)2/2+X(t)3/3 t...+(-1)kX(t)k/k t.... (9.14)
with
X(t) = eAteBt-1 = (I+At+A22,+...)(1+Bt+B22l+...) -1
.
Then we have a convergent power series expansion (9.15)
C(t) = log(eAteBt) _
1C.t3
,
Itl< (log 2)/(IIAII+IIBII)
where C(t) and all coefficients Ci belong to the Lie algebra A
49
0.9. Lie groups and Lie algebras In particular,
(9.16)
C1= A+B
C2= 2 A, BI
,
,
C3= 2[A,[A,B]]+
2[B,[B,A]]
The coefficient Cj is a finite linear combination of terms which are applications of (together j-1) operators adA or adB to A or B. Proof. We use a proof of D.Djokovic [Dj] (we owe to G. Hochschild) Let C(t)=log(eAteBt) , defined as composition of
the logarithmic power series and X(t)=' 1XjtJ
matrix coefficients cj1 of C(X) = log X = >
Note that the
.
1(-1)iXJ/j are con-
vergent power series in the N2 complex variables xjk , where X=((xjk))j,k=1,...,N whenever IIX(t)II<1. The latter holds for I
eIIAIItellBIIt_1=e(IIAII+IIBII )t_1<1
i.e.,
,
(IIAII+IIBII )t < log 2
.
Thus cjl are complex differentiable functions of the N2 complex variables xjk , which in turn are power series of t . This implies complex differentiability of the function C(t) for t as in (9.15). So C(t) is a convergent power series and (9.15) follows. We are only left with showing that the coefficients Cj belong to A . The first coefficient (cf. (9.16)) is easily verified by direct computation. For the general case we will show that C(t) solves an ODE ((9.18), below) which leads to a recursion for the Cj. ,
(Cm),Ck
We calculate that (eC)'e C = = 1=0 N(-!k,)l
(
1)n-m
1 (Cm),Cn-m (n-m)'m.
1
X00
x-n
=ten=1 n!=1(-1)
1
m
Ik
Here, (Cm)'= I_Oh C'Cm-k-1 (_1)n m-1(m+1)CkC,Cn-k-1
= min=01-k=0 CkC'Cn-k-1Zk
=k(-1)n-m-l(m+n
1) _
7t1- =0CkC'Cn-k-1
n-k-1= (_1)n-1-k(n-1) =0
(n)(Cm),Cn-m
I -II
1 namn
n-m(m)(C n m)'C n-m
the inner sum equals Tn
=
7
k
, where Zk =
, by induction. We get T n =
4_01
CkC'Cn-k-l(-1)n-1-k(nk1 )
(9.17)
' 1n!(ad C)n-IC' = (eC)'e-C = Vt'Vt 1 = A + eCBe-C
_ (ad C)n-IC' whence
Thus
50
0. Introductory discussions
ad using (9.9) and eC=eAteBt=Vt. With (9.12) we get eCBe-C= e C B The left hand side of (9.17) may be written as f(adC)C' with the power series f(ey)=
1n-1/n! =(ex-1)/x . Notice that
is a power series too. Applying it to (9.17) yields the desired differential equation ad (9.18) CO = g(adC)(A + e C B)
Expanding both sides of (9.18) we get Cl-
' 1(n+l)Cn+ltn
while the right hand side may be written in the form g(L1 tkadc )A + h(
k
tkadC )B
h(x)=g(x)ex
,
k
The coefficient of the k-th power of t , at right, is a finite linear combination of a finite number of applications of adC to j
A or B , where js k . In fact, the sum of all j in each such term equals k . We know that C1E A . If we assume by induction that C1,...,CkE A, it follows that (k+l)Ck+l E A , hence Ck+IE A
Also
Ck+l must be a linear combination of k applications of adA and adB to A or B , as stated. The proof is complete. Corollary 9.6.
and let A(u)=
Let All... ,A, be a basis of A (as a linear space), "J=1Ajuj
, ujE 1 . There exist v power series
fk(u,v) k=1,...,v in 2v variables u,v convergent for IuMvI« 0 with some e0>0 such that fk(u,v) = uk+vk + ... , k=1,...,v , and eA(u)eA(v) = eA(f(u'v))
(9.19)
juI,IvI<s0 ,
.
eA(u)eA(v) _ Proof. As in the proof of lemma 9.5 we conclude that eC(u,v) where C(u,v) is a convergent power series in u and v as IIA(u+v)ll
0.9. Lie groups and Lie algebras
51
its unit element I equals the Lie-algebra A . And, vice versa, if we depart from a general Lie-subgroup G of GL(&N), then define A as the linear space of all directional derivatives of curves in G starting at I and then, with this A, defining the above group G(A) , we get G = G(A) ,
.
First let us establish a coordinate chart for a neighbourhood of the identity I E =-G . With the notations of cor.9.6 define (9.20)
Y (u) =
eu' A, +u?. A2 ...+u, A,=
A(u)
ul<s
0
where EO>O is kept fixed. Then y(u) defines an invertible map of the ball BED {lul<E} onto a set UIC G containing I, and y(0) = I For general GE G similarly YG(u) = y(u)G defines a map of B. onto a subset UG C G with G=YG(0) E UG . Introduce a topology on G by using all the "balls" yG(BE) 0<ESE0, GE G as a basis. Then each ,
set UG is an open neighbourhood of G . Let UG fl UG, eA(u0)=eA(v0)Z
9.6 we get Z =
,
Z=GG'-1E G
,
eA(f(u-v)) 0 0
for some u0
,
v0 E BE
Then . Using cor.
, assuming E0 properly chosen.
Notice that f(u,v) = 0 can be solved for v near u=v=0 giving a function v=v(u) , v(0)=0 since the Jacobian flv(0,0)=I is invertible. It follows that also flu(u0,-v0) is invertible, so ,
,
f(u,-v)=f(u0,-v0) may be solved for v giving an analytic map vv(U) v(u0)=u0 , provided that u0, v0 are sufficiently small,i.e., that e0 is chosen sufficiently small. Conclusion: ,
Z = eA(f(u0,-v°) =eA(f(u,-v(u)) =eA(u)e-A(v(u)) for u close to uO.
Or, YG() = YG,(v(u)) for small lu-u0l showing that the map
YG'-1°YG : YG 1(UG(l;JG')
->
YG'-1 (UG"JG, )
is analytic. Thus indeed,
this imposes a manifold structure (and a topology) onto G making
it a Lie Group. Every finite product Thai , BiE A may be connected to I, by the path 17etBi, Osts1, hence G is connected. Evidently, the tangent space of G at I (i.e.,the space of all directional derivatives of curves in G starting at I) coincides with A . Thus, indeed, we established a 1-1-correspondence between the connected Lie subgroups of GL(2N) and the Lie subalgebras of L(&N),
Chapter 1. CALCULUS OF PSEUDO-DIFFERENTIAL OPERATORS
0. Introduction.
In this chapter we deal with the details of pseudo-differential operator calculus. We follow a presentation in a lecture of 1974/75 [CP], inspired by the local approach of Hoermander [Hr:],
dealing with operators on Rn, for didactical reasons. Replacing asymptotic expansions of [CP] by Leibniz formulas with integral reminder of 1.5 is an improvement we learned from R.Beals [Bi] who uses 'weight functions' more general than our (x)()ml of (3.2). Still asymptotic expansions are needed, and will be studied in 1.6. We will discuss 4 different representations of ido', referenced as a(x,D)=a(M1,D), a(Mr,D), a(M1,Mr,D), and a(Mw,D), the first two corresponding to the left and right multiplying of Kohn and Nirenberg [KN], and the others to a representation of Friedrichs [Fr3], and the Weyl representation. The reader who dislikes the infinitely repeated formal discussions has our sympathy. For other approaches to the same subject cf. ch.7 where the 4do's of certain symbol classes are identified as operators on H=Lz(&n), smooth under action of certain Lie
subgroups of U(H). Or else, cf. [ C1] and [C2], where regular and singular elliptic boundary problems are approached with tools of C*-algebras, avoiding entirely the ado-calculus. The calculus presented generalizes formal calculus of differential operators. We get a collection of Frechet algebras containing differential operators, with formulas for product and adjoint like Leibniz formulas, containing generalized inverses (so-called Green inverses) of their elliptic and hypo-elliptic operators, as seen in ch.II. The algebras are 'graded': Each of their operators has a differentiation (m,) and a multiplication (m:) order. We will get the same Fredholm theory in Sobolev spaces as *
provided abstractly, using C -algebras, in [C,],III,IV.
1. Definition oftpdo's. In this section (and occasionally later on) we will write a(M) for the multiplication operator u(x)-a(x)u(x). More generally, for
52
1.1. Definition
53
a pseudo-differential operator generated from a 'symbol'
E
C00 (1e3n),
with
ten
, we will write a(M1,Mr,D) to indicate
an order of operation: Multiplication corresponding to the x-(y-) variable is carried out to the left (to the right) of the differentiation D of the a-variable, hence "M " (hence "Mn') "(cf. cor.2.3). 1
C'(&3n) satisfying
Introduce the space ST of all ( 1 . 1 )
)x(k).((x,Y))X(1)IaI+I1Isk,
0((
with nondecreasing x(k),X(l)
,
(1.2)
j)
IYIs1
k,1=0,1,..., such that
= limj.(?(j) - j) = -0°
Here, and in all of the following, we use the abbreviation
(z) = (1 + IzI2)1/2 (particularly ((x,y))=(1 + x2 + (1.4)
,
z E 1em
y2)1/2) Another useful shorthand:
Ox = (2n)-n/2dx ,
0 _ (2n)-n/2d
measures over ten. The functions of ST are called symbols. For aE ST define a linear operator A=a(M1,Mr,D) called pseudodifferential operator (t,do) with symbol a by setting, with integrals over ien, (1.5)
(Au)(x) =
J4j
u(y)
y
The precise meaning of (1.5) is clarified in thm.1.1, below.
Theorem 1.1. The right hand side of (1.5) is well defined, for uE S, xE &n, in the following sense: The integral f ¢y exists as improper Riemann integral, for >en. It supplies a C(2")-function I. For fixed xE &n, I(x,.) E L'(&n), and f A defines a function v=AuE S=S(&n). Moreover, the map A:S-'S is continuous S->S.
Proof. Clearly
Jyiy
exists as descri-
u(y)
bed, since the exponential is bounded, while is of polynomial growth in y,for fixed x, so that a useful -estimate one uses the identity (1.6) ei (x-Y)
(x-Y)
=
m=0,1,2,..., with the Laplace operator Ay =
. A partial inte-
32 J
7
gration for fdy is applied over a finite ball. The boundary terms
1. Calculus of pseudodifferential operators
54
die out, as the ball tends to In, by (1.1) and uE S. This yields -2m
(1.7)
JAlY
In the following we shall say that a given expression is 'in the span' of another given set of expressions, if it is a finite linear combination with complex coefficients of these expressions. For example, the integrand of (1.7) is in the span of ei (x-Y)(Da a)(x,Y,t)u(R)(Y)
(1.8)
,
Ial + I1I s 2m.
Note that we get
(1.9)
Pap
using (1.1) and the decay of uE Sfr, where q may be arbitrary. Thus
(1.10)
I(x,T) = 0( (
)K(2m)-2m J((x,Y))"(0)(Y)-q
dy ).
.0) to A substitution y = (x) y- , dy = (x) ndy brings 'f 11 in (1 c=rr1J(Y)%(0)-q (1.11) (x) ,(0)+nffJ(Y)1(0)(1+(x)2lyl2)-q/2s c(x)X(O)+n, Therefore
(0)+n
(1.12)
and, with (1.2), it now is evident that (1.13)
J a
P1
exists as an improper Riemann integral
for all x.The estimate
x(2m) - 2m < -n-Ial
(1.14)
holds for large m. It insures that I(x,.)E L'
,
for all fixed X.
For a=o this proves the first assertion of the theorem. Let us clarify the dependence of (1.12) on the function u
Lemma 1.2. We have
(1.15)
II(x, )I s cllullk ( )K(2m)-2m(x)X(o)+n
0 with c depending on m and aE ST, but not on uE S, where the norms (1.16)
IIulik = suPlalsk,xE
R"' II(
x)ku(a),LOD
of S are used. This lemma is clear, since Iu(a)(x)IsIIuIIk(x)-k, xE In, Ialsk,
55
1.1. Definition
estimates u(a) by IxI-q, as the only way, u enters (1.12). Let us prepare a few other simple lemmata for thm.1.1. ST, for Lemma 1.3. For a E ST we get aaSY=DaDYDYaE ST and any polynomial P=P(x,y,l;). With x,a. of(1.2) for a, corresponding
functions for aaPy and P- a are is (j+ a + 3 I) , ?. (j+ Y I) , and is (j+L) , X(j+M), resp., with the -degree L and (x,y)-degree M of P. The verification of Lemma 1.3 is left to the reader. Lemma 1.4. The integral at right of (1.5) defines a Cc°-function
of x denoted by v . Moreover, the derivatives v(a) may be calculated under the integral signs of (1.5). Proof. Notice that is in the span of (1.17)
a + y =
By lemma 1.3 we have aDXa(x,y,l;)E ST. For an induction proof one only must show existence of the first derivatives and differentiability under the integral sign for 101=1. Let qE L'1L (B), supp cp
CC & . Using (1.12) and Fubini's theorem (twice) we get .
Jdxjq(xj)JdjdY (1.18) f
DO = -iaxj
where we set cp
X
[a,t]
,
the characteristic function of [a,t]. The
inner integral at right of (1.18) may be calculated explicitly. A differentiation of (1.18) for t will give the desired result.
Lemma 1.5. For aEST and every multi-index y we have
(1.19)
fdjdy
u(y)
r
_ (-1)YfdjdY u(y)
Proof. By the second identity (1.6), (1.19) is a matter of partial integration, carrying the differentiation from the exponential to
But the interchange of
fd
and fdy is generally im-
possible, so that we will require the argument, below. 1) It suffices again to assume jYl = 1, by lemma 1.3. 2) A conclusion as in lemma 1.4 may be used to verify that fdjdyaj{u(Y)ei(x-Y)a(x,Y,U)}=fda jfdyu(Y)eit(x-Y)a(x,Y,)
1. Calculus of pseudodifferential operators
56
3) The right hand side of (2) is zero, because boundary integrals vanish, by (1.12). Thus (2) yields (1.19) for IyI=1, q.e.d.
Proof of thm.l.l (continued). Let v = Au, u E S, as defined by (1.5). Lemma 1.4 implies (1.21)
xav(p)(x) = f41JoY
where the right hand side is in the span of 6+s=a, a+t=(3.
(1.22)
(x-Y) = in (1.22) we may integrate by Using Dxel parts, in the inner integral, to obtain expressions in the span of
fd (1.23)
6+s=a, a+T=p, X+µ+v=a Note We also applied lemma 1.5 and another partial integration that IF.-µIskEIs IaI, and ITI+IvIS ItI+IaI=lpi Hence lemma 1.2 and lemma 1.3 give an estimate of (1.22) by
fd
(1.24)
Accordingly xav(p)(x) also is bounded by (1.24). Applying this for all IaI s 1 for some integer 1, we get ,
(1.25)
Iv(p)(x)I s
)x(2m+IPI)-2m
cO(x)n+k(1)-lllulik
,
Here m , 1 are integers, but m must be chosen large enough to insure existence of the integral in (1.25), by (1.2). Also, co
and k depend on the choice of m,l and p, but not on u. Using (1.2) it follows that for 10 there exists k, c0 independent of u with
(1.26)
s 0OIIulik , u E S
IIAuII1 0
Clearly this amounts to continuity of a:S-'S and thm.1.1 is proven. 2. Elementary properties of y,do's.
Theorem 2.1. Let (2.1)
(u,v) = f n u(x)v(x)Ax , u,v E S
Then we have, for all a E ST , (2.2)
(a(M1,Mr,D)u,v) = (u,a(Mr,Ml,-D)v)
,
for all u,v E S
,
1.2. Elementary properties
57
with A"=a(Mr,Ml,-D), the 4do with symbol a Proof. Clearly a-(=- ST as aE ST. The statement is a matter of inte-
L1(&2n), vE S, by lemma 1.2.
gral interchanges. Get Hence (v,Au)
foJoyfpix ... =
f oyfRi f 41x ...
_
(u, A" v)
,
for A = a(M1,Mr,D) , with a 3-fold application of Fubini's theorem and a substitution (y,x,-l;) of variables, q.e.d. A" will be called the distribution ad joint (or dual) of the operator A, (to be distinguished from the 'formal Hilbert space ad joint' A*, formed with the inner product (u,v) = f uvdx ). Note: *
A = a(Mr,Ml,D)
(2.3)
Since A=a(M1,Mr,D) is strongly continuous S-'S, by thm.1.1, an extension B : S' - S' of A : S - S is given by
(2.4)
u E S'
(Bu,c)) = (u,A (p)
,
(P E S
,
with (u,cp), the value of uE S' at TE S. In other words, we may regard the ',do either as operator S-S or, S'-S'. Thm.2.1 gives AC B. Generally we will not distinguish in notation between the operators A and B but will regard a 'pdo as an operator A:S'-'S' with ,
AIS mapping to S. In fact, 'pdo's more or less will be regarded
like differential expressions in [C2],II with domain to be fixed.
Theorem 2.2. ST is an algebra of complex-valued functions under pointwise addition and multiplication, with (real) involution a-a; Proof. Evidently ST is closed under "+", """, and scalar product. Note that DaDyDY(ab) is in the span of (2.5), by Leibniz' formula: (2.5)
xy
(Da'DR'DY'a) (Da"DR11DYlob) x y i
,
a'+a"=a , l'+0"=(3
,
Y'+Y"-Y
.
In (1.1) we may assume the same x and X, for a and b. Using (1.2)
write x(j) = j - a(j)
,
?(j) = j - t(j)
and estimate the products of (2.5) by
,
t(j)
o(j)
(2.6)(x) IaI+ IRI-a(la' I+IR' I )-a( ICE" I+IR")( (x,y)) I? I-t( IY' I)-t( IY"I ) {t(1)+t(j-l)},
{a(1)+a(j-l)} , t'(j)=Min
Then define al(j)=Min lsj
le. j
We get the estimates (1.1) for c with x'=j-a', X'=j-t' instead of
1. Calculus of pseudodifferential operators
58
x,X. In particular we also get x'
00
->
%' - 00 , q.e.d.
,
Corollary 2.3. The symbol algebra ST contains all a of the form (2.7)
asN
ba(x)ca(y),a
where ba , ca E COO(&n) satisfy the estimates
A a(x) = 0((x) P) , Dc(x) = D((x)p)
(2.8)
with a constant p , independent of x (2.9)
A = a(Ml,Mr,D) =
,
a ,
R
.
x e &n
,
Then we get
ba(M)Daca(M)
a differential operator.
Proof.
(2.10)
The Fourier transform
u^ = Fu and its inverse u" = F lu
u" () = J91x
uES
u°(x) = JF1
satisfy (c.f. 0,(1.22)). (2.11)
FDjF-1 =
Mj
,
j = 1,...,n.
Theorem 2.4. Let a,b E EST be independent of y and x, respectively. Then the 1pdo's A = a(Ml,D) and B = b(MrD) may be written as ,
=fdye
(2.12) (Au)(x)=
for u E S , with the Fourier transform (2.10). Proof. The first formula is clear,,f using (2.10). Also, (2.13)
(Bu)(x) _
f4eix
JP1y a '
where the inner integral
is in L1(&n) , by lemma 1.2 (In this case I is independent of x). Hence we may write (Bu)(x)=I"(x). We know that Bu E S Therefore (Bu)^=(I`)^= I as proposed, q.e.d. .
,
Theorem 2.5. For a,b EST let
We have
c EST , and c(Mi,Mr,D) = a(Mi,D)b(Mr,D)
(2.14)
Proof. From Theorem 2.2 we conclude that c EST. Using (2.12), A(Bu)(x)=
ff
u(y)
u E S , q.e.d. = (c(Mi,Mr,D)u)(x) For a symbol a E ST consider the bilinear form ,
1.2. Elementary properties
59
(2.15) (v,Au)= (v,a(Mi,Mr,D)u)=
f9Ix fpig fyiy=f
for u, v E S . We have seen that
9I
JJTx9Iy
.Note that a
temperate distribution k E S'(&2n) is defined by (2.16)
(k,w) = J
w E S(&2n)
JoxPzy
By the techniques of sec.1 we get
(2.17)
(k,w) =
J<)
-2mf
P1xPlyei (x-Y) (1-Ox)m(aw)
For large in this is in the span of (2.18)
inner
(x,Y)
,
IPI+IYIs2m,
where (1.1),(1.2) give existence of the integral J. But J is (x,y)-continuous and O(((x,y))%(0)). Thus (J,DXW) define DXJE S'. We shall call k, defined by (2.16), the distribution kernel of A=a(M1,Mr,D), aE ST. Note kE SI(1e2m) is uniquely determined by
, u,v E S .
(v,Au) _ (k,v(gi)
(2.19)
Formally, with distribution integrals, we may write (Au)(x) = fk(x,y)u(y)piy
(2.20)
,
(A"u)(x) = fk(y,x)u(y)yiy
Proposition 2.6. We have (2.21)
k(x,y) = kA(x,Y) = a`'(x,Y,x-Y) = a'(x,Y,a)Ia=x-Y
in the following precise sense: The inverse Fourier transform aof with respect to its third (its 2n+1-st...3n-th) argument is a distribution in S'(13n) which may be written as a conS,(R2n), tinuous family pa =a3'(x,y,(x-y)+a) of distributions in
(2.22)
(P(Y,V) =
fdxdypO(x,Y)V(x,Y)
, 'VIE S(R2ri)
in the form
(2.23)
(8'',q)) = fdo(pa,V(,) , 'W0(x,Y)=T(x,Y,o+x-y) , AE D(13n) C(&n,S,(R2n))
with a distribution integral in (2.22) defining p0E while (2.23) is a Riemann integral with continuous integrand, of compact support. Then the distribution kernel k coincides with the distribution p0 of (2.22) for a=0 .
1. Calculus of pseudodifferential operators
60
Proof. Clearly a family p0E C(&n,SI(it2n)) is defined by setting
(2.24)
)'W(x,y) , PE S(&2n)
(P0,V) =
Indeed, we get pa in the form (2.17) with a replaced by aei6 Then a look at (2.18) (with this a) shows that (p(,,p) E
C0(ln)
We get fdo(p6,ypa)=
,(P)
confirming (2.23), where we have used a linear substitution of integration variable, and some Fubini-type interchanges. Q.E.D. The distribution kernel's singular support will be of interest in the following. In sec.3 we introduce restricted symbol classes with the property that the kernel k=ka for such a symbol a has 1e2n:x=y}.
singular support contained in the 'diagonal' {(x,y)E For general a e ST this is not true: Consider E ST. Then sing supp ka coincides with the set {(x,y):Ix-yI=1}, as follows from prop.2.6 0,lemma 4.1 and 0,(4.42). In the following let ST1={aE ST: aaa=0} be the set of aE ST independent of y. Following a general convention the class of all ,
pdo's A=a(M1,D) with symbol in a class P will be denoted by OpP . 3. Hoermander symbols: Weyl ydo's; distribution kernels. For differential operators A of the form (2.9) it appears that there are many representations as y,do's, due to the fact that
Leibniz' formula may be used to convert a product Daa(M) to a sum of products aa,(M)Da' and vice versa. In fact, it is clear that a differential operator (2.9) always may be rewritten in the form ,
(3.1)
A = b(M1,D) = b1(Mr,D)
,
only. with unique symbols bE ST and b1E ST depending on x, The same will be true for a iyido A=a(M1,Mr,D), if a satisfies stronger inequalities. In fact, a generalization of the Leibniz formulas will be developed, either involving a Taylor expansion type reminder, or an asymptotically convergent infinite series, as discussed in sec.'s 4 and 5. Later we will use only one representation A=a(M1,D) for all y,do's (also written as A=a(x,D)), following a general convention. But it will be convenient to have more
1.3. Hoermander symbols
61
general forms available. The restricted symbol class SSC ST below will have other desirable features: The distribution kernel kA of A, for an aE SS, (cf. sec.2) will have sing supp kAC {x=y}. OpSS will be an algebra. Also, as seen in 11,5, a 'pdo in OpSS will not increase the wave front set of a temperate distribution.. Apart from the representations A=b(M1,D), and A=c(Mr,D) resulting for A=a(M1,Mr,D)E OpSS, a third representation, of the form A=e((M1+Mr)/2,D)=e(Mw,D) is useful, with a third symbol e , and e(M1+Mr)/2,D)=f(M1,Mr,D) with e(Mw,D) will
be called the Weyl representation of a pdo. b(M1,D) and c(MrD) are referred to as the 'left (right) multiplying representations'. Let SS denote the class of functions a E COO(R3n) such that
(3.2)
o(()m,+S(IaH+IRI)-p' I?I(x)m2-P2IaI(y)m3-P3IRI)
DaaDyDya =
with real constants mj,pj,i satisfying (3.3)
0 < pj s
1
,
j=1,2,3, 0 s S < pl
.
Also, by SSm'p'S we denote the class of all aE ST satisfying m2 ,m3) , p=(pi p2 ,p3) , 6 , where only the first condition (3.3) will be required, and no longer S
(3.2) for a given m=(mt
since aaa=0. We then will set m3 =0, p3 =1 , or omit m3
,
p3 , writing
m=(ml,m2) and p=(pl,p2) as 2-component vectors only,ignoring the trivial estimates (3.2) for 5#0. We will mostly be concerned with y*, since it will be found that OpSS = Opih. We also define
fl {yi1'm,P,S
: m E 12 } = s(12n)
P,6 _ - U {'m,p,S : m E It2
}
Hoermander [Hr2] has introduced a class SP,, of local symbols consisting of C00(axln)-functions, with a domain 11C In using inequalities similar to (3.2). He defines SP,s by the estimates (3.4)
E In
,
x E K
for all a, on all sets KCC 11. A iydo A=a(x,D), similar to a(M1,D)
is defined for C0(11)-functions u by the first formula (2.12).
1. Calculus of pseudodifferential operators
62
Accordingly we shall refer to symbols in the classes SSm'P'b , Vh,,p,b as Hoermander type symbols SP,6 , SS , q* ,
From (3.2) it follows that
Im, I+Im: I)
(3.5)
recalling that pj Z 0. This implies (1.1), with K(k)= m1+bk, ?(k)= Im2I+Im3I . Moreover, (1.2) holds whenever 6 < 1 . Accordingly,
SS CST
(3.6)
Lemma 3.1. Let a
SSm 'p- '6-
,
SSm,p,o CST
E SSm'P'b
,
as 6 <
1
.
, b E SSm,P1'S', then we get a + b E
, a b E SSm 'P^ '6^
, where
(3.7) mj=Max{mj,mj}, 6-=6-=Max{6,6'}, mj=mj+mj, pj=pj=Min{pj,pj} Moreover, U {SSm'P'b:mj E 2}=SS001P'6 is an algebra under pointwise
addition and multiplication. All statements remain true if SS , in all expressions, is replaced by ih. The proof uses Leibniz' formula. It is left to the reader. Remark: Note that SS and bah are not algebras. In particular the
condition 6 < p1 needs not to hold for sum or product of two symbols with summands or factors of different p and O. Thm.3.2 below is central for the calculus of ydo's. Its proof will be given after some preparations, discussed in sec.5ff. Theorem 3.2. Let a E SSm'P'b, with m,p,b satisfying (3.3) . There exists a unique b E ohm .P b- and b1 E1yihm .P-,swith (3.8)
m1 = ml , m2 = m2+m3
.
P1 = P1
.
6- = b
,
p2 = Mini P2'P31
such that (3.1) holds for A=a(M1,Mr,D). Moreover, there also ex-
ists a unique b2E 0 m (3.9)
with m , p- , b- of (3.8) such that
A = a(Ml,Mr,D) = b2((Ml+Mr)/2.D)
Observe that c=b2((x+y)/2,1;)E ST, whenever b2 EE hm,p,b, 6<1, by an
estimate similar to (3.5). Thus c(M1,Mr,D)=b,((M,+Mr)/2,D) is well defined, but need not be in SS. We will find, later on, that, with m,p,b satisfying cdn.(3.3), the classes {a(Ml,D)}, {a(Mr,D)}
{a(Mw,D)}, where a E Vhm,Pb, are identical. Note that, for C = a((M1+Mr)/2,D), we get from (2.3) that .
(3.10)
C* = a((M1+Mr)/2,D)
1.3. Hoermander symbols
63
Note that, for C = a((M1+Mr)/2,D), we get from (2.3) that C* = a((M1+Mr)/2,D)
(3.10)
This points to one of the advantages of the Weyl type ipdo: If the
symbol a is real then the operator C above is formally selfadjoint On the other hand the 'left multiplying type' a(M1,D) has other advantages: If
a polynomial in l;, the resulting differ-
ential operator a(M1,D) is in the conventional form -coefficients at left from the differentiations. Recall, for Weyl type ?4do's, (3.11) a(Mw,D) =a((M1+Mr)/2,D) =c(Ml,Mr,D),
The restricted class SS produces useful distribution kernels: Theorem 3.3. Let a E SS. Then the distribution kernel k E S'(&2n) of A=a(M1,Mr,D) has its singular support contained in the diagonal {(x,y)E R2n: x=y} = O of R2n. Moreover, for fixed x (fixed y), the distribution k(x,.) (k(.,y)) equals a function in S(ln) outside any neighbourhood of x=y , and uniformly so for xE K (yE K) K any compact set (in fact, we have (y)alaXaYkIscaRYE as xE K as yE K , Ix-YIZE))Ix-YI-F ((Y)(1laXaYkISCapYE '
Proof. Referring to prop.2.6, and the discussion in [C,],II, note that a(x,y,.) E M hence & (x,y,.)E Sos (cf.[C,],II,thm.4.3). This implies the statement. Offering details (and independence from 00 [C,]), let wE CO(I2n ), w=0 near e. Then supp w has positive distance from O, and Ix-yI -mE C00(supp w). By (1.6), with a partial integration, one may bring (2.16) to the form (3.12)
(k,w) =
J0JfR1x91Y
From (3.1) we get
(3.13)
4ma(x,y,l;)
O((,)m, -2mp, ((x,Y))m:+m3
=
)
,m
If m is large enough then the integrand will be L1(R3n)
,
so that
the integrals may be interchanged, for (3.14)
(k,w)= fkwoxgly
,
k(x,y)= fs
(3.14) has convergent Lebesgue integrals, if m is large. Moreover,as m gets larger and larfrger, the formulas (3.15)
aaayk(x,Y) =
f41
aaaP els(x-Y)A a(x,Y,U)/Ix-YI2m
give the derivatives of k(x,y) (k is independent of m; the inte-
1. Calculus of pseudodifferential operators
64
that sing supp k C 0 . Very similar arguments will show that etc. Q.E.D. y alaIayklscaRYE, as x E K , jx-ylze ,
4. The composition formulas of Beals.
Proposition 4.1. For aE (4.1)
C0*(&2n)
a(Mr,D) = p(M1,D)
, bE C0OO(&3n) we have the formulas ,
b(M1,Mr,D) = q(M1,D)
with p,q E S(&2n) given by
iyrla(x-Y,t-j)
(4.2)
,
q(x,t)=JPlY9ll1e
Proof. Focus on the first relation (4.1), typical for the second. For u E S and a E C0OO(I2n) we get (4.3)
a(Mr,D)u(x) =
Jdyaz'(Y,x-Y)u(Y)
with '"y" indicating the inverse Fourier transform with respect to
the 2-nd (set of)variable(s) of by integral interchange. Write ay(y,x-y)=aa(x-(x-y),x-y)=c(x,x-y), i.e., c(x,z)=a'(x-z,z)= fg eil'za(x-z,T;). Clearly cE Cam(&2n), c=0 for large lxj, c(x,.)E S, for fixed x. Hence cE S(B2n); we may define p(x,U) fgtzc(x,z)e
ll;z
E S(>Q2n), and get c(x,z)=py(x,z), c(x,x-y)=pr(x,x-y). Accordingly, (4.4)
a(Mr,D)u(x) = J41yp (x,x-Y)u(Y) =
(x-Y) u(Y)
JR
Clearly the right hand side of (4.4) equals p(M1,D)u(x). Also, (4.5)
which becomes (4.2), after an integral substitution -t=rl
,
q.e.d.
Proposition 4.2. Let A=a(M1,D), B=b(M1,D) with a,bE C0"O(&2n). Then
given by
Then C=AB=c(M1,D), A*=a*(M1,D) with c(x,i;),
)e-iyq (4.6)
a*(x,s)
=
In prop.4.2 A* denotes the formal Hilbert space adjoint of A, as in (2.3), here of the form A*=a(Mr,D). Thus the second formula follows from (4.1) and (4.2). For the product AB write
(4.7)
(ABu)(x) =
Jo,t OYglrlgz
eirl(y-z) u(z)
65
1.4. Beals composition formulas
We may write a 4n-fold integral, the integrand being L1(R2n). Use (4.8)
and continue (4.7) as follows.
(4.9)
=
e1T1(x-z)
a
f
u(z)
Denoting the inner integral by c(x,il), (4.6) follows, q.e.d. In the trivial case of CO symbols formulas (4.1),(4.2),(4.6) and the calculation of the symbol achieve the transition M1 **M r
,
of operator product and adjoint. We will show that the formulas remain in effect for symbols in ST and STS if the Riemann integrals in (4.2) and (4.6) are replaced by a type of distribution integral we call a finite part - it resembles a concept of Hadamard [Hdl]. Let us still look at the Weyl-representation for Co°-symbols.
Clearly (3.10) gives a(Mw,D) =a(Mw,D), a formula like (4.6), but much simpler - it does not involve integrals. But the formula for the symbol of a product, now involves a 4n-fold integral: Proposition 4.3. Let a,b E C"(&n) (4.10)
,
C = c(M1,Mr,D) = P(Mw,D)
and c E C0OO(I3n). We have
Q = a(Mw,D)b(Mw,D) = q(Mw,D)
where p , q E S(R2n) are given by the formulas
(4.11)
q(x, )=f
(x-Y+z/4, -rl+t/4) b (x-Y-z/4, -n-t/4) a-i ('1z-tY) .
Proof. For (4.10)i depart from (4.3), writing c' (w+z/2,w-z/2,z) , with w=(x+y)/2, z=x-y. Defining p(w,l;)=d'(w,l;) =JFizd(w,z)e 12t, get Cu(x)=fslyp (x
,x-Y)u(Y)=P(MW,D)u(x). Again
pE S(1Q2n); interchanges are trivial, confirming (4.10)i-(4.11)i
For the second formula one writes Qu(x) = (ABu)(x) = fr(x,z)u(z)ylz
,
(4.12)
r(x,z) = Again write r(x,y)=s(w,v)=r(w+v/2,w-v/2), w=(x+z)/2,v=x-z,and get Q=q(Mw,D), with along lines used before. Then show that q assumes the form (4.11) after an integral
66
1. Calculus of pseudodifferential operators
substitution. No trouble with integral interchanges. qE S follows. Now we start with the discussion of a more general singular integral called the finite part (integral) since it resembles Hadamard's finite part (cf. also [Ci],II).This will be a distribution integral using partial integration, based on identities like (1.6)
(4.13)
a-iY'"
(Il)-2N(1_Ay)Ne_in
=
(Y)-2M(l_01)Me in
_
valid for all nonnegative N,M. Using (4.13) and some (legitimate) partial integrations, the first formula (4.2) assumes the form
in
)Ne
(4.14) =
f
aN(z,t)=(l-OZ)Na(z,t)
Using the other identity (4.13) we get, similarly, (4.15)
f 9jYPTqe
1(Y) -2M(l-0I)M(('il) (1-0I)M((,1)-2NaN(x-y,t-rl))
Here the expression (4.16)
is in the span of
IaI+IPI s 2M
.
If M and N are chosen sufficiently large then the right hand side of (4.15) is meaningful, as a Riemann or Lebesgue integral, even if we only require that aE ST1. Indeed, the expressions (4.16) are
0((l)-2N(x_y)X(2M)( _l)x(2N)) , with x,a, of (1.1), so that the integrand of (4.15) is 0((Y) -2M( x_y) a. (2M) (11) -2N(1; _,1) x (2N)). We assume xa0, x.20 in (1.2): they may be replaced by Max{0,x}
,
Max{O,a.},
X=0((x) k+(y) X)
also satisfying (1.1),
( 1. 2 ) . Thus (x-y) ; similarly for -i. Existence of the integral (4.15) follows from (1.2). The integrals in (4.2)2,(4.6),(4.1l) are treated similarly, using (1.1) with identities like (4.13), and partial integration. Note that all these integrals are of the form fdsdaeis w(s,a)
(4.17)
,
with an integral over &2m, and wE
If W(=-
(4.18)
,
asaaw(s,Q)
=
0((s)X(IaI)(a)%(jPj))
Cm(12s)
satisfies
X(k)-k + -oo , k - oo
we define the finite part(integral)p.f.f dsdae's w(s,a) as follows:
Use identities like (4.13) and formal partial integrations as above until an integrand in L1(R2m) is reached. Then define p.f.f as the Lebesgue integral of that integrand. That is,
1.4. Beals composition formulas
67
p.f.fdsdae'saw(s,a) =fdsdaeisa(a)-2M(1-Os)M((s)-2N(1_A(3)Nw(s,a){
(4.19)
dsdaeisa(s)-2M(1-A0)M((a)-2N(l_A5)Nw(s,a))
=f
for sufficiently large M,N, where it yet has to be shown that this definition is independent of N,M and the choice of first or second expression,at right.The latter will follow from the lemma,below.
Lemma 4.4. Let w E ST1(&m) (i.e., satisfy (4.18)), and let (4.20)
wj(s,(J) = w(s,(J)Xj(s)Xj(a)
,
j = 1,2,...
,
with Xj(s) = X(s/j) , where X E C00(e) equals 1 in Isi s 1 If the two expressions at right of (4.19) are denoted by I(w) _ IN,M(w) and JN,M(w) = J(w)
,
respectively, then we have
limjIN,M(wj) = IN,M(w)
(4.21)
,
limjooJN,M(wj) = JN,M(w)
for all sufficiently large N and M. Proof. If wj is substituted for w in IN,M (or JN,M) then Leibniz' formula may be used to obtain I(wj) for example, as sum of terms ,
2Masaaw(s,a),IaI+IYI52N,IPI+IOIs2M.
If N,M
are chosen such that the integral IN,M(w) exists,i.e., ?(2N) - 2N < -n
(4.22)
,
X(2M) - 2M < -n ,
then each integrand is of the form X(a)(s)X W )(a)O(((a)(s))-n)
This implies that all integrals tend to zero, except for a=4=0 using that cja)(s) = j-IaIX(a)(s/j) On the other hand, the integral for a = 0 converges to I(w) since cj(s)- 1 , sE &m q.e.d. .
,
,
The lemma below will be useful to differentiate finite parts under the integral sign.
Lemma 4.5. For a symbol w E ST1 the function
p.f.f¢y is C-(&2n)
,
and we have
h(P)(x,
)
= p.f.fOTY> 1
iyyl
The proof is a calculation: Just use (4.19), with M,N suitable
1. Calculus of pseudodifferential operators
68
for both above finite parts. Then confirm that the differentiation may be taken into the Lebesgue integrals,and also commutes with the two Laplacians there. Observe w E ST1 implies that STS
for arbitrary X, IA E & ,
x°
, °E
n .
Lemma 4.6. If in prop.4.1 we require only aE STI , bE ST , then the integrals in (4.2) still exist as finite parts, and define
Co functions p,q , with all derivatives of polynomial growth:
(4.23)
N
p(R)(x,
a,P)
,
)
Moreover, we have (4.1) satisfied in the sense that, for u E S
Proof. For a,b E Co we may differentiate p,q of (4.2) under the integral sign, getting p,q in the span of the expressions
(4.24) J91y91rle l
a(R)(x-y,
-r0
,
J91y911le-11aXayaOb(x1y1U)
y+S=a.
,
For general aE ST1, bE ST again construct sequences bj= bxj(y)xj(U), as in lemma 4.4. Let aj be substituted for a, and the partial integrations of the finite part be executed in (4.24).
As j-.oo get limits with integrands =0(((Y) (r;) ) -2N ( (x-y) ( -rl) %(2N)) , in the first case, and in the second, letting k=Ial+l0I. We were setting N=M, and x(k)=X(k), as always in the following. We get, with r = 2X(2N+1cI+IOI),
(4.25) and the same formula for q. Here pj, qj denote the p,q for aj,bj. The estimate O(...) in (4.25) also holds for pj, uniformly, since the aj satisfy uniform estimates (1.1). Clearly (4.1), in the form above, is true for uE S and aj, bj, pj, qj. For j-- get the desired relation for p and also for q, by a similar conclusion, q.e.d. We leave it to the reader to confirm that the corresponding statements are true for prop.'s 4.2 and 4.3. Summarizing: Theorem 4.7. If the symbols a,b of prop.4.1, symbols a,b of prop. ST1, ST ; STI STI 4.2 and a,b,c of prop.4.3 are chosen in ,
,
ST1, STI, ST, respectively, instead of in C" then the integrals in (4.2), (4.6), (4.11) still exist as finite parts. They define Co functions p(x,l;), res-
pectively, which are of polynomial growth in together with * * all their derivatives. Moreover, (4.1) , AB=c(M1,D) , A =a (M1,D)
69
1.5. Leibniz formulas with integral remainder and (4.10) still hold for u E ES
in the sense of lemma 4.6.
By silent convention we omit the finite part label at the integrals and write the composition formulas for symbols (which we shall call Beals formulas) in their forms (4.2), (4.6), (4.11), even for symbols in ST and ST1. Note that we are not getting the estimates (1.1) back for the symbols p,q, etc. Instead we obtained the weaker estimates (4.23) of polynomial growth. It is of interest to ask for subsets of ST1, for example, characterized by stronger inequalities, such that the composed symbols p, c, a q again satisfy the stronger
estimates. As a comparatively large such class we introduce
(4.26) ,pt = frt. _ {aE
(x) rtL' ),for all a,(3}
Here the pair m = (mi,mz) will be called the order of the symbol a and Ptm C pt will denote the class of all symbols of order m. ,
Theorem 4.8. Let 1I 11 "'r, 4% denote the classes of all a(Ml,D), a(Mr,D), a(Mw,D), respectively, with aE y,t. Then we have qT1 qT W =qT=OPVt, with transition formulas between representations of AE W given by thm.4.7. Moreover, LI' is an algebra under operator
r
multiplication containing its Hilbert space adjoints. The order m of AE LPl' (defined as order of its symbol) is independent of the representation used. Orders add when operators are multiplied. Thm 4.7 gives formulas for symbols of adjoints and products. For a proof we check earlier arguments. The stronger conditions imply the algebra property. Details are left to the reader. 5. The Leibniz formulas with integral remainder. The Leibniz formulas express a product P(D)a(M) of a differential polynomial P(D) and a multiplication a(M) by a C° function a as a finite sum of terms a.(M)P.(D), with multiplication and differentiation in the other order. Similarly a(M)P(D), (P(D)a(M)) We have a(x,D)b(x,D)=c(x,D), (a(x,D))*=a*(x,D), for differential operators a(x,D), b(x,D), with finite sums c(x, (5.1)
) = 0
7,(-i)'01/9! a(g(x, e
)
)
Other formulas express p,q of (4.2), or of (4.11), in that case. In the following corresponding formulas will be derived for
.
1. Calculus of pseudodifferential operators
70
-Pdo's. We focus on symbol classes 0m,P,B of Hoermander type. Actually the formulas follow for symbols in ST or ST1, and the tool of derivation is the ordinary Taylor formula with integral remainder. There will be a finite sum, as in (5.1), but also a remainder term, as in Taylor's formula. Unless the symbols belong to SSmp,b, 6
(5.2)
I OsN
pN (N+1)
a(O)(x,U)(-T1)O/O!
+
,
PN
1
I (-r1)0/0!P0,N IOI=N+1
' PO,N = 0
Substituting (5.2) into (4.2), (4.6), (4.11) one obtains the following typical collection of Leibniz formulas. Other formulas, for similar symbol transitions, follow just as easily. Theorem 5.1. Let a,b E ST1, c E ST , and let the symbols arl(x,), cl(x,) , cw(x,) , pl(x,) , pw(x,) a*(x,) be defined by ,
arl(Ml,D) = a(Mr,D)
(5.3)
,
cl(Ml,D) = c(Ml,Mr,D) = cw(MH,D)
,
pl(Ml,D) = a(Ml,D)b(M1,D) , pw(Mw,D) = a(Mw,D)b(Mw,D)
a(Ml,D)
,
= a (M1,D)
We have the following Leibniz formulas (with integral remainder):
arl(x,) = Lj=0(-iaxa)Ja(x,)/7! + (N+1)PN (5.4) 1
PN(x,) =
di(1-i)NIN!p.f.
91YF1Tle-iyr!(-iaX
N+1a(x-Y,- Tj);
a
0
cl(x,) =-J=II/j!{(-iay a )N+1c(x,Y, )}x=y + (N+1)PN
,
(5.5) 1
PN(x,U) =
dt(1-s)NINIp.f.f plyAe-iyr!(iay
0
cw(x, )=`7=01/j! (ia
a
N+1c(x,x-Y, -T1);
ay)Jc(x+Y/2,x-Y/2,)j
+ (N+1)pN
(5.6) 1
PN =
J
0
di (1-i)/N! p.f.f
p1(x,5)= (5.7)
01/j! (-ia
P1YAe_iy1(idya
N+1c(x+y/2,x-Y/2,-il);
ay)1(a(x,)b(Y,i1)jx=Y,=1 +(N+1)pN
1.5. Leibniz formulas with integral remainder
(5.7) pN (5.8)
1
0
_
(1-t)N/N! di p. f.
pw(x, )= =0,
pN M
1
0
ly"(-iay
ay-aX a,l)J(a(x,
-t) Ndt
e
71
b(Y,Y1) Ix=Y,
+pN,
21(zrl-yt)(2(asaK-ataa>> i N+1 a(s.a)b(t,K),
M=N+1, s=x-y,
a finite part integral.
(5.9)
(N+I)pN ,
a
with PN as in (5.4) , with a replaced by a Note that the term pN has been used in all formulas,to denote the remainder, without implying equality in different expressions.
Proof. In each case the equation at once is confirmed formally, by substituting (5.2) into (4.2), (4.6), (4.11), and using that
iy = ay(1 )"
(5.10)
=
(2x)n/2ayO(Y)
in the distribution sense. A partial integration must be used with (ax ay)1/1! =
(5.11)
aXay/8
I
e
1
For cw we must apply (5.2) onto the product ab, in 2n variables. The detailed derivation is technical and repeats the same steps over and over. Focus on (5.7), the others follow similarly. Using (5.2) in (4.6)1, (with c(x,x) called pl(x,x)) we get the terms
(5.12)
p.f.fgypiri e-i"1(-1)e/O!
For a,b E C0*(&2n) this is easily transformed into
=(-i) (5.13)
10
,
91Y9lT1 ay(e
=(-i)I0I/O!JP1YPlrl with N sufficiently large, but with the n-partial integrations
reversed to the status M=O. This still gives a well convergent integral, as long as a,b are Co . Recall that bN = (1-Ax)N b Here the ii-integral may be evaluated. It gives the Fourier
transform ((Y1)-2N)^=((,q)-2N)' =EN, the unique fundamental solution in S' of (1-A)-N. An expression for EN in terms of modified Hankel functions is known (cf. (0.24) for N=1; the method works for gene-
1. Calculus of pseudodifferential operators
72
s>0.) EN decays exponentially, as lxI-goo. Without ral Hankel functions one confirms easily that EN equals a function
in S, for 1xIa1. (c.f. [C1],II). Hence (5.13) is continued as =(-i)'°1/91 (5.14)
(-1)'01/01
confirming the first expression of (5.7), using (5.11), and for Cp functions only. For the remainder term, we look at
p.f.ff91y51rl a-iY11 (5.15)
_
iyrl
91Y91r1 a
(1-z)Ndia(e)(x,
0
1(l-i)Ndi
0
P1YP1r1
0
as a matter of Fubini type integral exchanges. Write (-rl)°e iyr) _ (-i)IOIaye-1Y11 and carry out a partial integration, for ,
n
r
0
(5.16)
[l
iai
IT
f
_,.... ,a%
(0)(x-Y,)
,
0
completing the discussion of (5.7), for a,b E COOO
For general a,b E ST1 one now may use sequences aj, bj E CO, as in lemma 4.4, and arrive at (5.7) by passing to the limit j
-> -,
using lemma 4.4 and a formula similar to (4.25). Details are left to the reader. In particular the occurrance of the factor r, in the argument of a does not influence the derivation. ,
6. Calculus of Vd2o's for symbols of Hoermander type.
The Leibniz' formulas with integral remainder of sec.5 may be of little use, unless we restrict symbols to the class SS of
sec.3. For symbols in SS or h the formulas of thm.5.1 imply corresponding asymptotic expansions of arl' cl' cw' p1' pw' a* The key is an estimate of the remainders pN of (5.4)-(5.9). , c E SSm",p"," b E y>hm, 1, 61 P with the 3 sets of paramaters satisfying (3.3). Then the remainTheorem 6.1. Let a E E - =0
S,
ders pN of (5.4) through (5.9) satisfy the following estimates:
73
1.6. Calculus of pseudodifferential operators oPPPl(
For (5.4):
)m.-(N+1)(p.-b)
(x)mz-(N+l)pz )
=of( )m."-(N+1)(p'"-b") (x)m2"+m3"-(N+1)p3°`.
For (5.5): For (5.6):
)m. "-(N+1)(P' "-b")
o((
X)M2 "+m3 "-(N+1)p-)
p-=Min{p2 ",p3 J"} .
For (5.7):
0
f(l;)m1 +m1 '-(N+1)(P.-b') (x)m2+m2'-(N+1)p2)
For (5.8): pN(x,U) = pj = Min{pj,p'}
,
b^ = Max{b,b'}
For (5.9): Same as for (5.4) Moreover, these remainders will be symbols of Hoermander type, of the order indicated by the above formulas. That is, PN E A'm-(N+1)(p-be'),p,b ,e'=(1,O)
for (5.4) and (5.9),
PN E 4* (MI "-(N+1) (P "411) M2 "+m3 "-(N+1 )p3 "), (P'
(6.1)
pN as for (5.5), with p3" replaced by p" PN E 4*m+m'-(N+1)(p-b'e'),p^,b^ replaced by b^,for (5.8)
,
P ),b ,
'
for (5.5),
for (5.6)
,
and the same with 6'
for (5.7)
.
Here we must require that b^< p,^, in case of (5.7) and (5.8). Then both orders of pN will tend to --, as N - - .
The proof proceeds similar as in sec.4. In each case we supply estimates, uniformly in s, of the integrand di only. For example, to estimate PN of (5.4), it suffices to consider
p.f.fdydrl a-h
(6.2)
1a(6)(x-Y,x-T1)
=fdydTl e
again with aN (1-AX)N a, for sufficiently large M,N. Use (3.2) for (6.3)
=o f dy(x-Y) mz -pz 1111 (Y) -2M l
fdll( -iTl) m. -p. 19 1+6 101(,n) -2N)
Here we also employed the estimate
(6.4)
((Tl)-R)(a)
=
o((Tl)-R-Ia
which is easily derived. Then apply the well known inequality
.
74
1. Calculus of pseudodifferential operators
(6.5)
(x) s(x-Y)s
=
0((Y)IsI)
,
valid for sE 1, to estimate (6.3) by (x) MI -PZ 10 I (
) m, - (PI -6) I 0 I
confirming the estimate for the remainder pN of (5.4). Similarly for the other 5 estimates.Also we may apply Lemma 4.6 to the remainder of (5.4), for
(6.6)
pN(R)
f 0(1 )N/NIf I1YAe iYn(-ia
-a
N+1a(a) (x-Y, -TI
)
Then the above estimate may be repeated for
(6.7)
)m,-(N+1)(P'-b)-P IaI+OIl1(x)m2-(N+1)p:-PZ IPI),
PN(a) = 0((
which confirms (6.1) for the first remainder. Again similarly for the other five remainders, q.e.d. Theorem 6.2. Under the assumptions of thm.6.1 the symbols arl' a* of thm.5.1 are in yah provided that, for cl, cw, pl, pw, the product symbols pl , pw we still require that 6^ < p, (which is automatically true if (m,p,S) = (m',p',S') ) More precisely, we get arl, a* E 0 cwEyhm ,P-,s m,p,S ,
.
Cl,
with m =(m, ,m2+m3) Apm+m',p^,S^
,
P =(PI ,P2
with p^
,
)
,
p2 =Min(p2 ,P3)
,
P1
, Pw E
6A as in thm.6.1.
The proof of thm.6.2 is a consequence of (6.1). Note that thm.6.2 contains the (so far unproven) thm.3.2 as a special case. Indeed, the symbol b of thm.3.2 is given by our cl c(Mr,Ml,D)=(c(Ml,Mr,D)*
while b1 may be obtained by writing b1(M1,D), taking adjoints. b2 coincides with ow of thm.6.2.
For uniqueness of b, b1, b2 note that a ,do A=a(M1,Mr,D) vanishes if and only if its distribution kernel (2.16) vanishes. For set w(x,y)=u(x+y)v(x-y), u,vE S. (2.16) yields
f
dsdteitta(s,t )u(s)v(t)=f dsd; a(s,, )u(s)vr'* (( ) =0 for all u,v E S.
This implies a=0. Thus the Weyl representation is unique. Similarly for the left (right) multiplying representations. Similarly thm.6.2 implies the following result.
Theorem 6.3. Each sethoo,p,b forms an algebra under operator multiplication, containing its Hilbert space adjoints, whenever p,S satisfy (3.3). The algebra product a°b may be defined by anyone of the representations a H a(M1,D), or, a H a(Mr,D) , or a H a(Mw,D). That is c=a°b satisfies c(Mx,D)=a(Mx,D)b(MM,D), x=l or x=r or x=w.
75
1.6. Calculus of pseudodifferential operators
The proof, in essence, is an application of (5.7), (5.8), and (5.9), as modified by thm.6.2. Details are left to the reader. In the following we tend to use the left multiplying representation A=a(Ml,D), and then will use the more conventional notation A=a(Ml,D)=a(x,D). For a class X of symbols let OpX={a(x,D): aE X}. For a class Y of ,do's let symbY be such that Op(symbY)=Y. For a symbol aE 11hm,p,s
1,2,...,
ajE yt j
,
and a sequence of symbols aj, j=0,
,
m+ =(mi,m2), mk-.-oo, as j-oo, k=1,2, we will
m ,p,S
say that a has the asymptotic expansion
a =
(6.8)
0
aj (mod Vh_.)
=0 aj
,
written as
(or simply "(mod S)"
,
if both orders of the symbol pN=a-e=0 aj tend to -oo, as N- (that
is,more precisely, if rNE h N j-
,
, with µN=(µN,µ2), where µ ,P,5 k=1,2). It is seen at once that µJ=mj+1 is the best possible
choice for µj . Also we must have m=m°
.
Using this terminology we find that all formulas (5.4)-(5.9) imply corresponding asymptotic expansions For example, (6.9)
(mod 0_,O)
a
as aE 0M OP 16, 6
It is clear that the asymptotic sums in (6.8) or (6.9) are not in general convergent infinite series. For future applications it is important that, for an v sequence aj of symbols of order mi--00,
an asymptotic sum may be constructed, by the lemma, below.
Lemma 6.4. Let ajE j=0,1,..., where p,5 needs not satisfy (3.3), but is independent of j with and where mJ=(mi,m2) ,
(6.10)
mk>mk>mk>
...
,
as jaoo
,
There exists a symbol a Ehm,p,S (6.11)
sN = a - 4=0 aj
k=1,2
such that
m = m0
E yh N M ,p,b
N=0,1,2,...
.
Proof. Indicating first the idea of the proof, let (6.12)
s 1/2
,
o )=l
,
as jxI+1fl z
1
.
1. Calculus of pseudodifferential operators
76
and let x=1-wE CD . Assume Os w,x sl in &2n j-- , to be For an increasing seqence {tj} , tjE I determined later, define
Assume wE C°°(1e2n)
,
w(x/tj,/tj)
(6.13)
Conclude that w.=0, as Thus is finite near
/2. Since t.->oo the sum in (6.13) is well defined. We get
a - f -laj
(6.14)
J=O
= -Lj=oxjaj + > N wjaj
where the first term at right is C0(22n), and satisfies (3.2) for any choice of parameters m,p,S. For the second term we get
(6.15)
IwjajIscj(1;}ml
(x)
sup{( )8 i(x)0, is JxJ+1i 12ttj/2}
By choosing tj sufficiently large the supremum in (6.15) can be made less than 2-3/cj . Using this in (6.15) , and then substituting (6.15) into (6.14) we get with 9S =md -m
9'Z
=mz -lZP
.
)mi
(6.16)
a - Lj=pai = 0((
(x)
Writing (6.16) for N+1, we notice that the highest term aN , at left is of the same order as the right hand side. Thus we get the first estimate (3.2) contained in (6.11). To get the other estimates we improve the choice of tj. For a
given a,4 and N let No=Max{jaI+I3I,N+1}. Write
(6.17)
- j=0a.
=a-`zX I-
,
and,
J
0
aXo sN =
Observe that o.E yah0,p,6, for every p,S with (3.3), and 0=(0,0). Thus also a.w4E1yah j and (6.15) follows for aaap(wja.) instead m ,p,S ml-1+Sjaj-piipi of-w a. with c., mN-1, mz-1 replaced by cj,a,, 13 3 m2 -p2jal respectively, with a constant cj,a,p depending on j, ,
a,s only. The first sum in (6.17) is CO and satisfies (3.2). The second sum is finite; its terms satisfy the proper estimate. To control the last sum, tj must be chosen according to (6.18)
(x)6'Z
s 2-j, as jaj+jPjs j.
This amounts to finitely many estimates, for each t, hence the selection is possible. In each term of the last sum we get I-1+1131 sN0sj, by construction of N0. Thus get the proper estimate,q.e.d. Lemma 6.5. Let {bj1:1=1,2,...} be sequences with the assumptions
77
1.6. Calculus of pseudodifferential operators
of lemma 6.4, for j=1,2,... Also, let bj1E 1hm-(j+1)µ,p,S' µaO, so that the asymptotic sums bj = 2;=0 b
E 0 m_jIl,p,6 again
jk
form a sequence of symbols with the assumptions of lemma 6.4. Then
(6.19)
0 bjk
-7=0
=O4=0bk, j_k
=
(mod O_
where infinite sums are asymptotic. Equality in (6.19) is under-
stood only up to a term in yah_.. Proof. All symbols bjl ,j+l=k , are in the same class yahm-kµ,p,S hence their sum is in that class. Thus the asymptotic sum at right is meaningful. Denote the left and right hand side of (6.19) by b '
and b'
, and write bN =
fj,k=ob.k
,
b'N =
b 1 71 j+lsN j
.
Note that
bN - b'N is a finite sum of symbols of order m-Nµ hence itself is of that order. Moreover, b' - b'N is of order m-Nµ , while b-bN =b -
4 =0bj + 4=0(bj - Vk=0bjk)
again is of that order. Hence
b-b' _ (b-bN) + (bN_b'N) + (b'N-b') is of order m-Nµ , for all N = 0,1,2,... It follows that b-b' E 0_0. , q.e.d. .
Lemma 6.6. An asymptotic expansion (mod S) of the form (6.8), with a E yhm,p,o
,
aj EOyh
, m = m0
,
etc. may be differentiated
mj,p,s
term by term . That is, (6.8) implies
a(P)=Oaj(a) (mod S
(6.20)
The proof is evident: All remainders rN are symbols, their derivatives also are symbols, of orders tending to --, as N-- . Problems. 1) A polycylinder C=S'x)en-l is described analytically by
its universal cover
1n=2x1n_ 1={
(x, , x° ) : x, E 1, x° Etn- l } :Write C°°(C) )
={uE C°O(&n) : u periodic (2n) in x, }, S(kn_l),j=0,1,..,uniformly
with
S(C)={uE COO(C) :
aX u(x, , . E
in x}. Show that a tpdo A=a(x,D)E OpVt0
periodic (2n) in x, may be regarded as a map S(C)-
COO(C), given by Au(x)=
with u j(x° )= 2a
_ooel3xl
2n fa-ljx' u(x,,x° )dx,
x°
)uj
and the Fourier transform
0
""" with respect to x° regarded as a 4d o on
.
&n-1
2) The operator A=a(x,D) of pbm.1 may be with operator valued symbol, acting on
1. Calculus of pseudodifferential operators
78
1n-1
functions on Rn-1 with values in C'°(S'): For
A(x° ,t") :COO(S' ) ,C
00
(S' ) by A(x" t" )u(9)=eljea(O,x" ,
define
j , where
2n
uj=
u(9)dO. Then write v(x")=u(.,x")
e-1
f oweix"t"
,
)v ( "),for VE S(Rn-1,COO(S' ))
and Av(x")
S(C). Show that,
for a symbol aE 9pto (and even for aE STS ) this operator A maps S(2n-1,C"(S'))
S(IIn-1,C,(S')), and continuously so when we equip + S(C)=S(1n-1,C°°(S')) with the Frechet topology of all norms must be 2nII(x")ku(a)Il 0 , k=0,1,..., aE IIn. (Recall that L
periodic in xi . ) 3) Investigate linear operators C00(S' B=A(x"
as
of pbm.1: Let Bu(xi )=;. fdyi el3(x' -y' )bj(X, )u(y,) , with
a sequence of functions b.(x,) , j=O,t1,t2,..., bounded in j and 2n-periodic (and C°°) in xi . First assume that bj(xt) is independent of j for j>>1 and j<<-1. Then B=b , (M)+b2 (M)H+K, with the Hil2n
bert transform H on S' (i.e., Hu(xi )=
dy, cot(x, -yi )/2 u(y,)
0
with a principal value integral) and with an operator K of finite
rank. Next assume that the limits limj.apj(xi)=Pt(xi) exist. Try for conditions still insuring the above form, whith more general K
4) Investigate the operators Bi
2)-1/2 and B2 =Dx Bi
2 -DX
in
the sense of pbm.3. 7.Strictly classical symbols; some lemmata for application. For many discussions, notably elliptic and hyperbolic theory it proves unnecessary to carry along the complicated classes of Hoermander developed for hypoelliptic theory. Let us introduce
,pm
= pm,(1,1),0 ,
setting p1 = p2 = 1
,
6 = 0
.
UCr = U
'_O, = n "m ,
m,
Symbols in y,cO,
,
and corresponding
operators in Opi,coo will be called strictly classical . Note that
(7.2)
'm C VIM C VSM C yitm
with the classes pt of (4.26) and 1,1
,
ps of ch.VIII.
The lemmata below are simple consequences of our calculus of
79
1.7. Strictly classical symbols Iyido's in sec.6. We are preparing them for later application.
It is clear that the classes Vtm and VCm , for finite at, are Frechet spaces under the collections of semi-norms, respectively,
x, E&n}
sup {ja(")(x, )fi(x)-m2
(7.3)
,
a,l EIIn
and
(7.4)
(a) a(P
sup
a,p EIIn
)I(x)-mZ+IaI(
)(x,
Similar sets of semi-norms for the classes t*m,p,b of sec.3 are suggested by the estimates (3.3),of course. Recall that a subset M of a Frechet space is called bounded if each semi-norm of a defining set remains bounded on M. Clearly
a bounded set of Pm is bounded in Vtm For a bounded set M of "m the set aE M} is bounded in ,tm -lal -ICI. Abounded set in
cur (Vtm ) is bounded in 4Cm, (Vt.,,) whenever
j=1,2.
Lemma 7.1. For bounded sets A={a}C cm, B={b}C yiam, the collection (7.5)
C
{c E 1ycm+m'° c(Ml,D)=a(Ml,D)b(Ml,D)
is bounded inm+m
a E A , b E B}
and the sets A* ={a*} of symbols of adjoints
and P = {p E i1cm : p(MrD) = a(M1,D)
Similarly , if
,
,c is replaced by it
Lemma 7.2. Let a E pcm , b e iycm,
.
,
a E A } are bounded in wcm in all the above.
Then we have , with e
a(M1,D)b(M1,D) - (ab)(M1,D) E OpyrCm+m'-e '
(7.6)
a(M1,D)
Also, if A = a(M1,D) (7.7)
E Opcm-e
a*(M1,D) - a(M1,D)
,
E
- a(Mr,D)
B = b(M1,D)
[ A, B] = AB - BA
Opicm-e
then
E Opyicm+m' -e
Moreover, if a, b range over bounded sets of lycm and VCm
respec-
tively then the symbols of the expressions in (7.6) and (7.7) 'c m-e, Vcm+m respectively. form bounded sets of Vcm+m'-a, V. m-e'
In fact, this is correct if we do not necessarily require that AC
yicm, and B C
Vcm,
are bounded, and a E A , b E B , but only that
respectively, for the 4 relations in (7.6), and (7.7),
(i ) {a0} bounded in tpCm-e,
,
and {b°} bounded in 1 cm-e.
,
80
1. Calculus of pseudodifferential operators
(ii) {a(8)} bounded in 4cm-e
(iii) {a(9)} bounded incm-e
(iv)
{a(e)} bounded in
,
Vcm-e'
and {a(0)} bounded in Vcm-e2
,
and {b(e)} bounded incur,-e,, and {b(0)} bounded in gym,-e2. For (i) ,...,(iv) the conditions are required for all 101=1
.
Remark 7.3: Using the 1-1-correspondence a « A=a(M1,D) between yucm and Opyscm, for x=c,t, one may transfer the Frechet topology of the
spaces 1xm onto their corresponding sets Op1xm. Then we may speak of bounded sets in Optcm for example. ,
Proof. Clearly (7.6) and (7.7) are consequences of thm.6.1, noting
that Wcm '*m,e,0' On the other hand all statements about bounded ness of sets follow by just going again over the proofs leading to thm.6.1, keeping in mind the boundedness of sets in Frechet spaces. For example, regarding boundedness of the set C of (7.6) it is sufficient to show that the constant of O(.) in thm.6.1, regarding (5.5), as well as the corresponding constants for the derivatives pN(a) of thm.6.2 of that remainder, all depend only on the bounds of the Frechet norms of a and b, in their bounded sets. This needs only be done for one N, say, for NO. Or even, one could estimate the first expression (4.6) of the Beals formula for the product symbol, in its finite part integral form (4.19), of course, using the principles of the proof of thm.6.1. Actually, going over that estimate again it becomes evident that indeed only bounds of expressions (6.2), for the symbols a and b, and corresponding bounds for the explicit symbols (7.8)
nm(x, ) =
mI (x) m2 E '4'am
and existing definite integrals over in
(7.9)
f(x) -rdx
,
r>n
as well as binomial and multinomial coefficients occur. We will not go over all the details again .
Chapter 2. ELLIPTIC OPERATORS AND PARAMETRICES IN In
In this chapter we focus at the main feature of our algebras They contain (generalized) inverses of some of their ope6 rators - called (md-)(hypo-)elliptic. For historical reason such an inverse is called parametrix or Green inverse. Speaking algebraically, we deal with inverses modulo some 2-sided ideal, either the class', of integral operators with kernel in S(12n) (K-parameOpWhp
:
metrix), or the operators of & with finite rank (Green inverse). Integral operators with kernel in S(&2n) are compact operators of L2(&n), as well as in every Sobolev space (cf.III,5). Thus a K-parametrix (Green inverse) of A will be an inverse mod K(H) (a Fredholm inverse) as well, under proper assumptions, giving normal solvability of the equation Au=f (cf. [C,],App.Al). For Oplhm p S 'elliptic' usually will be 'md-elliptic', amounting to ellipticity 'at and at jxI=co'-i.e., "d-" and "m-" ellipticity. This is parallel to discussions in algebras of singular integral operators ([Ci],[Cz]): For operators of order 0 'md-
elliptic' simply means that the symbol is #0 at 1x1+1V_o . An md-elliptic operator of order m will have a K-parametrix of order -m and existence of a K-parametrix of order -m is in ,
fact necessary and sufficient for md-ellipticity of AE Op'hm,p,S. However, it is possible for a non-md-elliptic operator to have a K-parametrix of order >-m. One such class of operators - the formally and-hypo-elliptic operators - is studied in sec.2.
The ellipticity concept can be localized, in the x-variable as well as in the 5-variable, and 'local parametrices' may be constructed under various assumptions. In particular, (formal hypo-) Rn ellipticity (for all ) over an open set 92 C implies hypo-ellipticity in the proper sense: All distribution solutions of Au=f are Coo(sl) whenever fE D' is Coo(ci) (cf. sec.3 and 4). For proper discussion of such facts we introduce the wave front set WF(u) of WE D' (sec.5), and show its invariance under spplication of ado's. In sec.6 we discuss (left)(right) md-ellipticity of a matrix of pdo's, together with certain partial inverses.
81
2. Elliptic operators and parametrices
82
1. Elliptic and md-elliptic pdo's. Historically a parametrix is an integral kernel with a singularity similar to that of a Green's function, which can be used to invert a differential operator, up to an integral operator with continuous or smooth kernel (c.f. E.E.Levi [Lvl], D.Hilbert [Hbl]). Hoermander [Hr2] used the term 'parametrix' for a local inverse up to an infinitely smoothening operator, for a ipdo. Here we will use
a type of global parametrix with similar properties. Let A,B be lpdo's with symbol in ST. We shall say that B is a K-parametrix of A if we have AB-lE Opyh-o. , BA-1E Op%ph-,O. Clearly then A and B are
K-parametrices of each other. We speak of a left (right) K-parametrix B of A if only the second (first) condition holds. A symbol aE Phm,p,6C 1ph will be called md-elliptic (of order
for all sufficiently large
m) if (i)
and (ii)
the function b=1/a, well defined for large
equals some
symbol in Vh-m,p,S' for large jxI+1g1. Then the yxio A=a(x,D) also
will be called md-elliptic. We will see that md-ellipticity is a property of the operator, insofar as, for an md-elliptic A=a(M1,D) =b(Mr,D)=c(Mw,D) all 3 symbols a,b,c will be md-elliptic. Proposition 1.1.
A symbol a E 1phm,p,8 C Vh is md-elliptic if and only if, with constants c,c'>O, we have
(1.1)
I
a
c(a)m (x)m2
,
as IxI+IsI a c'
,
Proof.If a is md-elliptic we get b=1/a for large and bE hence (x)m` ), using I,(3.2). This implies -m,p,S' (1.1). Vice versa, if a symbol satisfies (1.1) then a(x, w-0, as C w-1, as defi nes a CC°(&2n)-function satisfying the first estimate I,(3.2). By lemma 1.2, below, it satisfies all estimates I,(3.2), q.e.d.
Lemma 1.2. Let Oy6aE COO(Sc). Then (1/a)is in the span of
(1.2)
j)/a)]/a
{[IIi (a(,CE3
:
Ya3 = a
,
f3 _ (3
}
This lemma follows by induction. Proposition 1.3. For aE iphmIp,SC 0, let A=a(M1,D). Let A also be given as A=b(Mr,D)=c(Mw,D)=d(M1,Mr,D), with symbols b,cE lphm p,g and dE SSm 'p 'b C SS. The following conditions are equivalent: (i) b is md-elliptic
;
(ii) c is md-elliptic
;
(iii)
is md-elliptic (of order m ). (iv) A is md-elliptic.
)
83
2.1. Elliptic and md-elliptic operators
This follows from calculus of do's: The symbol a may be expressed as an asymptotic sum of b or c or d and their derivatives. We get modulo terms of lower order. Thus (1.1) holds for all or for none of these functions, q.e.d. Proposition 1.4.(a) AE Opyah is md-elliptic if and only if A* is.
(b) The product AB of two md-elliptic operators AE Opwhm,P,a, B E OpVhm,,P1b again is md-elliptic (of order m+m') ,
Again this follows from calculus of ,do's. Recall that a differential operator A =
ac`Da is called
IaJSN elliptic at x0 if its principal symbol (1.3)
a
aa a
N
is #0 for x=x0 and all E Rn with
In that respect notice
prop.1.5 below, which also motivates our terminology. Proposition 1.5. A differential operator A=a(x,D)E Opy1hm,P,S
,
m=(N,m2) is md-elliptic (of order (N,m2)) if and only if a p(x)m"
(1.4)
,
for all x , and all
1
and also, with suitable constants p, c>O,
(1.5)
for all x,g with lxi a c
Ia(x,l;)I a
.
Proof. Note that 'jxI+ii large' means that 'either lxi is large or is large' (or both). The first points to (1.5), the second, for a differential operator A = a(M1,D) translates into (1.4), N-1( x) m,) . using that the terms al
all are
By (1.5) md-elliptic differential operators are elliptic in &n. (x)-m-A even is 'uniformly elliptic' in ien. (1.4), in effect, is another ellipticity, with variables Theorem 1.6. An operator AE OptNhm.P.b
x,
,
reversed.
admits a K-parametrix in
Optph of order -m if and only if A is md-elliptic.
Proof. If A and B are K-parametrices of each other then the relation AB=1 and the uniqueness of the left representation implies 1= modulo terms of order -Ee, e=(1,1), E>O. Thus it follows that jabi a1/2, as ac, for large c. Accordingly, jal
a2 b ap( )m' (x)m' ,
IxI+I
I
ac, p>O, using that b is of order -m
m'). Thus A is md-elliptic (order m). Vice versa, let A=a(M1,D) be md-elliptic. Using a recursion and I,lemma hence,
84
2. Elliptic operators and parametrices
6.4 we will construct a K-parametrix for A. First, let b1 be a symbol such that ab1=1 for large by definition of md-ellipticity. If B1=b1(M1,D) then AB1-1=e1(x,D)
where order
e1 is of
Next set b2=e1/a, for large
correc-
ted to be C00(1 2n). Setting B2=b2(M1,D), get AB2+E1=e2(x,D)=E2 of
order -2a, where, again, we used lpdo-calculus for AB2. Together,
we get A(B1+B2)=1+E1-E1+E2=1+E2. It is clear now how to iterate: Next set b3=-e2/a, and B3= b3(M1,D), AB3+E2=E3=e3(M1,D), with e3 of order -3o, and A(B1+B2+B3)=1+E3, etc. Let B =7, Bj denote an
asymptotic sum. Then B-f=1Bj is of order -(N+1)a, by definition.
Also A(-1Bjj is again of order -No. Thus AB-1 is of order -No, for every N = 1,2,.... It follows that AB-1 must be of order --. Or, B is a right parametrix of A. Similarly construct a left parametrix B" of A. Get B =B AB=B (mod Opph-0), q.e.d. Finally, in this section, we show that Op4h-o,05(2n) = Lemma 1.7. The class Opih_.0 of ado's of order -oo coincides with
the class 'f of all integral operators (1.6)
Ku(x) = fk(x,y)u(y)gIy
,
u E S
with rapidly decreasing kernel k E S(&2n) Proof. We have seen before (cf. I,(2.16)) that the distribution kernel k of a yxlo is given as (1.7)
k(x,y) = a4'(x,x-Y) =
For aE ih-oo S(le2n) (1.7) is a well defined integral. The Fourier transform and b(x,y)- b(x,x-y) are isomorphisms S -S , q.e.d. 2. Formally hypo-elliptic yxlo's.
In sec.1 we found that a ipdo A of order m has a K-parametrix of order -m if and only if it is md-elliptic. It is possible, how-
ever, for A E C Wh to have a K-parametrix in OpWh m 1P'6 P'S with mj a-mj without being md-elliptic. Let aE iph C Wh. We shall say that a (or A=a(x,D)) is m,P,S
85
2.2 Formally hypo-elliptic operators formally and-hypo-elliptic if for
and (ii)
a=O((l;)
we have (i)
), with some m'=(m1
(x)m2
ml
), and (iii) for
all a,(3 the functions a((cc)/a satisfy the estimates
(2.1)
)-P'
a(a)/a = of(
In this case we shall call m'=(m1
m'
) an -inverse order of a
The symbol a is formally and-hypoLemma 2.1. Let a E m,PSC ph (ii) elliptic if and only if (i) a and a(c)/a are the functions of (i), for all a,(3, extend to symbols in 0m',p,S and Vh respectively, where .
,
ma
, (3
P,S ma',p
(2.2)
=
Proof. Trivially the condition of the lemma is stronger as that of the definition, because (2.1) is only the first I,(3.2), defining hma,p,p'S. We must show that also aXa
satisfy I,(3.2).
This follows from the lemma below easily proven by induction. Lemma 2.2. For all a,(3,x,X we have
00
span of the products
(2.3)=1(a(
)/a)
in the
3=163=a+k=103=(3+x , r=1,2,...
,
Again formal md-hypo-ellipticity is independent of the choice of left (right) multiplying representation. Proposition 2.3. Let A=a(M1,D)=b(Mr,D) with a, b E 0m,P,6C Vh Then A (or a) is formally md-hypo-elliptic if and only if b is Proof. From formula I,(5.4) we conclude that
(2.4)
j(i)191/9!
a is formally md-hypo-elliptic then the asymptotic sum (2.5)
c =
'01/e! a(e)/a e
is well defined, assuming that 1/a has been suitably modified for I xI +I
I < r!, to be C (&
2n )
This follows from lemma 2.1. Moreover,
the highest term in the sum (2.5) is identically 1, implying that the symbol c is md-elliptic of order 0 , by prop.l.1. One confirms easily that ca = b (mod iph_,O). Indeed, from (2.5) we get (2.6)
c -
/0! a(0)/a
E 1ih_(N(P'-S),Npz),P,S
2. Elliptic operators and parametrices
86
This may be multiplied by the symbol a (2.7)
ca
a(9) E
,
using lemma 2.1, for N=0,1,2,....
m-Np+NSe',p,S
In other words, we get ca = b (mod 1ph_,,) , using (2.4), as stated.
Using I,lemma 6.6, (2.4) may be differentiated, for (2.8)
(mod
8(i)I0I/0l a(6+a)
Then, repeating the above conclusion we see that
(2.9)
b/a
,
a/b E 0 0,P.o
E'iph
b(P)/a
Again, the functions (2.9) must be modified on some KCC I2n independent of a,P, before they are symbols as stated. (2.9) implies
1/b = (1/a)(a/b) E hm',P,S
b(R)/b=(b(a)/a)(a,b) E 0 «,(3
m
,p,S
so that b is formally md-hypo-elliptic. Vice versa, if b is formal-
ly md-hypo-elliptic then a may be obtained as asymptotic expansion in terms of the derivatives of b. The above conclusion may be repeated to show that also a is formally md-hypo-elliptic, q.e.d. Proposition 2.4. (a) AE OplphMIP16C Op1ph is formally md-hypo-ellip-
tic if and only if A*, its adjoint, is. (b) If A=a(M1,D),B=b(M1,D) E UmOpylhm,p,8C Opyah are formally md-hypo-elliptic then so is AB. (c) The formally md-hypo-elliptic symbols of (b) form a group under pointwise multiplication, for each p,S. (d) An md-elliptic symbol of 1phmIP.SC 0 is formally md-hypo-elliptic, inverse order -m. Theorem 2.5. A formally md-hypoelliptic A=a(M1,D)E Oplphm,p,S ad-
mits a K-parametrix B=b(M1,D)E Opom"p'S, for suitable m'a-m. Proof. As in the proof of thm.1.6 we focus on construction of a right K-parametrix. First set b1=a, and B1=b1(x,D). By assumption b1E 1phm,,PS, and the product formula yields .
(2.10)
AB1-1=E1=e1(M1,D)
,
e1= Ie ,0(_i)
e
a(e)bl(0) (mod
By lemma 1.2 the term a(e)bl(0)=a(e)(1/a)(e) is in the span of {(a(())/a)rI(a('j) /a)
:
YaJ=6}
.
Lemma 2.1 yields a(e)bl(e)E
Vh-Np+NSe',p,S' I0I=N. Accordingly el E y't2_p+Se',p,S has order less than (0,0), as in thm.1.6.
Next, in the recursion, define b2=-el/aE lphm'-p+Se',p,g Try to verify that B2=b2(M1,D) satisfies AB2=-E1+E2, with E2=e2(x,D),
e2E 1ph_2p+6e' , p , S
Indeed we get
2.3. Local md-ellipticity (2.11)
AB2+E1=e2(M1,D)
A term a(e)(el/a)(e) (2.12)
{
,
e2 =
10
for 101 = N
87
f0a(0)(el/a)(0) (mod
h-,0)
is in the span of
(a(e)/a) el(a) II.(aWj)/a)
:
a + I P3 = N }
again by lemma 1.2. Using lemma 2.1, confirm the proper conditions e2. Evidently the recursion of thm.1.6 works again, q.e.d. 3. Local md-ellipticity, and local md-hypo-ellipticity.
Recall that (formal) md-(hypo-) ellipticity of a symbol aE is a condition for large If a is modified in a set KCC R2n these conditions are not disturbed. y,h
,
city y
A parallel, though slightly different concept of md-elliptiintroduced in [C l],for the operators of some C -subalge-
bras of L(L2(&n)). Similar algebras are studied in [C2] for general noncompact manifolds c. Their operators also have symbols now defined over the boundary of a compactification of the cotangent
bundle T*S1 - for 11=1n for (x +
=o, at the 'boundary' of
R2n
The generators of such an algebra are yxio's as studied here. In [C1], ch.IV, a compactification Pnxien of RnxIn=R2n appears.
The symbol of an operator A in the algebra, called A, for a moment ffi=d(1QnxPn)=Pnxpn\R2n of PnxPn. is only defined at the boundary Then AE A is called md-elliptic if its symbol is #0 on M. md-ellipticity is necessary and sufficient for AE A to be invertible modulo operators of finite rank. The algebra A is generated by the special pdo's a(M) and b(D), where a,bE COO(R2n) are bounded and have
a(a)=o(1), bO)=o(1), as lxl-oo, for all a. len just is the smallest compactification such that all such a,b extend to C(P). This result on C*-algebras of pdo's offers some guide lines on the question of local md-ellipticity. Within A we speak of mdellipticity of AE A in a subset MC ffi if its symbol is #0 near M .
For fixed p,S consider AP,b, the closure of p'hO,P,b within
CB(R2n).
The maximal ideal space of this function algebra is a compacPP,6 tification of R2n - again the smallest to which all functions of ffiP,b=dPP s=PP,b\R2n again. Vh0,P,6 extend continuously. We set is md-elliptic if and only if Proposition 3.1. A symbol aE m,P,6 (the continuous extension to 2P,s of) the function b(x,l;) =
m, (x) mZ
00,P,S does never vanish on the set fP,6
The proof follows from (1.1) and prop.1.4, and md-ellipticity
88
2. Elliptic operators and parametrices
of (gy)m' (x)m'=
Vhm,p,b, using ospjst and Sao
.
Remark: A result of III,1 below implies that OpiihO'p's, for bsp,
is an adjoint invariant subalgebra of L(L2(&n)). The results on A sketched above, may be also derived for the norm closure Ap'o of OpPh0,p b, assuming pj>O, 0<6
Following the above lead, a symbol aEm,p,b will be called md-elliptic in a set MC 2p'b if (the continuous extension to &p,b) of is #0 on M. We reformulate this definition, avoiding reference to values of b: Definition 3.2.
Vhm,p,b is called md-elliptic in MC ffip'b if there exists a neighbourhood N of M in Bp,b with (1.1) in 2n for every KCC N. A symbol aE 0m,p,b is called formally mdKfl it hypoelliptic in M if (i),(ii),(iii) of sec.2 are valid in Kfl I2n
KCC N, with N as above. Again we call m' the inverse order of a The first part of def.3.2 is equivalent to our first definition of local md-ellipticity, because a continuous function is 340
at a point p if and only if it is bounded away from 0 near p. Def.3.2 still contains a reference to MC ffip'bC Bp'b. Also, Bp,b is of Stone-Cech type - its subsets are too general for our purpose. It seems practical to get restricted to a special kind of subsets M : The functions s(x), Ph0 generate a subalgep,b bra B of A for p sl, ba0. The maximal ideal space of B is a b, compactification of &2n again- it equals BnxBn, with the directional compactification Bn of Rn (cf. sec.5 or IV,1, below). Bn consists of &n and the ' infinite' set { oac° : x° E Rn x° I =1 } . The function s=(sl...sn)(x) provides a homeomorphism I B1={xE Rn,IxI<1}. 3n is characterized by the property that s(x) extends to a homeomorphism B"H Bi={jxjs1} The inclusion is an isomorphism, .
its dual is a surjective map t:B]BnxBn . Henceforth we will apply def.3.2 only for sets MC ffip'b of the form M=L_1M' with M'E a(BnxEn) Then we may choose the neigh.
bourhood N of def.3.2 in the form N=L-1N' where N' is a neighbourhood of M' in BnxBn. ffir,d does not enter the estimates of def.3.2. Thus we refer to M', N' only, i.e., in def 3.2 replace ffip,b by
M°=a(BnxBn), changing notation from M',N' to M,N. An aE h 0,p,6 needs not to extend to C(I0), but for symbols of differential operators and their parametrices we have that property in W below:
I° = a (BnxBn) = IQnxaBn U aBnxaBn U aBnxln the first term at right denoting WC ffi°
xE ien,
E aBn}=w. The set
will be called the wave front space, in view of an applica-
89
2.3. Local md-ellipticity
tion in sec.5. If M is a subset of W then we will drop 'md-' and refer to a symbol (formally hypo-) elliptic in M. For M of the form M=XxaBn, with XC 2n the operator A=a(x,D) and its symbol a are called (formally hypo-) elliptic in %, in agreement with conventions for differential operators. For more general MC W another notation is in common use: Such sets are called non-characteristic sets of the operator A and symbol a (cf. [T12], [Tr1], [Hr3]). The union Xp BnxaBn of the first two terms in (3.1) and Ms= asnxien, the third term, are called the principal and secondary symbol space, resp. For MC ffis we speak of m-(hypo-)ellipticity. In [CP] a symbol aE Vhm p,sC Vh (formally) md-(hypo-) elliptic over Htp BnxaBn was called (formally) d-(hypo-) elliptic. "md-(hypo-)
elliptic over Mp "just amounts to the conditions of (formal hypo-) aol ellipticity in some instead of a set IxI+IIaq. If a symbol is md-(hypo-) elliptic over M C M°=a(BnxBn), for ,
an open set M, then one may expect a local parametrix for the corresponding lpdo A=a(x,D). We mainly will consider the case MC W, noting that the case MC ffis may be treated similarly, reversing x, Instead of a local parametrix as in [Hr:] we construct a global i,do inverting mod C00in some set M of (formal hypo) ellipticity. Let be a symbol in Vhv,p,sC ph, for some vE &2. A sym.
bol aE 0 m,p,6 will be called formally md-hypo-elliptic with res pect to q if for xI+I xj some fl>0, we have (i) ay60 in supp cp ,
(iii) the restriction of a(R)/a to the set extends to a symbol in Vh m R ,P,S with ma'P of (2.2). If m'=-m we will call a md-elliptic with res(ii) TE 1hv+m',p,6 supp T
'
,
,
pet to T. Clearly a (formally) md-(hypo-) elliptic symbol is just a (formally) md-(hypo-)elliptic symbol with respect to T=1. be (formally) md6 (hypo-)elliptic with respect to T. Then we have ejE hv-m',p,S j=1,2, where m'=-m in the elliptic case such that A=a(x,D), Ej= Theorem 3.3. Let Tr= lphv P16
ej(x,D), (D1=cp(x,D)=y(M1,D),
(3.2)
AE1 = (D1 + K1
and let aE lPhm,p
,
(D.
=cp(Mr,D)
2
satisfy the relations
E2A = 02 + K2
,
,
K1,K2 E Op'tih-oo
.
Proof. 'Elliptic' just means m'=-m, hence needs no special consideration. We look at E1 only, taking adjoints for E2. With E2=B we once more go through the construction of the right parametrix of by (ii). Write thm.1.6 (or 2.5). Set b0=a, getting b0E v+m',p,S' (3.3)
AB0=(1+c0(x,D)
,
c0= J
J11
2. Elliptic operators and parametrices
90
where the asymptotic sum c0E hm+v+m'-p+6e',p,6' The next correction will be b1=-c0/a. Dividing (3.3) by a we get bl as an asymptotic sum of terms in the span of (a(0)/a)(a)(9)
(3.4)
It follows that bl E 0v+m'-p+Se' (3.5)
AB1=-c0(x,D)+C1(x,D)
,
101 Z1
I
and that
c1=-
j(-i)101/0!a(0)(c0/a)(9)
This then leads into the recursion bN =-cN-1/a (3.6)
BN = bN(x,D)
ABN =-cN-1(x,D)+cN(x,D) , CN 1.f-1(-0101/0!a(0)bN(0)
N=1,2,...
with
,
.
We show that, successively, bN, cN are of proper order
B. well defined, while the orders of C I
N
tend to
As instrument in this proof we use the lemma below. Proposition 3.4. Each symbol bN ,N=1,2,... of terms in the span of the expressions (3.7)
is an asymptotic sum
nj=!(x((33/a) : LjOal=L40PJ=101
where 101 a N , for bN
E
,
.
Clearly prop.3.4 yields bNE 0v+m'-N(p-Se')' hence cN abN+l v+m+m'-N(p-Se') has orders tending to -oo. The asymptotic sum 0bN b E E O v+m',p,6 is well defined and gives the desired e1=b.
Thus thm.3.3 is reduced to prop.3.4. We prove prop.3.4 by induction: It is true for N=1. (3.6) gives (3.8)
bN+l = -
j(-i)101/9!(x(0)/a)bN(e)
.
If bN is as in prop.3.4 we may reorder the double series by I, lemma 6.6, and get an asymptotic sum of terms in the span of (3.9)
(a(e,)/a)bN(e')
19112:1
where bN runs through all terms (3.7) is of the form (3.7)
,
for 101=N. Each term (3.9)
q.e.d. for 191Z N+1 We now apply thm.3.3 to a locally (hypo-)elliptic symbol: ,
,
Proposition 3.5. Let the symbol aE yMIP' 6C h be (formally hypo-)
elliptic in an open set MC W, and let TE yihv, p , 6 , T=_0 for (x,) O_ K with KCC NC BnxEn
,
as in def.3.2. Then the symbol a is (formally)
91
2.4. Formally hypo-elliptic expressions
md-(hypo-) elliptic with respect to q (so that thm.3.3 applies). Also apply thm.3.3 to the (formally) d-(hypo-) elliptic case:
Proposition 3.6. If aE 0m,P,b is (formally) d-(hypo-)elliptic, then it also is (formally) md-(hypo-)elliptic with respect to any n function where xE C (& ), x=1 for sufficiently large x=o in a spere ICI s rl, for sufficiently large i1>O. The proof for both prop's is immediate. It already was seen
that (formal) d-(hypo-)ellipticity amounts to (formal) md-(hypo-)
ellipticity for large IxI only, not for Problems. 1) Discuss md-ellipticity on a cylinder in the setting of 1,6, pbm's 1-4. Require the operator family C
00
(S') invertible for
i xA I + I ° I
L2(S' ), that
')II=
large; also, in the norm (III of MI-pt IaI+olII), a,(3 II
O((x^)mz-Pz
E IIn, as in I,(3.2)-(3.3). Discuss construction of a K-parametrix
B with 1-AB, 1-BAE4V (!fit as in 11,0, but with L(L2 (S' ))-valued kernels). 2) Show that the abovetf(with operator-valued kernel) contains noncompact operators of L2(S'x&n-1), but that KE Xwith kernel in K(L2(S')) is compact. 3) For mutual K-parametrices A,B, as in pbm.1, a K(L2(&n-1))-valued symbol may be defined for each, AB= I+K and BA=i+Kz, K.E. If and only if these symbols are invertible the operators A and B are Fredholm operators of 4) The generators [Cz ],VIII,(2.2) of the C *-algebra C there -i.e., L2(S'x&n-1)
on our polycylinder C=S'xRn-1, the multiplications by a(x)E COO(e) (2n-periodic in xi), and the operators A , ax A , axzA, ..., A = (1-A)-1/2 (on C) all are pdo's as in pbm.1 (verify!). Discuss the relation of pbm.3 above with [Cz],VIII,thm.2.6. 4. Formally hypo-elliptic differential expressions. Let a be a domain of &n, and let L = a(x,D) =
(4.1)
a (x)D'
IaTsN
a
be a differential expression with Co coefficients shall call L (or its symbol
:
(i) a(x,l;) (ii)
(iii) a(() /a
30 0 ,
as x E K ,
ICI Z 11
as x E K ,
for
i, p, such that
every compact set K C c there exist constants Mi,
(4.2)
E C'(S2). We
a a )formally hypo-elliptic if
,
ICI a 'n
,
as x E K , ICI a 1
.
(ii) is superfluous if L is locally of order >0 - i.e., L is
2. Elliptic operators and parametrices
92
of order Nx near x, with 0
, by (iii).
This yields (ii) locally, with M,=-Nxp, because aa(x) is bounded away from zero, locally, for suitable a. This also shows that M,= -p Min Nx s-p usually may be assumed negative For a constant coefficient L=a(D) of order >0 formal hypoel:
lipticity is equivalent to hypo-ellipticity in the sense of the definition in ch.0, sec.5. This was shown by Hoermander [Hr1],p.101. On the other hand there exist hypo-elliptic expressions with variable coefficients which are not formally hypo-elliptic. Here we use our parametrices of sec.3 for the following: Theorem 4.1. A formally hypo-elliptic differential expression is of (ii) can be chosen <0 (as true for L of locally positive order) then L-X is hypoelliptic for all XE M Proof. The second statement follows from the first, because for M,<0 we get hypo-elliptic. If M,
.
liml
uniformly on KCC
l
a
for
,
the theorem is best looked at in the light
of invariance of the wave front set of sec.5, below. We actually will use thm.5.4, below. Assume uE D'(cz'), fE C00(SZ'), for some
open Sz' C S2, and let Lu=f in Sz' . We must show that uE COO(Sc') . Fix a point x0E Sz', and balls B(x0,6)
,
B(x0,26) with center x0, radius
6, 26, closures C Sz'. Let TE Co, T=1 near x0, supp T C B(x0,6)
Define a global extension
a as follows: Let
XE C0"O([0,25)), x=1 in [o,6], x=o in [2O,00), 0stx(t)s2O. Define (4.3)
(4.4)
and that
a(x0+(x-x0)x(Ix-x0I)
,
)
,
a0E PcN.O C Vh(N,0),(P,1),O
for x, E in
,
a0 is formally d-hypo-elliptic in the latter space, with
inverse order M=(M1,0). By prop.3.6 a0 is formally md-hypoelliptic with respect to a suitable X(U), x=0 near 0 x=1 near -. By thm. ,
3.3 construct EE OphM
,O
with Ea0(x,D)=x(D)+K, KE Opih-o,. Clearly
a=a0, as xE B(x° ,6), hence cpa = cpa0, for all
assuming the
left hand side zero whenever a is undefined. Accordingly
(4.5)
cpf = cpLu = Lcpu + [cp,L]u = L0((pu) + [cp,L0]'yru
with another cut-off p, supp V C B(x,26), 0
=1 in supp [cp,L0]
93
2.5. The wavefront set Note that yfE CO(&n)C S, while puE E'(&n)C S'. Thus guE HS, for
some s, and h=[q,LO]VuE Ht, for some t. Clearly, h=O near x°, 0 since cp=1 near x causing the commutator to vanish. We get x in sing supp Eh , using the complement of sing supp h . Hence x° thm.5.4, below. Applying the operator E to (4.5) we get ,
(4.6)
epu+ (1-x)(D)qu + Kcpu = E(pf - Eh .
Here EcpfE S, since cpfE S. Also KfuE S, since TuE S', and K has or-
der (--,-oo) . Also, w=1-xE CO(&n)C S, thus w(D)cpu f 9Tyw' (x-y) ((Pu) (y) E S, the distribution integral existing, due to a E S, puE S'. Also Eh is C" near x°, hence Tu is C" near x0. Since T=1 near x0
we find that also u is C" near x0 Since this construction may be done for every x0 E fz' the theorem is proven. We made extensive use of hypo-ellipticity in [C,],II,IV. Hypo-ellipticity of L-?. especially of L.-ti is required to make the Carleman alternative work, as well as in general the construction of e.s.a.-realizations in [C:],II. .
,
,
5. The wave front set and its invariance under.Vdo's. We define the wave front set WF(u) of a distribution uE D'(sz) over a domain cC &n as a subset of the wave front space W=OT(fa)
,
already discussed in sec.3, for the special case ft=&n. For a gene-
ral fcC &n we define 1(fz)=fzxaznC &nxa16n=0r(&n)
.
It is common practice to interpret this space as T=fzxSn, i.e.,
(5.1)
T = W(c) = {(x,i;)
:
x E sz
,
E &n
,
ICI = 1 }
.
Similarly, for distributions on a differentiable manifold a one defines LT(n) as the "co-sphere bundle", i.e. the bundle of unit spheres in the cotangent space, with respect to a suitable smooth of a Riemannian metric. It is clear then that the unit vector pair as t - - , with x E =-n just indicates the direction tx where the real point (x,«1) is to be found. In other words, for a manifold sz the wave front space is properly defined as the subset ,
W of the symbol space $1(R)
,
as defined in [C2] ,VI, p.161.
One defines WF(u) as a subset of W by its complement WF(u)c= W\WF(u), as follows. Specify two types of cut-off functions called cp(x), and respectively. For x0E sz a cut-off cp is a C func-
tion, cpa0, y=1 near x°. For a "direction" o0E a>sn given by °E &n with 1°1=1 a cut-off
is a C(Zn) -function, "0, x=1 near
x=0 outside some neighbourhood of °, and, 'Wl&nE C"(&n). Also, x
94
2. Elliptic operators and parametrices
One equals some homogeneous function of degree 0 for large where x=0 near 0,
may think of a C(I)-function
=1 near oo
,
µ=1 near l;°
s
We define: (x0, 0)E WF(u)c (i.e., is not in WF(u)) , ,p , such that, with the Fourier transform 11^61
if there exist cp
,
ES
(5.2)
zero outside S2, a Note that Tu admits a natural extension to I distribution in S' hence has a Fourier transform in S'. ,
,
WF(u)c if and only if there exist resp., such that the ipdo C=1p(D)g(x)
Proposition 5.1. We have cut-offs cp,1p, near x and E OpipcO'-OO takes u into S
ip(D)cp(x)u E S
.
The proof is evident,since C=F-11pFg, and because the inverse
Fourier transform F-1 is an isomorphism of S onto itself. From the definitions of WF(u) and s.s.u we conclude:
Proposition 5.2. The singular support s.s.u of uE DI(n) is the projection of WF(u) to the first component Sz of W=S2xa18n s.s.u = {x: there exists F; such that (x,l;)E WF(u)}
(5.3)
.
Indeed, if xE (s.s.u)c, then a cut-off cp can be found with cpuE C0C S, so that (Tu) E S, -i.e. every WF(u)c. Vice versa if for some x every Wf(u)c, then (cu)^E S (i.e., guE S) fol-
lows for suitable T: The unit sphere is compact, and for 11;1=1 we
can find ((p"u)^ip"E S. A finite collection 1pj=,pj has supports cove-
ring the sphere, so that 1pj=vj/yvj is a partition of unity. Let be a cut-off near x with cpp
=cp
(P_
for all j (i.e. T 0 =1 on supp
(p- ), then ((p-u)^1pj E S, by prop. 5.3. Hence (cp u) (Jipj)E S. But it is possible to arrange for w--Y, y,jal outside a large sphere, and
have (o homogeneous of degree zero there. Adding a suitable C,(iQn)-
functions x we get w+xa1 in &n, and ((x+x)-1E T, the space of Coo(len)-functions with derivatives of polynomial increase. Thus (cp u)^ w+ E S, hence (y- u)^ E=- S, since 1-w/(w+x)=x/(x+w)E Co, and Coo
near x, or xE (s.s.u)c, q.e.d. C. Hence cp uE S, uE We have used the proposition below, also useful later on:
((p u)^E
95
2.5. The wavefront set Proposition 5.3. If we have ((pu)^tpE S , with a pair of cut-offs
q, 1p and a distribution u, and if qi cp=cp, for a cut-off qi , then we also get (qf u)^ V E S . Proof. We know that ((pu)^,p E S is equivalent to p(D)q>(x)u E S
and similarly for qi instead of T (5.4)
.
Write
'cp(D)cp u = ip(D)cp(x)gi u = qi ip(D)p(x)u + [1p(D),gi (x)]pu ,
where the first term,at right, is in S For the second term we use the asymptotic expansion of the calculus of ydo's : We get .
(5.5) p[W (x),cV(D)]=p(x,D), p=-
(0)(x) (mod S)
X01i9lp(0)
>0
Each term of p is =0, hence p(x,D)E O(-oo). The last term of (5.4)
also is in S, due to p(x,D)*=-[1p(D),gi (x)]cp(x)E O(-oo). Theorem 5.4. Let u E ES' in some 1phm,p,s
,
and let A = a(x,D) be a ido with symbol
, with the usual inequalities 0sO
(5.6)
Q.E.D.
and s.s.Au C s.s.u .
,
Proof. We are using As and O(s) of ch.III, here,
for convenience.
It is sufficient to prove the first inclusion (5.6), by prop.5.2.
We must show that
WF(u)c implies
WF(Au)c. For (x,t)
E WF(u)c we have 1p(D)gp(x)uE S with suitable cp,ip. It is found that
(5.7)
p(D)cp(x)Au = AV(D)cp(x)u + [C,A]u , C = 1p(D)cp(x)
,
where the first term at right is in S, since A takes S to S. uE S' is in some Es, sE 22, since S'=Uts. [C,A] is of lower differentiation order than A (and of multiplication order -o). In order to get the desired property for Au , we will iterate the proceedure as follows. Construct sequences of cut-off functions
such that q
(5.8)
)
Cju E S
,
j = 0,1, ...
, where Cj = 1pj(D)gpj(x)
.
Proposition 5.5. We have C0C =CC0 =C0 (mod O(-oo)), CjCj+1= Cj+1Cj= Cj+1 (mod O(-0°)).
(5.9)
Proof. We have CCO = 1p(D)cp(x)tpO(D)gO(x) = ip0(D)cp0(x) + +
1p(D)[cp(x),VO(D)]q)O(x)
Hence it is enough to show that But we have
the last term, called V for a moment, is in O(-oo)
.
96
2. Elliptic operators and parametrices
(5.10) V*=z(x,D)
S),
,
where again all terms of the asymptotic sum vanish identically Accordingly V* E O(-oo) , hence V E O(--) . This shows that CC0= C (mod O(-oo)) On the other hand, Similarly for CiCj+1 .
.
C0C = V0(D)T0(x)V(D)T(x) = 140(D)cp0(x) + VO(D)[TO(x),V(D)]T(x)
Let W be the last term. Consider the asymptotic expansion of W* W
=p(x,D)
,
(mod S)
le
which again implies W* E O(-oo) since all terms vanish, q.e.d.
Using prop.5.5, we may write CNAu = CN-1CNAu (mod S) = CN-1ACNu + CN-1[CN,A]u (mod S)
(5.11)
=CN-2CN-I[CN,A]u (mod S) = CN-2[CN-1'[CN,A]]u (mod S)
_ ... = [C,[C0,[C1..... [CN-1,[CN,A]]] ...]]u (mod S)
.
Here the N+2-fold commutator has multiplication (differentiation) order -- (ml-(N+2)(pl-O)) Accordingly the right hand vector of .
(5.11) is in HM'c., for every M, as N gets large. Thus CNAuE HM'co,
as N>NO(M). The q)k,Vk may be constructed such that
(Pk=1 in a fixed neighbourhood N of x (and similarly for Vk and a neighbourhood N' of ) . Thus we can find cut-offs cp" , 1p" with cp" qpk=f" "Vk=V" , for
all k . Using prop.5.5 we thus get (5.12)
C Au = C"CNAu E HM',, ,
for every M
C Au E S = fl Hs
This completes the proof of.thm.5.4. Corollary 5.6.
Let AE Opyih be formally md-hypo-elliptic with res-
pect to a symbol q' (5.13)
Morever, if q) (5.14)
.
Let uE S', and let Au=f, and cp(x,D). Then
WF(f) C WF(u)
,
and WF(4i) C WF(f)
.
(i.e., if A is formally md-hypo-elliptic),then WF(u) = WF(f)
.
The proof is an evident consequence of thm.5.4 and thm.3.3.
2.6. Systems of operators
97
6. Systems of ipdo's.
It is often convenient to consider a vxµ=matrix (6.1)
A = ((Ajl))j=1,..,v,l=1,..,FL
,
Ajl= ajl(M1,MrD)
of pdo's where we generally assume that aj1 E SSm'P'b C SS, m,p,b independent of j,l. Introduce a matrix-valued symbol and write aE SSm'P'b, and A=a(M1,Mr,D). In the present section, we will use all symbol class notations, such as ST, yah, hm,p,d also for vxµ-matrices to indicate that the entries are in the corresponding symbol class. The operator A of (6.1) will act on spaces of µ-vector-valued functions or distributions. All formulas of the pdo-calculus of ch.I remain valid, except those involving a commutator, such as I,(7.5). For md-ellipticity we focus on square (µxµ-) matrices of Idols We call A=a(x,D)E Opyahm,p,b (and its symbol a) md-elliptic if
E0 11m,P,6
(6.2)
is md-elliptic. A symbol aE SS (and its operator A) will be called
md-elliptic if
is md-elliptic.
Lemma 6.1. A square-matrix-valued symbol aE 1h is md-elliptic m,p,b if and only if for with some T>0 the matrix ,
a square-matrix-valued symbol bE
Vh-m
p,b
such that
as
Trivially the Hilbert space adjoint of an elliptic pdo and the product of 2 elliptic pdo's are elliptic again. We can expect a theorem like thm.1.6 for square matrices, and even a more general result for rectangular matrices. For the latter let us define as follows: If for a pair of (vxµ- and µxv-) symbols aE hm,p,b , bE Vh-m p,b ,resp., we have (for some 1>0) (6.3)
1 = ((6j1))j11=1,.,v' for all
then we will call a right md-elliptic and b left md-elliptic
rj
.
We then must have v:5µ. a is elliptic if and only if it is left and
right elliptic. b is then called a right inverse symbol of a, etc. be (right) (left) and-elTheorem 6.2. Let a vxµ-symbol aE m,p,b liptic, and let bE yh-m,p,b be a (right) (left) md-inverse symbol. There exists a (right) (left) K-parametrix E=e(x,D), eE 4h-m,p,6'
of the pdo A=a(x,D) such that
(6.4)
E'U (-m.-p.+6,-m2 -p2 ),p,6 '
2. Elliptic operators and parametrices
98
Vice versa, if a (right) (left) parametrix of A of order -m exists then A (and a) will be (right) (left) md-elliptic. We prove thm.6.2 like thm.1.6, constructing a right or left K-parametrix. If both exist they coincide mod O(-o3), as usual.
Many 'hypo-elliptic' arguments work as in the scalar case. For matrix-valued symbols there is another interesting case of constant rank for large to consider: Assume the matrix and that, moreover, aE yahmp,, allows a decomposition a = a0+a1
(6.5)
,
a1E 0m',p16 , m'<m ,
where a0 allows a 'partially md-inverse symbol' b0E 11h-,,,,b
the sense that
,
in
is a gxv-matrix with for large
(6.6)
where
is a vxv-projection matrix of the same rank as and
i.e.,
Clearly (6.6) and rank p0=rank a0 imply im p0 projects onto im a0, hence 1-p0 vanishes on im a0, i.e, (1-p0)a0=0. Introduce Then get g0-q02= also is a b0a0-b0a0b0a0=b0a0-bOp0a0=b 0(1-p0)a0=0, so that projection. Evidently rank g0(x,t)=rank p0(x,t), im p0 = im b0 Generally, if b=b0+b1, where b1E yh-m',p,S, with -m'<-m , we will call the symbols a and b partially md-inverse to each other. .
Theorem 6.3. Assume that aE and bE Vh-m,p,S are partially md-inverse symbols of each other (where p>0 pl>S). Let p=ab , q=ba Then and let A=a(x,D) B=b(x,D) P=p(x,D) Q=q(x,D) ,
,
(6.7)
,
AB - P = r(x,D)
,
,
.
,
BA - P = s(x,D)
,
where the symbols are vxv-matrix-valued and µ.xµfor some e>0 matrix-valued, respectively, and are in `Vh_Ee,p,S
,
The proof is calculus of ,do's. We cannot expect better para metrices making r or s of order -- , with improved B, etc. Matrix-symbols also occur in VI,IX,X, under different aspects
Chapter 3. L2-SOBOLEV THEORY AND APPLICATIONS.
0. Introduction. In this chapter we consider ipdo's as linear operators of
a class of weighted L2-Sobolev spaces over in . We specialize on L2-spaces and neglect LP-theory, because 'ydo's in general are
not continuous operators on LP-Sobolev spaces, for p#2 . To be more precise, general L2-boundedness theorems for xdo's are true for A=a(x,D)E OpiPh0,p,6
,
assuming pa0, 0s6!-.pi, 6341, but corres-
ponding LP-boundedness statements are false, except for p'=1. There is an extensive theory in LP-spaces of Sobolev and other
types (cf. Beals [B
Coifman-Meyer [ CM] , Marshall [ Mri ] , Muramatu [ Mml] , Nagase [Ng, ], Yamazaki [Ymi ]). 41,
In sec's 1, 2 we prove the L2-boundedness theorem, for 6=0, respectively. This result often is quoted as CalderonVaillancourt theorem. In sec.3 we look at weighted L2-Sobolev and 0<6s1
,
norms. Our class of spaces Rs=HS1,s: is left invariant by the Fourier transform, just as many of our ydo-classes. A y,do of order m=(mi,m2) is a bounded map Hsa Hs-m , for every s. For every m E 12 an order class O(m) is introduced - the operators S-S extending to operators in L(H ,H ) 0(m) is a Frechet space s s-m for all s under the norms of L(Hs Hs-m) ; O(0) and O(-)=U O(m) are algebras. A pdo of order m belongs to O(m). .
A refined Fredholm theory holds for (formally) md-(hypo-) elliptic Ado's. Such an operator admits a Green inverse- the equivalent of the integral operator of the generalized Green's function. This is discussed in sec.4. In sec.5 we prove that ado's of negative order (m,<0, mz<0) are compact operators Hs-CHs, for all s. 1. L2-boundedness of zero-order y,do's.
We refer to the class Vt0=lh(0,0),(0,0),0
of symbols here,
as introduced in I,(4.26), and discuss the following result. Theorem 1.1. An operator A=a(M1,D) E Oppt0 is bounded in H=L1(1n). and belongs to L(H). More precisely: The restriction AJH maps
99
100
3. L2-Sobolev theory
This result is quoted as the Calderon-Vaillancourt theorem (c.f.[CV1]). For independent proofs c.f. [CC], and VIII,thm.2.2, below. The very short approach below is due to Beals [B3]. Proof. Construct a partition of unity for &n, as follows. With the cube Qr {Ixjl0 in Q312. Define ncpa)-1/2.
cpa(x)=cp(x-a), aE 2n, and Wa a(
Clearly then
(3E2
(1.1)
Va E CD({xE &n:lx-aI<2,j=1,...,n})
a1Va2(x)=1
,
Va s 0
Given some symbol a E iVt0 we define (1.2)
Aa = aa(x,D)
We first show that AaE L(H). Since the Fourier transform is unitary we may show Ba FAaF1E L(H). But for aE ipt0, supp aC {Ixlsp}
FAF-lu(11) = J41Xe ixr1JR1
(1.3) = fol
u(U)J41x
Joie as
-T1.S) u(S)
(
with "I" with respect to x written as "u", and trivial integral interchanges. The integral operator at right of (1.3) is L2-bounded by Schur's lemma We get the operator bound
(1.4)
f
,Jdtla``(t-q,t)I
sup
:
,T1 Elen}
In particular, (
-''1) 2Nav( -l1,
)
= J
x a(x, ) (l-Ax)Neix(
-Tl)
=J1xaN(x, )eix( -q) , with aN = (1-Ax)Na
which implies
D((-rl)-2N IIaNIi
(1.6) hence
(1.7)
IIAIIL2
s
c IIaNII
,
2N > n ,
where c depends only on N , n and p . We need an improved estimate for the operators (1.8)
Aap = Aa'Wl3(M) = aap (M1,Mr,D)
aap=Va(x)WR(Y)a(x,)
101
3.1. L2-boundedness of zero-order operators
Taking the Fourier transformed operator A^ for A=a(M1,Mr,D) again, we get with a(x,y,l;) vanishing for (x,y) outside KCC &2n ,
fl-ixil
(1.9) A^u=FAF-lu(T1)=
JFTKely'{u(x)
Integrals interchange readily, as uE S. Therefore,
(1.10)
A^ U(T1) =
f91K U(K) fo1 alk
With the methods used to derive (1.6) one finds that = IIaNNII Loo
aNN = (1-Ax)(1-Ay)Na Note that (1.11) implies
f
(1.12)
0((K_,,) -2N)
IIaNNII LOO
Here it is important to observe that the "O(.)-constant" depends only on n,N, and the volume of the compact set K, the latter since an L1-norm has to be estimated by the corresponding Lc norm. Also,
-2N(-K) -2N= f
-2N, and f ((t) (2s-t) )-2Ndt
-(T1-x))
ItIfIsl + ItIJIsl = I1+I2
,
where 1
c(s)-2N,
c=f(t)-2Ndt
2
For the integral I1 the inequality implies 12s-tI22Isl-ItI2lsI so that the integrand is bounded by (s)-2N(t)-2N
,
i.e., we get
the same estimate as for 1 This shows that indeed (1.12) holds. 2 Next we transform the operator as follows. .
,
Aa(3
Aapu(x) = f91 JR1Y((1-A
)
M
eii;(x-Y) u(Y)
(1.13)
=amap(M1,Mr,D)u(x)
,
)M
am CEP
where the partial integration is justified as in sec.l.
Proposition 1.2. We have (1.14)
supp aap contained in the set Ix-als2n
,
IY-PI12n}
Moreover,
(1.15)
Ilaa(3NNIILo 5 c(a-R)-2M max
:Ia152M, ItI52N}
3. L2-Sobolev theory
102
where the constant c depends only on n, M. and N Indeed, (1.14) is trivial, by construction of the Va , while (1.15) is a calculation, using that (x)(K)/(x)=0(1) for all x .
Combining (1.12) and (1.15) we get
(1.16)
2sc
IlAapll L
(a_I)-2M[ [a]]2N,2M '
2N > n ,
with [[a]]k,l = max {Ila(a)IIL :
(1.17)
Itlsk
,
lalsl}
where c depends only on n,N,M. Choose the set K as product of the balls with center a,(3, radius 2n, its volume independent of a,(3.
For u,vE CO(Rn), ua eau, va=1av, 2M>n, we get
(1.18)
l (u,Av) l=
a S(ua,AaRvt) I S cl IIualIlIvnII(a-R)-2M a
I
s c2{ with
Iluall2 ;Ilvpll2} 1/2 = c2llull llvll
inner product (.,.) of H, where cj=cj[[a]]2M,?3N
j=1,2, with cj' depending only on n,M,N. In (1.18) we used (lull =
XIIuaII2, by construction of Va, and Schur's lemma in discrete form for 12-boundedness of (((a-(3)-2M)), 2M>n. Clearly (1.18) implies thm.1.1: Set u=ukE CO(Rn) where uk-> AvE S in H.
Moreover, we also have proven the useful corollary, below. Corollary 1.3. There exists c>0 only depending on n such that (1.19)
IIa(x,D)IIL2(,n) s c[[a]]NN, N=[n/2]+1 with [[.]] of (1.17).
Revisiting our above proof we notice that the same arguments apply for an A=a(M1,Mr,D) with aE SS00 (i.e., m=0, p=0, 6=0). 00 0 The Aa, are of this form anyway. Instead of (1.16) (1.17) we get '
,
(1.20)
2sc
IIAaPII
(a-p)-2M
L
[[[a]]] 2N,2N,2M
with (1.21)
[[[a]]]j,k,1 = max
:IaIsj,ItIsk,IkIsl}
.
LOO
Thus we have Corollary 1.4. All operators A=a(M1,Mr,D)
with a E SS
L2(len)-bounded. Moreover, we have
(1.22)
IIAIIL2 s c[[[a]]]N,N,N
'
N = [n/2]+1
.
0,0,0
are
103
3.2. L2-boundedness for positive delta
2. L2-boundedness for the case of O>0
.
Note that thm 1.1 gives L2-boundedness of all do's in the Hoermander classes 00.p,0 , with pj z 0 , but that it does not apply to the case S >0 . The theorem below asserts L2-boundedness for general m=0, and pjz0, plab, OsS<1. It was first proven in [CV Other proofs may be found in [ Ka2] , [ CM] . We discuss a proof after a scheme in [B3] which again is relatively short. 21.
Theorem 2.1. A pdo A=a(x,D) with symbol a Eiih06e',6 induces a bounded operator of H=L2(&n)
'
0<6<1
.
Proof. We use cor.1.4 and the following decomposition. For a partition wj, j=0,±1,±2,..., like Va but for n=1, i.e., (2.1)
wj(t) = w(t-j)
,
supp w C {It Is2}
,
yjwj2 = 1
,
wja 0
set xj=wj(log
i.e, Clearly xj has support near 2j. Since we get xj=0, as j<-1. The partition of unity Exj2()=1 has supports in concentric spherical shells:
supp xj C {2(3-2)/b s q) s 2(3+2)/s}
(2.2)
.
For A=a(x,D) with a satisfying our assumptions, we write
(2.3) (u,Av)=lj1(ujAjlv1). Aj1=xj(D)Axl(D). uj=xj(D)u, vj=Xj(D)v. We apply Schur's lemma once more to show that the matrix ((2-EIj-1I)) is 12-bounded, and prove an estimate of the form (2.4)
IIAj1IIL2
s c2 EIj-lI
,
j,l E Z
,
for some E>0
.
Then the argument following (1.18) will give boundedness of A We prove (2.3) for Ajl"=FAj1F-1 instead of Aj1 again. Here Get pj1=O unless Ajl"=Pjl=pji(M1,MrD) with 27-2s(x)6s2j+2
(2.5)
,
21-2s(Y)O 21+2
.
Repeating once more the partial integration of (1.13) we may write Pjl=pjl(Ml.Mr,D)
(2.6)
.
pj1=(x-Y)-2M((l-O )M pjl)(x.Y.
)
where e(x,Y)=(Y)26/(x-Y)2
_ and bM(x,
) = q)-21 (1-Ox)Ma(x,U)
3. L2-Sobolev theory
104
Notice that bME 00 ,Se',b again, for every M=O,1.....
.
Proposition 2.2. We have IIbM(-D,Mr)xi(Mr)II s c , with c independent of 1 for each fixed M=0,1,2..... Applying the Fourier transform we may prove IIbM(x.D)xl(D)II so that s c instead. Note also that O,6e',6 ,
.
E
E 00'6e'
.b
,
.
Proposition 2.3. The family gl(x,T)=c1(2-1x,210 is bounded in Vt0 Indeed,
21(lal-IRI)ci(R)(2-1x.210 g1((3)(x, )=
(2.7)
= 2l(lal-1pl)0((210 s(IPI-lal))
with 0(.) independent of 1. Also, x1(21 )=0 except if (210 6 differs from 21 by a factor between 1/4 and 4. Since qi contains the factor x1(21 ) we conclude from (2.7) that ql(Q)=O(1), with 0(.) independent of Thus cor.1.3 gives the uniform L2-bound for gl(x,D)=Q1 Note that Q1=VibM(x,D)xi(D)Vl-1, with unitary V1:u(x) .
2-In/2u(2-lx), so that IIQIII=IIbM(x.D)xl(D)II, proving prop.2.2.
Applying prop.2.2 for M=0 get IIAj1IIsc, c independent of j,l.
For (2.4) it suffices to look at Ij-lIZk0 with suitable fixed k0. Proposition 2.4. For Ij-llak0=5S (2.8)
e(x,y) s c 2 EI3-lI
,
,
and s=
6(1-6) we have
as (x,y) belong to the set (2.5)
with c independent of j,l. For the proof we use the estimate (2.9)
(x-Y) z I (x) - (Y) I
easily derived by an elementary calculation. We get (2.10)
O(x.Y) s (Y)2S/I(x)-(Y) I2= v. 41/Ivz 23l-'U, 21r'
I2,
with vj satisfying 4svjs4 and rj=1/6>1. For 2iI3-1IZ32, i.e., li-li a5S=k0 the right hand side of (2.10) is bounded by c41/4TIjsc41(1-3)=c4-hllj-lI, as j>l, and by
c4l/41l=C4-(r1-1)lI7-ll,
as j
latter symbol defines a uniformly bounded operator family, while,
3.2. L2-boundedness for positive delta
105
in supp k , we have (2.8), hence it seems that (2.4) should follow. For a real proof however we argue as follows: First show that
(2.11)
a as yO/O s c k+j=1R1
( Y) - 3 (x-Y) -k- I a l
Indeed, for IaI+It1=1, writing aaayv=j
for a moment, we get a(3= (log O)ias=26 (log( y) )iap-2 (log(x-y) )kR in the span of P and -SSlap
with T=( y) , S=(x-y) . For any " , t E,P+rj + (p4ap )(R
ment gives
,
and I E + I rl =1 we get (
in the span of rlK(a3,p3) im(y),63), where K(a,3)=
Sap, L(a,f3)=TaP, with product over all partitions a=jai+
31+76 3
t3 )ts 11 =
where we may set R=S,T,O. An induction argu-
, with multiindices
aj, (3 j , y j ,
,3
Finally note that aaT=O
as a#0. Thus we may assume ly3 =0, 2Ia3I=IaI. Let YIP3I=k, Y1631=1
and note that (x) (01') /(x) =0 ((x) - I a I) , as used earlier. Hence the above products are O(S-IaI-k T-1), where k+l=I(3I, implying (2.11).
Now we again recall the above unitary map V1. Look at (2.6), rewritten as
for M=1 , describing a symbol of Pjl (2.12)
,
p 1(x,y, ) =
The operator Wj1=V1-1Pj1V1 then has symbol (2.13)
From prop.2.3 we get describing a bounded set in Vt0. Notice that of (2.13) describes a bounded set E13-ll, with respect to the norm (1.22), while a[[[w.,]]]N,N,Nsc2 so that cor.1.4 gives (2.3). Indeed, axayaywjl is in the span of (2.14)
1U(v)
where -(x,y)=e(21x,21Y), wj(x)=X(21x),
, x+µ=a , X+v=P ,21x)
Note that aaaRE/E=21(Ia'1+1p1)(aaaI0/O)(21x,21y). For boundedness
in supp wjlxwe show that 21/(21y) yand 21(21x-21 y) are bounded, by (2.11). In supp wjl get 21/(21y)sc21/21/6sc. Also, 21/(21x-21y)s c21/12j/6-21/61, as near (2.10). With similar argument (and 1=1/6) sc21-13, as 1<j, so sc in both cases we get sc2l(1-r1) for 1>j, and In other words, the first factor of (2.14) is bounded. We already know the last factor bounded, since riE Vt0. For the second
3. L2-Sobolev theory
106
we get 6Xw=2IIL I1O((21x)-IRI)s121R1(1-hj), bounded for 1>j, but not for 1<j. The third factor is bounded by c2 61j-1I. Together we get the estimates for (2.4), via cor.1.4 , but only for 1>j .
For j
does not involve the support
restriction (21x)6, 2j, while prop.2.3 remains intact if that res-
triction is replaced by (21x)s21-5 a jxjs /4-5-4-1 . This means that we may replace the factor wj(x) in wjl by wj(x)0(x), using a cut-off z=0 for jxI2:2-6, =1 near 0. For j<
after (2.11) also remain intact with the new x-restriction. This completes the proof of thm.2.1. We also get Corollary 2.5. Under the assumptions of thm.2.1 there exists a constant c depending on n and S only such that for 0<6<1 we have
(2.15)
IPI)II
IIAllsc
with the L2-operator norm IIAII
IaIs2,I,Is[Z]+1}
:
L LOO
.
The proof is evident. Note that a result like cor.1.4 is easily derived for the case 6>0, following the above guide lines. 3. Weighted Sobolev spaces; K-parametrix and Green-inverse.
Define the spaces Hs (3.1)
Hs1 ,s2
'
for s=(si ,s2 ), as the classes
He = {uE S' : ns(x,D)u E H
where H = L2(&n)
,
and ns(x,) _ (x)sZs'E acs
.
For u,vE Hs we
introduce a norm and an inner product by
(3.2)
(u,v)s=(nsu,nsv)L2
,
Hulls=lInsulIiz
,
11$ = ns(x,D)
Note that the strictly classical 'pdo Its is invertible, as an
operator S -> S
(3.3)
,
or S'
s1= nS
-> S'
, with inverse
= n_s(Mr.,D) =
(D)-sl (M)-s2 E Op4c_s
In particular 11s is md-elliptic of order s
(3.4)
ns-1
,
and,
- r1s E OPVC-s-e
by calculus of ipdo's. From (3.2) conclude that II5:H5-*H is an isome
try Hs - H. In particular, Hs is a Hilbert space, for all sE It'
.
107
3.3. Weighted Sobolev spaces
We refer to the spaces Hs as (weighted L2-) Sobolev spaces over &n. In the special case s2=0 we obtain the ordinary (unweighted) L2-Sobolev spaces, commonly known as Hs - now with sE I First we discuss properties of unweighted Sobolev spaces, for the moment also called Hs sE & : For details (and a more general discussion involving L1'-norms) cf.[Ci],III, or [GT]. Write .
(3.5) As=(D) s , Hs={uE S' :AsuE H}
(u,v)s=(Asu,Asv)
,
hulls=IIAsull
,
1) Using Parseval's relation (0,(1.8)) we may write (3.6)
2s
Ilulls2 = JR1 Iu^
with the Fourier transform u^=FuE Lebesgue integral. In particular, He = {u E S'
(3.7)
where f exists as a
LIloc(&n),
L2(&n,(,)2sdU)}
: u^E
2) The spaces Hs form a decreasing set of subspaces of S' containing S, as s increases from -- to +- . Moreover, hulls is an increasing function of s, either on &, or on a half-line (--,a) 3) For s=k , a nonnegative integer, we get (3.8)
Hs = {uE S'
:u(a) E H
for all lalsk}
with distribution derivatives u(a). The norm hulls is equivalent to
llullk = {
(3.9)
l
This follows from (3.6)
(3.10)
llullk = Jdfslu^
a1sk
llu(a) ll2} 1/2
:
l2(1+7)2k
= l
(k)
ask a
afsk
[aJfd2alu" () 12
Ilu(a) ll2
with the multinomial coefficients (k) = k!/(a1(k-lal)1) .) 4) We have Sobolev's lemma For s > n/2 the space Hs is (compactly) imbedded into the space CO(&n) of all continuous functions over &n vanishing at infinity. We have the inequalities :
(3.11)
s csllulls
Hull
,
. )1/2 cs = ( J(S)-2sd
,
LOO
and (3.12) Iu(x)-u(Y)lsss(lx-Yl)Ilulls
,
6s=(2Jsin2(1t/2)(
)-2sd)1/2
3. L2-Sobolev theory
108
Moreover, if even s >n/2+j
for some integer j >0
,
then we
,
have a compact imbedding Hs-C03(&n) defined as the space of all uE CO such that also u(a)EE CO , for all lags j , and inequalities lalsj of the form (3.11), (3.12) are valid for all u(a) ,
.
,
Sobolev's inequality ( 3 . 1 1 ) follows by showing that the Fou-
is LI for u E Hs, s>n/2 (a matter of Schwarz' inequality). (3.12) is derived by estimating the Fourier transform rier transform u^
(cf.[C1],III ). The compactness of the imbedding is a consequence of the Ascoli theorem: Equicontinuity of Hs-bounded sets follows. The imbedding Hs-'CO., as s>n/2+j, follows from (5) below. 5) For u E Hs we have u(a)E Hs-lal
(3.13)
and
,
Ilu(a)Ils-IaI 5 llulls
This is a consequence of the inequality (3.14)
Ilu(a)lls - IaI=JIu^
JlU.
5 using that 6)The spaces
(3.15)
He = (1{Hs
:
sE &}
,
H-,, = U{HS
:
sE &}
,
have simple locally convex topologies (cf.[C1], ch.III). The space H. is contained in COOO(&n) , but contains S , while H-. contains E', but is contained in S'. All inclusions mentionned are proper. 7) Hs and H-s are mutually adjoint under the pairing (3.16)
(u,v) = (Asu,A-sv) = fuvdx
, uE Hs , vE H-s
defined by the second expression, involving the inner product of H=L2, using that As:H5-. H and A-s:H-s-a. H define isometries. Returning to weighted Sobolev spaces
H5--H
5I,sz
of (3.1) we
note analogously the following properties of Hs (i) We have
(3.17)
Hs = {uE S'
:
II cull
with the operator s*=(D) s' (M) 52 =ns(Mr,D)
.
:
II s*ull
,
Moreover, the norms
(3.2) and (3.18) below are equivalent. (3.18)
llulls = Il s*ull = II(D) S' (M)S' U11
ss*-I and s* s-1 beThis follows by calculus of ado's long to OpWcOC Op'pt0, They are L2-bounded, by thm.1.1, so that :
3.3. Weighted Sobolev spaces
csllulls s Ilulls s csllulls
(3.19) (ii)
(3.20)
109
We have Hs D Ht , as sjstj
,
j=1,2
,
although the norm Ilulls not necessarily increases, as sj increases. Moreover, the imbedding Ht - H. is compact, whenever i>sj
(3.20) follows from the estimate
(3.21)
Ilulls s Ilulls tz 5 1/C S. tz
s ct/cs. ,tz
Ilulls. tz
Ilullt .
using (3.2) and (3.19). or, Ilulls=ll sull=llLt(rc1ns)ull=llrc1 sullt
.
where Kst I11 E OpyPcs-t is a bounded (compact) operator of Ht, due to 17tKst1 '= $sr2 S-t( sltrlssl )= s-tLst, where LstE OpipcOC L(H), r,=( x) (D) E K(H), as rj<0 (cf. thm.5.1). (iii) For s1=ka0 (3.22)
,
k an integer, we have
Hs = {uE S' : u(a) E L2(&n,(x)2szdx),
1-15k}
and a norm equivalent to Ilulls is given by
(3.23)
la>sk
llI(x)szu(a)l
For an integer sz=j2-0 Hs consists precisely of all uE S'
such that xauE Hs =Hs.,0 (with the unweighted Hs,) for all lalsj. A norm equivalent to Ilulls is given by
(3.24)
ka ki
Ilxaulls
(iv) For sE &2, and k=(k. kz ), with integers k. , (3.25)
%+k = { uE He : xau (0 ) E Hs
,
as
l a l sk.
,
l
kz , l
l skz }
and Ilulls+k is equivalent to (3.26)
2
II lsk. , Ialskz
llxau(R)ll'
(v)(Sobolev's lemma): For s.>n/2+j we get a compact imbedding
(3.27)
Hs -> {uECJ(&n)
:
(x)szu E Cok(&n)}
.
(vi) Hs and H-s are mutually adjoint, under the pairing (3.28)
(u,v) = ( Su,II sv) = ,fuvdx , uE Hs , vE H-s
where (I'su,II sv) (with (.,.) of H) defines (u,v), uE He, vE H-s.
3. L2-Sobolev theory
110
and that (3.28) is independent of s That is, the continuous linear functionals on Hs precisely are given as v- (u,v), with some uE H-s and (.,.) of (3.28). This follows because S :HS- H and lI s:H-s.* H are isometrics.
Note that n =rI
,
(vii) We have
S = fl {Hs : sE 12} , S'=U {Hs
:
E N2}
.
Indeed, (v) gives fl HSC C°°(&n). Combined with (3.25) we get
fl HSC S, the opposite inclusion being evident from (3.1). On the
other hand, (3.28) implies that U H C S', with (u,(P)=(u,g), gE S, s uE Hs. The opposite follows from [C,],I,(6.3) (or [Schwl]). (viii) The space S is dense in every space Hs . Indeed, a sequence um-u may be generated, for uE Hs by setting um=xm*((Omu), X m nx(mx), wm=w(m), with suitable cut-off's x,w m The spaces Hs will play a central role. They allow a description of topologies on pdo-algebras in terms of L2-norms. Theorem 3.1. For a symbol a E tptm the corresponding yxlo A=a(x,D)
is a continuous operator H.
for a E0 m,Se',o
,
Hs-m , for every s E &2
Similarly,
.
O<6<1 , A is continuous from Hs to Hs-m
The proof is simple, due to our preparations: Since 1Is is an isometry H5-'H, one must show that
4.8 this sIE Oppc thm.1.1. We with the
s-mA $1 is L2-bounded. By I,thm.
is an operator in Op,t0, due to s-mE Op1cs-mC Opy)ts-m -5C Opy,t-s. Hence the desired Lz-boundedness follows from
Similarly for aE using thm.2.1 instead. now define a type of order classes: An operator A:S-S, property of thm.3.1 (i.e., A admits continuous extensions
) is said to have order m. The class of all operators of order m is denoted by O(m). Trivially AE O(m) admits a continuous extension AO:S'-IS' with
As:Hs->Hs-m, for all sE 8:2, and given mE &2
As=A0IHS. Thm.3.1 now may be expressed by stating that (3.29)
OP hm1p,6 C O(m)
,
for all M(=- &2
pa0
Ssp1
0s6<1 ,
We also define
(3.30)
O(_) = U{O(m) : mE I2}
,
O(-ao) = fl{O(m) : mE 1e2}
The proposition below is trivial. Proposition 3.2. The classes 0(oo) and O(0) are algebras under operator multiplication and 0(-a) is an ideal of both, O(0), O(co).
Moreover, we have
3.3. Weighted Sobolev spaces
111
O(m)O(m') C O(m+m')
(3.31)
.
The class 0(0) has the natural locally convex topology, induced by the collection { IIAIIs : sE 1Q 2} of all operator norms
IIAIIS = sup{ IIAuII
:
IIuOIss1} .
Proposition 3.3. The above topology is a Frechet topology. An equivalent countable system of semi-norms is given by (3.32)
Moreover, 0(0)
{
: kE Z2}
IIAIIk
.
, under this topology, is a Frechet algebra, i.e.,
it is complete, and the algebra operations are continuous. This follows from the Calderon interpolation theorem (in [C1],III the corresponding is shown for unweighted spaces Hs)
.
Proposition 3.4. The class O(-oo) of all operators of order -co
and, again, with the class ' of all integral operators with kernel in S(&2n) Proof. We have seen in II, lemma 1.7 that ih-00 is the class of all integral operators with rapidly decreasing kernel. Also, as a consequence of thm.3.1, get Op h- =Opit C O(-co). For the converse coincides with the class yad-,,
,
let AE O(-cc). Observe that Aapa,,,=MDRADR'Ma'E O(-co) for all a,13, a
a',4'. We must have A:S'->S, since A takes each Hs into f7ft=S, and S'=l.Ht. In particular A:H-.S, hence Au(x)=fkx(y)u(y)dy, uE H, with
some k E H, for xE >en, by the Frechet-Riesz theorem. Similarly, x (3.33)
Au(x) _ (kX,u)s
,
sE 12 , xE &n
with some kS E HS . Using (3.2) one concludes that (3.34)
Au(x) = fIlsfsix(y)u(y)dy
, u E S
with a distribution integral, where the temperate distribution (3.35)
kx
=
ensks E H_s
,
s E R2
must be independent of s , hence will be in (3.36)
Au(x) = fk(x,y)u(y)dy
,
It follows that
k(x,y) = kx(y)
.
It follows But the same conclusion also applies to all AaPa,s, by integration of (3.36) for large a, 3, a', I' that k must have .
derivatives of all orders, for x, y , and that
3. L2-Sobolev theory
112
±ifxaYa'a 4'k(x,Y)u(Y)dY
(3.37)
where the kernel has the same properties. Thus kE C0O(R2n), and
xaya'aOay'k(x,y)=O(1) i.e., we conclude that kE S(12n), q.e.d. Recall the concept of K-parametrix of a y'do of II,1. We now ,
recognize that a K-parametrix B of A is an inverse modulo O(-oo). The concept is meaningful for arbitrary AE O(oo) .
Finally we introduce a stronger type of inverse: Two operators A,BE O(oo) will be called Green inverses of each other if (3.38)
AB-1
,
BA-1
E F ,
where F C O(-oo) denotes the class of operators with finite rank. Notice that F is a 2-sided ideal of the algebras O(oo), 0(0).
Clearly a Green inverse is a special kind of K-parametrix. In sec. 4 we will show that an operator admitting a K-parametrix also admits a Green inverse. Clearly, if A E 0(oo) admits a Green inver-
se then it is a Fredholm operator of S' as well as of S
,
since
the Green inverse acts as a Fredholm inverse in both spaces (cf. [C1],app.A1, thm.2.1). More generally, if AE=-O(s) and BED(t) are Green inverses of
each other then r=s+ta0 follows. In case of r=0 - i.e., s=-t we have A:%-'Hm-s so B:Hm-s-'Hm continuous, for every mE &2 ,
,
that again A and B are Fredholm inverses of each other, as maps
Hm -HM-S . Hence then also A is Fredholm as a map Hm
m-s For general r=s+t>0 we may regard A as an unbounded operator Then B:Hm+t Hm is from Hm to Hm+t with domain dom A = Hm+r bounded, and we get BA C I+F , AB C 1+F' , with F,F' E F . .
This implies that the closure Ac of A:Hm+r dom A -.Hm+t is a closed unbounded Fredholm operator from Hm to Hm+t' for every m. We shall
discuss details in sec.4, below (cf. [CI],app.A1, sec.6). Remark 3.5. Note that A must have the same Fredholm index as a
map S-S
,
S' ->S'
,
Hm-*Hm-5 (or dom A = Hm+rC Hm -Ht )
Problems. 1) For the t,do's on the cylinder of I,6-II,3,pbms, if
IIL2 (S' )=0(1) , all a,(3, show L2 (S' x&n-1 )-boundedness of
IIA(a) (x°
A. 2) Define weighted L2-Sobolev spaces on the cylinder, using the
2:t
norms 0
f
dxi dx° (x°) 252
I (1-A)s' uI 2=IIuIIs, with properly defined po
wer of 1-A=1-aX,-A6. 3) Discuss H5-boundedness of y,do's as in (1).
113
3.4. Existence of a Green inverse
4. Existence of a Green inverse. In this section we will show that also a Green inverse exists if only an operator AE O(co) admits a K-parametrix. We have intro-
duced the concept of Green inverse in sec.3 as that of an inverse modulo the ideal F C O(-co) of operators of finite rank.
All results, below, (especially thm.4.1, thm.4.2, cor.4.3) are valid not only for a Wo with complex-valued symbol but even although we for a matrix-valued pdo A =((a.k(x,D)))j,k=l,...,v only discuss the scalar cases. Proofs extend literally. Actually, it is practical to prove a stronger result: An A with Green inverse has a finite dimensional null space ker A C S There is a complement T of ker A in S' and an operator B OE 0(-) such that B0Iim A inverts AIT . Here the order of BO may be chosen equal to the order of B Also we may choose T as the ortho-complement T= {uE S':(u,g)=0 for all TE ker Al , with the pairing (u,q) of (3.28). With this construction B0 has the properties of a special Fredholm inverse ([C1], p.259). In particular, ,
.
(4.1)
1
- BA = P
I - AB = Q
,
with projections P,QE F onto ker A and ker A*, respectively. Also, P, Q E 0(0) are orthogonal projections in H=H0 .
For a proper setting of (4.1) let A* be the adjoint of AE O(o) under the pairing (3.28). That is, A :Hm-s-> Hs is the operator satisfying (Au,v)=(u,A*v) for all uE Hs, assuming that AE 0(m) C 0(oo), with (.,.) of (3.28) . This defines A* for each Hs. All of them agree on S. By continuity they must agree wherever they are jointly defined. Thus we obtain A* as a map S'->S' with continuous restrictions S-S, %_s-H-S. Then AE O(m) amounts to the condition (4.2)
s-mA S1 GL(M)
,
for all sE &2
For the adjoint A* this translates into (4.3)
sl*A*ls-m = h1 sA*rcls E L(H)
,
s E H2 *
lE L(H), i.e., A E O(m). It folOr, replacing s by m-s, rIs-mA *rf s
lows that AE O(m) a A*E O(m), including the cases m=tom. Clearly, "*" has the properties of an involution 0(00)-O(oo). For a 'tpdo A
this adjoint is the formal Hilbert space adjoint
A*
of I,(2.3).
Theorem 4.1. Let AE O(m) have a K-parametrix BE O(m'). Then the
3. L2-Sobolev theory
114
operator A ,
as a map A:S' -> S' has the following properties.
ker A C S , ker A* C S;
(i)
(ii)
dim ker A < ao , dim ker A* < oo in A = If E S'
(iii)
:
;
(f,T) = 0 for all T E ker A* }
For fE He with (f,q)=O for cpE ker A* all solutions u
(iv)
of Au=f are contained in Hs-m,. There exists ps>0 such that
(4.4)IIff1s=IIAuIIskpsllulls-m,, whenever uE Hs-m (u,cp)=0 for gE ker A
with ps independent of u and f. Proof. For uE S with Au=O get (1+K)u=BAu=O, where KE O(-ao). Thus u=-KuE S, since an integral operator with kernel in S(12n) maps
S'- S. Similarly B*A*=1+K'*, hence ker A*C S, confirming (i). To show (ii) note that ker A C ker(1+K). But ker(1+K) is closed in He since uE Hs, (1+K)u=0 implies uE S. For an infinite orthonormal set ujE ker(I+K) - with respect to (u,v) of H - the sequence Ku j
has a convergent subsequence, in L(H), K being compact in H, its kernel in L2(12n) makes it a Schmidt operator. But uj=-Kuj cannot have a convergent subsequence, as an orthonormal set. Therefore dim(1+K)
,
,
In order to show the converse we will first establish (iv) and show that a solution u E Hs-m, exists for fE Hs, satisfying (4.4). To clarify the relation with the inner product of H we assume that m'=0 . This is no restriction: For general AE 0(m) , BE O(m') observe that (As* )(II m,B)=1+K , (rl m,B)(A1,)=1+K" , with
K"=r1m,K's,-lE O(-co). Also Au=f
,)(II m,u)=f
, and ker A* =
,)*
ker (
,
and IIuIIs-m,-IIn mulls . Thus all terms may be trans-
lated to the case m'=0: Replacing Ar ,E 0(r), r1 m,BE 0(0) with A,B
we get a pair A,B of mutual K-parametrices with AE O(r), r=m+m'a0, and must solve Au=f under the corresponding conditions. Now let fE S' be 'orthogonal' to ker A*. We must have fE Hs for some s, and now will proceed to show that uE Hs orthogonal to ker A exists solving Au=f, where u and f satisfy (4.4) (with m'=0) A further reduction of the statement is useful: we may assume s=0: Au=f ea( $A Sl)( $u)= sf, where rlsu, sfE H--H(0, while IIsA sl
E O(r) has K-parametrix sB s'E O(0). Also, (f,g)=0 for gE ker s*
a (sf,p)=0 for V=II scp satisfying (sA sl )III sA*ffsrl_s(p = 0 Thus we now are reduced to an operator equation Au=f in the Hilbert space H , where we may regard A as an unbounded operator
3.4. Existence of a Green inverse
115
with domain dom A = Hr r=m+m'a0 The operator A satisfies AB=1+K BA=1+K' with K,K'E O(-oo) . Let us next prove (4.4), or, ,
.
,
IIAuII a pllull for all uE Hr don A with u 1 ker A*
(4.5)
.
Suppose this inequality is false. Then there exists ujE Hr , uj 1 ker A o , j- . But we get IIujII=1 , with IIAujII ,
(4.6)
(lull s II (1+K' )uhl + IIK'uhI = IIBAuII + IIK'uII s cllAull +IIK'ulI
,
for all uE Hr . Since K'E K(H) as used before, there must be a convergent subsequence of K'u (we assume K'uj convergent itself). ,
It follows that
j-K'ul11_0, j,l-
.
Thus,
ujE Hr, uj-u (in H), 11u11=1, Auf. 0. Hence uE don Ac , with the closure Ac of the unbounded operator with domain Hr, as discussed.
But we also have IIAu jlls+r_' 0,
Ilu-ujlls-
0 for ss0, s+rs0, where now
A:Hs+r_Hs is continuous, so that u is in the null space of A:Hs+r H. It was seen that this null space is independent of s, hence =ker AC S. Since uj-u in H and uj I ker A in H it follows that u 1 ker A, hence 1=(u,u)=11u110=0, a contradiction, proving (4.4). The adjoint A*E O(r) also may be interpreted as unbounded A*:dom A*=Hr-> H. For this operator we have (u,Av) =
(4.7)
(A*u,v)
for all u,v E dom A = dom
A*
- Hr ,
by definition of A*. In other words, A and A* are in adjoint relation in the Hilbert space H : The Hilbert space adjoint AA of A ex-
tends A*. We get ker AA =ker A*. Indeed, ">" is trivial. Let fE ker AA : fE H, (f,Av)=O for all vE H Then A*f=0 wth A*:H-H_r . Or uE ker A*C S, ker A* independent of s, so that ker AC ker A*. We conclude that Ac is an unbounded closed Fredholm operator of the Hilbert space H since (4.6) implies that in Ac is closed. It then is well known that in Ac =(ker A)1 (cf.[C,] app.A1,6,7). Notice finally that Ac equals the restriction to don Ac of
r
.
the operator A:H-H-r. Indeed, ujE Hr don A, u j-*u (in M), Au j- Acu (in H) implies ujE H uj-u (in H) Auj-> Acu (in H-r), hence Acu = Au since A:H -, H-r is continuous. This completes the proof of (iii) and (iv) of thm.4.1. Q.E.D. ,
,
Theorem 4.2. An operator A E=-O(m) admits a Green inverse Bo if and only if it has a K-parametrix B. If BEO(m'), then also B0 E O(m').
Also, B0 may be chosen in such a way that (4.1) is valid. Proof.
A Green inverse is a K-parametrix. Thus we must construct
3. L2-Sobolev theory
116
a Green inverse whenever a K-parametrix exists. Applying thm.4.1 by setting B u=0 for uE ker A* , and B 0 = we define an operator Bo 0 * 0 (AI(ker in in A =(ker A )1 . Every u65' is in some Hs, and AO)1)-1
we have a unique decomposition u=v+w, vE ker A, w 1 ker A. Thus the above defines a linear operator S'-'S'. Clearly B0Au=u for u 1 ker A, and B0Au=O for uE ker A. But AB0u=u for uE in A =(ker and AB0=0 for uE ker A*
A*
In other words, (4.1) is valid.
.
On the other hand, (4.4) implies (4.8)
IIA lulls_m, s
PS P
IIuIIs for all uE H. , u 1 ker A*
for the abstract inverse A 1 of A:{uE Hs+m: u I ker Al
-> Hs
Actually, we get (4.8) for Ac-1 with the closure Ac of the unboun=H +H , and Ac-1 is defined in all ded operator A:dom A =H s-ml A*}.s+m-r m of {uE H : u I ker By our definition of B (4.8) implies IIBOuIIs_m s=
JIBOVIIs_m,SpsIIvII -csllulls, where u=v+w, wE (ker A* ) 1
is
the above unique decomposition, and where we used that Ilvllss cOIuIIS
uE Hs, c independent of u. Indeed, with an orthonormal base k (in H) of ker A* we have w=YWj(Wj,u)
s IIuIIs+ JIlrojIlsllrojII_slluIIs
,
,
v=u-w , hence IIv1Is
so that c=1+7,Ilrojllsllacjll_s may be chosen.
Clearly Q= 7,cp j) ((pj and P=j) (Vj (with an orthonormal base {W j}
(in H) of ker A ) belong to 0(0)
.
The proof is complete.
Corollary 4.3. The operators B and B0 of thm.4.2 differ by an operator in O(-oo)=Opph_oo only. Thus a Green inverse of a (formally) md-(hypo-)elliptic tpdo always is in the same symbol class Vt'm,p,6
of ch.Il
as the K-parametrix constructed there.
Proof. Indeed, that Green inverse is a special K-parametrix, and we observed earlier that, as inverse modulo the ideal O(-oo), a
K-parametrix is unique up to an additive term in O(-c)
.
We conclude this section with the remark that a Green inverse of a pdo in essence has properties very similar to the integral operator with kernel equal to the generalized Greens function of a boundary value problem: It constitutes an inverse modulo operators of finite rank; its distribution kernel has singular support at the diagonal x=y only (as a consequence of I, thm.3.3). Problems. 1) The operator B: COO(S')-Coo(S') given by u.yeij°bj(O)uj
3.5. Hs-compactness
with uj=,1nf a liWu(p)dp assume that
,
117
as in I,6,pbm.3, is L2 (S' )-bounded if we
=0(1) (in j and 0
,
for each k=0,1....
. [De-
rivatives of all orders are not required, but how many are needed for a proof similar to that of thm.1.1?] 2) Investigate the C*-subalgebra of L(LZ(S')) generated by all operators B as in (1). Show that its finite Fourier transform is the subalgebra A of 12(Z) gen-
erated by the shift operator and all diagonal matrices with bounded coefficients. 3) Does X have compact commutators? 4) Answer the same questions if the condition of pbm.1 is modified by requesting that 0((j)-pl) in 0 and j , with the finite difference Vaj=aj-aj_1 and some p>O for all (some?) 1 and k. ,
,
5_HS-compactness of ipdo's of negative order. We shortly discuss an often used compactness result. Theorem 5.1. A (matrix of) 'tpdo's A=a(M1,Mr,D), with aE SS and or-
ders mj satisfying m,<0, m2+m3<0, is a compact operator Hs- Hs, for every sE &2. Moreover, for general mj, A: Hs Ht is compact
whenever m, <s, -t, ,
mz +m3 <s2 -t2 .
Especially, A=a(x,D)E 0pO m,p,s
p,>0, pz>5, is E K(HS,Ht) whenever mj<sj-tj
Proof.
,
j=1,2.
Observe that Q,=(x)_6(D)_F is compact H-'H, as E>0 (cf. [ Ci ]
III,lemma 8.1, for example - but other proofs are known. Details:
The operator (D) -E may be written as convolution (x) -E ° * , where (x)-E
(5.1)
pE(x) =
(x) = cE'vlxl(E-n)/2K(E_n)/2(Ixi) ,
with a constant cE,n, and the modified Hankel function K0. (5.1) is verified using techniques as in ch.0, sec.4, for construction of the fundamental solution of 0+kz Note the function (5.1) is .
L'(1m) The kernel gE(x,y)=(x) EpE(x-y) may be approximated in the sense of Schur's lemma by Schmidt kernels qE E Lz(&2n). Then (i) the operators Q£, QEu(x)=fgE(x,y)u(y)dy H . Second, we have lIQE-QElI - 0
,
j->oo
,
are compact, from H to
, by Schur's lemma, so that
QEE=- K(H), since K(H) is closed under uniform operator convergence.
With Lz-compactness of QE we get the statement at once: Compactness A:HS-* Ht means compactness of sIAllt=(1 IAnt+Ee)QE:H4H'
where the first factor is bounded, by III,thm.1.1 (or thm.2.1), assuming the inequalities of thm.5.1, by calculus of pdo's. Thus indeed A: Hs-Ht is compact if these inequalities hold. Q.E.D.
Chapter 4. PSEUDO-DIFFERENTIAL OPERATORS ON MANIFOLDS WITH CONICAL ENDS
0. Introduction.
In the present chapter we will focus on pseudo-differential operators on differentiable manifolds. We assume either that 11 is compact -then our theory will not differ from others - or that Q
is a noncompact Riemannian space with conical ends. While the Fourier transform and, correspondingly, the concept of Fourier multiplier a(D) = F-la(M)F is meaningfull only for functions or distributions defined on ten
,
the kind of ydo's we intro-
duced has a natural environment on a type of differentiable manifold, to be studied. The reason: Our ipdo's of ch.2 are invariant under a type of coordinate transform (discussed in sec.3) while Fourier multipliers do not have this property. In sec.1 we discuss distributions on manifolds. A special type of 'S-manifolds' is preferred, allowing the definition of a class S(c) of rapidly decreasing functions. The linear functionals on S(cz) will be our temperate distributions. For simplicity we
will consider only manifolds a allowing a compactification W to which the Co structure can be extended - making r° a compact manifold with boundary. In essence then S(a) will be the class of functions over n vanishing of all orders on an . In sec.2 we will introduce 'admissible' charts, cut-off's, partitions, as well as admissible coordinate transforms, all designed to give S(c) similar properties than S(8n) In particular a Riemannian metric is introduced, making n a space with conical ends. In sec.3 we prove invariance of pseudo-differential operators under admissible coordinate transforms. Sec's 4 and 5 generalize the calculus of pdo's to spaces with conical ends. In particular, we again get md-elliptic and formally md-hypo-elliptic operators, defined by their local symbols.Results of ch.'s II and III regarding K-parametrix, Sobolev spaces, order classes, Green inverse, etc., all generalize almost literally. Similar results, on coordinate invariance as well as and-el.
118
4.1. Distributions and temperate distributions
119
liptic operators on manifolds, were discussed by Schrohe [Schrj] in somewhat different setting. Regarding application to differen[C2] where similar C*-algebras tial operators we point to [CDg1] ,
of singular integral operators on L2(a) are considered, with corresponding results, but abstract proofs. 1. Distributions and temperate distributions on manifolds. In ch.0 we were discussing distributions on an open subset n of In Presently, we first will extend the distribution concept .
to differentiable manifolds, compact or not. For a C00-manifold a of dimension n (always assumed paracompact -even with a countable atlas) the space D=D(a)=C00(n) is well defined, and the convergence cpj-0 in D(O) just as well: One requires that (i) supp cpjC K, with a set KCC sz independent of j.
We get KC U .
,
a finite union of charts gill and Tj=Folroj , with a
partition of unity Twl = 1 in K
,
supp w1C Szl
,
w1a0 , w1E
C000
(n)
One also requires (ii) that wlrpj- 0 in D(SZ1), as j->0 in the coordinates of the chart Szl
for 1=1,...,N. We define a distribution uE D'(n) to be a continuous linear functional over D(n) where 'continuous' means that ,
,
(u,(Pj) -> 0 whenever y j - 0 in D(n)
(1 . 1)
.
With this definition a conceptual difficulty arises if we attempt to interpret a function fE LIloc(a) as a distribution. For S2C in we defined (f,(p) ffgdx , taking advantage of the existence of a distinguished measure on 2n - the Lebesgue measure as Haar measure of the group In For a manifold a there no longer is such a dis.
tinction, but we may construct a positive Co measure dµ, locally, dµ=xdx on a paracompact manifold n , and then define O
,
(f,P)=f
(1.2)
fq)dµ
, cpE D(SZ)
SZ
for fE LIloc(R)
establishing the analogous imbedding L1loc+ D' This dependence on prior choice of a measure may be avoided ,
by defining distributions as linear functionals on a properly topologized space of signed C0(SZ)-measures - expressions of the form cpd1t, yE D(Sc), in our terminology. But we are tied to the use of
Sobolev norms anyway, where we use a distinguished measure. Hence
4. Operators on manifolds with conical ends
120
we follow our habit of [Cz], and always assume given a pair {sl,dµ} of a manifold st and positive C-measure dR on sl, so that dµ, dis-
tinguished by other uses in our theory always is on hand. Then we may use (1.2) to define the imbedding Llloc-' D'(a). Often ds=dS will be the surface measure of a Riemannian metric on sl
.
Most concepts obviously extend, such as restriction of a distribution to an open subset, (singular) support, definition of Lu, for a differentials expressions with smooth coefficients, etc. A differential expression L on SZ (with CC coefficients) is defined as a linear map L:C00(sl) - C°°(n) with Lu=
sa u in deriva-
tives dX of the local coordinates and C coefficients aa for u with support in a single chart. The adioint differential expression is defined as the expression L*:C00(9l) - Cw(c) satisfying
(1.3)
fLuvd s =
f uL*vdµ
,
for all u,v E C00(n) with uv E CO(sl).
The dual expression L" of L is defined by setting L f=L'f, fE D . For a general distribution uE D'(sl) one then defines Lu by setting
(1.4)
(Lu,(p) = (u,L c))
,
for all qp E D(a)
.
One derives the familiar facts on E'(11) the space of compactly supported distributions, on regularization, and once again ,
may introduce the standard topologies on D' and El (cf. [C1],I,6). For an extension of S(c) and S'(n) we introduce a type of manifold looking conical at - . The main feature of fE S=S(&n) is the behaviour of f for large lxi. In 11,5, looking at wave front sets, we introduced a type V of directional cut-off and regarded f belonging to S in a conical sector only if ifE S for suitable such ' . A general change of coordinates can destroy the property of a function u to belong to S (or 1puE S). Accordingly, a special transforms as structure on sl, allowing only special admissible coordinate changes is required before a space S=S(s1) can be defined, with general properties of S()Bn). Such structures
were introduced by Schrohe [Schr1,2] and were called SG-structures We only look at a restricted class of such n. First define the diffeomorphism s:1en- B1, B1=B1(0)={yE 2n:IyI<1}, by setting (1.5)
y = s(x) = x/(x)
,
Introduce [ y] = 1- y Z
x = s-1(y) = y/ 1- y
,
then,
,
= t(x)
4.1. Distributions and temperate distributions
(1.6)
Y = x/( x)
,
x = Y/[ Y]
121
[y] = 1 /( x)
,
Proposition 1.1. The map (1.7)
u(x) - v(Y) = u(s-1(Y)) = (uas-1)(Y)
defines a bijection S(&n) - COOO(B1).
In prop.1.1 COOO(B1) denotes the class of Cx(B1)-functions vani-
shing with all their derivatives at aB1. CO Remark 1.2. A function fE CO (B1) has a natural extension to in f=0 there. The extension v is CQ(&n), with supp vC Vice versa, for vE COO(e) with supp wE BC we have wIB1E Bi=(B1)clos.
Proof. If vE CO"(B1) then v and all its derivatives vanish of infinite order at lyl=1, by Taylor's formula. Thus we get
v(a)(y)
(1.8)
=
0(((1-1y12)k)
,
for all k=0,1,...
,
and all a
For a=0 this implies u(x)=v(y)=O([y]k)=O((x)-k), k=0,1....
.
But u(a)(x)=aa(v(x/(x))) is a linear combination of terms k
v(R)(
(1.9)
)II ay
(xjk/(x))
,
IRlslal
,
lYk = «
X
by induction. Get aX(xj/(x))=O((x)-IY I) , easily confirmed. Each term (1.9), hence
ua is 0((x)-k)
Vice versa, let u E =-S
,
for all k, i.e., uE S
and v(y) = u(y/[y]). Note that
(Y/[Y])2=1+Iy12/(1-IY12)=1/(1-IY12)=[Y]-2, hence (Y/[Y])=[Y]-1
(1.10)
u(a)(Y/[Y])
O((Y/[Y])-k)
=
For a=0 it follows that v(y) E CO(B1)
.
=
O([Y]k), for all k
.
For arbitrary a one again
uses the chain rule to express v(a)(y) as a linear combination of
u(P)(Y/[Y])nay j(Y1 /[Y]), IPIslal,
(1.11)
j=a
,
similarly as (1.9). Here we get aY(yl/IYI) = O([y]-21Y1-1), and y (1.10) still implies all terms (1.11) to vanish, hence v(a) =0 Thus v E COOO(B1) follows and prop.1.1 is proven. at IYI=1 The above suggests the concept of an S-manifold as the interior a of a smooth compact manifolds no with boundary. Then define the space S=S(S2) of rapidly decreasing functions on sa by setting .
S(a)=CO'(1 )= {all uE C°(Sto) vanishing of infinite order on an}. no
is a compactification of Sz, just as >sn of &n in II,(3.1).
4. Operators on manifolds with conical ends
122
aSt0 is called the infinite boundary of a; its points at infinity.
We will insist, in the following, that a0 be a compact Cmanifold with boundary. sz0 will be useful for function classes
other than S(c). On the other hand, for defining S(a) this is not required. For example we might allow sz0 as a cartesian product of finitely many manifolds with boundary, or allow 110 to have corners
vertices -i.e., at some boundary points we have charts in such Cartesian products depending on j. with Ij=I or I.=le+=[0,oo) xln=R2n Notice that 2 is (should be) an S=manifold, but the product sz0xsz0=BixBi no longer is a C* manifold with boundary. As ,
in prop.1.1 one shows that we get a bijection S(&2n) H COOO(B1xB1)
as well, induced by the map sxs:R2n - BlxBl Thus, for a-I n we could use as compactification nl =Bi (in 2n dimensions) or szO=BixBi. We choose the first and keep it fixed,
for the following (note the exception at the beginning of sec.4). Before discussing temperate distributions on a general S-manifold a we look at the transfer of the Frechet topology of S(In) onto CO(B1) under the bijection of prop.l.1. The seminorms
(1.12)
Ilullk = sup{II(x)ku(a)IIL°°(Rn)
:
Ialsk}
k=0,1,2,...
,
generating the topology of S , are equivalent to the'seminorms (1.13)
vk(v) = sup{II[y]-kv(P)II
o,
L (B.)
:
, k=0,1,2,...
I1Isk}
in the sense that (1.14)
Ilullks ckvl(k)(v)
,
and vk(v) s ckllulll(k)
, k=0,1,2,...,
with ck, 1(k) independent of u,v. This is confirmed looking at the proof of prop.l.1. On the other hand, since all v(p)(y )=O as Iy I =1, Taylor's formula with integral remainder (I,(5.20)) implies
(1.15)
v(y) = (N+1)
O J1(1-i)Nv(0)(y°+i(y-y° ))di
71
IOI=N+1
For O#yE Bi let y° = y' (1.16)
,
0
so that I (y-y° )6 I s y-y° j16 I s [y]K. Then
v(y)= Max{IIv(0)II
L (Bi )
:I8I=N}.O([y]N+1)
,
N=0,1,2,...
.
Similarly for derivatives v(a). Combining (1.14) and (1.16) we get Proposition 1.2. Under the bijection S(zn) H CO (B1) of prop.1.1 the Frechet topology of S (induced by (1.12)) is equilent to that of COOO(B1) as a subspace of COO(B?), induced by (with Ck=Ck(B?)) (1.17)
IIvII k = Max{IIv(a)]IL°°(B
:
)
jalsk}
,
k=0,1,2....
.
123
4.2. Distributions on S-manifolds
As a consequence of prop.1.2 a temperate distribution uE S'
=S'(1n) defines a distribution vE D'(B1) , by (v,V)=(u,cp)
where
,
yE D(In), V(y)=q(x). We get an injective map S'(&n)->D'(BI). Actually, v defines a functional on the subspace CO"O(B1)C C00 extending to w:COO(B?)-T ely,
,
(Bc 1)
by the Hahn-Banach theorem. More precis-
v:C000
(B1)-'T is bounded with respect to some Ilvil
k
Then Hahn-
Banach, applied for the B-space Ck(B7) gives a continuous functional on Ck whose restriction to COO(B?) gives the desired w In turn we may interpret w as a distribution zE D'(1Rn) with
supp z C B 1 by setting (z , (p) _(w, c)(Bc) , qE D(e). Thus v may be obtained as restriction v=z1B1 of a distribution zE D'(&n). Vice ver-
sa, a restriction v=zIB1 of ZED' (,n) clearly transforms to uE S'. The last arguments may be repeated for a general vE CO"O(S)C C (fl0 ): The space of continuous linear functionals on CO (SZ) under
the Frechet topology of
C00
(SZ°) coincides with the space DA(11) of
restrictions z I Sz of zE D' (SZ, ) , with any Co manifold Sz' M Sz°
.
Then we define S'(SZ), the space of temperate distributions on Sz by setting S'(c)=D°(SZ). We will work on some details in sec.2. Later on we will introduce ipdo's on S-manifolds. 2. Distributions on S-manifolds; manifolds with conical ends. We return to the discussion of S-manifolds as interior n of a manifold with boundary c° In order to make the space C00O(St) .
look like S(n) we will use a special kind of chart only. First, instead of relating the charts of n° to coordinates defined in open subsets of a half space, we use charts (for Sz°) of the form
(2.1)
- of (U°) C Bi
of :U°
with a homeomorphism of between the open sets U° C sz° and oi' (U°)
Then an S-admissible chart of sz is of the form (2.2)
w:U - w(U) C &n
,
where U=U°flsz
, w=
s-1.(0 1U)
with an sz°-chart {U0,0} as in (2.1), and s- =t of (1.5)
.
The S-
structure on Sz is induced by an atlas of S-admissible charts. An interior chart UCC Sz of the noncompact manifold Sz is Sadmissible, since we may define U°=U, of =so (o. A general chart w:U
-.w(U)C ien is S-admissible if and only if the map s,w:U- B1 extends
to a diffeomorphism of :U° - V° between open subsets where UC U° C Sz° , sow(U)C V° C B1 , and where U and so w(U) are dense in U° , V1,
4. Operators on manifolds with conical ends
124
respectively, and U°n sz=U. For a given S-admissible UC sz the chart
U° C fe
is given as U° = U U (Uen ac° ), with the closure U0 in n' of
U . In particular, {U°,&} is uniquely determined by {U,w}
.
Note the 'shape' of a neighbourhood of an infinite boundary point p°E ast°
in 'admissible coordinates': It contains a 'lense'
{ ly-y° l<e , I y < 1 } , with jy°
j =1 ,
y° image of p°, in B1-coordinates
-y°kss}. After a rotation asss2} sume y°_(1,0,...,0). The neighbourhood system NE={(1-{x2+ sa, is equivalent to ME={1 se}, where x=(x,,x ). ME is the (x) I intersection of the solid hyperboloids of sz°. With s-1 this goes to NE={j
(2.3)
Xi 2 2t%' (1+p2) , Xi >0 , %= 2' E F E
with p=lx 1, and,
(2.4)
p2s62 (1+xi2 )
,
6=tea
(fig.2.1). In other words, the set NE is contained in cut-off cone
(2.5)
CC yo ,T1
={xE In: IxI>
l where we may choose r1==1(E) 40 as e-0 tains a CCy°,,,
,
IXI
Vi Ice
.
Y° 1s11 }
,
11>0
versa, any set Na con-
for a smaller 1>0. This describes the neighbour-
hoods of our infinite points. The sets (2.5) form a base at p° An S-admissible coordinate transform w,X-I:x(U n V)->w(U n v), .
for S-admissible charts w:U- In , x:V' In , must be of the form (2.6)
x=w,X-1=s-1°x° , s , with x° = w ° °X° -I :X° (U° n V °
)
- of (U° n V°)
a diffeomorphism between open subsets of B1 For investigation of maps of this form we focus on the Jacobian matrix P(x)=ay)/ax=((ay,/axl))j,1=1,...,n of a global coordina-
te transform q=t°rp°s, t=s-I with some diffeomorphism W:Bi HBi . Write I=((6jl)), pqT=((pjgl)), for p,qE In, (j=row, 1=col.-index): (2.7)
at/ax --[y(I+t(x)t(x)T)
,
as/ax =
x)(I-s(X)s(X)T)
It follows that P(x)=((x)[ V(s(x))})-I(I+t(V(s(X)))t(v(s(x)))T)T(I-s(x)s(x)T)
with 111=arp/as(s(x)) . Here A.(x)=y = 1
(2.8)
,
hence
P(x) _(1Si )1/2(I+t('V(s))t('U(s))T)11J(s)(I-ssT)
4.2. Distributions on S-manifolds
125
Taking determinants we get JT(x) = Idet P(x)I =
(2.9)
)3/2Idet
II-(s
'(s)I
using that det(I+t(x)t(x)T)=1+t(x)2=11x2. Therefore is (ex2 1 11-ss tends to) a smooth positive function on B1. Moreover, = 1 distance(s,I) E C'(Bc) (with t=aB?) follows, ause 1+ vp gistance(V(s),1') 1 1 p: B1 « B1 must map r to r . (Near s° with Is°1=1 introduce local coordinates v=(v1,...,vn), setting vl=dist(s,1'), with coordinates 2,...,vn of IS on 1'. In such coordinates the diffeomorphism V is represented by an n-tuple of funtions V.-
(v). Clearly V1-=0 as v1(s)=0. Thus
,
where dist(V(s(v)),r)=
''1-/v1E C
,
and it is
clear that this quotient is >0 since yr is a diffeomorphism.) This result may be extended, as follows. ,
Q(s)=Pos-1=P(t)=(acp/ax)(-"
Proposition 2.1. The function
), sE
B1, extends to a Cco(B?)-function, nonsingular for all sE B1 Proof. First of all, the scalar factor up front in (2.8) is C"O(B1) and the matrix q(s)(1-ssT) has entries in C"O(B1). Introducing an orthonormal base SO ,...,s`" of &n varying smoothly with sE B1 near
some point of r such that s°=Is (2.10)
,
we may write
I- ssT = (1-Is12)s°s°T + L1 sjsjT
hence
(2.11)
W(s)(I- ssT) = (1-Is12)(w(s)s° )s°T +
_i(tY(s)s.
JT
Applying the matrix t(V(s))t(V(s))T=1- s V(s)?Q(s)T to the terms V -7 CC'(B1)-matrix, at right of (2.11), the first term gives a using that 11,,(x)2 E COO(B1). The same is true for the other terms, since q(s) must map tangential vectors at sE t to tangential vectors at p(s)E r. or, p(s).W(s)s3(s)=0 as IsH=1, since p(s) is normal to r at V(s) Also, q(s) = ('(s).'(s)s3(s))p(s) is Coo near IsI=1. Thus the quotient 1-V s also is Coo near IsI=1 Q.E.D. There exists a finite S-admissible atlas. Indeed we may choose a finite atlas of the compact c° and construct the corresponding charts {U3. ,w.} of (2.2). Moreover, for a partition of .
.
(with supp XOC UO) a correspon-
unity P,9=1 subordinated to 3
J
7
J
1
ding partition subordinated to {u wj} is defined by %j=(X9j )os 3 j, SSO,(1,1,1),0' Notice that, in admissible coordinates, we have %a)=0((x)-Ial) for all a, (T extended 0 outside a(U) in i.e., 3
3
len). The same is true for the restriction x=x°In of a C (St°)-func-
126
4. Operators on manifolds with conical ends
tion x° with support in a single chart U''. We will call such {X
and x an S-admissible partition of unity, and cut-off function, respectively. (Often we drop "S-admissible" for "admissible"). With an admissible partition {Xj} and finite atlas {Uj,wj} we define a Riemannian metric on st by setting ds2 = 71j% jdx2
(2.12)
,
with the Euclidian metric dx2 of wj(Uj)C ten, in each summand.
From prop.2.1 it is evident that an admissible coordinate Rn into a change cp:Rn -10 converts the Euclidean metric dx2 of metric of the form 2 hjl((s(x))dxjdxl where hjlE Cm(B?)
,
and the
matrix ((hjk)) is defined and positive definite still at asz°. Thus
in admissible coordinates, the metric tensor ((gjl)) of (2.12) has the same properties - gjkos-1 extends as a Cc'-function to the infi-
nite boundary of the chart, and the matrix is >0 there. and admissiProposition 2.2 For each infinite boundary point p° ble chart p°E ,)C w(U) in there exists a cut-off-cone CC y°=w°(p°) such that (i) the Riemannian distance d(p,q) of the metric (2.12) is equivalent to the Euclidean metric in CCy"Yl in Y,
,
the sense that c , x-x° l s d (p, p° ) s c x-x° l for all x, x° E CCyo and corresponding p,p°E sz
;
(ii) the Riemann space fl (with metric
(2.12)) is complete; (iii) after a linear transform the metric is approximately Euclidean, insofar as gjl(x)=gjl+sjl,
=const. in CCy°
,
with E jl->0 as
(iv) gs 1
is C
,
and
the metric is positive definite, even at an' A complete Riemannian space n with above properties - e.g., .
st admits a compactification 9?°, a compact Co°-manifold with bounda-
ry; a finite atlas of n exists with charts derived from charts of st° in the sense of (2.1),(2.2); in a neighbourhood of a point of ,
an' =n°\n the metric of sz has the properties (i),(iii),(iv) of prop.
2.2 - will be referred to as a space with conical ends. Note that this term is used synonymously to 'manifold with S-structure' or 'S-manifold', 'S-space', in the following sense: An S-manifold becomes a space with conical ends by introducing a metric of the form (2.12) (or any other Riemannian metric with properties (i)(iv)of prop.2.2 - called a conical metric). Vice versa, a space with conical ends is just an S-manifold together with a conical metric, not necessarily of the form (2.12). In the following we always assume a space with conical ends, and then choose the sur-
4.2. Distributions on S-manifolds
face measure dµ=dS
127
(=det((g]k))1/2)dx) of the conical metric ds2= for our distributions and Sobolev spaces
= jjkdxjdxk as
In particular we introduce the Hilbert space H=L2(SZ,dµ)
.
As a side remark it may be mentioned that the quasi-Euclidean metric ds2 has a corresponding Beltrami-Laplace operator A . Using the comparison triple {Sz,1-A,dµ} in the sense of [C2],V one may generate L2-comparison algebras with complex-valued symbol, containing a core of Wo's of the kind to be studied below. Remark. For a Riemannian manifold sz introduce d(x)=d(x,x°) , xE S1, and d(%)=sup{d(x):xE X}, X a subset of Sz , with the geodesic distance d(x,x°) and a fixed point x° . An end Sz' of a is defined as
a subdomain of Sz with compact as1' and d(i1)=- such that no decomposition n' =SZ, Uiz exists with a, M bounded and d (Sk, ) =d (SL )=w (think of the two "ends" of a cylinder). (Actually, two such Sk`, called S1'
,
all
, define the same end if c' \SZ" and Sz"\SZ` are bounded.) is isometric
Such an end will be called conical if (a suitable) Sk'
to an end of an S-manifold, as above. It is possible that some ends of a Riemannian space Sz are conical, others are not. Also, one may consider cases like the cone x1=V(x2+...+xn , x1>0 , having the concial end{x1>11, but
a conical tip {x1<1}. Our theory, below, will not apply then. But
the conical end and conical tip may be separated, using algebra surgery, similar to that in [Cz],VIII, as not to be discussed here (cf. the problems of chapters I, II, III, IV). We summarize, below, also discussing some trivial additions: a)Temperate distributions and rapidly decreasing functions are introduced for a noncompact manifold S1 which is the interior of a compact manifold 91° with boundary, where acz°=0 is permitted, giving the special case of a compact s . We define S=S(S1)=CO00 (S1° ),
S ' =S' (Sk) =D° ( c ) _{ uE D' ( c ) : u=v Sk with vE D' (Sk` ), with an S1'
=}
b) S(Sk) and S'(n) assume their conventional looks only in special coordinates: An S-structure, making 91 an S-manifold is in-
troduced by declaring certain charts, coordinates, coordinate changes etc. as admissible. Admissible charts, atlantes, cut-off's, partitions of unity, all are obtained from charts,..., of nO, using the map s(x)=In-B1 of (2.1), (2.2), ... :
.
c) A neighbourhood base of an infinite boundary point in admissible coordinates is of the form (2.5). d) An S-manifold sk possesses a distinguished type of Riemannian metric ds 2 called conical metric. In admissible coordinates ,
4. Operators on manifolds with conical ends
128
ds2 has properties (i)-(iv) listed in prop.2.2. A manifold with conical ends is defined as an S-manifold c with distinguished conical metric. Write d(p,q) for the geodesic distance, and d(p)= =d(p,p°) with a fixed point p° and variable p in the following. The distinguished measure dµ on S2 is introduced as the surface
measure of the distinguished conical metric. We have dµ=x(s(x))dx in admissible charts, with dx in local coordinates and KE C,(U°). Also, globally, we have dp.=(d(p))n-ldv with a positive Co measure
dv on the compact S2° e) We have
,
and (d (p)) = 1 +d (p )1
S(S2)= {uE C00(St): (Xu)ow lE S(len)
(2.13)
for all w,X}
,
and
S'(S2) = {uE D'(S2): (?u). (x lE S'(ien) for all w,X}
(2.14)
,
where, in each case, w:U-w(U)C In and X denote an admissible chart and subordinated cut-off function (e.g., supp ?C U) f) The space (2.15)
LpI01(a) = {uE Llloc(n):
d(p)-kuE L1(SZ,dµ) for some k }
is naturally imbedded in S'(S2) by
(u,) = J
(2.16)
u(pdµ
, TE S(n)
92
g) Define L(S2) as the space of differential expressions L:C00 (11)->Coo(S2)
such that in every distinguished chart w:U->w(U)C In
L is represented by a PDE Maaaa with as of polynomial growth -that is, Xaaow l E T(&n), with T(In) of[C,],p.28, the space of S'-multipliers (cf. [Schwi]) for every admissible X, supp XC U. Then ,
L:S'(SZ)
-> S'(S2) for all LE L
,
and such L is a contionuous map.
h) We have
(2.17)
S(a) = {uE C00(92): Lu=O((d(p)) -k) for all LE L
,
ka0}
i) The topology of S(S2) is the Frechet topology of C00(92°)
where k=0,1,..,
it may be generated by all seminorms II(d(p))kLull L 00
and where L ranges over L(c) (or a suitable countable subset). In S'(S2) we use the topology of weak convergence. (Note that the
inductive limit topology based on S'=USAS is available as well.)
4.3. Coordinate invariance of operators
129
3. Coordinate invariance of pseudodifferential operators. It is trivial that Apdo's on in transform into pdo's on in
under a linear change of coordinates, of the form x=mx'+p , where
and m=((mjk))j,k=1,..,n is invertible, mjkE & PE &n a E ST , and That is, if Au=a(M1,Mr,D)u=f, where u,f E S(ien) ,
u(mx'+p)=v(x')
,
f(mx'+p)=g(x')
,
then we get
(3.1)
where ImI=Idet ml, ff for a moment. With =m (3.2)
t=(m 1)t,
we get
t 'MY')
g(x')= Joxlfol
generally is Or, g=b(M1,Mr,D)v, where a(mx+p,my+p,m in the same symbol class as For more general local coordinate changes and local ivdo's a
we refer to Hoermander [Hr2]. A very elegant proof may be found in [Fr3]; its idea seems due to Kuranishi (unpublished?). Using this technique Schrohe [Schr3] proved a result for a class of global transforms and pdo's on In of our general kind. Here we will use the same 'Kuranishi trick' again, for a subcollection of OpST. We only admit coordinate transforms T:1n-1n with the property that sog0s-1= V extends to a diffeomorphism B +.B1 . Clearly then the diffeomorphism T:2n-jkn extends to a homeomorphism En+ En. The homeomorphism may be used to carry over the manifold structure of Bi to the compactification in of &n . Thus we may regard En as a compact manifold with boundary. The above type of coordinate transform is precisely the class of homeomorphisms EnH En preserving this manifold structure. Given such g:1n-2n and a 'pdo A=a(M1,Mr,D) , aE SSm,P,b let again f=Au, withffu,fE S. Let g(x)=f(g(x)) , v(x)=u(g(x)). We get (3.3)
g(x)=fob
where Jg(y)=Idet((aTj/ayk))l
.
Let us split the expression at right of (3.3) into two parts by inserting a partition 1=x(s(x)-s(y))+w(s(x)-s(y)) under the integral signs, where co(z)=0 for lzIse, w(z)=1 for jzj2-2e>0. Writing
correspondingly, we first consider g2. First look at g2 in the old coordinates. With f2(x)=82(6(x)), 6(x)=g 1(x), we get
g=g1+g2
(3.4)
,
f2(x)=JRiJRye1(x-Y)w(s(6(x))-s(6(Y)))a(x,Y,)u(Y)
4. Operators on manifolds with conical ends
130
With the inverse function t= (3.5)
l=s.0.s-l we have s.O=l
s
hence
,
f2(x)=fo1
We use the identity
The inte-
grand vanishes near x=y. An N-fold partial integration gives
f2(x)=fO1
(x))
foy.il
(3.6)
with aN(x,Y,) = (-A It is clear that this partial integration is legal, similar as the ones performed earlier. The function 1, as a diffeomorphism B1+B1, satisfies an inequality It(x)-t(y)Iaplx-yI , with some p>0 Hence .
the integrand in (3.6) vanishes for p sp . In other words, we may assume that Is(x)-s(y)Iap . Observe that s-l(x)=t(x)= x (3.7)
T(x) = ((tilxl)) =
,
where [x]=V . We get
implying T(x) to be real symmetric, T(x)a
t(x)-t(y) = fd-r
=(I+t)(t)
l 3((6il+titl))
UT-x))) ={
1XI
, xE B1. Notice that
fdtT(x+t(y-x)}(y-x) = S(y-x),
where SaMin{ th , [ y] 1. We get 1 =(t (x)) , hence It(x)-t(y)Ia(Min{(t(x)),(t(y))})Ix-yl, x,yE B1
,
or,
(3.8)
Ix Y1
Min{(x),(Y)}
s(x)1s(Y)I
,
x,y E&n
Thus we may assume that (g)2NMax{X2N(),x2N(Y)}
Ix-YI-2N 5
(3.9)
,
Assuming aE SS., P'6
,
fi(t)= I
p1>O, and N sufficiently large the
integrals f dgf dy in (3.6) may be interchanged, for
f2(x) = f9iyu(y)k2(x,y)
,
with integral kernel
(3.10)
k2(x,Y)=
Ix-YI
f
where N is arbitrary (sufficiently large), and where since the i-order of aN is m1-2Np1 < -n
,
fd
exists,
as N gets large.
4.3. Coordinate invariance of operators
131
Proposition 3.1. The kernel k2(x,y) of (3.10) as well as its transform k2(c(x),cp(y))J (y) = k2IT(x,y) under the coordinate (P change x -* T(x) are inS(&2n) . Hence the operator K2 defined by K2u=f2 , and its transform under g are qdo's of order Proof. The integral in (3.10)2 may be written as a4(x,y,x-y). Clearly vaN(x,y,z)E Coo(&3n\{z=0}). Indeed, aN(x,y,z)= ,
(- I2M)aN+M(x,y,z) where aN+M decays better as M increases, so aN+M admits more and more z-derivatives, as M-- . Similarly,
(3.11)
aUdyaZaN(x,Y,z)=O((x)m`(Y)
)
,
as Izk s>0, x,y,zE &n,
with 'O(.)-constant' depending on N, but not on the mj. Therefore,
(3.12)
aaayaN(x,Y,x-Y)=O((x)m:(Y)m')
,
as x,yE &n
,
Ix-YIae>0
For the factor k3 in front of the integral in (3.10)2 we get
(3.13)
aaayk3(x,y)
=
O(?.21(x-y) Max{?.11(x),?J1(y)})
where, of course, ri=3N may be choosen arbitrarily large. Here (3.14)
Max{X1(x),X1(y)}=?1(x)Max{1,(
F1} s kY'(x)(x-y)11
using the well known inequality. Accordingly, (3.15)
aXayk3(x,y) = O(X1(x)X1(x-y)), for all ri>0
Combining (3.12) and (3.15) we indeed get k2E S(&2n) The remainder of prop.3.1 then is a consequence of prop.1.1 and prop.2.1 The map (x,y)-(s(x),s(y)) takes S(12n) to CO-(BixB?), which is preserved by (x,y)-(lp(x),i(y)) The Jacobian determinant defines a C00(RO)-function J(os-1 , by prop.2.1. Q.E.D. .
After dealing with the part of A=a(M1,Mr,D) belonging to w(s(x)-s(y)) we now turn to the other part, (3.16)
gi(x) =
(Y)v(Y)
foI
where X=X(s(x)-s(y))=0 for js(x)-s(y)ja2E
.
Write
1
g(x)-(P(Y)= 0 di{atcP(Y+i (x-Y))}= M(x,Y)(x-Y)
(3.17
where
M(x,y) =
dx(aq)/ax)(y+i(x-y)) 0
If x and y are close then clearly M(x,y)zt aye/ax which is an invertible matrix. We have (T(x)-cp(y))= Then an inte-
4. Operators on manifolds with conical ends
132
Mt(x,y)
gral substitution
,
for fixed x,y, close together, may
be used to convert (3.16) into the form of a ipdo-integral again.
Having employed our partition we may assume now that js(x)-s(y)js2s , where s>0 is arbitrary. The question is whether this condition is sufficient to guarantee a global such integral
substitution, valid for all x,y, with js(x)-s(y)Is2E Write M of (3.17) as M(x,y)=
.
dtQ(s(y+t(x-y)) with Q(s)= 0
a(p/ax( s ) E C'(B1) (by prop.2.1). The curve {s(y+t(x-y)):t E &}=T
is a half-ellipse with center 0 and vertices at
and 7
X_Y z= vector to the point of the line {y+t(x-y)} closest to
easily seen. The arc {s(y+t(x-y)):Ostsl} connects s(y) and s(x) on r. It is contained in the ball Is-2(s(x)+s(y))Is2ls(x)-s(y)Iss. The COO(BI)-function Q(s) is uniformly continuous and invertible
Thus indeed E may be chosen small to insure invertibility of M for all (x,y) as js(x)-s(y)ks2E. Carrying out the integral substitution as indicated yields (3.18)
(x,y, )v(Y)
a
detM(x,y) a(y(x),T(Y),M
, with
X(s()-s(Y))
.
The following observation about the new symbol a" is useful. Proposition 3.2.
For any function b(s)E COO(B?) the composition c=b,s is a symbol in SS0 More precisely, we have (1,1,1),0 (3.19)
c(a)(x) = 0((x)-ja1)
,
for all a .
Indeed, this is a consequence of the chain rule, and the fact that s(a)(x)=O((x)-kaI) for all jal ,
.
Looking at the -derivatives of a of (3.18) it appears that the parameter pl remains unchanged under our coordinate transform. An x- (or y-) derivative may land on the M t(x,y) inside the
t-argument of a 1 factor 0 (( x)
,
or anywhere else. In the first case we get a
in addition to the already existing ones. In the other cases we get a factor 0((x) -p: (l;) S ) or 0((x) 1-p,
(All this for an x-derivative).
It follows that a E SSm,p,, with 6°= Max{S,1-pi}. To verify this we use prop.2.1 when differentiating for the x in g(x): acp/ax
is of the form needed in prop.3.2. Also, bounds of the form 0
, with p.>O
,
and pjsl
,
j=1,2,3.
4.3. Coordinate invariance of operators
133
Assume that cp:Rn->1¢n with inverse 9:&n-&n has the property that s,(P,s- 1
extends to a diffeo(with s(x) (X = ) a map VTI T-7 morphism B?H B1 of the closed unit ball B1. Then the linear operator A" = T(PIATT , with A=a(Mi,Mr,D) and T u(x)=(u.T)(x)=u(T(x)), u E S, is a pdo again with symbol a E SSm, 66 =max{6,1-p.}. Up s to an additional term in S(22n) the symbol a" is given by (3.18). Usually we will tend to apply thm.3.3 for operators of the form a(x,D) a(Mr,D) a(Mw,D) , with aE ih These are special cases of A=a(M1,Mr,D) in thm.3.3. However, thm.3.3 then will give an A =a (M1,MrD) which must be converted to b(x,D), c(Mr,D), e(Mw,D), using I,thm.6.2, assuming that S°
P,
,
.
,
1-p. 2. We therefore have Corollary 3.4. Let aE tphm1 p s with pz >O
,
p. >2 , S
.
Then the
coordinate transform of thm.3.3 takes each of the operators a(x,D) a(Mr,D) a(Mw,D) into an operator of the same form with symbol in 111hm, p , 6A , S° = Max{ O , 1-p. } . The new symbol, up to terms in m'<m , is given by a with a" of (3.18). Problems. 1) For n=1, consider the transform t-x=q(t)=et, a diffeomorphism & .&+=(0,o) (and a group isomorphism). 'p(t) and its in,
verse t=6(x)=log x ) do not satisfy the cdn's of thm.3.3, but they take A=a(M1,Mr,D) to a i,do A if only supp KxKxl, KCC P ( KCC R+ in case g). 2) The Mellin transform is defined as M=FTT with the Fourier transform F and Tu(x)=u(cp(x)), c of pbm.1. Express M and M like Fourier integrals. Show that M is the FourierPlancherel transform of the group &+: Define c=a © b =fa(y)b(y)ay, for a,bE L'(I,dx), and get Mc=(Ma)(Mb). 3) Consider Riemann space with metric ds2=y,(Sjk+ni nk)dxi dxk
,
R2*=R2\{0}
as
ni=xj/Ixl (a co-
ne Z isometric to {y=jxj}C 13 ={(x.,x2,y)} ). Show that this cone is mapped conformally onto the cylinder {-oo
tric 2dt2+d62 by the map x=(xl,x2)- (t,8), where t=loglxl, O=arg x arg x = arc tan(x2/x1). 4) Use the diffeomorphism between cone and cylinder of pbm.3 to install a natural class of pdo's on Z. In particular, the pdo's should be the global coordinate transforms of the "cylinder pdo's" introduced in II,3,pbms 1,2. They should be
"local tpdo's", in the sense ?.At is a t,do for X,pE C 00 (Z). Try for a concept of K-parametrix, for md-elliptic operators to be defined especially with an operator-valued symbol at the conical tip.
4. Operators on manifolds with conical ends
134
4. Pseudodifferential operators on S-manifolds. We start our discussion with a generalization of I, thm.3.3. Proposition 4.1. For the special S-manifold S2=1Qn let 0,?. be admis0°=00s-1
and 7v°=X s-1 extended sible cut-off functions such that to B1, have disjoint supports. Then, for any AE Op1Uhm we have 0A?. E Opph-00 , an integral operator with kernel in ,
S(12n).
Proof. By I, thm.3.3 the distribution kernel k(x,y) of A has singular support at x=y only, thus the kernel K(x,y)=k(x,y)0(x)X(y) of 2n). 0A?, is C'(3k
In fact, k equals a function in S for xE KCC &n and large lyl, and vice versa. Moreover, looking at I,(3.15), using that Ixkklx-yl as xE supp 0 , yE supp k 1y1"Ix-y1 ,
lye >> 1
,
,
,
.
IxI
we indeed find that xyy6aadyx(x,y) = 0(1), for
all a,(3,y,S - i.e., xE S(12n)
q.e.d.
,
Now assume that 92 is a Riemann space with conical ends, in-
a compact manifold with boundary, as in sec.2. We will use admissible cut-off functions now, defined as 0=01os, with terior of S2°
,
supp 01 C U° with a chart o
:U° -w° (U° )C B1 of S2°
,
and s of (1 .5) .
The above relation between the cut-off 0 and the chart U will be expressed by writing 0 ® U . The same notation will be used for another relation: Writing 0 (> O X , or , 7v m 0 (4.1) indicates that 0=0°,s X=X°,s where 00 X° are cut-off's of ,
S2°
,
with support in a chart U°
,
,
and 00=1 near supp X°
The following is evident: Just look at s?
.
instead of sz
Proposition 4.2. For a given admissible cut-off function 6-:* U we may construct an infinite sequence 0=0 0<10 01 ® 02 ® ... U. <XJ First define the class e =LS-- LS-,,(S2) of integral operators
(4.2)
Ku(x) = J k(x,y)u(y)dµ(y)
, uE S(S2)
(or uE S' (92), with a distribution integral) with kernel k(x,y)E S(S2xS2) = COOO(S2xc2) (note the remark above (2.12)). We will regard
dt as the class of gdo's of order --, as in case of S2=ien
.
A pseudodifferential operator A on n then is defined as a continuous linear operator A:S(S2)-> S(92) such that, given any admis-
sible cut-off function X and chart w:U-Ten ists A,=a,(x,D)E Optphm p S (4.3)
,
with 7`GQ U, there ex-
, KXE r with
as x
A(Xu) = KXu + (AX((Xu)0w I))°w , uE S(S2)
where we assume m,p,S independent of k and w:U->&n
.
w(U), and
135
4.4. Operators on S-manifolds
Note, we assume m=(m. ,mz)
,
p=(p.,pz)
S independent of X ,U.
,
As usual, m1E &, Ospjsl, OsOsp., but for a local calculus of ipdo's and coordinate invariance we require pj>0, 1-pisO
such yxlo's for a given m,p,b will be denoted i LSm,p,O. Write LSm =U LSm,p,s, union over p,b with p>O, 1-p.sO
will be focus of interest. We thus define LCm LSm,e,O
'
LC. =U LCm.
Note that (4.3) is of strictly local nature, insofar as ku as well as (AX((Xu)ow 1)ow have support in a single admissible chart U. Thus it is practical to regard w in (4.3) as an identification of the points of U with those of w(U) - i.e., as the identity map, which may be omitted in writing. Then (4.3) reads
A(Xu) =
(4.3')
AX(a.u)
.
Note, we also get
A(au) = Kku + k° AX(Xu)
(4.3")
with any other admissible cut-off ? o
,
%< 0 ko Q U . Indeed, we get X< 0U9 with the admissible chart . By definition we get (4.3'), possibly with another but ak=O for x Uo, hence (4.3") follows. On the other hand, if (4.3") holds for some X Oko Q U we may use the symbol in place of to get (4.3'), for equivalence (4.3') ca(4.3") From now on we always assume AE LS - i.e., O
Uo={Xo=1}into U =wI Uo
If we have both XQ U, 0<) U, for two cut-off's X,O
,
then
A),O=a),(x,D)XO+KXO=a0(x,D)XO+KOX, showing that (ak-a0)(x,D)OXE(f . If X(p)*O, O(p)*O
for some pE ft° then prop.4.1 implies existence of an admissible cut-off x with x=1 near p such that ,
,
S(12n)
(4.4)
Clearly (4.4) expresses a uniqueness property of the local symbol. Note that LSm,p,5(&n)=OpVhm,p,s
,
assuming p.>O, 1-p.sb
First let A=a(x,D)E Opiph. For X.ZJ?o Q U
,
w=idI U ,
construct
%QXo
.
,
7
7
,
4. Operators on manifolds with conical ends
136
AE LS" p,6(1Qn) choose U=& n, w=id. to show that AE Vh
m,p,6, For general sa we clearly get LS-,O(sz)C LSm,p,6(a). For general
AE LSm and finite admissible partition of unity EX j=1, % <E)Uj get Au=2 AXju=y, XA), (Xju)+Ku , uE S , KE LS-00
(4.5)
aj
just by repeated application of (4.3). Vice versa, a yxlo AE LSm may be constructed by assuming AX =AjE OplyIhmIP.b in (4.5) as arbi7
trary i,do's. Before confirming this let us prove the following.
Proposition 4.3. For admissible cut-off's 0=0°os
,
?=1,.°os, if
(supp 8° )fl(supp x, )=Jd, then OA), E LS-00 , for every AE LSm . Proof. Construct a finite admissible atlas and partition of unity Eoj=1 wj:Uj_wj(Uj)C In such that for each j either %.<10 U. ,
Ujfl supp
,
or U9fl supp 0° = jd
.
29(0A), A +K )k
29(A),
OA?.
With this partition write
The sum may be extended over j with Oflsupp o, 7 for 0X vanishes. In the left-over terms we get
0, else the facsince J
then k jk = 0 , so that 6AX = E0Kj). E LS-00, q.e.d. For {Uj,wj}, A. j, as above, and general AjE OpVhm,p,S write (4.6)
A= 3
3
Let us prove that A E LSm, p , s . We trivially get A: S (SZ) ->S (c) continuous. To confirm (4.3) let A., X. be as in (4.3"). For each j the
product U j is an admissible cut-off with supp(.Aj)C Ufl Uj
.
The
coordinate transform w o : w . (Ufl u w(Ufl u.) is of the form x=wow 1 = s-1x° os with a7diffeomorphism x° :w.( U° fl u .) -* of (U° fl u .) 0
7
7
J
J
We may assume the partition wj refined so far that x° differs little from a linear map. Thus we may assume x° extendable to a diffeomorphism B .Bi . Hence x extends to a diffeomorphism In E-,&n Thm.3.3 (or cor.3.4) may be applied to transform Aj to the coordi-
nates of the chart U resulting in a 'do Aj° E OpPhm,p,S precise, in the formulation of (4.2) we have ,
(4.7)
(A1((X), ju) "wjl))owj = (Aj°((AAju)oW
with properly extended diffeomorphism wowjl
. To be
))ow
but (%Xju)owil has its support within wj(Ufl u.). Using prop.4.1, write the right hand
side of (4.7) as (0 1Aj° ((..ju)ow 1))ow + Kj
,
K1E LS-.(&n), with
an admissible cut-off 0j, supp 0 j C U. By prop.4.2 the integral
137
4.4. Operators on S-manifolds
operator Kj has a kernel of the form v1(x)u2(y)kj(x,y), with admissible cut-off's vj, supp v.C U. Returning to the notation of (4.3)
-(4.3") it follows that indeed each (X9AXjX) transforms to a term of the form (4.3"), proving that A of (4.6) belongs to LSm.p.S. Clearly a differential expression LE L(n), L(c) of sec.2,(g) defines a pdo LE LS for some N (the order of the expression) provided that the local coefficients as in admissible coordinates satisfy aa(P)(x)=O((x)f-P= I(3I) ,
for all a,3
As for differential operators a global symbol in general is not defined, for an AE LSm,P,s although we have a well defined local symbol
ax is not unique, since
(i) any term c(x,l;)E S(12n) may be added, and (ii)
is more
or less arbitrary for x outside supp X. For a differential operator LE L(n) the local symbol coincides with the polynomial a(x,i;)= where in local coordinates L=Eaa(x)Da, D=iax It is clear that we will seek properties of operators in LS
similar to those of Opo in earlier chapters. We summarize corresponding facts in thm.4.4. Proofs are straight extensions, and will not be discussed in detail. Theorem 4.4. The classes LSm
each given p?0, 1-pi <S
.
P.S
form algebras for
for
More generally, we have
LSm,P,S.LSm',p,S C LSm+m',P,S
(4.8)
,
LCm.LCm,C LCm+m' .
LSOO, LSm One finds that LS-,Ois an ideal of LS0 , m=0,0 PS All spaces are invariant under the involution "*" defined by the ,
.
Hilbert space adjoint (with respect to (u,v) f 1vdµ ). Calculus of 'ado's holds locally: For AE LSm,P.SC LSm , BE
LSm,,P.S the local symbol cx of C=ABE LSm+m,,P,S is expressed by the asymptotic Leibniz formula (of near any point pE St° with x (p) x0. The symbols ax,ax of A E LSm.P.SC LS and its adjoint * A are related by an asymptotic I,(5.1)3 (with x= ). In detail,for
admissible X.J?o ®U, near pE c° with X(p)x0, a....... of (4.3"),get (4.9)
cx=
i
6
9
)bx(e)
a
ax=
Kl;
{xax}(9) (mod
h-,O)
6 9 where "f=g (mod (Z) near p" means (f-g)xE(; with a cut-off x near p. Local symbols are unique: ax=aO (mod()near X(p)xO*O(p), cf. (4.4). The left multiplying representation of (4.3") may be replaced by Weyl (or right multiplying) representation without changing LS. As next important point we shall look into existence of a
parametrix, for A E LS
.
Here A,BE LS are said to be parametrices
4. Operators on manifolds with conical ends
138
of each other if AB-1
,
BA-1 E LS-00 . This problem is quickly res-
olved, using our local parametrices of 11,3. For a finite admissi-
ble partition 1=1k let
as in prop.4.2.
Given AE LS write AR=K j+A11 µ
,
µ=X.
. Assume Ap to admit a left K-
parametrix B. with respect to the symbol X.
Write Aµ Aj
.
Notice we also have
.X2A
.
,
so that BJA11=A..+Kj
= X, X. + K j ,
since
E Op,ph-co , by prop. 1 . 1 . As a consequence, ? BjXjA = .B .X .A .k3 + .1B .??A(1-X3 ) _ c +K
,
333 J J J
using prop. 4.3 with A=T,?
J J JJ J j j X=1-X.. Assuming all left parametrices B. to belong to the same m'2:-m, it follows that B=l?.jBj?j E Alm,,P,b is a left K-parametrix of the operator A E LS . Involving adjoints to possibly convert right into left parametrices, we have proven: LSm . For an admissible partition X>,.j Theorem 4.5. Let AE =1 , ?.i 61 ®Uj with admissible cut-offs ej and charts w:Uj-.&n let the local operators A. of (4.3') admit left (right) K-param,
j
etrix in 4*m',P,B
,
(with inverse order m') with respect to Xj
,
for all J. Then A admits a left (right) K-parametrix of order m' A pdo A E LSm,p,6 C LS is said to be md-elliptic (of order m) if for every admissible cut-off 6 ®U , w:U -> ien an admissible chart, there exists Oo (E>6 Bo o` U such that the local symbol ,
a0,
,
is md-elliptic (of order m) with respect to 0
.
Theorem 4.6. An md-elliptic 'do AE LSmp S C LSm admits a parametrix B E LS-m,P,6 . Vice versa, if AE LSa,P'6 C LSm admits a K-parametrix in LS-m,P,b then it is md-elliptic. This theorem is an immediate consequence of thm.4.5, as far as sufficiency of the condition is concerned. For necessity one must use local calculus of pdo's .
Theorem 4.7. Suppose AE LSm,P'OC LSm has the property that, for some m'a-m and some admissible finite partition 1=EXj , %j<0ej® wj:U.-in U. , with admissible ej the local operator A0 of ,
,
J
(3.3) is formally md-hypo-elliptic (of inverse order m') with respect to %j. Then A admits a K-parametrix BE LSm,,P's of order m' This theorem is an evident consequence of thm.4.5. Problems. 1) The operators B: COO(S' ) -C"O(S' )
of III,4,pbm.1 are ydos
in LS(S'), under proper assumptions on the sequence bj(0) of periodic functions. 2) Consider y,do's A with operator-valued symbol for fixed x, , is a pdo in LCo(B), for a A(x,i), where
139
4.5. Order classes and Green inverses
smooth compact manifold B. As in 1,6, pbm's 1-4 and 11,3, pbm's 1C=Bx&n properly defined. We 4 such operator should act on S(C) ,
,
should get a t,do-calculus, a K-parametrix construction, and, gene-
rally, results analogous to those in the problems mentioned, including those of C in [Cz],VIII. 3) Reflect on a theory of 'ado's on a more general Riemannian manifold st which has conical and cylindrical ends both, as well as conical tips: In a subdomain sic containing a single cylindrical end, stc should identify with a neighbour-
hood of the right end of the cylinder of 1,6, pbms.1,2,3 (or of a
more general cylinder MxRk , M a compact manifold). Near a conical tip nz should identify with a neighbourhood of the tip x=0 of the cone Z of sec 3,pbms.3,4 (or a more general such cone (0,o)xM, M compact). There should be an operator-valued symbol at a cylindrical end as well as at a conical tip, but not at a conical end (cf. [CDg]). 4) Show that the operators A of pbm.2 are tpdo's on the non compact manifold C=Bx&n insofar as, for cut-off's (o,x with support in a chart UCC st , we have wAx a ipdo on &n uE S'(ft)
,
5) For a distribution 12 a manifold with conical ends, define the concepts of .
WF(u) - the wave front set - and ZF(u), looking at ZF (Xiu) of xi <s)Uj 11,6 and VI,7, with an admissible 1=1k wi:Uj->&n A wave fron space W(c) may be introduced such that WF(u)C W(st). Show that W(R) may be interpreted as the cosphere bundle S*(c) i.e., the bundle of unit spheres in the cotangent space, with respect to any (conical) Riemannian metric on st What would be the corresponding space Z(a) for ZF(u) ? ,
,
.
,
.
5. Order classes and Green inverses on S-manifolds. On an S-manifold sZ introduce the weighted Sobolev norms
HAS = {lllxiulls} 1/2
(5.1) with l=?
uE S' (st)
,
s=(s ,s2 )E &2
a given fixed finite admissible partition of unity sub-
ordinated to an admissible atlas {U,}. To be precise, we abbreviated IIxiu]IS=II(xi IIns((), ju)owj1)H1 2 n). The (L2-Sobolev-) L (R
space Hs=Hs(s2) consists of all uE S'(c) with finite norms Ilull s Generally, when writing hulls we imply h1uh1s
,
i.e., uE Hs
.
Again we regard the maps wj as identifications, hence k u i
as functions on &n (extended 0 outside wi (Ui)). Notice that *
(5.2)
hlxiuIhs =
s
Sk.u)
,
as uE Hs
4. Operators on manifolds with conical ends
140
with the pairing III,(3.28), or, with the inner product of L2(in), as uE H2s, where Qj=Xjrls E Oppc2s Hence we get .
s
(5.3)
11ulls=(u,P2su)
P2s=DcjQjkj
,
,
xj=dµ/dx
,
for all uE H2s
with the inner product (u,v) f uvdµ of H=L2(n,dt). Clearly P2s is as follows by comparing (5.3) and a ido , P2sE LS2s,e,0 = LC2s (4.6). Moreover, P2s is md-elliptic of order 2s, as confirmed by ,
looking at the local symbol: At each point it is a finite sum of transforms of the non-vanishing symbols of Qjkj, all of them locally md-elliptic and nonnegative, so that they cannot cancel each other, looking at formula (3.18)
.
It follows that P2sP-2t and
P-2tP2s both are md-elliptic of order 2s-2t, by calculus of pIdols. From estimates III,3,(3.19),(3.21) we conclude at once that (5.4)
HtC Hs
,
and 1Iu11ss cs,tllullt
as s s t , uE Ht
,
.
Also one at once confirms III,(vii), i.e.,
SHS(n)
(5.5) S(cz)=
,
S'(c) = UHS(c)
,
as well as formulas similar to (3.22) through (3.27) in ch.III In detail, we have (5.6)
Hs={uE S'
:
k u E Hs(in)
,
j=1,...,n}
.
i
This validates III,2,(i)-(v),(vii) for Hs(n), just as for n=ten
Similar arguments imply that, again, S is dense in every Hs. It should be desirable to obtain an isometry H ->H
,
just as
s of (3.2) for 92=1n. Note that P2s:dom P2s = H2s* H=L2(n,dg) may be regarded as an unbounded hermitian positive definite operator of H , assuming sja0 It is found that the positive square root of the Friedrichs extension of P2s (or its continuous extension to Hs ) provides this isometry. Instead of engaging in an argument to show that this operator is a yxlo in LCs we prefer to use the operator PsE LCs instead. While Ps generally is not an isometry we will show that it is an isomorphism Hs H H at least. .
Indeed, it already was seen that Ps:Hs - H . In particular, IIPsu112 = (5.7)
11 D,j'/2s/2 a'jull25 cI11ns/2 s/2Xju112
s cy'llsXjull2 = cllulls , uE Hs using III, thm.1.1 on ns/2s/2 sl E
to show that Ps:Hs-H is
4.5. Order classes and Green inverses
141
continuous for every s. As md-elliptic operator Ps admits a K-parametrix Q_S, a i,do in LC_s. If uE H, Psu=O, then 0=Q_sPsu=u-Ksu, Ks E LS_00,
=::,, u=KsuE S =*- (u,Psu)=7'11
u=0, showing
ju=0
s/2Xju112=0,
that Ps:HS-H is 1-1. In H let fE H be 1 to im Ps. Then (f,PS(q)=0, cpE S
0=(f,PSQ-sp)=(f,(1-LS)i)
,
i E S
where LSE LS_Q. Hence
,
(f-LSp,i)=0, iE S f=LspE S b0=(f,Psf) =7' IIns/2kjfII2
f=0
.
It
5H5 defines an follows that P5:H5-H has dense image P5. Then Ps1:P-' unbounded operator from H to Hs with dense domain. Ts=PsP-5E LC0 TS is bounded Proposition 5.1. For WE 12 we have -1 in H=L (S,dµ) and has a bounded inverse Ts :H-H. Moreover, E LC0 is a yido as well, and thus is a special K-parametrix of T5. .
Proof. We already noted that TsE LC0 is md-elliptic, hence has a K-parametrix RSE LCO . As t,do's of order 0 TS and RS are L2,
bounded. For example, IITsufl=112rs, j?ju+KuUISCIIuII+jIITS' jull , where III,thm.1.1, or III,thm.2.1 may be used to show IITB'jullscIlull
Next confirm that ker TS =0, and that im TS is dense in H, where we mean the map TS:H-H. Indeed TSu=PSP_SU=O yields uE S, using the K-parametrix RS, then P-SUE S=>P-Su=0 u=0, so TS is 1-1. Similarly, using Rs, show that (f,Tsp)=0 for cpE S implies f=0, hen-
ce im TS is dense. Existence of the (L2-bounded) K-parametrix RS amounts to existence of an inverse mod K(H). Thus TS is a Fredholm operator (cf.[C1],App.Al,thm.4.8), it has closed range. Since ker TS=O and im TS is dense in H , TS is invertible, Ts1E L(H). But we
R5T5=1-L5, with K5, LSE
have
This implies RS= SubstiOr, TS =RS+Ts Ks, and T5 =Rs+L5T5
T5 -TB KS=Ts -L5Ts tute the second into the right hand side of the first: Ts1=Rs+RSKS +LSTs1K5. Here RSE LSO, while R5K E LS_,0. The third term also belongs to LS_0 , since we get IILsTs KsullmSClIIT51Ks1II5C2IIKsuII5c3IIUIIm' .
for all m,m'
.
It follows that Ts E LCO
.
,
q.e.d.
Using prop.5.1 we show that P E L(H H) admits a continuous Ps1 inverse. Indeed, T5=PsP_S implies = Ps5Ts1 where we regard the right hand side as a composition of Ts 1E L(H) and P_5E L(H,H5). ,
To confirm the latter note that IIP-sulk=2Iln 5(XjP-5u)II2=
7,11n s(
s11 /2?.ku)112 The point is that, for fixed j, all
s/2
KkA`k
4. Operators on manifolds with conical ends
142
terms r1,2Ps/2?.ku must be taken to the coordinates of the j-th
chart, using thm.3.3. and only within a neighbourhood of supp XjC insuring L2-boundedness Uj. They will give operators of order s ,
of each term of the sum. Hence IIP_Sullsscllull, i.e., P_5E L(H,Hs),
implying PS1E L(H,HS)
So, indeed, we have the result, below.
.
Proposition 5.2. The operator PS of (5.3) are 'ado's in LSS exists a y'do Q_SE LS_S
,
PSQ_S = Q_SPS = 1
(5.8)
We have Q_S:H->HS continuous, i.e., P
There
S
: H
S
.
« H and Q_S:R HHS define cs>0 we have
isomorphisms between Hs and H . With constants cs
(5.9)
.
acting as an inverse -i.e.,
,
c5IIPsuII s (lulls s csllPsull , u E RS
Using the isomorphism PS we can prove HS-boundedness of ipdo's: Theorem 5.3. A i,do AE LSm satisfies (5.10)
llAulls_ms cSllulls
,
for all u E HS
Proof. We have llAulls_ms cfl(Ps_mAQ_s)Psufl
,
cs independent of u.
, where PS_mAP5 E LSO is
L2-bounded, as seen above. Thus IlAulls_msclIPsullsc'llulls, q.e.d.
In particular thm.5.3 insures that the pdo's PSE LSS and Q_SE LS_S are operators of L(Ht,Ht_S) and L(Ht,Ht+s) respectively, for every tE !2
not only for t=s or t=O as known earlier. Next we again introduce a pairing between RS and H_5 by
(5.11)
,
,
(u,v) = (PSu,Q_SV) _ J uvdµ , uE Hs , vE H_S
.
sz
Here PSu, Q_sE R, hence the middle term in (5.11) is well defined, as inner product of elements of H. It is clear also that (u,PSU)= =(PSu,u), (u,Q_Su)=(Q_su,u), as uE S, so that (5.11) is meaningful. With such preparations we reintroduce order classes, setting (5.12)
O(m) = {AE L(S(c))
:
PSAQ_S E L(Hm,H)
,
for all SE &2}
.
Or, O(m) consists of all AE L(S) such that for all sE 12 A extends to a continuous map Hs'Hs-m. Then, clearly, LSmC O(m) for all in. We again define O(±ao) by III,(3.30), and get LS_00=0(-oo). The concept of K-parametrix is meaningfull not only for y,do's, but also for general A E O(ao). A Green inverse of AE O(w) is defined as a K-parametrix BE O(oo) with 1-AB, 1-SAE O(-oo) of finite rank.
We now can repeat every line of argument of 111,4, showing
143
4.5. Order classes and Green inverses
that a Green inverse - and even a special Green inverse, with all properties III,(4.1), exists if and only if a K-parametrix exists, for AE 0(00). This holds, because all arguments there are abstract, using only the system of Hilbert spaces Hs, the pairing (5.12),
and the operator s, here represented by Ps
:
Corollary 5.4. All statements of III,thm.4.1, thm.4.2, and cor.4.3 are also valid for operators of our present order classes 0(m).
Moreover, they still hold for operators between sections of admissible vector bundles (the latter denoting the restriction to St of where evidently Hs-norms and Hs-spaa C"o-vector bundle over ? ,
ces and pdo's may be introduced for sections, in the same way). As a final remark, we mention without detailed proof that a system of equivalent norms is given by (5.13)
11ulls = II(d(x)) S' (1_0)
sh /2u11
, u E S(n)
,
with the Beltrami-Laplace operator A of the distinguished conical metric (or any conical metric): With constants cs, cs we have
(5.14)
csliulls s Ilulis s cs11u11s , u E S In particular, the system of spaces Hs and the order classes
0(m) are independent of the partition of unity and atlas chosen. As an argument leading into (5.13), (5.14): For s.E Z, s,>O, (5.14) follows immediately. For other st>O use an interpolation argument, as for the proof of III,prop.3.3. For si<0 one uses calculus of [do's in combination with the above. Let us not forget to mention that 0(0) is a Frechet algebra again, with topology induced by the operator norms in L(Hs). Use Calderon interpolation as in III,prop.3.3 to show this.
Problems. 1) Connecting to the problems of IV,4, introduce order classes and Green inverses for ,do's on manifolds with all three, conical and cylindrical ends, and conical tips. Note, there will be a difference in technique of constructing a Green iverse, once a K-parametrix exists: md-ellipticity is not enough ([C;],VIII).
Chapter 5. ELLIPTIC AND PARABOLIC PROBLEMS
In this chapter we take up the lead of ch.0, sec.4, with regard to elliptic and parabolic problems. There we applied the Fourier-Laplace method to free-space problems of elliptic equations, and to evolutionary half-space problems of the (parabolic) heat equation, all with constant coefficients. We covered Dirichlet and Neumann problems in a half-space, for elliptic equations. With the tools developed in I, II, III, IV we now can give a similar "Fourier-Laplace treatment" to much more general variable coefficients elliptic and parabolic problems. This may be done in "free space" (that is, in &n , or on a smooth compact manifold n or on a noncompact n with conical ends - but without the presence of boundary points). Such results are special cases of theorems on Green inverses of pdo's already discussed, but they will be summarized (in more general form) in sec.1, below. If Sz is compact we
need ellipticity, else md-ellipticity of the operator. Not only (md-) elliptic operators on a complex-valued function but even maps between crosssections of vector bundles are considered. Note that there is a different approach - a functional analysis approach - to these theorems, not using ado's at all. Elliptic theory, in its beginnings, was developed for 2-nd order equations. Such 2-nd order theory is of dominating importance for many physical applications. The Laplace, Helmholtz and Schroedinger operator each has its own well developed theory. One finds that virtually all results we state for &n can be reached by focusing on the C*-subalgebra A of L(LZ(&n)) generated
by the multiplications a(M):u-au , aE C(Zn) (cf.II,3) and the operators D.(1-A)-1/2=5., defining S.=s.(D)=F-Is.(M)F as Fourier mull
3_1/2
tiplier or else (1-A)
J
7
7
as inverse positive square root of the
unique self-adjoint realization of 1-AaO, A=a
([Cl],III,IV). 7
Similarly, in case of a general Riemannian manifold with conical and cylindrical ends, and conical tips (cf. IV,4,5, and the D(1-A)-1/2 problems of IV,4) we may generate such C*-algebra from with the Beltrami-Laplace operator A of n under the metric discus-
144
145
5.0. Introduction
sed for such manifolds, with DE D# and a(M) . aE A# , with suitable classes A# of functions and D# of FOLPDE's on a . Such ap,
proach to elliptic theory is discussed in detail in [C1] and [Cz]. Here we look at this in sec.10, only to clarify some questions comparing both the pdo- and the C*-algebra approach. The C*-algebra approach is of importance not only for elliptic but also to hyperbolic theory (of ch's VI and VII):Conjugation with the evolution operator eiLt of a first order hyperbolic equation defines an automorphism of the above algebra A . The dual of this automorphism will be a Hamiltonean flow in symbol space determining propagation of singularities, as in Egorov's theorem. In sec.2 we start focusing on the general elliptic boundary problem for smooth boundaries under Lopatinski-Shapiro type boundary conditions, but only for compact szii' CC &n, for simplicity.
First we discuss common facts, such as reduction to the case of a homogeneous boundary conditions or a homogeneous PDE. In ch.0 we used a reflection principle to convert a Dirichlet or Neumann problem over a half-space into a free-space problem. Similarly, for a general boundary problem, here we will ex-
tend all functions from MX into all in
and the elliptic opera-
,
tor to an md-elliptic operator on in (sec.3). The boundary problem will become a 'Riemann-Hilbert type problem' involving (genuine)
distributions. We get a type of distribution best classified as 'multi-layer potentials'. In potential theory one uses a singleor multi-layer ansatz, using the same kind of distribution for solving a boundary problem. Here we find that every solution necessarily is a sum of a Co function and certain multilayer potentials. Actually, in sec. 4, we discuss a result we call boundary hypoellipticity: Just as uE C°° for fE C"o follows for a distribution u solving Lu=f for a hypo-elliptic L inside a , we will show that, if f admits a certain asymptotic expansion near the boundary so must u satisfy the same kind of expansion, under proper assumptions on L (i.e., hypo-ellipticity, and that r is non-characteristic for L).This will be very useful not only for the elliptic problem but also for parabolic problems, later on. In sec.6 we then discuss existence and uniqueness (again in finite dimensional degeneration- i.e., normal solvability) of the elliptic boundary problem, if the boundary conditions are of 'Lopatinski-Shapiro type'. Of course we use the "multi-layer ansatz". We only look at the simplest nontrivial case: A single even order equation. But the generalization to operators (of even or odd or,
der) between vector bundles should be fairly evident; also exten-
146
5. Elliptic and parabolic problems
sion to boundary problems for subdomains of an S-manifolds should involve no new ideas, under proper assumptions on a and r . The multi-layer ansatz (for a homogeneous PDE) will convert the boundary conditions into a system of pdo's on r. Ellipticity of that system will amount precisely to the L.-S.-conditions. Actually, for existence and uniqueness considerations we need two different such elliptic systems, one n/2xn/2-system, the other nxn . In sec.7 we turn to the parabolic initial-boundary problem.
A parabolic problem is evolutionary - we assume an equation au/at= Lu with a differential operator L under initial conditions at t=0. The evolution operator U(t)=e1tL is well defined as a semigroup. For a parabolic problem the operator a/at-L is hypo-elliptic. For existence of the semigroup we assume standard results, such as the Hille-Yosida theorem. Boundary-hypo-ellipticity (sec.4) will be usefull to secure C00-solutions instead of distribution solutions. An efficient use of results such as the Hille-Yosida theorem in sec.7, will require investigation of R(X)=(L-X.)-1 of an elliptic operator, as will spectral theory of elliptic PDO's. Thus, in sec's 8-9, we look at spectral theory and R(?.) for elliptic L. n = s2 = compact manifold We only consider the cases a--in ,
,
,
manifold with conical ends, nCC 1n a domain with smooth boundary. Extensions to (i) systems of equations (ii) differential operators between sections of vector bundles of equal dimension, (iii) exterior boundary problems, (iv) Riemann-Hilbert type problems are fairly evident, but are left for the reader to deploy. Existence of a Green inverse for operators defined with boundary conditions is discussed as auxiliary result (thm.8.3), extending results of sec.6. Mainly we focus on compactness of the resolvent, under proper assumptions. Results on selfadjointness and dissipativity of differential operators (making thm's 7.3, 7.5 applicable) are discussed under various assumptions (thm.8.5 for n without boundary, thm.9.1 for LYE 1n with boundary). In all cases proper deployment of the results is left to functional analysis - insofar as the spectral theorem , the Hille-Yosida theorem, or existence of an orthonormal base of eigenfunctions for self-adjoint operators with compact resolvent is not discussed in detail. In sec.10, finally, we discuss the C*-subalgebra A of L(H), H=L2(1n), generated by our algebra OpCo of ch.I. The point: A°
A/K(H) is a commutative C*-algebra isometrically isomorphic to C(ap), with the boundary aP of a certain compactification 1 of 12n A corresponding result holds for the norm closures in LZ(st) of LSO.p.6 6
5.1. Elliptic problems in free space
147
1. Elliptic problems in free space; a summary. In the present section we summarize (and trivially extend) results on elliptic equations, previously discussed in the more general context of pseudodifferential equations. "Free space" is interpreted as "no boundary" - a compact manifold is a free space.
Theorem 1.1. Given an elliptic differential operator of order N, on a compact C"-manifold n (of dimension n) without boundary. That is, in coordinates of a chart UC St, for uE Cp(U) (1.1)
Au(x) =
aa(x)Da
, aaE Cm(U)
IaIsN with (1.2)
a
a
0
,
as
xEU
differentlia_lN
Assertion:(1) The
equation
(1.3)
Au=f
f e C00(n)
,
,
. That is, (i) the homogeneous equation Au=O admits at most finitely many linearly independent solutions; (ii) there exists a solution uE Cm(n) of Au=f if and is normally solvable, for uE C°°(s2)
only if f satisfies finitely many linear conditions of the form (1.4)
fndu ju = o
,
j=1,...,ja
.
with any positive Cm measure dµ on a and certain C"(f)-functions
Vi [In fact, the Vj may be chosen as a basis of ker A* , with the adjoint A* of A with respect to the inner product f.uvdµ]
.
(2) The operator A:C0O(St) * CO'(S) admits a special Green inverse G E LS-N = LS(-N,0),(1,0),0 , the latter as in IV,4, with (1.5)
GA=1-P , AP=I-Q
,
where P is a projection onto ker A, annihilating some (arbitrarily chosen) complement of ker A , and Q a projection onto an (arbitrarily chosen) complement of im A annihilating im A . Proof. The compact manifold a is a special case of S-manifold studied in ch.IV - there are no conical ends at all. Since there is no infinity, the classes LSm,p,s are independent of m2, p2; We and LCm in case of p,=1 , b=0 thus write LSm,,p ,6
1
5. Elliptic and parabolic problems
148
Also, since there is no infinity, md-elliptic means the same as 'elliptic' - i.e., cdn.(1.2) above. Thus the existence of a K-parametrix in LS-N follows from IV,thm.4.6. The special Green inverse G may be constructed as in 111,4 (cf.IV,cor.5.4). Once we have G the normal solvability (1) is immediate. Thm.1.1 generalizes immediately to the case of an elliptic differential operator A:I->Y with X,Y the spaces of C'° sections of two vector bundles E,F over sz with dim E = dim F = ro . We will not discuss details of the proof of thm.1.2, below: They are just ,
,
formal extensions of earlier discussions. Theorem 1.2. Given an elliptic differential operator A:I-Y , with X,Y as described above, A of order N , on a compact Cc manifold a of dimension n without boundary. That is, in local coordinates of a chart UC Sz
for uE I with supp u C U , we have (1.1)
,
,
where
now aa(x) are roxro-matrix-valued, while (1.2) is replaced by det
(1.6)
0
, xE U ,=1 .
Then the assertions of thm.1.1 hold again, as follows: (1) The equation
Au=f , fE Y
(1.7)
is normally solvable
codim im A
,
,
as u E I - that is, dim ker A
.
(2) There exists a special Green inverse G ,
roxro-matrix of pdo's in LC_N
,
locally an
such that (1.5) holds.
Proof: See the remarks above. If the manifold cz without boundary no longer is compact, but, at infinity, still is conical - in the sense of ch.IV , then our theory of earlier chapters yields results of similar structure but only if (i) the space C"O(sz) is replaced by a smaller space C000
X(n) still containing all of (ii) the coefficients as of (n) (1.1) and their derivatives satisfy growth restrictions at 0 (iii) we supplement the ellipticity condition (1.2) (or 1.6)) by ;
a condition at infinity (m-ellipticity).
A useful choice for I(c) is the space Ham, a fl(Hs (tz) : sz =a} for any aE 1. Then, to get a normally solvable operator A:X-X , we may choose aa(x) as restrictions to U of functions in the symbol
class 0o,e,o
,
whenever the chart U gets near a point of c0
(That is, the full local symbol
a
asN
a
.
belongs to
5.2. The elliptic boundary problem
149
.) Finally, instead of (1.2) we ask for
VC(NIO)
(1.8)
as
x
rl ,
with some c,rl>O
That is, A must be ad-elliptic of order (N,0) More generally we may regard operators A:X->Y .
,
%=Hm'a(n)
Y=HOOT(S), or even let %,Y be spaces of smooth sections of S-admissible vector bundles of the same dimension ro, with components in x=a,t, again. Let us summarize this in thm.1.3, below, again a consequence of IV,thm.4.6, and 111,4 (i.e., IV,cor.5.1).
Hoo, x ,
Theorem 1.3. Given an md-elliptic differential operator A:X->Y with the spaces %, Y of Co sections on two S-admissible vector bundles E, F, dim E =dim F =ro, with components in HOO,a and Hoot
respectively. Let A be a differential operator , A E LC(N,x) where x=a-r and let A be md-elliptic of order (N,K) That is, in the coordinates of an S-admissible chart UC St, we have (1.1), ,
,
aa(P)(x) = 0((x)x-IPI)
(1.9)
and
.
(with the ro xro -matrix norm
(1.10)
Ia(x, )I=I
asN
1
.
1
,
,
for all P
,
and some c,rl>0)
a
as
Then the differential equation (1.11)
Au=f , fE Y
,
is normally solvable, for uE % - i.e., dim ker A
< oo
Problems. Consider the paraboloid P:xi=x:2+x32 and the one- and two-shell hyperboloids B+:xI2-x:2-x32=t1 in &3
.
Let A be the Bel-
trami-Laplace operator of the Riemannian metric induced in P and B+ by the Euclidean metric of &3. 1) Show that the natural S-structure of 23 discussed in IV,1 also induces an S-structure in B. :An atlas of admissible charts may be obtained by restricting admissible charts of 1R3 to B. .) With this S-structure, 1-A is md-ellip-
tic. 2) Discuss the corresponding facts for the paraboloid P . 2. The elliptic boundary problem. In this section we start a discussion of the elliptic boundary problem on compact domains with smooth boundary.
5. Elliptic and parabolic problems
150
Consider a compact subdomain SEC 1n with boundary r=anC &n where r = U rj is a finite disjoint union of compact smooth n-1Bj=bj(x,D), j=1,.. dimensional submanifolds of &n . Let A=a(x,D) ..M, be differential operators of orders N, Nj, with smooth coefficients, defined near rtlf and near r , respectively. Assume complex (globally defined) coefficents, for the moment, - i.e., ,
(2.1)
A=
IaI-N
where aaE C°O(N)
aa(x)D"
,
, baE C"(Nj)
ba(x)Da'
B. = J ,
ttU'CC N , Ft
N j, N,N1C in
For given functions fE C0O(c&f) , PjE C°'(r) one seeks to find a function uE C°°(nLX) satisfying the equations
(2.2)
Au=f , xE Stld'
,
Bju=pj
, xE r ,
j=1 , ... ,M
.
The problem of finding u for given f, Tj (or the discussion of existence and uniqueness of such u) is called a boundary problem; the first relation Au=f is called the differential equation, while the relations Bju=gj
,
required for xE T only, are called the
boundary conditions of the problem.
We recall the definition of COO(nLr) and C°°(T) : uE C°°(c f) extends to a C0O(N)-function for some czLfCE N .
Similarly for r .
Example 2.1. Consider the Laplace operator A=A = 4=1a2
(i.e., J
N=2)
, with M=1, B1=1 (i.e., B1(u)=u). The boundary problem (2.2)
with this choice of A and Bj is called the Dirichlet problem of the Laplace equation. It is well known in potential theory, and has many physical applications. Example 2.2. Problem (2.2) with A=A as in Ex'le 5.1, but B,=BM=I,= Ivjax
(where v=(v1,.... Vn) denotes the exterior unit normal of r)
is called the Neumann problem of the Laplace equation. It has many applications as well. More generally, for (2.3)
B1=BM = av+h , hE c°O(T)
,
(one often assumes h(x)>0 ) one gets Hilberts boundary problem of the third kind, known for problems of heat conduction for example. Physical applications are not restricted to 2-nd order equation problems. For example, the case of N=4, A=A 2, under various choices of Bj (such as B1=1, B2=BM av
) describes a loaded elastic
151
5.2. The elliptic boundary problem
plate over n , under various restraints at the boundary 1'
.
Often a different problem is of interest, of the form
(2.4)
Au=f
, xE
In\S2 ,
B ju=gyp j
, xE r ,
j=1 , ... , M, uE B.,
,
where now A and f are defined (and Coo) in the noncompact closed set In\S
. Here Bco denotes a linear set of functions defined near
infinity, containing all uE C*(In), so that "uE Boo" amounts to a
(set of) conditions at infinity. Also, to arrive at a well posed (or normally solvable) problem one must replace the condition fE C00(ln\n) by a stronger one "fE C00(In\R)rW00 with some space I
satisfying the above conditions for B0.
.
(2.4) is often called an exterior boundary problem. As example consider A=A , B1=BM l
, Boo {uE Coo(ln): u(x)= O(IXI)}, leading
to the exterior Dirichlet problem of the Laplace equation. The examples given all use an elliptic expression A=a(x,D). Here we used A=D and A=D2 . Other examples, using a general ellip-
tic A=ajk(x)axjaxk +A1 with Al of first order, ajk(x) real, C" ((ajk(x))) a positive definite nxn-matrix, are easily quoted. For the moment we shall focus on (2.2), also referred to as interior problem, assuming that A is elliptic in Said' . A fortiori, we assume that A extends to an md-elliptic expression on In : A=a(x,D)
,
a={cIN} , where c=
1aIsN
(2.5)
c
as E tc a N,m2
is md-elliptic of order (N,m2) for some m2 EE I (2.5) will be convenient but is not essential. It trivially holds after restricting A to any small ball, hence will disappear if the theory is drawn up for general manifolds with boundary, as we ,
shall not do here. Note that, under (2.5), interior and exterior boundary problem both are meaningful, once B.0 and %00 are fixed.
If, for a moment, we admit an empty domain n=y!
,
then the
exterior problem fits into our earlier theory of md-elliptic problems on In : Choosing B. = Bs - for any given fixed s=(si,s2), problem (2.4) becomes normally solvable if we require in addition
that fE B
s-(N,m2) -%..
ces Boo
%00
,
,
Accordingly we tend to think of similar spa-
also for non-empty Sa and r .
One will expect that the Bj must satisfy certain conditions before solution of (2.2) or (2.4) can be attempted. A (somewhat complicated) set of conditions called Lopatinski-Shapiro conditions - will be introduced below. Then we will prove normal solvability of (2.2) : u exists for all f, Tj satisfying finitely many
152
5. Elliptic and parabolic problems
For f=cpj=O the space of solutions is finite dimensional. In other words, with an elliptic A and a system {A,B1.... BM}
satisfying the Lopatinskij-Shapiro conditions, the linear map
(2.6)
AxB1x...xBM : C00(Sllf) -> C"0(aLf)xC00(r)x...xC00(T)
defined by u - (Au,B1u,...,BMu) is a Fredholm map. Apart from existence and uniqueness we will pursue a question parallel to hypo-ellipticity, as in II,thm.4.1. For open SIC In
if uE D' (n) has CuE CO(n) then uE
C00
(12), by hypo-ellipticity of C.
In the line of solving boundary problems, a similar result,
with Slid' instead of sI would be useful. Of course, D' (Slid') is not defined. Using ideas of Melrose [Me1] we replace D' (SII.[') by a space of extendable distributions. Then this result indeed extends, since existence of certain asymptotic expansions is preserved. We shall refer to this as boundary hypo-ellipticity. As already mentioned, a boundary problem (2.2), in its abstract setting, amounts to the problem of inverting (or Fredholm inverting) a linear map of the form (2.6). However, (2.2) proves equivalent to either of the two simpler problems arising if one either assumes f-0 on s or Tj-O, j=1,...,M , on F. That is, either
(2.2')
, xE Slid'
,
Bju=Tj , xE 1,
Au=f , xE Slid'
,
B ju=0
Au=O
,
j=1 , ... ,M
.
Or,
(2.2")
, xE I'
,
j=1,. .. ,M
.
Clearly, if (2.2) is (normally) solvable then so are (2.2') and (2.2") each. Vice versa, let (2.2') be (normally) solvable. Assume Condition S: The differential equation Av=f admits a solution vE
Cm(nid') for each fE C00(slld') . Moreover, a unique such v can be assigned to every f, such that v=Xf becomes a linear operator (In other words, A: C00(nLr) - C°°(SllF)
admits a (linear) right inverse). Given fE Cx(nLr) and a solution u of (2.2) we define w=u-v. Then w solves (2.2') with ypj replaced by pj-Bjv. Since (2.2') is normal ly solvable we get a solution w if only cpj-BjXf satisfies finally
many linear conditions, translating into finitely many linear Clearly also the null space of (2.2) coinciconditions for f,cpj des with the null space of (2.2'), hence it is finite dimensional. Next let (2.2") be normally solvable. Assume Condition BS: For an arbitrary selection of TjE Coo(r) there exists a function vE C (slid') such that Bjv=Tj on r , and, again, .
5.2. The elliptic boundary problem
153
a linear right inverse of v - (Bjv) may be constructed. Then, for given f,qj and a solution u of (2.2) set w=u-v again. Then w satisfies (2.2") with f replaced by f-Av. A solution ,
w exists if f-Av satisfies finitely many linear conditions, etc. When are cdn.S or cdn.BS satisfied? For cdn BS there is a simple answer: Observe that the normal derivatives of order 0,1,..
... of a
C'O(szl.d')-function v may be arbitrarly prescribed, and that
the equations Bv=c)translate into a system of differential equations for (vO,...,vN-1) where vj=O vjT on the boundary r . Cdn. BS simply says that this linear system can be solved for every (Pi, and, moreover, that a linear right inverse can be constructed. One may check with the examples and find this trivially satisfied. For cdn.S we use that C is md-elliptic hence has a special Green inverse G (III,thm.4.2): CG=1-P, P a finite dimensional projection onto some complement of in C, annihilating in C=(ker C )l ,
Let gE C0(1 n) extend f to &n : g=f near c&E', gE CO. Clearly
(2.7)
Pz= Dj(ajz)
,
zE S'
,
with a basis {a.} of ker C , C" the distribution adjoint of 1,3, and a basis {(3j? of the complement of in C, bi-orthogonal to {aj}. Try to select Rj such that (3j=O near s F. We only must construct a set {y } of linear independent functions yJ.E S such that yJ.=0 in j B. ={ I x I s,q} , for some sufficiently large rl , and that no linear com
bination y of yj satisfies (y,al)=0 for all 1. Let us first assume that we have "unique continuation" of solutions of Cu=o, a property common to all second order (and much more general classes of) elliptic equations (cf. [As1], [Cu], [Hr3],ch.28). We will not discuss such results here. In particular our present use is not essential for discussion of L-S-theory here Under unique continuation no solution a of C a=0 may vanish outside BI, except a=0. Thus a basis of ker C is linearly independent in X,1=&n\BI
:
The matrix ((JX aja1dx))=Z is nonsingular.
I
For a suitable cut-off x with supp x C X1-£ X=1 in X1 and yj=Xaj we get (((yj,al))) close to Z thus nonsingular as well. ,
Thus yj give the desired functions. For P of (2.7) with aj
,
(3j as constructed and the correspond-
ing Green inverse G let w=Gg. Then Cw=CGg=g-Dj(aj,g)=f for x near szld'. Thus a v solving Cv=f near nL-r exists: Set v=w in N(szLf) Moreover, the above construction has given a right inverse of A
5. Elliptic and parabolic problems
154
as required, and cdn. S always holds, under above general assumptions, assuming that C"u=O has the unique continuation property. Remark 2.3. Note that the last argument is superfluous if C has
a fundamental solution -i.e., a special Green inverse which is also a right inverse. Examples 1 and 2 above satisfy this condition - the fundamental solution cnlx-y,2-n (or c2log1x-yl) is known A transformation of the general (2.2) to a form similar to (2.2') will be essential, in the following. Since the above discussion undesirably depends on unique continuation we note the proposition, below, tailored to fit the proof of thm.6.3. Proposition 2.4. Pick any basis yl,...,yR of a complement of in C (We may choose the yj as CO(ln)-functions.). With the above Green 0
inverse G of C let z=Gf', f'
00
an extension of fE C-(nLf) to CO(In).
Then u- =u-zlnLr transforms (2.2) to the following problem:
(2.8)
Av=y
, xE Snl,r, B,P=1l j=(,j-Bjz
, xE r ,
with yE M=span {(3IfLf}, a fixed finite dimensional space, with I3 of (2.7). We may arrange f->f'
such that
is a linear operator.
The proof follows our discussion around (2.7). 3. Conversion to an In-problem of Riemann-Hilbert type.
We enter the discussion of solving (2.2) (or (2.4)) by setting up relations for the extended functions v , g of u and f
(3.1)
v=u in StLF , g=f in nLr v=g=O in In\ (Stld')
.
For a function zE LIloc(Stlr) we denote the "zero-extension to ln"
by z" , so that z" =z in c&f, z" =0 in In\S2\r. Thus g=f" , v=u
.
We have Cv-g=O in a and in In\S2. In general v,g are discon-
tinuous on r, although clearly v,gE Lpol(In)C S'. Looking at Cv-g= h (interpreting the derivatives of C as distribution derivatives), (3.2)
Cv-g = h E E'C S'
,
where supp hC r .
A distribution with support in a single point aE In is known to be a finite sum of derivatives b(a)(x-a) (Schwartz, [Schwi],thm.35) Similarly, a distribution with support on a smooth hypersurface locally is a finite sum of distributions o ,rE D'(In) of the form
(3.3)
(6k,r,cp) = (V,avqIr) gE D(n) , with some 1NE D'(r)
5.3. A Riemann-Hilbert problem
155
where av denotes the k-th normal derivative at r (l.c.,thm.37). Therefore h of (3.2) must be such a sum. Actually, for an uE C0O(nLF)rW we get such a representation ,
for h explicitly, as a result of Green's formula. To be precise we introduce (t,v) as new coordinates on s near r , t denoting the foot point of the perpendicular from x to r , and v the distance from x to r , with negative sign in the interior of sl For a suf.
ficiently small neighbourhood NT of r the footpoint t of x is unique, so that (t,v) are useful coordinates. Then ak simply denotes the k-th partial derivative for v, - derivative along t=const. Let gE D(ln) be a testing function, and let uE C00(cZLf), Lu=f, and v, as above. For the distribution derivative Cv=c(x,D)v we get
(c(x,D)v,(p) = (v,c(Mr,-D)f) = Juc(Mr,-D)cpdx S2
The right hand side may be integrated by parts, for (3.4)
(Cu ,q) =
ffcpdx
+
jaj+1PjSN-1
Here yap are COO(T)-functions depending on the order of partial in-
tegration, dS denotes the surface element on r . Perhaps it is useful to introduce the coordinates (v,t) in the second integral:
(3.5)
(h,q)=(Cv-g,
IaI+1P +j+ksN-1 T We still mayy integrate using that COO(r) grate by parts along r Accordingly, all t-derivatives may be assumed on apu . We proved:
Proposition 3.1. Given uE C'm(Rlf), f=a(x,D)u=Cu . Let v=u
,
g=f-,
as above. Then, with differential operators P1 of order 1 on r (3.6)
h, (p)
=
I f(PN_1_k_.avu)avpds , wE D(in) j+ksN-1 T
where the P1 have smooth coefficients. Note that (3.6) amounts to (3.7)
h =
kke =Ub k ,T , V
PN-l-k-javu , 6.,T of (3.3)
hence it gives the desired decomposition of h explicitly.
,6,T
Remark 3.2: For a distribution TEE DO(in) of the form T=Y
j
with SN
I, of the form (3.3) and a finite sum, the distributions J,
T E D'(T) are uniquely determined (as long as we define av by
Sv J,
means of the above coordinates (v,t)). Indeed, for any given inte-
5. Elliptic and parabolic problems
156
ger ka0, and cpE D(1'), a function P°E D(kn) may be constructed such
that
jk on 1'
(3.8)
,
j=0,1,... . Then clearly
('!'k,(p)
showing that indeed the ,yk
,
(T,q°)
hence the 6k are fully determined.
Proposition 3.3. Suppose for any u, fE C0O(fz(X) and their zeroextensions u-, f- we have (in the sense of D'(len))
Cu = f" + h , supp h E T .
(3.9)
Thus Au=f in fz, and h coincides with the distribution (3.7). Since u, Indeed, we trivially have Cu =f in the open set fz .
fE C"(fz) derivatives in Cu In are classical, hence we may write Au
=f in n. Taking closure in fzld' get Au=f in c&f. By above construction, get (3.7) for h, with the precisely determined yak, q.e.d.
The discussion shows that, for u,fE
C00
(nLr)
the PDE Au=f
,
is exactly equivalent to (3.9) with h of (3.7). For a boundary problem (2.2) we must solve Au=f, hence (3.9), for given f, where uE
VO' V1' ... VN-1 E C"(F) are to be determined. The right hand side of (3.9) belongs to S'(2n) Therefore
C"(fzlf)
'
.
(3.9), for given y,j , may be solved with the methods of ch.II (or V,thm.1.1). The md-elliptic operator C admits a Green inverse G We get CG=1-P where P may be chosen as in (2.7). The distribution .
(3.10)
v = Gf- + Gh
will solve Cv=f-+h if and only if P(f-+h)=0 (i.e., (aj,f"+h)=0 for a basis aj of ker C-). Then the general solution (in S') is
(3.11)
v = Gf" + 'j_IGo
Vi
+ ,jwj ,
tyjE C" (r)
with X jE C, and a basis wj of ker C. For arbitrarily given f, +yo,. 'VN-1 one will not expect such v to vanish outside fall' - i.e., v is not a zero-extension of some u. In view of the above our pro-
blem of solving (2.2) thus appears reduced to the following:
Problem 3.4. For given fE C"(c&E'), T .E C" (r) determine w0" "VN-1 E C"(F) such that (i) (f"+h, a) =0
aE ker C"
,
and that, for
suitable X. the function v of (3.11) satisfies (ii) v=0 in In\a\T; (iii) vIfz extends to a function uE C"(r&F), so that v=u ; (iv) we get Bju=Tj, xE t, j=1,...,M - i.e., u satisfies the cdn's of (2.2)
Actually, v of (3.11) is a temperate distribution only, now to be examined for its properties. Evidently we have wjE S . The
5.4. Boundary hypo-ellipticity
157
of differentother two terms involve the operator GE OpVc-N,-m2 as we know. From II, iation order -N< 0. G sends Hs to Hs+(N,m2) ,
,
2.5 we know that the ado G leaves wave front sets and singular supports invariant. Also, the distribution kernel k(x,y)=g,(x,x-y) g=symb(G) is equal to some function in S(&n), for fixed y and large lxi
(and uniformly so for yE RCL 1n). We have proven:
Also, v is equal to a v of (3.11) is C'(kn\r) function in S(In) for sufficiently large lxJ Proposition 3.5.
.
.
Indeed, sing supp(f"+h)C r, and Gf" f 121. k(f +h)dy, nLr(= 1n. We will make a detailed discussion of Gf" and G6 ,1 uj near their singular support r in sec.4, below, and then return to pbm. 3.4. In essence one finds a system of pile's for the y,j of pbm.3.4. This The vj will be called multilayer potentials (induced by G) .
notation is suggested by considering examples 2.1, 2.2 again. For C=A use of the fundamental solution G of 0,(4.12) instead of the v, are known Green inverse is natural. The two distributions vo in potential theory as single and double layer potentials. They are used to solve the Dirichlet and Neumann problem for C=A . With a result of sec.4, we get an improvement of prop.3.5: ,
Proposition 3.6. 1) For fE
C00
(c&f) we have w=Gcp" E
over the N-th derivatives w(a)
,
JaI=N
,
CN-1 (Rn)
. More-
jump accross r, but are
otherwise smooth: For every a the restrictions of w(a) to 12 and to
Rn\(f2LF) extend to functions in C00(c&F) and COO(In\12), respectively 2) For 'tpE C°°(r)
,
vj=GSA is a function in
and j=O,...,N-2 the multilayer potential CN-2-j(Nn) For all derivatives (a) the .
restrictions to n and &n\(nu) extend to functions in CO0(nLr) and in C00(&n\ft)
,
respectively, j=0,...,N-1. (That is, all derivatives
of order s N-1-j may be expected to jump on r) Proof. We will get C(Gf") = f" + s' Cvj = 6j + s" , s',s"E S. .
,
Then thm.4.4 implies the statement; details are obtained by comparing the particular asymptotic expansions of f" and fi,.
4. Boundary hypo-ellipticity; asymptotic expansion mod av In this section we focus on matters of
boundary hypoellip-
ticity, as mentioned in sec.2, and used in prop.3.6. With notations of sec's 2 and 3 focus on PDE's of the form (3.9), i.e.,
(4.1)
Cv=f" +2_0S
j
where vE D'(1n). We will mainly be interested in the case p=N-1
5. Elliptic and parabolic problems
158
fE C"(MX), 1V E C"(r), attempting to solve pbm.3.4, but results will be interesting for general p and ii.E D' (r) and fE C"(n) with
f- E
(4.2)
Llloc(&n),
as will be assumed henceforth. Define the spaces
Z ,=j wE D' (&n) : supp wC nil , wE C" (n) } ,
(&n
Y
%szm l loc
)
Let us use the coordinates (v,t) introduced in sec.3, where vE & , T E=- r . They are valid only in some neighbourhood N of r . E>O. Since r is compact N contains some set NE=IExr , IE={jvjsE} Here the neighbourhood NE of r may be regarded as a subset of the supp x C IE, cylinder n'= &xr as well. With a cut-off function x ,
,
x=1 near 0 (4.3)
,
and w=xv , equation (4.1) takes the form
Cw=f"x+ fj=o(-)javs(v)e)Vj+ a. , XE CO({va0}), a.=O as v<E/2,
with the Dirac measure 6 on &={-oo
and the distribution ten-
,
sor product (We were using the fact that v(=- C"(&n\r), by hypoellip-
ticity of C). We get v=w near r={v=0}. The behaviour of v near r may be studied by looking at w of (4.3), while (4.3) may be regarded as a PDE on the cylinder n' (Assume C extended to Ill for example by freezing the coefficients: In these coordinates we get ,
.
(4.4)
C = Lj=oCN-j(v)ay
with PDE's {Ck(v) : vE [-e,E]}, of order sk. Extend Ck(v) to I by setting Ck=Ck(±E) as vaE and vs-E, resp., smoothened near tE
.)
We work with (4.3) and n° = {v>0} C sz' =&xr instead of (4.1) and n. Correspondingly, %I° and Yre are defined using &xr instead of &n Clearly CO(v)= c0(v,t) of (4.5) is a function. By ellipticity of C we have c0(v,t)#0 on n'= &xr . This condition alone - in fact only the property of r being noncharacteristic for C cO(0,t) # 0
(4.5)
(on all of r)
,
i.e.,
,
is sufficient for our present discussion. We assume C hypo-elliptic (not necessarily elliptic) apart from (4.5). By assumption f is C"(nld')
.
It admits a Taylor expansion in the variable v
q=0(ayf(O,t))j!
:
Jrq(v,t) 7
f(v,r) _ (4.6)
+
q
V
(aq+lf(O.t))(vq dx
rq(v,t) _
,
q=0,1,...
, vaO
0
For the extended function f" we get an expansion valid in a 2-sided neighbourhood of r={v=0} by introducing xj(v), j=0,1,... (4.7)
xj(v) = 0/j!
,
as va0
,
xj(v) = 0
,
as v
.
5.4. Boundary hypo-ellipticity
159
For N=0,1,2,...,
f- (v,t) _
(4.8)
oavf(O,t)xj(v) + rq (v,t )
vE T
,
Bilks
(4.8) may be The xj(v) are homogeneous of degree j in v expressed as an asymptotic expansion mod av (at v=0), in the sense of [C,]1II,3 For fixed t=t° E T we have .
:
f (v,t°) _
(4.9)
=0 ayf(O.t° )Xj(v) (mod av)
at v=0
,
In details, the terms of (4.9) are homogeneous of degree pj=j-+ 00 j=0,1,...
(XjE Hj"', speaking in terms of [C.],II,(3.3) ).
,
r-25_,
We have
by well known pro-
av(p(O,t° )xj=rq E Cq(St`)
perties of the remainder rN . Equivalently, for N=0,1,2,... there exists M such that rk E CN for all kxM . The terms of (4.9) are Lpol hence also are homogeneous distributions. In [C,],II,3 we were discussing more general such asymptotic expansions, allowing arbitrary homogeneity degree pj - positive and negative, and even complex - but with certain restrictions for degrees pjE Z pjs-n , and allowing certain non-homogeneous terms for integers pjkO . Here it is practical to extend in a somewhat simpler way: For j=1,2,... define the distributions ,
X_j = avc0 = av-1b
(4.10)
Observe that then, for k=0,1,..., aYXi - Xj_k
,
jE Z
,
(k+ )I
vkX
Xk+j
jZ0
(4.11)
jsk vkX_j= (_1)k _lk Xk_j j>k =0 also for j<0 are homogeneous distributions of degree J. All xj We shall say that a distribution T E %., allows an extended Taylor expansion (mod av) at v=0, (abbreviated "ETE") written as i
,
,
,
,
,
T = 0p
(4.12)
where pE Z
,
and yjE DI(T)
X? yj (mod av) at v=0 ,
j=p,p+1.....
there exists NO=NO(M) such that
,
if for every M=0,1,...
yjE CM, near v=0, as Na
N0. The largest such p will be called the order of the expansion. as ja0 , while Actually we always will require that yjE C°O(T) ,
the yj
,
j<0 will be allowed as general distributions in D'(c)
This definition proves useful at once: (4.3) may be written as
160
5. Elliptic and parabolic problems
Cw =
Ixj® y. (mod av) at v=0 , with
(4.13)
yj=avf(O,t)
,
ja0
as j<0
,
It is evident that the 0-extensions of C'O(aLr)-functions
are characterized by the nature of their asymptotic expansions: Proposition 4.1. A distribution g(=- D'(a') with supp g C {v2:0} and
sing supp g C {v=0} is Coo({vz0}) if and only if there exists functions yj(t)E C°°(r) such that
g='.0 xj®yj (mod av) at v=0
(4.14)
Then we have yj=avg(0+,t) c G G Proof. One direction was shown above. Vice versa, if the expansion ,
(4.14) exists, then it is evident that gj{v>0} extends to a q.e.d. CN({vZO})-function, for every N , hence is C00({vz0}) ,
Proposition 4.2. 1) An extended Taylor expansion may be differentiated term by term - for v as well as t of arbitrary order. ,
2)The product of
,
where wE C00(al) and f allows an extended
Taylor expansion, possesses an ETE with terms explicitly determined by Cauchy multiplication of the expansions of f and of =(oIc0 )"
using (4.11). (Note that of has an ETE , by prop.4.1.) The proof is left to the reader. A consequence of prop.4.2:
Proposition 4.3. If u allows an extended Taylor expansion then so does Cu , where C is any differential expression as in (4.4). The matters of boundary hypo-ellipticity, as well as prop.3.6 now are settled by proving the following. Theorem 4.4. Let C of (4.4) be hypo-elliptic with smooth coeffici-
ents. Assume c0(v,t) of order 0 ( i.e., a function), satisfying (4.5), but the Ck(v), k>0, are differential expressions in t only, of arbitrary (finite) order, with coefficients depending smoothly on i and T. If, for uE DI(al) we have Cu=fE %,o and if f allows an ETE of order p at v=0 , with smooth negative coefficients, then so does u. Moreover the expansion of u is of order p+N . ,
Proof. Assume we have supp u C {v2:0} and (4.15)
j=p x3Otp j (mod av)
at v=0
As a first step we will construct a distribution of the form u0=
5.4. Boundary hypo-ellipticity
161
xp+N® Yp+N such that the extended Taylor expansions of Cu and Cu0 have the same term of order p. By (4.11) we get a Taylor expansion Ck(v) ' YICkj)(O)vj/j: . Hence
(4.16)
CN-j(v)avuO = xp+j®CN-j(O)Yp+N+ ... (mod av) at v=0
Taking the sum (4.4) one finds that (4.17)
Cu0 = xp0(c0(0't)Yp+N) + ... (mod av) at v=O
To get the same lowest order terms of (4.15) and (4.17) we choose (4.18)
Yp+N = wp/c0(0't)
.
(Note (4.5) - i.e.,T is noncharacteristic for C). After constructing u0 we form w1=u-u0. Clearly the expansion of Cwt is of order p+1. We may repeat the process to construct u1=
xp+N+1® Yp+N+1 such that Cwt and Cu1 have the same lowest order term, so that C(w1-u1)=C(u-u0-u1) has an expansion of order p+2. By iteration we get u0,u1,..., such that C(u-u0-...-udyl)
,
for each
M, has an expansion of order p+M . Finally notice that, formally, the infinite sum u0+u1.....
is well defined. Write uk =
(4.19)
=p+k+Nx j0 Y j
.
Then
IZ=Ouk= cp+N®Yp+N+ xp+N+10(Yp+N+Yp+N+I) +...
the factors of xl at right being well defined finite sums. Setting (4.20)
Yp+N+j= Yp+N+j+Yp+N+j-1+ "' + Yp+N , j=0,1,...
.
we may write the formal sum (4.19) as (4.21)
2k=Ouk = V=p+Nx j0 Y j We show that (i) there exists vE D'(n'), supp vC {va0} with
(4.22)
v = r=p+Nxjo Yj (mod av) at v=0
and that (ii) we have (4.23)
C(u-vq)E Cq , q=p+N,p+N+1,... , where vq =p+Nxjo Yj
Constructing v of (4.22) means writing v=Dcjxj0 Yj
, with suita-
5. Elliptic and parabolic problems
162
ble cut-off's xj(v) near v=0. With a technique similar to that of I,lemma 6.4, we show that xj may be chosen to converge weakly in D'(S2'), and to satisfy (4.22). Then (4.23) follows easily.
It follows that C(u-v) has an ETE with all terms vanishing. C00 This means that C(u-v)E (c'). By hypo-ellipticity of C this imp-
lies u-vE C"(S2° ), implying that u=)j0 yj (mod av) at v=0. Q.E.D. Corollary 4.5. Every solution vE D'(Rn) of equation (3.11), with
fE CO0(Stld'), vjE C" (r), has vIS1 extending to a
C00
(Stld')-function.
The proof is evident.
Remark 4.6. Prop.3.6 follows from thm.4.4 and remarks following it Remark 4.7. The technique may be used as well for nonsmooth Cv=z= assuming that f,'j belong to suitable Sobolev spaces.
f"+` j_16v 7
Remark 4.8. Thm.4.4 is of local nature:If its conditions hold only for a subregion R of r , then (4.15), valid only for (v,x)E 1QxR implies a corresponding ETE for u , also valid only in RxR Indeed, all discussions of the proof extend literally. 5. A system of pde's for the y,j of problem 3.4.
We return to our pbm.3.4, the general elliptic boundary problem. After sec.4 we know that vE D'(len), satisfying (3.11) is C"()en\t) and has interior and exterior limits for all aav on F
If v is to satisfy cdn's (i)-(iv) of pbm.3.4, then the exterior (interior) limits must vanish (satisfy the boundary conditions. In the present section we will analyze these conditions, and convert them into a more explicit form. We shall see that a system of pde's on r results, which is normally solvable, under certain conditions - the Lopatinskij-Shapiro conditions already mentioned. First of all, let us assume cdn.S (of sec.2) satisfied. In other words, then it suffices to focus on (2.2') - i.e., assume f=O in SZId'
.
This simplifies (3.11): Up to the additional solution
D,,jwj of Cu=O, v of (3.11) must be a sum of multilayer potentials.
Thus, clearly, the well known multi-layer Ansatz from potential theory appears to be justified. We have proven: Proposition 5.1. Every solution u of the boundary problem (2.2')
5.5. A system of operators
163 i
is representable (mod ker C) as a sum v=
of multi-layer i
potentials, with certain viE Coo(r)
extension of v l Sa to Said'
.
That is, u is the continuous
up to an additional term in ker C
,
.
Remark 5.2. Note that the behaviour of v in the outside of QLr is unessential. If z is a solution of an exterior boundary problem, using the same C and r , but possibly different boundary operators Bi (but with f=0 in &n\(Mr)) , then the LIloc(R")-function w=u (in 12) , w=z (in &n\(RLr) ) , undefined on the null set r ,
solves Cw= 4_i63
, hence
, with certain y,iE C"(Sa) i
v = e4_oGbj
(5.1)
+ w , wE ker C
,
p1E C°O(r)
J
This suggests that -possibly- a solution u of (2.2') has man
rep-
resentations as u=vin, with v of (5.1), and different sets of 1yi.
To prove existence of u - a solution of (2.2') - it is enough to find Vi such that v of (5.1) (with certain w) satisfies the boundary conditions (2.2'), using derivatives from the inside of r .
In details the Vi must satisfy (with interior derivatives)
+Bkw = (Pk , k=1,...,M _ i Then u , the zero extension of u=vl will also satisfy (5.1) but with redefined Vi , as we have derived initially in sec.3.
(5.2)
",J=IBkGOI
Moreover, u-, of course, will satisfy the additional boundary conditions (in addition to the interior conditions (2.2)) (2.2)1
avu
Li='(-iDv)kGbj +avw= 0 , xE G , k=O,...,N-1 J
with derivatives from the exterior - since u =0 outside Said'
Thus, for uniqueness, we may show that the N+M conditions (2.2) (with yi=0) and (5.2)t, imposed on N functions Vi imply Vi=0 We now will attempt to express terms of the form BGOV , with a boundary operator B - such as Bi of (2.2) or B=av of (2.2)1 ,
with interior or exterior derivatives, in the form BGb,=QW , where Q must be studied. Q will be a pdo, and (2.2'),(2.2)1, after modification, systems of pdo's. Fixing some y,i, (2.2') alone is converted to an elliptic system with Green inverse. Thus we will settle existence and uniqueness (prove normal solvability of (2.2')). The manifold structure of r , and the fact that r=ac implies
existence of a coordinate transform, for each xoE r , mapping a
5. Elliptic and parabolic problems
164
neighbourhood Nx° of x° onto a neighbourhood of 0 in the half space ie+={xja0}. r is covered by a finite number of such neighbourhoods.
Using a partiton of unity {xj} subordinate to such covering {N j} x we split: 1 = DCjV , supp Kjy C Nxj
,
and write v=auk , vk=Bb lp .
Our above coordinate transform (for N j chosen sufficiently small) x may be extended to a global transform meeting the requirements of IV,3. In the new coordinates vk transforms into a linear combination of multilayer potentials with respect to the half space &+ with boundary y given by {x,=0} = &n-1
. That boundary is noncom-
pact. However, we now may assume the density 1V to have compact support. G, of course, will transform into a 1,do of the same pro-
perties. Returning to our old notations, we now will have
(5.3)
a--jx=(v,t )E &n:v>0}
,
I'=&n-1={x=(v,t
)E &n:v=0} , 1E CO (r)
v= G is a special Green inverse of CE OpVCN,ma
,
an md-elliptic
differential operator of order N For our boundary conditions (2.2) we also need derivatives Dav IaIsN-1 , where .
,
(5.4)
Dav = (-i)kDaGDv(6(v)QV(t))
, 'WE C0(17)
For the calculation of (5.4) we approach O(v) by a 6-family: E>0 , with as E-0 , weakly
Choose any family 6E(v)E=-
in D'(&) - for example 6E()=6i(E), E>0, with 6,E C'(&), fbtdv=1. By continuity of the tensor product and the operator B we then get Dav = (-i)kDaGD;(O(v)QV(t) =(-i)k limE-0TE (5.5)
P=DUGDiE Opwc(IaI+k)el -m
TE= DaGD1(SE (v)
Clearly 6E0 1V E CD(1n) C S(e), hence (with x=(v,t)=(x,,x°))
(5.6)
TE(xI ,x°)=
J
tfoKeiT(x°
K)14(tJove ivxi6e.
(v)p(x.(v.t))
with p=symb(P), 6,.=Fourier transform of 6E We have written _ (v,t)=(l,,t), y=(y,,K), and used I,(1.5) with some Fubini-type in.
tegral exchanges, due to 66^E S, p of polynomial growth.
Introducing the n-l-dimensional symbol (with parameter m)
5.5. A system of operators
(5.7)
QQ
PE
J9iveix, v
(XI ,x"
be. (v)p(Xi ix" 'V'V
we get TE= PE (x, ,x° ,Dxs )1. We get 6E^ S^ =, as E-0, hence expect (5.8)
x° . ' ) =f dveix, vp(x, XA 'V' V )=K111? (xl x° ,xI
possibly with a distribution integral. Actually, (5.7) amounts to implying the convergence )O E , for fixed x° RE=p(x, x° D X, in S'(&). Formally we thus expect (5.8) for each fixed
(5.9)
ikD v(v,t)
= Yo (v,t.D,C)1U ,
PO (v,t. ' ) =
with the inverse Fourier transform ,3' for the 3-rd argument. Let us analyze the special form of
(and its derivatives for x, of arbitrary
Proposition 5.3.
order) may be written as. homogeneous asymptotic sums (5.10)
p(x,
)
- Lj=0pj(x,
(mod ICI) at
)
in the sense of [Ci],II,def.5.2, and uniformly so for xE K CC in any K . Here the pj are rational functions of with coefficients smooth in x , homogeneous of degree -N-j+Ial+k in 1;. Specifically,
e.=(1,...,0)
(5.11) rj(x,U)/cN+1(x,U)
, rjE Wcjm-je+(Ial+k)e,
of degree sNj-j+Ial+k with coefwhere the rj are polynomials in ficients in VcO'jm -j denotes the principal and where part polynomial of (i.e. the homogeneous part of degree N). Asymptotic convergence mod ICI at 1_oo means that ,
(5.12)
0(
uniformly for xE KCC In
,
all K
,
as RZRo (q),
and with all
-derivatives.
Proof. G is the operator B constructed in II,thm.1.6, up to an additional term in 0p5(1Q2n). In the notations used there,
b = IL;I
bj (mod pc-.) - where b0=c (large IxI+
) while, for j=1,2,.., the bj are recursively defined : (e)
(5.13) bj+1=Dl6bj(9)c
K0=(-i)I()I/01 for 1:5101:5N, =0 otherwise ,
5. Elliptic and parabolic problems
166
again. At the moment we have x varying over a For our present bounded set, so, 'large a'large ICI notation we use g=symb(G) and replace bj by gj in (5.13). for large
.
'
Proposition 5.4.
We have (for xE KEE In
gjE span{iII(c(aj))/c)
(5.14)
:
,
ICI large
j s YIajl = IIPJI s jN}
The proof is immediate, using (5.13) and induction.
Notice that a term at right of (5.14) is of the form
jjII(c ( R j) /c)
:
Such a term is a symbol in Vc-m-ke
7,I aj I =j 113 j I =k .
m=(N,m2), e=(1,1). Asymptotic convergence of Igj
means that
,
in our case,
as RaR°(q) and xE K, hence coincides
with asymptotic convergence mod ICI, as defined. Terms of (5.14) are rational functions of l; with coefficients coo in x. Denomina-
tors are powers of (5.15)
By reordering the expansion we get L1,=0Pj/e7+1
g =
(mod ICI) at
where pj are polynomials in
p0=1
, PjENj_j,Nmz_j
of degree s Nj-j. Write c=Lj=0cj,
with homogeneous cj of degree j. The md-ellipticity of c implies
(5.16)
IcN(x,° )
I
zp>O, for all 11;11=1 , xE K
.
We get
(5.17)
c
cN(x,
{1+cN"(x,
°)11-1 +...+cN(x, °) ISI-N}-1
)
cN(x, IF
where the yj are homogeneous of degree 0 Uniform convergence of the series 1+y1II-I+... in (5.17) for large ICI and xE K is evident, using the geometric series +y 1-y+y2-+... Again, a study .
.
of the coefficients yj, from expanding yj=(C -...CNshows that (5.18)
c =
c
N
Oqk/ck
degree gkskN-k
The series converges uniformly for xE K and large ICI with all (term by term) derivatives. A fortiori we have asymptotic convergence mod Ii;I. Similar expansions result for all powers c-I Sub.
5.5. A system of operators
167
stituting (5.18), etc., into (5.15), and reordering we get the desired expansion (5.12), for the case k=a=0 For general P=DaGDX we use I,(5.7), giving the finite sum .
_
P=Y
(5.19)
101 1 )
In other words, the symbol of P=DaGDX is a finite combination of (x,U) Since we know that (5.12) may be differensymbols g tiated term by term, and since multiplication by a power of .
does not change the asymptotic convergence, nor the structure of the expansion terms, we get (5.12) in the general case. In particular the term is as stated, as a comparison shows. Q.E.D. Here we return to formula (5.9), so far instituted only for-
mally. Substituting from (5.10), taking the Fourier transform "I" term by term, we must deal with the distribution integral
(5.20)
f dxeivxpj(v,T,K,
Nr2_npJ
°)
where pj = rj/cN+1 is a homogeneous rational function of degree k+Ial-N-j in the variables For fixed the pj are rational functions of the single variable K Their poles, in the complex x-plane, are independent of j - just the roots of the denominators cN+1 i.e., of the polynomial equation .
0 .
(5.21)
Since C is elliptic there are no real roots of this equation, by (5.16). Assuming first that (5.20) is a Lebesgue integral, one will tend to evaluate it by a complex method: Assume v#0 - i.e., x is not on F. Depending on the sign of v we will integrate over a semi-circle in the upper or lower half-plane: For v>0 (v<0) the eiync=eiVRex.e-VImK function will decay as Im K -> oo (Im x -> -00) Let the roots of (5.21) be called x1 Then it is clear that .
f
,
K
(5.22)
-
mx <0 ResK 1
as v>0, as v<0.
1
So far we assumed a Lebesgue integral. However, in general, the term pj=rj/cN+l is a linear combination of terms
where
°e/cN+1 is L1, or at least improperly Riemann integrable, as v#0.
Thus (5.22) holds for these terms, while ".xq" amounts to "Dq" under Fourier transform. Differentiation for a parameter may be ta-
5. Elliptic and parabolic problems
168
ken under the residue. Therefore we get (5.22) in the general case Looking at (5.9) we will be interested in the two limits (5.23)
ikD v(±0,t)=limva0ikD v(v,t)
as y»0
,
and ip<0
resp.
,
With (5.22) we get the following result.
Proposition 5.5. We have
(5.24)
ikDav(t0,t) = n±(t,D.)i , iE CD(ien-1)
where
is smooth in
(5.25)
n±(T'b° )
j=Onj'+(T,S°) (mod
=
and satisfies
for all I
°
I
) at °=oo
with
III, >0 Resx pj(0't'x' °) 1
(5.26)
1
n7'-(t'°) _ Y'Imx <0 Resx pj(0't,x,°) 1
1
all K uniformly in t and for partial derivatives of all orders. In particular, the nj'*(t,°) are smooth in as 1;°340 and positive homogeneous of degree -N+Ial+k-j+1 for TEE K= ten-1
,
,
,
,
Resxlpj(0't'x'V) = JIx-x11=E pj)dx
(5.27)
with small a>O
.
Proof. Note that the roots in
of (5.21) are continuous
as long as x°340 . Moreover, these roots are homogeneous
of degree I in "
p>O
:
.
Using this, and that pj are homogeneous of degree -N-j+Ial+k we find that the pj'±(t,t°) are homogeneous of degree -N-j+lal+k+1 in ° , by an integral substitution in (5.27). Furthermore, the
nJ'$(t,°) are smooth in t,t° as well, since they may be represented by an integral over a countour containing all roots in the given half-plane, where the countour may be locally kept constant. As for the asymptotic convergence in (5.25), the remainder
Pq = p-2gp .
=
0(1
1_R)
as qZq° (R)
,
and I x I -R={ I ° 1, +xz } -R/2
hence pq (p-2gpjyp = O(fdKltl-R) = O(jt°j-R+1)
.
Similarly we may
argue for partial derivatives. Finally, regarding smoothness of
we look at the ex-
169
5.6. Lopatinskij-Shapiro conditions
pansion (5.15) (which, in fact, is mod S(R2n), not only mod ICI). so that That is, the remainders are O(()) not only 0(1 1_R) ,
the "a"-remainders are arbitrarily smooth. Regarding The same residue calculus may be applied - but remember, the polynomial But may have real roots, in a bounded set IKI51 we really have xpj/cJ+1 with x=O as lxlsq Thus the path of the Fourier integral may be laid clear of those roots generating a remainder involving an integral over a C°° function on a compact (pj/cj+1)0,
.
.
,
countour - a
C00
-function. For fypj/cJ+1
along a path y clear of
,
the roots of c -but coinciding with I for jxI>21 we may repeat the above residue argument to get smoothness of the term. Q.E.D.
6. Lopatinskii-Shapiro conditions; normal solvability of (2.1. It is time now to return to the two systems of equations (5.2) and (5.2)1 seen to govern our boundary problem. Writing Bk bk(t,Dx)=
(6.1)
k ba(t)Dxa
,
k=1,...,M
NksN-1
,
in our boundary coordinates of sec.3, using (5.9), (5.10), (5.11) and (5.24),(5.25),(5.27), equations (5.2) assume the form (6.2)
1=1,...,M
k=0glk(t,Dt)Vk = W1-B1W where
1 ba(t) n+'k,a(t, °)
.
aIsN (We write n' of (5.24) as n`'k,a .) In view of (5.25) the glk are smooth in and have asymptotic expansions
(6.3)
glk =
(mod
1=O
glkj=(-i) k
I
)
at °
ba(t)nJ'+' k
sN Here the nj,t,k'a are given as sums of residues, of the pJ,=pj,k,a in the upper and lower half-plane, respectively. For every j,k,a and the poles of are the roots of Looking at (5.11) we first seek an explicit form for glk0 I
(6.4)
glk0-
mx1>OResxl(-ix)k(
Ia sNl ba(t)(x. °)a)/cN(O.t.x. °),
simplifying to (6.5)
dx(-ix) kbl(t,(x
fy
+
ixi
°
5. Elliptic and parabolic problems
170
in Im x >0
with a positively oriented countour y+=Y+ rounding all poles there. Similarly we get (6.6)
fy+dx6(-ix)kIIalsN1
glkj =
,
sur-
a(t)(rj,k,a/cN+l)(O,i,x,
with the polynomials rj=rjk,a in =(x,°) of (5.11). Note, the rj,k,a are of degree sNj-j+Ial+k. Thus the polynomials in the nume Arguing as for prop.5.5 rator of (6.6) are of degree s Nj-j+k+N1 we find that the glkj are (finite) sums of homogeneous functions with largest degree -N-j+k+N1+1 . After reordering we thus get a homogeneous asymptotic series, and the following result: .
Proposition 6.1. The system (5.2) of boundary conditions translates into a system of pseudodifferential equations
=OGlkk = p1-B1w , 1=1,...,M
(6.7)
with pdo's Glk on r . After a local coordinate transform near a boundary point onto the coordinates (v,i) of sec.3 the symbol allows a homogeneous asymptotic expansion
glk(t, °) _ 7j Oglkj(mod
(6.8) where
glkj are homogeneous of degree -N+N1+1-j+k in ° , and Moreover, the highest order terms are given by
smooth in t
(6.9)
1) at 1;°=oo
.
glk0(':' ° ) =
Here bl,N 1(t, °)=
°+x n)
fy+ +
1
N
bla(T)l;a is the principal part polynomial
-' °) In (6.9) we used the notation n=(1,0,..,0) )- 1 1( =e,= interior unit normal. is the above countour. For existence of a solution of (5.2') we must find density funcof
tions V0 .. VN-1 solving the system (6.7). Here we leave the term
B1w , wE ker C , undetermined, noting just that they are Coo(r)functions, belonging to a finite dimensional space. In matters of uniqueness we also will translate the system (5.2)1 into a system of 4do's. A similar procedure may be applied: We now replace B1 1=1,...,M, by Da 1=0,...,N-1 and interior by exterior derivatives. We get the following: ,
,
,
Proposition 6.2. The system (5.2)1 is equivalent to a system of pseudo-differential equations of the form (6.10)
`k=OHlkPk = D1 w
,
1=0,...,N-1
171
5.6. Lopatinskij-Shapiro conditions
Locally, in the boundary coordinates of sec. 6, the symbol hlk of Hlk allows a homogeneous asymptotic expansion
withtpdo's Hlk on I'
(6.11)
.
hlk(i,
)
_ Ylj 0 hlkj
(mod °) at °=o
where hlkj is homogeneous of degree 1+1-N-j+k. Specifically, (6.12)
k,1=0,..,N-1.
(-i)kf7 dK
To prove uniqueness - or rather the finite dimension of the null space - we must show that (6.7) and (6.10) together, imposed on a set of density functions Vj imply Vj = 0 (allow only finite ly many linearly independent solutions). Again the right hand side ,
of (6.10) is C00(I') and belongs to a finite dimensional space.
We now are equipped for a result on the boundary problem (5.2)'
.
For simplicity assume N even, and M=N/2
.
We impose
Condition L-S:
(The Lopatinski-Shapiro conditions) (i) In the boundary coordinates of sec.3, for every &n-1 the equation ,
(6.13)
0
, KE C
admits exactly k=N/2 roots 1=1,...,N/2, in the upper half-plane Im K >0 counting multiplicities. (ii) For every 0A°E In-1 the polynomials ,
(6.14)
b1, N1(t,l;°+Kn)
,
1=1....N/2
are linearly independent mod c:+ M =nImK >0 (K K1) 1
Theorem 6.3. For an even N the elliptic boundary problem (5.2)' under Lopatinski-Shapiri conditions, is normally solvable.
Proof. The point is that, under cdn.L-S, (a) we have the matrix of symbols of rank N/2 and (b) the matrix of symbols of rank N/2 as well and (c) both matrices together of rank N More precisely, under cdn.L-S, the polynomial c+(K) is of (precise) degree M=N/2. By the Euclidean algorithm we may write ,
,
(6.15)
.
1(t,°+Kn) = a1(K)cN(K)+pl(x)
(31(K)=b1,
degree pl<M
N
with uniquely determined polynomials pl(K) of degree <M . cdn.(ii)
of L-S means that the M polynomials p1,...,pM are linearly independent. Looking at (6.9) we find that
5. Elliptic and parabolic problems
172
(6.16)
fy+dK(-1K)kpl(K)/cN
glkO -
since the integrand involving al(x)c+(K) is regular inside y+, so that the integral is 0. Linear independence of the pl means that (6.17)
pl(K)=f4_lP1jKJ
P=((plj))1=l'..,M,j=1,..,M-1 invertible. '
Focus on the matrix ((glkO))1=I,..,M,k=O,..1M-1=Z Z is invertible we must show that
.
To show that
Y=((JY dK KJ+l/cN(O,t
(6.18)
is invertible, since, essentially, Z is the product of Y and P But, indeed, Y of (6.18) is invertible: Suppose not, then there exists a polynomial 9(x) of degree < M , not =O Jy
dK 6(K)?(K)/cN(K) =0
,
such that
for every polynomial X(x) of degree <M.
However, 9(x)/cN(x) must have poles inside y+, since of total order M there, while s is only of degree <M
has poles
aN , hence
cannot cancel all poles. If x0 is a pole of O/c then set T,=II(K-xl), N the product running over all other poles of i/cN . This choice of X(x) is permitted, but the integral cannot be zero now - it will give the residue of a holomorphic function with a (genuine) simple pole at xO . Thus a contradiction results. This verifies (a) above, giving a corner of the matrix of rank M . Note that the same argument shows that any MxM-section (6.19)
((g
1=1,..,M,k=R,...,R+M-1
1k0
is nonsingular
,
for all T
for R=0,1,..,M,
,
1;° 340. Also, note that (b) follows
from a similar argument: just note that c-(K)=-Zmx
<0(K-xl) also
1
is of degree M, and that our matrix P above resembles the matrix (6.12). Regarding statement (c): Suppose we have (6.20)
1=1,..,M
Ik=0glkOzk 0
and "k=0hlkOzk 0
1=0,..,M-1
Repeating the above argument, this means existence of a polynomial X(K) of degree
J
y+
dKX` 0=0 0N
,
j=0,..,M-1
,
and J
y-
dx
cN
KJ=O
,
j=O,..,M-1
Thus it follows that X/cN is regular inside both curves y+
.
Since
a non-zero polynomial of degree
5.6. Lopatinskij-Shapiro conditions
173
1/cN it follows that %mo, i.e., z0=...=zN-1=0 Note we only used half of the matrix hlkO and have shown that the square matrix .
'
(6.22)
(g1kO " ..'gMkO'hOkO " ..'hM-1k,O)k=o,..,N-1
is nonsingular for every t°#0 For the proof of thm 6.3. it now will be necessary to convert the nonsingularity of our matrices (6.19) and (6.22) into ellipticity of corresponding systems of equations. In that respect is should be noticed that the principal symbols glk0 and hlk0 are homogeneous in t° , but not of the same degree. To remedy this we use the operator A=PL-'2 , with P=Ps of IV, (5.3), on r , with its inverse A71=Q'2 , both i,do's in LCt1 as discussed in IV,5. Again, we might choose instead AA' as (inverse) .
,
square root of the second order elliptic differential operator (6.23)
L = jw(I-0)w ,
with (1-0) in local coordinates, and a subordinated partition{w} of a finite atlas on r . This choice indeed will work, but will require an additional effort we tend to avoid. In (6.7) and (6.10) introduce (6.24)
Vk=A kwk , k=O,...,N-1
.
AN-1-N l Also, multiply the 1-th equation (6.7) by and the 1-th AN-1-1. equation of (6.10) by After this modification we arrive at a new pair of systems oftpde's, now all of order 0, and in the new unknown functions wk. The nonsingularity of (6.19) and (6.22) now indeed imply that (a) the modified system (6.7) is elliptic of order 0; (Ii) the modified system (6.7) and the first M equations of ,
the modified (6.10) again form an elliptic system of N equations in N unknown functions. Green inverses may be constructed. Conclusion: (1) Equations (6.7) are solvable if cpl-Blw satisfy finitely many linear conditions - i.e. the cpl satisfy finitely
many conditions, since the B1w are finite dimensional. (2) Equations (6.7) and (6.10) together , with zero right hand sides, have at most a finite dimensional solution space. Again since the B1w are finite dimensional this means that there can be at most a finite dimensional set Vk solving both (6.7) (with (p k=0) and (6.10). This completes the proof of thm.6.3. Problems: !)Set up a generalized set of Lopatinski-Shapiro conditions working for an elliptic system of R equations in R unknowns,
5. Elliptic and parabolic problems
174
Then generalize under proper boundary conditions, just as (5.2) thm.6.3 to this case. Hint: Virtually all discussions of the four preceding sections generalize immediately. However, it now may be .
practical to leave a part of cdn's L-S in a form using matrices of complex integrals. 2) Same as (1) for an even order PDE on a comppact C -manifold R with boundary. 3) Same as (1) for a PDE mapping between sections of vector bundles over a of equal dimension (This is completely formal). 4) Same as (1) for an exterior problem:
(6.25)
c(x,D)u=f
,
outside r ,
Bju=cpj on 1'
,
if
,
f- E=- S,(ln)
,
where again if f" denote the zero-extensions (=0 in a, this time). 5) Same as (1), for a Riemann-Hilbert type problem: (6.26)
c(x,D)u=f in In\r , u,fE S'(ln)
,
Bju=(pj
,
j=1,...,M .
where now Bj = Bj,+ +Bju_ with.Bj$ containing interior (exterior) derivatives only, resp. 6) Same as (1) for a noncompact domain S with "conical boundary r ", in the following sense: The homeomor-
phism s(x)=7 of IV,1 maps MT onto a submanifold of B1 with closure a submanifold with boundary of B?. (Then the cpj must satisfy
suitable conditions at -. For example, in a chart of an infite point, after a coordinate transform onto a piece of a half space we might require yjE $(ln-l) - or only S'(ln-1) ? . Also, of course, we then must require if f- E S,(ln) .
,
7. Hypo-ellipticity, and the classical parabolic problem. We now want to focus on some problems with features similar to those of the heat equation - example (4.3) of ch.0. Generally such problems will be of the form of an abstract Cauchy problem (7.1)
au/at = L(t)u + f(t), t>0
,
u(0) _ q)
,
where L(t):X -* % denotes some family of linear operators, depen-
ding on t , and where f(t) is a functions of t taking values in X. Also, cpE I , and the solution u(t) takes values in % . The theory of abstract equations of this form, in a vector
space 8 with topology is well investigated ([Kal]
,
[ Fal]
). one
speaks of an evolutionary problem. The abstract differential equation (7.1) often is called an evolution equation. Some formal comments on evolution equations: First consider the case of a homogeneous equation -i.e. f.0. If a unique solution
5.7. The classical parabolic problem
175
exists for all qE X then a family {U(t):ta0} of linear operators U(t):X-X is defined by setting U(t)T=u(t) with the solution u at t for the initial value cp . Actually such a family exists for the initial-value problem at an arbitrary point T instead of 0: (7.1) T atu= L(t)u , t>T u(T)=q Let U(T,t) tit be the corresponding family. Then, for general f(t) defined for ta0 the solution of (7.1) formally is given by .
,
,
,
t (7.2)
u(t) = U(T,t)W +
U(T,t)f(T)dr 0
It depends on properties of the family U(T,t) to be derived whether the integral in (7.2) exists, and a derivative atu(t) is meaningful. However, assuming that this is true, and that conventional rules hold, it is confirmed at once by formal differentiation that u(t) of (7.2) satisfies (7.1). The family {U(T,t) t2T} is called the (family of) evolution operator(s) (or solution operators). For constant L(t)=L and f(t)=O the solution formally may be written as u=exp(Lt)q. We get The collection {U(t)=U(O,t)=eLt tao} then has the properties of a semi-group: :
U(T,t)=eL(t-T)
:
(7.3)
.
,
U(0)=1=identity
,
U(t)U(s)=U(s+t)
,
s,t2:0
,
as follows from the fact that then U(T,t)=U(t-T) In the cases considered here % will be a space of functions or distributions on some domain or manifold sa and L(t) a family of elliptic differential operators on 11, of the types studied in sec's 1-6: 2 is a "free space", as in sec.!. Or else, n 3 in has .
,
smooth boundary -then we define L(t) by a boundary problem (2.1)". That is, L(t) is an unbounded operator L(t):dom L =Y - %=COO(n), where (for a domaih 11cc Wen with smooth boundary t)
Y = dom L = {uE X: Bju=O
,
j=1,...,M}
, we define ,
(7.4)
L(t)u = A(t)u
,
as u E Y
,
with a system {A(to),Bj}, for a fixed to, of PDE's fitting (2.2)" Generally, A(to) is assumed elliptic. But stronger conditions are needed. First we assume that at A(t) (as a PDE in (t,x)) is hypo-elliptic. Second, we require conditions insuring (7.1) to have a unique abstract solution in some Hilbert or Banach space A. Such solution u(t)=u(t,x) then will turn out to be smooth, either by hypo-ellipticity or by cor.4.3, because it proves a distribut-
5. Elliptic and parabolic problems
176
ion solution of the PDE atu-A(t)u=f (under proper boundary cdn's. There are well developed abstract tools available, for this 2-pivot approach: For example we have a detailed abstract semi-
group theory (cf. [HP]
, [ Yo 1 ]
, [ Ka 1 ] ,
[DS,]). Also, an abstract
theory of evolution equations [Ka5]. For simplicity we stay with semi-groups - i.e., L(t) of (7.1) is independent of t. Definition 7.1. A strongly continuous semi-group of a Hilbert or Banach space I is defined as a 1-parameter family {U(t):tZO} of
U(t)E L(I) which is strongly continuous in t (i.e., limt-toU(t)u= U(to)u in I , for every uE I , toZO), and satisfies (7.4). The Hille-Yosida-Phillips theorem [DS1],p.624, states that a semigroup always is linked to a closed linear operator A: dom A -I dom A C I called the infinitesimal generator of {U(t)j, where ,
Au=dU(t)u/dt(O) = limt..0,t>0((U(t)-U(0))/t)
,
(7.5)
for uE dom A = {all WE X for which the limit exists}
.
If AE L(I) is a bounded operator, then we have U(t) = Lj=O(At)3/j! = exp(At)
(7.6)
Then U(t) is uniformly continuous in t - and even extends to an entire function of t, defined on T. In the general case (7.6) no longer makes sense, although one still might like to write U(t) =exp(At). The theorem describes the class of closed operators occurring as infinitesimal generators of a strongly continuous U(t): Theorem 7.2. (Hille-Yosida-Phillips) The class of infinitesimal generators of C0-semigroups coincides with the class of closed operators A such that (i) the resolvent R(X)=(A-X)-l exists for a half line {X>Xo}, (ii) for a constant M>O independent of
(7.7)
IIR(X)JIJ s M(X-Xo )-j
, a,>ko
j = 1,2,...
,
we get .
Then we have (7.8)
U(t) = limj->Oo(1-tA)-j =
t>0
and, vice versa, (7.9)
R())
f dte-ktU(t)dt
,
Re X >ko
0
For TE dom A we have u(t)=U(t)cpE C1([0,00),I)
.
177
5.7. The classical parabolic problem
One will recognize well known formulas for the exponential function as basis of (7.8) and (7.9). Note also that, under (7.7), formula (7.8) defines the resolvent in an entire right half-plane. Looking at (7.1) we will focus on the following two cases: Case (F)
:
L=A=a(x,D) with an md-elliptic differential ope-
rator on a manifold sz with conical ends. (or a compact manifold st
- then A needs only to be elliptic)
.
Case (B) L is an operator of the form (7.4) involving a domain S with smooth boundary, an elliptic differential expression :
A=a(x,D), and Lopatinski-Shapiro-type boundary expressions Bj In order to qualify thm.7.2 we will look at (7.1) in the case where % is a Hilbert or Banach space. Choose X=H=L2(sz), in either case. Then (7.4) defines an unbounded operator of H - in case (B), .
but this operator is not closed. Its domain is dense in H, and its adjoint L* exists (its domain contains (n), dense in H, hence is dense). Thus the closure L** exists. We may look at the problem C000
(7.1')
u(t)E CI([0,oo),H)
Similarly, in case (F) dom L = CW(a)
(7.10)
au/at = L**u
,
,
ta0
,
,
u(0) = tp E Y .
we introduce the operator L by
,
Lu= Au , u E dom L
Again, dom L is dense in H, dom L* contains C00(st), thus is dense; 0
the closure L** exists, and we may look at Now, suppose the closed operator L** ditions of thm.7.2. Then indeed a solution the form u(t) = U(t)q , U(t) = exp(L**t) .
(7.1') with this L** of H satisfies the conof (7.1') is given in Indeed, (7.3) implies
au/at(t) = limh,0,h>0 U h)-1 u(t) = L**u
(7.11)
in particular, because (7.8) implies U(t): dom L** -, dom L** In other words, for cpE Y=dom L C dom L**, u(t)=exp(L**t)T
be
longs to C1([0,00),H), and solves (7.1'). Moreover, then even u(t) of (7.2) with cpE dom L**, fE C([0,00),dom L**)
(under graph norm
of dom L**) is C1([0,00),H) and solves (7.12)
a
t
u = L
**
u+f
,
t2-0
,
u(0)=c
,
by a calculation. Particularly this holds for yE Y, fE C([0,00),Y). Also one shows uniqueness of such u as a solution of (7.12).
We first focus on case (F). A solution u of (7.12) also is a distribution solution of the PDE au/at-Au=f in (0,oo)x11
(7.13)
fu(-app/at-A"V)dxdt=
f (atu-L**u)4dxdt=(f,tp)
,
.
Indeed,
y,E C0-((0,00)xst).
5. Elliptic and parabolic problems
178
Thus, if fE C'([O,oo)x12) and at-A is hypoelliptic, get u(t)=u(t,.), C00
where u(t,x)E ((0,o)xn): u is a classical solution of (7.1). In fact, we can put to work our boundary hypo-ellipticity theorem (thm.4.4): The solution u(t) of (7.1') is CI([0,00),H). It follows at once that v=u°=u tZ 0 , = 0 t
,
,
(at A)v=6(t)0 u(0)+f = 6(t)O cp + f
(7.14)
,
Clearly the differential expression C=at A satisfies (4.5) (with n+1 variables v=t , i=x). Using thm.4.4 it follows that v admits an ETE of order 0 hence in the sense of distributions on $x12
.
,
it follows that uE C00([0,o)xc)
. We have proven thm.7.3, below.
Theorem 7.3. In case (F) assume the differential expression C=at A hypo-elliptic. Let the closure L** of the differential operator L defined by A in the domain dom A = C0 (St) satisfy the assumptions of thm.7.2. Then there exists a solution uE C00([0,o)x12) of the Cauchy problem (7.1), for general pE CD(c) , fE C'([0,co)xf) supp fC [0,00)xK, KCB n . The solution is unique if a is compact. Proof. After the above, we only comment on uniqueness: The solution of (7.1') indeed is unique, by thm.7.2. If 11 is compact then
u(t)=u(t,.) of thm.7.3 also solves (7.1), hence must be unique. Remark 7.4. It is well known that such solution needs not to be
unique if a is non-compact. Of course, the condition at x=- that H=L2(St) u(t)EE C1([O,00),H) insures uniqueness as well. Now let us look at case (B). If the semi-group U(t)=exp(L**t) ,
,
exists we again get a unique solution of (7.12), for TEC00(Stl.d') fE C0o([0,o)xnLF)
,
,
in the form (7.13). Using a localized version
of thm.4.4 (cf. rem.4.8) it then may be concluded, as in the free case, that u(t,x)E C0o([0,co)x(S& ')) - assuming that (i) at A is hypo-elliptic, and (ii) A is noncharacteristic along r : Theorem 7.5. In case (B), assume that L** is infinitesimal generator of a C°-semi-group. Assume (i) and (ii) above. Then (1.1) admits a unique solution u(t)=u(t,x) E C00([0,o)x(SZId'))
,
for every
qE C'(OUG), fE Coo([0,oo)x((&F)), q and f(to ,.) satisfying the boundary conditions for all to (-= [0,-) . Finally we ask: Will our results on elliptic problems, of sec's 1-6, give control on cdn.(7.6) for L** above: Clearly (i), (ii) of thm.7.2 only involve = R(?) for L**. Elliptic (L**-X)-I
,
theory will supply a Fredholm inverse, even a (special) Green inverse for L-? In case (F) we need md-ellipticity of A-X, for :
5.8. Spectral and semi-group theory
179
case (B) we need A elliptic and cdn's L-S for Bj (cf. thm.8.3). Still, looking at a variety of examples one finds that md-ellipticity of A-% alone in case (F) or ellipticity and cdn. L-S for case (B) though it might be helpful, is not enough. The insight of the present section is the fact that (i) hypoellipticity of C=at A and (ii) the non-characteristic property for r with respect to A (or (0,00)xr with respect to at A) reduces the Cauchy problem (for case (F) and case (B)) to a study of the resol** -1 vent (L -a.) inverse of the closure of L with respect to H. Commonly one refers to an evolutionary problem (7.1) with hypo,
elliptic at A and well defined semi-group U(t)=exp(L**t) as a parabolic problem. Often one requires also that U(t) is analytic (cf. Yosida [Yol], also Fattorini [Fal]). As application we mention: Example 7.6. (7.15)
au/at=
k=1
a/axj(ajk(x)au/axk) + f
t20
,
,
u(0,x)=g ,
either in free space (xE &n) or in a bounded domain f with smooth boundary r. In the latter case one will need a Dirichlet condition (u=0 on r) or another self-adjoint boundary condition. Assume the symmetric matrix ((ajk(x))) real and >0 for all x (and smooth). In case s--&n one assumes that L-% (of (7.10) is md-elliptic of order (2,0), for a.>0. This requires ajkE 1)c0 (7.16)
ajk(a)(x) =
,
i.e.,
O((x)a') , x E In
,
all a
and A uniformly elliptic in In . Under such assumptions the operator C=at -A indeed satisfies cdn's (i) and (ii) Moreover, L** then is self-adjoint (admits a spectral decomposition) and semi-bounded below (sec.8,9). Thm.7.2 is applicable for A=L**. Hence thm.7.3 (or thm.7.5) apply. .
8. Spectral- and semiaroup theory for pdo's. In this section we shortly consider spectral theory of an unbounded closed linear operator L:dom L-> H with domain dom L C H
of a Hilbert space H. Generally, L will be a realization of a differential expression A in the well known sense, where A=a(x,D), a (locally) of the form (1.1), is given either on In , or on a compact manifold it, or on a noncompact a with conical ends, or on a subdomain a CC In with smooth boundary as in sec.2. In each case we set H=L 2(n) - but methods will apply to prop-
180
5. Elliptic and parabolic problems
erly defined L2-Sobolev spaces as well. A realization of a PDE A on a is defined as a (closed linear) unbounded operator L of H ex-
tending the "minimal operator" Lo (with L. u--Au as uE dom Lo =C00(a) ) For aCC &n with smooth r we assume L the closure (in H) of an operator of the form (7.4) (that operator is called L' here). Equation L'u=f (for uE dom L') then corresponds to the boundary value problem (2.2") , while L=L' In all other cases we set L=LO .
i.e., L is defined as the closure of the minimal operator Lo We only consider the elliptic case: For a compact a or the .
case of an aCC &n with boundary assume A elliptic and A, B1.... BM Lopatinski-Shapiro, respectively. For a noncompact St (with conical
ends) we assume A md-elliptic (of order m=(N,m2)), mz20, and the coefficients of A such that AE LSm,p,0 , with p,=1 p2 >0 ,
.
The spectrum Sp(L) is defined as complement of the resolvent set Res(L) _ {XE 0:(L-X)-lE L(H)J. Control of Sp(L) (and of spectral theory of L) depends on criteria for existence of (L-%)None of our above criteria of elliptic theory gives existence of a precise inverse of a pdo, but we have a generous supply of results giving a Green inverse. Let us first omit the case cc in with T,0 Note that, for m2>0 in case of conical ends (in general for compact 2) not only do we have A md-elliptic, but we even have A-X md-elliptic, for every XE M Thus a Green inverse Gk of AX=A-X exists for every XE T . Moreover, Gk may be chosen such that .
(8.1)
(A-X)GX=1-PkerA.
GX(A-X)=l-PkerA.* ,
with finite dimensional orthogonal projections PkerX of H Theorem 8.1. In case of 12=ten, or n compact, or n noncompact with
conical ends assuming that m2>0, if at least one XoE $ exists with (8.2)
ker(L-X0)=0
,
im(L-X0) dense in H ,
then L is of compact resolvent. That is, Sp(L) is a discrete set, at most countable, with no finite cluster point. The resolvent R(X) = (L-X)-1 is a compact operator of H , for every XE Res(L) Proof. First we note that ker(L-Xo)=ker(A-Xo)={u6S'( °°:(A-%o )u=0}.
Indeed let LX=L-X. Clearly LXu=O means existence of ujE C- with uj ->u, Auj->0, as j--. Convergence in H implies convergence in S'.oAl so AX:S'->S' is continuous. Thus Aku=O in S' follows. Since OE C and LX is hypo-elliptic we conclude that uE Coo. Moreover, from III
thm.4.1 we know that uE S, since A% is md-elliptic. Vice versa, every uE S with AXu=O belongs to ker LX. By our assumption it fol-
5.8. Spectral and semi-group theory
lows that ker Axo={0}. Hence PkerA
181
=0, and (8.1) yields GXoAXo=1.
,o
Specifically, for u=AXo -Iv we get GXo V=(G%O A%o )AX -Iv=AXo -Iv,
vE im(Ao -Xo ). Since in LXo is dense the same holds for im(Lo =Xo ) , since LXo is the closure of Lo -Xo . Therefore (L-Xo )-I is bounded. )-1=Gxo
Its domain is H (since it is closed). Moreover, R(Xo)=(LX in all of H. Moreover, GX,E O(-m) is a compact operator of H, by III,thm.5.1. Thus indeed L is an operator of compact resolvent,
and the remaining statements follow from standard functional analysis (cf. Kato[Kal], 111,6,8 or [DSI]) Q.E.D. .
Corollary 8.2. Theorem 8.1 holds also for fCE &n, r#0: If A is elliptic and (A,B1,...,BM) satisfy cdn. L-S, then (8.2) for at least one Xo implies L to be of compact resolvent. Proof. This depends on the construction of a 'Green inverse' for the operator L' (i.e., the operator L of (7.4)). inverse should be redefined for operators acting domain of &n : We will take this in the sense of generalized Greens function, defined as integral cial Fredholm inverse of an operator of the form
The term Green on a bounded the well known kernel of a spe(7.4).
Theorem 8.3. Under the general assumptions of sec.2 let A be elliptic. Assume that (A,BI,...,BM) satisfies cdn.L-S. Then there
exists an integral operator G. : (8.3)
Gxu(x) = f
a
C°O(s LF) ' dom L'
gk(x,y)u(y)dy ,
,
u(=- C'(fl
)
such that
(8.4)
GXLXu=u-PXu , uE dom L1, LkGkv=v-Q%v
, vE
C00(slld')
where P., QX are projections of finite rank onto ker L' and a complement of in L' in C0O(nLF), and where the integral exists as a Le-
besgue (or improper Riemann) integral. We call gX a generalized
Green's function of L
We get gX(x,y)E
C00
(Slxcl\{ (x,x)E Slxsa}). At x
=y gX has the same singularity as the distribution kernel of the Green inverse of C-X, C as in (2.5), as in sec.3. Calling that kerC00
(S1xSa). Moreover, nel h (x,y), for a moment, we have gX-hXE C (Slid')dom L' extends to a compact operator H+H. GX :
Proof. We may assume X=0, since A and AX=A-X have the same principal part, hence AX is elliptic ((AX,Bj) is L-S) if and only if A ((A,B.)) is. Given a uE dom L' solving L'u=f - i.e., a solution 003 uE C (Said') of (2.2"). Let C be the md-elliptic expression of sec.3
5. Elliptic and parabolic problems
182
extending A to Rn, and let G be a Green inverse of C, as in sec.3. With the procedure of sec.3 we convert (2.2") into (3.11). Assume ker C = ker C"=O for simplicity. Again we are only interested in achieving equality in (3.11) inside a , and it suffices to set , then solve the system (with interior derivatives) VN/2= " 'VN-1=0 (8.5)
`j/U-1
BkGO
=
k=1,...,M=N/2
Bkf"
j
With the procedure of section 6 - using the ipdo's A defined near
(6.23) we set Vj=A iwj N-1-N 1
A
.
,
and multiply the k-th equation (8.5) by
One obtains an elliptic system of order 0 for the wj N-1-N Hw = -(A B1Gf")1=1,...,N/2
(8.6)
N-1-N1 With a Green inverse K of H we get cu=-K(A B1Gf") +Pw , P of
finite rank. Here K will be a ydo of order 0. Again assume P=O, just to clarify the idea of proof. In detail, we get
(8.7)
u = Gf" +
/2-1 4=0
GSA
Vi
, xE n , where
N- 1-N
'W j =
4/2A7 jKjkA
k[ BkGf" ] T,
int
where Klk are certaintpdo's on r , of order 0 The second term in (8.7) is of the form Zf with an integral .
operator Z on tt with C0O(ttxa)-kernel f=0 near r, get f " E C00(&n) , a'Gf" _
:
JOa
Let xE a. For fE C0(n),i.e g(x,y)f (y)dy, with the
distribution kernel g of G. The integral exists as Lebesgue proper Riemann) integral. a g is C" except at x=y, where it singularity of order N-n-Ial. The A3 may be written as j-th differential operators with 0-order yido's as coefficients -
(or imhas a order for
N-1-N example A=A1P1, Ps of IV,(5.3). Hence Kk1A 1B1Gf =27kaaaGf
with 0-order ydo's Jka , and a sum over laIsN-1 For xE T , yE supp f the kernels ax (x,y) are C00(T) . Hence we get .
(8.8)
A kKk1AN-1-N1 B1Gf"(x) = Jvk1(x,Y)f(Y)dy
I vk1E C (tuts)
Combine this with f
(8.9)
GO
(x) =(-1)J
aj g(x,Y)W.(Y)dSY
r vY
to conclude that the second term in (8.7) is as described.
5.8. Spectral and semi-group theory
183 k(x,y)=O(Ix-yIE-n)
An operator Ku(x) f k(x,y)u(y)dy with
is bounded as a map L2 (St)- L2 (I'), whenever E>2, by a variant of Schur's lemma (We leave the little exercise to the reader). Conclu N-I-N
sion: A bounded map L2 (.Q)-L2 (t) is given by f-' A kKklA
1B1Gf- ,
and also by the maps f- Vj defined by the second line of (8.7).
Finally look at the maps L2 (I')- V (a) induced by 'y» GS, defined explicitly by (8.9). We may assume ksN/2-1
jl.dSyaXavg(x,y)V(y) where the kernel also has a singularity of or-
der N-n-k-Ialal-n. These operators are of the form adjoint to tho-
se L2 (SZ)- L2 (I') used earlier. Thus they are bounded as well, by the same Schur-type argument. Assuming Na2,
defines a boun-
ded operator of H, for every k of the sum (8.7). The operator even is compact, by III,thm.5.1, since VX is bounded, for every first order differential expression V (i.e., we may write X as a finite
sum D j(LjX) with QjE LC-1). The kernel of G has the corresponding properties, hence the two lines of (8.7) define an operator of the form (8.3) with all proper-
ties of thm 8.3 (for ?=0). Let us next admit P#0 in (8.6). P may be chosen as an orthogonal projection (in L2(n) onto a space %C COO(T) with dims
additional Fjcj) (Xj
,
Kit XjE COO. Theorem 8.3. is established. (We
still assume ker C =ker C-=0
,
but its removal is a technicality.)
Proof of cor.8.2. We get II(1-Px)uUI=IIGxLx(1-Px)uIIs cIIL'(1-P,)uII
uE dom L' (8.10)
.
Taking closure it follows that IILull a cliull
,
for all u 1 ker L'
, uE dom L
5. Elliptic and parabolic problems
184
This implies ker L = ker L' C COO(c& '). Clearly we may take closure
in (8.4) - i.e. substitute L' by L there. Similarly as in thm.8.1
we conclude Gko Lko u=u, uE dom L, G. v=L,,o
v, vE in Lko . Lko -1 is
closed hence im Lko=R, R(X)=Lk-1 exists (compact) at k=Xo. Q.E.D. Definition 8.4. The operator L is called hermitian if (8.11)
(u,Lv) = (Lu,v) for all u,v E dom L
L is called formally dissipative if (8.12)
(u,Lu) + (Lu,u) s 0 for all uE dom L Both conditions may be verified for Lo
(in case s2=&n,
St com-
pact, n with conical ends) or for L' (in case 12CC &n) only: (8.11) ((8.12)) holds for dom L if and only if it holds for dom Lo - or
dom W. Indeed, L is the closure of either L. or L', respectively. In the cases without boundary these conditions may be expressed by direct reference to the differential expression A : 'Hermitian' means that A=A* (with the Hilbert space adjoint A* of A). 'Dissipative' amounts to (u,(A+A*)u)z0 for all uE C"(cZ) i.e., the minimal operator of the expression A+A* is positive. Such expression A will be called self-adjoint (dissipative) respectively. For a boundary problem L' we also need the above conditions for the expression A . But, in addition, a postulate on the boundary conditions results: Certain self-adjoint or dissipative boun,
dary conditions (with respect to a given expression A) are to be defined, such that L' is hermitian (dissipative) if and only if A is self-adjoint (dissipative) and the boundary conditions are self-adjoint (dissipative) with respect to A . the discussion For the general case of an L-S-system (A,B with self-adjoint (dissipative) expression A the discussion of general self-adjoint (dissipative) boundary conditions is quite complicated, and will not be attempted. A simple example will be Self-adjoint L-S-conditions the case of the Laplace operator L=0 with real coefficients all will be of the form :
(8.13)
bo avu+b, u=0
with real-valued
bo, b,
, xE T , ,
bo 2 + b 2 =1
We exclude the case n0Z &n for the remainder of sec.8. A result similar to thm.8.5 for boundary problems is found in sec.9. Theorem 8.5. Assume 12=&n
,
n compact , or n noncompact, with coni-
185
5.8. Spectral and semi-group theory
cal ends. Let A be md-elliptic of order (N,mz) with N>0 , m2a0 Assertion:1) If A is self-adjoint then L is not only hermitian but even a self-adjoint operator of R=L2(f): It admits a unique orthogonal spectral measure (cf. [C 2],I,thm.3.3) If mz>0 then L is of compact resolvent. It admits a complete orthonormal set of eigenfunctions {cpj}j=1,2,... , with (pE S(c) .
.
2) If A is dissipative then L is infinitesimal generator of a semi-group Ut = etL E L(H) tZO, in the sense of thm.7.2. ,
Moreover, the operators Ut are contractions : we have
(8.14)
IIUtII = 1
.
Again, for m2>0 L is of compact resolvent.
Proof. The assertion is equivalent to showing that for X=tia (for ?=E>O) the equation (A-?)u=O has no nontrivial distribution solution u with uE H . But we know that for md-elliptic A+X every such u belongs to S Note that A-A for X=tEi (?.=E) indeed is md-elliptic for small s>0 due to m:a0 and since we get real (or Re for self-adjoint (dissipative) A . .
,
,
On the other hand, we clearly get SC dom L 0 = Im((L±Ei)u,u) = e1IuJI2
,
u = 0
hence (L+1,,)u=O ,
Thus
.
for A self-adjoint,
(8.15)
0 = Re((L-a)u,u) s-EIIu1I2
u = 0
,
for A dissipative.
This argument shows that indeed im(LtEi) is dense in H for A selfadjoint (since every uE H, u I im(L±Ei) will be a distribution solution (in HC S') of (A;ir)u=0 u=0) . Also (Ltai)u=0 u=0 by an argument as in (8.15), and, moreover II(LtiE)ulla£IIuII
implies that R(;ai) = (L.ai)-l E L(H) exists. By a well known result (cf. [C 2],I,cor.2.3, or [Ka,],V,thm.3.16) this implies self-adjointness of L in H - i.e., L has the desired spectral resolution. If m:>0 one may apply thm.8.1 with ?.o=±Ei: L is of compact resolvent. This implies discrete spectrum of L and the orthogonal set {Tj} .
,
On the other hand, for dissipative L we similarly get exi-
stence of R(E) _ (L-E)- E=-L(H) This implies and, IIR(E)IIsE-1 (7.7) with M=1 by a simple additional argument. Hence thm.7.2 applies, and we also get (8.14) and the compact resolvent - the Q.E.D. latter for m2>0 only - from thm.8.1 again, with Xo=a ,
.
,
.
5. Elliptic and parabolic problems
186
9. Self-adiointness for boundary problems. We discuss an extension of thm.8.5 for boundary problems, for some examples like the cdn's (8.13) for L=A, and the self-adjoint case. These matters are very technical. Perhaps we point out the high lights and leave complicated calculations to the reader. For an A as in sec.2 , nCC &n (9.1)
,
consider the sesquilinear form
Q(u,v) = f dx(Auv-uAv)dx , u,vE C"O(r&f)
In coordinates (v,t) (sec.2) for u,vE CO'(nUP) let uA=(aju)1=0..N-1
v°=(ajv)j=O,...,N-1' for xE r. By partial integration get
(9.2)
Q(u,v) = (Qu°,v° )I.
where (u° ,V°) =f dS u°Tv° r r
,
with an NxN-matrix Q=((gjk(t,at)))j,k=O...N-1 of differential expressions Qjk gjk(t,at) in t Here Qjk 0 for j+kaN, and Qjk is of order N-1-j-k for j+ksN-1. Specifically the 'anti-diagonal terms' Qjk=gjk(t) j+k=N-1 , all are functions. They are bounded away from zero if A is elliptic. Note iQ is a formally selfadjoint .
:
expression on r , we get Q(v,u)
(9.3)
Q(u,v)
(Qu° vA )I. = -(u° QvA
, u,v E CO0(c f)
for all uA vA E C°°(r)
, hence
.
For simplicity consider the case where all other Qjk are of order 0 as well (A=A is such a case). Also assume that the boundary operators Bj all involve normal derivatives only - i.e., in the coordinates (v,t) they may be written as a set of M=N/2 linear conditions on the vector u°, without any differentiation for t, of the form bjTu°=0 , xE G, j=1,...,M. (Again this holds for (8.13).) Then Q=((gjk(t))) is an invertible matrix of functions (as lower right triangular matrix with nonvanishing antidiagonal). The boundary conditions Bju=O j=1,..,M, of (2.5 translate into ,
(2.2)4
u°E Zx ,
for all x E r ,
with a linear subspace Zx of TN (all uA with bTu°=0)
,
at xE r
The property of a given fE C0O(ar) to satisfy the condition (9.4)
(f,Au)=(Af,u) for all uE Cw(sz1J') with Bju=O, j=1,..M=N/2,
translates into a condition of the form (9.5)
f°(x)E (QZx)1 = Zx , for all xE r
5.9. Self-adjointness for boundary problems
187
Clearly (9.5) may be described by 'adjoint boundary conditions' (9.6)
= 0
Sjf°
,
j=1,...,N/2
again. Then L (defined by (A,Bj)) is hermitian if and only if the boundary conditions (9.6) are equivalent to the conditions Bju=O. Theorem 9.1. Assume A elliptic and (A,Bj) to satisfy cdn. L-S . Let the differential expression Q of (9.2) be of order 0 (i.e.,
an invertible NxN-matrix of
COC
functions on r) and let the boundary operators Bj only contain normal derivatives. Assume L of sec.8 hermitian. Then L is self-adjoint - it has a spectral resolution - and is of compact resolvent as well. There exists a complete orthonormal basis of eigenfunctions. Proof.
,
We argue similarly as for thm.8.5: Given fE H with (f,(L'-z)u)=O
(9.7)
for fixed zE C
,
,
all uE dom L'
.
Self-adjointness of the closure L=L'** follows if we can show that (9.7), for z=±i, implies f=0, ([C2],I,cor.2.3, or [K1],V,thm.3.16) Now (9.7) implies that (A-z)f=0 , xE 1t in the sense of distribu,
tions. Since A is hypo-elliptic conclude fE C00(f2). Let f- and C be
as in sec.3. We get (C-z)f- = h , supp hC r . The distribution h
must be of the form h=f=OS
,
j
y, jE D' (r)
, S
j
of (3.3).
In fact, we claim that R=N-1. Indeed, let v(x) - so far only defined near r - be extended into a C00(&n)-function, also called v=v(x) . For any uE C00(nUP) conclude that vNu=vE dom L'
,
since
clearly v°=(av(vNu))=0. Thus it may be substituted for u in (9.7), implying f dxf(A-z)(vNu)=0 for all uECOO(fLf). In turn, this yields
(9.8)
(f-,(C-z)(vN(P)) = f of (C-Z) (vNcp)dx=O for all cpE D'(&n),
with C and the "zero-extension" f- of f (=0 outside f&f) defined
as in sec.3. In other words, we get vN(C-z)f-=vNh=O in D1 (,n) A calculation shows that (,N6 j,,P) =('4', av (vN(P) r) r =0 as jsN-I but =j!/N!
as jaN
.
Thus it follows that Vj=0, jaN.
Observe that we obtained a more explicit description of
the distributions ,y j in (3.7), but only in case of fE COO(c2Lf ) With our matrix Q of (9.2) we get
5. Elliptic and parabolic problems
188
(9.9)
j=p
_ (Qfg°)r , q ED'
,(p)
J as a careful comparison of (3.4) - (3.7) with the above shows. We were assuming that the matrix Q contains no differentiations but simply is an invertible matrix of smooth functions on r. On the other hand, we can meaningfully define the vector f° _ =(avf)j=0,...,N-1 also for our above f E H : We get (9.10)
(C-z)f
h = `J =O6 J
where C is noncharacteristic on r . With the same iterative method as in sec.4, conclude the existence of distributions fj E D'(r) ,
j=O,...,N-1 such that, locally, near r , in the coordinates (v,s), (9.11)
(c
- z)(f"-
j
jED'(r)
j=a
3=0
Clearly these fj are uniquely determined. A close examination shows that then f°=(fj)j=0,...,N-1 , may be substituted in (9.9), above, under our present assumptions. In other words, we get
(9.12)
((C-z)f-,(p) = (Qf°,q ) r ,
(pED'(1Rn)
.
Introducing some complex conjugates we conclude that
(9.13)
0 = ((C-z)f ,(p)=(f,(A-z)gPIcj)
q>E D(1n)
(Qf°
Or, in other words, equation (9.8) reduces to (9.14)
(Qf ,u°)=0 for all u° with bjTu° = 0
Or, recalling self-adjointness of the boundary conditions, one concludes that we must have the boundary conditions Bjf=O satisfied for f in the following generalized form: We have (9.15)
Bjf = bjTf° = 0
,
j=1,...,M=N/2
.
We also have (A-z)f=0
and may rederive (3.11) in the present case (with G replaced by Gz). Thus f- again is given by a sum of multi-
layer potentials, only the y,j no longer are C" , but VjE D'(r). Thus the cuj=n-jy, must satisfy a homogeneous elliptic system of Wo's. Since ellipticity implies hypo-ellipticity, the Vj are
C00(ct) after all. Hence also the fj are C"
,
and f E C" (c&F) fol-
lows. In other words we get fE dom L'. Then, however, we must have (9.16)
0=Im(f,(L-z)f) = (Im z)J1f112 f=0, if In z #0. Q.E.D.
189
5.10. C*-algebras
10. C*-algebras of y,do's; comparison algebras.
In this section we start from the following observation. From III, thm.3.1 we know that Opyic0 is an algebra of bounded operators
on H=L2(Kn) (in fact, a subalgebra of L(RS), for every s=(s1,52)).
Moreover, this algebra contains its (Hilbert space) adjoints, by 1,2 and 1,6. For each s the norm closure of Oppc0 in Hs will define a C*-subalgebra (called As) of L(HS). Commutators in Optpc0 are in Opyic-e
,
as we know (cf.I,6). Thus III,thm.5.1 implies com-
mutators to be in K(Hs), sE &Z. It is known that the compact operators of a Hilbert space I form a norm closed ideal K(X) of L(X). We claim that our algebra As contains K(Rs) for every s ,
Indeed, OptpcOC As clearly contains the operators (10.1)
sj(M)
,
sj(D)
,
j=l,...,n, where sj(x)=xj/(x)
,
and sj(M)u=sj(x)u(x). The pdo's (10.1) generate a (unital) subalgebra Cs of As, a comparison algebra in the sense of [C2],V,1. Thus As D Cs contains K(H5), by [Cz],V,lemma 1.1. (The proof is simple: An irreducible C*-algebra containing at least one compact operator #0 always contains K(H) (Dixmier [Dxl], cor.4.1.10). Thus we prove Cs irreducible and get O#CE Kf1C (details cf. [C.],p.130,thm.1.1.). Quoting another result on C*-algebras: The quotient of a C*algebra by a closed ideal is a C*-algebra again. Accordingly, the quotient As °= As /K(R s) is a commutative C *-algebra. Let us focus on A, = AO. By the Gelfand-Naimark theorem ([DS2],IX.3.7 or [C.],p320,thm.7.7) A° is (isometrically isomor.
phic to) a function algebra C(ffi) with a compact space X. One will ask about the nature of the space ffi and the isomorphism A"-C(ffi). ,
These facts are intimately related to normal solvability of the equation Au=f, for AE A. First, by a theorem of Atkinson ([C.].,
p.271,thm.4.8) an operator AE L(H) is Fredholm if and only if its coset A"={A+K(H)} is invertible in L(H)/K(H). Second, A", as a C*subalgebra of L(H)/K(R), contains its L(H)/K(H)-inverses: A"E A" is invertible in L/K if and only if it is invertible in A', by another well known result ([C.],p.322,1.7.15). Hence Au=f (for AE A) is normally solvable if and only if the continuous function a associated to A° by above isometry is invertible in C(ffi): a(m)#0, mE ffi
In other words: For an AE A let us introduce a continuous function a=OAE C(R), called the symbol of A (relative to A) using the map A- A- =A/K(R)-.C(I). Then the above chain of arguments gives
Theorem 10.1.
An operator AE A is Fredholm if and only if its
5. Elliptic and parabolic problems
190
symbol aA(m)
,
defined on It , does never vanish.
As in [ C, ] , [C: ] we call II the symbol space of the algebra AO Asking about the nature of I and the homomorphism E:A-C(D1) we suspect a relation between 'algebra symbol' aA and '1pdo-symbol' a(x,l) =symb(A) for the generators OptpcO. These questions were discussed in detail for the algebra C with generators (10.1) by E.Herman and author, using methods unrelated to tpdo-theory ([CHe1], thm.36 -the algebra C is called T there - (cf. also the algebra called 4s in
[Ci],p.135/136,pbm's 1-4). Clearly the 4do-symbols of (10.1) are (10.2)
for A=sj(M)
for A=sj(D).
,
The functions si(x) are continuous over &n. Recalling the directional compactification Sn of II,(3.1), note that the sj even extend to functions in C(Bn) - actually, Bn is described as the compactification of in allowing continuous extension of sl,...Isn Clearly the ido-symbols (10.2) of the generators (10.1) of C extend to continuous functions over the compact space (10.3) n} BnxBn = Moreover, by the Stone-Weierstrass theorem, the si(x),sj( ) generato the algebra C(BnxIn) - they strongly separate the space (10.3). .
The result of [CHel] regarding C may be stated as follows. Theorem 10.2. The symbol space ffi_ IC of the algebra C is (homeo-
morphic to) the boundary (cf. 11,3) of the space (10.3), i.e., (10.4)
ffi°= a(BnxIn)=BnxaIn U aBnxBn =
IxI+jtj_.}
Moreover, the symbols of the generators are the restrictions to I of the continuous extensions to BnxBn of their pdo-symbols: (10.5)
as,(M)=si(x)
BnxBn
as
,
,
Accordingly an operator AE C°, with the algebra CO finitely generated by the operators (10.1), is a Fredholm operator of L(H) if and only if its pdo-symbol
a
9 > 0 for all
1/11 ,
for some ii chosen sufficiently small.
The last statement was inserted because it may clarify the relation between thm.10.2 and II,thm.1.6: We may rephrase (10.6) is ad-elliptic (of order 0). By II, by stating that AE C°C thm.1.6 this is necessary and sufficient for existence of a K-parametrix (hence also a Green-inverse) in Op4co .
191
5.10. C*-algebras
On the other hand, AE C° is a finite sum of finite products of the generators (10.1) hence aA is the corresponding sum of The sj(M) and sj(D) do not all comproducts of sj(x) and mute, but their commutators are in OpVc-e , hence their pdo-sym-
bols vanish at lxI+11I=c . Since sums and products are finite the statement aA#O on 111 of thm.10.1 indeed is equivalent to (10.6). In other words, for AE C° the abstract discussion leading to thm.10.2 gives a very similar necessary and sufficient criterion on existence of a Fredholm inverse. [In fact, similar considerations for the Cs guarantee existence of a Green inverse as well if and only if A is md-elliptic of order 0 (cf. [Ci],p.149).] Let us now state a result for the algebra A. The symbol class VCo forms a sub-algebra of CB(RnxIn), the set of all bounded continuous functions on Rnx1n. The closure (i,co)c (under sup-norm of n n CB(RnxIn is a (unital) commutative C *-subalgebra of CB(R xR ). Thus (Vco) c is a function algebra C(P) with the compact space P=
Pe,O of 11,3. The space RnxIn is densely imbedded in P, as the space of all maximal ideals mxo o= for given (x°,1;°)E Rnx1n. Or, P is a compactification of It is known that a 1-1-corresponce between compactifications of R2n and C*-subRnxln.
algebras of CB(22n) containing 1 and C0(12n) exists: Each compactification defines a subalgebra - the algebra of its continuous functions. Each subalgebra defines a compactification - its maximal ideal space. This relation is order preserving. The largest - called Stone-Cech compactification - belongs to CB(Rn) itself. The smallest is the 1-point compactification - it belongs to Our P will be 'larger than' EnxEn since pCo contains the generators s(x) j , sj() of C(EnxEn). The map P->EnxEn is given as
I+CO(12n).
,
dual of the injection C (EmX
1) -+ ( tWco ) C
.
(For details cf. Rickart
[R1],III,2; Kelley [Ke1],ch.5,p149; [Ci],IV,lemma 1.5, and p308f.) Theorem 10.3. The symbol space of A=A0, and As, sE &2, is given by (10.7)
1 = Z(A) = M(As) = at = P\(RnxIn) = fle,0
Moreover, for A=a(x,D)E Op14)Co
.
(cf.II,3).
the symbol aA equals the restriction for all sE P2
to I of the continuous extension of a(x,l;) to P ,
Proof. The last sentence of thm.10.3 describes a homomorphism t: since C=c(x,D)=a(x,D)b(x,D)-(ab)(x,D)E Op1c-e has c= symb C =0 at xI+l I=oo, hence tC =0. We tend to show that t extends to A, with ker t =K(H), hence the induced t` :A/K.C(aP) is an isome-
try. This implies that at and 2 are homeomorphic, proving thm.10.3.
5. Elliptic and parabolic problems
192
Proposition 10.4. The homomorphism r satisfies the inequality
(10.8)
inf{IIA+KII:KE K(H)} Z IITAII , AE Vco
with L2 -operator norm IIA+KII and sup norm IIT All Proof. Note that ITAI, the restriction to aP of Ia(x,l;)I extended to P assumes its maximum a=IItAII at some pE P. There exists a sequ-
ence
p ,
or
IxJI I
E 1n where Ixjl+I jl-o0 . Either we have or both. We may assume ,
3Is11 are mutually disjoint. Pick pE uj=c)j° , u=cp` , go,
II(vIIL2=1, and let
with
"= inverse Fourier
transform. Then the uj=eix u form an orthonormal system of H. Let b(x) be a bounded C00(I )-function. With "=" to be defined write
ti
IlbAujll = IIbJF
(10.9)
IIbfol
First consider the case where Ix3I remains bounded. In the compactification Enx)gn of It
where I° I =oo
,
1x0 I <00
2n
will converge to
. We have the estimate
s
s
eclx-x°I+clx°-xil+Ej where the second and third term at right tend to zero, as j-o0 For e>0 pick a ball Ix-x°I< a/c=ry and conclude that
(10.10)
a-E-E j
,
as Ix-x° Is rj
Accordingly, choosing bE CO({Ix-x°Isi}) (a-E)Ilbujll - Ej
.
Next, to control "u"
,
,
,
Ej-0 as j-*oo
we get lla(x,gj)bullz
we get
(10.11)
s CA O - 0
Furthermore, bAuj=Abuj+Cuj , with CE K(H)
.
,
Since {uj} is ortho-
normal it weakly converges to 0, hence Cuj+0 strongly. All in all we conclude that, for any KE K , we have
(10.12)
II(A+K)bujll Z (a-E)llbujll - Ej
,
Ej- 0
,
as j.00 .
This for all E>0, hence for E=0. With "inf" over K get (10.8) so far only if xj is bounded Here Ilbu jll=llbull#0. As j- co get IIA+KIIza-E .
In the leftover case Ixil-oo we follow the same argument, but
193
5.10. C*-algebras
now require a more precise evaluation of the error terms. For sim-
plicity of calculations we discuss only the case n=1. Choose (p(l;)=1
,
0 elsewhere. We get u(x)=2S1x x , uj=eix ju.
as
Now we have al s
sc/(xj) I x-xj I + E j , hence (10.10) now holds with I x-x° I srl replaced by I x-xj I srl(xj) according to b.=1 in We now will pick b=b. depending on j IbjIs1 bjE C0"0({Ix-x3l
,
c(xj)-1/2 s Ilbjujll s C(xj) -1/2
(10.13)
,
,
j=1,2....
.
A choice is bj(x)= b((x-xj)/(xj)), bE Co(IxI
(10.14)
O(Ej(xj)-1/2)
Ilbj(x)f
and that
(10.15)
II[bj,A]ujIl = o(Ej(xj)-1/2)
Ej = 0(1)
,
,
as j-oo
.
Indeed, we then may introduce v3= bjuj(xJ)1/2 with C>IIvjiI2c>0. One
easily shows that vj- 0 weakly, so that IIKvjll- 0 for compact K Repeating the arguments of (10.12) one gets hence II(A+K)vjIIZ (a-E)Ilvjll-Ej again proving (10.8). Let us offer some details: For (10.14) look at
IIAvjllz(a-E)Ilvjll-Ej,
,
f
1
J
-1
-.1
o(.) as Ix-x31si1(x3)
J1
,
j-ioo
1
implying (10.14). For (10.15): We get [bj,A] Z (bj,a)(x,D) with the Poisson bracket (bj,a)=bjl -bjlxa,, = P(x,l;) ajx .
Clearly IIP(x,D)ujll=llbjlx
-"I
plral,(x,l;3+rj)11=o(Ej(l;j)) as desired.
For the error term expressed by "-" we involve I,(5.7) for N=1. From I,(5.7) and I,thm.6.1 we get the error as IIYj(x,D)ujll with 1
,rTl)blxx(x-y)
0
where
O((l;j)-2)
. Thus the error term obeys the same
194
5. Elliptic and parabolic problems
estimate, and we get (10.15). For the weak convergence of vj obsince serve that, for cpE C0(1), we have (v.,cp) = 0 for large j ,
supp v C{ I x-xj I s i(xj) } where 11<1 may be assumed so that I x I a oo , as j-o . since Ilv Its C and Co is dense in H we I xj I -ri(xi)
get
I (v j,(P) I s
I (v j,fl
I
+ CIIro-VII
To continue with the proof of thm.10.3 observe that (10.8) implies the map t":Opipco" C(at) to have a continuous extension to A"_.A/K
,
since the left hand side of (10.8) is just the norm
II A` II. Then t' induces a (continuous) algebra homomorphism AV.
C(aP)
. Likewise t itself extends to a continuous homomorphism
and t--rlox with the canonical injection We claim that tv defines an isomorphism between Av and C(dk)
.
This indeed will establish thm.10.3, since then the dual t°' will give a homeomorphism I « at , the latter being the maximal ideal space of C(aP)
.
Proposition 10.5. An operator AE Op1co is Fredholm in L(H) if and
only if TO 0 in all of at . Proof. Clearly TA#O on all of at means that A is md-elliptic (of order 0) Such an operator will have a Green inverse BE 0(0) by II,thm.1.6 and III,thm.4.2 Clearly, B is a Fredholm inverse .
.
in L(H) as well, so that B must be Fredholm. Vice versa, let AE
Opyico be Fredholm in L (H) . Then we have
=A+
1 [ A, s] S s] E O(-1)CsK(H) since s1[A,
Fredholm too, for all sE &2
1 Ails
.
as in 111,(3.1). Recall that
Here Ill = ns (x, D) = (x) $2 (D)
s:Hs- H is an isometry between Hs and H. The above means that not only is A:H-H Fredholm but also A:Hs For clarity let us denos te the latter operator by As Then As:Hs->H5 is Fredholm for all s and we have A0=A . All operators As have the same index, v(As) _ dim ker As - dim ker AS . However, as s, increases, for fixed s2 one finds that dim ker As decreases while dim ker AS increases Since the difference V(A5) is constant, they both must be constant (Here As denotes the adjoint of A. with respect to the pairing .
.
(10.16)
(u,v) = (n-s(x,D)u,ns(x,D)v)Lz
, uE Hs , vE H-s
not with respect to the usual Hilbert space pairing (u,v)S
we have AS:H-s H-s dent of s
.
,
,
. Thus
in fact, As=a(Mr,D) with a symbol indepen-
This explains why dim ker AS increases with increasing
sj.) The fact that ker* A C Hs, dim ker A. =const then implies that S
ker As C S , and ker As C S as well. Both spaces are independent
195
5.10. C*-algebras
of s
For a more detailed discussion cf. [C,], ch.III
.
.
It is clear then, that a special Fredholm inverse B of A, constructed as in 111,4 - i.e., AB=1-Pker A* BA-1-Pker A ' also '
defines a Green inverse of A. Thus, if A:H- H is Fredholm, it follows that a=symb(A)E 1pco is md-elliptic, hence rA#O on at. Q.E.D.
Finally, in order to show that t" is an isomorphism, assume
that AE A and that VA(p° )#0 for some p°E at . Then t * (p)ac>0 in A A Np0
,
a neighbourhood of p°
in at . (Note that clearly t is a
*-homomorphism.)
Observe that t maps onto a dense subset of C(al) (containing But a *-homomorphism u:X-Y between C*-algebras always defines an isometry u°:(X/ker u)-*Y (cf.[Ci],p.323,thm.7.17)
the set "ola1 )
.
Therefore its image is necessarily a closed subalgebra of Y . In our case it follows that im t = C(aP) since im t is dense in C(a1) There exists a function XE C(al) such that x=1 near p° supp X C N(p°) , while Xko Conclude the existence of XE A with ,
.
.
TX=X . Then the operator XA*A+(1-X) = Z has cZ=Xt * +(1-X)#0 on A A
all of at. There exists a sequence Z jE OpyCo , Z j- Z. Then tZ j+ tZ.
For large j we get tZ ai>0
.
By prop.10.5 Z. is Fredholm. There
exists a Fredholm inverse Y j with tY =1/zZ = fltY
j
(10.8) we get an inverse mod K(H)
J
j
11:z
1/rI
. Using
called Wj with JlW111s 2
,
,
and
71
a Fredholm inverse Vj with JIV1Ils 1
This implies that Z is Fredholm as well (cf. [C,], corollary on p.267). .
But if A were compact, so is XA*A compact. Hence 1-X then is a Fredholm operator. The latter is impossible: Then we get a Fred-
holm inverse BE A and CBz(1-X)=1 on all of at =1
,
=*O-rB(po )(1-x(po ))
a contradiction. Thus VA=O implies A compact, and t" must be
an isomorphism. Theorem 10.3 is established for s=0. For general s observe that As= s1A , using the isometry 5:H5-> H . Also,
s
s1AIIs=A + K , KE K(Ht), for all t, showing that AE Op1NCo has the
same symbol for all s. Q.E.D.
Chapter 6. HYPERBOLIC FIRST ORDER SYSTEMS
O. Introduction.
We now focus on hyperbolic theory. It the present chapter 6 we look at a first order system of pseudodifferential equations on
&n of v equations in v unknown functions, of the form (0.0)
au/at = iKu
,
K=k(t,x,D)
,
with a vxv-matrix of symbols k=((kjl)), kjlE 1Ce, for fixed t. In essence 'hyperbolic' means that either the matrix self-adjoint, or at least has real eigenvalues, both mod loco
is .
In sec.1 we discuss a symmetric hyperbolic first order system of PDE's, using a method of mollifiers, after K.O.Friedrichs. The case of a ydo K(t) may be treated by a similar technique (sec. 2). Here we use the weighted LZ-Sobolev spaces of ch.III. In sec.3 we will look at properties of the evolution operator U=U(x,t). We find that U(i,t) is of order 0, while atagU is of order (p+q)e. In sec.4 we discuss strictly hyperbolic systems, no longer symmetric hyperbolic, by the method of symmetrizer. In sec.5 we discuss a global Egorov type result for a single hyperbolic equation, proving existence of a characteristic flow. Actually our flows are of more general type, called particle flow, using a generalized principal symbol of K no longer homogeneous in i . This flow is related to the family A H U 1AU of automorphisms of the C*-subalgebra A of L(H) generated by Opilco (V,10), where U= U(x,t) is the evolution operator: The restriction of the flow to is the dual of the induced automorphism of A A/K(H). In sec.6 we discuss the action of these flows on symbols. Sec.7 deals with propagation of singularities and maximal ideals of A. ,
General theory of hyperbolic N-th order (systems or) equations will be reduced to the case of i-st order systems, as already prepared in sec.4. This will be discussed in ch.7. 1. First order symmetric hyperbolic systems of PDE. We enter the theory of hyperbolic (pseudo-)differential equations from our discussion in 0,6. There it was seen that a
196
197
6.1. Systems of PDE's
single linear first order PDE of the form au/at=Lu with real-valued bj(x)
,
,
L=2 bjaxj +p
,
essentially is equivalent to an ODE along
a set of curves in (x,t)-space. The curves were called the characteristic curves. They were given as solution curves of the system of ODE's dt =-b(x)=-(bl(x),...,bn(x)) If u(x) is no longer a scalar, but a vector function, and the equation a system of equations, then the simple approach of 0,6 no longer works - although the essence of it can be saved if there is only one x-variable - cf.[CH],II, ch.V, for example. We regard it as a principal merit of yido-calculus that it will restore the relationship between the PDE and its 'characteristic flow', i.e. the system of characteristic curves. First we study abstract existence and uniqueness results. For a first result, for PDE's only, look at the problem a
tu + iKu = f t E I , ,
u(0) = y
,
where (1.2)
L
= iK
= Lj=1a3ax
+ a0
aJT(x) = aJ(x)
,
j = 1,...,n
J
is a first order vxv-matrix of differential expressions in x with real symmetric principal part, and coefficients independent of t. Let L be defined on &n, and let a.E C,(&n), aj
,
aj1x1 bounded.
The adjoint expression L* of L is given by (1.3)
L* = -iK* = -L + R(x)
,
1(x) =
a0T+a0-2aJ jxj
Clearly it also has bounded coefficients. We regard (1.1) as an ODE in t with dependent variable u taking values in the Hilbert space H=L2(&n,tv), assuming that f(t)E-= H , for every t. It then is natural to interpret L as an unbounded
operator of H, defined in a suitable domain dom L C H . In this setting problem (1.1) suggests that we look at the family {eiKt} possibly generated by the infinitesimal generator L=iK. Formally L+L*=(3(x) is a bounded matrix function. Thus L+y is formally accretive (dissipative) whenever the constant y E I is large (small). In other words, we have, correspondingly, 2Re(u,(L+X)u) = (u,Lu)+(Lu,u)+2X(u,u) = (u,(L+L*+2?.)u)) (1.4)
_ (u,((3(x)+2a.)u) Zo (or s0) for all uE C00(e)
6. Hyperbolic first order systems
198
By V,thm.7.2 the operator L will generate a strongly continuous group {E(t) = eLt = eiKt : t E 1e} such that dom L = {u E EH : d/dt(E(t)u)lt=0 exists in H }
(1.5)
and that u(t) = E(t)u u(t)E C1(R)
(1.6)
,
,
for u E dom L
du/dt - Lu = 0
satisfies
,
,
,
t E le
,
u(O) = u ,
provided that we can verify the assumptions for all real IXIay
.
The t-derivative in (1.6) exists in strong convergence of H. We get E(t)E=- L(H), and, with operator norm 11.11 of H, M of V,(7.7), (1.7)
IIE(t)11 s MeyI tI
t E & .
,
Then the inhomogeneous Cauchy problem (1.1) may be written as (1.8)
at(E(t)u(t)) = E(t)f(t)
An integration then solves (1.1)
:
, tE & , (Eu)(0)=p
For f(t)E C(I,H) , TE H, we get t
(1.9)
u(t) = E(-t)T +
E(t-t)f(t)dt 0
Note that (1.1) and (1.9) are equivalent only if tpE dom L
Theorem 1.1. Let L=L** be the closure of the minimal operator L0 (with domain dom LO=CO(>en)) induced by the expression L . Then,
under our above formal assumptions on L and its coefficients, L and K=-iL satisfy all of the above properties, so that the group E(t) is well defined, and the Cauchy problem (1.1) admits the solution (1.9) for every f E C(le,H) TEE dom L For the proof we need the lemma, below. ,
Lemma 1.2. Under the assumptions of theorem 1.1 the closures of the minimal operators of L and its adjoint expression L* are mutually adjoint unbounded closed operators of H . (This often is stated by saying that the concepts of weak and strong L2-solution coincide, under the assumptions given.)
Assuming Lemma 1.2 correct we next notice that (1.10)
Re (LOu,u)I =ZI(Lu,u)+(u,Lu)I s y(u,u) , u E CO(In)
hence for real ? we have (1.11)
(X-y)(u,u) s Re((L0+%)u,u)
Taking closure we get
,
u E dom L0
199
6.1. Systems of PDE's
(1.12)
(k-Y) IIull2 s IIull II (L+X)ull
u E dom L
,
which implies that (L+X) is one to one and has closed range
(1.13)
II(L+? )ull 2 IX-Y I
IIull
, u E dom L
and
for all A>y
,
.
However, in (1.10) we may replace L0 by the minimal operator of the adjoint expression. Since its closure is the adjoint of L, and the estimate (1.13) also we get exactly the same properties ,
for the adjoint operator L* of L . It follows that also (L+X)* is is one to one, and has closed range. Accordingly L+% is 1-1 and onto, and has an inverse R(7,)=(L+k)-1 E L(H) for all X>y .
,
Similarly we get existence of R(k)=(L+?,)-1,for all .<-y. For real ? satisfying IXI>y we get II(L+X)-1IISI
-Y
This implies the .
assumptions of the Hille-Yosida theorem, and thm.1.1 is proven. *
Proof of Lemma 1.2. Let L and M denote the closures of L0 and L0 respectively. Clearly L and M are in adjoint relation. That is, (1.15)
(Lu,v) _ (u,Mv)
,
for all u E dom L , v E dom M ,
as follows for u,v E CO(n) by definition, and then for general u,v by taking closure. In other words, we have L C M* , and are left with showing that M* C L Let f E dom M* i.e., .
fE H ,
(1.16)
,
(f,Mu) _ (g,u)
for all u E CD(ien)
,
for some g E H
(defined as g = M*f) Then we must show that f E dom L and that g = Lf In other words a sequence fkE CO(c) must (in H) be constructed such that fk + f Lfk - g as k ' o . E>0 , by setting Introduce the convolution operator JE ,
,
,
,
(1.17)
JEU(x) = JwPE(x-Y)u(Y)dy
with some cp E C"(&n)
,
,
gE(x) = E nT((x-Y)/E)
supp p C {Ixls 1}
It is clear that fE=JEf+f (in H) as E->0
,
,
q);0
, jpdx = 1
,
(P even.
for every fE H . Indeed,
for continuous f with compact support one finds If E-fl(x)=lfroE(x-Y)(f(x)-f(Y))dylSMax{If(x)-f(Y)Itlx-YIsE}+0, E+0,
while supp (f-f) stays within a fixed compact set, implying that f,-)-f in H . On the other hand, JE=V£1JlVE with the unitary (dilashows that IIJEII=c is independent tion) operator VEU(x)=En/2u(Ex) of E . Also C0(In) is dense in H ,
For fEH satisfying (1.16) let hF--VFfE=tPEJEf, with VF=cPE^ ^(Ex), as in (1.17). Then h. -)-f in H holds as well. Also, hEE C0
6. Hyperbolic first order systems
200 for every E>O
.
To prove lemma 2.1 we must show that also LhF-0g
But we get LhE _ WELfC + [L,pE]fE
,
.
where the commutator [L,iE] is
a multiplication by the matrix Ejaj(x)T4jx (Ex). The latter tends J
to zero uniformly on &n. Hence [L,'tp2]fE-*0, and we must show LfE+O.
Let us write a3(x) = ((aJkl(x)
))
,k
and introduce the vector (PE,x (y) For the k-th component
(bkj(PE(Y-x))
((PE,x(y) )j=l,...,m = of LfE we get (summing over all double indices)
(LfE)k(x) _ J(akl(x)axj+a01(x))rPE(Y-x)fl(Y)dy
(1.18)
= J((-ayjagl(x)+agl())(P E:X(Y))fl(Y) dy
=(L*Te,x,f) + Tkf with Tkf = J(ayj(aql(Y)-aql(x))+(agl(x)-agl(Y))TE"X(Y)fl(Y)dY But (1.16) implies (1.19)
(L*(PE,x,f)
= (rpE,k,g) = (JEV)k
As e - 0
this converges to g (in H) Therefore the lemma is established if we can show that Tf-> 0 in H , as s -> 0 For the latter we first note that 1ITHH is bounded for all e >0. Indeed this follows because the first derivatives of cpE(y) are .
O(E-n-1)
, while the support is in a ball of radius E and the factors a(y)-a(x) are O(E) It follows that the integral kernel
tE(x,y) satisfies Schur's condition fltE(x,y)l{ay s c for all x,y. On the other hand if fE Co(&n),then a partial integration, removing the derivative from qE(x-y) implies that Tf-+0 in H , as O. Combining the two facts we conclude that Tf-0 for all fE H, q.e.d. 2. First order symmetric hyperbolic systems of t,de's on ten
In this section we shall work on the Cauchy problem for a system of pseudo-differential equations over &n of the general form ,
(2.1)
au/at + iK(t)u = f(t)
,
t1 5 t 5 t2
For convenience we assume a compact interval
,
u(0) = qp
.
0 E I = [tilt 21-
6.2. Systems of operators on Rn
201
We assume that K(t) are matrices of ,do's in the sense of ch.I (2.2)
K(t) = k(t,M1#D) = ((kjl(t.M11D) ))j,1=1,...,v
'
where the functions are symbols, for every fixed t. Again we regard (2.1) as a first order ODE for u taking valu E Hs (of III,(3.1)) ues in a space HSO CV of all u=(u1,...,uv) ,
j
For simplicity we write Hs instead of HSO Cv . We assume K(t)E Opipce
e=(1,1), introducing the class VcV of vxv-matrix-valued
,
symbols with entries in ipcm. By III,thm.3.1 get K(t)E L(Ht'Ht-e)
for all tE 22. In general we do not get KE L(HS) but may interprete K as an unbounded operator of HS with domain HS+e, e = (1,1).
We consider initial values T E Hs and require f E C(I,Hs). The following precise assumptions on the are imposed: ,
(2.3)
}-I+lalk(a)(t'x'
) E C(I,CB(&2n,Lv))
a, PE=- IIn
,
and
(2.4)
(x)
E C(I,CB(&2' ,iv))
lal
,
a, PE IIn
Here 1v denotes the algebra of complex vxv-matrices,and we write CB(X,Y) for the space of all bounded continuous functions from X to Y. Also IIn is the class of all n-multi-indices a = (a1,...,an), aj E II II = {0,1,2,3,...} and we have written K*=k*(t,M1,D) for the Hilbert space adjoint K *(t) a yido with symbol given by I,(5.9). Alternately we may express (2.3) and (2.4) by writing kE C(I,4ce) and E C(I,i,cO) using the Frechet topology of tpcm Clearly (2.4) implies that the skew-symmetric part 1/2(K(t)* K (t)) of K(t) is of order zero hence bounded in every HS This ,
,
k-k*
.
.
motivates the notation "symmetric hyperbolic system" of ipde's. As
a special case we may choose K as a differential operator with the properties of K in (1.1), (1.2), but K may depend on t here. In the functional analytical sense the solution of (2.1) no longer involves a group eiKt ,since now K(t) depends on t. While abstract existence results (cf. Kato [Ka5]) could be used we find it practical to use the technique of lemma 1.2 again (cf. [Tl1]).
Theorem 2.1. (2.1) admits cp E HS , and Remark. As a
Under the assumptions (2.3), (2.4) the Cauchy problem for each a unique solution u E C(I,Hs) f1C1(I,Hs-e) II2 is arbitrary. f E C(I,HS) where s E consequence of uniqueness the solution u of (2.1) is ,
,
independent of s, if gE S=fHs, and fE C(I,Hs) for all s. Then u also takes values in S, and u , a u are continuous in every HS t
6. Hyperbolic first order systems
202
Proof. The key of the proof is the so-called energy estimate (prop.2.3) We first shall solve (2.1) for a "mollified" operator KE(t) instead of K(t) where KE(t) are bounded operators of Hs Under such conditions Picard's method of succesive approximation may be used to solve the Cauchy problem. A limit s - 0, using the Arzela-Ascoli theorem, will provide the desired solution of (2.1). As in section 1 we choose a regularizer (or mollifier) .
is _ ('WE*) = it (ED) , e > 0 , with a fixed C000( ,n )-function ii,, supp yi C { l x l < 11, V 2t 0, f ypplx =1, and VE(x) = E-nV(x/e) . Observe that yr E S . We introduce
(2.5)
(2.6)
KE(t)=V (sM)K(t)yr(ED)=kE(t,M1,D),
'(Ex)yr(e )
Since yr E S, it is clear that the matrix kE has all its entries in
wto, using (2.3). Therefore KE(t) are bounded operators of every Hs, for s>O, tE I. In fact, we get KE(t)E C(I,HS), for E>O. Thus the Cauchy problem (2.1) E (with KE instead of K in (2.1)) admits a unique solution u=u8E C1(I,Hs) for each e > 0 ([Dd1] ) .
,
Letting e - 0 we attempt to construct a uniformly convergent subsequence of u5(t) with limit function u(t) solving (2.1). We shall need more properties of JE and KE(t) in that respect. ,
Proposition 2.2. a) The family J. = V (ED) 0 e. e s 1 defines a bounded function [0,1] - L(Hs) for each s E 12 , which is norm continuous in (0,1) and strongly continuous at 0. We have J0=1. ,
,
,
If V is choosen as an even function , then also all JE are hermitian symmetric in the Hilbert space H = H0 . b) The family KE(t) defines a bounded function [0,1] x I L(Hs ,Hs-e) which is norm continuous in (0,1] x I
continuous in [0,1] x I . We have K0(t) = K(t)
,
,
and strongly
t E I
.
Proposition 2.3. (Energy estimate). Let u(t)E C(I,HD) f C1(I,Hs-e) y , f(t). There exists for given s s a constant c independent of s,s,q and f such that
be a solution of (2.1) 6
(2.7)
,
,
te-oillf(z
Ilu(t)IIs s IlaPllse°Itl + e°ltl
sgn t)Ils ds
0 of H$ A proof, based on the symmetry of K(t) expressed in (2.4), and some commutator relations of 1,7 is postponed to the end of sec.2. Note that the energy estimate (2.7) implies that
with the norm II.II
(2.8)
IIuE(t)Ilss c,
,
0 s E s
1
, t E I,
203
6.2. Systems of operators on Rn
with c independent of s and t Let us first assume that y and f satisfy the assumptions of thm.2.1 for all s The uniqueness of the solution us contained in Picard's theorem then implies that us is independent of s and that us E CI(I,HS) for all s. Using (2.8) and the differential equation (2.1)s it follows that .
.
,
,
(2.9)
IIuE(t)Ills IIKE(t)u,(t)Ils+ Ilf(t)Ilss cllus(t)Its-e+ Ilf(t)Its sc.
We also were using prop.2.2. Similarly, (2.10)
s cslluE(t)lls+e
II(M)(D)uE(t)Ils
all with constants independent of t and s
s
c
In (2.10) we used that the norm II(M)(D)ulls is equivalent to the norm Ilulls+e (cf. 111,3). .
The operator C=((M)(D))-I is compact in Hs, for all s, by III thm.5.1. Thus each set Cc {uE Hs IlC- Iullsc} is conditionally com:
pact in Hs: Each sequence ujE Cc has a convergent subsequence. The family {us
0sss1} satisfies the assumptions of the Arzela-Ascoli theorem. There exists sj>0, such that us (t) :
u(t)E=- C(I,H5), uniformly in I. Under the present restrictions on
q and f this holds for all s. Using another Cantor diagonal scheme s may be chosen such that this convergence holds in every H , s EjZ2, and then for all Hs, sE I . In particular, we get uE C(I,S). Next one confirms that u(0) = limjusj(0) = cp limjuE (t) = f(t) - i lim Ksj(t)us (t) ,
j
(2.11)
= f(t) - iK(t)u(t) Since us
j
in S and every Hs
and its derivative uE
j
uniformly over I
converge uniformly the limit
function u(t) is in CI(I,HS) for every s a u and u solves the Cauchy problem
uE
and in CI(I,S) = (2.1)
Also
(2.1)o
Now consider the general case of cE Hs, fE C(I,HS), for fixed s. We may construct sequences cpjE S, fjE C(I,S) such that cpj-*q,
fj(t)-af(t) in Hs, uniformly in I. For example one might choose
(2.12)
cpj
= tr (M/j)q (D/j)q
,
fj(t) = 'iV'' (M/j)'W" (D/j)f(t)
,
with i as in (2.5), using prop.2.2. Then, letting uj be the solution of (2.1) with cpj and fj, it follows that uj-ul solves (2.1) for cpj-cpl and fj-fl. Thus the energy inequality (2.7) implies
(2.13)
Iluj - u11lss c{Ilavj- cpllls+ MaxtE Illfj- f11ls} - 0
,
j,l- °°
6. Hyperbolic first order systems
204
Thus {uj} is a Cauchy sequence in C(I,H5), and u =lim ujE C(I,H5). Similarly u(=- CI(I,HS-e)
,
and u satisfies (2.1), proving existence.
Uniqueness of the solution is a consequence (2.7).
Q.E.D.
We now are left with the discussion of prop.s 2.2 and 2.3. We first discuss a few commutator properties of the 4do's involved, referring to the machinery prepared in 1,7. 1) The families {Le--V (EM) 05s51} and {JE=FLEF}, are bounded in L(H0) in L(H0) Both have strong limit 1 as E-0 ,
:
.
and
,
,
(Indeed the family of functions IV (ex) (Ex) = yr(0) = fgIx p(x) = 1
2) The family {K(t)
:
:
05651 } is bounded,
,
pointwise, for all x .)
t E =-I } is bounded in OpVice
.
(This
follows directly from (2.3).) 3) The family {K(t) - K*(t)
t E I } is bounded in OpPcvO
:
(Again this is a consequence of (2.4).)
4) The families 1[n j E ]/E } and {[ s,LE]/E}, where 0<e5l, and = (M) S2 (D) s' , with fixed s E H2 , are bounded in OpPcs-e Indeed we may apply I, lemma 7.2 onto ,
s
(ED)](D)-s' (M)-s2
noting that
((x)s') (0) E
= [(M)S2 Ir (ED)](M)
s2
O, s2 -1' while { (p (Ex) (0)/E : 0sss1} is bounded in W si-1.0' as 101=1. For the latter one confirms that
(2.14)
)-1=O((
)s cEl (1+E
a
with 1 = 101
,
for any (o E S . Similarly for LE
5) The family {[As,K(t)] t E I } is bounded in OPiycg This follows from (2.3) and I, lemma 7.2 again.) 6) The families {[K(t),JE]/E } and {[K(t),LE]/E } where 0 < E s 1 and t (=- I are bounded in OpipcO Indeed we again may apply I, lemma 7.2 using that E ipce2 k(0 :
.
,
,
:0< E s 1} is bounded in Vc-e,
E-=Vice,
for 101= 1.
7) The family {[JE,LE] 0 < e s 1} is bounded in Opipc-e (Proof similar to the above.) :
Proposition 2.4. We have (2.15) I(U,K
where the constant
with the inner product (u,v) s of Hs Oses1, tE I, sE &2, uE S,
cs is independent of e,t,and u.
Proof. Let v=su=(M) '(D) 'u. Abbreviate K(t)=K
, JE =J, LE =L, IIs=II:
(KEU,u)s - (u,KEU)s =((K-K*)v,LJV) +(Kv,[J,L]v)+([K,J]v,Lv)
+(v,[K,L]Jv)+([I],LKJ]u,v)+(v,[LKJ,II]u where [II,LKJ] = (2.16)
= [ 17, L] KJ + L[ II, K] J +LK[ H, J]
,
6.2. Systems of operators on Rn
205
(.,.)=inner product of H. We assumed J£=J9 (V even) and used that (2.17)
(lulls = II s'vIls = I1vII2 = 11V110
For the first 4 terms at right we use,in this order,(1) and (3), (2) and (7), (1) and (6), (1) and (6). For the last two terms, the commutator split as stated, use (1),(2),(4),and (1),(5),q.e.d. Proof of proposition 2.3, for u E C1(I,S). First let the solution u of (2.1) 9 be in C1(I,S). Then µ(t)=llu(t)II, is C'(I,le). We get dµ(t)/dt = µ (t) = (u (t),u(t))8 + (u(t),u (t))s (2.18)
= -i((K9u,u)s - (u,K9u)s) + 2 Re(f,u) s
Thus (2.15) implies I 1A
(2.19)
(t) I s Cellu(t) II$ + 11f (t) 112 + HUM 112
=(cs+1)µ(t) + IIf(t)112
,
tL(0) = IIcvlis
The Cauchy problem (2.19), for an ordinary differential inequality may be integrated in the usual way. It follows that
(2.20)
µ(t)e-ct
- (Irons s foe
ctllf (z ) Ilsd
proving (2.7) for t a 0 . Similarly for t s 0. In the general case uE C(I,H,) fl CI(I,Hs-e) it no longer is evident that µ(t) is differentiable since the right hand side of (2.19) no longer is meaningful. The discussion below shows that µ' E C(I) with (2.19) still exists. Then (2.20),(2.7) follow again. Instead of µ we first form µb(t)=(u(t),J,Lbu(t))8, tE I, 6>0. Abbreviating as in (2.16) again we note that JLu E C1(I,S). The inner product of Hs is trivially extended as (g,h),=(A,g,A,i gE S' , hE S . Hence we may carry out a differentiation as above: ,
(t) = (u ,JLu)5+ (u,JLu )5=-i((K9u,JLu)5-(u,JLK9u)5) (2.21)
+ (f,JLu)5 + (u,JLf) 8
.
We get restricted to the case e = 0 since for e > 0 we may keep 6 = 0 and carry out (2.16), using the extended inner product. For e = 0 consider the term, with M=H6=JL=J6L6 , v= su=Ihi, (Ku,J6L6u),- (u,JSL6Ku)5 = ((K-K*)v,Mv) + (Kv,[11,M]u) (2.22)
+([II,K]u,Mv) +(v,[K,M]v)+(v,[MK,f]u) + ([I1,K]u,[II,M]u)
,
6. Hyperbolic first order systems
206
by a calculation.As 6-0 the right hand side of (2.22) converges to
((K-K*)v,v) + ([II,K]u,v) + (v,[K,II]u) = p(t)
(2.23)
,
. Indeed,the second,fourth and sixth term,at (5) and (4) ,respectiright,tend to zero,using (2) and (4) ,(6) vely. The remaining three terms of (2.22) converge to to the terms in the order of listing,using (1) and (3) ,(1) and (5), of (2.23) uniformly for t E =-I
,
and (l)and (4) for [MK,rl] = M[K,IT] + [M,rl]K - [K,rIJ in L(HS,H0) respectively. Each convergence is uniform, for t E I Using this in (2.21) as 6 - 0 we find that (2.24)
limb_.0µb (t) = p(t) + 2 Re (f,u)s uniformly for t E I
Since µb and µ(t) is C1(I)
p'6*
both converge uniformly it follows that the limit
and lim p'b.
= µ
Hence we indeed get (2.19) esti-
mating the terms (2.23) with (3),(5). This completes the proof. 3. The evolution operator and its properties. As a consequence of the existence and uniqueness theorem of sec.2 we observe that, for every fixed t E I cp - u(t) defines a linear operator U(t): He
,
the assignment HS , where u(t) de-
notes the solution of (2.1) for f e 0 This operator U(t) will be called the evolution operator (or the solution operator) of the Cauchy problem . As an immediate consequence of (2.7) we find that .
IIU(t)roIIS
5
IIroHe
,
E Hs , t E I
.
Moreover from thm.2.1 it follows that the operator family { U(t) :t E I } is strongly continuous in L(Hs) while the derivative U (t) = dU(t)/dt exists in strong operator convergence of L(HS,HS-e) and is C(I,L(HS,HS-e)) under the assumptions stated. Similarly, under the assumptions of thm.1.1, the evolution operator of (1.1) will define a strongly continuous group in L(H). With thm.2.1 U(t)=sU(t) exists for every sE &2, and we get sU(t)q)=s,U(t)T whenever gE He f1 HS,. One obtains a well defined continuous map S'-3-S', of order 0 in the sense of 111,3. We denote it by U(t)E=- O(0). Clearly U' exists in S' , and U'E O(e). Also, ,
(3.2)
U(t) +iK(t)U(t) = 0
,
t E I
,
U(0) = 1 = identity.
Somewhat more generally we may consider the Cauchy problem (3.3)
du/dt + iK(t)u = f
,
t E I
,
u(t0) = g
,
207
6.3. The evolution operator
with given z0E I. Then I no longer must contain 0. This problem re-
duces to (2.1) if the new variable t+t0 is introduced. Consider
(3.4) v+ iK(t+t0)v = f(t+t0) ,t E I-t0 = [tl-t0,t2-t0], v(0)=q. Clearly v solves (3.4) if and only if u(t)=v(t+to) solves (3.3).
Thm.2.1 applies to (3.4), which is of the general form (2.1) with all assumptions satisfied also for the coefficient K(t+t0) and f(t+t0). Hence for each f E C(I,Hs) there exists a unique solution u E C(I,Hs) fl C1(I,Hs-e), solving (3.3). Thus get a more general
evolution operator U(t,t):
mapping Hs-'H5, sE &z ,
atU(t0,t) + iK(t)U(t0,t) = 0
(3.5)
,
t E I
satisfying
, U(t0,t0) = 1
.
If K(t) is independent of t we get U(t0,t) = U(t-t0) = e-iK(t-to). so that U(t0,t) is determined by the group U(t) similar as in thm.1.1. For reasons similar as above we have ,
(3.6)
IIU(t0,t)Ils
5
cs
t,t0 E I
,
.
Also U and aU/at are strongly continuous for each fixed to. Regarding dependence of U(t0,t) on t0 we consider the dependence of v in (3.4) on the parameter t0. We strengthen our assumptions (2.3)(2.4) on k=symb(K(t)) by requiring that also kE C'(I,,ae), i.e., (x)-1+Ial{
(3.7)
E C(I,CB(&2n,Lv))
Let us return to the mollified equation (3.4)E, i.e., (3.4) with K replaced by KE of (2.6) . The coefficient KE(t+t0) is C1(I,L(Hs)) , with the set I no {(t,t0) E &2 t0 t + t0 E I } :
,
.
Classical theory of linear ODE's with bounded coefficients implies that vE(t,t0) solving (3.4)E for f=0 is C1(I,L(Hs)). Moreover, atovE=WE solves the formally differentiated Cauchy problem wE + iKE(t+t0)we+ iKE (t+t0)vE = 0 , WS(0) = 0
(3.8)
where vE is given as solution of (3.4). Assuming first again that q E S the technique of sec.2 may be repeated. One finds that using (3.7). This implies d/dt(IlwElls) = O(lIwEII$ + Ilvalls+e) ,
,
(3.9)
IlwEll2 s ecltIIlvell22 +e
5
e2CItIIIVP112+e
and cp with c independent of E,t,t0 The family {vE(t0,t):0<E51} is equi-continuous; it has bounded first partials for t and t0 It is bounded, and maps into a conditionally compact set. Thus Ek, k=1,2,..., may be found, Ek-0, vE converging to v(to,t)E C(I,S), .
,
.
k
208
6. Hyperbolic first order systems
in every Hs. Also atVF k=-iKE (t)v£ (t) converges in S, hence atvE k k solves
C(I,S). Moreover, the function z5(t)=(v(tO+S,t)-v(t0,t))/S z5*(t)+iK(t+t0)zO(t)=-i(AK/O)v6
(3.10)
,
z5(0)=0
,
with v (t)=v(t0+S,t) , AK=K(t+t0+S)-K(t+t0). For the difference p=zS-zS, S S' >0, with A'K=AK, for 6=6' we get ,
,
,
")+(AK/S-A'K/S')v ', p(0)=0
p +iK(t+t0)p=-i(AK/S (v S-v
(3.11)
,
t+to +S where AK/S=S-1
K (x)dx is a bounded family in L(H,,H,_ )
t+to
since K'(t), tE I, is bounded in OpWce. Moreover, AK/6-K'(t+to)-0 in L(Hs'HS-e), by (3.7) and III, thm.1.1. Also, v -v -'0 in each
Hs, since vE C(I,S). The right hand side of (3.11) tends to 0 in each Hs, as uniformly in Z. Using (2.7) on (3.11) get p-0 as 6,6'-0, in each Hs, uniformly for (to,t)E I, hence ato vE C(I,S) exists. Taking limits in (3.10) conclude that w = ato v satisfies
w + iK(t+t0)w +iK (t+t0)v = 0
(3.12)
,
= 0
w(0)
.
Clearly (3.12) implies that w is continuous, so that vE C1(I,S). Finally, if T E HS we again construct T.= LJp E S Tj - T in Hs and then will get 11v-v100s -* 0 11v -V1 "s-e -> 0 as in the proof of thm.2.1, with vi the solution for Tj . Using (3.12) we also get flw-wills-e - 0 for wi = at Vi Consequently, ,
,
,
,
.
0
v(t0,t) E C(I,Hs) (1C1(I,Hs-e)
(3.13)
Similarly u(t0,t)=v(t0,t0+t)
,
the solution of(3.3), satisfies
u(t0,t) E C(IxI,HS) f C1(IxI,Hs-e)
(3.14)
.
Thm.3.1, and cor.3.2, below, now can be left to the reader:
Theorem 3.1. Under the assumptions of (2.3), (2.4), and (3.7) on the symbol of K(t) the evolution operator U(t,t), t,tE I, of the Cauchy problem (2.1) has the following properties. (i) U(t,t) S'->S' is an operator of order 0 it continuously maps S-S, and Hs-Hs, sE ie2. Its first partials atU a U exist in t strong operator convergence of L(Hs,Hs-e), sE &2. They are E O(e). in (ii) U(t,t) is bounded and strongly continuous over IxI L(HS) s E H2 atU and a U both are bounded and strongly contit t0,t E I s E H2 nuous in L(Hs,Hs-e) :
;
,
,
,
.
,
209
6.3. The evolution operator
(iii) For t,T,x E I we have (3.15)
U(t,t) = I , U(t,x)U(t,t) = U(t x)
U(t,T )U(t t)
,
1
(iv) U(t,t) satisfies the two differential equations
(3.16)
atU(t,t)+iK(t)U(T,t) =0 , aTU(T,t)-iU(t,t)K(T) =0, t,TE I,
(v) The operator U(T,t) is uniquely determined by properties (i), (ii), (iii), and one of the differential equations (iv).
Corollary 3.2. The evolution operator U(T,t) is invertible in S S' and in every Hs for T,t E I Its inverse is given by U I(T,t) = U(t,t) . Moreover, the family ,
.
,
V(T,t) = U 1*(t,t)
(3.17)
=
U*(t,t)
, t,TE I
,
is the evolution operator of the adjoint equation's Cauchy problem V'+ iK*(t)v = g
(3.18)
t E I
,
,
v(t) = 1 ,
where "*" means the L2(mn)-adjoint. For fE C(I,Hs) the unique solution of (3.3) is given by t
u(t) = U(T,t)tp +
(3.19)
U(x,t)f(x)dic
T
Next we notice that higher T,t-derivatives of U will exist if higher derivatives of K(t) are assumed. For example, let
symb(K) = k E C,(I,4ce)
(3.20)
.
Then it is clear that K E C*(H H ) for all s. Accordingly it s s-e follows from the differential equations (3.16) that a U and a U T t have first order partials for t and T in L(HsHs-2e) in strong operator convergence. Thus the second order partials of U for t,T exist and are strongly continuous in L(HsHs-2e)' Using this in (3.16) again we find that the third order partials of U exist in
L(HsHs-3e)
,
etc. By iteration one finds the corollary, below.
Corollary 3.3. Under the assumption of (2.3), (2.4), (3.7), and (3.20) U(T,t) has partials of all orders existing in strong operator convergence of S and S'. We have ataiUE O((j+l)e). That derivative is strongly continuous in every L(HSHS-(j+l)e)' sE &2 Remark 3.4. It should be noticed that diffentiability in thm.3.1
and cor.3.3 may be strengthened if we replace (3.7) or (3.20) by
(3.21)
kE C'(I,i4ae,)
210
6. Hyperbolic first order systems
(or, resp., kE C1(I, yice,)
, where a°=e'=(1,0) or a°=e2=(0,1)
)
With the stronger assumption get aTU, atUE 0(-e°), existence, continuity of atU,ayU in strong convergence of L(HS'HS-e)' (for C1), and atazUE O(j+l)e°),corresponding existence, continuity (for c0o). Proof by reinspection of (3.16), under the new assumptions. 4. N-th order strictly hyperbolic systems and symmetrizers. The hyperbolic equations encountered in ch.0 were of order 2. Correspondingly we plan an existence and uniqueness theorem for an N-th order hyperbolic system (4.1), below, with a detailed theory
t. E I, let
to be worked out in ch. VII . With I=[ ti , tz ] ,
(4.1)Lu=1=0(-i)iAN_j(t)(at)ju=f(t), t(=- I, dlu/dt1=W1, t=t0, Osl
on &n, with smooth coefficients, and A0(t)=1 . Again (4.1) will be regarded as an ODE for u=u(t) taking values in L2(]en) or He(&n). Using a standard method of ODE convert (4.1) to a first order system. In correspondence with rem.3.4 use a comparison operator ,
(4.2)
=(x), °=Z.
A 1=Ae,-l=nee=ne° (x,D)=(x)(D), a°=e,
Note that Ae,E Opyc_e,. With the new dependent variables uj = AjDtu
(4.3)
j = 0,1,...,N-1
,
the Cauchy problem (4.1) (i.e., (5.7)) is equivalently written as (4.4)
v + iK(t)v = g(t)
,
t E I
v(t0) = V
,
where we have introduced the (column-) vectors (4.5)
v=(uO,...,uN_1)T,
"((p0,...,(-iA)N-1(PN_1)T'
g=(0'...,0,iAlf)T,
and the square matrix of yido's 0 0
(4.6)
K=A
-1 .
1
0
0 .
0 .
NO
AN,A AN_1A'' ,
.
-1
.
.
.
...
. .
.
...
0 0
. .
. .
.
0 N- 1 ,A
I
AIAN.
Clearly u=u0 solves (4.1) if and only if v solves (4.4). K of (4.6) is a matrix of tpdo's. If (4.1) is a single equation for
211
6.4. N-th order strictly hyperbolic systems
one unknown function then (4.4) is an NxN-system. In general, if (4.1) is an vxv-system, then (4.4) will be an NvxNv-system. We desire to match the system (4.4) with the assumptions of thm.2.1, or better, thm.3.1 and cor.3.3 (or rem.3.4), but find it generally impossible to achieve the symmetry condition (2.4). All other conditions translate into natural conditions for equation (4.1): We assume that symb(K(t))j1E C"(I,wcea), with the 3 choices
of ea corresponding to (4.2). This means that (4.7)
E
Or, as a possibly stronger condition
Coo(I,1jea)
we require
aa, j(t,x) E C'(I,Vc(0,(N_j)fz ))
(4.8)
,j=I,...,N
,
f = (fl ,fx)
ea
Let us be guided by the case of (4.1), for scalar u(t,x) (i, e., v=1), being constant coefficient hyperbolic with respect to t (ch.O,sec.8, ex'le (e), or VII,1). We then think of ea=e', A--%(D), The symbol a of the operator matrix KO=AK then has the property that,up to lower order terms, det(a+z)=-7=O ZN_jaj(g)Xj(g)
where PN(t,g) is the principal part polyno-
mial of L=P(Dtlj )= of PN(X,
N_j(Dx)Dt. If L is hyperbolic then the roots
) are real, as
O. L is strictly hyperbolic if and only
if they are real and distinct. For ItI=co this happens if and only if symb(Ko) has real and distinct eigenvalues.
Accordingly, from now on, we focus on a system (2.1), where (4.9)
KO(t) = AfK(t)=k0(t,x,D) + x(t,x,D)
,
with k0E 'CE C°°(I, cve), the vxv-matrix having v real and distinct eigenvalues for all (t,x,t)E IxMe,0 ' with ,
f2 IxI+fi
with the compactification Me,0 of 11, 3, ea=(f, ,f2
We then will speak of a strictly hyperbolic system of type f=eA
.
An N-th order vxv-system (4.1) will be called strictly hyper bolic of type ea if it leads to a strictly hyperbolic system (4.4) We will rederive all results of sec,s 2.3.4 when the symmetry condition is replaced by a weaker condition, using "symmetrizers". A (global) symmetrizer is defined as a one-parameter family {R(t)=r(t,Mw,D):tE I} of zero-order 4 do's, r(t,.,.)E C0O(I,ta0), such that, in the Hilbert space H, we have R*(t)=R(t)ac>0, where R(t) is md-elliptic, uniformly over I, and that
6. Hyperbolic first order systems
212 (4.10)
(RK)*(t)-(RK)(t)=K*(t)R(t)-R(t)K(t) E C'(I,OptpcO)
.
Here "*" denotes the H-adjoint. The operator R(t), in other words, is a bounded hermitian and positive definite operator H - H. Since we use the Weyl representation R(t)=r(t,MM,D) of 1,3, it is no loss of generality to assume hermitian and zc>0 Symmetrizers were first studied by Friedrichs and Lax [FL]. For an application similar to ours cf. Taylor [T1]. We achieve a simplification of the method of [T1], using prop.4.1, below. .
Proposition 4.1. Given a symmetrizer R(t) = r(t,Mw,D) (satisfying above assumptions). Then the (unique) positive square root S(t) of , defined for every bounded positive self-adjoint Operator
R(t) H-36
,
is a pdo of order 0 again,together with its inverse S-1(t)
Moreover, S,
S-1
are md-elliptic, and we get S
,
.
Coo
S-1E
(I,Optpc0).
Proof. Write S(t)=Af dX\/X(R(t)-X)-1 as a complex curve integral r with a suitable branch of VT and a curve r in the upper halfplane surrounding the spectrum of S(t). The resolvent (R(t)-X)-1 is a Green inverse coinciding with the K-parametrix of II,thm.1.6, mod O(-oo). In particular, r may be laid such that R(t)-a, is md-elliptic for tE I, ?E r. The terms of the resulting asymptotic expansion of the symbol of the integrand are analytic in X and smooth in t. The 0-order term yields s0(t,x,U)=2nf
r
t
x%
-
,
using a
harmless interchange of integrals. Lower order terms contain higher powers of r-X in the integrand, but are well defined symbols of proper order. One gets an asymptotic sum
j defining a
symbol of S(t) (mod O(-co)). Then all other properties follow.
Notice that prop.4.1 legalizes the transformation (4.11)
S(t)u(t) = v(t)
,
t E I
,
for the evolution problem (2.1). That problem is taken to atv+iK"(t)v = g(t)=S(t)f(t) , tE I
,
v(0) = i=S(0)g ,
K _(t) = S(t)K(t)S-1(t) +iS (t)S-1(t) Clearly K"E COO(I,Opyicf)
.
, by prop.4.1. Moreover, we confirm that
6.4. N-th order strictly hyperbolic systems (4.14)
K-*(t)-K
213
(t)=S-l(K*R-RK)S-l_i(S-lS'+S'S_l)E C'(I,Oppc0)
using calculus of tpdo's. For a =e all assumptions of thm.2.1, thm.
cor.3.2, cor.3.3 now hold for K-. We have S, S I O(0), they map Hs-Hs, and S-S, and S'-'S'. For e°=e we even have the stronger conditions of rem.3.4. The result is summarized in thm.4.2, below. 3 . 1 ,
Theorem 4.2. Let the system (2.1) of ic1o's satisfy kE Cm(I,Vcv,)
and assume that there exists a symmetrizer R(t), satisfying above conditions. Then the statements of thm.2.1, thm.3.1, cor.3.2, cor. 3.3 still hold. Moreover, for e°#e, the solution u of (2.1) satisfies atuE Hs_je, (not only E Hs-je), as f(t), gE Hs . Also, for the solution operator U(t,t) we get aiatUE O((j+l)eA) Now we will show that a strictly hyperbolic system (2.1) (or (4.1)) of type a°=e, e',or e2 in the above sense, always has a global symmetrizer, so that thm.4.2 applies. Indeed, we have (4.9), where the eigenvalues of are real and distinct uniformly over in the compactification Ie'0. It follows that k0(t,x,t;) is diagonalizable for and that its eigenvalues may be arranged for ,
(4.15)
t-'2(t,x'U) < ... <
Moreover, we must have
2Tt1>0, j=1,...,v-1, f2 IxI+ft
(4.16) with a constant rt1 > 0
,
independent of t,x,l;
.
krl0.
By well known
perturbation arguments for matrices it then follows that the
functions R j are coo for t E I
, f2 I x I +fi 11; I ZTI0 . The point is that the spectral projections pj(t,x,t) of the
symbol
defined for f2JxI+f1JlJ 2 rt0, may be extended to
symbols pjE C"(I,tv), and that then R(t)=r(t,MM,D) with the symbol rO+r_e
(4.17)
for a suitable term r_e, will define a symmetrizer. Note r(t,x,l;)
of (4.17) is symmetric modulo lower order. Its first term r0 is in vertible, since r0u=0 implies pju=0, j=1,...v, hence u=lpju=0. We
have k0=D'lpl
,
with pTplT=pTSj1
hence r0k0 X jpjTpj
,
, using that pjpl=pjoj1 . Also,
and k0=q,1p1T , we get 0r0=Xjpj Tpj=r0k0
,
for
6. Hyperbolic first order systems
214
tpco then we If we can show that can hope to use calculus of Wo's , and a proper choice of r_e to get the desired properties for R(t) First the eigenvalues µj(t,x,t) are solutions of the charac=0, with ©, a polyteristic equation ®(? nomial in x with highest coefficient (-1)v, and other coefficients in tvc0. Its roots are distinct, by assumption, and are bounded by the matrix norm of k0(t,x,t) which is bounded above. For the first derivatives one finds
tE I,
.
µj(a)=-(0(a)/© (4.18)
=1,j'1(µj(t1x1t)-µl(t,x,t)) (
which is found to be 0(( )-JaJ(x)-JPJ) , using condition (4.16), and the boundedness of Rj Similarly, for the higher derivatives, including t-derivatives, one finds by successively differentiating (4.18) recursively that t,j(P) = so that .
O(x-JaJe'-JPJez(x,U))
indeed µ.E C (I,'tpa0)
.
,
In all the above we were always assuming
f21xI+f. 4 1kn0, and all estimates are derived only for those x, For eA=e it is clear that any extension of the restriction of Rj to JxI+ItIX210 to a C00 (Ixle2n)-function will satisfy the same estimates, thus give symbols in C°°(I,tvc0) equal to the µj, for large
IxI+lgl. For a°=e' use a cut-off x(t)
,
=1 in ItIa2rI, =0 near
111-n} , and use µjx instead of µj . Similarly for aA=ez For the pj we have a complex integral representation (4.19)
i/2xJ
-µj (=Y1' /2 with the constant n1 of (4.16). Here (4.19) may be differentiated under the integral sign, where the integration path may be kept
constant during differentiation. It is clear at once that we get C0o(I, Vcyo) , after modification outside J f2 Jx J+f. I t I x2rI} _ E11 . p jE For e°#e again choose xpj as pj, with x as for µj. We summarize: Proposition 4.3. For a strictly hyperbolic system of type a°, with
e° = e
,
e'
, ez
, and k0E
symbols in C°°(I,'Wa0)
,
Cm(I ,'grc0)
, the eigenvalues R j of k0 are
and the spectral projections pj are symbols
in Cm(I,Vcvo), both after suitable modification outside some Ix 2" ,i.
Theorem 4.4. Let equation (2.1) be strictly hyperbolic of type e° and assume that k0E C0O(I, Then there exists a symmetrizer, and thm.4.2 is applicable.
6.5. Particle flow
215
Proof. For a°=e let r be given by (4.17) with r-e to be determined For e°#e add (1-x) with or x(x) to r0 at right of (4.17), res p ectively. Observe that R O=2(r0(t,x,D)+r0(t,x,D) * ) has the proper form, by calculus of pdo's , while R0 is a bounded self-adjoint
operator of H by III, thm.1.1. Evidently R0 is md-elliptic of order 0 , and so is RO-y 0sys1/n , because, for zE &n , we get ,
z r0z = 1ipjzl2 an
(4.20)
lIzl2
Thus we conclude that the self-adjoint operator RO(t)-y is Fredholm, for y
C00(A,4v-,0) ,
and we may define a global R(t), for all tE I , as
a weighted mean, using a suitable partition of unity 1=(j(t) This completes the proof of thm.4.4. 5. The particle flow of a single hyperbolic ide. In this section we again consider the Cauchy problem (2.1) but assume v=1, i.e., a scalar problem (5.1)
u +ik(t,M1,D)u = f
,
t E I
u(s) = cp
,
,
with a compact interval I and a complex-valued containing t C00(I,Pce) symbol kE The conditions (2.3), (2.4) then amount to ,
,
.
(5.2)
k1 real-valued, k0E C(I,'pc0)
Under these assumptions thm.2.1, and thm.3.1 apply. For j=0,1 we assume kjE C"(I,y1aje) and then have cor.3.2 and cor.3.3 as well. Assume a°=e; the cases e°#e mean stronger assumptions, to be studied later on. We will get back to systems in ch.9. In this section we focus on the result below, regarded as one of the focal points. Thm.5.1 will be called Egorov's theorem, although what we present will be an amended global version of the
result of Egorov [ Egl] .
216
6. Hyperbolic first order systems
Theorem 5.1. Conjugation A - At't= U(t,t)-IAU(t,t) with the evolution operator U(t,t) of (5.1) constitutes an order preserving automorphism of the algebra Op4v of pseudodifferential operators as defined in 1,7, for all t,% E=- I. In with symbol in e = yic00 other words, for A = a(M1,D) E OpVcr , rE 12 we have ,
,
At ,.t = U(t,t)AU(t,t) = at,t(Mi,D)
(5.3)
,
with a certain symbol at't E Vcr , for each t,t E I The proof of thm.5.1 will be prepared by discussing a few consequences of the theorem providing hints towards the proper .
approach of proof. First consider the case r=0. In V,10 we have discussed the C*-subalgebra As of L(HS) obtained as norm closure of Op,a0 in L(Hs). It was seen that the symbol space of As -i.e., the maximal ideal space of the commutative C*-algebra As/Ks - is given as boundary Me'O of 12n in the compactification Pe,O allowing continuous extension of symbols aE VcO. We trivially have U(t,t)KSU(t,t) = Ks
(5.4)
, KS=K(HS)
From thm.5.1 we conclude that U(t,t)ASU(t,t) = As
(5.5)
Accordingly there is an induced automorphism of the commutative C*-algebra AS/KS a C(Me,O). It is well known that every such automorphism of an algebra C(Me'0) of continuous functions must have the form y(x) - q(v(x)) , with a homeomorphism v: Me,O `-'Me,O The map v is defined as dual of the automorphism. Thus we have v' = vi't
(5.6)
,
v': Me,O H Me,0 , t,t E I
,
a family of homeomorphisms induced by the automorphisms (5.5) with (5.7)
at't = aovt,t , as IxI+I
I=o ,
for all aE pcO
.
In other words, the map a-at,t of the symbol of A=a(x,D) under the automorphism A->U 1AU of thm.5.1 is given by a certain fami-
ly of homeomorphisms, as far as the values of a at - are concerned. Perhaps this justifies the idea to attempt construction of at't over all of 12n from a family of diffeomorphisms (5.9)
:
vt,t
12n , 12n
, t,t E I
,
extending continuously onto the compactification Pe,0
, where
6.5. Particle flow V1
(5.10)
t,t m
217
vt,t Me,0
'
while we expect U IAUa(aovt't)(x,D). Let us assume that
(5.11)
vt 't = identity = id
,
vi
'K
= vt'xovt 't t,x,t E I
,
because the corresponding relations for v' follow from (3.15).
Actually (5.11) points to the fact that vt't is a flow generated by a system of differential equations (details later on). Our condition that vt,t extends continuously to the compactification 1e,0 amounts to the statement that
aovt't E Vc0
a E lc0
(5.12)
(We shall see that even a E tpcr implies aovE 1Jcr
,
for all rE 1R2. )
An explicit construction of vt't and a proof of thm.5.1
results if we assume that the symbol at't of (5.3) is of the form
(5.13)
at = aovt,t + rt,t
rt,t E yic-e
.
For simplicity of notation assume t=0 and write at = a0,t , vt = v0't , v't = V1 U(t) = U(0,t) . At = AO,t 0't From thm.3.1 we know that U(t) hence At admits a strong ,
,
t-derivative in L(Ss H ) s-e (5.14)
Using (3.16) we find that
At =U (t)AU(t)+U(t)AU (t)=iUI(t)[K(t),A]U(t)=i[K(t),A]t.
Notice that the commutator [K(t),A] of two Wo's is readily evaluated in terms of 'pdo-calculus. From I,(5.7) we conclude that
(5.15)
i[K(t),A] = (k,a) (Ml,D) + Rt , Rt E Opyic-e
,
with the Poisson bracket (5.16)
(k,a)
Assuming ( 5 . 1 3 ) ,
(5.17)
I j=1
(k(e)a(8) - a(8)k(0))
and in addition that rt
at (x,
)
.
exists and rt E la_e
=
a symbol of order -e It is practical to replace k(t) by the real-valued k1(t) of (5.2), effecting an error term (k0,a) of .
order -e again. Applying the chain rule to the left hand side and (5.14) to the right hand side of (5.17), with (5.18)
x, E &n
218
6. Hyperbolic first order systems
a comparison gives the relations (5.19) xt =k
IN
t
I, x0=x , 0
modulo terms of order -e again.
We try to satisfy (5.19) precisely, not only modulo -e, observing that (5.19) gives a Cauchy problem for a nonlinear system of 2n ODE's in the 2n variables This is a hamiltonian system with real-valued Cm'-coefficients kll , kllx. Locally, for a small the solution of (5.19) exists uniquely, and depends smoothly on the initial parameters by basic results on ODE's. The local solution may be continued into all of II2nxi provided
that we can derive apriori estimates insuring boundedness of over I, for any given finite x,g. Such estimates will be aXa t(x,) E obtained in sec.6. Hence for a,1E IIn, C00(Ix&2n) , by standard results (cf. [CdLv], for example). Note that the same argument for U(i,t) instead of U(t)=U(O,t) will lead to the same system (5.19) of differential equations, but with initial conditions at x, not at 0. Thus we get (5.11) as an consequence. Particularly, vt,T,vt't = identity, so that vt't is indeed a family of homeomorphisms (even diffeomorphisms) of &2n onto itself. This family will be called a particle flow of equation (5.1) Clearly the particle flow is not uniquely determined by (5.1), since the symbol ki is only fixed up to an additional term of order -e. However, it is unique at lxI+lgl=oo , as will be seen. Theorem 5.2. For any given real-valued k1E pce the flow vs't defined above (following the solution of (5.19) through (at t= t) to t=t) induces a family a - at't = aovT' t of automorphisms 'm H yicm for all m. Moreover, we have at,tE C'(IxI'Vcm), mE I The proof will be discussed in sec.6. It is evident, after thm.5.2, that the map aa,,t aov,,t also provides an automorphism of the sup-norm closure of the function algebra c0 the maximal ideal space of which defines the compac,
,
tification 2e,0. The associate dual map of this automorphism will
be a homeomorphism defined as the continuous extension of v't onto Pe'O. The restriction v'T't to Me,O=ape,0 will be uniquely determined by the system (5.1). This map coincides with the homeomorphism v't,t'Me,O-Me,O previously discussed (cf.(5.6)). Proof of thm.5.l. From (5.11) we get at(vt',0v,'t) = 0 From the ODE's (5.19), using the Poisson bracket of (5.16), we get .
(5.20)
atxt'i + (kl,xt't) = 0
,
atgt'T +
0 ,
6.6. The action of the particle flow
219
with (,) of xt"t (x,l;) taken componentwise. For an arbitrary smooth a(x,g) and (5.21)
atat,t(x,
at.t=aovt,,L we get
) + (ki,at'z) = 0
as follows by using (5.20) in
(5.22)
)atxt,t +a,, (xt't
)at t.t
For a E Vcm let us use the operator (5.23)
Pt't= U(t,t)At't U(t.t)
Clearly At't E
.
Ate.,=at,t (x,D)
,
at,,=acv t',t
C00
(IxI,L(HS,Hs_m)), by 111,3, and thm.5.2. Since
atU(t,t)E C(IxI,L(HS,HS_e)) in strong operator topology, by thm.
3.1, we get Pt't E C1(IxI,L(Hs'Hs-m-e)) in strong convergence and
(5.24)
atPt 't = U(t,t ){ (atat,t) (x,D) + i[ K(t) ,At't] }U(t ,t) = Vt
by (5.14). Using (5.15) and (5.21), and calculus of pdo's, and UE 0(0) (by thm.3.1) we find that VtE O(m-e). Also Vt is strongly continuous in t as a map Hs-Hs-m+e Hence Vt is integrable, and
Jt
(5.25) Pt,t Pt,t = U(t,t)At,tU(t,t)-A =
doVV E L(HS'HS-m+e)
T
This confirms that at least (5.3) holds modulo an additive term in 0(r-e). To get more precision note that Vt=U(t,t)WtU(t,t), with Wt E OpVcm-e , hence (5.25) assumes the form t (5.26)
U(t,t)AU(t,t) -at't(M1,D) -
dK U(K,t)WKU(t,K) t
Clearly (5.25') may be iterated. Using (5.26) on Wk expresses U(K,t)WKU(t,K) as a sum TK+RK, with TKE Opyicr_e, remainder RK in 0(r-2e), as in the integral of (5.26), etc. Thus, for all N, (5.27)
with Tt
(5.28)
U(t,t)AU(t,t) = at,t(M1'D) + "l=1Tt t + XN E Op"cr_Ne
XN of order r-(N+1)e
Then we get
00 U(t,t)AU(t,t) = at't(M1,D) + Yj l Tt't +
X00
with X°'E Oppc_o,, and an asymptotic sum at right, using I,6. Q.E.D.
A generalization of thm.5.1 to hyperbolic vxv-systems was discussed in [CE]. We will discuss this in ch.9 for a larger and more natural algebra of Wo's (called Opts), rather than for Opypa.
6. Hyperbolic first order systems
220
6. The action of the particle flow on symbols. In this section we again look at the particle flow {vtt} In particular we prove thm.5.2. Assumptions are as in thm.5.1. In order to simplify the calculations let us write
cP=cP(x,)=VP(x,,t,i)=tt(x.)
(6.1)
Writing k1=k , for simplicity, we then have (5.19) in the form (6.2)
f'=kl (f,g)
,
cp'=-kix(f,g)
Proposition 6.1. The functions f
(6.3) ,
f=x , T= at t---r
.
cp satisfy the apriori-estimates
,
0
for t,t E I
,
,
0
&n , with constants c,C independent of
Proof. One derives the following inequalities for (f)'
(f)' = f.f'/(f) = kl .f/(f) = O((f))
((P)'
,
:
,
(6.4)
((p) * = cp cp * /(cp) =-k I x . w/(Cp) = 0((q)) ) Integrating (6.4), under the initial conditions (f)=(x) at t=v we get
log((f)/(x)) = O(1)
(6.5)
,
log(((P)/(l;)) = O(1)
,
((p)_(
)
.
(q) a 1 this indeed give (6.3), q.e.d. Note that prop.6.1 gives the apriori estimates required for
Since (f)
,
continuation of the local solution of (6.2) into all of I for any choice of initial values x,l; . As noted before, we then also get continuity of derivatives aiatf(R) aial(P(s) of all orders ,
,
j,l,a,p
,
as t,tE I
,
1n
Proposition 6.2. For j,1=0,1....., and a,pe ffin we have
(6.6)
aiatf
(a)=0((x)1-IUI()-IaI) ,
a C°-function (6.7)
consider the composition
b(f(x,5),g(x,t))
The derivative c(R) is in the span of the terms
221
6.6. The action of the particle flow
(PXvb)(f. ))njri f03 nlsl
(6.8)
where Iajl+If3 I21
,
Ia'lI+IP'lIa1
,
Yaj+2q'l=a
2Pj+D "=p
,
t
are to be and where the tensors V V b and vectors f(a) x (PI T with 0 contracted in arbitrary order of indices, f with This proposition follows by induction. For the proof of prop.9.2 we also use induction and first assume j=l=0 and a+t3=1 Note that the case j=l=a=p=0 is a matter of (6.3). Differentiating (6.2) we obtain ,
.
f(a). = k (6.9)
(3)
f(a) + k
f(a)=x(a)
(a)
Ix (1)
(P(P) --klxxf(R) - klx(P(3)
(1)
(0)
,
as t=t
,(P)_ ,
(P)
Here we have written klxf=4Ixjfj , but
fj
,
etc.
j Multiplying the first equation (6.9) by (cp)/(f) we get, as matrix,
(p)/(f)f(R)I
klgx
(6.10)
(a)
-k l X'( f)
(P
The 2nx2n-matrix in (6.10), called P1
-k l xi; I I I a, , has bounded coefficients,
since kE pce. Let the vector at right in (6.10) be called p=(pZ) The left hand side in (6.10) is not p', but we get
(6.11)
P. ._
hence (6.10) yields the linear ODE p'=Pp with a matrix P differing from P1 only in the upper left nxn-corner. There we find the additional term -((f.kl,)/(f)2+((p.klx)/(p)2) also bounded in x, ,
so that P also is a bounded matrix. We get (6.12)
IPI2'=2p.p' = 2p.Pp = O(IPI2)
This may be integrated for log (IpI/Ip(s)I)=O(It-tI)=O(1). Calculating Ip(t)I from (6.9) we get (6.6) for j=1=0, IaI+I3I=1
.
Proceeding in our induction proof, assume (6.6) true for j=1=0 and IaI+I1Isr-1, and consider a pair a,3 with I(xI+I3I=ra2 . We claim
that the vector p at right of (6.10) now will satisfy
(6.13)
p* = Pp + q
,
p(t) = 0 , q =
(qZ )
6. Hyperbolic first order systems
222
with the above matrix P
,
and with q'
,
(6.14)
q2
in the span of
(0)
111=1 (R4)
and of
(Qs+IQs'k(f,T))n3=1 f(3) 1Z1=1 T(e)
(6.15)
respectively. In (6.14) and (6.15) we assume that s+s'sr, and 1sIajI+I3jI
traction of V V k with the products of vectors is in arbitrary order of indices. Relations (6.13), (6.14), (6.15) follow by repeated differentiation of (6.9), using (6.8). The induction hypothesis implies that by direct examination of the terms (6.13), (6.14). Then (6.13) implies = O(IpI2+IgI2) IP(t)I2=0 . This may be integrated Ip12.
1
for
completing the proof for j=1=0. Next we look at t and t-derivatives of f,q . First it is evident for the vector of (6.13) that also p'=O((x)-IPI(t)1-Iai)
just looking at (6.13). This gives (6.6) for j=0, 1=1. In fact, we may differentiate (6.13) for t The right hand side then contains only terms involving the 0-th and 1-th t-derivatives of .
f(a and cp
already known to satisfy (6.6)
finds that also p''=0((x)-IRI(1;)1-IaI)
Using this, one
In fact, the procedure
may be iterated to get the same for all atp . This implies (6.6) for j=0 and all 1=0,1,2,...
.
Finally let us look for t-derivatives. In that respect it is convenient to notice that (f,tpj=(xtt'tt) satisfies the system (6.16)
f'=-k1 (f,q)
,
(p'=klx(f,(P)
,
"l" = "a/at"
Indeed, relations (5.20), (5.21) indicate that, for fixed the curves vt,t(x°,°) are the characteristic curves of the single first order PDE (5.21) in the unknown function This equation differs from (5.17) only by a sign, hence it follows that vt,t must be the 'reverse flow' satisfying (6.16), just as vt,t satisfies (5.19) . This clearly implies (6.16). However, (6.16) implies an ODE like (6.13) with ""ll replaced by for the same vector p with components ((,)/(f)f(R)=p'
It follows at once that (6.6) holds for 1=0, j=0,1,2,..., for reason of symmetry. But mixed t,t-derivatives may also be estimated in this way: For example, writing p'=Qp+r, we get p''=Qp'+ y(a)=pz.
223
6.7. Propagation of maximal ideals
Q'p+r'=QPp+Qq+Q'p+r', where the right hand side consists of terms already estimated. Hence p''=atatp may be estimated. Similarly for all mixed t,t-derivatives, completing the proof of prop.6.2. Note that now the proof of thm.5.2 is a matter of induction. First of all it follows from prop.6.3 that ai't aovt,t=a(f,(p)6Vcm for all t,t E I , whenever aE Vcm : All terms (6.8) obey their proper estimate, using (6.6) . Moreover, also the derivatives ajd1at't exist in the Frechet topology of "m , so that at'tE
C (IxI,tUcm) (6.17)
.
For example,
atat,t = alx(f,(P)f' + a, ro' =
using (6.6), prop.6.1, and the estimates for the symbol aE Vcm . Similarly for all higher derivatives, by induction. 7. Propagation of maximal ideals and propagation of singularities. We have noted before that the particle flow induces a class of homeomorphisms v.t:2 1 , of the space 1=2e,o i.e., the maximal ideal space of the C*-algebra A=AO of V,10. In particular vtt is defined as restriction to M=aP of the continuous extension to P of vtt defined by (5.18),(5.19). The algebra symbol aA of tt ,
A t=U(t,t)AU(t ,t) is given as (7.1)
= 'A°vtt tt for every AE A. Similar for AE As the Hs-closure of OpycO It is interesting to note that the above facts immediately translate into a result concerning propagation of wave front sets - and, correspondingly, propagation of singularities - under hyperbolic evolution. The key is the following result (cf. [Tl2]). Theorem 7.1. For any distribution uE 5'(ien) the wave front set WF(u) (as defined in 11,5) is given as ('A
,
(7.2)
WF(u) = fl(char(A,C): AE C° , AuE S}
,
with the characteristic set of A relative to C of V,10 defined by
(7.3)
char(A,C) = {mE W : aA(m)=0}
Proof. Suppose a(x°
#0 for some (x°
.
)E W , A=a(x,D)E CO
,
with AuE C'(An) . Then it follows that, with respect to some cutoffs (Px° (x) 4k-0 () , as in 11,5 , we must have A md-elliptic with
224
6. Hyperbolic first order systems
respect to
in the sense of 11, 3. By II,thm.
3.3 we then get a local left parametrix Et Ki ,
such that EiA=q(Mr,D)+
Ki E O(-oo) , Et E Op4c0. It follows that (tk, ((px° u)" )" =E, Au-K, uE
S , hence
W\WF(u) , by II,prop.5.1.SSIn other words, we
have "C" in (7.2). Vice versa, if (x° ,
°) E W\WF (u) ,
then there
exist cut-offs as above with i(D)W(x)uE S , by II, prop.5.2. It is
clear then that B=(p(D)g(x)E C°
,
char CB . Hence we also get ":"
,
does not belong to
while q.e.d.
Thm.7.1 first prompts us to define an A-wave front set
(7.4)
WF(u,A) = fl{char(A,A): AEOp(Nc0 , AuES}
where
(7.5)
char(A,A) = {mE W(A): UA(m)=0} W(A)=i-1W
with the wave front space surjective map L:Me,O - I of 11,3
= b-1(&nxa1n) of A , using the
Clearly we get
.
LWF(u,A) C WF(u)
(7.6)
.
We also may define the larger sets ZF(u,A) = fl{charmd(A,A) : AE Oprquo: AuE S}
, with
(7.7)
charmd(A,A) = {mE ffi(A)
aA(m)=0} .
:
Note that ZF(u,A) (and ZF(u,C) = ZF(u) defined similarly, with reference to C instead of A) also addresses singularities of uE S' at x=-. In fact, ZF(u)J WF(u) U WF(u^)
(7.8)
.
Theorem 7.2. Consider the solution u(t)=U(t,t)q of the Cauchy pro-
blem (3.3) with f(t)=O . We have (7.9)
WF(u(t),A)= vt(WF(q),A))
Similarly , for f(t)#0
,
,
ZF(u(t),A)=v,t(ZF((p,A))
the sets at left in (7.9) are contained
in the unions of the sets at right with the sets ZFJ(U(K,t)f(K),A) =vxt(C K between t and t JZF(f(K)A)),
Proof. Observe that ZF(u(t),A)=fl(charmd(A,A):U(t,t)AU(t,t)TE S} where charmd(A,A)={m:OA 0}={m:OAtt(vtt
(m))=0}=vtt(charmd(Att'A) )
Therefore, ZF(u(t),A)=vtt{charmd(Att'A):AttuES}=vtt(ZF((p,A) Similarly for WF(u(t),A)
,
and for ZF}(U(K,t)f(K),A). Q.E.D.
225
6.7. Propagation of maximal ideals
Corollary 7.3. Let k0(t,x,l)=symb(K0(t)) of (4.9) allow an asymptotic expansion k0 Dc0j (mod ICI) at 1;=-, as in V,(5.10). Assume kj 1 1;I=1, xE KG: &n, all K. Then we also get formulas (7.9) for WF(u) instead of WF(u,A)
E Co°. for tE I,
.
For a proof we only must show that (i) CO
in thm.7.2 may be
replaced by the algebra B of all AE OpipcO with symbols having asymptotic expansions, and (ii) that conjugation with U(x,t) lea-
ves B invariant - assuming the asymptotic expansion on k0 while (iii) the particle flow leaves M invariant. Details of these (rather formal) discussions are left to the reader. ,
Remark 7.4. A propagation law for the generalized singularity sets ZF(u) may be derived as well, under proper assumptions on k0
.
Problems. (1) Do the algebras VOS and 'L (cf.VIII,5) have commutators in K(LZ(&n))? (2) For the L(L2(in))-norm closure A of an algebra of (1), give a useful definition of WF(A,A) and ZF(A,A).
Chapter 7. HYPERBOLIC DIFFERENTIAL EQUATIONS
0. Introduction.
A polynomial P(x) =
I
sNaaxa
, aaE E (and the PDE P(D))
is called hyperbolic with respect to a vector hE in if (i) PN(h)#O (ii) for some real to we have P(t+ith)#0 as t E=- &n, tsto.
The above definition was given by Garding [Gal]. Its importance for the Cauchy problem of P(D)u=f in the half space x.h2:0
,
with data given at the hyperplane x.h=0 becomes evident if we try to apply the Fourier-Laplace method of ch.0,4: Let h=(1,0,...,0). Taking the Fourier transform in (x2,...,xn) and the Laplace transform in xl will convert P(D)u=f into P(t+ith)u^=f^, where the imaginary part t of the t1-variable must be sto, for some to, in accordance with the rules of the Laplace transform of ch.O,(3.14). Thus (ii) insures that we can write as tsto. Specifically, for f=6/xn, the inverse transform (P( +ith))"=e defines a fundamental solution of P(D). An analysis shows that (i),(ii) insure existence of e as a complex integral. Moreover, a=0 outside a cone lxlscx,. This leads to Garding's theorem: Theorem 0.1. A differential polynomial P(D) admits a fundamental solution eE D'(&n) with support in a strictly convex cone OE h C {0} U {x-h>0} if and only if P is hyperbolic with respect to h. For a distribution e with the properties of thm.0.1 and a general fE D' with support in the half-space x.hzO the convolution product u=e*f E D' is always defined, and supp u s {x.hz0} ([CI],I, thm.8.1,for example), and P(D)u=f. In fact, the noncharacteristic Cauchy problem of the half space x.ha0 is well posed if and only if P is hyperbolic with respect to h (written as "PE hyp(h)"). For a detailed discussion of such results cf.[Hrl]. For PDE's with variable coefficients we do not know results of similar precision. Existence theorens assume strict hyperbolicity (cf. VI,4) which is not a necessary condition. Still we discuss some useful algebraic properties of hyperbolic polynomials in sec. 1 below, focusing on the cone h of time-like vectors. In sec.2 we
226
227
7.1. Algebra of hyperbolic polynomials
consider strictly hyperbolic (differential) polynomials. The class s-hyp(h) of strictly hyperbolic polynomials consists precisely of all PE hyp(h) with only simple real characteristic surfaces (i.e., the PE hyp(h) of principal type). "s-hyp(h)" is a property of the principal part of P, insensitive against lower order perturbations A PDE a(x,D) with variable coefficients is called s-hyp(h) at x=x° if a(x°,D)E s-hyp(h). For an a(x,D), strictly hyperbolic over a domain SIC Rn the vector may depend on x. If is can be chosen constant, we speak of a normally strictly hyperbolic a(x,D). It is convenient then to assume h=(1,0,...,0) and distinguish the time direction h. This leads to sec.3, considering an N-th order vxv-system, looking like an ODE in t with coefficients PDE's in x.
We prove existence and uniqueness for a relativistic and a nonrelativistic normally strictly hyperbolic Cauchy problem. In sec.4 we simplify the assumptions, essentially by linking strictly hyperbolic differential operators with the strictly hyperbolic i,do's of ch.VI. In sec.5 we discuss the 'hyperbolic features' - region of dependence and influence, and finite propagation
speed. In sec.6 we derive a local existence result, and apply it to hyperbolic problems on a manifold. 1. Algebra of hyperbolic polynomials. We follow [Hri], discussing some algebraic properties of hyperbolic polynomials. No differential operators will get involved. Proposition 1.1. A homogeneous polynomial P = PN is hyp(h) if and only if (i) P(h) # 0 , and (ii) for each fixed 1; E Rn the equation P( + th) = 0 has only real roots. Proof. Since P = PN the equivalence of conditions (i) is evident.
For a homogeneous P E hyp(h) let P( + rh) = 0 . Using that P is homogeneous we also get P(aT+ath) = t)h)+i(Im t)a) = 0, for all real o. Since P is hyperbolic, we get a Im t zr0, for real a, implying that In t =0. Hence all roots of are real. Vice versa if all roots are real, we get for all real t#0 implying (ii) with t0=0 . Hence P is hyperbolic. ,
Corollary 1.2. A hyperbolic homogeneous polynomial has real coefficients, except perhaps for a common complex factor # 0 must equal Proof. We have Thus .
the product of all the (real) roots of real, for all real This implies that .
hence must be has real coef-
228
7. Hyperbolic differential equations
ficients, since all its values, for real l;, are real.
Proposition 1.3. If P is hyp(h)
,
then P is hyp(-h).
Proof. We have PN(-h)=(-1)NPN(h)#0 , i.e., condition (i) for -h. Consider the polynomial of the single variable t
P(T) = P(l+ith) = PN(ih){tN +
(1.1)
of exact degree N. We must have
with the N roots
tj( ) of p(t)=0. Also, pl(t) must be a first degree polynomial in
, i.e., a linear function of 1. We have Re tj ar0 for all , thus Re pl(t) a Nt0 For a linear function this implies that Re .
I Re tj s Re p1 -(N-1)t0
const. Accordingly, Re tk = Re p1 -
j7q--
is also bounded from above. Accordinglg we also have p(t) # 0 for all t E Rn and t s t1 q.e.d. ,
,
Proposition 1.4. The principal part PN of a PE hyp(h) is hyp(h). Proof. We note that PN(t) = limlal_.,,,, with Q0(t)=a NP(at).
Let qa(t) =Q0(g+ith) , then also lim qa(t)=q(t)=PN(l;+ith) all t
Suppose q(t)=0
,
for
There must be some complex root to of qa(t) = 0 101 > 00 with lim t a = t But this implies aNga(ta) = P(ao+itaah) = 0 so that a Re t a a t0 Since .
,
for some real t
.
.
,
,
.
,
Re to -> t this is possible only if t=0 . Accordingly,
as t is real and # 0 Therefore PN is hyperbolic. (Note that the highest coefficient of qa(t) is PN(h)(i)N again, hence is independent of a . Therefore qa is close to q on some sufficiently large circle Jtj=a0 so that Rouche's theorem supplies the root to .) .
,
The converse of prop.1.4 is false: A polynomial P with PNE hyp(h) needs not be hyp(h). For discussion see the proof of prop. 2.6. Given a hyperbolic homogeneous PN , the polynomial P=PN+R , deg R
7,IR(a)( )I2
1PN(a)(,)12
s c
a
(=-in ,
a
We will discuss this in sec.2. A polynomial P (or a differential expression P(D) ) will be called strictly hyperbolic (or Petrovsky-hyperbolic) (with respect to a real vector h3A0 ) if (i) PN(h)#0 and (ii) for each E 1n ,
linearly independent of h , the equation
has real and
distinct roots. We use the notation s-hyp(h) instead of hyp(h) for strictly hyperbolic polynomials or expressions. Note that
7.1. Algebra of hyperbolic polynomials
229
"s-hyp(h)" is a property of the principal part PN. The lower order terms do not enter the definition. We will show in sec.2 that "s-hyp(h)" implies "hyp(h)". Thus PNE s-hyp(h) has the property that all P=PN+R , deg R
.
,
,
.
,
is non-void. Also, h is a cone: If ?.E h, and p>O, then PN(pll+Xh)=
pNPN(rl+hk/p)=0 implies that 2=p(k/p)
tivity, for the special hyperbolic expression D2+...+Dn-Di
,
this
cone h plays the role of the cone of time-like vectors.
Theorem 1.5. For P E hyp(h) the set h = h(P,h) is an open convex cone. Moreover we have P E hyp(rl) for all rl E h The proof requires some preparations. Proposition 1.6. The set h equals the connected component of h in the open set {TIE &n: PN(rl)#0}gym . Moreover, h is starshaped with
with respect to h : For riE h the straight segment rjh also is in h.
Proof. For rl E h we get PN(q+th)=PN(h) llj=1(t-t j) , and all t are < 0 . Hence PN (rl) = PN (h) n(--c ) # 0 . We already know that h is j
open. For an it on the boundary ah all roots tj still are s 0 , but at least one must be zero, so that PN(rl)=0
.
Hence ah is a subset
of am . To complete the proof it suffices to show that h is star-
shaped, since then it must be connected, and cannot be a proper subset of the component of m containing h; otherwise some boundary and =X.rl+µh, point of h would be in the interior of m. Let TIE h X,p > 0 , X+µ=1 . Write PN(t+th)=PN(q +(t+µ)h)=XNPN(rl+h(r+µ)/a,). This shows that for a root t of PN(l+th) = 0 the expression t' _ (t+µ)/,% must be a root of PN(q+th)=0 , hence is negative. Hence also t=1v'-p< 0 It follows that tEh , so that h is starshaped. ,
.
as Re t < t0, Proposition 1.7. For q G h we have Re a s 0 E 1n , with t0 for h as in the definition of hyp(h). Proof. This evidently is true for Re a=0, by definition of hyp(h). Consider the roots of the polynomial as Re t < tO . As t varies these roots must vary continuously, since the highest coefficient of the polynomial is indepenBy our above remark none of the roots may cross the dent of t ,
.
230
7. Hyperbolic differential equations
imaginary a-axis so that the number of roots of q(a) = 0 within Re a < 0
stays constant, as t varies within the half plane Re t
< to. Therefore it suffices to show that q has no roots with Re a < 0 whenever t is real and ItI is large. Assume t#0 and write a = tµ so that q = 0 takes the form ,
,
. We know that all roots of PN(h+1x11) = 0 are
0
negative real, since 11 E Eh . Consider the family of polynomials
rt (µ)
(h+1111)) = PN(11)RN + ... . rt (14) = PN(h+µr1) . Accordingly rr (ti)
lim -,oo
Clearly we get = 0 implies Re µ < 0,
since µ must be close to one of the roots of PN(h+µr1) = 0 as ItI is large. In other words, for t<0, with large (ti all roots a = rr of q(a) = 0 must have positive real part, q.e.d. Proof of Thm.1.5. Let us apply prop.1.8 for real a,t , where t=ea, a > 0 assuming to < 0 We get ,
,
,
.
(1.3)
0 ,
as a < t0/E
and a cone. It follows that P E hyp(11+eh) ,
,
,
.
wla instead of 11 one concludes that P E hyp(11) Finally, to prove convexity of h that P E hyp(1l3)
,
Z; E &n .
since h is starshaped e > 0 But h is open, for small a>O . Applying the above for
We know that PN(1l+ah)#0 , due to 11+EhE h hence contains 11E = 11-Eh
,
,
let rl1
,
for every rl E h 112E h. It follows
j=1,2 . Moreover, we get h = h(P,11J)
,
j=1,2,
since h and the 113, all are in the same connected component of the set m. Since h then is star-shaped with respect to 111, the segment from 111 to 112 must be in h so that h indeed is convex, q.e.d. ,
As an example consider the 'wave operator' a2xl -a2x2-...-a2xn This operator is hyperbolic and strictly hyperbolic with respect to the "time-like" direction h=(1,0,...,0) Indeed, we get .
I2-1;12 .
I
hence
The two roots are real and distinct except if t-=0 We have P=PN homogeneous, and PN(h)=1#0. The cone h of time-like vectors is given by as follows from prop.1.6. .
I
.
2. Hyperbolic polynomials and characteristic surfaces. In this section we shall investigate the relation between "hyp(h)" and "s-hyp". It turns out that the set s-hyp(h) consists precisely of all PE hyp(h) with simple characteristics. Here the concept of characteristic (surface) must be understood in the sense of ch.0,8. There we already defined the concepts of simple and
231
7.2. Hyperbolic polynomials
multiple characteristic. Presently we look at these definitions under the aspect of constant coefficient expressions. A differential polynomial P(D) is said to be of principal type if PNI (U)#0 for all 0#tE &n . For a principal type expression all (real) characteristic surfaces are simple. Consider a surface cp(x)=W(x°), near x0, where PN((PIx(x0))=0 (cf. ch.0,(8.3)). If h E In is any vector and i; = Tlx(x0) then the polynomial p(t) = has a root at t = 0 If h can be found such that 0 is a simple root of p(t) then the surface is called simply characte,
.
ristic at x0. We have given the same definition for characteristic surfaces in ch.0,8. The multiplicity at x0 is defined as the lowest multiplicity of the root 0 of p(t) atteinable for various h For a real root of let so that (pIx t ,
and PN(cIx) = 0 It follows that, for a differential polynomial P(D), every real root t of the principal part polynomial PN gives raise to a family x.T;=c of real characteristic hyperplanes, with normal vector t . Vice versa, for an arbitrary real characteristic surface t of P(D) the tangent planes, at all points x0 E T , and their parallels, all are real characteristic (hyper-)surfaces. Now let P(D) be of principal type. Clearly Taylor's formula .
yields
Hence if
is a real
root of PN 0, then we may set h=PNI (U)#0, and get p(t)-rPNI which clearly has a simple root at t=0 . Vice versa, for an
expression P(D), let
at some t#0 . Using Euler's for-
mula for the homogeneous function PN we find that NPN(t)=t.PNIt(l;) =0 Thus we get p(t)=O(t2) for any choice of h. Or, the surfaces .
with normal t are multiple characteristics. We have proven:
Proposition 2.1. A constant coefficient operator P(D) is of principal type if and only if all of its real characteristic surfaces are simply characteristic. Elliptic operators (in the sense of ch.0,sec.8(d)) have no real characteristics, hence cannot have multiple real characteristics. Thus an elliptic operator always is of principal type. Proposition 2.2. An operator P(D)E s-hyp(h) is of principal type. &n. We know that Proof. Let PE s-hyp(h), and let for PN(h)#0 so that =?.h is impossible. Hence is linearly indepenhas a simple dent of h, and PE s-hyp(h) implies that root at t=0. Hence the planes normal to 1; are simple characteristics. It follows that all characteristics are simple, q.e.d. ,
Theorem 2.3. We have hyp(h)C s-hyp(h). For a polynomial PE hyp(h)
7. Hyperbolic differential equations
232
we have PE s-hyp(h) if and only if all real characteristics of P(D) are simple. In other words, s-hyp(h) consists precisely of all PE hyp(h) which are of principal type. We prove thm.2.3 in several steps. Proposition 2.4. Let P=PNE hyp(h) be homogeneous, and let the polynomial R of degree
first, observe that PNE s-hyp(h) must be of principal type, by prop.2.2. Conclude that PN is homogeneous of degree N-1, while for all ICI=1. It follows that, with some co>O, (2.1)
aIPNa)( )I2Z IPNI
(
for all E In
)I2.
However (2.1) implies that (1.2) holds for all R, deg R
s
c I a with some constant c, for all E &n. One confirms easily that this defines a partial ordering compatible with the algebraic structure of the ring of polynomials. Using Taylor's formula, we get 7, IF(a)(,)I2
a
(2.3)
for all
F (
E In
,
)
= { lIF(a)(,)I2}1/2
rl E En
(+)
,
, with a constant c(1) independent of
Accordingly, F®G implies translations F,q( )=F(
s c(T1) F
.
F1. J G,q, for the (real or complex)
..... Evidently F<0 G implies that the order of F cannot exceed the order of G Moreover, let G=GN be .
.
homogeneous and F=Lj=1 Fj , with homogeneous parts Fj of degree j,
then F<0G implies that Fj®G for all j. Indeed it follows at once that also for all t>0, hence that For a collection t0,...,tN of distinct positive numbers the van der Monde determinant is different from zero, and we get (2.4)
/k=0
G
Thus indeed, we get Fj
,
j = 0,...,N
7.2. Hyperbolic polynomials
233
Let PN E hyp(h). From thm.1.5 we conclude that h+Re i; is
time-like for sufficiently small ICI as I1Is 2Vn-b >0
,
. Accordingly,
E ten. Using Cauchys estimate lal-times, on the
analytic function f(t)=PN(t+i(h+t)) of n complex variables i;, get (2.5)
IPNa)(t+ih)/PN(t+ih)l s(c/O)lal1
max 5
with c independent of
.
For every fixed t
,
the polynomial cp(t)
= f(tt) cannot vanish, as ItIs2, assuming Il;i Is O
i.e., IzIsVno.
,
ltjlz2 with the roots ti of q . Hence the Max at the right hand side of (2.5) is estimated
This gives (p(1)=T(0)r)j=1((tj-1)/s
,
by I(p(1)/q)(0)Is(1/2)N, as I?Isvno. We have proven the following.
Proposition 2.5. If P=PNE hyp(h), then for all a and E In we have
and a
c independent of
(2.6)
If P<MOPN then we have seen that Pj
,
j=O,...,N-1
where again c is independent of t . Using homogeneity we get
IP j(g+ith) I=1t I iIP
Isclt I j1 PN(t/t+ih) I =clt I j-NI
Let R = PO+P1+...+PN-1
then we get Therefore P = PN + R satisfies the estimate (2.7)
,
IP(t+ith)l Z (1-cltl-1)IPN(g+ith)I # 0
,
as t < -2/c
This establishes prop.2.4.
Proposition 2.6. Suppose all polynomials P with principal part PN are hyp(h). Then PN is of principal type, and PN is s-hyp(h) 0 Proof. It is clear that PN must be hyp(h) so that .
,
has only real roots, and PN(h)#O If PN is not strictly hyperbolic then for some ? linearly independent of h, there must be a multiple root T. of To prove prop.2.6 we will show .
,
that in this case not every P=PN+Q, deg Q sN-1, can be hyperbolic.
We may assume to =0. Then the polynomial has degree N-k s N-2 Now we will show: If, under above assumptions, P=PN+R is also must have degree hyp(h) then the polynomial p sN-2. For a general R of degree N-1 or less, this can not be true. .
7. Hyperbolic differential equations
234
Therefore not every PN+R can be hyp(h), and prop.2.6 follows. Consider the equation as a polynomial equation in 2 variables a,t With x=,11 one may write it in the form xr(x,a) = 0
(2.8)
r(x,o) =
,
with a polynomial r(x,o) in x, deg r sN-1. At x=0 we get the equation PN(i;+ah)=0 , which, as we know, has the k-fold root a=0, ka2. From Rouche's theorem we derive the existence of N roots aj(x), j=I,...,N of (2.8), near the roots of for all IxIsO. These roots may be organized into "Puiseux series" -i.e. power series in x1/1 , with suitable integers 1>0 In particular each ,
.
of the k roots approaching 0, as K-0
,
has an expansion
I amxm/1Ms
Let s be choosen such that as is the smallest coefficient #0 then Clearly szO since the series vanishes for x=0 If 0<s<1 ,
,
.
ta(1/i) sast 1-s/1 =I asI It I Yexp{ i(arg as+y arg t )} , <1
.
where 0
For real t we may set arg t =vn , with an integer v . Here v
may be choosen such that arg a + vyn is not a multiple of n, since s must be i2. Then we get It Im aj(l/t)I . However, as ItlI
Imtaj(I/t))
we have
so that Imtoj(1/s)
remain bounded by definition of hyp(h). Thus we get a contradiction unless ral It follows that a(x)=0(Ixl) , as 1(90 for at .
,
least k a 2 of the roots aj , while all others are 0(1)
Finally observe that Xj(t)=ra(1/s)-1 are the N roots of for sufficiently large Itl It follows that .
O(ItIN-k)
that f(0) = PN(h)4=1(_? (t)) = p -(T) = P(tl;+h) must have degree s N-2
Corollary 2.7. If P E s-hyp(h) time-like vector Tl E h
,
,
. Hence indeed
q.e.d.
then also PE s-hyp(rl)
,
for every
.
Proof. The cone h of time-like vectors clearly is determined by the principal part PN only. Also we know that PN+R=P is hyp(,q) for every R of degree N-1 or less, because each such P is hyp(h) hence also hyp(q), using thm.1.5. Therefore P E s-hyp(rl) follows from prop.2.6, q.e.d. Clearly thm.2.3 is a consequence of prop.2.4 and prop.2.6. The fact that, for PNE hyp(h) the condition P®PN is implied by PE hyp(h), was proven by Chaillou [Chu1] and Svensson [Sv1] ,
.
235
7.3. The hyperbolic Cauchy problem
3. The hyperbolic Cauchy problem for variable coefficients. A linear differential expression L=a(x,D), as in ch.O,(8.1), will be called strictly hyperbolic with respect to h, O'hE &n, at is s-hyp(h). If this holds for x°, if the polynomial in all x°E cC Rn, where the vector h is allowed to change with x°, then L is called strictly hyperbolic in si. We shall mainly focus on the case of h being independent of x0 in some domain st. Then we call L strictly and normally hyperbolic with respect to h , in sz We will write the latter property as LE s-hyp(h)=s-hyp(h,st). For a general set M we say LE s-hyp(h,M) if LE s-hyp(h,st) for open sz M. In this section we will assume h to be one of the coordinate .
vectors. This is no restriction, in view of the fact that "s-hyp" is a property of the principal part LN of L , while the principal
part transforms like a contravariant tensor of N variables when new independent variables are introduced. A linear transform x=my with constant matrix m takes a(x,Dx) into a"(y,Dy) with principal part polynomial a N(y,rI )=aN(my,m-Try), where m -T=(m -1)T. Choosing
h°=mTh=(1,0,...,0) will make the transformed expression a"(y,Dy) s-hyp(h°) In fact, for a more general transform x=m(y) the constant matrix m may be replaced by the Jacobian matrix J=ax/dy .
.
Then h-=j Th will depend on y. One may use this - under proper con-
ditions - to transform an expression L=a(x,D), strictly hyperbolic
with variable hx, onto one, L
which also is normally hyperbolic.
Of special interest will be the case of a second order
strictly hyperbolic expression L (3.1)
L =
written in the form
ja jkDjDk + lower order terms
The principal part polynomial is given by
(3.2)
,
ajk = akj
.
We get
with
ajk a common complex factor). Cdn. "s-hyp(h)" then means that tor). "s-hyp(h)" then means (i) a2(h)¢O, (ii) >0, as t#yh. Here (ii) yields for tE N={h}l. Assuming a2(h)>O we get the form a2 negative definite over N . Hence, up to a nonvanishing complex factor all second order s-hyp-operators are of the 'classical' form: The principal symbol matrix ((ajk)) has one positive and n-l negative eigenvalues.
236
7. Hyperbolic differential equations
Rn+l = in Euclidean space We now assume h=h°=(1,0) OE Rn {(t,x): tE &, xE Rn}, and an expression L=a(t,x,D) of the form ,
(3.3)
,
L= aa'k(t'x)DaxDt=4=OAN-k(t)Dt'
1-1+k:-.N
AN-k
IaIsN_kaa,kDx
with principal part
(3.4) LN
aa.k(tx)DxDt-4=OBN-k(t)Dt, BN_g
aa.kDk.
IaI+k=N
IaI=N-k
Let sz=IxRn, I=[ t] ,t2], and aa,kE COO(R) , for all a,k. For complexvalued aa,k
the expression (3.3) is s-hyp(h0) if and only if k=0
(3.5) a0,0(t,x)'0,
has real distinct roots,
IaI+k=N
as polynomial in z, for all (t,x)E Sz, and all O
E &n, by a simple
calculation. It is convenient to replace the first cdn.(3.5) by a0O(t,x)=1(=AN(t)), as can be achieved by a normalizing factor. So far, in this chapter, we assumed complex-valued coefficients. However, the results, below, are valid for the case of a vxv-system as well. From now on we assume that aa,k(t,x)E Lv, with the algebra Lv of all complex vxv-matrices. A vxv-matrix-valued expression is called s-hyp(h°) if the determinant of its symbol is s-hyp(h°). For L of (3.5) this means that a0'0(t,x)=identity, and
7
(3.6) p(i)=det(
has Nv real distinct roots
IaI+k=N Rn+1 , 0
E Rn for all (t,x) E # This condition ties up with the conditions on L in VI,(4.1). There we were transforming to a problem of the form VI,(4.4) with intent to apply thm.4.4, requiring "strictly hyperbolic of type e or a"'. The latter is identical for xE KCC gn with our present "s-hyp(h°)" - while at x=w the conditions do not match. We focus on the local "non-relativistic" Cauchy problem (3.7)
Lu = f
for given
,
(t,x)E S
,
atu
as t = t0
,
j
f (defined in Sz) and Tj (defined for t=t0E I), assumed
CO-functions. The function (distribution) u is to be found. The term 'non-relativistic' expresses that the initial conditions atu = are imposed at a given constant time to. It will be practical to also consider "relativistic" problems with initial conditions on a more general surface EC c We find that the study .
of relativistic Cauchy problems is only a matter of a 'space-timetransform' -i.e., a transformation of independent variable. Recall the cone h = ht,x of time-like vectors. h is convex, h°E H, and L
7.3. The hyperbolic Cauchy problem
237
of (3.3) is s-hyp(rl) at (t,x), for every 1E Ht,x (cor.2.7). A smooth hypersurface I C St will be called space-like if its Since h surface normal is a time-like vector, at each (t,x)E S .
is convex and WE h, a space-like hypersurface has a unique projec tion to t=0. Thus it is given by an equation of the form t=©(x).
Assume such space like EC S2 given, ©(x)E C°°(len) and O(x)=t° E I for large jxl. The relativistic Cauchy problem (for E) is defined by
(3.8)
Lu=f
,
(t,x) E Sz
avu=gj
,
,
(t,x)E I
,
j=0,...,N-1.
Here av denotes the normal derivative to S, av defined as in V,3. 00 We choose the normal pointing to increasing t. Assume f, WjE Co given, defined on Si and E respectively. ,
The relativistic Cauchy problem may be reduced to the nonrelativistic problem by a coordinate transform. For simplicity we only discuss the case where the cones ht,x hx are independent of t (for example, if L or its principal part are independent of t). Let us consider a transformation of the form t' = t'(t,x)
(3.9)
,
x' = x
(t,x) E SZ=Ix&n
,
inverted by t = t'(t,x') , x = x'
(3.10)
,
(t',x') E S2'
defining a diffeomorphism S2 «12' , which takes the surface E onto
the plane t'=0 (i.e., require that t'(O(x),x)-0 ). One then verifies that the cdn's avu=ypj of (3.8) transform to cdn's of the form in (3.7), with modified data. On the other hand, the DE Lu=f transforms to L'u=f', (t',x')E c' of the general form (3.3). We get
(3.11)
aN'(xaI,te') = aN(xa(xa),Jt"I)
,
J = ((axa'j/axa1))
where, for a moment, we are introducing the notation xa=(t,x) xa' = (t',x') In the Jacobian and with aN(xa,ta) = aN(t,x,t,t) ,
.
matrix J the row index is j and the column index is 1 . One calculates that Jl;a' = (t'Itt',t,'+t'Ixt') = (0,t') + t'h(t',x'), with ,
(3.12)
h(t',x') _ (at'/at,at'/axl,...,at'/axn)(xa(xa'))
Thus 0=det aN'(xa',l;a') = det
aN(xa(xa'),(0,t'+t'haxa))
.
has real
distinct roots t' if L is s-hyp(h) for h of (3.12), all (t',x'). so that t(t',x')=t'+O(x'), and get h= Let t'(t,x)=t-8(x) as (t,x) =(1,V8(x')) independent of t'. Clearly h is normal to I E I. Thus indeed there we get h E ht,x since I was assumed space ,
,
238
7. Hyperbolic differential equations
like. Now, if ht =h is independent of t then h(t'x')E ht,x' for all t,x , and the transformed problem is s-hyp(h°), and nonrelativistic. Clearly the set a' contains a smaller sk"=I"xlen. Note that the same transformation still works if h is time dependent, if it is assumed that S is not only space-like, but, afortiori, has its normal at x in the intersection fl(ht,x:tE i}= =hx. Clearly hx is a convex cone with nonvoid interior, for all x. Theorem 3.1. Assume L of (3.3) in s-hyp(h0), and, moreover, assume that the transformed problem VI,(4.4), with the matrix of VI,(4.6) is strictly hyperbolic of type e or e'. Then there exists a unique of the nonrelativistic solution u(t,x)E C0O(I,Hs) for all sE k2 ,
,
Cauchy problem (3.7), for every f and ypj E Co .
The proof is an immediate consequence of VI,thm.4.4. We observed that the condition "strictly hyperbolic of type e°" coincides with (i) "s-hyp(h°) on all compact sets KC 11 " and
(ii) an additional condition obtainable by modifying L outside an arbitrarily given compact set KO If S is a surface as above, then it follows easily that a transformation x'=x , t'=t-8(x) will .
not influence cdn.(ii), since we were setting ©(x)=t° for large lxi. As a consequence we have the result, below. Theorem 3.2. Let L satisfy the assumptions of thm.3.1, and let EC dint be a space-like surface with unique projection onto the plane t=0, and parallel to t=0, for large I xI. Let T, =Min © , s2=Max e , taken over In. Assume that E is not only space-like, but even that its normal is contained in the cone hX fl(ht,x: (t,x)E M}, with the "slab" M={z,-t,St-O(x)st:-t2}. Then there exists a unique solution u of the relativistic Cauchy problem (3.8), defined for (t,x)E M 00 and each f, E CO , such that the function u(t-0(x),x)=u (t,x)
satisfies u" EjC0o([i, -t, ,t3 -t2 ] Hs) for all sE N2 The two above results, thm.3.1 and thm.3.2 will be improved in sec.5, below, after discussing 'finite propagation speed'. 4. The cone h for a strictly hyperbolic expression of type a°. In sec.3 we already used the fact, that an expression L of the form (3.3) is s-hyp(h°,K), h°=(1,O), for every K(= Ixxn, whenever the first order system VI,(4.4) is strictly hyperbolic of ty-
pe e or e'. Discussing this in detail, note the symbol of the operator KO AK, K of VI,(4.6), is of the form k0+K0, where the xOE
7.4. The cone
239
C'(I,yrcf_e), while the matrix k0(t,x,t) coincides with the matrix
in VI,(4.4) except in the last row. All elements in rows 1 to N-1
equal 0 or 1, as in VI,(4.6), but the elements of the last row are (n-e°(x,U))N-jaN_j(t.x,U)
(4.1)
with
aN-j
,
j=0,...,N-1
lal
in the order listed. We may choose k0 and x0 above for the decomposition VI,(4.9). Then the condition "strictly hyperbolic of type e° " implies that has real and distinct eigenvalues µ1<µ2<...
Under the present special conditions, the matrix k0 and its eigenvalues even extend to continuous functions over {Ixffi°xZn: W-1.1, with in of 11,3, and the compactification RA allowing continuous extension of all functions a(x)E ip0 . Here we assume that
aa, j(t,x)E
(4.2)
(N-j)(e°-e') as
Accordingly, for every jt°j=1 , the limits limpj(t,x,p°) -Rj(t,x,«x°) exist, and define Nv real-valued continuous functions over Ix>2i^x{jV°j=1}, assuming mutually distinct values at each point of this compact space. A calculation shows that tj= -µj(t,x,° °) are the roots of the equation
(4.3)
det(
7
v-ltjbl
j+1 =N where bN-
7
7
Ian=N-j
(t,x, ° )) = 0 are the principal parts of
J
the aN_j , and we have assumed b0=1 again. Also, we have set v=
(x) for e° =e
,
and u=1 for e° =e'
.
Note that (4.3) may be rewritten as (4.4)
p(tu) =
det
Y,
bl(t,x,°)(tv)j = 0
j+l=N
with the polynomial p(T) of (3.6)
,
taken at
In other
words, the roots of (3.6) are given as the numbers -vµj We have proven the following result. Proposition 4.1. a) Assume that the coefficients of L of (3.3) satisfy (4.2), and that the equivalent first order system VI,(4.4) Then is strictly hyperbolic of type e° , where a°=e or e°=e' .
7. Hyperbolic differential equations
240
the differential expression L is s-hyp(h°,K) for every KC: Ixin Moreover, we even have L E s-hyp(h°,Ix&n) , uniformly, insofar as the minimum difference between two roots of (3.6) is bounded below .
by c° >0 for all
as e°=e' , and by c° (x) , as e° =e. Further-
1 ° 1=1 ,
more, the roots of (3.6) are bounded by a constant Co as a°=e'
and by Co ( x) as e° =e
,
for all tE I, xE en,
I T;° 1=1
.
Remark. Note that a different choice of k0 and x0 in VI(4.9) does will not influence the above derivation, since the limit give the same polynomial as in (4.3) Note that our knowledge about the roots of (3.6) gives the following information about the cone h=ht,x of time-like vectors. .
Proposition 4.2. Under the assumptions of prop.4.1 the cone ht,x for L of (3.3), at (t,x), is given by (4.5)
&n , i>-v(x)
where u(x)=nea-e, (x,i;)=1 as a°=e'
lµj(t,x,°°l
,
,
l)
j=1,...,Nv)
=(x) as a°=e
Corollary 4.3. If a°=e' , the cone ht,x contains a cone {t2Co 1
1}=
=ho for all (t,x)E Ix& If a°=e then, at least, ht'x contains the cone 1} for every (t,x)E Ix&n Here Co is some positive constant independent of t and x . All above statements are evident, after our remarks. Now assume that an expression L as in (3.3) is given, with aa,k satisfying (4.2). For all (t,x)E Ix$n, 1V°1=1, let equation (3.6) have real and distinct roots, bounded by Co, with differences bounded below by co, just as in the statement of prop.4.1. .
,
hlx1={tZC°(x)1
.
Proposition 4.4. Under above assumptions the system VI,(4.4) is strictly hyperbolic of type e' .
x0
,
(4.6)
Indeed, we will just make the same decomposition into k0 and as above. The negative eigenvalues ij=-pj then again satisfy
1
(
k+l=N with aN-j of (4.1) or (3.3). We may wite this as (4.7)
p(t;t,x,y)+
7, -)a=0 Tk ( )l-lalaa,N-l(t,x)( k+l=N laI<1 where denotes the polynomial of (3.6). For 1;=«1° equation (4.7) assumes the form and the above
assumptions then state that its roots are real and distinct. On the other hand, the polynomial in z of (4.7) has its highest coef-
7.5. Regions of dependence and influence
241
ficient equal to 1, independent of Thus the roots may locally be arranged into continuous functions. They are distinct at the compact set IxX°xaBn, hence they will stay real and distinct in some neighbourhood Bn, But this is precisely the statement of prop 4.4. Q.E.D.
5. Regions of dependence and influence; finite propagation speed.
Let us again focus on the Cauchy problems (3.7), (3.8). We can significantly improve on the existence results thm.3.1 and thm 3.2, insofar as it may be shown that "uniform s-hyp(h°)" is sufficient - it will imply the condition on the system VI,(4.4). Also the solution u(t) is CD (&n) for each fixed t, not only in S=f1HS To simplify the assumptions let us assume that the conditions of thm.3.1 are satisfied for all compact intervals I=[ti,tz]. Consider the adjoint L* of our expression (3.3), defined as L* =
(5.1)
I Ial+ksN
a,k(t,x)
Dxa'Dt
Proposition 5.1. We have L E s-hyp(h0) if and only if L* is s-hyp(h0) Moreover, the cones ht'x of time-like directions for L L* and coincide, at each (t,x) and a surface I is space-like for L if and only if it is space-like for L* For L with complex-valued coefficients we know that "s-hyp", the cones ht'x and "space-like" are completely determined by the principal part of L. Cor.1.3 implies that the principal parts of L L* and differ only by a nonvanishing complex factor y(t,x): We get real-valued, for some g(t,x), hence the prin.
,
.
,
cipal part of L* evaluates as aN gb=(g/g)aN. Since "s-hyp" implies "hyp" this proves the proposition for complex-valued expressions
L. For general matrix-valued expressions we still have aN (aN)T
,
and det aN = (g/g)det((aN)T) follows because det aN is a hyperbolic polynomial. Thus again the proposition follows, q.e.d.
In the following let us assume that L L* both satisfy the assumptions of thm.3.1. Consider a region no C Rn+l as in fig.5.1. In detail, the boundary an0 is piece-wise smooth, and consists of ,
two parts, E1 and E2, both space like surfaces, with equations t=91(x), j=1,2.
7. Hyperbolic differential equations
242
The surfaces E1 intersect at positive angle in some smooth compact n-l-dimensional surface. Moreover,
we assume 81 extended over all of In, and that the extended surfaces t=01(x) t=t, satisfy the assumptions of
thm.3.2. Assume the compact region ito defined by
Fig. 5.1
rte= {(t,x) :xE &n, 81(x)sts82(x)} .
(5.2)
Lemma 5.2. Suppose a solution u of the relativistic or nonrelativistic Cauchy problem ((3.8) or (3.7)) with f.0 is defined in the
compact region rto
,
and assume that u=ult=...=cat 1u=0 on I, rl9rto
Then u vanishes identically in all of rto
.
Proof. Note that the relativistic Cauchy problem (3.8), for L* rather than L , and for the surface E2, admits a solution v . Here we assume that Tj=0, but will allow an arbitrary C function f Existence of v then is a matter of thm.3.2. Note that u and v both are C (rto)
. By a partial integration we get f
(5.3)
u fdtdx = fuTL*vdtdx = 0
RO
Indeed, we get Lu=f=O, by assumption. No boundary terms may appear
because the Cauchy data of u and v vanish on It fl912o and on E2 (8a respectively. Thus the partial integration gives 0
.
,
Since this is
true for arbitrary smooth f with compact support, we conclude that
u=0 in r)o
, q.e.d.
Proposition 5.3. Assume that L, L* both satisfy the assumptions of thm.3.1. Let E:t=8(x) be space-like surface satisfying the assumptions of thm.3.2, and let u be the solution of the relativistic Cauchy problem of thm.3.2, for the surface I and for data f,Tj where fE CQ(IxIn) for every I , and qvi of compact support on I Then, if E° is any space-like surface t=01(x) , xE 1n , with the ,
,
property that supp f, and supp Tj all lie above I°, in (t,x)-space then u vanishes identically in the set 8(x)stse1(x) (see fig.5.2). Proof. This is a matter of constructing "lenses" onto which lemma
7.5. Regions of dependence and influence
243
5.2 may be applied. First one notices that, for each fixed t the function u(t,.) has compact support: For example, one may work with a surface E° carrying a spherical bulge, such as
t=to +Max{
x 2 -2a.x ,b}=& (x), with 0
0 The surface t=tj outside E° t=tj-1 inside E° t=o' otherwise then, after smoothening, may be seen to satisfy the assumptions of thm.3.2 for existence of the ,
.
,
solution v for the adjoint problem, as in lemma 4.2. Then an argument as in the lemma will give u=O below that surface. Q.E.D. Theorem 5.4. Let L of (3.3) have its coefficients satisfying (4.2)
Assume that L is s-hyp(h°,&n) uniformly, in the sense that the roots Tj of equation (3.6) are real and distinct for all (t,x)E
In",
1n, while, on 1 1=1, t, ststz , we have Ii - IsC° , Ii j-sll>
independent of Assume E to be a space-like surface satisfying the assumptions of thm.3.2. Then the relativistic Cauchy problem, for the surface E, and arbitrary smooth data c° with O
f,Tj, where yj are compactly supported while (supp f)MxlnO Rn+1 C0' for all I( 1, admits a unique solution uE (1n+1). Moreover, we have (supp u)fl(Ixtn) compact for all compact I.
Proof. Let us set a°=e'. Then confirm that L as well as L* satisfy the assumptions of prop.4.4. Accordingly, the corresponding first order systems VI,(4.4) are strictly hyperbolic of type e'=e° and ,
we have the assumptions of prop.5.3 satisfied. Thus our assertion follows. Q.E.D. Remark 5.5. It might be interesting to note that our above discussion will still work in the case where we require that (4.2) holds for a°=e - i.e., we have (5.4)
aa,j(t,x)E
W(N-j)e2
as jajsN-j
.
7. Hyperbolic differential equations
244
However, we then must require the roots of (3.6) to be bounded by C. (x) and their differences bounded below by co (x) , co >0
, as
In addition, one will require the same conditions on the roots of (4.7), valid not only for large but also for all iii, as long as lxi is sufficiently large. We will not discuss details. We now may compare our results of thm.5.4 with those achie-
ved by the Fourier-Laplace method in ch.0,4. Note that use of ydo's has been eliminated, except in the proofs. For a Cauchy problem as in thm.5.4 we have finite propagation speed: The solution u(t,x) stays =0 for sufficiently large lxi as long as t stays ,
within a compact interval. There is a "region of influence" For a change of data - within a compact set K - the "influence" on the solution u(t,x) will be felt only inside a conically shaped region, expanding from K towards increasing times. There is a "domain of dependence": The values of u at or near (t°,x°) depend only on the values of the data within a conically shaped region with tip at (t°,x°), expanding backward in t. For a point (t°,x°) one can define a precise region of dep:
endence RD (t° , x°) and region of influence RI (t° , x°) as the backward part and the forward part of the envelope of all space-like surfaces through (t°,x°). This envelope will have the normal cone
of the cone hto point (t°,x°)
, x°
as a tangent cone, at (t° , x°)
.
Hence, at that
it may be decomposed into a forward and backward part. The regions RD and RI also may be described by the bicharacteristics of L, in the sense of ch.0,8, at least for v=1. Or, it may be linked to the "particle flow" of VI,5,6, generalized to systems of the form VI,(4.4) - of. ch.IX . ,
6. The local Cauchy problem; hyperbolic problems on manifolds. Notice that thm.5.4 implies an existence result for a local hyperbolic Cauchv problem. Assume first that, in a set a--IxB3
with BY =BY (0)={xE In: I X
,
I=[ t1 , t2 ] , we have a differential ex-
pression of the form (3.3) defined. Assume that LE s-hyp(h°,S) and let the coefficients aa,j(t,x) be COO(n) . Assume a space-like surface EC n int Then ask for existence of a function u(t,x), def,
.
ined near I satisfying Lu=f (near X) and atu_c on E, j=O,..N-1, where f and yj have support within IxB1 To solve this problem we will modify and globalize the ex,
.
pression L as follows. Letting L=a(t,x,Dt,Dx) mial
a
a,k
, with the polyno-
(t,x) %k , and with a function 6(p)E-=
245
7.6. The local Cauchy problem
0(p) 0s9(p)<2 in (0,m) =0 as pa2 . Then define L° = a°(t,0(jxj) f ,Dt,Dx) , tE I, xE Bn Observe that L=L" within IxBi . Also the coefficients of L° are
C00((0,00)) satisfying i(p)=p as pE [0,1]
,
,
there. In
independent of x as jxja2 - we have L° = a(t,0,Dt,Dx)
particular, L° is uniformly s-hyp(h°,Ixen), in the sense of prop. 4.4. We may modify the surface E outside IxBi to obtain IA satis-
fying the assumptions of thm.3.2, at least close to E. Since E=E° within IxBi we may extend the initial conditions imposed there by In this way we obtain a Cauchy setting Tj=0 f=0 outside IxBi problem satisfying all assumptions of thm.3.2. The solution exists and its restriction to IxBi will supply a solution of the local problem. Rn Now, for a general set n=Ixn° , where n° CC we may solve the corresponding problem, using a partition of unity. Assume L of the general form (3.3), defined in Ixf with an open set SE J nc Construct a finite covering of and let it be s-hyp(h°,Ixn-) n by balls BE(xk) with the property that B3E(xk)C si and a par,
.
,
.
,
tition of unity
=1
,
supp wk C BE(xk)
.
The solution of
(6.1) j=o,...,N-1 Lu=f , near I atu=4Pj at I where I C Ixn is a space-like surface, and where supp g C Ixn supp y)jC n , may be obtained as a superposition of the solutions of (6.1) for f « wkf cj H wk(pj to be regarded first as a problem within IxB3E(xk). The solution of the latter problem may be extended zero along E for a small time interval, because it, also will vanish within IxBE(xk) outside the region of influence. The problem within IxB3E(xk) may be translated and dilated into a problem in IxB3 . We have proven the following result. ,
,
,
,
Theorem 6.1. For a differential expression L of the general form (3.3) with smooth coefficients defined in Ix12 , and s-hyp(h°,Ixft ) there, and a space-like surface ECC Ixn , where ncx sz the Cauchy problem (6.1) admits a unique local solution, defined in a sufficiently small neighbourhood of I ,
.
Remark: The local uniqueness simply is a matter of an argument like that in the proof of lemma.5.2. Next we notice that the concepts "s-hyp(h)", "s-hyp(h,n°)" are meaningful for differential expressions defined on a differentiable manifold n. Since we have coordinate invariance we may interpret h as a vector of the cotangent space T*ax° , at a point x°, and define L to be s-hyp(h) if this is so in any set of coordina-
7. Hyperbolic differential equations
246
tes. If we are given a field of cotangent vectors h=h(x)E T*nx for x E n°, an open set Ca, then we shall define LE s-hyp(h) _ s-hyp(h,nz°) if LE s-hyp(h(x)) , at each xE n' . We then will have a cone h C T*nx of cotangent vectors ar x - it is the cone h of x the differential polynomial of L at x, in any set of local coordinates; note that the polynomial is now defined on T*nx, and that its principal part is defined as well. It determines hx as a subset of T*n . A space-like surface EC n will be defined as an n-1dimensional submanifold with normal at xE E contained in hx. Here we will assume that L is s-hyp(h,n ) with some domain sE C n , and nthat XC n° Then again we may formulate a hyperbolic Cauch problem: For compactly supported fE CO(n) and (fjEC0(E), 0 N-1 (where L is of order N) find a solution uEC (n') of c
.
Lu=f , xE n'
(6.2)
,
ah(x)u=Wj , xE E
,
j=O,I,...,N-1
where n' C :re is some (sufficiently small) open neighbourhood of E,
and with the j-fold derivative ah(x)u (for t, at t=0) of the function u(x+th(x)), in some fixed set of local coordinates. Theorem 6.2.
The solution u(x) of problem (6.2) exists and is
unique, up to the choice of n'
.
The proof is by localization, exactly as for thm.6.1. Notice that we have admitted vxv-systems, in the above Correspondingly we may allow an expression L mapping between v-dimensional complex vector bundles on n , in thm.6.2, and then specify data f (pj which are sections in vector bundles. ,
Clearly the notions of finite propagation speed, region of influence and dependence RI and RD extend literally as well. The case of a second order scalar operator again merits special attention: This will be related to relativity theory. We have seen that, locally, up to a nonvanishing complex factor, a second order expression L is of the form (3.1) of rank n and signature n-2 .
, with a quadratic form
Chapter 8. PSEUDO-DIFFERENTIAL OPERATORS AS SMOOTH OPERATORS OF L(H)
.
0. Introduction.
In this chapter we look at a different way to introduce pseu-
dodifferential operators. From their introduction in ch.I - using I,(1.5)- they appear as a technical device inspired by the FourierLaplace method. It may be surprising that there is a natural defi-
nition of pdo's at least for some special symbol classes: Certain algebras of yado's appear as "smooth" subalgebras of L(H). For exam2n), 2n)=pto the class of ple, the algebra (1 with CB'(1 OpCB00
E
C00
(3k
2n)
, bounded with all derivatives, coincides with the class
of operators AE L(H) which are both "translation smooth" and "gauge smooth". That is, with the family {Tz:zE Rn} of translation ope-
rators TZu(x)=u(x-z), and the family {M:tE ien} of "gauge multipli cations" Mu(x)=eitxu(x), we get T-ZATZE C(&n,L(H)) and M-,AMtE 00 n C (& ,L(H)). Notice that both Tz and Mt are groups of unitary operators on H, representations of the Lie-group In in U(H). 2n), In terms of ch.I, a subalgebra of L(H) by (1 III,1, is the class of translation and gauge smooth operators. There are other natural actions of Lie groups on H=L2(Hn), such as the dilations u(x)-u(yx), with y340, or the rotations u(x) -u(ox) , with a rotation o:&n..&n, or the linear group u(x)-u(mx) with a nonsingular real matrix m. If we require dilation and rotation smoothness, in addition to translation and gauge smoothness we obtain a more practical algebra (called Oppso) with a calculus of '4do's and symbol decay similar to Optpc0. With the linear group we get a still smaller algebra, called Opy,10 Similar symbols were studied by Weinstein and Zelditch [WZ]. We will give a completely independent presentation of these facts, introducing lpdo's again, without reference to ch.I. In parOppt0=OpCB00
.
ticular, we will work with symbol classes 1tm, Vsm, and V1m exclusively, in ch.9 and ch.10 below, derived from the natural classes
1pt0, ,s0, and V10 These classes are similar, but not identical to yicm. They allow a (slightly more complicated) calculus of pdo's. In sec's 1,2,3 we are concerned with the 'IDO-theorem' describing the new characterization of Opipt0. In sec.4 and 5 we discuss the classes Opps0 and Opipl0. Sec's 6 and 7 are concerned with symbols in Vxm for general m, with calculus of ipdo',s, and decay
247
8. Smooth operators of L(H)
248
properties of the new symbols. Sec.8 deals with corresponding Liealgebras and Beals-Dunau-criteria. An earlier version of our theory was announced in [C<] and distributed as lecture notes of a seminar at Berkeley in 1984, but remained unpublished. 1.
pdo's as smooth operators of L(H).
Before giving a natural introduction to the concept of pseudo
differential operator we make some observations concerning smooth ness of continuous linear operators on the Hilbert space H=L2(&n). ien is an abelian group under vector addition. The characters of this group are the functions e1 , E &n, the character group is 2n again. For zE In define the translation operator TzE L(H) by Tzu(x) = u(x-z)
(1.1)
For each character
,
Mt=eitm
we define the gauge operator Mtu(x) = eitxu(x)
Note that Tz and M (1.3)
u EH .
Tz+z' = TzTz,
by
.
are unitary operators of H . Moreover, , Mt+V = M t M V ,
for all z,z',i;,t'
,
which expresses that z - Tz and - Mt each define a unitary representation of the group ien on the Hilbert space H .
We want to emphasize: The differentiable manifold In with above group structure, is a Lie group, but the representations (1.1) and (1.2) are imperfect insofar as z->Tz and -Mt are not smooth as maps &n, L(H). For fixed uE H the functions Tzu=f z, M u=
are con-
tinuous but need not be differentiable ( u(x)=sgn(x)e IxI, for example). For uE CO we get fZ, 9,E C00(0,H) but not in general. In fact, it is known that the operator families Tz and M are strongly continuous in their parameter z or t
,
but are not
norm continuous: Choosing u E H with support in a ball Jxlsa (or Ix-x°lss with x°(?-%')=n ) with small e one confirms that (1.4)
IITz-TZIII a
,
IIM-MII a
,
as z34z'
, K '
.
Similarly, for a given operator A EEL(H) the families (1.5)
AZ'o = TZ-'ATz
,
AC't = M1AMt , z,t E In
are strongly continuous, but in general not uniformly continuous.
249
8.1. Smooth operators of L(HO)
The pseudodifferential operators within the algebra L(H) will be defined as those operators A E L(H) for which both familiesAz,O (1n,L(H)) and AO,. are not only norm continuous, but even are 00 C , L(H) being equipped with the norm topology. Let us denote the class of all such operators by 1I0T. That is, t1CT = {AE L(H): Az,0
, A0't E Co(&n,L(H))}
.
We will show in the following that every operator A E tICT C L(H)
has a unique representation in the form (2n)-nfd
Au(x) =
(1.7)
where
a complex-valued function in C00:partials b(a) of all orders are bounded}
called the symbol of A . (1.7) holds for uE C,(ien), but determines the operator A since Co is dense in H. Even for uE Co the existence of integrals in (1.7) will be a point of discussion. Interestingly, the existence of a representation (1.7) with symbol function aE CB00(1en) characterizes the operator class IICT
:
For any aE CB"(le2n) the integrals in (1.7) exist for all uE Co(&n) and Au(x) of (1.7) belongs to H. The operator A:C0(&n)->H thus defi-
ned extends to a map E L(H). Moreover, A belongs to iIGT
In the remainder of this section we make a few observations, preparing the detailed discussion of the result mentioned above. First, observe the 'commutator relation' T-zMtTzu(x) = eit(x+z)u(x) = e1ztMtu(x) Or,
T-MTZM-l
(1.9)
-
e1z
Thus the subgroup of the unitary group U(H) of H generated by all Tz and Mt z,t E 1n consists precisely of the operators ,
(1.10)
,
Ez,t,T =
ei((P+xt)Tz
,
TEE & ,
z,tE &n
.
This group is called the Heisenberg group. We denote it by GH. Get (1.11)
Ez+z',t+t',T+w'-t'z
Thus GH is isomorphic to the group obtained by equipping the space gh=lenx&nx$1={(z,t,T):z,tE &n,WE $1=&/2nZ} with the group operation (1.12)
unit = (0,0,0)
.
8. Smooth operators of L(H)
250
Clearly GH with the manifold structure of gh is a Lie group. Our above condition describing 'WT is equivalent to the following: (Note that AZ't For AE L(H) let AZ't = EZ,t,W IAEZ,t,(P .
is independent of cp .) Then we have
Az,t E C00(gh, L(H))
(1.13)
.
Indeed, (1.13) trivially implies the condition of (1.6).
Vice versa, note that (1.9) implies (1.14)
AZ,t = T-zAO,tTz = M-tAZ,OMt
If (1.6) holds then use that Az.t,ii =
showing that
Az,
t for fixed z
,
Ao,t,ii
Az,,tii = IIAz'0- Az,'011
,
is continuous in z for fixed t and continuous in
and either continuity is uniform in the other
variable. This clearly implies norm continuity of in both Az,t variables. Similarly (1.6) and (1.14) imply existence of (1.15)
aaAZ,t= T-za AO,tTZ
'
aRAZ,t-
in operator norm topology, uniformly in the other variable. Since aaAO' and aPAZ,O are still C"(&n,L(H)) , we may repeat the above conclusion and find that both functions (1.15) are in CB(gh). By a standard argument we then conclude that all mixed derivatives as Az't exist (and are CB(X2n)) as well. Indeed, this needs to be shown at z=t=0 only, in view of the group property. 00
Thus look at XAz,t with a cut-off function XE Co , X(z,t)=1 near 0. For u,vE H consider Let p(a)=a Zap p. We get
E E =L2(QyQy=I(z,t):Izjlsy,
j=1,..,n}, for large y, and
all a,(3. Parsevals relation for Fourier series implies that the (multi-)sequence {q0'.: O,vE Zn} of Fourier coefficients of p sat-
isfies Oag0,v
v1g0,,u E 12 = L2(Z2n)
,
for all a,P
.
Thus also
,
Oav (1+jOj2+Ivjz)kE 12 for k=0,1,..., and all a,(3. It follows that of 111,3. By k,0 Sobolev's lemma we thus get p(a)E Ck(I2n), for all k. Looking at p p(R)E Hk, for all a,(3, with the Sobolev space
Hk--H
in dependence of u, v we get
II p (a) ii , 11p(p)11- c ii u li ii vii , with norms Following this in Hy and H, resp., and with c undependent of u,v estimate through the above argument - Parseval, Sobolev, etc.,..., .
we find that also iip(a) ii k s ciiuii iivli
,
for each
This implies
C
(%Az,t)(R) E L(H)
,
and even ii(XAz,t)(Pa)iis c
.
Then one concludes
251
8.2. The pseudo-differential operator theorem
that aZ,PAZ,t=(AZ,t)(a) exist in norm convergence, for all a,1 Thus indeed (1.6) and (1.13) are equivalent. Secondly, we want to examine existence of the integrals in (1.7). Note that the inner integral fdy exist trivially, as a pro-
per Riemann or Lebesgue integral, and equals where u E CO(&n) . Moreover, it is well known that u^ is a rapidly decreasing function. That is u^(t) = and the for every k=0,1,2, ... same is true for the derivatives u^(a) Since a(x,l;)E CBO'(12n) is bounded, and lelt(x-y)1=1 it then is clear that also the ,
.
,
outer integral
fd
exists as improper Riemann or Lebesgue integral
and that that there is a bound for the integral independent of x In order to show that the right hand side of (1.7) defines a function in H we prove that xaAu(x) also is bounded, for every a . Indeed xa =((x-y)+y)a is a finite linear combination of terms (x-Y)OYY
,
3+y=a , while
Again the inner integral equals (2n)n/2eixt(xYu)^, hence is a function in S , while has all derivatives bounded. It follows that a partial integration may be carried out, leaving no boundary terms at infinity, giving =
0(1)
Thus indeed xaAu(x) is bounded for all a , hence (1+IxI2)nAu is bounded, and Au E L2 In fact, integrals of the form (1.7) already have been investigated in 1,1, under weaker conditions on the symbol 2. The 'MO-theorem.
As prepared in sec.1 we will now prove a theorem linking the concept of pseudodifferential operator, introduced by formula (1.7) to the class WT of continuous operators on the Hilbert space H = L2(1n) which are 'smooth' with respect to the translation operators, and the 'gauge transforms' u -> eitxu (or, equivalently, gh-smooth). Here we were using the standard representation
8. Smooth operators of L(H)
252
GH of the Heisenberg group gh, i.e. GH={Ez,.,(P $1 gh = 2nxlnxl1 = 2/(2nZ) , where
gh}
, with
,
(2.1)
(Ez,(Pu)(x)=el(T+1'x)u(x-z)
,
H
Especially,all translations Tz:u(x) - u(x-z),for a constant vector z, and all "gauge transforms" u(x) - ei(T+tx)u(x) for constants
t E In belong to this class. For A E L(H), a continuous operator on H , we are interesGT -> L(H) obtained by conjugating A ted in the function Az,t gE It
,
:
with the operators
Ez,.,T Ez-
(2.2)
1
,t,TAEz,t,g
Az,t =
Note that Az,t does not really depend on T since the constants elq) commute with all of L(H) . Hence Az,t may be regarded as a function GT'- L(H),where GT' denotes the quotient of GT modulo its normal subgroup {ei(P :cp E 2} Clearly GT' is isomorphic to 12n as the composition law (1.12) shows. We may write (2.3)
Az,t
and henceforth regard
EZ1tAEz,t
EZ,t ,
=
Az,r, as a function 12n -> L(H)
We restate the essence of our discussion in sec.1 as a theorem, below. Here, for a general class X of functions over 12n
and class Y oftUdo's we write (2.4)
Op X = {a(x,D)
:
a E X}
,
Symb Y = {a
:
a(x,D) E Y}
.
Also, for a single function we will write Op(a)=a(x,D) Symb A = where A= Op(a) = a(x,D) denotes the formal pseudodifferential operator (1.7) belonging to the symbol a E X . ,
Theorem 2.1.
Let Vt0 = CBCO
(I2n)
denote the class of all complex-
valued with partial derivatives a5a)(x, all orders bounded over 12n Then we have
)
aXa a(x, ) of
.
(2.5)
Opi,t0 = {A E L(H) : Az, E Cco(12n,L(H) )} = '1GT
.
Here the partial derivatives of Az,t are assumed to exist in the convergence of the operator norm 11.11 of L(H)
(2.6)
IIAII = sup { IIAufl/IIuhl
defined by
: u E H , u# 0 } , A E L(H)
.
The ydo A of (1.7) is defined for u E C0OO(2n) by integrating in the
253
8.2. The pseudo-differential operator theorem
order stated, and for general u E EH by taking continuous exten-
sion from the dense subspace CO .
With pseudodifferential operator notation we have (2.7)
=
E Z ,t
using that (eizDu)(x)
u(x+z)
=
.
With (2.7) we may differentiate (2.2) formally, for (2.8)
azjAZ,tIz=t=O = i[Dj,A]
,
,
atjAz,tJz=t=0 = -i[xj,A]
with formal commutators between the operator A E L(H) and the unbounded differentiation or multiplication operators Dj and xj Similarly the higher derivatives are related to iterated commutators between A , Dj ,and xl This provides a link of thm.2.1 to a slightly different characterization of more general classes of pdo's as operators S - S with well defined iterated commutators with differentiations and multiplications (cf. R.Beals, [B2]) Looking for a proof of thm.2.1 we first will discuss the fact that, given a symbol a E CB"(le2n), the formal expression A = a(x,D) of (1.7) defines an operator in L(H). Actually, this is already done (cf. III, thm.1.1), but we offer another proof here, with an approach first published in [CC]. (It may be a bit more complicated, but it fits the present general approach.) Theorem 2.2. An operator A = a(x,D) with symbol a E pt0= CB'°(1e2n)
is bounded in H = L2(&n) . More precisely: The operator (1.7) extends to a continuous operator H- H
of
Proof. It turns out that we need only boundedness of 'a few' derivatives of the symbol, for this proof. Consider the case of n=1, for simplicity. The case of general n may be handled analogously. For m=1,2,... introduce the function (2.9)
ym(t) = e-t tm-1/(m-1)!
,
as ta0
= 0
,
,
as t<0
.
One checks easily on the following facts: Proposition 2.3. We have ymE Cm-2(2)fl C"(2\{0}). The derivative ym (m-1) still is piecewise continuous with a single jump of magni-
tude 1 at t=0 . We have ym(j)(t) = O(e-JtI/2) (2.10)
(at+1)mym = 6
,
,
as t#O
.
Moreover,
254
8. Smooth operators of L(H)
with the Dirac distribution 6
.
(In other words, ym is a fundamen-
tal solution of the differential operator (at+1)m ). Ck(Hn) For k=0,1,2,... let CBk(&n) be the space of with bounded derivatives up to order k including. Also, define
x, Elk .
(2.11)
Corollary 2.4. Every function over &2 with (ax+1)2(6+1)2a =b E CB(1 ) may be written in the form
(2.12)
a(x,t) =
f
x,t E 2
Then aE CB2(1Q2), with derivatives given by differentiating (2.12) under the integral sign.
We also introduce the 'pseudo-differential operator' Q = q(x,D) , where of course q is not in i,t0 , but still the right
hand side of (1.7) with a=q defines a function Qu(x) for every so that u E CO(&). In fact, Qu(x) = y2(x) F-ly2Fu(x) ,
(2.13)
1
Q = Y2(x)Y2(D)= x1y2(x) ((l +ix)Z *)
It is clear from (2.13) that Q E=-L(H)
.
In fact one finds at once
that Q is an integral operator with y2(x)/(1+i(x-y)) kernel xi2 E L(& 22) - i.e., Q is of Schmidt class. Moreover, we may write (2.14)
Q=V*U
,
V=(1-ix)-1(1-ax)-1
,
U=(1+ix)(1+ax)Q
where both operators U and V are of Schmidt class. Indeed, V and
U have integral kernels in V(11), given by (2.15) vx(Y)=1+ix y1(x-Y)'
µx(Y)°xi(1+ix)(1+ax)((1+i(x-Y))2Y2(x))
Now thm.2.2 will follow from prop.2.5, below. Proposition 2.5. For a with b=(1+ax)2(1+a )2aE CB(&2) we have
(2.16)
IIAuIIL2
5 cIIuIILz
, c= 2n[[[U]]][[[V]]] IIbJIL'(R2)
with the Schmidt-norms [[[U]]] =
'
x
'
L
2
, [[[V]]]= IIv(y)It
Proof. Use (2.12) to write
(2.17)
f dzdtb(z,tj91l
,
q(x-z,-t)eixu. ()
,
L
2
255
8.2. The pseudo-differential operator theorem
the integral interchange being legitimate, for uE Cp, hence u^E S. The inner integral may be written as q(x-z,D-l;) , using (1.7). For an operator A=a(x,D), aE lt0 (general n) one finds that (2.18)
iz
AZ'Ou(x) =
= a(x+z,D)u(x),
where a(x+z,D) denotes the 4do with symbol a(x+z,t),of course, with the constant parameter z E &n. Similarly, (2.19)
A0, tu(x) =
a(x,D+t)u(x)
ffl
Combining (2.18) and (2.19) we get Az,t = a(x+z,D+t)
(2.20)
Clearly (2.20) holds for a=q as well. Thus (2.17) may be written
Au(x) = fdzdl;b(z,l;)Q-z,-tu(x)
(2.21)
.
This leads into the following estimate for the inner product (.,.) I(Au,v)I=
JI(Q-z,-tu,v)Idzdt,
IIbII
L
where I(Qz.tu,v)I = I(UMtTzu,VMtTzv)I s i{IIUMtTZUll2+ IIVMtTzvII2}.
Let w=TZu, keeping z fixed. Using Parseval's relation we get fdtlIUMtwII2= fdjdxl(Vx,MMw)12 = fdxfdtlfdyµx(Y)w(Y)eiyl;I2
= 2nfdxdYlFAx(Y)121w(Y)12
Recalling now the meaning of w : fdzdtIIUMtTZU112=2nfdxdydzlµx(Y)I2Iu(Y-z)12=2n Ilull2fdxdylµx(Y)12
Similarly,
f dzdi;IIVMtTzvII2 = 2nIIvII2fdxdylvx(Y)I2 Summarizing, we have derived an estimate of the form (2.22)
I(Au,v)I s i{S2llull2+ T211vII2}
with constants S,T. Here we let v=v
, u,v E Co1
run through a sequence in CD Noting
with LZ-limit Au/S2, concluding that IIAuIIs STIIuII , uE CD
.
256
8. Smooth operators of L(H)
the precise values of the constants S,T we get (2.16).
Q.E.D.
The case of general n = 2,3,... follows analogously: Define and the corresponding operator Q=q(x,D) Also, the We get (2.12) with b(x,l;)=rlj=1{(1+ax )2 (1+a 7
7
U and V are tensor products of the 1-dimensional operators, their Schmidt norms are products of 1-dimensional Schmidt norms. Corollary 2.6.
(2.23)
IIAII
In the general case of n dimensions, we have
s cnsup{ Irf'1{(1+ax )2(1+a j )2}a(x,U) I 3= j
2 n L(R)
len},
where x +ix I2ffdxdyl(1+ix)(1+a){
(2.24)
}I2}1/2
+ixx( y
)
Next we prove that OP,tOC WT. From thm.2.2 it follows that, for AE OpVto, all derivatives of Az,t exist in norm convergence of L(H). For example, the difference quotient
(2.25)
VT1,jAz't _ (a(x+z+1e7,D+1) - a(x+z,D+t))/1
,
ri # 0
is a 4do with symbol
V
3
1-1(a(x+1e7,
)
1
-
f 0
dt alxj (x+t%1ej,
(using (2.20) where z=t=O, without loss of generality). Therefore, (2.26)
dt
O
(Vq.ja)(R)(x') =
But the uniform boundedness of all derivatives of a(x,t) implies that each derivative a(R)(x, ) is uniformly continuous over 12n 1
J
a(x,g) = h
(2.27)
Xa(x+th,t) = lhlO(1)
0
for example,where
0(1) is bounded for all x,h, E 11
(2.28)
a(R)(x,
) = O(Ihl+lll)
Thus the integral at right of (2.26) equals (2.29)
00111)
with 0(.) independent of (2.30)
Thus (2.23) implies
lim1,0 1lV1.jAz.t
alz(x+z,D+t)II = 0
7
Similarly for all partial derivatives, by induction.
Similarly,
8.3. The other half of the theorem
257
3. The other half of the IM- theorem. Note that we have proven one half of thm.1.2 :Every pdo in Opit0 is contained in the class WT. To attack the other half we must answer the following question: Given AE L(R) with Az,t E C_ how to define a symbol aE ,to such that A=a(x,D) ? We will give a in terms of A, first in the case A=Op(A)=a(x,D) formula for
E OpyrtOC WT, constructing a left inverse for the map a- Op(a). Again let n=1 first. Departing from (2.12), given aE Vto, we get ix
with
(1+ax) 2(1+a
writing
(3.2)
) 2a(x,i;)
,
as product of 2 Schmidt kernels:
p(x,U)4dl1 TF+-f 77 {(1+a,
In fact, both kernels
(3.3)
Y
2
and
(3.4)
u(x,rl) = (1+a1){(1-if1)2Y2(x)Y2(11)eixrl}
are L2(12)fl L1(1e2), as easily checked. Using (3.2) write (3.1) as
(3.5)
fdxdTju(x,rj) f
using a Fubini-type interchange of integrals. We introduce
(3.6)
w(x.Tl) =
f
and the integral operators U , V , W with kernels u(x,y)
,
v(x,y)
and w(x,y), respectively. We get W=VF*V, with the Fourier transform F. Note that wE L2(&2), so that W is Schmidt. For two Schmidt operators U,V consider the inner product of their kernels (3.7)
trace(VU*) = fdxdyu(x,y)v(x,y)
.
It is known that (3.7) may serve as a generalization of the matrix trace, introducing a "trace" for products of Schmidt operators,
hence the notation. With this notation (3.5) assumes the form (3.8)
a(z,t) = trace((Bz,W)U*)
,
8. Smooth operators of L(H)
258
where we used (2.20) for the tpdo B=b(x,D) with symbol b of (3.1).
Next we notice that (2.20) implies (3.9)
B = (1+aZ)2(1+at)2Az.0z=t=O .
This shows that a(z,t) of (3.8) is well defined for a general operator A E TILT In fact, for general A E TILT we have .
Bz, = (1+az)2(1+aO 2Az,t E
(3.10)
C00 (I2,L(H))
.
The product map (A,B)- AB for AE L(H), BE S(H), is continuous in A,B, in the norms of L(H) and S(H)= Schmidt class: We have
(3.11)
[[[AB]]] s IIAII' [[[B]]] , [[[BA]]]5 IIAII [[[B]]]
Limits may be taken inside the norm, hence a of (3.8) E CB°°(12).
In other words, our left inverse S:A - a (with a defined by
(3.8)) of the map 0: tpt0 TILT (with Oa = Op(a)) is a well defined map TILT-> 1Ut0.'To show it to be an inverse of 0 we prove that SA=O implies A=O. Indeed, then S is 1-1, and S(OS-1)=SOS-S=(SO-1)S= 0, hence OS=1 as well, so that S and 0 are inverses; S maps WT onto and we have the full equivalence of the conditions of thm.2.2. At this time we return to n dimensions: Define the constant coefficients PDE P = p(az,at) by Tyto
(3.12)
p
=
J=1(1+aZ7)2(1+atj)2
= p(az'at)
For A E L(H) with Az,t E C1(&2n,L(H)) define Bz,tE L(H) by (3.13)
Bz,t = PAZ,t = P(az,at)Az,t
With B=B0,0 one findsothat BZ,t=E-1 BEZ,t ,(i.e.(2.2) holds for B) Z,t We still have BZ,tE C (& ,L(H)), by (3.13), so that BE TILT. Let (3.14)
b(z,t) = Pa(z,1) = p(aZ,a,)a(z,l)
and note that we get (3.1) again in the form (3.15)
a(z,t) =
1P 3=1{Y2(xj)Y2(j)}
with
All earlier arguments of sec.3 may be repeated: With the 1-dimensional (3.3),(3.4) define Un, Vn on L2(1n) with kernels
(3.16)
un(x,T1)=
j)
,
E L2(12n)
259
8.3. The other half of the theorem
and an operator Wn with kernel
(3.17)
wn(x,1) =
1=1w(x j,rl j) E L2(1e2n)
Again Un, Vn , Wn are of Schmidt class. With an analogue of (3.7)
as definition of trace we then get a(z,1) = trace((Bz,tWn)Un*)
(3.18)
valid for all aE 1ptO
,
and A=Op(a), with BZ,t of (3.13). The right
hand side defines a map S: CWT -> iitO, a left inverse of 0 . Now we prove that S is 1-1
.
Let w(x,l;) E D(&2n)
. Then we
claim that a function T(x,g) E D(E2n) can be found such that (3.19)
ix
Jdzdl;
=
Indeed,using prop.2.3 or cor.2.4 we obtain cp in the form (3.20)
Let the right hand side of (3.18) vanish identically, for some A. Using (3.7) we get (writing U for Un again, etc.)
(3.21)
0 for all z,l;
=K n
where we write the kernels u(x,1)=u1, v(x,ri)=v1 as families of vec
tors u1, vIE H, depending on the parameter i E &n, using the inner
product (.,.) of H . The n-dimensional (3.2) assumes the form (3.22)
while (3.21) may be written as
(3.23)
0
using that EZ'tF
(3.24)
*
= e
iztF
Et,-z
Multiply (3.23) by T of (3.20) and integrate: Jdj may be pulled out; its inte grand is
L1(,2n)
Thinking of BF* as an integral operator, our attention will be directed to the kernel .
(3.25)
k
:
Use the definition of Ez't, (3.22) and (3.19) for
8. Smooth operators of L(H)
260
r
f
(3.26)
fdzdtp(z,l;)eizte-itxe
=
Jdzdt(p(z,t)e
ix4q(z-x,t-t)
-
To use (3.26) we first assume B to be an operator of finite
rank. Let BF* = DO) (,j
,
bj
,
cj E H . Also let
with (real-valued) functions x, iV E D(&n). Then, by a calculation, dzdtd>1y(z,t)(Ez,tuq,BF*Et,-zvn)
= (3.27)
_ (X,BF*'W) For general B EW T with (3.21) (or (3.23)) focus on Bj=P1BPj, Pj denoting the orthogonal projection onto span{q1,...(Pj}
with
some orthonormal base {ypj: j=1,2,...} of H . Such Bj is of finite
rank, so that (3.27) holds. The left hand side of (3.27), for Bj, converges to that with B, hence to 0, (by (3.23)): We get B-B j=
Q1B + PjBQj , Qj= (1 - Pj) at left Denote the factor of in (3.27) by X(B) for a moment. By Schwarz' inequality we get .
,
(3.28)
IX(PjBQj)ls [[[u]]]'[[[(1-Rj)V]]],
This expression goes to zero, as j+-, for fixed z,t, and it is bounded in z,t. The first is true because, letting xj(y) be the expansion coefficient of the kernel v(x,y) in the orthogonal expan sion with respect to the base 1P j=Et ,-z*Fp j, we have [[[V]]]2 = 1JIxj(Y)I2dy<-, so that [[[(1-R1)v]]}2 = Lj=nJlxj(Y)12dy
0.
For the boundedness we use that IX(PjBQj)Is [[[U]]}[[[V]]]. Similarly for X(QjB). By Lebesgues theorem the integral at left of (3.27) (with B=Bj) goes to zero. Hence
(3.29)
(x,PjBF PjtV) + 0
j+
for all x,'tU E D($n)
.
It follows that B=0. This gives A=O, in view of the result below. Proposition 3.1. For an XE WT, assume that either (1+aZj )XZf,.=O or (1+a .j )Xz,.=O, for some j, and z=t=0. Then it follows that X=O.
Proof. These relations are true for all z,t, if they are true for
261
8.4. Smooth operators z=t=0, by (2.2). Using that XZ'tE C(&2n,L(H)) conclude that 0
(3.30)
0 =
etdt = X +
f0(1+a )X
-
t
tej,0
(1-a )etdt = X
f X
te3,0
-
t
by partial integration, from (1+aZ )X0,0 =0. Similarly the other. 3
This proves prop.3.1, and completes the proof of thm.2.1. 4. Smooth operators; the V*-algebra property; pdo-calculus. To summarize our accomplishment in sec's 1 to 3: Let us call
an operator A E L(H) translation smooth if the family Az,O= and gauge smooth if the other Tz-1ATz of (1.5) is C"(&nL(H)) family A0' = M4-IAMB. of (1.5) is Then the essence of thm.1.1 may be reexpressed as follows. ,
Theorem 4.1. An bounded operator A on H=L2(&n) is both translation smooth and gauge smooth if and only if it is a pseudo-differential operator with symbol CB"(Rn), with Au(x) expressible by (1.7) for u in the dense subspace D(&n) of H. Here the symbol a of A is given by formula (3.8), for n=1 (or by (3.17) for general n) W=(2n)n/2FV , U and V denoting the integral operators with kernel (3.4) and (3.3) (or (3.15)). The trace is defined by (3.7), and the (Green's) function yj(t) by (2.9) Several comments are in order, to illuminate this result. First, it is trivial, that the class VT of operators which are both, translation- and gauge- smooth, forms a subalgebra of L(H), .
just using the common rules of differentiation. In fact, the algebraT is *-invariant - it trivially contains its adjoints, since A* = A z,0 * , A* 0,t = A0, * , and IIB II; 0 implies IIB*II=IIB z,0 3 * 3 3 The *-algebra 11-UT no longer is a C -subalgebra of L(H), but it carries a Frechet topology induced by the natural norms of CB00: II
(4.1)
- 0.
IIAIIk= IIAz,tIICk= sup{IIayaPAz'tII:z.lE 1en,Ia+SIsk}, k=0,1,2,..
The norms (4.1) establish 1113T as a Frechet space. The limit A of a
sequence A.E 1T has A z, =lim(A.)z
in the norms (4.1), implying ,
AZ'. EC-(3k 2n,L) so that AE WT. Involution and algebra operations are continuous, for similar reason, looking at C0O(1Q2n,L). There-
fore WT , with this topology, is a Frechet-*-algebra. Note that the same topology may be just as well described in terms of the symbols: For a E Vt0 we introduce the norms (4.2)
&2n,Ia+PIsk}, k=0,1,..
8. Smooth operators of L(H)
262 If A = Op(a)
,
then
(4.3) aZa Az.t-Op(a(a))z,t by (2.20) and (3.8)
.
,
a(a)(z,t)=trace(p(azat)aya Az,tWU*),
By (2.23) and the formulas of sec.3 we get:
Proposition 4.2. Let A=Op(a). There exists a constant c depending on n only (expressible in a form similar to (2.24)) such that (4.4)
IIAIIks cilPalik, P=rIJ((1+azi)(1+at7))2.
IIalIks
The topologies on 1IGT=Op(Vt0) of (4.1) and (4.2) are equivalent.
As another remarkable fact: If A E qGT possesses an inverse in L(H), (i.e. if there exists an L2(&n)-bounded operator B=A I with ABu=BAu=u for all uE H), then we have AIE IIGT as well: We get AIz,0=Az,0-1E Coo, and may calculate the derivatives in the usual way. For example, azj(A Iz,O)=-Az,O-1(azjAz,O)Az,O-1
,
...
.
We summarize: Theorem 4.3. The class 1IGT = Op(pt0) is a Frechet-*-algebra under either of the set of norms (4.1) or (4.2). Moreover, tIGT C L(H)
contains all its inverses (with respect to L(H)). Algebras with the properties of thm.4.3 were investigated by B. Gramsch, who uses the notation 'p*-algebra' for a *-subalgebra of L(H) containing all its L(H)-inverses. In many respects, notably regarding the Fredholm property of its operators, a 1V*-algebra
behaves like a C*-algebra. We notice that the resolvent R(a.)= (A-X)-I (in the sense of H=L2(&n)) is a well defined operator of 4I0T as well, for every
point X of the resolvent set Res(A)
an open subset of C assuming that A E 407 . In fact, the complex derivative dR(X)/dX _ ,
,
-R(X)2 belongs to qGT as well. Moreover, for the difference quotient Q = I(R(X+h)-R(X)) we get = Qz
h{(Az,t->v-h)-1-(Az,t-%)-1}.
One finds easily that limh-OaZaQz, exists, for each z,t, uniformly as z,t E 2n. Accordingly, the resolvent is complex differentiable within the Frechet-*-algebra 1IGT, defined on Res(A). Commonly, for a (complex-valued) holomorphic function T(X), holomorphic in a connected neighbourhood N of the spectrum Sp(A) of A, and a simple closed contour r containing Sp(A) in its interior, one defines q(A) by the 'Dunford integral' (4.5)
p(A) =2n
existing in 11GT since its integrand is continuous. We have proven:
263
8.4. Smooth operators
Proposition 4.4. For A E WT and a function cp with above properties we have y(A) E=-SGT The symbol of the resolvent R(X) .
E 1VT is holomorphic in Res(A), as a function of ? taking values in yto with topology of (4.2). The symbol of T(A) is (4.6)
-2n fr Extensions of prop.4.4 are possible, applying to the case
where the contour r touches the spectrum in a single point, or r may run through - on the extended plane, or that AE OpVtm, m#0 Also, the same questions will arise for other symbols. The symbol a of an operator A E 40T plays a role similar to an integral kernel. In fact, we may write (1.7) as (4.7)
Au(x) =
u" = Fu
showing that AF* has integral kernel a(x,U)eix = The question arises for a composition formula, linking the symbols a,b of A, B E WT with the symbol c of C=AB, similar as well known for integral kernels. Such a formula, involving a 'very singular' integral, called finite part, was discussed in 1,4: (4.8)
1Y1dydq
c(x,t) =
The integrand clearly is not L1(R2n) and the integral was defined by a special proceedure (cf.I,(4.19)). Similarly, a formula linking a=symb(A) with a =symb(A ): We have (cf.I,(4.6)) ,
a
(4.9)
1C2f
(x-y,t-11)e
iyndydl
where again the integral is a finite part. Different formulas are known for differential operators (4.10)
A = a(x,D) =
Ia
sNaa(x)Da , aaE C'(&n)
We know that A of (4.1) may be written in the form (1.7) with (4.11) a polynomial in t with coefficients depending on x . Indeed,
(4.12)
(Dx.u)"
,
(DXu)" (U) = eau" (U)
,
J
whence (4.13)
Au(x) = a(x,D)u(x) = xnfdfdyei(x-y)a(x,)u(y)
8. Smooth operators of L(H)
264
A differential operator A of (4.10) never belongs to i1GT, except, perhaps, if as 0 for a#0 For differential operators we have composition A=a(x,D), B=b(x,D) with symbols .
and adjoint formulas given by the Leibniz' formula of differentiation, expressed as follows : AB= C= c(x,D) and A =a (x,D) again
are differential operators with symbols given by (4.14)
c =
(YT 11 a(Y)(x,
)b(Y)(x'
)
, a-=
(Y
1 a(Y)(x,
)
where the sums at right are finite, since a is a polynomial in We have seen in 1,5, 1,6 that (4.14) generalizes to yado's of rather general symbol classes, giving Taylor-Leibniz-type composi.
tion formulas, involving as sum like (4.14) but also a remainder term. This was seen in 1,5 for operators with symbol in it, and even in ST1. However, a Taylor-Leibniz formula proved to be useful only if the remainders - given in form of finite part integrals become small, and an asymptotic expansion results. In that respect the symbol class ipt proved to be impractical: we always were working with yrhMIP16 , where both pj>0 and 0s
,
Here we refer to the groups On of rotations o:&n- &n and of dilations a:1Rn_&n, with an orthogonal nxn-matrix o (of determinant 1) and a a> 0 , of I. Define the groups of unitary operators (4.15)
00u(x) = u(ox) , of On , Sau(x) _ Vu((3x) , aE
B(+
.
An operator AE L(H) is called rotation smooth (dilation smooth) if (4.16)
0 *AO0 E C'(OnL(H)) 0
,
Sa*ASa E C'(&+,L(H))
,
respectively. We denote the class of operators AE L(H) with all 4 above smoothness properties by 4I0S and will investigate this ,
class in sec.5 below. Clearly WS C'IGT= Op(it0)
,
so that i1GS
consists of pseudodifferential operators with symbol having all derivatives bounded. We will see that the symbols of 4GS have
derivatives decaying at ferentiation increases.
,
and stronger so, as the order of dif-
Instead of requiring dilation and rotation smoothness, we even might require smoothness of AE L(H) with respect to the class of all maps u(x)-u(gx) , where g=((gjk)) is a real invertible nxnmatrix (in addition to translation and gauge smoothness). This de-
fines yet another class LIGLC'WTC'WT of pdo's, studied in sec.5.
265
8.5. Operator classes
5. The operator classes 'TVS and WL , and their symbols. As already observed, the classes 1IGS D WL of sec.4 are contained in 1IGT, thus consist of ipdo's (1.7) with symbol in CB00(1
The additional smoothness of
Az,
2n)
will lead to stronger conditions
for symb(a), to be worked out next. The maps for a general matrix gE GL(&n), together with the translations u(x)-u(x-z) and gauge transforms u(x)-eiz we will write as of sec.1 generate a larger Lie-subgroup of u(H) ,
(5.1)
GL =
(g,z,1,(P) E gl}
where Tg,z,;,tu(x) = (det g)
1/2 ei(l;x+(P)u(gx+z) ,
u E H ,
(5.2)
gl = {(g,z,1,(p): g E GL(In)
,
in
z,i;E
,
(PE S1}
,
The group operation in GL (or gi) is best decribed by introducing the linear maps g: ten->ien and X: &n->>e, for a given (g,z,t,(p)E gi C &nZ +2n+1 by setting g(x)=gx+z, T,(x)=i;x+T. Write (5.3)
Tg,z,t,e(x) = T(g,?)u(x) = (det g)1/28i%(x)(uog)(x)
We get (5.4)
T(g,X)T(g',).') = T((g,A)o(g',X'))
,
T((g,l)-1) = T(9r,X)-
where (g,A.)=(g,z,t,(p)E gl
and (g 1,-Xog 1)
(5.5)
.
For a pair of maps g(x)=gx+z, X(x)=1x+q intoduce the matrix (5.6)
M = Mg, =
0,g,z 00
'
,0,1
The group operations (5.5) correspond to matrix multiplication and inversion -i.e., (g,X)o(g',a.') H M'M, (g,X)-1 , M 1, by a calculation. Thus (g,?.) -M defines an isomorphism between GL and the Lie-subgroup of the linear group GL (&n+2) consisting of all matrices (5.6). Actually we must calculate mod 2n in the variable cp
,
i.e., work with the quotient of the matrix group modulo the subgroup of all (g,z,t,cp)=(1,0,0,2kn), to obtain an isomorphism. Speaking in terms of Lie-groups we find that (5.6) describes
266
8. Smooth operators of L(H) .
a representation of the Lie-group gl in GL(&n+2) We introduce groups GS and gs from the unitary maps used for definition of '1t3S : GSC GT consists of all Tao,z,t,w , with a rotation o and 00, 0 0=1, det o =1 . A matrix representation of gs is given by
(5.6) again, since gs is identified with a subgroup of gl We thus have introduced a chain GT=GHC GSC GLC U(H) of subgroups of the unitary group U(H) , where GX , X=T,H,S,L, all are .
unitary representations of Lie-groups gt=gh, gs, gi . Note that gx, x=h,t,s,1, are connected. For each gx we also have the finite
matrix representation (5.6). Clearly, the relation between gx and its corresponding Lie-algebra (called ax) is described by the discussion of ch.0,9, using the representation (5.6). It is suggested that the representation GX of gx on the infinite dimensional space H is governed by similar principles.
Such principles stand behind the discussion of 4L and WS, below. - This is why we decided to present sec.9 of ch.0. However, this central part of Lie-group theory will not be needed, below. To characterize the symbol classes yr10 and ipso belonging to
IL and 'S we assume that A = a(x,D) is given with a E
CBm(&2n),
and ask for the symbol (if any) of the operator (5.7)
Z(g) = Z(g,A) = Rg-IARg
,
Rgu(x) = u(gx)
.
Note that the function ug(x) = u(gx) E S has the Fourier transform (5.8)
ug"(x) = (det g )u(g t) = (det g -)u -t(x) g using the abbreviation g = g-1, g-t = (g-1)t . Accordingly, (5.9)
,
Aug(x) = xn(det g
Another transformation of integration variable yields t
Z(g)u(x) =Aug(g-x) =(det g-)xn elxg a(g
t )d
(5.10)
With another substitution of integration variable we get:
Proposition 5.1. For A = a(x,D) E WT the operator Z(g,A)=R
gARg
is in t1GT again. We have (5.11)
a(g-Ix,gt )
Note that the symbol (5.11) indeed belongs to
lpt0=CB°
Now we can evaluate linear group smoothness; it amounts to
267
8.5. Operator classes Z(g) E C'(GL(len),L(H))
(5.12)
with derivatives existing in norm convergence. Clearly (5.12) implies the existence of the partial derivatives (5.13)
)ll...(agnn ) "nn Z(g111 ...gnn) at gjl=6jl a
(a
gil
for all integers ajl 2:0
.
Formally we may execute the differentiations of (5.12) by just differentiating the symbol. For example, using that (5.14)
(g-1)
ag
-g-lhpgg-I
_
,
hpq= ((6pi6gj))i,j=l,...,n
Pq
hence agjl (g
ix)mlg=1
= -(hjlx)m = -6jmxl
(5.15)
agj1
(hlj)m = slmmj
we get agjla(g
(5.16)
= 4=1(-alx (x,U)6jmxl+ a,
m
m
(x,U)slm
)
j
Our attention is directed to the folpde's (5.17)
ejl = jal-xlaxj ,
j,1=1,...,n ,
since (5.16) formally implies (5.18)
agjlZ(1) = (Ejla)(x,D)
.
Proposition 5.2. For A=a(x,D) ET , u E S and x E &n we have (5.19)
agjkZ(g,A)u(x) = Lp=lgp.(Epka)(g
where we again abbreviate g 1=((gpq)) Note that (1.7) may be differentiated under the integral sign .
for whatever parameter tihe symbol a might depend on as long as uE
S (hence u^E S) and the derivatives of the symbol a (for the parameter) are of polynomial growth in (x,g), uniformly in the parameter. We will omit the function uE S in (5.19), derive a formal re-
8. Smooth operators of L(H)
268
lation for the operators, to be checked for polynomial growth.
We have Rhgu(x) = Rgu(hx) = R9Rhu(x), so that (5.20)
Z(hg,A) = Rh_,g_,AR9Rh = Z(h,B)
where B = Z(g,A)
,
.
With Dk=derivative in the direction of (the nxn-matrix) k we get DhgZ(g,A) = {E{Z(g+Ehg)-Z(g)}}IE. (5.21)
0
{E{Z((1+Eh)g,A)-Z(g,A)}}IE. 0 = DhZ(1,B)
Setting h=h3k=((Sjp6lq))p,q.1,...,n get DhZ(1,B)=agj1Z(1,B) _ by (5.18). The left hand side is
(Ej1b)(x,D),
{£{Z(g+Et)-Z(g)}}IEa 0 = tpgagpgZ(g,A)
,
tpq (hg) p4 sjpglq
Together we have (summing over indices occurring twice) (5.22)
6jpglgag
Z(g,A) = glgagJq Z(g,A) = (Ejlb)(x.D)
Pq
where (Ej1b)(x,U) =
(g Ix.gtU)gpj
p
p
= {gr jglp'lraTpa(Y.1) - glrgpjYraypa(Y.TI)} IY=g-' x,rt1=gtt Summarizing we get (5.23)
(Ejlb)(x,
) = grjglp(Erpa)(g-lx,g 5)
Substituting (5.23) into (5.22) the formal relation (5.19) follows Going again through the arguments, starting at (5.21), find the polynomial growth condition satisfied. Thus the formal argument without u(x) leads to a rigorous proof of (5.19).
Notice that (5.19) may be written in the form (5.24)
agjkZ(g,A) = Lp=IgpjZ(g,(Epka)(x.D))
as a consequence of (5.11). This shows that (5.19) (or (5.24)) may be iterated to obtain all partial derivatives (5.13) of arbitrary order. These all exist as linear operators S- S, as AE SGT . Moreover, derivatives (5.13) are linear combinations of b(x,D) with
(5.25)
b = (Ejj 11
Ej:l:...)a
8.5. Operator classes
269
finite application of the ejl to a, with coefficients rational in the coefficients of g, with no further assumptions. Theorem 5.3. Let the symbol class V10 be defined as the set of all a E CB00(1e2n) such that all finite applications (5.25) also are Then we have GL = Opyi10 contained in CB"O (&2n) .
.
For WS we state a similar result, using the folpde's 1ljl- Ejl-Elj=
j)+(xjaxl-xlax,)
,
j,1=1,...,n
,
(5.26)
100
=lepp = "1n=1,pa,p
Theorem 5.4. Let the symbol class Vs0 be defined as the set of all a E CB00
(1e2n)
such that all finite applications of qjl to a are in CB'(&2n) again. Then we have WS = Op1s0 Before discussing the proofs we note: .
Proposition 5.5. Introduce (in addition to (5.17),(5.26)) (5.27)
Ep0\p0= axp
EOp=T10p =
'
app ,
p =1,...,n .
Then V10 and "0 consist precisely of all aE C00(1 2n) allowing arbitrary finite application of ejl (for i,10) and qjl (for ips0), as
bounded functions on &2n, for j,1=0,...,n with sjl or Yljl defined.
For the proof of prop.5.5 we note the commutator relations (5.28)
[a p0'Ej13 _ -6pltjo
'
[Eop'EjlI = bpjE01
,
p,j,1=1,...,n,
verified by a calculation. (5.28) shows that, a product (5.25) may be written as a combination of products with and Ep0 pulled top out to the right. Thus the condition of thm.5.3 implies that of prop.5.5, while the reverse is evident. Similarly for thm.5.4. Proof of thm.5.3. It is clear from remarks around (5.24) that all derivatives (5.13) exist in convergence of S if only A=a(x,D)E SGT (i.e., aE tpt0). Moreover, the derivatives are linear combinations of b(x,D) with b of the form (5.25), as noted. Thus they be-
long to L(H) - even to 11GT - if aE ipso, with a uniform bound on compact regions of GL(&n). This implies that A=a(x,D), aE tps0,
must be GL(&n)-smooth. For example we get (5.29)
t dt(d z (g(i)))u
Z(g(t))u - Z(g(0))u = 0
d-C
,
u E S
for a smooth curve g(t). Here the integrand I(i) satisfies III(t)II sclluII, with L2-norms, and c independent of u and t. First conclude
8. Smooth operators of L(H)
270
from (5.29) that Z(g) is continuous in g. Similarly all derivatives (5.13) are continuous as maps GL(&n)->L(H). Knowing this we omit
uE S in (5.29) and divide by t. As t-0 get existence of the partials in norm convergence. Iterating we confirm GL(&n)-smoothness. Vice versa, if A EEL(H) is GL-smooth, in addition to translation and gauge smoothness, we get A=a(x,D) with aE wto. As in sec.1 one shows that T(y,X)-1AT(g,?)=A(g,?.) has continuous mixed partials for z,t,g. All derivatives (5.13) are translation and gauge smooth, hence belong to OpWto. Hence a=symb A E 4w. , q.e.d.
The proof of thm.5.4 is analogous to the above. For dilation and rotation smoothness focus on the derivatives along the curves (5.30)
g(t) = tes
,
0 < t < ao
,
s=((sjk)) = - ((ekj))
with the matrix exponential function es=l+s+s2/2l+..
,
It is known
.
that a rotation o may be written in the form o=es, with skew-symmetric s Thus, the derivatives (5.13) must be replaced by .
(5.31)
a ((at)00(as
)
a 12... )Z(tes)
,
at t=1
,
s=0
sT=-s
,
12
where only derivatives for sjl with j
4(Eppa) (g x,gtD) - (T 00a) (g x,gtD) = Z(tes, (T100a) (x,D)) using (5.19). similarly, for j < 1 (5.33)
ae
Z(es) _
q(as
jl
,
jl es)Pgagpq
((g))
For s=0 we get (5.34)
as jeels=0 = hjl- hlj i
,
hpq = ((s
8vq )) NAP
µ,v=1,...,n
Accordingly, using (5.18), (5.35)
asj1 Z(es)I5=O = (Eji-Eij)a(x,D) = ('njla)(x,D)
For an analogue to (5.19) we repeat the discussion leading to (5.23) as follows: A modification of (5.21) yields (5.36)
asjlz(eso,A)Is=O = asjlZ(es,B)Is.0 = (Tljib)(x,D)
with B = b(x,D) = a(otx,o D) . Using (5.23) we get
(5.37)
(rljib)(x,U) - (ojrolp(Erpa) - olrojp(Erpa))(o
t
x,ot)
,
8.6. The Frechet algebras =
271
0jr0lp(Tlrpa)(o x,ot5)
Accordingly, we have the following substitute for (5.24): (5.38)
as.1Z(eso,A)1s=0 = f,p=lojr 1pZ(o,Arp) , Axp=rlrpa(x,D)
.
We need the Campbell-Hausdorff formula of ch.0,9, quickly rederived as follows here: For sE N(so), a neighbourhood of so write
(5.39)
es = e7 (s,so )es°
,
(1-ese-so )m .
a. = log(eseso ) = _yn=1
Clearly ?(s,so), is well defined (and holomorphic in the coeffi-
cients s jl
,
so jl ), as long as
t _X orthogonal we get e =a
1-eses0 1
<
1
. Since a=e8es0 is
so that X , hence Xt = log(e-k) _ -% For is skew symmetric (and real-valued, for real-valued s,sO) fixed so the function %(s,sO) has a local inverse at so . We get ,
.
(5.40)
asj1Z(es) I s=so =
Or, with holomorphic functions xjlRV(s), j
(5.41)
Z(e%e$0 ) IX=0 µv
as.1Z(es)=;,
,
vpz(es,(Tlpga)(x,D))
Note that (5.41) is the desired substitute for (5.24); It may be iterated, showing that derivatives of arbitrary order (5.42)
)a12....)Z(es)
((as 12
may be expressed as linear combinations of
(5.43)
Z(es, (Tl j, 1, Tj j, 1, ...a) (x, D)
with coefficients holomorphic in sjl. The proof of thm.5.3. now may be repeated with 1jl instead of Ejl, for a proof of thm.5.5 (Note that (5.32) way be iterated as well).
Q.E.D.
6. The Frechet alaebras Vx0, and the Weinstein-Zelditch class.
We already stated that the symbol classes s0 and V10 consist of locally classical symbols. More precisely we have the following Proposition 6.1. For any open set QCC In the sets 1yx JQx&n, x=s,1, are subsets of restrictions of symbols aE ix0 to Qx&n={xE
of the Hoermander class S0,1,0(Q) ( 6 . 1 )
a(Pa)
= 0((
) - I a I )
(cf. [Hrz]). That is, we have ,
as xE Q CC Q
8. Smooth operators of L(H)
272
Proof. It suffices to look at i,s0, since V10C iNs0. Then we get xE Q, aEl,s0.
(6.2)
Since gjla = 0(1) as well, we get (6.3)
00a =
ja Ja = 0(1)
kjla =
,
7)a = 0(1)
Multiplying (6.2) by l andj and adding we get (6.4)
j =
Dl?jla =
jko0a -
or, (6.5)
0(1)
i.e., aI
,
0(()-1)
=
7
3
This chain of arguments may be iterated, to prove (6.6)
xE Q .
Starting with
a(3)
instead of a we get (6.1), q.e.d.
Let us work out a global result, with the same argument. Proposition 6.2. For a E C0O(&2n) let
(6.7 )
a00 = 100a
,
apq = Ylpga
ap0 = np0a = a ,
i xp ' a0q =
1
a
, y2)
pq E
2
13 Ix
=aI q '
p , q = 1,... , n .
Then there exist symbols y 1pqj (=-ype -e
(6.8)
ll0ga
q=0
Y
pq apq '
al
e 2 -.1
such that
n 2j fp, q=Oy pq apq
i=
Here we recall the symbol class Vam , at = (ml,m:) , defined by
(6.9)
{a E C°D(12n)
:
O((
)m1-ICI I(x)M2 -ICI) }
For symmetry reason it suffices to discuss only one of the identities (6.8). For the second one write (6.2), more explicitly, as (6.10)
apax a=a00+ ?pap0 p p
(
,
)a=apq+(xpag0 xgap0)
_ q 1 1
ptl
p
q
p
Then (6.3) may be replaced by (6.11)
p
p(apj+xpaj0-xIap0)
pap0)+
p
by the same derivation. Dividing (6.11) by we obtain a formula of type (6.8) for alb , noting that the quotients 7
8.6. The Frechet algebras
xp/() 2
,
q/(
)2
,
2
273
all are in VCe2 -e,
Q.E.D.
.
As a consequence of prop.6.2 we get the result, below. Theorem 6.3. For a E 1Vs0 we have the estimates -IRI s j s IaI
(6.12)
,
for all a,f
with O(.)-constant independent of x, but not necessarily of a,(3 Note that (6.12), for. IaI+ISI=1, is a consequence of (6.8), since a E 1Vs0 has all apq bounded. For general a,s we use that apq E rVs0, again, so that we may iterate (6.8). Q.E.D.
The class Zn of all symbols a satisfying (6.12) was introduced by Weinstein [Wel] and Zelditch [Zel]. Thm.6.3 amounts to
(6.13)
1V10 C 1Vs0 C Zn .
Let us observe that all three classes 1Vs0
,
1V10
, and Zn are
symmetric in x and . Therefore in prop.6.1 x and may be interchanged. We also get an x-decay of derivatives on compact a-sets. The inclusions of (6.13) are proper. For example, let n=1 and x
(6.14)
,
where x(t), w(t) E COO(&), x=1 for tal, =0 near 0, w=1 near t=1, =0
in It-11 a 1/2 . Clearly all derivatives of a
,
for x and t
,
are
bounded. Moreover, a and all its derivatives vanish, as X is outand s 2x side the interval (1/2,2) -i.e., unless we have xs 2T ,
both. In other words, we have (x) s
s 2(x) in supp a
and
Accordingly all (positive and negative) powers of are bounded in supp a . Hence we also have all bounded. In particular, it follows that aE Z1 On the other hand it is clear that talk-xalx = -x sin x as x= , IxI a 1 Thus s11a is unbounded and a E iV10 follows. Next we focus on topological and algebra properties of rVx0. .
,
.
Proposition 6.4. The classes tVx0, x=t,1,s, are algebras under the
'composition product' a°b defined by (a°b)(x,D)=a(x,D)b(x,D), as well as the 'pointwise product' (ab)(x,t)=a(x,D)b(x,t) .
Proof. By definition 1Vx0 are algebras under ",". Pointwise we get
(6.15)
E jl(ab) = (Ejla)b + a(Ejlb)
,
rlji(ab) = (rijia)b + a(rljlb).
By (5.19) or (5.35) a,bE iVxo implies Ejla, Ejib, rljla,
rl j1bE 1Vx0
hence Ejl(ab) (or rljl(ab)EWt0. Similarly for iterated Eji. Q.E.D. It is natural to introduce on 1V10 the Frechet topology of
8. Smooth operators of L(H)
274
the countable class of semi-norms (6.16)
Ile pNgN ..... eplg1 all Loo =IIaII
with epq
of (6.1)
,
for p,q a
ep0 = 1p0 = a x
(6.17)
1
nK
79=(p1,...' N), K=(g1.... ,gN)I
and
,
EOq = 110q = a q ,
p '
(e 00 undefined)
Similarly, for Vs0, using the jjl instead of the ejl. Or, for Vt0, using only the expressions (6.17). On the other hand, for Op*0 GX, one may use the semi-norms
(6.18)
IIVaa(x,D)II = IIIIaIIIIa
,
, and VaA denoting the partial deat the unit of the rivatives of A , or A or Az,t t 6101z, g,z,t group. Here a will be a multi-index of the dimension of the group.
with the L(H)-operator norm II'II
,
,
Proposition 6.5. The topologies (6.16),and (6.18) are equivalent. Proof. Consider the case of V10 . We know that (6.19)
VU(a(x,D)) = aa(x,D)
, as = IIlaisp q a J J
with some selection ePjgj
,
corresponding to the multi-index a
From cor.2.6 we conclude that
(6.20)
IIIIaIIIIa = Ilaa(x,D)II s c IRl7ly ISn Ilaa(0)IILoo
where the terms of the sum at right are all norms of type (6.16), since the include the expressions (6.15). ejl On the other hand, for =WS we have a=symb(A) given by the trace formula (3.18). For b=1 e a we get B=b(x,D)=VA with 1 pjgj
the corresponding combination VU of ag
az and a. on Agizi. at pq q p (g,z,l;)=(1,0,0), as follows from (2.20) and (5.19). Use (3.18) for
(6.21)
IlbIIL s c
,
IPI7IYIs2nII(Vta)(a)(x,D)II
Again, the terms of the sum, at right, are semi-norms occuring in (6.18) This shows indeed that, in case of V10, the topologies are equivalent. Similarly for is0. Q.E.D. .
Theorem 6.6. The classes i1G%, X=T,S,L, are V -subalgebras of L(H),
under the operator product and the topology (6.16) or (6.18).
Proof. Prop.6.4 and prop.6.5 make
0
(or WX) Frechet algebras.
275
8.7. Polynomials in x and Sx
The adjoint invariance and inverse-closed-property follow from A*g,z,S
(6.22)
= (Agrzr0 *
I
(A 1)g,z,l; _ (Agrz10
-1
.
is differentiable in L(H) then so are the right hand siIf A grzrt des in (6.22), assuming that A IE L(A), in the second case. Q.E.D.
7. Polynomials in x and ax with coefficients in W% . So far,in this chapter,we were dealing with zero order pdo's only. This restriction now will be removed by defining the classes with coefficients
,x of all polynomials in the 2n variables x,
in px0, x=t,s,1. Also,
xm, m=(m,,m:), with integers mjz0, denotes collection of all such polynomials of degree sm, (sm2) in (in x) For general mE 12 we define Vlm , x=1,s, by setting
(7.1)
{a E C00(1 2n)
y)x
a(x, ) E px0 }
(x)
:
Note that we have
(7.2)
(x) = s0(x) +
jsj(x)
,
s0(x) = (x)-1
,
sj(x) = xj/(x)
and a corresponding formula for (l;), where the sj(x) and sj(g) are
in Vc0C *0, for x=t,1,s. Thus for integer mjzO an aE Vxm of (7.1) may be written as a=ao (x) m'(1;) m' , a polynomial (of proper degrees) with coefficients in wxo, showing that the two definitions agree. Note that ryxm C 4x , even if mj are not integers, since we always can write a E rpxm as a = ( a0( m, -M, (x) m2 -Ma ) ( ) M, (x) Mz )
with a0E VxO
,
and with integers O<Mj2mj, so that the expression
(.) is also in px0. We get tpx=U(Vxm : mE ILI } as well. Clearly the Vx are graded algebras under pointwise multiplication; we have m m' m+m' Moreover, the Frechet topology may be carried to Vxm using (7.1). As seminorms on *m use (6.16) for ao of (7.1). The above definition of Vxm uses the 'weight functions' (x)= (x)=(1+Ix,2)1/2 again. The symbols of *m are said to have multiplication (differentiation) (total) order ma (m,) (m). Proposition 7.1. The class Vsm consist precisely of all aE C"(&2n) such that, with xm(x,l) of III,(3.1) i.e.,
(7.3)
, m = (ml rm2) E
nm(x,U) = Q )
&2
we have the estimates (7.4) for all products 1"1=1I1pJqJ of 3
(7.4)
(rl =111pj,ja)(x,U)
=
x,l E &n .
276
8. Smooth operators of L(H)
Similarly, '1m is described by (7.4) with Y1
µv
replaced by ERv
Proof. Consider x=s only. If aE Vsm , then an-l=bE VsO. Hence
llpga = (1lpgb)nm + (b(npgnm)/nm)nm
(7.5)
Note that npgnm/nm E yicO. Thus Y1pga=O(nm). Moreover, (7.5) implies
that
pgaE Vs m, since ,cOC
s0, and since Vs0 is an algebra. Hence
11
we may iterate, for (7.4). Vice versa, let (7.4) hold for a. Then (7.6)
1lpgb _ n-mllpga + b(iipgn-m)/n-m)
= 0(1)
,
using that r1 pgn-m/n-m E V00 C Vs0. Apply 1rs to (7.6) for 11rsrJpgb= O(1)
,
etc. ...
It follows that bE s0, so that aE ism , q.e.d.
Proposition 7.2. Let aE 4wm, x=s,1, mE 1k2. Then, for all a,1, E'Wxm+r(e'-e2)
(7.7)
,
as -dal s r s 1PI
,
r = integer.
Proof. First let JaJ+JPJ=1. Prop.7.1 shows that 1xm is left invariant by arbitrary applications of spq or tlpq' resp., hence Vxm for aE Wxm. Thus (7.7) holds for r=0. On the other hand, (6.8) expresses by apq with coefficients in r=t1, r(e'-e2) correspondingly. This gives (7.7) for r=t1. Formulas (6.8) may be iterated, proving (7.7) for arbitrary a,P and r. Q.E.D. Remark 7.3. Note prop.7.2 contains thm.6.3. Generalized classes sm' with Zn,O=Zn , may be introduced, replacing O(.) in Zn,m
(6.12) by O(nm+r(e'-ez))
For a symbol aE Vx, x=s,1, we will define the ipdo (7.8)
A = a(x,D) = xxaaa.P(x,D)DR
as
However, one just as well might write the same operator A as (7.9)
A = Db a,R(x,D)xaD(
,
with (other) coefficients ba,P(x,U)E i,x0. Or also in any other order A=F'aa,,(x,D)DPxa=YjtaDRda,P(x,D), etc. This follows from (2.8):
The reason for this fact can be found in formulas (2.8)
:
For A =
For A=a(x,D)E IWXC IIGT, X=L,S,T, we have [xj,A], [Dj,A]E Opy,x0 In fact, for x=s or 1, the commutator symbols have better decay properties: Using (2.20) we get (7.10)
az1A0,0 = alx.(x,D)
,
atJA0,0 = al j(x,D)
,
8.7. Polynomials in x and 8x
277
so that [xj,A]=a3(x,D)=ial,j(x,D)
(7.11)
,
satisfies stronger estimates of Weinstein-Zelditch type
O(((x)/() )1)
(7.12)
,
-IRI-1 s 1 s lal
s 1 s lal+1 -l O(((x)/( ))1) Similarly for higher order commutators and higher derivatives. Hence yw is an algebra under a°b of prop.7.4, since in a product of 2 expressions (7.8) one may unite the x- (D-) powers at
left (right). With calculus of ipdo's, below,one also confirms that
i.e., 1Vx is a graded algebra, under 16" . m+m' , Next we look at calculus of pdo,s, for the algebra OpVs, si-
vxm°1m'C
milar as in 1,6 for 0M,P,o We have yixmC y,tm, hence may use the Leibniz-Taylor formulas I,(5.7) I,(5.9) for product and adjoint in Opts. The question remains whether (in what sense) the remainders decay, and about significance of the terms of the expansion. ,
In view of applications in ch's 9,10 we focus on (7.13)
KA , AK , [K,A]
where aE yixm, kE Cm
K = k(x,D)
,
, A = a(x,D)
,
m,m'E &2, x=s,1. We get a Leibniz formula
with controlled remainder, but decay of order not quite as fast.
Proposition 7.4. We have
KA
=
j
4=0 I
(7.14)
AK = 4=0 l
(-i)j/8! (k(e)a(8))(x,D) + RN =j
ej
(-i)J/e! (a(8)k(0))(x,D) + RN
[K,A] = Lj=1(-i)j/j! (k,a)j(x,D) + RN where k(8)a(0) ,and a(8)k(0) (7.15)
,
and the 'iterated Poisson brackets'
,(e)(k(e)a()-k(0)a(e))
(k,a)j =
,
(8) = j!/e!
lef
belong to Vxm+m'-re'-r'e2' for all r,r'=0,1,2,..., r+r'=j, while R for all r,r'=0,1,..., with r+r'=N+l m+m'+re'+r'e2
NE
,
Proof. Note that k(e)E 1lcm'-je'
' a(A)E
m+r(e' -e2)
,r=0,...,j
as lel=j, by prop.7.2. Their products are E 1Vsm+n'-(j-r)e'-re2' proving the first statement. In the remainder of I,(5.27), i.e., _
(7.16)
PN (N+1) IAl N+1(01)
N+1
1
P 0,N' Pe,N
(1-t) Ndx 0
IO,N(x,lj,t)I
8. Smooth operators of L(H)
278
with the 'finite part integrals'
(7.17)
I may be written as an improper Riemann integral, namely (7.18) I =JI1YP1Tje i1(Y)-2M(1_A1)M((Tl)-2M
with aM = (1-Ox)Ma, and the usual formal partial integration. In 1,5, by a somewhat difficult interchange of limits, we proved (7.16), (7.17), (7.18). Formally this is Taylor's formula with integral remainder. Moreover, we saw that the derivatives I(a) are obtained by differentiating (7.18) under the integral sign. We discuss RN only, noting that RN, RN may be treated analogously. We need an estimate for I. They are in the span of integrals (7.18), with replaced by (7.19)
a ,
-rq)
P'+3" = P
We have aNE Vsm' again. Hence, using estimates (6.9) for k and (7.7) for aN , we arrive at an estimate for I( by a sum of terms
7.20) fdYdTl((Y)(TI)
-tq)m' -N-1-Ia,
where -la"I s r s N+1+I3"I rr
.
J dy(Y) -2M(
(7.21)
x-Y)m12-r(
)mi+r,
Here we can estimate
x_y) M'2 r
= 0((x) m' Z -r)
JdTI(1I)-2M(-zTl)m, -N-1-Ia' I = O((t)mI
-N-1-Ia, I )
using I,(6.5). Note (7.21) holds uniformly in t, 0sts1. The estimates remain true if. a'=a"=13'=13"=0. Hence,
(a) (P)
.22) (7.22)
O( nm+m,-re
-r e : ,
)
,
r+r
= N+1
for all multi-indices a,3, giving the corresponding for the symbol rN of RN . These estimates are weaker than the stated ones. But, (7.23)
(-i)IOI/61k())a(6) + rR 71 N+1:I0IsR where the sum at right is E Vx m+m,_re,-r,eZ, r+r'=N+1, for all R= N+1,N+2,... For fixed r,r', r+r'=N+1, let R=N+2j, with j>O. Get R+ 1=(r+j)+(r'+j), hence rNEtm+m'-re'-r'ez-je' by (7.22). Thus,
(7.24)
rN =
=111pjgjr2 j '
1
1
E
Vtm+m'
-re' -r' e2 , as N' s j
279
8.8. Characterization by Lie algebra
Thus all products of sj factors are O(nm+m'-rg'-r'e2)' for 4, hence also for rN. Since j is arbitrary get rNE Xm+m'-re'-r'el as required.
Q.E.D.
Proposition 7.5. We have
(7.25)
Opyxm = {(x)m:A(D)m' : A E WX = Opix0}
and, more generally, for arbitrary sE 12 (7.26)
Op*xm = {ns(x,D)Anm-s(x,D) : A E WX }
Theorem 7.6. The space Op*xm is characterized as set of all AE T-1AT belO(m) such that for all T=T of (5.2) the operator g,z.,.W 1 ongs to the subgroup GX of GL AT defining a C *-map gx- O(m) in the Frechet topology of O(m), as discussed in 111,3. Moreover, A-T-
,
a sufficient condition for AE OpVxm is that (i) AE L(Hs'Hs-m) C00 (ii) T-1AT is (gx,L(HS, Hs-m) for just one sE 12 We leave the proofs of prop.7.5 and thm.7.6 to the reader. .
8. Characterization of '% by the Lie algebra.
In [B 21 , [ Dn1] , [Ci ] , IV, 9-10 one introduces 'derivatives' of a linear operator AE L(H), as commutators with Dj or M1. If all such derivatives exist (in suitable topology) then A is found to have a symbol in a class like CBOO or y't'O,P,b - i.e., A is a pdo.
There is a link between that approach and our present one: It points to the relation between the Lie group GH and its Lie al-
gebra, as discussed in ch.O,sec.9. If a Lie group G is represented as group of invertible matrices on some BN, the 'tangent vectors' at IE GC GL(IN ) form a linear space A of NxN-matrices, containing its commutators - a representation of the Lie algebra of G. The connected component of I in G then is the set of all products of eA for AE A , by the Campbell-Hausdorff formula. Now focus on the 'representation' A -> EZ,t,q),
of our Lie group gh of (1.11),(1.12) on the «-dimensional space . Actually, two representations are involved: We represent ei((P+K)TZ=ei(T+Mi;)eizD on H (as gh (=&nx&nxSl) by (z, ,g)- EZ.t, unitary operators). That representation generates the above A-> AZ,t as invertible maps L(H)-L(H). Following the lines of ch.0,9 we formally get a tangent space at (z,t,q)=O for the first representation as the linear span (called AT=AH) of the folpde's L(H)
280
8. Smooth operators of L(H)
(8.1)
axj
i
,
,
ixj
,
j = 1,...,n
.
The (8.1) are unbounded operators of H. A precise definition will specify the domain - that of the infinitesimal generator of
iD.t
ixjt},
respectively. These groups consist of unitary operators of H, thus their generators are skew-selfadjoint each having an orthogonal spectral measure. Notice that the precise domains of the operators (8.1) are not identical. Linear combinations no longer are skew-self-adjoint. However, as easily seen, the self-adjoint relizations of 1, Dj ixj and their linear combinations zD+jx+? , are unique. Declare as common domain of 1, Dj , Mj the space S. Then AH indeed is a linear space; for GAM the closure (in H) is skew-selfIs
3 }
,
{elt}
,
,
{e
,
iD
**
adjoint, and e
is unitary. Using ch.0,sec.6 we get **
(8.2)
Ez,t,W+(z. )/2
.
Note that (8.2) supplies an analogue to the Campbell-Hausdorf formula for our present infinite dimensional representation. GH and AH are related exactly as G and A of 0,0. Our second representation A- Az,t of gh on L(H) supplies another representation of the Lie algebra AM , as the (also unbounded) operators on L(H) formally given as linear combinations of ,
(8.3)
ad ax
,
ad ixj
,
j=1,...n, with (ad X)A = [X,A] = XA-AX .
j
Indeed, let (8.4)
then, formally,
dt(e
iLtAeiLt) _ -i(ad L)A
Formula (8.4) seems to imply that A- Az't is C1 as a map H2n -
L(H) if and only if (ad aj)A , (ad xj)A E L(H), moreover, that A Az't is C if and only if finite products of ad ax ad x1 take ,
7
A to a bounded operator. In turn A- Az' is Co. if and only if AE OpVto , by the 'tpdo-theorem. Both facts suggest thm 8.1, below. Theorem 8.1.
An operator AE L(H) belongs to Opipto if and only if
the operators (ad x)a(ad D)OA: S -* S' all have their image in H and extend to bounded operators of H , where
(8.5) (ad x)a=(ad x )a' (ad x2 )a:..., (ad D)A=(ad Di ) P' (ad Di ) P2 ... Here (ad xj)A= xjA-Axj trivially is a map S-'S'. Similarly
281
8.8. Characterization by Lie algebra
(ad axj)A. Higher order commutators (ad x)aA are sums of terms
IhcATbc, with only one A between products The of x j. Again they are S- S'. Thus the statement is meaningful. The proof will be a verification of above relation between differentiability and existence of commutators in L(H). Details are left to the reader. Clearly the map (g,?.)of (5.2) defines a unitary representation of the Lie group gl on H. Note that the map is just the restriction to gtC gsC 91 of T above. For GT , GS GL the Lie algebras AT , AS , AL are the (unat e=(1,0,0,0) bounded) directional derivatives of T(g,X)=T ,
of gl, in directions allowed by the subgroup. In this way we get AL and AS as the real linear spans of (respectively)
xlax
(8.6)
+ bjl/2
,
ax
ixl
,
,
i
,
j,l = 1,...,n ,
J
7
and (8.7)
4=1xkaxk +n/2
x1ax J-x ,az1 ,j
J
Again declare S as joint domain of all operators (8.6),(8.7). The exponentiations eLt, as in 0,6, for LEE AX, define operators of GS. We again get a Campbell-Hausdorff formula, of the form
exP (iD Y
(8.8)
)-T 9,z° .V ,
'V
,
where g=eY , z° =ei (y ) z
and where (8.9)
DY,z,t,(p
= Dxgaxp+16pq) + 2zpaxp +iYpxp + i(
For LE AS we get (8.4) again. The algebras ibGX may be characteri-
zed by the Lie algebras A% , just as in thm.8.1 for X--H
:
Theorem 8.3. For X=T,S,L, the class W% is identical with the set of AE L(H) allowing arbitrary finite application of ad L, LEE AK, in the sense that II(ad Lj)A:S-S' maps S-*H and extends to L(H). The proof is similar to that of thm.8.1, and is omitted. Problems: (1) Verify formula (8.8) in details, using ch.0,sec.6. (2) Question: Can you relate the proof of the real Campbell-Hausdorff formula (0,lemma 9.5) offered in 0,9, with pbm. (1) above? ,
Chapter 9. PARTICLE FLOWS AND INVARIANT ALGEBRA OF A SEMI-STRICTLY HYPERBOLIC SYSTEM; COORDINATE INVARIANCE OF Oppx .
0. Introduction. We use the natural symbol classes ipx, x=1,s, in this chapter.
First we ask for invariance of OpVz under global coordinate transforms. Each TVX is invariant under conjugation with TE OX. For a subgroup these are linear coordinate transforms: For GT get the translations ezD, for GS the 'similarities' Tao,z,0 (all distances multiplied by a constant), for GL get all linear substitutions. Another subgroup gives 'gauge transforms' (conjugation by eitx) The question about more general coordinate (or gauge-) inva-
riance of iIGX='ThX may be phrased as follows: LIGX is the set of AE L(H) with At eLtAe-LtE C_(R,L(H)) for every LEAX In new coordinates y=q(x) the folpdes LE AX will transform to other folpdes L"
forming a Lie algebra AX-, where Lu=f H L" v=g
,
v=uo cp 1, g=f o q
1
Clearly eLt transforms to eL t, as solution operator of atu=Lu etL Ae-tL E C"(&,L(H)), assuming atv=L"v. Thus "AE 'IGX" means A that u(x)-'u (x)=u((p(x)) defines an isomorphism of H. Thus, in new coordinates the property "AE VGX" transforms to smoothness under certain eLt, with more complicated LE AX-, depending on Vice versa, since a coordinate transform is invertib'le, "AE 4GX" (that the transformed operator is in LIGX) may be expressed as smoothness of for certain M - the transforms of LE AX back to the old coordinates. For coordinate invariance we must
t
.
eMtAe-Mt
show that eMtAe_Mt E COO(R,L(H)) for the transforms M of epq or Tlpq.
Observe that our folpdes L=epq (or =ilpq) were either skewsymmetric, or, at least, have L+L a multiplication of order 0. We shall require that condition for M, as an assumption on the coordinate transform cp. Then eMt is the solution operator of a hyperbolic equation atu=Mu , of the form studied in ch.VI.
We always will require ME O pyce
M+M E OpVcO
so that the
existence theorems of ch.VI are applicable. For a detailed discussion of coordiate invariance cf.sec.4. But observe that coordinate invariance appears as a corollary of a study of invariance of Vxo
under the particle flows of the hyperbolic symmetric equation atu= Mu . For eMtAe_Mt E C"O(R,L(H)) we first prove at aovtE C°°(R,VXo)
a=symb(A)E ix0
(sec.1 for x=1, sec.2 for x=s). The discussions 282
,
9.1. Flow invariance
283
are parallel to VI,6, but use a different approach: at solves the PDE atat (k,a), k=-isymb(M). Repeated application of spq (or r1 pq generates a system of PDE's. We use this for boundedness in t of rkpjgjat (or rblp7q at ), and, more generally f p.gja1at, etc. In sec.3 we apply our result to show that At=eMtAe-MtE CO'(&,L(H)). We prove this for a rather general class of hyperbolic t'ide's a u=iKu with KE Oplyce, K-K E Under proper assumptions on K
t
we get eiKt Se-iKt=fix, as for Oplyco in VI,5. In fact, Optpxm= e
iKt Opyucme -iKt
and even a iKtAe -iKtE C (&,OpVxm), as AE Opyurm
This invariance under particle flows and invariance of under conjugation is the second topic of the present chapter. Thirdly, we will look at particle flows for systems - in sec's 5,6,7. focusing on the algebras Opyix. Here KE Opice will be a vxv-matrix of pdo's. To include the Dirac equation (ch.10) we require the system atu+iKu=O to be semi-strictly hyperbolic of a generalization of a concept of VI,4: The eigenvalues of (the essential part of) k=symb(K) must be real, of constant type e°
,
multiplicity, but need not be distinct (sec.5). It is no longer true that 1W%v (or Opxv) is invariant under m conjugation with a-iKt. Rather e-iKtAeiKt remains a 4 do in Optxv m only if A belongs to a certain subalgebra P=P(K) of Opwxv. The existence of this invariant algebra P gives raise to physical speculations in case of the Dirac equation, to be looked at in ch.10. A necessary condition for a yado AE OpPxm to belong to P(K) is modulo that its symbol matrix a=symb(A) commutes with k=symb(K) terms of lower order. While investigating invariance of P we will ,
find one particle flow for each of the distinct eigenvalues of k(x,l) The flows satisfy VI,(5.19) with k replaced by X .
.
For the Dirac equation we get 2 particle flows, for electrons
and positrons, resp. The flows will describe the exact relativistic motion of the particle under the potentials imposed. For 1-dimensional systems we get a theory parallel to that of VI,5,6. For v>1 the different parts of the symbol matrix will propagate along different flows. Roughly, an a(x,t) commuting with will split into 'diagonal boxes' corresponding to the eigenvalues of k. Modulo lower order each box propagates along its flow, undergoing a similarity. For details cf. sec's 5,6,7. 1. Flow-invariance of VI..
In this section we assume a scalar real-valued symbol k=k1E
9. Particle flow and invariant algebra
284
yce. Let 'fit be the flow of VI,(5.18), for this symbol k. In VI,6
we proved that vt:142n-12n satisfies at=aovtE yicm for aE 141cm . Here we prove the same for i1, under stronger assumptions on k. From VI,(5.17) and 0,6 we know that at is given as solution of
(1.1)
a
ta t
+ (k,at) = 0
,
t E & , a0 = a
,
with the Poisson bracket (.,.)_(.,.)1 of VI,(5.16). Note that Our con(1.1) is a Cauchy problem for a first order PDE in ditions on k will insure that at solving (1.1) gives a symbol atE
11
for all tE & , a E 1
,
.
q a is bounded for finite
Assuming WE V10 we know that lk
ij such products. We apply epq (or a finite product) to (1.1), for a pgrst=ers e system of equations in a a a etc. Then we pqat pq t , will derive apriori estimates showing that apgt, ... remain bounded in t,x,l . As a preparation we note the following. pqt=e
,
Proposition 1.1. (a) The expressions epq, of VIII,(5.17),(5.27)
may be written as Poisson brackets (1.2)
epga = {epq,a} = Epgl
Epgjx a,,
"Ix
with the functions (1.3)
Epq = -
as p,q 2
Ep0
= p , EOq =-xq
(b) For Poisson brackets we have Jacobi's identity
(1.4)
(f,(g,h)) = ((f,g) h) + (g,(f,h))
,
f,g,h E C"(1n)
Using (1.1), (1.2), (1.3), and Jacobi's identity, we get (1 .5 )
a tapgt + (k, apgt) + ((E pqk) , at) = 0
.
Suppose that the folpde Lpq:a - ((Epgk),a) may be written as a linear combination of E00=1 and the other ejl , with coefficients in tpc0. In other words, assume that, for p,q=0,...,n, (p,q)O(0,0), (1.6)
=arrr,s=0XpgrsErs
'
where ?pgrsE "0
Then (1.5) assumes the form (1.7)
atapgt+ (ko'apgt) + r D,pgrsarst =0
p,q=0,...,n, (p,q)#0. '
Now (1.1) and (1.7) may be thought of as a linear system of (n+1)2 ODE's for the unknown functions apq along the charac,
285
9.1. Flow invariance
teristic curves of the common principal part of all equations, That is, along the particle flow vt , above, we get a linear homogeneous system (with Xpgrs=X pgrs°vt (1.8)
rP,
Also we of course have the initial conditions (1.9)
apg(x,U)
,
at t = 0
.
)) is bounded in all variables Since the coefficient matrix ((k pgrs it follows that apqt are bounded in x, , over t E It compact t-intervals (cf. the detailed discussion in VI,6). We get
x, E 1n ,
,
(1.10)
y(t) _ 2 1a pg (t)12 y(0) = E 1a Pq I2= 0(1) IaIscy pq pp c independent of t,x,g, which may be integrated for y=O(ect). Thus (1.11)
,
at, ax at, a
P
at, Epgat
pa
q
qat
xgaxPat' p,q=l,.... n,
are bounded in every compact t-interval, as In Clearly the procedure may be iterated, under proper conditions on k. Once more, apply epa to (1.7). Write Epoepgat=apgpot : .
(1.12)
atapgpat + (k'apgpot) +
(Epok,apgt)
rs(Ep(?'pgrs)arst = 0
+
In the last term we use that XpqrsE pcOC VIO
,
VcO . For the fourth term we get
implying epcpgrsE
%parsersapgt=P'parsapgrst
r relations (1.5), (1.7) get using (1.6) again. With this, and the
(1.13)
at a pgrst +(k'apgrst)
+ n7,
Xpgrsxxpaanxpat = 0
p,q,r,s=0,.... n, again with X pgrsnxpOE tpcO. Again the 'vector'
(a pgrst)p,q,r,s=0,..n satisfies a system of ODE's in t along the Again it follows particle flow with coefficients bounded in that the apqrst are bounded for x,1E &n, tE ICC 1. We even may dif-
ferentiate for t, deriving initial conditions for (atapgrst)t=0 from (1.13), for yet another system with bounded coefficients. Thus t-derivatives are bounded as well. We proved, for m=0
:
Theorem 1.2. Assume that the real-valued symbol kE Vice satisfies
condition (1.6). Then at aovtE COO(2,i10), for every aE 1lm, mE V.
9. Particle flow and invariant algebra
286
For general m we use VIII, prop.7.1: i41m is characterized by rkpjqja=O(nm)
,
for all such products. For aE V1m , at as above,
again get (1.1)
(1.7) (1.13), etc. The coefficients k, Xpgrs etc., are bounded as before. However, the initial values Xpgrspa ' are Thus, the resulting estimates now will be ,
,
atapgt
(1.14)
'
atapgrst '
...
,
= O(nm'v-t(x,))
However, in VI,6 we proved that the right hand side is This completes the proof of thm.1.2 for general at. e satisfying (1.6) will be denThe class of all symbols kE oted VXne. Note prop.1.3, below, describing a subset VX e of of Vane. For another such class see sec.4.
Proposition 1.3. Let *e be the class of kE Vce with
(1.15 )
k (x , ) =
gkpgxp q+kd+km , kdE
kmE 'yice2
, kpqE a
The (real-valued) functions of VX form a Lie-algee bra over M (k) under the Poisson bracket product (kl,k2), contaiThen V), eC V%ne
ning the a Jl of (1.3), and we have epq: VX e)Xe The proof is a calculation. 2. Invariance of Vsm under particle flows. In this section we will extend the results of sec.1 to Vsm. We introduce analogues to PXe, )Xne: Let We consist of all (2.1)
kd + km ,
x00x +
with xjlE T, and symbols kd, km, as in (1.15). Let Vane consist of
all bE Vce with bpq=llpgb satisfying (2.2)
(bpq' ) - 4, q=OY pgrs'1rs ' Y pgrs E VC0 'W?ne is a proper subset of yice: For n=1, 14 and Vkne, as easily seen. Clearly VaeC V)`e and yiaeC Vane
be in ryane, and let Theorem 2.1. Let the real-valued symbol of (0.2) a E ilsm Then, with the characteristic flow vt = and at = aevt , we have atE C00(1k,4-m).
The proof of thm.2.1 is analogous to that of thm.1.2. Define (2.3)
,100
rljl=
j,1=1,...,n,
7101=-x1,
287
9.2. Invariance under particle flows
similarly than the ejl in sec.1, such that
1jla = (rljl,a) , j,1=0,...,n ,
(2.4)
r110 = x1 , '1O1
171 as Proposition 2.2. The (real) linear span of the functions well as the class yPe , are (real) Lie-algebras under (.,.) The proof is a calculation again. More general symbols in Vane are used in the proof of thm.4.1 For hyperbolic systems in sec.5 we discuss the following: ,
.
Proposition 2.3. Let aE ism, r=(p,p), pE &,
(nra) = Pnr-2e(
(2.5)
Then
7' 71P (11pga) + 1100(1100a)) p
The proof is a calculation
nr a) = P( x) (2.6)
p-2p ja l 2
= Pnr-2e pq(xpxgalxq -
xpgq(xpalx
pq
- P( ) P(x) z
q-xgalxp + al, q- gal, P
tjl = I jln-e
,
I, j
gxpal p)
)
+ 7, xg g(xpalx -tpal,
pq
which implies (2.5). For a more general class of functions in " (2.7)
P-2D[ja
p e
p
we introduce
j,1=0,...,n .
Proposition 2.4. The class ipn0' of all functions c of the form (2.8)
o(tjl) + cd(x'U) +
with a polynomial O in too' t12 " " ' with complex coefficients,
and cdE yic_e2 , cmE Vc_e , is a subalgebra of 1Uco. For cE V(mo ' we have b=cneE Vane, and 113 jq jb=nec" , c" E yiane j or all products. Proof: The algebra property of yian0' (under the pointwise product) is trivial: Clearly l;j1E qco , hence a polynomial in l;jl also be-
longs to c0 (Note that y,c_e, + ''c_e2 is an ideal of Vco). Now let (2.9)
b = x8O(t) + bd + bm , bd E VC e' , bm E "e2
.
We introduce a multi-index-notation, writing
(2.10)
O(t) _(a,a = 'Yan_Iale rla
a
a=
'1=0tj1 J1 , YaE T
9. Particle flow and invariant algebra
288
with an (n+l)2-dimensional multi-index a = (ajl)
.
The next obser-
vation is that every application r1NJ_ rl b = b" again is of the -1 pjgj form (2.9) Indeed Ipgbd and rlpgbm have the same property than .
bd and bm, respectively. Also, calling the first term in (2.9) b0,
we get b0 E "e , hence Ip0b0 E ice, (2.11)
loon Re= µ(nµe-2e2
rl0gb0 E "e2
fljlnµe = 0 '
nµe-2e')
.
One obtains
j,1=1,...,n ,
where µ E & , by a calculation. Thus we get
(2.12) 11jlb0 = 4=0 la -vYa(('1jln(1_v)e) rla + n(l-v)e (rljlla)). for j=1=0 this term is in For j,l s 1 the first term vanishes hence will go together with bd+bm . We must focus on ;
ice, + 4-e2
(2.13)
,
n(1-v)a
_E Pq
nj11a
=
(1-v)a 4q=1a'pgla
(rljl,ylpq)
with Epq having (p,q)-component equal 1, all others zero. By prop.
2.2 (rl jl,rlpq) is a linear combination of the rlrs. Thus njlb0 is of the same form as b0 except for an additive term in vice, +yice2
.
Hence ihlpjgj b is of the form (2.9). We are left with showing that
(b,.) is a combination of the rlpq with coefficients in Pc0 The terms (bd,.)
,
.
(bm,.) are handled as those in prop.1.3,
using VIII,prop.6.2. For one of the terms of b0 we get
(2.14)
(a (1-v)e
11a,a) =
a) + n(1-v)e zpgyl
a Epgrlpga
.
The last term indeed is a combination of the gpga, coefficients E tpc0. For the first term apply (2.5), with r=(1-v)e. Q.E.D.
Define Vanes nm-eyiane, and yianm'=nm-evane' , for m--Re. Then
yianm consists precisely of all cE 1pcm with c " =rl lpjglc satisfying (2.15)
(C-
..) =
jl 1jl
.
Y jl
E
m-e
as is easily verified, using (2.5). Moreover we note that Van0 and Van0' are algebras, with the pointwise product. For Van0' this
was part of prop.2.4. For c,d E Van0 one finds that
(2.16)
(cd,.) = c(d,.) + d(c,.) = l(cOJl+dY jl)1jl
with evident notations, where we get cbil+dyjl Eilio_e for (cd)" , which gives the algebra property.
Proposition 2.5. Every continuous root
.
Similarly
of
289
9.3. Congugation with eiKt (2.17)
ON = 1
0(X) = L=09 XJ
,
8j E po 0 '
may be extended to a symbol XE yxt0 , with all Thl
Pjgj
j=0,...,N-1,
XE V= O, pro-
vided that for large IxI+ItI all roots have constant multiplicity and their mutual distance is bounded away from zero. with constant multiplicity p we can write Proof. For a root
i0'(t)/0(t)
(2xip)-1f
(2.18)
,
where E>O is fixed -say, as half the minimum distance between two roots. But we have expansions (2.19)
(6 j ,
)
_ lY jpg1lpq '
Y jpq E "-e ,j=0,1,...,N
.
Hence, with polynomials Ypq 2'jpgXJ,
(2.20)
(0(t),.) = lYpq(t)Iipq
2YPq'(tA pq
Substituting into (2.18) one obtains (2.21)
( X, ) _
jl1jl , Xjl =
(2xip)-1 f tdt (Y 3 1l0-ely jl)/02
with path of (2.18). The path is kept constant during differentiation, the integral being locally independent. For t on the path,
(2.22)
Ie(t) I = Iri(X j-t) I a EN > 0
,
since by assumption the path has distance aE from all roots X.
This allows to estimate Xj1 uniformly in x, , for large IxI+hi. kjl=O(x-e). Since 0 and 0' are Vc0 , but Yj1E IPc_e we get The formulas for Xj1 may be differentiated under the integral sign ,
for Xj1E tpC-e. Also we get expansions for rhlp jq X : Apply rj pq to J
(2.18), using estimates as above. Corollary 2.6. Let k E tVaxpe
(2.23)
.
Q.E.D.
Then we have
for all a E Vsm , m E &2
(k, a) EVsm+(µ-1)e
The proof is a consequence of (2.5) and (2.15). 3. Conjugation of OptVx with eiKt
,
K E Op1ce
With invariance of the classes Vxm x=1,s under the particle
9. Particle flow and invariant algebra
290
flow of kE'W%ne or kE Vane established we now prove the invariance of OpPxm under conjugation with eiKt . The results will be similar in form and principle to VI,thm.5.1. Existence of the group {eiKt} of operators S-S (or or with the spaces of 111,3) is insured by VI,thm.3.1. It would be easy to carry the results to the case of a solution operator U(t,t) of u'+Ku=O, with K=K(t). In thm.3.1 we work with k0E 1 e, focusing on a simple nontrivial case. Slightly weaker conditions on k will be needed in sec.4, to prove coordinate invariance (cf. cor.3.4). A result for systems, requiring the classes Wne is discussed in sec.5. Theorem 3.1. Let K=k(x,D), where k is of the form k=ko+x, with a
complex-valued xE0 , and a real-valued koE
a
,Then,
for every A=a(x,D)E '1m , me &2 ,we have Ate-iKtAeiKt E Op1lm tE I. Moreover, we get Ata(t,x,D) with z E 11-e, n yl-e2
(3.1)
Also, At(-= C00(it,Op1xm)), and atAt=(-i ad K)JA .
Proof. We retrace the proof of VI,thm.5.1, taking special care of the remainders, due to less perfect pdo-calculus of VIII,7. Let P(t) = eiKtat(x,D)e iKt
(3.2)
where aE Vxm, x=1,s, ata,V t At=at(x,D). Here we use thm.1.2 (or thm.2.1) to assure that a E yurm and at(x,D) is meaningful. Note t that this is the only direct reference to x=s or x=1 . We get
dP(t)/dt = eiKt(i[K,At] - (k,at) (x,D))e iKt
(3.3)
,
where the operators are S -> S . By VI,thm.3.1 at(eiKt) exists in L(Hs,Hs-e) and by sec.1 (or sec.2) exists in yucm ,
atat-(k0,at)
Hence dP/dt exists in L(H 'H ), for all s, and (3.3) is valid, S s-m-e using III,thm.3.1. Now we use our calculus of ,do's. VIII,prop.7.4 yields
(3.4)
[K,A]=cA(x,D)
,
where (k,a)E 1)xm+e-re'-r'e2' r+r'=j, and the remainder RN has symbol in Vxm+e-re' -r'e2 ' r+r'= N+1, if the first N terms are used.
It is important that, for our present k, we even get (k,a).E
Vxm-re' -r' e2 ' r+r' = j-1 , hence RNE Op Xm-rei -r' e2 1 r+r' =N . For x=1,
j=1 get
1=-(Egpa) =-cgpaE yrxm, while (kd,a) , (km,a) belong
and Vxm+e2 -re' -r' e2 l resp. With r=1 (r=0) get all terms E Vsm, as stated. Similarly for j=1, x=s. For j>l the
to Xm+e' -re' -r' e2 '
291
9.3. Congugation with eiKt first sum (1.15) gives 0, since Epglxx epgl
-0. The
-qpglxx-q pgI
other terms in (1.15) go as for j=1. To summarize:
Proposition 3.2. Under the assumptions of thm.3.1 we have
cA
(3.5)
3
4=1(-i)7/j! (k,a)j+rN
,
r 3E N
m-re'_r,e2
where the j-th term belongs to Vx
,r+r'=N,
m-re,_r,e2, r+r'=j-1.
Applying prop.3.2 for N=1 we get
(3.6)
nvxm_e2
(k,at)
icA At
Note that (3.6)implies )(x))-1/2).In fact, we get yE 'ptm_e/2. Moreover, we know from sec.1 or 2, and the calculus of ydo's between y'ce and ix that y is continuous as a map
Him-e' (or 4w m_e2 ). Actually, we even get it COO: One knows from sec.1 or 2 that at is C" , and the asymptotic expansion (3.4) (or Leibniz-Taylor-expansion (3.5)) may be differentiated arbitrarily for t. Then prop.3.2 may be applied to the differentiated expansion. Hence we even get y E C00(I,Vt tdie
P(t) = A +
(3.7)
J0
m-e/2
). Again this gives
iKty(t,x,D)e-iKt
Or,
e-iKtAeiKt
(3.8)
= at(x,D) -
ftdieiK(t-t)Y(t,x,D)eiK(t-t)
0
This is a decomposition like (3.1), but the reminder (second term) is not in iyido-form. It is norm continuous in L(HS'Hs-m-f)' f d't(e_iKtAeiKt)
=e', e2, e/2. Also,
exists in L(HsHs-m), by (3.8).
(3.8) may be iterated: Noting y satisfies (3.6), and even yE C00(&,Vxm_f), f=e', e2
we get, setting
,
(t,x,D)- f dx'eiK(x' x)W(x,,t)eiK(x x
(3.9)
0
with a ydo W = w(x',t,x,D) obtained by substituting y for a into (3.7) . Clearly it follows that wE C00(12,1Vxm-re' -r'e2) , r+r'=2 using (3.6) . Also, yx E C00(1e2,'wxm_f), f=e' , e2 . With (3.9), for x=t-t, in the integrand of (3.8) we get (3.10)
e-iKtAeiKt = at(x,D) + zit(x,D) + W1t '
where Wit E L(HslHs_m+re'+r'e2)
C0D(H,1xm-f) , (3.11)
a iKtAeiKt
e2
,
while the symbol 1;1t is in
. An N-fold iteration then yields
f=e'
= at(x,D) + 4=1zjt(x,D) + WN
,
9. Particle flow and invariant algebra
292
with symbols zjE C"(&,"m-re,-r , e2), r+r'=j
,
and with a remainder
in O(m-re'-r'e2) , r+r'=N+1 . Moreover, the remainder will have all derivatives for t of order up to N+1 existing in norm convergence of every L(Hs'Hs-re'-r'e2)' since it is an N+1-fold iterated integral over a bounded, strongly continuous integrand.
Finally we take the asymptotic sum (3.12)
zt = Lj=lzjt (mod Vt-00)
It exists, because tjtE yet-je/2 follows as above. Then e-iKtAeiKt - at(x,D)-zt(x,D)=Vt clearly is of order --. Its t-derivatives of
all orders exist and are of order -- again. Thus Vt vt(x,D), vtE (1Q,S(& )). Also all t-derivatives of the asymptotic sum z we get a -iKtt AeiKt exist in 4wm-f, f=e', e2. Writing zt+vt =at(x,D)+z(t,x,D), which amounts to (3.1). In particular, we have
Cl2n
verified all the properties stated for z in thm.3.1. With differentiability established we clearly get atAt=
-i[ K,At] =-i(ad K)At, where [ K,At] E Opyurm again. This may be iterated, to complete the proof of thm.4.1. It will be useful for sec.4 to reexamine the proof of thm.4.1 for a possible generalization to the case where the first sum of (1.15) (or (2.1)) is replaced by a linear combination of terms
(x)a(s())T;q , with
s(x)=(X)
closed unit ball B?C 1n
(3.13)
,
k= (x) q
,
and a function cE C" (B?), with the
as in IV. In other words, let
+km+kd, kmE '°e2 , kdE yice1 , aqE C"(Bi),
Let V%Ke be the class of such functions. Clearly yueC V*eC yip,' e' Proposition 3.3. We have V%xeC 'Xne. In fact, not only (epgk,.), but also (k,.) is a combination of the ejl, with [c0-coefficients.
Proof. It suffices to focus on the case
A calculation shows that epq:w7,,Key Vx1ce
.
Thus we only must show
that (c,q.) =2' pq a pq . But (al, . ) =aa -26a =aa xq-ja lxle ql xq q lxl q1 +Y,alx xlax . Here alx E pco, so look at sax -yalx xlax =(x)-laax 1 q 1 q 1 q q
+xl(sla(s)-((x)a(s))Ix
)axq.
Finally, ((x)a(s(x)))lx1=(sl(y)M (s()
1
+OAlxm(s(x))(slm slsm(x)), so that 2lxl(sla(s)-((x)a(s))lx1)=
- m,l xlalxm(s(x))(6lm °lsm)=(x)2,slall(s(x))
.
Q.E.D.
293
9.4. Coordinate and gauge invariance
Looking at prop.3.3 and the proof of thm.3.1: In the case x= C00
1 we clearly get at well defined again, and
(&,1V1m)
E E C
.
Also,
(k,at)=r pgspgatE lP1m again, if we weaken the assumptions, requi-
ring
koE.tVXxe only. Formula (3.6) remains true as well: We need
(k,a)2=
again. First
for
jl
observe that alx j=sja(s)+sjalj(s)-sjTalm(s)sm is a sum of function
a(s(x). Thus we may look at Ta(s(x))lx1 gaxla
+Y,O(s(x))lxixlaxga 1Vc-e2
J
j
=20(s(x))1xlegla
j
. We get a(s(x))lxl=yalm(s)(slm slsm)/(x)E
, hence the first term is in Vlm-e, n vim-e:
For the second
term we get p1°lm(s)(blm slsm)=omalm(s)/(x)2 , Using VIII,(6.12) we get the proper estimate for (3.6) for the second term as well. We have proven: Corollary 3.4. In the case x=1 the statement of thm.3.1 is still
valid, if "ko E iV%o " is replaced by "ko E iVXxo "
.
Indeed, the iteration and asymptotic sum leading to (3.1) no
longer depend on "ko E 1VXo ". 4. Coordinate and gauge invariance; extension to S-manifolds.
We now will look into coordinate invariance of the two alge-
bras 1I.L and Y14S
and more generally, Op1V1, Opys. Given a global
inverted by 1V: transform x=q(y) with a diffeomorphism sp:&n- In &n- In :We have sp(x(x))=x(W(x))=x, xE &n. Assume (the components of) p(x) ((a(P,/axl(x))) _ x(x) are Coo(&n). The Jacobian matrix ,
,
J
J(x) is nonsingular, together with ((axj/axl(x)))=J1(T(x)), xE &n Consider the substitution operators TT, TX defined by (4.1)
(Te)(x) = u((p(x)) = (uo(p)(x)
for functions u over In Assume 0 H If a folpde .
(4.2)
L =
Yaj(x)axJ + a0(x)
,
,
(Txu)(x) = u(X(x))
s ldet J(x)Is c:
,
, with con-
aj E C00(In) ,j=0,1,...,n
9. Particle flow and invariant algebra
294
is given, then we get a transformed folpde L" = TgLTx
(4.3)
L=
j
-
+a"0(x)
j(x)ax
. We get
a"0 = T,,a0
,
Y,(axj/ail)((p(x))al(T(x))
a j(x) =
The exponentiations eLt and eL t are related by = TtPeLtTX
eL" t
(4.4)
since the PDE atu=Lu transforms into atv=L"v , with To prove coordinate invariance of iIGL or t1GS we will use thm.3.1 or cor.3.4, with the ,do K=k(x,D)=-iM , M as L- in (4.3), p replaced by x=q 1, with L=ixq xgax +2Spq, in case ofWL. ax P
For WS we use L=ixq
ax p
,
x
-xgax
a
p x
p
q
and Ex3 +Z . We must
,
p
J verify that the transformed expressions satisfy the assumptions. as outlined in sec.0. We consider S-admissible transforms, as in
IV,3, only: With s(x)=5, s:I -B1, of IV,(1.5), assume that
(4.5)
tP = s-lo'ylos
x=c
with diffeomorphisms p:B1 u.Bl
,
1
s-lo eos
=
O:B1 a B1
,
inverting each other.
Under this general assumption, if it can be shown that T(P LIGXTX C iIGX
,
for %=L or X=S ,
it follows at once that also TTOp7UxmTx C
Opyurm
because we already know from IV,thm.3.3 that T(,OppcmTx C
OpVcm
For AE Op xm we have B=AII EE Optpx0 ='WX. Then TWBTXE tIGX
-M follows. Also, T BT =(T AT )(T P_ T ), where the second factor X belongs to Oppc mC OpVx_ Also T P T is inverted by T P T E M x OpVxm. Hence T(P ATx = (TgBTx)(TTPMTx)E For thm.4.1 below we must focus on the expressions M=Mpq (4.6)
MOq ixq(x) . Mp0= Irojlx (x(x))ax J
p
p,q=1.... n
Mpq 2xq(X)Tjlxp(x(x))axj + ZOpq in case of x=L, and MOq , Mp0
,
I
j
Mpq Mqp, I:sp=qsn, for X=S.
Here we first apply IV,prop.2.1: With our assumption (4.5) on 9, we have TjIx os-IE C"(B?) hence Tjlx (X(x)) _ and x=s-IoOos P p with p o(s_0l Oos)(x) =(((Pjlx os- 1 )o6)os(x) = (s(x)) a. =TjIx P ,
,
,
295
9.4. Coordinate and gauge invariance
ajp=(cPjjx os-1)oOE C°'(Bcl) p
On the other hand, (x)-1cp(x)= yr(s(x)){(x) 1- V(s(x)) 2}-1
,
}1/2
I-S, E C"(B?) where the second factor may be written as 1, -y(s) 1 just as in IV,(2.8),(2.9). This shows that Mpq ikpq(x,D)+x0, where 2
,
(4.7)
k0g
(x)a0gos(x),
kpg
are symbols of tVXxe of sec.3, so that cor.3.4 applies. We proved
Theorem 4.1. Let the diffeomorphism q:&n -in and its inverse function x:&n -In be of the form (4.5) with s(x)=`xl, and diffeo-
morphisms y,:Bl-.Bl, O:Bl -Bl, yp°8=id. Then the transforms T and Tx (4.1) leave i1GL and every Opyrlm, mE &2, invariant. That is,
(4.8)
TTi1GLTx=TxtIGLTgi1GL , TT OptUlmTx=Tx0ptVlmTgOpyilm , mE &2
Theorem 4.2. Assume that the diffeomorphisms T,x satisfy p(x)=gx+w(x) , x(x)=g 1(x)+v(x)
(4.9)
, gE GL (&n)
, w,vE
"o
Then (4.8) is valid - i.e., TT and T leave OpyJlm invariant. MoreX over, if in addition we have goo , 0
L= e iµ(x)Leiµ(x)
(4.10)
=
p(x)µlxp(x)
L + i
P
Applying our arguments to the generators ixq, ax , xgax + 00
26pgLpq again we now must prove eMtAE-MtE C
where M=MpgLpq+Rpq , tOq 0
, tLp0=iR I
xp ,
P p &,L(A), for AE t1Gx,
µpq
I xp.
IAIX
Theorem 4.3. Let the 'gauge function' µ(x) be real-valued and let
a (x) E yice2 (4.11)
. Then Op1Jixm, mE &2 ,
X=S ,I, are invariant under
A -* e-'R(x)AelR(x)
.
The proof follows from thm.3.1, for m=0. For general m we
still must show that a i L( D) se' E Vcs , sE I . We leave this to the reader; the proof is not difficult. Finally, after discussing coordinate invariance, it is clear that the class OpVlm of natural yxio's may be considered on a mani-
fold with conical ends, just as earlier Op*m'.,s (or OpVcm).
9. Particle flow and invariant algebra
296
geneRepeating the discussions of IV,4 with W1m instead of rates classes called LLm of 'natural pdo's' on an S-manifold 11 On the other hand, we do not expect invariance of OpWsm under general S-admissible transforms (4.5), although, evidently, thm.4.2 implies invariance under local transforms extended as a multiple of the identity outside a compact set. Our use of Opps in Dirac theory might profit from introduct.
ion of Op1s(sz) for an n coinciding with &n outside K E 11
.
Problem: Give a definition of Opps(11), for a as above, using a par
tition 1=Exj with x'=1 near -, and also, by smoothness properties. 5. Conjugation with eiKt, for a matrix-valued K=k(x,D)
.
In this section we take up the question of extending thm.3.1 to the case of a vxv-matrix K = k(x,D)=((kj1(x,D)))jl1=1,...,v of of pdo's Kj1 = kj1(x,D) . We shall give preference to the class from now on, but note that most arguments will also work for 'Pl. Generally we assume kj1E Vae (cf. sec.2). Instead of requiring yes
,
atu -ik(x,D)u = 0 , u=q at t=0
(5.1)
,
to be strictly hyperbolic of type a°, as in VI,4, we will accomodate the Dirac equation by allowing multiple eigenvalues, but of
constant multiplicity for
similarly as in [CD] With comparison operators A=nea(x,D), a1=e,e',e2, as in VI,4, we call equation (5.1) semi-strictly hyperbolic (of type e°) if
(i) We have
.
kea=neako, KE Wcea-e
,
with entries of the vxv-matrix kea of the form (2.1):
(5.2)kea jl-
,q=oKpg11P +kdjl+kmjl,
KpjE T' kdjlE tee' ' kmjlE
e2
where all but kdjl or kmjl vanishes in case of a°=e', e2, resp. (ii) The vxv-matrix
is diagonalizable for all fzIxl+
sufficiently large q0. All eigenvalues Xj f1 I Zrl0, of kea are real, and of constant multiplicity vj (independent of x,i;), as f2 JxJ+fi ICI arl0 Well known perturbation arguments imply that the X j are Co functions of x and The matrices kea and k0=kean-ea have the same spectral projections, and the eigenvalues of k0 are given .
by lAj = kjn-ea . We may arrange for
(5.3)
vl+...vP=v, f2 lxI+fi As our third condition, we assume that
ZtIO
297
9.5. Conjugation for a matrix-valued K 260
(iii)
,
f2JxI+f1Ifl 2110, j=l,...,p-l
with some positive constant 60 independent of x, . We only will admit the weight functions ne, mentioned, although others, such as nel+ne2 might be useful as well. Mainly e°=e' will be of interest, for the Dirac equation. Theorem 5.1. If equation (5.1) is semi-strictly hyperbolic of type e° then the solution operator eiKt of (5.1) exists as an operator in 0(0). Moreover, the partial derivatives (5.4) exist in norm topology of L(Hs,Hs-je,). For every sE &2, and j=0,1,2,..., we get
at(eiKt) = KjeiKt E 0(jf)
(5.4)
(5.5)
r_e(x,) , r_eE 1Va_e
using the spectral projections to of f2 xl+f1 22rI0 , extended to symbols (with coeffivients) in Vco. The discussion follows the proof of VI,thm.4.4, with small amendments, left to the reader. The point is formula VI,(4.19), i.e., I
(5.6)
1x1+1 V >110,
describing the spectral projections in >qo . Then the Pi must be properly extended to H2n and r_e must be chosen to satisfy the cdn's of VI,4. ,
Proposition 5.2. The eigenvalues Xe, , f2 IxI+fk Ill>2io , of ke, extend ty symbols X jE lPone (cf. sec.2). Proof. The eigenvalues µj solve the algebraic equation (5.7)
"J=0
61(x,U)ptj = 0
where the coefficients Aj are polynomials in the coefficients of
k0
.
If e° #e then we just get µjE Vc0, Xj=µjne, E Vce, , by an argu-
ment as for VI,prop.4.3. For e°=e apply prop.2.4 for 01E 1on0', hence also µjE Van0', by prop.2.5. Note that the assumptions hold, by (i)-(iii) above. Thus we get Xj=neµjE vane, as stated. Q.E.D. The following example shows that the class Oppsm of all vxv-
matrices A=((A )) of ,do's A. E Opts is not invariant under the m conjugation A-*Ate-iKtAeiKt . Let (5.8)
f
-()J
'
(0
0)
9. Particle flow and invariant algebra
298
Then clearly k E °e , and a E
(5.9)
e-1KAe1K =
But a calculation shows that
\
(0
0)
,
with p = -21(D)
21(x) E Vt0 but p 0 Vs0 Clearly P = p(D) p( ) = e On the other hand, as a simple consequence of thm.3.1 we get -iKt iKt e are diagonal matrices for As E OpVsm if and we can large IxI+IgI. If only a and k commute for large IxI+l ,
I
expect (3.8) with an error term Zt (the integral at right) of lower order, not necessarily a y,do (cf.also [T13]). Thm.5.4, below, shows that even the error Zt becomes a 4do
E Op sm-e° ' e° =e` , e2 ,
if only a suitable 'correction' of lower
order is added to the symbol a(x,i;) commuting with k0(x,t) (that
is, if we let a=q+z, with [k0,q]=0, and suitable lower order z
).
From now on we always assume that (5.1) is semi-strictly hyperbolic of type a°. We shall see that a subalgebra P=PKC Opsv can be defined, essentially by the property that its Ado's remain 4c1o's, of the same symbol type, when conjugated with eiKt For mE H2 we define f PKm as the class of all A=a(x,D) E OpVsm such that At of (5.10) belongs to Opysm, and, moreover that
At = e iKtAeiKt E C-(&,OpjsV)
(5.10)e
For e° =e' (5.10)e°
,
,
in case of a°=e.
or e° -e2 we replace (5.10) a by atAt E C (H,OP1S jj = 0,1.....
Then we define, for a°= e (5.11)
,
e'
P = PK =
U
, or e2
.
,
mEV PKK
Proposition 5.3. The class P is a graded algebra invariant under ad iK (as in VIII,(8.4)), i.e., (ad iK)A=at(eiKtAe-iKt)It=o
)
Especially, if AE Pm, BE Pm , , then AB E Pm+m' Moreover, if AE Pm,
then (ad K)A E Pm-(e-e ,)' Moreover P0, with the Frechet topology
of VIII,6, is an (adjoint invariant) i*-subalgebra of L(H).
Proof evident. Clearly B=(ad iK)A exists for AE Pm, and Bt =e iKtBeiKt=_At E C00(H,Ovm-(j+1)(e-e°) Theorem 5.4. Let (5.1) be semi-strictly hyperbolic of type a°. (1) For each vxv-matrix-valued symbol q with
(5.12)
q E iism
, [k0(x,),q(x, )] = 0
.
1x1+1 U Z iii
there exists a symbol zE 1ysm_e such that A = a(x,D)
, with a=q+z
9.5. Conjugation for a matrix-valued K
299
is an operator in Pm . That is, in particular,
At = e-itK AsitK E OpVsm
t E & , At = at(x,D)
,
(5.13) atat E C'(L,'V)S _7(e-f))
j=0,1,2..... (2) Vice versa, if A = a(x,D) E Pm is given .
,
,
then there
exists a decomposition (valid for all t E & ) (5.14)
at qt+zt
ztEVSm_e
)]=0
,
,
2yl0
Moreover, the decomposition (5.14) may be differentiated for t for the corresponding decomposition atat atgt+atzt of the symbol v aja of ((ad. )iA) In particular, ajz Vs m-j(e-f)-e t t iK t t t (3) If, for any q with (5.12), the symbols z1, z2 both satinstead of m . isfy (1) then z1-z2 satisfies (5.13) for m-e We shall lay out the proof of thm.5.4 in the remainder of ,
.
,
this section, and will finish it in sec.7, after discussing some auxiliary results on a commutator-differential equation in sec.6. (x,t) will be imporThe particle flows of the eigenvalues .
tant.
Even in the strictly hyperbolic case, when all eigenvalues
are distinct, we will have to solve a succesive sequence of matrix commutator equations, for careful alignment of the correction z. Moreover, for multiple eigenvalues, not only will the symbol flow along these flows, but, in addition, there will be a 'similarity action' within the eigenspaces of k(x,T;). For details see sec.6. For a different interpretation for the Dirac equation see ch.10. Discussion of (3). Prop.5.3 implies that z3(x,D)=(z,-zz)(x,D)E E Pm-e . Thus, indeed, z3 allows a decomposition (5.13) for m-e. Discussion of (2). First let a°=e. For an AE Pm we get At at(x,D), atE C"(2,ivsv)
.
Then again (5.1) yields At +i[K,At] = 0
(5.15)
,
t E & , AO = A .
(We first get (5.15), applied to uE S , due to differentiability of a*iKt in L(H H s-e) (thm.5.1). Then, since all operators invols ved are ydo's, we get (5.15) as an equation for Ado's.) Using calculus of ,do's we may translate (5.15) into symbols :
(5.16)
0 = at + i[k,at] + (k,at) -i/2(k,at)2 + ...(mod S)
Here the Poisson brackets are formed with matrix multiplication : (5.17) (a,b)=alb
bix-bl
alx , (a,b)j=
Ie=j8'(a(e)b(e)-b(e)a(0))
9. Particle flow and invariant algebra
300
The commutator [k,at] normally does not vanish. However, the discussion of VIII,prop.7.4, carries over with the same proofs,
except there will be an extra term [k,a] in the third formula
VIII, (7.14) . Moreover, as in sec. 3 , we have (k, at) jE "m-rei -r' eZ ' r+r'=j-1, under our assumptions on k and at. Thus (5.16) implies
[kf,at] = rt E 1Vsm
(5.18)
.
We regard (5.18) as a commutator equation for the unknown matrix function at . Here we know that (5.18) admits a solution. With Xi and pj as constructed, we get (5.19)
at = L4=lp.3
+
atp.3
(Just note that at Ppjatpl
j7lpjrtpl/(kj-kl)
= qt + zt
and that pj[k0,at]Pl =(1j-X1)pjatpl
,
leading to pjatpl and the solvability as j#l = pjrtpl/(Xj-X1) condition (5.20) (which here holds automatically).) ,
,
(5.20)
pjrtpj = 0
j=l,...,p
,
.
Notice that (5.19), with qt and zt equal to the first and second sum, respectively, gives a decomposition (5.14), as desired, using
that p jE 1Vc0
,
(X j-T,)-lE 1Vc-e
,
(possibly after a correction for
Xj are small f:IxI+f,ll;I), using (iii) above. Moreover, since pj independent of t , we may differentiate (5.19) for a decomposition of the symbol of Bit using Bj=(ad iK)1AE Op1Vsm-j(e-e,) ,
,
(5.10). This proves (2) for e°=e.
Next consider e° =e' , noting that the case e° =e2 handles reversed. Again get (5.16), where now we have kf=kdE 1Vice, , however. Also, atE 1Vs -e: , by (5.10)e' Hence,
similarly, with x and (5.21)
(k,at) j E 1Vsm+e'
-re' -r'e2
r,r'2-0
,
r+r'=j
The proof runs exactly parallel to the proof of prop.3.2. (One gains the slight advantage of an improved multiplication order because the term km is missing.)
Now, assuming that k=ke, +K
, xE 1'e' as required by (i) of
our e'-semi-strict hyperbolicity, we get (5.18), with rtE
But we now also have Xi E 1Vice,
,
instead of Xi E 1Vice
.
Therefore
the second term at right in (5.9) still is a symbol in 1Vsm-e again, and (2) follows for f=e3 as well.
Discussion of (1). First assume a°=e. For a q satisfying (5.12),
9.6. Commutator equations
301
if z E yism-e exists with A=q(x,D)+z(x,D)E Pm, then get (5.16) for
for at qt+zt. Separating symbols in (5.16) of order m from those of lower order (with (5.12), and cor.10.5), get
(5.22)
i[ kf , zt] + qt+ (kf, gt) + i[ K , gt] = 0 (mod 'Wsm-e, fl yisvm-e2
(In particular we know from (2), already proven, that ztE Vsm-e Also, the only additional term of (5.16) to be taken into (5.22) for a relation mod Vsm-e will be -i/2(k,q )2. We will not use this
term, at the expense of a weaker (5.21), resulting in slower asymptotic convergence of the series for at, similar as in thm.3.1.) Vice versa, we start the construction of a symbol z with the attempt to solve (5.22) exactly, not mod lower order, assuming qt E 4wm given. Rewrite (5.22) as commutator equation for zt
[kf,zt] = -[K,qt] + i(qt + (kf,gt)) = cpt
(5.23) Assuming that (5.24)
(Pt
is known, for a moment, (5.23) implies
zt = Llplztpl + i j7, pj (qt+(kfgt)+i[K,gt])pl/(),j-kl)
;dl
This solution is valid only under the condition (5.20), i.e., (5.25)
pl(gt + (kf,gt) + i[K,gt])pl = 0
,
1 = 1,...,v'
.
In sec.6 we will investigate commutator equations of the form (5.18), with solvability condition (5.20) as in (5.25). In sec.7 construction of zt and the proof of thm.5.4 will be completed. 6. A technical discussion of commutator equations.
In this section we consider an equation of the form
[b,w] = d
(6.1)
,
for general vxv-square matrix valued symbols where b,d are given and w is to be found. Assume that b, d are C" for R C &2n, and (t,x,t;)E 2xcz Let the matrix b be diagonalizable, have real eigenvalues, and eigenspaces of dimension independent of x, . As used before (sec.5) the eigenvalues A, may be arranged as C"(c)-functions: a.. is a simple root of 3 q(p-1)(A,)=0, with q(A)=det and the multiplicity p of X. Hence aq(p-1)(Ai)#0. The implicit function theorem yields )vj locally. The real distinct eigenvalues may be ordered by size, giving globally defined C"(a)-functions. For X=Xj E C°D
,
,
9. Particle flow and invariant algebra
302 (6.2)
(b(x,t) -
p = i/2nJ
µ)-1
dµ E C"(n)
It-?(x, )1=e
with a=e(x,t) > 0 sufficiently small, is the eigenprojection. For solvability of (6.1) it is necessary and sufficient that p(x,t)d(t,x,t)p(x,t)=0, x,t E a , for each eigenprojection p of b. (Indeed, if (6.1) is solvable, then pdp=[pbp,pwp]=[Xp,pwp]=0. Vice
versa, if the latter holds for all pj=p, then (6.1) is solved by (5.19) with Xj , pj j=1,...,p, the distinct eigenvalues an pro,
jections of w (and at replaced by w). The first sum in (5.19) is an arbitrary matrix commuting with b. All solutions of (6.1) are given by (5.8).) We plan to solve (6.1) for d of the form (6.3), with given (t-depedent) symbols g, d^, and d
Assume g=g(t,x,t) commutes
with b(x,t), for all (t,x,g) E lxtz, while d^ (t,x,l), d" (t,x,t) are C"-matrix functions, and (.,.)=(.,.)1 is the Poisson bracket.
d = g + (b,g) + [d^ ,g] + d"
(6.3) For given X
,
,
g = atg
.
be
p let T1(x,i;),...,q) p(x,l;)
a bi-orthogonal pair of bases of the eigenspace S=S(x,l)=im p(x,l;)
(such that 1cp1=6 ill for all (x,t)E t , and that also V j,VE C00(c) )
b"=((% 1 )) of Consider the local matrices Y=((Yjl)), S^=((6jl)), g, pd^p, pd-p, (all leaving S invariant): In detail, y-y(t,x,l;),
6^ (t,x,i) (6.4)
,
S" (t,x,l) are defined by
pgp-
Yj lp j 1
j,l with Pjku = cpj('y u)
, Pd"
P=
, u E CV , Vk =
6j1Pjl ,
Pd P
sj1
for all (x,t) E tz
Proposition 6.1. The condition pdp=0 with d of (6.3), and [b,g]=0 for all (t,x,t) E 2xf , translates into a differential equation for the restriction g0(t,x,t) = g(t,x,i;)IS(x,l;) of g onto the
eigenspace S of
(6.5)
.
, of the form
p(gp)'PIS+[v,g0]
+d"0=0
,
d"0 -pdpIS
,
where
of = at + %Itax - xIxat , v = (pd^P + P(PItbIx)P)IS Moreover, with the matrices y , 6A, 6" of Y 0 , d^ 0=pd^ p I S , 6-0 (6.6)
with
n
,
9.6. Commutator equations
(6.8)
a=((e jl))
,
O jl=-Vi'(p1
The restrictions g0 of S
.
,
303
l
+
d^0
d"0 ,
, 11111=d/ds.
clearly are linear operators
Prop.6.1 implies that, under above assumptions, cdn. pdp=O
involves only g0, not the restrictions of g to the other eigenspaces, hence a 'decoupled' system of p2 DE,s results for each eigenspace of b, where p=dim This is a system of PDE with same
principal part, translating into a first order system of ODE along the flow of the common principal part, in the sense of sec.1. Let us first focus on (6.5). In this discussion we again consider the full set of eigenvalues ?1,...,kp of multiplicities pj, with projections p=pj and gj=gpj=pjg . From pkgr 0 , k#r, we get ,
(6.9)
0
PkgrPk = -PkgrPk = 0 ' PkgrlxPk =
, k#r .
We have g = 29k ' b = DrPr , hence
Pk (6.10)
Pk
= T 1 + T2 '
where (6.11)
T 1 = Pk( -M g) pk =
using (6.9). Also, as j#ky6r#j (6.12)
,
Pk using gjlxPk
= -gjPklx ..... 0
Pk(Pr,gj)Pk =
.
Similarly, as j # k = r , using pkpkl = (6.13)
Pk(Pk'gj)Pk =
Pkl gj(1-Pk)Pklx =0.
On the other hand, for j = k # r , and j=k=r ,
Pk(Pr'gk)Pk =
(6.14)
Pk(Pk'gk)Pk = Pkl(1-Pk)gklxPk - Pkgkl(i-Pk)Pklx
= PkIpkIxgk - gkpklPkIx = LPkpklPklxpk'gk] Accordingly, using (6.9) again, (6.15)
T2 = Dr 1PkPkIPrlxPk'gk1 = LPkPklblxPk'gk]'
9. Particle flow and invariant algebra
304
Then (6.10), (6.11), (6.15), and (6.9) imply that indeed (6.5) is equivalent to pdp=0 with d of (6.3) , and [b,g] = 0. Next a calculation, not given in details, will show that the "covariant derivative" Pkgkpk of gk in the space Sk has the matrix
(6.16)
Yk' + [01,Yk}
,
_-
01 = ((,*(Pr))
with respect to any biorthogonal base {(Pj}
is the matrix of gk . Accordingly, (6.5)
,
(('U'j*(Pr))
{ipj} of Sk, where Yk
is equivalent to (6.7),
with 0 defined by (6.8) q.e.d. Clearly the system (6.7) of pk ODE's has a unique local solution satisfying y=y0, at t=0, defined near each for small N2n t. Let it be assumed that a = ,but that only a local base pair ,
yj , V, exists, in subsets St1C 12n
,
covering
12n.
Then solutions
in overlapping sets S2 j, for different bases, remain compatible as
long as they are jointly defined. From the well known properties of ODE's we conclude existence of a unique solution g =Y_g j of our
matrix commutator problem, assuming a Note that d/ds just is the derivative (1.3) of the function X(x,l;) (instead defined for all t if we assume that X prop.2.5 ). We then get existence and
at t=0. given value along the Hamiltonian flow of k0 ).This flow is well E y)Qe,
for example (of. uniqueness of ,
for all (t,x,l;) E I2n+1
Theorem 6.2. Let the assumptions of prop.6.1 hold for f21xI+f-II
>rjo , and let bE yiae,
,
d^ E C'(1t,
-e), d- E C"(1,Vsm+e° -e)' For
e°=e assume in addition that the coefficients of b are E Wne. For 00 e°e even atd^E CO(&, c(1+1)(e°-e))' atdoE C00(&'y'sm+(1+1)(e°-e))
Then, for every g E tpsm a unique
gtE C (Ii sm) exists such that for all p=pj of b we have (for large
(6.17)
[b,g]=0
,
p(g +(b,g)+[d^ ,g]+d" )p =0 , tE & , g=g0 at t=0.
We also have atgt E Proof.(Assume
1 = 0,1,2.....
.
large,where needed.) We know existence of a unique solution g of (6.17).In order to show that g has the stated symbol properties we note that (6.5) and gj = pjgjpj imply (6.18)
gj + [vj,gj] = pjgj+gjpj-pjd pj ,
This may be interpreted as another set of ODE's for the v2 coefficients of gj. There is a unique solution of (6.18) under the init-
ial conditions of (6.17), defined in2n+1 N. This must be the same
305
9.7. Proof of theorem 5.4 gj , also solving (6.5).
C'(l.'iVcf_e)
We have p;=(? j'Pj) E 'cf-e'
Therefore (6.18) may be interpreted as a vzxvz-system of first order ODE's for gj" = govt of the form (6.19)
dgj"/ds +
cj(s)
s E R , gj
,
90Pj
, at s=0,
g." being interpreted as a v2-component vector with entries wile cI-E C00 (&,'1cv2e,_e) and the v2-vector cj has entries in C (R,1Vsm+eA-e)' By our remarks a global solution of (6.19) exists for all t. Also, prop.2.5 was used, for the above. It is a matter of deriving suitable apriori estimates to show that gj"E C"(R,Psm) hence also gjE C"(R,ism) etc., proving the theorem. ,
For example, let us introduce the norm (6.20)
Ia12 =
E Iajr(x, )I2(1+IxI2) j,r
From (6.19) one obtains the differential inequality (6.21)
1m2
Idlgj
/d81 5 i1TIgj"Im
Isi s T
with a constant riT=sup
with a matrix norm Then (6.21) may be integrated, using the initial conditions, for
IgJ Im S ril(e)
(6.22)
with locally bounded ril(s)
.
,
sER,
Then (6.19) implies
k
(6.23)
Iatg Im-e+e°s T12(s)
One may differentiate (6.19) for t to derive estimates for dgj-/dt recursively. Also apply (a finite number of) ijl of (2.3) to (6.18) to obtain a similar system of ODE's for r elri with 1 coefficcients in ps0
,
Jp p following the proof of thm.1.2. This will
give estimates for all expressions Id/dsI=lri
1 g-I2m, 7pp 7
completing
the proof of thm.6.2. Details are left to the reader. 7. Completion of the proof of theorem 5.4. In all discussions of sec.7 a restriction with sufficiently large rio, is assumed wherever needed.
Returning to the proof of thm.5.4,(1) , we observe that (5.24), together with [kf,gt]=0, i.e., (5.14) are of the form of
9. Particle flow and invariant algebra
306
(6.17), so that thm.6.2 applies (presently with a°=e). Thus qt is determined by its initial value qo, by property [q,ke]=0 and cdn. (5.20) (i.e., (5.24) for (5.22)). By thm.6.2 such symbol exists, as solution of (5.22), belongs to C00(L psm). We also get
i.e., in the form (5.23). First set plztpl=0. (5.23) implies ztE C°O(&,yrs _e) , similarly as in the proof of thm.5.4 (2). Note however,that at qt+zt does not satisfy (5.16) precisely, but only mod j=1,2. To improve our choice of zt we will set up a recursion, for a sequence of improvement symbols. An asymptotic sum will give the total correction symbol. First let (7.1)
Wt = -e-iKtq(x,D)eiKt + gt(x,D) + zt(x,D)
and note that Wt + i[K,Wt] = Rt = rt(x,D) , tE 2 , WO = ZO = z0(x,D) (7.2)
rt=zt+i[ x , zt] +(k, zt) -i/2(k, zt) 2+...+(K, qt) -i/2(k, qt) 2+... . Note that
In particular, we have used (6.22) and [kf,gt] = 0
rt E
J=1,2
,
and z0 E 'Wsm_e
We now may solve the linear inhomogeneous differential equad/dt(eiKtWte-iKt)=eiKtRte iKt tion (7.2): It follows that hence, ,
eiKtW a-iKt = -Q +eiKt
t
(S2t + Zt)e-iKt = Z0
t iKt Oe Rte -iKi
di
which implies
aiKt(Q+ZO)eiKt
(7.3)
= Qt+Zt -
foe-iKTRt_SeiKT
dv
Start the recursion by attempting to write [kf,zt]=irt, with zt to be found. This is a commutator relation as studied. It can be solved only if rt satisfies a cdn. (5.20), generally not to be
expected. But recall that the first term of (5.23) was set 0 so far. We may replace zt by ztzt+zt , where zt commutes with kf but otherwise is arbitrary. Substituting this into rt of (7.2) we get
rtrt+zt
with rt denoting
our old rt formed with zt rt = (K,zt) -i/2(k,zt)2 +
Here we write rt = rt + rt , with and improve the above 'recursion "' ansatz' by attempting to find zt solving the commutator equation
(7.4)
[kf,zt] = in
,
.
rt rt+zt +(kf,zt)+i[K,z°]
,
[kf,zt]=0
exactly of the form (6.1), (6.3) again. For j=1,2 we get rtE With initial value z0=0 use thm.6.2 to get z E C"(l1S _e) to solt ve the compatibility conditions pjrtpj=0. Then (5.19), with the
307
9.7. Proof of theorem 5.4 first sum set 0, gives a solution of (7.4) of the form
(7.5)
zt=
PjirtPl/(%j-Xl) E C00(R,Wsm-e' -e)(C°°(!,Wsm_ez _e)
.
After fixing the symbol z E C°°(R,Wsm_ej) we also have a well defit ned r2E CO0(R,Wsmj+j'-2 , using prop.5.3. j'ez) e, Set rt [kf,zt1+rt in (7.3), and 'integrate by parts': Write ,
2 + ...), rt = rt -i([k,zt2 ] -i(k,zt)+(-i)2/2(k,zt)
(7.6)
with rt=rt+i ([ x, zt] +i(k, zt) +-...) EE
fte-iKT Rt-T aKt dt
e-iKtZ2eiKt
CO0(R,VPSm-je'
_j' ez
j+j' =2. Get
toe iKtR _t eiKt dr +if to-iKz [ K, Zt_t ]e iRr
_ Z2
t
0
+f 0 to-iKtR2
t-t
eiKtd
where we introduced rrt It-zt instead of the old rt. Substitute in (7.3) (we now have Zt+Z
instead Zt but Z0=0 by construction). Get
eiKt(Q+ZO+ZO)eiKt = Qt+Zt+Zt+Zt - fe-iKTRt_telKtdt
(7.8)
Cm(R,Wsm-je'-j'Jezz)
In particular, we have rt E ,
zt E Cm(R,VSm_ je, _ j ' ez)
j+j'=2
,
and
j=0,1,2.
It is clear that this proceedure may be iterated. In the next step we seek for a representation of rt as a commutator. Again this requires (a) redefining the symbol zt of (7.5), adding a suitable CO0(R,Wsm-je'-j'e2)-function zt , commuting with ke,, and (b) in the redefined (7.8), using zt+z2 splitting rt into a lower order term rt , and another term rt fitting into thm.6.2.
The point is this: The zt again will have to be of order m, j+j'=2 since it must solve (6.5). The rt will be of
je' -j'ez
,
this order, with j+ j' =3 , and the zt, solving [ ke, , zt] =rt^ , will be of order m-je'-j'ez-e , j+j'=2 , which is even lower. After solving that commutator equation the integral in (7.8) is treated as that in (7.3) earlier. The splitting corresponding
to (7.6) gives symbols of the proper order, and z3. (going to the next remainder), is of proper order m-je'-j'ez, j+j'=3. Thus get
(7.9) a iKt(Q+ZO+...+ZN)eiKt=(Qt+Zt+...+Zt)
f
trite-iKERN_teiKT
where generally
(7.10)
zt
,
rt E COO(R,'WSm_ je' _j I G2 )
,j+j'=1 ,1=0,1,2....
.
9. Particle flow and invariant algebra
308
Clearly the integral in (7.9) is E C°°(" Vs _je, _ j , e2)
,
j+j' =N
Next, the asymptotic sum zt Fzt is well defined, and we get (7.11)
a
iKt(Q+Z0)eiKt = Qt + Zt + Wt
'
with Wt E O(-a), W0 =0, and Wt -E C'(&,Optpsm), using (7.9). We get
A=Q+Z0E Pm, At=Qt+Zt+Wt . We have proven (1) of thm.5.4, for a°=e. Notice that we were selecting all the initial conditions for the successive 'commuting parts' of the zt as 0 This means that .
the decomposition a = q+z0 for t=0, coincides with the initial decomposition of thm.5.4,(2). In particular, we get q0 of (5.14) ,
equal to our initially given symbol q of (5.12) On the other hand the lower order commuting parts of the zj are not all zero,in general. Hence the decomposition qt+zt+wt =at suggested by (7.11) in general is not of type (5.14). In fact, .
for general q E Vsm , the symbol zt + wt may not be of order m-e but only will be of order m-ej j=1,2. The reason is, that the ,
'commuting part' plztpl of zt will have to go together with qt , to make a decomposition (5.14) This suggests that the commuting part of at is no 'clean propagation' of q , according to the differential equations (6.5). The above discussion of thm.5.4 (1), a°=e, works again (with .
the following amendments) for eA =e' (and hence also for e ° =e2) . , but for a different reason: The right hand side of (5.22) now is in Vsm_e2, but division by gets this into Vsm_e again. qt of (5.24) has qtE C0'(&,'U)svmle2 C00(N,Wsm-lee First, zt of (5.23) again is in C°D(&,1sm_e)
_e)' and, atgtE
)sv_e2 . Moreover, atztE
)'
by a calculation similar to that in [CE]. The corresponding effect is observed on higher iterations. For example, the symbol rt of (7.2) now is of one order -e2 better since zt and [x,zt] and (kf,zt) all are better. Therefore the result follows just as well. Details left to the reader. Q.E.D. ,
,
For a possible later application we summarize the special observations in the proof of thm.5.4 (1) as follows. Corollary 7.1. For any A E -=PP a decomposition (5.14), valid for
tE 2 , may be constructed as follows: Start from any such decomposition a = a0 = q0 + z0 at t=0 and then set (7.12)
gjt:Sj -> Sj
qt = Lj=1 gjtpj ,
,
Sj = im pj
309
9.7. Proof of theorem 5.4
Here the linear operators qjt of the j-th eigenspace Sj of ks, are given as unique solutions of the first order system of PDE (7.13), changing with x and with covariant derivatives in ,
(7.13) Pj((gjtPj)It kjI
(gjtpj)Ix+?jIX
satisfying the initial condition qjt = gtlSj , at t = 0
(7.14)
and vj :Sj - Sj are defined by (7.15)
vj = (iPjfPj - PjPIkflxPj)ISj
In "local coordinates"-that is if (p1,...,cp
,
Pi
Pj is a
and
local bi-orthogonal basis of the eigenspace S (x,g) if Kjt = C" in some open set a of
, defined and is
the matrix of qjt, with respect to that basis, then, for (7.13) and (7.14) are equivalent to
(7.16)
Kt it +
[O 1 ]
`7I Kt Ix
=0
KO = Kl
,
1,
,
where 0j is the sum of the matrix of vj and ((-(Xj1Vr*) s))
For the qt thus obtained one next writes
(7.17) Here
zt=z0+i jI p.(qt+(kf,gt)+i[K,gt])Pl/(Xj-Xl)
,
[kf,zt]=0.
zo=0, zt lzjtpj, zjt:SjlSj given as solution of
(Pj(C),(zjtPj)))ISj + [vjzjt] + (7.18)
7
+ (Pj(zt+i[K,zt]+(kf,zt)+(K,qt)-i/2(kf.gt)2))ISj = 0 with the differential expression of the "j-th particle flow"
(7.19)
CX
j=
ax - kj lxat
at + 7,.j
.
l
The above holds only for large f2lxl+f,ll;l, but the symbols obtained, defined for large may be extended to 1e2n
to get symbols in 1Vsm, 'Wsm-e' "m-e' n vsm-ez ' respectively, and, (7.20)
at = (qt+zt) + (zt+wt)
,
a decomposition (5.14), where wtE yrsm_je,_j'e2' j+j'=3 may be det-
ermined up to order -- by a recursion, similar as above.
Chapter 10. THE INVARIANT ALGEBRA OF THE DIRAC EQUATION
0. Introduction. Consider the motion of a charged particle in an electromagnetic field with potentials V(x), A(x)=(A,,A2,A3)(x) The particle is thought magnetic field W as a small charged
sphere of mass m and total charge e spinning about an axis through its center of gravity,
B=curl A electric field
i
and moving along some
orbit x(t)=(xi
x2
E=grad V
x3 ) (t) .
In the sense of classical Physics the
/
0/
particle experiences forces from the fields E and B determining
Fig.1
I
its motion, once an initial location x° and velocity v° are given. The spinning charge makes the particle a magnet (generates a mag-
netic moment) which experiences a twist from the fields B (and E Thus the particle is acting like a spinning top with an angular momentum trying to turn it. Assume the velocities large to implicate special relativity. Let c denote the speed of light. The motion is described by a pair of systems of ODE. With location x(t) and magnetic moment j(t) at time t we get (with "x" denoting the cross product of 3-vectors, also in the following) -x.
(0.1)
t
J.
c
= -E +
C(x'xB)
as equation of motion for the particle, while j(t) satisfies
(0.2)
t M 1-x'2/c2 B"xj = 0
where
(0.3)
B" = B + (1/ec)(1+ 1-x'2/c2 )-1x'xE
denotes the magnetic field 'seen by the particle' (even if B=O,
310
311
10.0. Introduction
the relatively moving electrostatic field E is experienced as a magnetic field). Clearly E = E(x(t)), B=B(x(t)). We work in terms of special relativity. x' in (0.2) and (0.3) may be substituted with x'(t) from (0.1), so that the combination (0.1),(0.2) is a nonlinear system in the variables (xj(t) , jj(t))j=1,...,3 Also M is a coupling constant, involving the quotient of the magnetic moment and the mechanical moment of the rotation. It is given only after making some assumption about the geometric '
distribution of charges and masses of the little sphere. In somewhat greater precision one will have to also expect an
effect of the magnetic moment on the motion of the particle: If B changes rapidly there will be different forces acting on north and south pole of the spinning magnet, resulting in a non-zero combined force on the particle. We have reason to first ignore this "Stern-Gerlach-effect", as a lower order quantity. In terms of quantum mechanics, assuming that the particle has 'spin 1/2' - such as an electron or a positron (or certain mesons) the physical description is very different, given by the ,
Dirac equation, a system of first order PDE
.
The physical state of the system is no longer described by initial position and velocity coordinates and initial spin, but rather by a unit vector
1V(x) = (1V1 (x) ,1U2(x) ,1V3 (x) ,1V4(x) )T
,
x=(xi ,x2,x3)
in the Hilbert space H = L2(t3,M4) of squared integrable 4-vector functions with complex coefficients. (The inner product is written
as (V, (0) = fF, 7,lVj(ojdx
,
and the norm as
II pII=(V,1 )1/2
.)
The (bounded or unbounded) self-adjoint operators acting on a dense subdomain of H are called observables. For example the x1-
coordinate of the particle is an observable, given by the unboun-
ded self-adjoint operator p(x)->x1 !,(x) with domain {i E H :
x1i. E H1.
'Measuring' an observable A for a physical state 1V will in
general not produce a precisely predictable result. Rather one will measure one of the eigenvalues of A with a certain probability (assuming that A has discrete spectrum). If {cPj:j=1,2,... } denotes an orthonormal base of eigenvectors with corresponding eigenvalues X1,,2,... then we may look at the expansion ,
(0.4)
1V =
=1aj(pj
aj = (g)j,1V)
Iaj12 =
11V112
=1
.
The measurement of A in the state 1V will produce the result
10. The invariant algebra of a Dirac equation
312 >`j with probability jai12
.
Observe that the sum of all probabili-
ties is 1, as expected, and that the statistical expectation value
of the measurement is given by (0.5)
Lj=1XjjajI 2 = (yr,AV)
for , E don A
,
If A has continuous spectrum one can talk about the probability of
measuring a value in a given interval A=[a,b]. This probability is given by IIEAVII2 with EA the value of the spectral measure on the interval A . The expectation value still is given by (0.5). After measuring the observable A , the physical state i' in general has changed: If the eigenvalue Xj has been measured,then the new physical state will be the corresponding eigenvector Tj A successive measurement of the same observable A will produce the same result T,j with certainty (probability 1), as the expansion (0.4) for "Tj shows. However, measuring another observable B will transfer the state into an eigenvector of B, hence a following measurement of A no longer will produce T,j with certainty, except if the operators A and B commute. Then the expansion of Tj into eigenvectors of B will lead to values of B corresponding to simultaneous eigenvectors of A and B for which both may be measured in any order, always producing the same result. The first momentum coordinate pl is given as (the unique self-adjoint realization of) the differential operator -ir
(0.6)
xipl-pixl = ifi
,
the Heisenberg uncertainty relation: Successive measurement of the observables cannot eliminate a certain minimal uncertainty. The total energy of the system is given as such an observa-
ble, denoted by H. This observable has a special significance. It determines the 'time propagation' -substitute for (0.1) and (0.2). The law of conservation of energy requires that a measurement of H must produce the same result at any time. The represention of states and observables by vectors and operator of H is not unique, of course. The choice has been a matter of convenience as well as of individual preference. Time propagation may be described either by the Schroedinger representation: Observables are constant in time, but physical states change. Then the change of the state V0 in time will be
determined by the differential-equation-initial-value-problem (0.7)
radii/dt + i Hy, = 0
,
0st
,
p(0) = Vo
.
10.0. Introduction
313
Or, equivalently, by the Heisenberg representation: Physical
states are constant in t, but observables change, according to
A - At
(0.8)
eitHAe-itH
=
.
Observe that U(t)=eitH is a unitary operator, defined by the spectral resolution, and is the solution operator of (0.7). The quantum mechanical problem corresponding to our above classical problem is described by the following energy observable: H = V(x) + mc2Il + c
(0.9)
..1aj(-iaa/axj - 2Aj(x))
.
Here V(x) and Aj(x) are the potentials as above, m,e,c,a are mass, all a2, a3 charge, speed of light and Planck constant. Also R are certain constant 4x4-matrices. We are used to work with ,
«j = i(_aoj)
(0.10)
j=1,2,3
o-IJ
with the 'Pauli matrices' aj and I defined by (0.11)
a1-
-i 0)
a2= to 0J
'
,
a3= to-!J
,
I = to
011
For convenience we also define the 4x4-matrices (0.12)
lij = fa oii
j
'
pj = ro3a, j)
,
j=1,2,3
and observe that the following (anti)-commutator relations hold: (0.13)
aj2=1,
a1a2=-a2a1=ia3, a2a3=-a3a2=ia1, a3a1=-6163=io2
,
hence ajak + akaj = 26jk, pjpk+pkpj = 26jk, Paj+aj(3 = 0 (0.14)
ajak + akaj = 2ojk, µjµk+l'kµj = 26jk, fl2 = 1 (3a j= iµj
,
a1a2= ip3
,
a2a3= 141
,
a3 a1= ip2
Note (0.15) below, for the formal 3-vector a = (x1'(y 2'°3) and arbitrary formal vectors a,b, components commuting with a (a)
(0.15)
(b)
a(aa) = a + iaxa ,
,
(aa)a = a - iaxa
((ja)(ab) =
(c) [ (axa) , (b a)] =(axa) (b a)-(b (3) (axa) =2i{ (aa)b-(ab )a} . The (system of) differential equation(s) (0.7), with H of (0.9), is called the Dirac equation. Clearly the Dirac equation is
314
10. The invariant algebra of a Dirac equation
a symmetric hyperbolic 4x4-system, in the sense of VI,2. Moreover, unwe shall see that it is semi-strictly hyperbolic of type e' ,
der proper assumptions on V and Aj . Accordingly our results of IX,5 apply. We are getting an invariant algebra of ydo-s. From now on let us assume the potentials V, Aj to be bounded C ()e )-functions. Note, we exclude singularities like that of a
Coulomb potential. This appears as a serious handicap of the theory, but may, at the contrary, prove to be one of its major points. It then follows easily that the differential expression H is formally self-adjoint, and, moreover, that the minimal operator (with domain C0(&3)) admits precisely one self-adjoint realization In other words, there is a unique self-adjoint energy observable (the Hamiltonian) induced, and our theory is well set. The above scheme indeed is capable of explaining physical
observations around our model of fig.1 with such amazing accuracy and detail, that it survived, in spite of the fact that some strange contradictions or paradoxa also were noted. In earlier work ([CD], [CF]) we attempted to show that our invariant algebra of IX,5 may explain some of the paradoxa. One might even go beyond this and ask the question whether the 'perturbation symbols' we discuss in sec.5 could explain other physical phenomena so far only derived from quantum field theory or gauge theory. In sec. 1 we propose a modified observable concept. In sec.2 we discuss a link between invariant algebra and Foldy-Wouthuysen transform. Sec's 3,4,5 give details of the invariant algebra in case of the Dirac system (0.7),(0.9). In sec's 5-9 we focus on the 'correction symbol' first for 'standard observables' then in general, discussing some finer details. 1. A refinement of the concept of observable. To motivate an application of the theory of
ydo's, let us
So far, focus on the concept of observable introduced in sec.0 L2(H3T4 any bounded or unbounded self-adjoint operator of H = .
(having a spectral resolution) qualifies as observable. On the other hand, the observables of real interest normally turn out to be either multiplications (like the x1-coordinate, we mentionned) or differential operators - such as H above, or the momentum etc. We will meet other observable, given as pl = -iha/ax ,
.
1
"standard observables" later on, and mention that multiplication by a (proper) matrix may occur as well, such as the spin observa-
315
10.1. The concept of an observable
ble, commonly defined as multiplication by pj of (0.12)
,
j=1,2,3, with pj
.
on the other hand, in many respects the admission of all unbounded self-adjoint operators as observables appears inconveniently large and the question can be asked whether it is useful to get restricted to a special class of observables. We choose to suggest a certain condition of continuity or smoothness on observables, to be discussed now:
Note that for the remainder of this section it is not significant that physical states are 4-vectors. We might just as well assume that H - L2(&3) (with complex-valued functions), or, more generally, that H = L2(&n,,v) for arbitrary n,v = 1,2,... First consider a bounded operator A E L(H) . We propose to ,
.
admit A as observable only if it is norm continuous under translations. In more detail, it is well known that the operator family Ah = T-hATh , h E lea , with the translation operator Th = eihD (Thu)(x) = u(x+h), u E H , in general is not norm continuous in h in the Banach algebra L(H) , but only 'strongly continuous' For each fixed u E H the family uh = Ahu is continuous as a map from &3 to H . However, this continuity may degenerate if u varies over a bounded set of H , it may not be uniform. Example: let ,
:
A = multiplication by a discontinuous function a(x) = xl/Ixl1 = sgn(xl)
like a(x)
.
If we ask for differentiability in the parameter h the situation gets worse: A derivative ah Ahu as limit of a difference quoj
tient exists only under special assumptions on A and u .
We impose as condition (t) for a bounded observable A that its family Ah is not only norm continuous, but even norm-Cw : All partial derivatives ahAh exist in norm convergence. Observe that this condition may be very natural: Given the inherent inaccuracy of space measurements an observable can have little significance, if it changes strongly under very small translations. (Although of course this may be an idealization just as the concept of derivative - interpreting velocity as time derivative amounts to a strong simplification, but the built-in limit in praxis never can be carried out, but will have to be replaced by a difference quotient). Next, we impose the same condition also in phase space:
The Fourier transform (1.1)
Fu(r) = f
Pox = (2n)-3/2dx
10. The invariant algebra of a Dirac equation
316
defines a unitary operator of H 'diagonalizing' the momentum obser
vable (i.e., pj _ -iax j is transformed into the multiplication
operator u^ (x)
xJu^ (x) ).
Our quantum mechanical measuring process is invariant under unitary transforms: If i, is a state and A is an observable, as above, and if U:H-H is a unitary operator then we get exactly the same probabilities and expectation values if we replace i, and A by respectively. In sec.0 the position obseryf = U, and A" = UAU* ,
vable was diagonal (was a multiplication operator). If we apply the Fourier transform then the momentum operator gets diagonal. Position and momentum play a dual role in much of the theory. Hence it appears natural to impose the same "translation smoothness condition" (i.e.,cdn.(-u)) also in the momentum-diagonal form.
A simple calculation shows that this amounts to the following Ah=eihxAe-ihx gauge-smoothness condition (y): The operator family is norm-Coo in the parameter h (i.e., Ah E C-(&3,L(H)) ).
[Indeed, (FAF*)h = T-hFAF*Th =
FAhF*
,
as easily checked]
Remark 1.1. Both conditions (z) and (y) (i.e. translation and gauge smoothness together) amount to the condition of norm-smoothness of conjugation within the Heisenberg group HG : The map g -> A(g)=
g*Ag is
C00
(HG,L(H)). Recall that the Heisenberg group (or its stan-
dard representation within the unitary group U(H) of H) consists of all unitary operators of the form g - ei(P eil;xeizD=ei(T+tx)Tz (PE It, z,t E 13 with operator multiplication as group operation.
At this time we recall the concept of pseudo-differential operator (qdo) with symbol in CB°°(13): The space of symbols Vt0 = CB°°(13) is defined as (1.2)
Vt0 =
E C0O(16): all derivatives are bounded}
(we mean all partial derivatives in the 6 variables x and and of all orders). For a "symbol" E Vt0 define the operator A - op(a) =a(x,D) by setting r
(1.3)
(Au)(x) = (2n)-3fd
)u(y)
The integrals at right exist in the order stated whenever u E S = S(13) , the space of rapidly decreasing C'O(13)-functions, and the function v = Au then is in S again, as seen by a technical but straight-forward calculation (cf. I,1, VIII,1). Thus A of (1.3)
is defined as a (continuous) map S -> S , whenever a E CB'O = y,to The significance of the last remarks become clear if we
317
10.1. The concept of an observable
look at the following result:
Theorem 1.2. Every pseudodifferential operator A = a(x,D) with symbol a E Vto extends to a continuous operator of L(H) (from the dense subspace S of H), also denoted by A . Moreover the class of all such L2-bounded pseudo-differential operators precisely coincides with the class of all bounded observables satisfying cdn's (t) and (y) above (i.e., the class of A with A(g)E C'(HO,L(H)))
.
Furthermore, for the latter type of A the corresponding symbol is
uniquely defined by the formula (1.4)
a(z,t) = (2n)ntrace{Q_*P(az,at)Az,t} e_itxeizDAe-izDeitx
where and where P(az,at) is A2.t= any differential polynomial (constant coefficient differential and atj with the property that it admits expression) in d =a1a z j
,
zj
a fundamental solution q(z,l;) of polynomial growth, and such that the pseudodifferential operator Q_ = q(-x,-D) is of trace class
in the Hilbert space H .(Such a polynomial is given by p(z,t) (l+z2)n(1+12)n for example, as may be seen. The corresponding fundamental solution q then is given as the product of Bessel )n). potentials inverting This result was discussed in [CL]. Actually, it coincides with VIII, thm.2.1, of course, except for some variations in the symbol formula. ,
(1-Az)n(1_A
The conclusion we derive from thm.1.2 is that, with our above restriction of translation and gauge smoothness imposed on bounded observables, we automatically arrive at the consequence that a bounded observable must be a pdo and that a self-adioint ,
,do AE Opyito automatically is a bounded observable satisfying the translation smoothness and gauge-smoothness condition.
Notice that we are lead into a set of axioms selecting better observables which seems elegant, but has several defects. First, most standard observables are not bounded (such as energy, position and momentum, for example)
.
Second the class Optpt0 of
ydo's with symbol in0 has severe shortcomings, as far as standard theory of 'tpdo's is concerned:
These lydo's need not to obey
the asymptotic calculus of 'gdo's valid for classes of symbols with
derivatives decaying at We address the second objection first: If in addition we also impose a "rotation smoothness condition" (cdn.(p), below) .
and also a "dilation smoothness condition" (cf. cdn.(S), below).
10. The invariant algebra of a Dirac equation
318
then we get a smaller class of bounded observables, and a class of 4 do's allowing a global calculus of ydo's, in a form to be specified.
Condition (p) (rotation smoothness): For an orthogonal 3x3-matrix o define the unitary operator S E L(H) by u(x) - u(ox)=S0U(x) 0
Then the family A(o) = S*AS0 is norm-Coo over the orthogonal group 3 of IQ3
Condition (S) (dilation smoothness): For a positive number 6>0 define RS EL(H) by setting Rsu(x) = 61/2u(6l). Then the family ROARS is norm-C00 over R+ Note that cdn's (p) and (S) are Fourier invariant: They mean the same, if imposed on the position-diagonalized or the momentum diagonalized representation, because the Fourier transform leaves the corresponding subgroups of U(H) invariant. All above cdn's (T),(y),(p),(S) together may be expressed by using the Lie-subgroup GS of U(H) of all operators of the form (1.5)
g =
ei(peizDeii;XR6SO
, T E & , z,t E &3
,
o E 03
.
Proposition 1.3. For a bounded observable cdn's (s),(y),(p),(S) together are equivalent to the property that the function A(g) _ gAg , defined over GS is C0O(GS,L(H)) with norm topology used in L(H) ,
,
.
We denote the class of operators used in prop.1.3 by IWS and then have the result below (cf. VIII, thm.5.4). Theorem 1.4. The class WS precisely consists of all yido's A =
a(x,D) with symbol in the class ls0 , where Vso denotes the class of symbols in yito such that application of a finite number of dif-
ferential expressions 1jl of (1.6), below, repeated in arbitrary orders always gives a function of Vto
(1.6)
`1oo=
qjl=( A ,- la
xjxj +(xi axl-x1ax,)
,
j,1=1,...,n
One confirms easily that the functions in Vso have the "classical symbol property" on compact sets: For any compact set K C In and all multi-indices a , 1 we have (1.7)
aaaPa(x,U) =
x, E &n , x E K
There is an analogous condition with x and 1; reversed. In fact, global such conditions (degenerating at infinity) hold cf. VIII,6.
319
10.2. The invariant algebra
Next we introduce the class ips of all polynomials
aa,p E 4s0
(1.8)
.
in the 6 variables x,t and observe that Vs as well as the class 'IS = Opts = {a(x,D}:a E ps} are algebras (cf. VIII,7). For a of the form (1.8) the operator A=a(x,D) is of the form (1.9)
A =
with A,,, = aa,0(x,D)
.
However, the commutators [ Dj,A] and [ xj ,A] for A E OpVs , handle
rather well (cf. VIII,7), and we also may write (1.10)
A
with different coefficient symbols as . A Leibniz-Taylor formula as in 1,5 holds in OpVt, with the class 'tVt of polynomials with coefficients in ipt0, but the remainder may not be well behaved. On the other hand, for symbols in is estimates for the remainder ,
can be derived, resulting in asymptotic expansions, and a calcucf. VIII,7).
lus of i4ido's, (although degenerating for
In the following we shall require all observables considered to be members of 'IN = Opps (Note, however, that further restric.
tions will be introduced in sec.2, below. Introducing the classes Vsm (for m = (ml,m2)
,
mj=0,1,2,..,)
as the polynomials (1.8) of order m1 in l; and order m2 in x, and
the operator classes LISm = OpVSM , we observe that Its = Um lsm, and that a characterization of tINm by conditions like our above 'smoothnesses' (t),(y),(p),(6) is possible with the (polynomially
weighted) L2-Sobolev spaces of 111,3, i.e.,
(1.11)
Hs
= Hs ,s2
=
{u ES': (x)s2(D)s1u EH}
.
Theorem 1.5. Let s=(s,,s2) be given fixed. The class LISm (for any m=(mi,m2)) consists precisely of all operators in L(HS'HS-m) all with with the norm-Coo properties (u) (y) (p) (6) ,
,
respect to the operator norm of L(H
S
,
,
H s-m ) instead of L(H)
The proof will be omitted. 2. The invariant algebra and the Foldy-Wouthuysen transform. As stated above, an observable will be a (self-adjoint) operator of uN , from now on. But it will be seen later on that the
10. The invariant algebra of a Dirac equation
320
algebra ills is not in general invariant under time translation.
Assuming states constant and observables to change by conjugation with a-iHt, we need At=eiHtAe-iHt to remain in W for all t before we can admit A E '11S as an observable. In fact, we tend to require a norm smoothness of the time translation of an observable similar as posted for space and momentum translation. To be precise: For an L2-bounded A we require that (x)3a3At E C'(l,L(H))
(2.1)
,
j=1,2,....,
and, moreover, (2.1) is not only required for At , but also for every g-derivative of At(g) as in sec.1. (over the group GS For general A E Wm we require (2.1) (for all the derivatives ) mentioned) with L(H) replaced by L(H H as in thm.1.5. s s-m The above condition will be referred to as 'cdn.(t)'. Clearly cdn.(t) implies that AtE tI{S for all t. Thus At has a symbol at(x,l) Equivalently we also may express the same cdn.(t) in terms of the symbol at For aE Vsm this takes the following form. Condition (t'): We have At = eiHtAe-iHt = at(x,D) , with atE s and (with e2= (0,1) ) we have ,
,
.
.
(2.2)
at at E PSm-jet for all t E & , j=0,1,2,...
.
Moreover, if we apply any combination (of arbitrary order) of the differential expressions 1jl of (1.6) to at we obtain a symbol bt E Vs still satisfying (2.2) m .
The class of all A E iI{Sm (for a given m) satisfying cdn (t)
(or equivalently (t') ) will be called P Pm (since it depends on m the 'Hamiltonian' H ). One trivially notes that (for fixed H) Pm C Pm, as mjsmj' j=1,2 The union P = UmPm is a graded alge,
.
bra, left invariant under conjugation with a-iHt It will be called the invariant algebra of the Dirac equation (or, more generally of the Hamiltonian H ). .
Let us now analyze the significance of cdn (t). The special form (0.9) of the Dirac operator is of crucial importance, in this
respect. We know that V(t) =
e-itH,F
is given as the solution of Explicitly we are given the Dirac equation (0.7) with V(0)=4i' an initial-value problem for the partial differential equation .
(2.3)
avlat + i{V(x)+mc2s+ Note that (2.3) is a symmetric hyperbolic first order system
of
4 partial differential equations in 4 unknown functions. Its
321
10.2. The invariant algebra
principal part has constant coefficients. The existence and uniqueness of the solution of such a system is a well known fact (discussed in detail in VI,2). We will require as additional condition that (for j=1,2,3, and all multi-indices a) we have
(2.4)
V(a)(x)
=
O((.)-IaI)
O((.)-IaI)
,
Then H = h(x,D) is a pseudodifferential operator in Op", (e'=(1,0) ) with symbol h and pricipal symbol h" given by (2.5)
h
h-
the symmetric hyperbolic
property of (2.3). Under (2.4) we have hE ipce, so that HE Opyice, On the other hand, the hyperbolic system (2.3) is not strict,
ly hyperbolic: The matrix of course has real eigenvalues, but they are not distinct. A quick check shows that the eigenvalues of are =tI I. For X960 they both are double, hence have constant multiplicity. has the diagonalization The hermitian matrix-function (2.6)
+(x,
)=1(x,
with a unitary 4x4-matrix 11=12=X+
(2.7)
,
) =
and eigenvalues
13=14=X-
,
X+=V(x)tcfr
,
where fr m2c2+n2
('+' denotes the adjoint 4x4-matrix, 'a*' the Hilbert space adjoint's symbol - a(x,D)*=a*(x,D), in all of ch.X.) for all x,1; E &n.
In 1,7 we defined the class of symbols ypcm, m=(ml,m2)E &2, by
(2.8) with (
"'cm = {aE CO: a(T) = O(( )m1 IOI(x)m2 ITI),6,'rE Z+ )
=
1+ i;
1 ,
a(0) = aeaXa. Now we claim that,under the
conditions (2.4), we have X±E yice,, e'
= (1,0) . Moreover, the
entries of T are global symbols in Vc0. In fact the 2-fold eigenvalues Xt have the two-dimensional orthogonal spectral projections a0=me/(m2c2 + a2)1/2. t= /(m2c2+n2)l/2
(2.9)
are A calculation confirms that X. as well as the entries of let V3, yi4 be yr2 be the first two columns of p+ in yc0 . Let ,
V1,
the last two columns of p-
,
and let Tj = V Pj/(1+a0) l/2
.
Then
10. The invariant algebra of a Dirac equation
322 (2.10)
cp(x,t) =
is the desired unitary matrix satisfying (2.5) and again has coefficients in ta0, by a calculation. We write yicm for the class of all rxr-matrices with entries in cm, so that pt, VE V00. In matters of the invariant algebra P it should first be noted that for a single first order hyperbolic equation we get an Egorov-type result, saying that cdn (t) (or (t')) holds for (cf. VI,6). For a semi-strictly hyperbolic system of every A E ills
the type discussed in IX,5 we found that the invariant algebra P is properly contained in 46 , however. In essence, membership in P depends on the symbol of A to commute with the symbol of H , at least modulo lower order terms. Such a result was proven in IX,5. In order to apply these results we just have to confirm that the Dirac equation (2.3) is semi-strictly hyperbolic of type e'. Indeed, we have the explicit eigenvalues (2.7) of the (complete) symbol of H , and it is readily checked that IX,5,(i),(ii),(iii) are satisfied, if only the potentials V and Aj satisfy (2.4).
(Note that the classes Voxe .... simplify to just yrce, , as &=e i
.)
One finds that most standard dynamical observables (such as
position coordinates xj, momentum coordinates pj=-iax
, angular
j
momentum coordinates (xxp)j etc. all belong to 'IS Surprisingly none of these operators belong to P. They all need (normally small) additive corrections to become members of P. To illuminate this fact we proceed stating a pair of theorems. From now on we will distinguish in notation between complexvalued and 4x4-matrix-valued symbols: Let Vtm , VSm , etc. be the 4x4-matrices of symbols in Wtm, VSm , etc. Note that (2.4) implies ,
that E Vae, , hence H E Opy,ce,
.
e'=(1,0), for h and H of (2.5).
,
Evidently the algebra P contains the identity, and the Hamiltonian H as well as all polynomials in H. In fact, it is clear that every AE LIS commuting with H - in the sense that eiHtA-AeiHt =0 using the operator product of O(oo) satisfies At A=const. ,
,
hence will belong to P . For example, the resolvent R(A)=(H-A)can be shown to belong to 1IS-e, as A is non-real: Clearly the symbol h(x,l;)-A =g(X-A)q)+ is md-elliptic of order e' - one easily finds that It has a special Green inverse, in the sense of III,thm.4.2 (or V,thm.1.3). Since it is invertible ,
in H - due to self-adjointness of H - it follows that R(A) must be that special Green inverse. Hence R(A)E 11S-e, implying that ,
323
10.2. The invariant algebra
On the other hand, the operator eiHt does not belong for O#XE C0 to W , hence not to P either . The operator eiHtx 0 R(.t)E P_e'
.
,
shifts singularities (cf.VI,7) while AX E Oppc , for AE X16 leaves wave front sets invariant, by II, thm.5.4. The result, below, in essence links the algebra P to the Iido's with symbol matrix commuting with h(x,l;).
Theorem 2.1. (1) For every AEPm, of the form A=a(x,D), the symbol a allows a decomposition
(2.11)
a=q+z
,
z E y)sm-e, q E 4 s 4
[h(x,i;),q(x,1;)] = 0
the latter for all x,t E R3 of symbols commutes R3 satisfying with the symbol h(x,l;) at least for all x,T; E xl+Z y > 0 then there exists a symbol z E ys _e such that (2) Vice versa, if a matrix qE yes ,
,
A = a(x,D) = q(x,D) + z(x,D) E Pm . A2 E Pm (which must have a decomposi(3) Suppose A 1 tion (2.11) both) have the same q, but different z (=z1,z2), resp. ,
Then b=z1-z2 allows a decomposition (2.11) with m replaced by m-e. Thm.2.1 is just IX,thm.5.4, in case of the Dirac equation. In the next result we are going to construct a unitary operator of H and a pseudo-differential operator which decouples the Dirac equations modulo a term of order --. The fact that this can be done for finite orders - not order- - is well known. The corresponding unitary operator is called Foldy-Wouthuysen transform.
Theorem 2.2. There exists a unitary operator X:H-H of the Hilbert space H = T4xL2(&3) , which also is a 'pdo with symbol in yPc4 such that (2.12)
X=X(x,D)
,
X=(p+cu
, wE 1c4e
,
cp as in (2.10)
and that the substitution u = Xv (and multiplication from left by X 1) brings the Dirac equation to the form au/at+ i(A + T)v = 0
(2.13)
Here r is a (4x4-matrix of)
.
,do(-s) in O(-oo), while A vanishes in
its upper right and lower left 2x2-corners. Moreover, we have (2.14)
symb(A) = diag
(mod "_e2
)
,
e2 = (0,1)
with diag( ) denoting the diagonal matrix with entries listed.
10. The invariant algebra of a Dirac equation
324
A proof of thm.2.2 will be discussed in sec.6, below. Thm.2.1 and thm.2.2 are related by the following argument: First, for a hyperbolic system of the form (2.13) consider the e-i(A+I')teiAt. operator Zt Confirm that Zt'=e i(A+T)treiAtE 0(_o), C00
and, in fact, Zt'E
(2,O(-oo)). Accordingly Zt-1=WtE COO(2,0(-oo)) e-i(A+I')t=e-iAt(1+Wt)=e-iAt+Vt
by integration, and we get
It follows that At
VtE
, where
ei(A+r)tAei(A+I')t=eiAtAe-iAt +
+Qt = Bt + Qt , where QtE
Coo(2,'IS_00) . This shows that PA PA+r' while thm.2.2 and *a simple argument i.e., PH=XPA+rX . It follows that imply that AE PH . X AX E PA+r ,
PH = XPAX
(2.15)
On the other hand, it is clear that PA contains all matrices A = lC C} with B,C E 'IS2 = Op y,s2
(2.16)
while A = (Q P0) EPA requires that
i?+(x,D)t
(2.17 )
e
Pe
-iX_(x,D)t ,
e
iX_(x,D)t -iX+(x,D)t Qe
E 'IS
to satisfy differentiability conditions of type (2.1). Using (2.1)
for AE O(m) and j=1 yields (x) (X+-X_)(x,D)P-(x)[P,X_(x,D)]E O(m), with ad-elliptic (x)(X+-X_)(x,D) having a K-parametrix E , while
(x)[P,X_(x,D)]E 0(m). Thus PE !O(m)C O(m-e). Continuing with (2.1), for j>1, get PEO(-oo). A similar argument shows that also
QE 0(--)
.
In other words, we have (2.18)
PH = {XAX*+F: A of the form (2.16), FE OpWs40}
3. The geometrical optics approach for the Dirac algebra P. We mention again that thm.2.1 is a special case of IX, thm. 5.4. the proof of which is discussed in IX,5,6,7. In the present section we want to reexamine the formal geometrical optics of this proof in the special case of the Dirac equation. The point will be to derive explicit formulas for the successive perturbation terms, to be evaluated for their physical significance. We will settle for explicit formulas of the first and second step, since calcula-
10.3. The geometrical optics approach
325
tions for higher orders transcend into the unmanageable. for some AOE Pm Changing notation, we write (with some mE 12). The function A=A(t) with values in O(m) assumes values in OpVsm 5m, by assumption. Differentiating u(t)=A(t)u, A=eitHA0e-itH,
for u E S C fl(dom Hk :k=1,2,...} yields atu(t) = i[H,A]u . Accordingly the ,do-family A(t)=at(x,D) satisfies
A' = dA/dt = ata(t,x,D)
(3.1)
=i[H,A]
with the time derivative existing in Op's4m-e2
,
and in norm conver-
gence of L(HS'HS-m+e2). Hs of 111,3, using III, thm.3.1. Similarly it is seen that time derivatives of all orders of A(t) are in P. In fact, we get a,ktA(t) E Pm-ke2,as A E Pm , for all k
In terms of symbols, equation (3.1) assumes the form
(3.2)
a' = i[h,a] + (h,a) -i/2(h,a) 2 + ... (mod i,c-o,)
.
Actually, for all common observables we examined, the symbol a belongs to icC Vs . A subalgebra PC C P of Oppc4 and spaces ,
PCmC Pm of Oppc44 may be defined by imposing a condition (tc), like
cdn.(t) above, but with Vsm replaced by Vcm. Then a result like thm.2.1 holds for AE PCm , with all perturbation symbols in yicRC VSR, µ=m-e, etc. Actually, all proofs are the same, but simplified since yido-calculus in Oppc is somewhat simpler. The corresponding result is discussed in detail in [CD], cf. thm.2.1 there.
Since we have yicmC yism (and the topology of 'l'cm is stronger
than that of Vsm ) it is clear that PCmC Pm , for all m . For a q E yicm commuting with h any sequence of correction symbols to ob-
tain an A=Q+ZE PCm also is a valid sequence of correction symbols in the sense of thm.2.1, above, -i.e., for Pm. This explains why we choose here to assume that all correction symbols are in instead of tpsµ for the corresponding R. Of course, a more general choice of zE Vsm-e \Vcm-e is possible, we note that in (3.10), below, the choice of ct , at t=0, is arbitrary. ,
This explains why we will work in the algebra PC, for our symbol calculations. In particular, instead of (2.2) we assume
at at Ecm-je2
(3.3)
restricting the choice of z
,
,
t E & , j = 0,1,2,...
but not the possibility of choice.
The first term at right of (3.2) has order m+e'. all others have order m-e2 or less (assuming AE PCm). It follows that p=[h,a] E,)m-e 4 2 for all t. This gives (2.11) with ,
326 (3.4)
10. The invariant algebra of a Dirac equation
q = P+aP++P_aP_
,
z = p+ap +p ap+ = z++z_
,
where clearly [ h, q] =o , [ h, z+] = (? -A.T) z+= 2cf rz+= P+PPTE 46 4m-e2 as stated in (1). hence z = (p+pp_ pip+)/2cfr E tcm-e Note that we have achieved a decomposition (2.11) for all t a 0, not only for t=0. Moreover, since p+ are time independent it follows that a = q + z is the decomposition (2.11) for the
operator A E Pm_ez
.
Therefore we get z'E Vcm-e-e2
and (2.11) in (3.2) already proves (1)
(3.5)
[h,z] _ -i(q'-(h,q) )
.
Using this
: we get
(mod Psm4 _e-e2
)
Vice versa, in order to discuss the iteration leading to a proof of thm.2.1 (2), let q E lsm be a given symbol commuting with h(x,l;) in the sense stated. If an A (of thm.2.1(2)) exists, then At is well defined for all t, and we must have the decomposition (2.11) in the form (3.4), for all t as well. Moreover, also the commutator relation (3.5) will follow for all t, z=zt , q t To obtain a first approximation for qt and zt let us regard Such (3.5) as a sharp equation, not an equation modulom-e_e2 an equation is solvable if and only if its right hand side w=w(q) = i((h,q)-q') satisfies the two relations .
(3.6)
p+w(gt)p$ = 0
,
for "+,+" and "-,-"
,
and all t
If (3.6) holds then we get (3.7)
z = h(w)/2cfr + c + + c_
h(w) = p+wp- - p wp+
,
where ct are arbitrary symbols leaving S. = im p+ invariant and equal to 0 on S{ (i.e.,c+ = P+c+P+ , c_ = p c p_ ) Let us first assume that is a multiple of the identity in each of the two spaces S$ This means that .
.
(3.8)
with complex-valued symbols K. E tpcm . Apparently this is the case
if we seek for a z fitting the position or momentum observable. We then may assume that (3.8) holds for all t because that 'Ansatz'
will supply a valid symbol From (3.8) we get q' = x+ + K_*p_ since p+ are time independent. Accordingly, (3.9)
Pq *P+
= 0
327
10.3. The geometrical optics approach
and we may replace w(q) in (3.7) by w0 = i(h,q) (3.10)
,
for
z = ih((h,q))/2cfr + c + + c_ = z0 + c+ + c_
Note that (3.10) no longer contains time derivatives. Anticipating a later discussion, note that the symbols c+ at t=0 remain arbitrary, subject to restrictions stated. They must satisfy certain differential equations, but initial values are free. Note that (3.6) (for all t) leads into a pair of Cauchy problems of first order PDE, where xt are given for t=0: (3.11)
Kt.
-
0
,
We have discussed this in IX, prop.6.1, in more general setting. For our present special case it may be derived from (4.1) and
prop.4.1, below - we have q=x+p++K p_=g++g_ for "+" in (3.6) we get 0=g+-p+(h,g+)p+-p+(h,g_)p+, where the last term vanishes, by (4.2)2, while the second term equals p+()`+'g+)p+, by (4.2)1, using ;
that the commutator in (4.2)2 vanishes. Finally (4.1) may be used to convert into (3.11)1. Similarly one obtains (3.11)2. Solving (3.11) means solving the Cauchy problems for the characteristic equations (cf. ch.0, sec.6) (3.12)
x* = -X,1
Clearly (3.12) is a Hamiltonian system of 6 first order ODE The
in 6 unknown functions x(t), fi(t) with Hamilton funtion
field of solutions of (3.12) defines a Hamiltonian flow (3.13)
vt_t:12n
12n, T(t)=9tt(x0,
of diffeomorphisms 12naa12n where
0)
is defined as the unique solution of the Cauchy problem (3.14)
f' _ ?.tl
(f,W)
f(0)=x0 ,
T*=-?.lx(f,ro)
,
,
The solutions of (3.11) simply are the functions constant along the solution curves of (3.12). Hence, for a given x+ define Ktt = Ktovtt (that is, x , ,
=xt(xtt(x, ), , Then K±(t,x,')=x+t(x,l;) solves (3.11), since x t,vt-t Ktovt_tovtt =x. is indeed constant (independent of t). Also xt0 x,, so that
Ktt indeed solves (3.11).
+?
In other words, the initially (for t=0) defined symbols "flow" along their Hamilton flow (with Hamilton function defining the symbols Ktt xt(t,x,U) We have :
10. The invariant algebra of a Dirac equation
328
K+'t Kovtt
(3.15)
Kt(t,x,S)= K(x+t(x.S),
,
for all times t. Clearly this determines a function In VI,6 we have shown that not only the flow Vt as a family of homeomorphisms 26i. &6 is well defined, but also that (3.8) and
(3.15) indeed define a family of symbols in ipcm with the proper
dependence on t (i.e. COO()e," m) )
.
After determining q for all t we obtain a first correction symbol z for all t given in the form (3.10). Since we were ,
,
neglecting lower order terms the symbol q+z will not be a precise
solution of (3.2), but we now may go into (3.2) with
a= q + z + s
(3.16) with q
,
a' =q' +z' +s'
,
z of (3.8), (3.10), and (3.15), and with sEm-2e to be determined. Using [h,q]=0 and (the sharp) (3.5), we derive that
(3.17)
,
[h,s] = -i(z'-(h,z)) + (1/2)(h,q)2 (mod VOm-2e-e:) '
as another commutator equation of the general form (3.5), again to be considered as a sharp equation, not only mod m-2e-ez As a condition for solvability we now get (3.6) with
(3.18)
w = w(q,z) _ -i(z'-(h,z)) + 2(h,q)2
In (3.18) we have z=z0+c++c_ , with z0 of (3.10), where ptzOpt=O, hence p.z;pt= 0. Thus the solvability condition assumes the form (3.19)
ptw(c+)pt + ptw(c_)pt
with w(c) = -i(c'-(h,c))
+pt(i(h,z0) + 2(h,q)2)p = 0
.
As discussed in IX,thm.5.4, under more general assumptions, (3.19) amounts to a pair of first order PDE - equivalently, ODE's along the flow (3.15) - for the symbols c. so far undetermined. After setting the initial values of ct, at t=0, the first correction z of (3.10) is fully determined. From (3.17) we then get ,
(3.20)
s = sO + d++d_
,
sO = h(2(h,q)2-i(z'-(h,z)))/(2cfr)
with dt so far free. For the next iteration carry a=q+z+s+r into (3.2) with above q,z,s and a new correction rE Pam-3e' The construction of a sequence of corrections and asymptotic sum a=q+z+r+s+..., leading into a valid A=a(x,D)E PCaC Pm was discussed in IX,5,6,7. Presently we want to look into explicit calculations of only the full first correction symbol for the special case
10.4. Some identities for the Dirac matrices
329
of the Dirac equation (2.3), assuming that the potentials V , Aj are symbols in pc0 first focusing on (of posik Xk tion and momentum), and other (scalar) standard observables. ,
Note that our above iteration ignores self-adjointness. Even if we start with a self-adjoint q(x,D), the above construction
will not necessarily give a self-adjoint a(x,D). Basing our calculations on the Weyl-representation of ipdo's (cf. 1,3), rather than than the left-multiplying would be a cure. However, the algebra PC and its spaces PCm are adjoint invariant. After obtaining an
AE PCm by our iteration we just use the selfadjoint AA E PCm We close here with the observation that the Hamiltonian flows vet of (3.13) indeed are the particle flows of electron and positron in the following sense: If we eliminate from the 6 equations (3.12) we get the system of 3 second order ODE's r mx' ±+ (3.21) 1 = -grad V + 1(x'xcurl A) = -E + !(x'xB),
l 1.r
in 3 unknown functions. These are the relativistic equations of
motion (0.1) we departed from, for a particle of negative and positive charge, respectively. Indeed, with X.,=V(x)±cfr equations (3.12) assume the form
(3.22)
x' _ ±cn/ m=c2+nz
_ -(E(x) m fr1x)
Solving the first equation for n we get (3.23)
n=
-eA =±mx/ 1-x'2 /c,
For A=0 we get fr1x 0 and (3.21) arises by differentiating (3.23) and equating the result with the second relation (3.22). For general A an additional calculation is required which we skip.
Note that the momentum variable for the case "-" is - , not In fact, one of the classical difficulties of Dirac theory arises if one tries to link (3.11) (3.12), for "-", to a varia,
tional principle in the sense of classical Hamilton Jacobi theory. We shall see later that the equations of motion (0.2) for
the spin also can be "derived" from (2.3), if a general matrix q not reducing to 1 in S. is allowed. 4. Some identities for Dirac matrices. First we provide a list of useful identities around the Dirac equation. To simplify notation we write p+ p p-=q (while the prim-
10. The invariant algebra of a Dirac equation
330
arily given symbol q of thm.2.1.(2) will be written as x). Also 1+ X (but we may fall back to the old notations if they are clearer). For a summary of conventions used cf. (4.3), (4.4), (4.5), below. First we discuss some matrix results of IX,6, in the present special case. Formulas have analogues with p and q interchanged. For ""'_ "Ix" or "l," we have pp'p=-pq'p=p'qp=0, etc., hence pp'p = qq'q = pq'p = qp'q = 0
.
Proposition 4.1. For any symbol g with pg = gp = pgp =g we have
p(h,g)p = p(A,g)p (4.2)
_ (X,g) + gp(%,q)q + q(k,q)pg -
q(h,g)q=0 . Note that (4.2) holds the derivation of (3.11) from (3.6): For (q of (3.6) now called x) x = x+p + K _q with scalars K. we get
)p = x+.p - p(h,x+p)p - p(h,ic_q)p
0=
The last term at right vanishes, using (4.2), with p and q interchanged, since g=x-q satisfies the (reversed) assumptions. For the second term we use the first formula (4.2) with g=x+p, noting that the commutator vanishes, while pg'p=x+'p. The second form of the first (4.2) follows since (3.19) amounts to an ODE along the classical orbits: We get pw(c_)p = 0 by the (reversed) second relation while pw(c+)p = (at + k ax - kIxa )c+ + terms not involving derivatives of c+. The above term is the directional derivative along I
the classical orbits. Thus (3.19) indeed translates into an inhomogeneous linear first order ODE along those orbits. Proof of prop.4.1. For the second relation (4.2) we have
q(h,g)q =
glx)g Here the first term produced by each of the parantheses vanishes since it contains the factor pq=qp=O. The second terms also vanish since they contain the factors gglxq =-ggglx and gglsq -gggl (note that qg=gq=0, hence q'g=-qg', qg'q=-qgq'=0). We organize the remaining terms at right in the form ?T + T_ where ,
T =
(using (4.1))
ggltplxq =
(using qg'=-q'g) = 0 , using p'+q'=0, due to p+q=1 The term T_ vanishes .
10.4. Some identities for the Dirac matrices
331
for analogous reason, proving the second relation (4.2).
For the first relation we have
p(h,g)p = k Ixp+X-Ixq+kplx+k glx)p pgl due to the where the second term in each paranthesis produces 0 factor pq=qp=0. For the third and fourth term we use p'=-q'. ,
These terms of both parantheses give (using (4.1)) (using qg'=-q'g) (using q'=-p') Therefore the first form of (4.2) follows. For the second form we =(%-a._)[ppl plxp,g]
write pg'p = g' - pg'q - qg'p (noting that qg'q = 0 ) = g' + g(pq'q) + (gq'p)g (using g'q=-gq'). This achieves the desired form, proving prop.4.1. Now we discuss a list of identities for 'Dirac matrices'. For aj
pj cf. (0.10) f. We use the abbreviations
µj
- cA , fr m2 c2 + n
n=
(4.3)
h0=h/c-V = me 3 + a- n
of
1
,
= n/fr
a0=mc/fr
rI0= h0/fr a013 + a i; = p - q ,
,
p = 2(1+rl0) = 2(1+a013+a.l) where p
,
,
q = 2(1-T10) = 2(1-a0R - a. )
q are orthogonal projections, and form a decomposition the spectral resolution of h and h0: ,
p+q=1
,
p+=p
,
q+=g
p2=p
,
,
q2=q , pg = qp = 0
,
(4.4)
h0=frp-frq , h=Xp+X q , X,=Vtof r . For a general 4x4-matrix x we write f=f+ (4.5)
,
h=h- , where
f+(x) = pxp t qxq , h,(x) = pxq t qxp
The following formal 3-vectors of 4x4-matrices are useful: w (4.6)
c
= a - rtot = a - (a0(3+a t)i; w = fr C=(m2c2+n2)a + h0n
,
,
6c= a0µ + pxt
,
0 = mcµ + pxx
We discuss the following list of identities (4.7)-(4.19): (4.7)
(4.8)
paq =
2( ,+ i0c)
pap = pt
,
,
qap = 2(wc- i0c)
gag = - qt
,
10. The invariant algebra of a Dirac equation
332
(4.9)
ppq =
(4.10)
pRq = Z(a0Oc- (t' Ft)t) + 2(0t - a0a)
(4.11)
ppq = 2(t2P - t(P.t) - a0µxt + iaxt) _ (qpp) +
(4.12)
-a12 +
2(R - a0710) - Rt _ (q3p)+ , ppp = a0p
The 'operators' f+ and h+ clearly act on the 4x4-matrices as projections (onto the spaces of matrices commuting and anti-
commuting with h0
,
respectively), and we get
f++ h+ = 1
(4.13)
,
i.e., f+(x) + h+(x) = x
,
for all x
.
In the same sense (regarding f,h,etc. as operators on matrices), f+= f2 = f+ , h+ = h2 = h+
(4.14)
It illuminates the role of we and 0c above to note that (4.15)
h+(Oc) = 0c
,
h+((00) = c
while h(a) = iOc , h(Oc) _ -iwc
(4.16)
,
h((oc) = h(a) = iOc
Also note (4.17)
h(µ) = i(Rt-a0a) , h(p) = iaxt
,
h(1) = -ilrt
Furthermore:
(4.18)
pc= peq = paq = 2 ( c+ iOc), qc= q cp = gap = (c-Oc) , 2
p0c pOcq = 2(Oc- iwc) Or, in terms of w and 0
, q°c= gOcp =
2(Oc+ ic)
:
pw = 2(w + ifr0)
q(0 = 2(w -ifr0)
,
(4.19)
pO =
2(O - iw/fr)
,
qO =
?(T +iW/fr)
First we verify (4.7)-(4.8), using (0.15): 2pa1 = a1+ a0sa1 + t1a12 + t2a'2a'1 + t3a3a1
333
10.4. Some identities for the Dirac matrices
= al +t1+ a0µ1 + i(p2t3-p3t2) confirming the first component of the vector equation ,
2pa = a + t + i(a0µ + pxl) = a + t + i9c
(4.20)
(the other components follow analogously). In (4.20) we may take the self-adjoint part: pa + ap = a + i;
(4.21)
.
Applying p from left and right, using (4.4), confirms the first relation (4.8) (the second follows similarly). Then (4.20) yields 2paq = 2pa - 2nap = a + l; +i0c -t(1+90) = a - got + i9o= c+ i9c, confirming (4.7) (the second relation is the adjoint of the first) From (4.7) we conclude (4.22)
h+(a) = c
, h(a) = i0c .
Then the evident identity h(h(x)) = h+(x)
(4.23)
implies h(iOc)=h(h(a))=h+(a)=wc , or, (4.16). Also, (4.15) is a consequence of the projection properties of h, i.e., h3=h eic. For a confirmation of (4.12) focus on its first component: ,
(a t)a1(a t) _
(a1t1+(al;)-)a1(all;l+(at)")
_ t a1 - a1((at) )2 + 2t1(at)a1t2
=
(with (at)-=a 2t2+a3l3) (where ((at)")2=t2+l3) (using (0.15) again)
Again the other components follow similarly. Next for (4.9) proceed as for (4.7)-(4.8)
2p(1= (I+a0) -it'µ , p1+Rp=f +a0 onto which p may be applied from left and right for the second relation (4.9) . Then also the first follows. For (4.10): Multiply (4.20) from right by 1 for (4.24)
2pµ = (µ -(pp)xt) +i(Rt - a0a)
This implies (4.25)
2h(µ) = 2pµ - 2µp = 21(f3
- a0a)
Applying "h" to this: (4.26)
h+(µ) = h(h(µ)) = ith(R) - ia0h(a) = t(µ't) + a09c
10. The invariant algebra of a Dirac equation
334
Using that (4.27)
pxq = 2(h+(x) + h(x))
,
for any x
we get the required formula (4.9). Next, for (4.11): 2pp=s+iaxl; , with a self-adjoint matrix s. Thus the second formula (4.17) follows. As above we then get h+(p) = ih(a)xt _ -Acxl; , and (4.11) follows. Es Finally, (4.18),(4.19) follow from (4.7),(4.8), and (4.27). 5. The first correction z0 for standard observables. Using the identities of sec.4 we now are equipped for an of the explicit discussion of the 'anti-commuting part' first perturbation symbol, as defined by (3.10). It will be more difficult to also comment on the propagation of c$ of (3.10).
These symbols may be set zero for t=0, but the might propagate into nonvanishing quantities. We get restricted to the case of a symbol tcm commuting with which equals multiples of the identity xt in the two eigenspaces S. of h . Such symbol may be written as x=x+p+x q, with complex-valued xt Introduce ox=2(x++x-) , sx=2cf (x+ x-) .
Proposition 5.1. The symbol z0=2cfrh((h,x)) of (3.10) is given by
(5.1)
z0 = ih((p,(Yx)) - icfrh+((p,Si)+) - ioxE.h(pit)
where we have written (p,S)+ = pli;.slx
E
+ pIx.6I
Proof. First assume that x=0 . Write x as xp h=chi cfr(p-q), by (4.3). We must compute z0=
,
'
= VIx
assume V=0
27r
,
.
Then
h((fr(p-q),xp))
Calculating (fr(p-q),xp) we get (fr(p-q))'=fr'(p-q)+fr(p-q)'
, and
(xp)'=x'p+xp'. Substituting this we need only consider terms with one derivative on p or p-q and the other on fr or x , due to (p-q,p)=0 and h((p-q)p)=0 . Thus we get (fr(p-q),xp) = T1 + T2 = (5.2)
where
.
10.5. The first correction
335
h((p-q)plx)=p(p-q)plxq-q(p-q)plxp--h+(plx) so,
,
Also,
h(T1) =
h(p(p-q)lx)=2pplxq=(h+h+)(plx)
Summing we
so,
h(T2) =-frKlxh+(plt) -frxltsh+(plx) + frh((p,K))
get
h(T1+T2) =-f2(fr)l h+(plx)-f2(i)lxh(plt)+frh((p,K))
.
,
.
In
our present case we get 2aK=x=cfrSK, hence z0=2f h(T1+T2) = -icfrh+((p,SK)+) +ih((p,ax)) r
(5.3)
in agreement with formula (5.1). Now we may do the same calculation with xq instead of xp . It is left to the reader to confirm that the same formula (5.3) follows, now for 20K=x=_cfrOK. Summing over the two formulas we get (5.3) for K=K+p+K q , still with V=O.
Finally, to account for a nontrivial V(x) we evaluate implying h((V,xp)) _ Similarly, For k=k+p+x q we thus get
h((V,icq))
(5.4)
2f r h((V,K)) =--(K+ r
=-iESKh(pix)
Adding (5.3) and (5.4) we get (5.1) in the general case. Q.E.D. It is useful to offer (5.1) in a different form. Recalling
(4.3) we get 2(p,a) - (rj0,a) - (a0,a) fl + (i;,a) a , where (t/o) _ (n,a) + a (a0,a) . Thus 2(p,a)-(a0,a){1+a$-} + f (p,a) . Or,
fr
0
(p, c) =
r
0
(p-q) +
0
(n,o) .a , h((p,a))
r
Conclusion:
ih((p,ax)) =2f (aKn) .0
(5.5)
r
With exactly the same argument we get (5.6)
E.a , -icfrh+((p,sx)+)=-
-iB
n,Sx)+.uc
For the last formula we used (5.7)
h+(a)=ih(Oc)=uc=a-(a03+al)l
Also, we get (p,sx)+
a0
.
(a0'sx)+(p-q)+-(n,S)+ and h+(p-q)=0, etc.
r
336
10. The invariant algebra of a Dirac equation
Summarizing: Corollary 5.1. Formula (5.1) may also be expressed in the form
(5.8)
z0
fr(oK,n) Dc+ r- E.Ac- 2c(OK,n)+.wc , E = Vlx
0C and we of (4.6). The hermitian symmetric part of this symbol is 1
(5.9)
z0 ,S 2
+ OKE}.Ac
fr{(°K,n)
Let us now look at the explicit form of this perturbation for some common dynamical observables. To get an idea of orders of magnitude we reintroduce fi#1 The Dirac equation (0.7) with H of (0.9) coincides with that for n=1 if we replace m,V,A by m/n,V/n, A/n respectively. Accordingly the corresponding substitution in :
,
(5.8) and (5.9) will lead to the corresponding terms for general n One finds that the very same formulas will remain intact, except that now must be replaced by n . In all details, we get .
1
z0,S2f r {(ax,n) + WOKE}.6c
(5.10)
n=n
CA
, fr m2 az +n2 , aSmfr
,
,
, where now
Oc=aOµ+pxi;
M=fr, aK=2(K++K-) , OK=Tr(K+-K_)
with ict denoting the scalar symbols in the two eigenspaces of h For space, momentum, and angular momentum, electrostatic potential
mechanical momentum, relativistic mass, energy, particle density we all have initially x+ x_ OK, 6=0 - the initial observable is a multiple of the 4x4-identity. However, as time progresses, this will not remain true: We know that x+ will propagate along different particle flows. Still it is interesting to note that an initial correction is needed. (We decide for the moment to set c+=0, at t=0). We get the following list of corrected observables. All of them are stated for t=0 only. In all of them we use the term kc
2f
r
ec = 2f (aOµ+pxt) r
(5.11)
Position
(5.12)
Momentum p=-inax
fr
'
,
a0
,
l; of (5.10).
x = (x1'x2'x3) : xcorr = x - kc
:
c 'cjAj1x
pcorr = n
j
(5.13)
Angular momentum k=xxp
:
kcorr
c
0j(xxAjlx 7
10.5. The fast correction
337
(5.14)
Electrostatic potential V(x): Vcorr V-Xc H
(5.15)
Mechanical momentum p-oA
(5.16)
Relativistic mass M=H-V :
(5.17)
Energy H :
(5.18)
ncorr !BxXc , n- cA' E .
Hcorr =
Particle density 6=6(x-x0) : Scorr
The last observable (5.18) only has a formal meaning since the Dirac function 6 is not a symbol in tpc In (5.11) -(5.18) x corr 6corr each denote the corrected symbol, not the yido. We have added only the first correction, while an infinite number of lower and lower order corrections remain to be worked in. Perhaps it might be time well spent if we now look at a possible physical interpretation of our mathematical results. From the view point outlined in sec.0 this will amount to a study of the spectrum - or the spectral theoretical properties - of the self-adjoint operator A=a(x,D) = x(x,D)+zS(x,D)+... where x(x,D) is one of the classical observables of (5.11)-(5.18). These x(x,D) .
,
are self-adjoint operators of H , with their domain properly defined. The newly proposed observables consist of x(x,D) with an additive perturbation (zS(x,D)+..). We thus should study the influence of the perturbation on the spectrum, the (generalized) eigeniunctions, even a properly introduced spectral density function, etc. The mathematical difficulties liable to occur in such investigations are well known. Moreover, it is clear that our mathematical approach does not define a unique corrected observable - an additive self-adjoint operator of order -- remains completely arbitrary. Such additional perturbation PE O(-oo) clearly is a compact operator of H . Any P=2y1)(y2 , with a finite sum and
S(&n)
will qualify. An addition of such perturbation could produce any finite number of arbitrarily prescribed eigenvalues and eigenvectors. There are examples of compact perturbations with very singular effects on the spectral resolution of a self-adjoint operator. Such effects are physically undesirable or impossible. In fact, the cut-off procedure used in constructing the asymptotic sums of I,lemma 6.4 appears physically undesirable.
10. The invariant algebra of a Dirac equation
338
Our hope to derive significant physical information perhaps should depend on isolating spectral properties invariant under since we will have perturbation by PE O(-oo) - or even by PE K(H) to leave higher order perturbations (beyond zs(x,D) ) unattended anyhow. As a first such concept the essential spectrum comes to mind - the set aess(A) of -dimensional eigenvalues and cluster for points of the spectrum. It is known that aess(A+C)=aess(A) ,
,
all compact C. (cf.[CI], p.279, thm.6.2., for example). More refined discussions might be interesting but are beyond the scope of this book. In this section we confine ourselves to more general observations about the operator norm of the first perturbations listed. Note that in all cases (5.11)-(5.18) the correction to the symbol carries the factor Xc which has the dimension of a length. From the differential equations (3.11) of the particle flow we get
= Lfr
(5.19)
=
x'
a0 = 1-x'Z/c2
Using this we get
kc = 2f {pxc + µ I-xZ /cz }
r
2cµ - t c 2 ' as x'<< c
The number ic= ma Z 4x10-11 cm is known as the Compton wave length It often is regarded as a lower limit of measurable length. Our correction term -Xc for the observable x of (5.11) - the position observable - agrees well with this empirical fact: when measuring position, there will be a term z XC(x,D) not in diagonal form, with expectation value depending on the physical state, but of the order of magniture of tC More precisely, p., µ3 . have matrix norm 1. With tC as above, 3x10-13 and the 'classical electronradius' pC= = cm write .
(5.20) Xc= tCµC, µc I
-pCA/e)-2{µ+px(tC -pe/e)}, (z) -1+ z Z
.
It is not hard to estimate the L2-operator norm of µc(x,D) using III, thm.l.l. There we were not interested in a numerically useful bound. Estimates given in [cc] on the basis of our proof in VIII,2 might be handier, for present needs. There we prove that (5.21)
IIµC(x,D)IIL2 s Coll(1-Ax)(1-e
C0=4x,[[L]]
with the trace norm [[.]] of the inte ral operator L on 13 with kernel L(x,y)=e-ixyt(x)s(y) , z(x)=e 1x1/lxI (cf. [CC],p.124) .
The trace norm may be easily estimated. For example, we obtain L=
339
10.5. The first correction
4nMN , with Schmidt class integral operators M,N , having kernel
g(x)/2-lxl Thus
L] ]:5
t(x-y)
i(y)(1-Ax)e-ixy-(x)/2
and
,
,
respectively.
[[[M]]][[[N]]] with Schmidt norms at right. By a
, [[[N]]]s10n
crude estimate, [[[M]]]s4x
,
so cos4V <4.
This and formula (5.20) (which allows easy calculation and estimates of x and t-derivatives of µc) shows that, indeed, the norm In fact, for A=0 we get the simpler PC11 is of the order of is for general A we still note invaestimate II%c(x,D)II 2=IIkc(t)II .
;
L
Loo
riance under change of unit of lenth: In (5.21) we may replace II(1-0x)(1-Aµc(x,t)IIL by inf{II(1-POx)(1-P )Fic(x,t)IILo:O
order of magnitude sl
L00,
II(pc lxx)/eII
L
have
Then indeed the correction to of the position observable should be of the order of LC. .
Looking at the correction of momentum in (5.12): For vanishing potentials A the first order correction z0 vanishes. Generally we will get a first perturbation -hp cDAcj(Ajlx/e) with norm of order :c, assuming the above condition, regarding IIAjIx/ell
.
Note,
Loo 11
p=4 ax is the canonical momentum, defined as variable canonical to
x
in the sense of Legendre theory. As A#0 the mechanical momentum of (5.15) is the standard physical observable. It also has the more meaningful correction term BxXc of order Lc whenever IIBII LOO ,
eIIBIxII
IIBIXXII o, are of order L
L
co
.
Similarly for (5.14), (5.16),
with the magnetic field strength B replaced by the electric E. In other words, the norms of all above correction terms are small or negligible as long as the potentials V(x) , A(x) and their gradients do not assume excessive values. On the other hand, a potential wall, such as V(x) = V0 sgn(x1), or suitable approximation, having slope near x1=0 comparable to 1/Lc produces a perturbation E Ac of the observable V(x) which seems no longer negligible. This fact might serve as an explanation for the so-called Klein paradox of Dirac theory (cf. [Sm]) . Similarly, a Coulomb potential V(x)= -e2/lxj is inadmissible of course, due to its singularity, but will become admissible, af-
10. The invariant algebra of a Dirac equation
340
ter "capping" the singularity, in an ever so small neighbourhood of zero. Then, however, one will have to account for the perturation term FAc of the observable V(x) . To make it negligible, One would have to arrange for a "capping" - replacing V(x) by a E small, in such a way that V1x=E« X. smooth function in Ixl%E Physically one might argue that the potential V of a hydrogen atom might not well be measurable very close to the nucleus - the first electron orbit is about 100 times larger than Lc . Actually since the mathematical requirement for the observable V(x) seems one even might to require the presence of the term E.ac near 0 ,
,
ask about the forces such an additional potential might produce. Note that forces other than that produced by E = grad V are known to act on a particle getting very close to the nucleus. It might be interesting to investigate whether such forces can be explained from our above correction of the potential V. So far we have not looked at the angular momentum observable
kcorr of (5.13). Again, this observable is of particular interest in case of A=O. In particular it leads to the common definition of the spin observable. It is known that the operator J=k+Zp commutes with the Hamiltonian H whenever A=O and V is rotationally symmetric. Indeed, look at If V is a function of jxj only, then we get [k1,V]=o since kl contains only angular derivatives. Thus [J1,H]=c[J1,H0] , HD h0(x,D) ,
,
Clearly [J1,P]=0, hence [JI,H]=yc=l{icak[kl.axk]+2ic[pl. ]axk} Here [kl,ax
]
and [pl,ak] may be calculated, using (0.10), (0.12)
k
and (0.13), and A=O. One finds that [J1,H]=0 . Similarly for J2 J3 This formal commutativity indeed implies a spectral theoretical one (i.e., the resolvents and spectral families all commute), as we will not discuss in detail. Commonly this is used as a motivation to interpret the (constant-matrix-valued) vector 2p as the spin observable, so that 3 becomes the total angular momentum, satisfying a conservation law: eitHJe-itH=J Since J and H commute, Jt is independent of t. Note, we have JE Opipce, and JE PCe follows trivially, since it const. With kcorr of (5.13), j=symb(J) we get ,
.
(5.22)
j=kcorr+2s
,
a = f(p)=fr2(m2czp + (n p)n + mcµxn),
assuming A=O, but not necessarily rotational symmetry of V. Indeed
10.5. The first correction
j(x,)=ftxx+p=k li 2 corr
r
2
lc r
corr 2
s
2
341
cx
2
. Here 2
I
Using 1=IzI2+a0 we get t=p+21\.cxt=a0p+(rt)t+a0µxl;, in agreement
with (5.22). Finally, to show that t=f(p), note that f(p)=p-h+(p). From (4.11) get h+(p)=pPq+(pPq)+=ItI2P-(Pt)t-a0µxt . Hence t=f(p). In the frame of our present discussion it thus is natural to interpret the matrix-valued symbol s=2t E y'c0 as the (symbol of
the) spin obervable. We then get the angular momentum conservation law for J=kcorr(x,D) 2 (x,D) as above. Note that t=f(p)=p+pp++p_pp_ commutes with h. Thus the new ,
spin 2t will be a basic observable (def.7.3), qualifying for the correction procedure of thm.2.1: A correction zE Vc_e will give a symbol of a z(x,D)E PCO. On the other hand the classical spin symbol 2P does not commute with hence cannot be corrected. There might be a physical objection: The spin is supposed to measure always t2, while the operator t(x,D) -or 2(t(x,D)+t(x,D) ) has continuous spectrum, different from tl. To counter this argu-
ment with physical reasoning: For t = c. << 1 we will have t ar p so that one still might expect to measure t2 except for relativistically large velocities. (There is a similar contradiction with the Stark effect's Schroedinger operator. It has purely continuous spectrum, while, of course, spectral lines are observed.) The above first perturbations are only those encountered for t=0. As time progresses, the symbol xt will each flow along their corresponding particle flows. Clearly they no longer will assume the same value at each x,t , even though they did initially. Thus we now must focus on the second term of (5.10). Write it as ,
(5.23) T2 = 2Ax.Vpt/Et
Ax=x+-x_ , Vpt=E. c , Et cfr m2 c4 +c2 n2
Clearly Vpt is the correction of the observable V of (5.14), containing the 'noncommuting Compton wave length' Xc. Again this may be estimated in norm and found negligibly small, assuming E<<1/Lc As a common experience, many negligibly small quantities have non-negligible time derivatives. This prompts us to consider time-derivatives of our new observables. (3.6) and (3.9) implies
(5.24)
x'=p(h,x)p + q(h,x)q = f((h,x))
.
Here x=x+p+x q . By (4.2)3 we get p(h,x q)p=q(h,x+q)q=0
(5.25)
,
hence
x'=p(h,x+p)p+q(h,x q)q = p(?,x+p)n+q(X_,xq)q ,
10. The invariant algebra of a Dirac equation
342
and similar where (4.2)2 was used, noting that for q , since xt are scalars. Finally, using (4.1), (4.3), we get
(5.26) x'= {,x})p+(%-,x-)q=2((X,x+)+(X-,x-) We reintroduce ax, Sx with xt=axtcOxfr ,
Xt=vtcfr , and get
Hx' _ ?E.axI+2f sn.(n,ax)(MCP+a.x)
(5.27)
r
-
+ 2 2 n.(n,b
r
E. (frSx) i (mcfi+a.t)
in (5.26) we had 1=1 , but (5.27) checks for general 8
,
if the
quantities of (5.10) are used. Let us use (5.27) to calculate A' of (3.1) for x=x - clearly this should be the velocity observable v = A' We get (at t=0)
v = f 2 (mc2 I3+ca a) =
(5.28)
r
Clearly the and with (5.29)
o f (p-q)
r
components of the vector v commute with each other as well. As eigenvalues one obtains
tnf
^-
, as x' << c
t
r
In turn, for the velocity (5.28) we have xt=tnf , a=0 r
2nf . Substituting this into (5.27) one finds (at t=0) r
(5.30)
a = v' =-2- (eBxl + (E-(E r
with xt = f (eBxt t (E-(Et)1)
x+p + x_q ,
r
with the electric and magnetic field vectors E and B , as a calculation shows. In particular, using (5.19), we get (5.31)
xt =-m 1/c2{cBxx't(E- c (Ex')x')}
corresponding to the classical law of motion of a particle of (relativistic) mass m/ 1-x'z/cz under the electromagnetic force t(E-c (Ex')x') cBxx' Actually, the above could have been derived more directly: We know that x=x+p+x q, where x=x0ovtt with vt of (3.12). We get (3.11) as propagation law. For xt=x this gives (5.26). In fact, (5.31) follows for all t, not only t=0. We offer this derivation to show that there is no "Zitterbewegung", and that the paradox of velocity components, in classical Dirac theory, does not exist.
343
10.6. Proof of the Foldy-Wouthuysen theorem
A similar argument shows that there no longer is a difficulty with the current observable. It was argued before that the difficulties are resolvable by means of the Foldy-Wouthuysen transform of thm.2.2. Possible our axiomatic necessity for corrections could be a more natural way. 6. Proof of the Foldy-Wouthuysen theorem.
Let us next discuss a proof of thm.2.2. One might expect that the desired unitary operator I will be constructed as asymptotic sum prj , Ij=Xj(x,D)E OpVc4je , by an iteration similar to that of the proof of thm.2.1. Assume that X = Xj(x,D)E Opipc4je
have been constructed, such that
,
j=0,..,N,
4=0 Ij = I satisfies
(6.1) I*I=1-r , I*HI=A-A
-Ne-e,-Ne-e2,e',resp.,
with a pseudo-diagonal matrix A. Here and in the following 'pseudo-diagonal' (or, yrdiagonal) means that the upper right and lower left 2x2-corners - called A0R and ALL - of the matrix A vanish. We shall construct Sa=SN+IE O 4Ne-e such that Y=I+Sa satisfies (6.1) for N+1. By induction we then get an infinite sequence {Ij} with IO,...,xN satisfying (6.1) for every N=0,1.... The asympto.
tic sum jXj then satisfies (6.1) for N=co. It will provide the de-
sired unitary operator after a final correction in OpVc40 *
For N=O let Xo q? , I0= cp(x,D) , with T of (2.10). Then k+' ,_, ._) -A0 , AOE Op1Uc4e2 , by calculus of yddo' s ,
IOHX-diag
confirming (6.1) for {*0}. For the above as we get f' "E= OpVc_2Ne-2e C Op" 4 Ne-e' and I n=om n, mod O(-Ne-2e). Hence (6.1) for X at N and Y at N+1 implies (6.2)
I*Sa + fa*I
=r
and, in terms of symbols, using
(6.2')
(mod Op)c4 (N+2) e ) ' (I-4))*nE O(-Ne-2e)
qp+co+(g) +(o)+ = qp+w+w cp = y
,
(mod O(-(N+2)e))
this yields .
It is possible to satisfy (6.2') sharply - not only modulo O(-(N+2)e) : Just choose +w = I+i0 That is, t=r(g+w-w+p)-s+ ,
(6.3)
w = q(y/2+iO)
,
0 +=0, 0E Vc4Ne-e
.
10. The invariant algebra of a Dirac equation
344
Having thus Y*Y-1E Op1c4Ne-2e satisfied, we seek to determine 0
to also get the second relation (6.1) for Y and N+1 - i.e., Y*HY= X*HS+S2*HX+X*HSt+c*Hc
= A.N+1-AN+l with yr-diagonal AN+1E O(e')
and
,
general AN+fE O(-(N+1)e-e2) . Here we have n*Ht2E O(-2Ne-a-e2), and %*Hx=A-A=AN-AN , ANE O(-Ne-e2) . One may assume that the operators AN , AN both are hermitian, implying that the symbol 6N of AN sat-
isfies 6N+_6 E O(-(N+1)e-e2 ). Also, again St*H(X-D)E O(-(N+1 )e-e2 ), hence the symbol of rZ*HI equals w+hcp (modulo yrc4
. All
(N+I)_e2)
together, we obtain the relation a*H(P + *Hc = (AN+l_AN)+AN (mod O(-(N+1)e-e2) and, for symbols, we get - with Xj= symb(Aj) ,
(6.4)
+hw + w+hp = (XN+1-XN) + 6N (mod yrc4(N+i)e-e2)
Here we substitute from (6.3), noting that Y=YN, bN are given, but
the hermitian symbol 8 E Vc4(N+1)e still is arbitrary. We get
(6.5)
+hw+w+hcp =1(Y+i8)+(. -i0)1 4(1Y+Y1)+i[1,8]=(?N+1-4'N)
+0N
4 using (2.6), where )`N+1 and 0 are to be found. mod Uc _(N+1)e-e2 Again we regard (6.5) as a sharp equation, after still replacing 6N by 2(SN+6 N) (6.6)
.
That is we seek to solve
[1,8] =2{(1Yx_ON)+(1YN-ON)+} - i(?N+1.?N)
A hermitian symbol 0 solving (6.6) will exists if and only
if XN+1 is chosen according to kN+1-kN 0'
where v'
,
u4 J
v° are
,
v24J
the yr-diagonal corners of Z-2(W+W+) , W=1YN SN
,
i.e., Z= v' .
Then we will have 8= (e 0,
e2 04
, with arbitrary hermitian 0'
and 02 =0' + satisfying (X + X_)02 = v2
(6.7)
+v
0
'
0 l
1
04
In other words, we get i
0 v2
0
2)
with arbitrary skew-hermitian 0'
.
,
0\{
+0
4J
04 (-= "2 (N+1)e and vj from the matrix Z above. In particular, 0 obtained this way is hermit,
ian since the matrix iZ clearly is skew-hermitian, giving v3 _U2+ Evidently we have W=lyN-SNE O(-Ne-e2), giving u E O(-Ne-e2), hence v2=v2+E O(-(N+1)e) , exactly as required.
It is now clear that the construction of %j is possible for
all j=0,1,2,..., and we may form the asymptotic sum %_
.
`7-0%j
345
10.6. Proof of the Foldy-Wouthuysen theorem
Then it follows at once that (6.1) is valid for all N, for the redefined X . In other words, we now have A,T of order -oo. We plan to make another correction of X such that it actually becomes unitary. First we conclude that 0 E Op1yc0 is a (bounded)
Fredholm operator of B, by II, thm.6.2 and III, thm.4.2, for any choice of potentials Aj and V, since the symbol q of (D is uniform-
ly non-singular at lxI + Iii = oo. Moreover, if the potentials are zero, then 1 is unitarily equivalent to a multiplication operator by the Fourier transform F.
It then follows that O is invertible
(even unitary). In particular, T has its Fredholm index equal to zero. For general potentials we introduce a family p(s) of symbols sV , 0sss1. A calcultion corresponding to the potentials sAj ,
shows that q(s) is continuous in 0sss1, as a map into the Frechet space 1pc0 , with the semi-norms induced by (6.6). Hence the corre-
sponding operator family D(s) is norm continuous, by III, thm.1.1, and index((D)=0 for potentials satisfying our general assumptions.
Observe that X=DFC , with C=
X1E Op1pc_e compact in L(B),
by III, thm.5.1. Thus X is Fredholm as well, and index(X)=0. Hence dim ker X*X=dim ker XX*. We get 1-XX*=TE O(-o) 1_IX*=V E O(-e). ,
Proposition 6.1. The null spaces of X and X* are contained in S . Proof: For u E S, = fl(B5:sE 12} with Xu=0 we get u =-1'u E S, since
T is of order --, hence takes S'to S (cf. III, sec.4). For u E S' with B*u = 0 we get u = -Fu = (_r)ru , r = 1,2,..., hence uE Bs-Ne for some s and all N=1,2,..., which implies u E S , q.e.d. Let us choose a basis wi,..., r of ker X*X , and a basis VIP .... Vr of ker XX* (both orthonormal), and let
(6.8)
X- = X+St
, n--Y, (o j) (1pj
Clearly we have n E Opyic400 .
, with {(o) (1p}u=(V,u) w , uE B.
In fact,
(6.9) where indeed the symbol is in 1Nc400 Note that X- of (6.8) is an invertible operator,since
(6.10)
*
X- *X- = X*1 + st*a + St*X + X*sa = X X + Dpi) (Vi
,
where the two operators at right are self-adjoint and positive definite in (ker X*X) 1 and ker X*X , and zero in ker X*X and (ker X*X) 1 , respectively. It follows that X-*X- is self-adjoint and positive definite in H , hence it is invertible. Similarly one
10. The invariant algebra of a Dirac equation
346
concludes that %"%-* is invertible, hence S' also is invertible. Since I is determined only up to operators of order -- we may regard tt as another correction of I, and hence we may assume without loss of generality that I, as constructed in this section, is
invertible in H. We then may introduce (6.11)
(1-r)-1/2
r^= (X*I)-1/2_1 =
_1 =-2r/n
N
-r+% 2)d%
0
Proposition 6.2. P is an operator of OpyJc4 -00 Proof. With another change µ =arctan X of integration variable get n/2 (6.12) (1-rcos2µ)-1 cos2µ dµ r^= -217/n 0 .
This is now an integral over a finite interval with norm continuous integrand in H. In fact, we observe that 1-rcos2p is continuous in norm convergence in every Hs, for Osµsa/2. Since it is invertible in H it also is invertible in every Hs - actually, that inverse is a Green inverse in the sense of 111,4. Hence the integrand is norm continuous in Hs and the integral exists in every Hs in norm convergence. Thus the integral is in 0(0). Due to the q.e.d. factor r in (6.11) we then get r^E O(-oo) ,
We have (I+Ir^ )*(I+Ir^ )=(1+r^ )I*%(1+r^ )=(X*I)'h I*1(%*I)'h =1 . Hence Xr^is another correction Ir^E Op" 4O, of %, which will effect
that I is invertible and %*%=1 The proof is complete.
.
In other words, Z now is unitary.
Let us ask for an explicit form of the first terms of the resulting asymptotic expansions (6.13)
x = sy
(I) = Lj=OXj . symb(A) = X0
We started setting x0=g , making, as a possible choice,
(6.14)
( 1+ao -µt) , X0= 1 = V+cfri
x0=
6 of (6.1) for Looking for X1 x1 , we calculate symbols y N=O. By I,(5.7), (5.9) we get (modulo 0(-2e) or O(-a-e2), resp.) ,
(6.15)
,
y=1-symb(gg(x.D)*(p(x,D)) = i(Tl+ 6=1-symb(W(x,D)*HT(x,D)) = i(yl+
cp+g1
(l1x)
hq)+Tj+(hT)jx+T+hjtTjx)
Recall, that we agreed to choose 6 hermitian. A calculation shows that 6 of (6.15) satisfies 6-6 using This has the effect that we still have (6.6) and (6.7) true, but
10.6. Proof of the Foldy-Wouthuysen theorem
347
must replace the second formula (6.14) by 0 = 1 + 2 Ix
(6.16)
= V+cfrP 2frlAklxjtjtk S
Confirm that y of (6.15) is hermitian - we have Y-Y+=i(c+(P)Ixt=0
.
Calculating Y and W=Y1-S, introduce X=T+qIx, E=c+cpIt, and get
Y=i!Ix , W = Y1-6 = -ill ItX-!1Ix + [=-,1]X}
(6.17)
.
Indeed, the expression for y follows from (6.15). With 1=cp+hcp get ro+hItTIx} _
w=i{4p,+Tlx1-qI+(hy)lx ro+hltTIx}=i{(pI
1pIxl-TI+(T1)Ix
Notice that X and ° are skew-hermitian, due
to X+X+=(g+T)Ix 1Ix=0 E+9+=1I=0. We get Substituting this we get (6.17) for W For the further evaluation it now will be necessary to cal,
The proof of prop.6.3, below, is postponed to the end of this section. culate the two matrix symbols X=cp+pIx and r=p+(pI,
We have
Proposition 6.3.
X=
-i I x`k ' n - 2fr
e
(6.18)
where
E _
+ao txa
tE,
_e
I
_4(-(tk/fr)InjAjlxk
(tk/fr)Inj={fr(akY}i,=-(tj/fr)(ak 1+a0tk)
tk fr
-T =f r2 {? jak
kj-tktj) fr 1+ao Ylal(ajl-tjtl)tk} (a0 )t jtk +T s jk -23+ao jtk +tko j}
a+a0
+I+ao jtk fr(I+ao
=fS{(tkoj-tjak) +
(6.19)
Et
t - a -T+-a0
'
First calculate (2)UR 2("Ix)UR Here
.
Sjk - 1+aa'
(at)tjtk}
-4cfrr{a. (Bxl;) +
(2)UR = (2)LL
.
It follows that
1_ t+ao FAj Ixktjtk}
We also need (2)UL-(S)LR= VE/fr)Ix=cY-(Ek/fr)InjAjIxk , for the hermitian ,p-diagonal part of cp+xl . Here we get (Ek/fr)I,
_{ (txa)k/(mc+ m2 c2 +n2 )}Inj
r
1+ao
( (ej-t jt)x(y)k- fr (1+ae ) k =
= f r2 (1+a° ) (ejxa)k -f r2 (i a. )2 tj(txa)k . Accordingly,
10. The invariant algebra of a Dirac equation
348
(6.20)
(2 )UL (2)LR 4fr- T
1+ao
(OB) +4of 's
2 LF jA jlx(,xa)
Next we calculate W = y1-S=-i1I X+i!1IX i[8,1]X=W1+W2+W3
Get W1=
2f Z
ZEIk, hence Z1=l(wl+wl)-
r
hence
Similarly, W2-2f E.(LE2 DjIx tjli -E) k r k
J
e =If-r E.E - 2fr I
2
-0,k
j I xk
j EkS
K'KZ
' where
x' x' )
2fr(1+ao ) (Et) (ta) , while
K
0' x4
Thus Z 1 +Z2-
10, +2f r (Et) 01
}=2fr(1+ao )E. (l;xa)+2fr(1+ao !(AklxjA j Ixk)t j(txa)kR
(6.21) I 2fr( +ao ) {E.(txo)
+ e!.((1,x0)xB)3}
=2fr(1+ao ) {E. (l;xo) + e(l a) (BH)P} -2fr (1-ao) (Ba)I
Finally, [ =-,1] =ic E0i 0) , hence W3=2fr
kIx
J
t j (e - ) k' Therefore,
4r
Z3)UL -(Z3)LR
+B2[t3'il]+B3[il't211where [r2,t3]=2i{a1
.
1[Z 2' 3]+
l+aoa.(!x(ezt3-e't2))},
etc., so that (6.22)
(Z3)UL -(Z3)LR =2fr{a0(Bo)+I+ao(al;)(Bt)}
Also, (Z3)UR -(Z3)LL 4fr
(6.23)
klxj[tj'Ek]+
.
A calculation yields
t.(ejxek)
[tj,tk]+=tjtk+T)jj=2(bjk-tjtk) ' [t j,Ek]+
(We have used (0.15), in this connection.) Therefore, (6.24)
(Z3)UR -(Z3)LL
2frz.EJlklxjejxek
=
A =
(131;)
From (6.21) an (6.22) we first get an expression for the
matrix
(6.25)
ro,
v,)=U of (6.7)
U =2fr{1+a-E.(!xp) - e(Bp)l}
10.6. Proof of the Foldy-Wouthuysen theorem
349
using (6.16). Accordingly, we get formula (6.27), below, for X1 On the other hand, for x1 we get the (more complicated) expression (6.26) Indeed, looking at (6.7), with 0'=0°=0, we first assemble (6.19) and (6.20) yield the hermitian part I -of g+x1 ,
.
:
.2
=rcy-T {(Bxt)'µ -I+ao(Bp) +(2+a,
D(t)P +(i+
Next, for the skew-hermitian part, using (6.7), (6.21), (6.23),
c r v' 0)
=4-Cr
a 0) (Bt)
r (E[t -Tao
.
Taking the sum
we get (6.26). Let us summarize:
Theorem.6.4. We have - setting C(t)=
(6.26)+xi=4o r
+4ep r
(++ao
-
{i(Eµ)+eµ(Bxi;) -I+ao(i(E1;)(tµ)+e(BP))}
(µt) -(Bt) (_0
with arbitrary skew-hermitian 01E (6.27)
jAjIxt, D(t)=D. jAjlxxt
00 1
+(2+
y,C?e)_ez
,
3 (PD(T))} +(00'(0)4) j=1,4, and
1 p.(Ext) - e13(BP)} -2f C(t) 1= V+cfrfl + 2f r {- r
The last term in (6.27) is skew-hermitian, hence will not appear in the corrected observable. Up to terms in O(e'-2e) we get (6.28)
A = V + 2cfr(Nw,D)P +(2f {1+a.P.(Ext) - e1(Bp)})(Mw,D) r Physically, one should regard A as a corrected energy obser-
.
vable, cf. [IZ], (2-82), for example, where such expression is obtained with the third kind Foldy-Wouthuysen transform, evaluating terms. Note, the symbol (f+x1 belongs to (the expansion of) a unitary but nonselfadjoint operator, not an observable. Its physical interpretation might be more meaningful in Weyl representation. Some ingredients of the symbol x1 of (6.25) will show up in the coefficients of the differential equation governing time propagation of "corrections" - as seen in sec.7, below. Finally we discuss a proof of prop.6.3. First we neglect real .b-c=uI multiples of the 4x4-identity matrix I , and write b=c 4 4,
uE I x+x
l
From (6.14) get+(P nc a 2+
ao
x+x I xL
, x=1+ao -iµt , where
nl=L (1+ao +iµt) (-ao tl-iDL (6jl-tjtl))=f-{-ao tliµt-i(I+ao )µl
Using (0.12) and (0.15)(b) conclude that (µi)2=ItI2 is a multiple of I4. By the same formulas,
(µ0
so that x+xlnl.fr{-(1+ao)(µl-
tl)+(Pxt)1}
10. The invariant algebra of a Dirac equation
350
confirming (6.18) modulo a real multiple of 1. Both, q+q
if
1,
and
,l are skew-hermitian. Hence we have equality, not only "_",
since no real multiple of 1
4
except 0 is skewhermitian.
Q.E.D.
7. Non-scalar symbols in diagonal coordinates of h(x,t).
We return to consideration of observables in our algebra PC= PCH , for the operator H=h(x,D). Looking at the first perturbation symbol of (3.7) we have calculated z0 in sec.5, for a variety of well known observables, for t=0, as well as for general t. We even developed a general formula (5.10), valid for x-x+p+ +x p- with scalars x$ The calculation of z0 for the general case where xt are operators on St remains to be done. Also, nothing was worked out for the two symbols ct of (3.7). We know of course that one may set x*=0 at t=0 - as perhaps physically indicated. On the other hand, for ta0, ct will have to satisfy cdn.'s (3.19), i.e., a pair of differential equations. They will be determined as solutions assuming the given initial conditions at t=0. We will now look explicitly at these equations. The formulas will look more transparent if we work out the form of the quantities of sec.4 in a coordinate system of T4 having the symbol matrix diagonal at all x,g. A unitary matrix symbol cpE Wc0 diagonalizing h(x,t) was defined in (2.10). We get .
(7.1)
p(x.l;)
(1+ao - iµ t)
2
In effect we are performing a "zero-order-Foldy-Wouthuysen transform at the symbol level".
According to (2.6) we get (7.2)
h
= (P +hq
=
l
0-J
,
p = *+pcp = (0 0) , q =
q)+qcp - (o 01)
.
and for matrices g,f satisfying the assumptions of prop.4.1 in the form pg = gp = p , of = fq = f generally gf- are of the form ,
(7.3)
g = Co o,
f- = (o v)
All matrices in (7.2), (7.3) are 4x4 , with 0
.
,
1
, u , v , all
denoting 2x2-blocks (0-matrix, identity matrix). For a 4x4-matrix x let generally x- = (P +xcp . We ask for the
10.7. Nonscalar symbols
351
explicit form of the transformed matrices of the quantities of (4.5)-(4.19). To simplify the calculation note that x - I+a0-iw and x-I , differing from p by a scalar factor, may be used instead of p : We have -(2+2a0)-1x+
1 + K= X -1 KX =2+2ao x xx
(7.4)
Thus, using (0.15) again, we get (2+2a0)a = (1+a0)2a + Here we employ (4.12) for (2+2a0)a = ((1+a0)2 + t2)a +213(1+a0))t
(2+2a0)a + 213(I+a0)i; - 2(a t)t Dividing by (2+2a0) we get
a" =a - l+ao t
(7.5) Next, (2+2a0)13
(7.6)
= (1+a0)213 13-
-1;213
.
Or,
=a013 -g .
With control of a and 3" we first get
T10 =P -q =13
(7.7) Then,
(7.8)
We
=a-
= a - T10
Reintroducing the matrices
(7.9) (t1,t2,t3) , t = a of (6.18) we may write (7.10)
0
=
W
-11TIo
it
-it 0,
Then we get
e0 = -i(h(c))
(7.11)
= (0
O)
and, using (4.7), (7.12)
(P«q) = 2(c+i9c) = f0
i)
(qap) = 2(c-i9c) = f it 0, prop.4.1
Next we translate (4.2) of
to the new coordinates.
Using the identities of sec.4, we get (using pp'p=0) (7.13)
x - ?_ 2cfr
Recalling that r10=h0/fr p-q, hence ph0q=0, using (4.7), we get
MO 'q = (f )'Ph0q + f Ph0'q = of Paq = 2f (wc+i9c)
r
Using (4.3), we get
r
r
r
10. The invariant algebra of a Dirac equation
352 (7.14)
PPlxg
(wc+iec)j
fr7-3xjl
=4fr7,jnjlx(wc+iOc)j
Looking at the transformed quantities (7.15)
(PPI q)
we get, setting
(7.16)
2fr7-jxjl
(PPIxq) =
(0 Oj)
f
-
xjlxt0 OjJ
Tjl=fojt1 0)
xllx)Tjl=
rl= (X-%_)(PPItPIxP) =-C (xji
11xjTj1
Calculating as for (6.17), for any symbol g"= f0 0j, we get
(?.,g) = (),,g) + [r3,g]
(7.17)
with the matrices X=T+WIx
,
,
r3= XXI -EXIx
+cpl of (6.18). With this informa-
tion we change coordinates in (4.2). Using (7.2) and (7.3) we get
(7.18)
(P(h,g)P)
_ (0 0)(h,g) (1 0)
a matrix having only the upper left (2x2-) corner different from zero. Accordingly, in transforming (4.2), we are only interested in the upper left corner bUL of every matrix b occurring. We get
(P(h,g)P) UL = ((?,g) ) UL + ([rl,g] )UL
(7.19)
Cor.7.1, below, is the result of a transformation of (4.2)
- as well as its analogue for q and f"=(0 v) - i.e., (7.20)
q(h,f)q = q(k_,f)q
to diagonal coordinates, using (7.16), (7.17), and corresponding formulas for q . We also have linked the "coefficients" of (7.19) to the matrix Z of (6.7) , as occurring in thm.6.3. Corollary 7.1.
(7.21)
For matrices g, f with g""= 1a 0{, f-=(0
(h,g) UL (X,u) +i[ZUL,u]
,
v{
we have
(h,f)-LR= (?._,v)+i[zLR,v]
where Z=2(W+W+) denotes the hermitian part of the matrix W of (6.17). That is, using formula (6.24), we have (7.22)
ZUR
2fr{1+aOE.(txa)-e(Ba)}
, ZLR
2fr{1+aoE.(txa)+e(Ba)}
10.7. Nonscalar symbols
353
Only the second formula (7.22), and the occurrence of W of (6.17) with its hermitian part, still need to be discussed. We get
(4.2) as (h,g) UL (p(h,g)p) UL=(X,u) +
[rUL,u],
with r=r1+r3, rj
as defined above, and notice that r3UL i(W1+W2)UL , with Wj as defined after (6.19). Similarly, comparing (7.16) with W3 , below (6.20), we get rIUL Accordingly iWUL , with W of = iW3UL rUL (6.17). This confirms (7.21)1 , but with Z replaced by W . Observe also that (WI+W2)UL is hermitian, while the skew-hermitian part of W3UL is a multiple of the identity, hence may be omitted in the commutator of (7.21).- Confirm that [tj,i1]+ 2(aj1-l;jt;l), whence 2` .IX 1[tj,v1]+ is a multiple of 1. This confirms (7.21)1. For
(7.21)2 we first look at (7.20), where
-)(qql glxq)
=-2cfr(gpl,plq) _ -2cfr(gpl p) (cIPJ .q) =-2cfr(ppl,q)-+(pplXq) Application of (7.15) will give a formula for r1LR confirming r1LR iW3LR . Similary, a commutator with r3LR=i(W1+W2)LR occurs in the formula for (%_,f)- corresponding to (7.17). We get (7.21)2. Now we return to our solvability condition (3.19). We noted ,
that pw(c-)p = qw(c+)q=0. With (7.21), recalling that c+ =f0 0,
c -=fp Y_), and that after 11-11 all information is in "UR" or "LL",
(7.23)
Y*-(X,Y)-i[ZUL,Y] _
UL 2(h'K)2 UL
Y-*-()- Y-)-'IZLR,Y_l _ (h,t0) LR 2(h'K)2 LR We still may substitute from (7.22). Also, recall that (7.24)
Y,
= Y'- (%,Y) = at(Yov_t)
is the derivative of y along the classical orbits, in negative time direction. Similarly for in the second equation, with the positron orbits. Hence we get (7.23) in the form
Y` +i[
Y
(7.25)
- (%,Y) + i[ (G+' u),Y] = UUL '
Y_ +i[ (G .0),Y J= Y_'-(X Y_) + i[ (G-. a),Y_] = ULR with (7.26)
G. = 2f {eB r
and
±J
T+a 0
(txE)}
'
10. The invariant algebra of a Dirac equation
354
U=(h,z0)" - 2 h,x)2
(7.27)
It will require a lengthy calculation, to obtain U (cf. sec.8). The initial symbol x=0 Here we focus on a case, where U=O ) satisfies all assumptions. It may be regarded of o r t=0 d e r m. We get ( 3 . 8 ) with x$=0, and xt=0, all t. Also o f t hence the right hand side of (7.25) vanishes identically. Prescribing yO±E 4cm_eat t=0 starts an iteration, for a perturbed symbol z+s+..., where z and h commute, since zo=0. In effect we obtain a symbol a=0+z+s+... =z+s+... of AE Pm-e with highest order symbol z satisfying all assumptions of our earlier q=x, for m-e instead of m. However, we no longer impose (3.8), In other words, if we replace m-e by m then we have a construction of z according to thm.2.1 (2) as before, but without the condition (3.8) imposed on the starting symbol K. The time propagation xt`gt :
(for all x,T
,
+ft of x satisfies Kt
( pt Y_ ), y,y
satisfying (7.25) with U=O.
It is interesting to invoke one further transformation: Every 2x2-matrix b=((bjk)) has a unique representation in the form
, b0 trace b , b3=2(b11+ b22)
b - b0 + 7, bjaj (7.28)
b2
1(b12+b21), b1= 2i(b12- b21) In fact, b is hermitian if and only if bj are real. It is unitary if and only if b0E T and the 3-vector b^=(b1,b2,b3) E T3 satisfy
(7.29)
lb012+ 1b^ 12 = 1 , Re(b0b^) +
b^xb^ - 0 .'
2 Actually, we are dealing with a well known representation of the
group U2/U1=U2/{eie} in 1x13
:
Lemma 7.2. Every unitary 2x2-matrix is of the form
where 0 E 1, b0E 1, b^ E 13
b2+ b^ 12 = 1
,
Let us write our symbols y,y_ in the form
(7.30)
Y=YO+Y" . (I
Y_=gO+Y_ a , Y0,YDE T
,
Y"
, Y_ E
T3
Introducing this into the differential equations (7.25) yields y0' + y^' a - i[ (G a), (y^ a)] = 0 , where (0.15) (b) implies i[G a,y^ a] = 2a (Gxy^)
,
i.e., (assuming U=0),
=0
,
Y
Y 0,
- 2Gxy ^ = 0
,
(7.31)
Yp = 0 , y_
- 2G_xy_ = 0
355
10.7. Nonscalar symbols
This yields the time propagation of a general symbol, not necessarily a multiple of 1 on St and also the full connection to the classical theory, as described initially in sec.0, as follows: ,
Definition 7.3. A basic observable is defined by assigning a point wise self-adjoint symbol x(x,t) E Vs4 (usually in %pa ) which commutes with the symbol for all x, . \\
For a basic observable we have x"= f0 Y_{. Also, by thm.2.1
there exists a perturbation symbol z such thatJJa=x+z is a symbol of an operator in Pm resulting in a 'time propagated' symbol at
= xt+zt , where xt may be defined by setting xt =fptY_,t{ , with the solutions yt ,
of the differential equations (7.25) with
y
-,t
U=0, and initial values y
, y_ at t=0.
Then we have proven the following result:
Theorem 7.4. Let xEVcm be a basic observable, not necessarily scalar in the spaces S. Then, to obtain the time propagation xt of .
x without perturbations write x"=(Y Y_) with y
,
y
,
Y Ec . Expand
y_ in the form (7.30). Then set Y ot-y0' y t=0 constant along
the particle orbits (of electron and positron, respectively). Also, let the 3-vectors y^ and y _A propagate along these particle orbits as a magnetic moment does - i.e. let y ^ t , yt^ satisfy
(7.32)
Y^
+ c 1-x /cz {B+eC(1+ 1+x'2 /cz )-1x'xE}y^
and a corresponding equation for y_^
(7.33)
0
, namely ,
y_'+ c 1{B_ (1+ 1+x'z/cz )-lx'xE}Y_ = 0
Then xt is given by xt in the same form as x- , with yt , Y-,t Note the coupling constant M = me . It involves the so-called Bohr Magneton ma in view of the fact that we are setting i=1 here. Note that, in sec.5 we even established the electron spin as a basic observable. - In classical theory, it was thought that the spin is given by the matrix-valued vector p , while in (5.21) we got the commuting symbol 2t to represent the spin, in view of the correction needed for the angular momentum. Note, the components of i = ppp+qpq are not scalar in the spaces SJ. , hence form nontrivial examples for thm.7.4.
356
10. The invariant algebra of a Dirac equation
One might thus ask whether all physically reasonable observables must be (related to) basic observables. Looking at the need of classical Dirac theory to "fill" all states belonging to X_ and to regard positrons as "holes" there might be an advantage to this interpretation: Simple electron-positron transitions, with noncommuting part of lower order, are OK. Others are 'unobservable'. 8. The full symmetrized first correction symbol zS(x.U . We return to our task of obtaining the full first correction symbol of (3.7). Rather, as noted in sec.5, only its symmetric part is of interest, so we shall focus on zS .
In sec.7 we have reduced the conditions for c$ of (3.7) to the form (7.25), in our diagonal coordinates. Then we investigated only the case of a vanishing right hand side, to obtain thm.7.4. From the time-propagated operator A=x(x,D)+z(x,D)+s(x,D)+.... of thm.2.1(2) we only need As = 2(A+A*) , where AS- 2(x(x,D) + x(x,D)*) + zs(x,D) + ss(x,D) + ...
.
Thus we are only interested in called the symmetrized first correction symbol. By (3.10) we have zs = z0,s + c+,S + c-,S
where c+,S
(os
=
0)' c_,s -
(0
,
5), with yt,S=22(Yt,S+Yt+S), Yt Y
solving (7.25). Taking adjoints in (7.25) get
(8.2)
Y+
Y± -i[ (G_' a).Y+] = ULR
-i[ (G+' a),Y+] = UUL
Taking the average of (7.25) and (8.2), (8.3)
yS' -i[ (G+' a),yS] _ (US)UL '
yS -i[ (G_ a),ys]
(US) LR
Proposition 8.1. For zS of (8.1) we choose z0's according to (5.9).
Then ct,S are described by conditions on the 2x2-matrix functions YS=YS =C +,S UL' and ys c_,S LR, in diagonal coordinates of h(x,l), as in sec.7: ytS are solutions of the differential equations (8.3) with arbitrarily given self-adjoint initial data y0t , and
(8.4)
US (h,z0)s -(2(h,x)2)s Next we will calculate US of (8.4). As a preparation we state:
10.8. The full symmetrized first correction
357
we have
Proposition 8.2. For any (4x4-matrix) function
(ab)" = ab- + [=-,b"]
(8.5)
(axb)- = axb" + [X,b"]
,
!=cp+p1 of (6.18). The proof is a calculation, similar as for (7.17).
with X=(p +CPIk
,
We shall use (8.5) to calculate (h,zo)-S
(8.6)
(h,x)
,
-
= (hid) (zolx)
for h"=1p 0-)
(hix)
For z0- we use (5.8) and (7.10), (7.11):
(8.7)
z0 =2fr(SKE + (oK,n)) ( T 0) + 2(OK,n)+(_t of
where we introduce t =
('of Note that
z0 UR ' v - t Q - Z0 LL
, v = T- Q=t (R-S) , where
t = T- P--r- (R+S) (8.8)
R =2g (SKE+(o ,n))
r
,
1C
S = C(6K'n)+ 2
Using (8.8) on the expression (h,zo)-, we get , setting
(8.9)
E =-2ifr
Ss c klxEk , v= aklxtk E fE
ag Z0
(8.10)
vl +[E,v]
I
vv-iv I L
xz_ 0 =EE
ti-vt]J Ix +C6 L]
vt-vv j
vlx+[S'v]
On the other hand, by a calculation,
ict
X
(8.11)
(8 xh)
_
l '-ICJ = c(-
- ictI
(ah)
-icV, x
i,
itll
icy
it
X_ I
-E
-
eyAk l x -it
)k
Hence,
(h,zo)
It+T[t,t])v,
UL
(8.12)
(h.z0)"LR -ict(t Focus on " UL " (8.13)
,
(Xt-v )+ic(vl +1[E,v] )v+a_Ix(1t-v1) and write
(h,zo)-UL = T1+T2+T3
, T1{"I
T2=iR{t[S,v]-[s,t]v} , T3=iC(tvlx-t Iv)
358
10. The invariant algebra of a Dirac equation
First we look at
(8.14)
T 1=2f (Zv-L Z)
r
,
l
v-?,' xt = -T- (E+eBxt ) = -t Y
by a calculation. For V =(V1,V2,V3), possibly operator-valued, let
V° =V -ra
(8.15)
0
In particular t=a° , and V W° =VA-W, for arbitrary V,W. We get 2ifrT1=-(Yt) (Qc)+(Pr) (Yt )=-(Y°a) (Q°a)+(P°a) (Y°a)=2Y° S° -21o.YA xRA .
using (0.15)(b). Only the symmetric part is of use. Get (8.16)
T1S
-o.(E+eBxl;)°xR°
Next look at T2= 2--c From (0.15), (1+a0)e=txo get r
T2 =(1+a0)t[s.v] a FAklxjtj[(txo)k.(Q°a)]=-2iAklxjtj{(to)(Q°)k -(l;Q°)ak}. Here we use the formulas (b°t)= a0(bt)
(8.17)
,
for all b
and
Aklxjajak =
(8.18)
div A ,
easily derived. Again we neglect skew-hermitian summands, and write a-b if a-b is skew-hermitian. We get it (to) =i(ej°a)(to) -a.(ej°xt)--(txa)j and itjak iajok= jlj(to)akConclusion: =1l+aotj(txo)k -
(8.19) T2. 2c klxj(Q°)k(l;xa)j+2aoc(tQ)Dlklx7{iojak+ll+aotj(txo)k}.
With (8.18) we get (8.20)
2eT2=(1+ao )AklxjQk8
jEk}-ao (Qt)(Bo).
J_
Similarly, 2eT2 2e(1+ao )[E,t]v =-i2Aklxjpj° (ta)(ekdo)+iao (Pt)Dlklxjajtk (8.21)
--(1+ao )2AkIxJPj°Ek+iao (Pt)
klxjakok-ao
implying
(8.22)
YiT2--(1+ao )2AklxjPjEkao (Pi;)(Ba)+PtD'klx3{t jEk aotkE j}
Subtracting (8.22) from (8.20) we get
359
10.8. The full symmetrized first correction
(8.23)
(1+ao )e T2 -7c (TZ-T2)=2ao
)2Aklxj{Pjek+Qke j} lklxj{(Pz)tkEj+(Qt)tjEk}
.
To write this more conveniently, interchange indices and use that 7_(Aklxj_ljlx )MjNk = M.(NxB)
(8.24)
,
k
for arbitrary formal 3-vectors M,N, with commuting components. Get T2=f r
(8.25)
+ao
{gaol
+ao
(Ba)+FAklx3(PjEk+Qk nj)
2{((aoQ-P)t)AkIxj+((aoP-Q)t)Ajlxk}tjEk}
+l1
One may entirely avoid explicit magnetic potentials A, and
only use B=curlA and 6=
Aklx Ek . By a calculation not offered
cj
j
in detail (8.25) then assumes the form T = 2
(8.26)
a {2(R6)-2(Rt)(to)} fr
-j---{ (Bt)(Qa)-a2(Pt)(Ba)- (Bt)(to){2(Rt)-211-ao0(St)}}. 2 +a Of r Let us observe that T2=0 if the magnetic potential A is constant.
We next focus on T3 = icrvIx -
T4 + T5
(8.27)
=i{c((QAj)Ixks +
e((AjIx1 (Pk
) I
)'}okoj
= iZjkokaj
Therefore, (8.28)
(T3)S = a1(Z23-Z32) + 02(Z31-Z13) + 63(212-Z21) = a-T(Z) First examine the (symbolic) 3-vector T(Z) defined by (8.28)
for a general (symbolic) 3x3-matrix Z=((Zjk)). We will abbreviate T(((Zjk)))=T(Zjk) Note that .
(8.29)
T(bjck) = bxc , T(bjlxk
-curl i
for 3-vectors b , c, and that T(X)=O for any symmetric matrix Z. Let us first focus on Z = W = ((Wjk)) (8.30)
Wjk (((Q°j)Ix)°)k (Q°j)Ixk -1+a0(Qj-l0tj)Ix1ltk
10. The invariant algebra of a Dirac equation
360
Observe that (8.31)
T(W) =-curlxQ° -1+a0T(Qjlxltllk)
where we used that T((
Ma 0 )lxll;j;lt;k)
since the argument
= 0
matrix is symmetric. We introduce i;jlxk-car{Ajlxk r;jtlAllxk}
(8.32)
into the last term of (8.31), noting that the second term of (8.32) again yields a symmetric matrix, thus T=0 :
T(W) =-curlxQ°
Zr(
1+a0T(Q jlxli;li;k)
T(A7lx1tlt;k)
Or,
(8.33)
T(W) =-curlxQ° + 11+a0 1(lxQlx1) +f
++a0
S2
S1
S3
We will seek to simplify this formula, using the terms intro S1+S2+S3 , with duced earlier. Remember, we will need
Sj generated by the j-the term at right of (8.33). Here we get (8.34)
cf
r
r
1
(I+ao)tlAklx ek +f (1+a0)(l6) 1 r
hence S
(8.35)
Next focus on S1 (8.36)
3
=
fri+a0
.
:
)lxx = R1+R2+R3
curl xQ° = curlxQ - Ocurlx -(1
From (8.29) and (8.32) we get curlx
=-cfrB +cfrt1(A1lxxt) =-cfrB
cfrt1(txAlxl)
(8.37)
Alxl)) Application of (8.24) and (8.34) thus yields
a (curlxl) =-cf (Be) -f r
r
-cf {(gB)(ta)-(1-a0)(aB)} r
Or,
(8.38) and,
a
--2-{a2(Ba) + (B1;)(t;a)} -f (l+a0)(l;
r
r
)
10.8. The full symmetrized first correction
Iao(Ba) + (Bt)(ta)} + r
o
fr
361
(Qt)(tb)
For R3 we have R3 ( +a0
OQj(tjlxxt)
(a
aolx cfra0tlAlIx , tjlx -cfr(tjtlAlIx - AjIx) hence,
-O(Qjlxxt)
Efe
- e r-f+a_(tiAllxxt)
cf r (Qj`9jlxxt)
+11
0
The sum of the first and third term equals R4=
(8.42)
(1 a
)2cfr(txt1AlIx)
Recall that (8.43)
cf
2)(Ba)
(1+ao)f (tb)+ r C-yr
r
It follows that
(8.44)
a- R4= -L
Qt (tb) -cfrll
(Qt) (Ba) T(1 )2(B?)(ta)
1
+1+a0ofr(a Q7Ajlxxi;) + a R4
and, (8.45)
a R3=-1+aoa (t
we get
For the second term at right of (8.45) (called (3- R5
11
e
Alxj))
=-fr(QS) +cfr(Ajlxk Aklxj)QjEk =-f (Qb) +1+a of r 0 r Summarizing we get
{(Bo)(Ql;)-(Qo)(B?)}
(using (8.24))
(J- R3=- Oo'(yjlxxt)-fr(Qb)
+ocfr{(Bo)(Qt)-(Qo)(Bl;)}
(8.46) o(rb)-cfr
f 1S
Q+ao)
where the third and fifth term still allow a simplification: They {ao(Ba)(Qi;)-(Bi;)(Qa)}. All together we get give the sum +1+a 0
cf
r
362
10. The invariant algebra of a Dirac equation
(T4)S c(S3+S2+S1) =fr1 - (l;
+1+a
xQ)
)
) jI xxt) f r(QS)-1+a o-{a0(Ba)(Qt)-(Qa)(Bt)} r
o
frl +ao-(to)
- fr(1
(Bl)(to) - fr(Qt)(tb)
fr1-{a2(Bo)
-
- (Bt)(l;a)}
In this sum term 1 and 7 cancel, and terms 8 and 10 simplify. Also denoting U = curlxQ , for a moment, the sum of terms 2 and 4 give -1+a0a't1(tx(QIxl-RlIx)
_- +ao((U?)(lo)-(1-a0)(Ua))
It follows that (T4)S =-ca0 o (curlxQ) - 11+a (to) t'(curlxQ) 0 (8.48)
+ P
T+a f (Ql)(Bo){a0+a2} o r
r
+1+a0fr(Q(,)(Bt) + fr(Qt)(Bt)(ta){1+a0(1+a0)2} Or,
(8.49)
11+a0(to) t'(curlxQ)
(T4)s
(Bt)(tc)}
aa0
Next we turn to the expression
(T5)S (8.50)
=-ea'T(Ajlx 1(Pk )I 1 ) +T+-0 a T(tmAmIx 1 (Pk )I 1tj).
Here (8.29) implies that
(8.51)
M1=,AIx xPI 1
1
- Alx x(1 1
t)
0
I
1
= M3 +M4
while (8.52)
I M2= l+a0MIX1 aPl lxt -I
I
Here,
"4= -(1 0 I l AIxl (l;xa) - pao a AIxlx{el-?t1}
e 1+a0)(
One finds that xel = -(Ba) 1
Also,
xttl=
,
1
e(1+a0)(tb)
363
10.8. The full symmetrized first correction
(t6)+S- cPkl
of r Pa0
_
Pa
C
r
(tb) +e-1_
r
0
ltkbl+fr
oPk(ek-ttk)'b
0 Pkll 1t
+e 1+a
0
kal
Together we get (8.53) lr
i
+-r(to) +-..{(Po)-(Pt)(tb)} + ePklt kbl r
r
0
1
Using that txt=0 we get M6= -(1
Aklxltk(el-tt1)xt
Pa0 Pa
=-f
r
Aklx tk(el4) 0
1
So,
a'M5 frPao Aklxltkel' (txa) = frloAlIxktkEl o(tb)-Pr(1P+a
-frl
)22{(1-a0)(Ba)-(Bt)(ta
Summarizing we get
-so.M1-eo.M6- +aOAklxllka'
(T5)S= (8.54)
+r T_{-(Bo) e(1+a0)(tb)} f {(Pb)-(P,)(,b)}
-(fib) + ?-rT+a +r_ a (tb) --1. r r r
r
0
0
0
0
{(1-ao)(BC) -1 (Bt)(ta)} 0
Or, simplifying some terms (there are 4 terms with (Pt)(tb)),
(T8)5 K1 -
xPI1 )
(8.55)
1
-e -1
r
a } f-{(PS)-2a0 -{a0(Ba)+(Bt)T o
r
(Pt)}
o o with K1 = sum of the second and third term at right of (8.54). We simplify K1:
e K1=
11
=11
e +a0Pkl
Setting pjk=1+a Pjl 0
tm(txa)k} 1Amlxl{lk(txa)m-
Aklx 1
K1=Y jkIt
and r1= txa for a moment we get 1
1k k1j} = (Y23-Y32) (12'13-t3'12)
+(Y31-Y13)( 3T11-t113) +
This shows that
(Y12-Y21)(t112-t2v11)
10. The invariant algebra of a Dirac equation
364
(la)t-
1+a 0O
(PIE1xAIx1) - e(1-a0)a- (PIE1xAlx1)
Summarizing we get
(T5)S
-
1
(8.56) frl+a0{a0(Ba)
(Bt)) f {(PS)-2a0 1
+1
r
0
Finally then, combining (8.49) and (8.56): (T3)S = (T4)S + (T5)S
-ca0a- (curlxQ)-
(8.57)
= - +a0(ta)l'(AIx1xPIt1)
T+-curlxQ)-ea0a- (AIx1xPI1)
+ f {((Q-P)8)-(Qt)(t6)+2a0
)}
r
0(t;
Pa
Qa
(Bt)(ta)+ I (Qt)(Ba)} 0 We then finally can combine formulas (8.16), (8.26), and (8.57) +e-{a0((P+Q)t)(Ba)-
r
(Bt)(ta)+a0
o
0
Proposition 8.4. We have
(8.58)
((h,z0)-UL) S TI,S+T2,S+T3,S
=-f a.(E+eBxt)AxRA -2f {(RS)-(RR,)(l;S)}
r
r
e {(Be)(Qa)-ao(Pl;)(Ba)-(Bt)(ic){(Rt)-I ll+a to (SW)}} o r
Q) 11+a0(ta)(t'(Al
+fT{2a0(Rl;)(Ba)+(
1)
xPlt1) -
-{(2S6)-((Qt)-2a0 0))} o(Qt)(Bo)}
+a0
Finally we are left with the second term in (8.4). Looking at the second Poisson bracket (h 'K)2 , note that hIxx=0 (8.59)
,
hence
(h 'X)2 = -xltjtkhlxjxk.
In diagonal coordinates we get (assuming u,v multiples of 1, abbreviating u;j=ul
,
u;jk= ul
j
"It
jk
,
etc., inside the matrices):
t
+ad "j)(a
(a
=(atj+ad
k
j
k
+ad "k) o v,
)ru;k icbKrk1 i l_icOKZk v;k J
365
10.8. The full symmetrized first correction
ic(SKtk);j ) l-ic(SKtk);j v;jk
_ fu;jk
[
1
E
ft EJj' T )j'
ic6Ktk f-u'v;k
]
vva vp XX p,
where (using anti-commutator brackets [b,c]+=be+cb)
vX ==u:jk-2froK[t j.tk]+' UP =ic(SKtk);j+ifr j(frOK):k+2frOK[E j.tk]
%X=-ic(Sk7x);j-i
r
j(frsK);k-frSK[EI'tk]' XP =v:jk+2T"SK[tj'tk]+
To evaluate this we use (6.23). Also, frl+a0{(2
(8.60)
Sjk)(at) - ajtk}
and, using (0.15)(c),
(8.61)
(Sjk--t0jlk) - aOtkaj}
(1+a0)[ E j,tk] = 2i{
It follows that 9-6K(6
vX = um
-tjtk)
(8.62)
+tj6Kk }
up = - XX = is{tk6KI
.
l
j
In particular we notice that
icSKtkl +2f SK[E j,tk] + j r by a calculation.
i-tj6xfrl k r
o
Next we need
hlxjxk = Vlxjxk
(8.63)
+ chOlxjxk
only in form of n_ tA(x), where h0 involves the variables x and In fact, h0=mc(3+an is a linear function of a. Accordingly, _e
hOlxj =
e llxjholnl = c llxjc`i
hO (xjxlt
= -8a A l xjxk
Using (7.5) we get (8.66)
hlxjxI
= Vlxjxl- eAIxjxl
Notice that both matrices F =
it
-it iT
and J = hlxjxk
are of the
10. The invariant algebra of a Dirac equation
366
general form (8.67)
e
F=
-ift
if g)
'
J=
l-iq irJl f
with real scalars e,g,p,r and f,q C E13 . We need the expressions
l(+2FJUL)s 2j1(. (ep+(fu)(gt))S = NUL
2(h,x)2"UL)S =
(
(8.68)
(-2(h'K)2 LR) S = 7'j1(2(gr+(ft)(gt))S = NLR
Using (0.15)(b) we get
NUL =a q° xf° , NLR =
(8.69)
a. q° xf°
Checking with (8.62) and (8.66) we find that
(8.70)
f° =cek° SK I
j
+ceJ° bK I
k
,
q° = -e A I X X
°
,
1
hence (8.71)
NUL= NLR=_2ecOKl
where we write Alxj
etc., for a moment.
,
=A
Here the first term at right, equals T1 = -2ecOKl
a Blx j
j
while the sum of the remaining terms is evaluated as
-(1;BIxj}
T2 =
0
It follows that (8.72)
(-(tBIXj)I
NUL, NLR
Combining (8.58) and (8.72) we finally get Theorem 8.4. For a basic observable K=K+p+K q , scalar in St , the first symmetric correction symbol zS is given in the form zR =z O'S
+c+,S+c-,S , where
+,S-=[e
c+,S
0)
, with yS solving the differen-
tial equation
(8.73)
YS -i[ (G+' a),YS] =-2ecj
OKlj{ao (Ba) IX j
j
-f--a.(E+eBx%)°xR° -2f {(RS)-(Rl;)(t6)}
r
a 0fr
r
0
367
10.9. Some final remarks
1ao(ta)(t'curlxQ) -
-1 o(to)(t'(Air
Pr{(2S6)-((Ql)-2a0
(B1;){ao(Qt)-(P1;)} + I (Ql;)(Bo)}
+a0
r
)(l;6)}
0
The matrix c- S satisfies a similar relation, along the line of (8.3)2 , where we will not work out (US)LR in detail. 9. Some final remarks.
We return to the differential equation (7.25) (or rather (8.3)(or (8.73))) where now also the inhomogeneous term (the right (lxE))
hand side) is under control. In particular, G : , = -
is given in terms of E and B , while (US)UL is given as in (8.73), with P,Q,R,S of (8.8), S,v of (8.9). As a first remarkable fact we record that
u = (US) UL a- u^ In other words in the expansion (7.28) of (US)UL in terms of Pauli matrices and 1 the term b0 is absent: it vanishes identically. Accordingly, writing (8.3)1 in the form (7.31), we get (9.2)
yo' = 0
, y" - 2Gxy^ = u"
The first equation still is homogeneous, hence we may set yo=0 i.e., we do not need another correction which is a multiple-of 1 in S. The initial yo will stay constant along the particle orbits On the other hand, the equation for the classical motion of the R3-vector y^ satisfies the inhomogeneous differential equation (9.2)2. Note that - under our present assumptions - the basic observable x=aK±icfrSK is a multiple of 1 in S+, for all t, hence its vectors x+^, in the expansion (7.31), vanish for all t. However, as (9.2)2 indicates, the corrresponding perturbations y+" in general will not remain zero, even though the theory allows us to prescribe them arbitrarily at t=0 - or at any other t. A question of stability arises: If we prescribe y"=0 at t=0, can it happen that y^ grows large as t becomes large? This may be looked at under two aspects. First, we already noted in sec.7, in the case of u^ =0, the law of propagation of y" coincides with the classical motion of a magnetic moment under the given fields, along the particle orbits. We now should have such a motion under
368
10. The invariant algebra of a Dirac equation
some "magnetic space charge", described by the vector u^ Second, we might return to the 2x2-matrix-equation (8.73). Write it as .
(9.3)
Y' -i[ g, Y I = u
,
g=o G+ , u=a u^
.
Introduce the evolution matrix v(t,i) of the 2x2-matrix-system
v'=igv , t,rEI , v(i,i)=1
(9.4)
.
It then is trivial to confirm that solutions of (9.3) satisfy
(9.5)
t
Y(t) = v(t,t)Y(s)v(r,t) +
dOv(t,O)u(O)v(9,t)
Z
Note that v(t,s) is a unitary 2x2-matrix, for all t,i, since g is hermitian: Just confirm that v 1+(t,i)=w(t,t) satisfies (9.4) as well, and apply uniqueness of the solution of (9.4) . Thus the matrix norm of the first term in (9.5) stays constant, while the t
matrix norm of the second term is bounded by
Iu(O)IdO , an aver
rage of the inhomogeneity along the particle orbit, from r to t. Generally one will expect the unitary matrix v(T,t) to oscillate along these orbits, hence the second term should remain smaller. It would be a matter of investigating special well studied observables now - like those of sec.5 - to check about physical
consequences of our present approach. Of course we must remember that we only look at the first correction, with an infinity of other corrections to follow. The expression at the right hand side of (8.73) appears to be quite complicated, and might be hard to interpret. Thus, perhaps it is useful to look at it in the special case where the magnetic potentials A(x) vanish identically, just to have a less com-
plicated formula. We then get B=6=O, hence
(9.6)
u^ _ -f E xR° - ca0curlxQ
r
- +a0 (t curlxQ)l;
with R,P,Q of (8.8). Specifically, 2R =- °x
(9.7)
5KE) IX
and, using that fr is independent of x, hence XIx=E , and that the curl of a gradient vanishes, (9.8)
Therefore,
curlxQ
= -21 curlx(uIX 6KXIx) = 2fr6KIxxE
369
10.9. Some final remarks
(9.9)
°^=2-n r
E°x(°Kix-6
E)°
2f ra0 6K xxE 2fr1
0
t. (sxlxxE)
-2f r2 H A X aK I X -2fra0Sx I xxE -f 110t- OK I xxE) Here we use another general formula valid for arbitrary a,b E (or also symbolic 3-vectors with commuting components):
(9.10)
a°xb° = a0axb +l+a t. (axb)
C3
.
0
Indeed, we have a°xb° = axb -1+a
P1 P2 0
where (1+a0)P2= tx{(al;)b-(bl)a} =1;x(tx(bxa))
=tx(txc) = (ca)t - (1-a0)c
setting c=bxa
,
Thus we get P1-P2= axb - (1-a0)axb
q.e.d.
Combining (9.9) and (9.10) we get (with K+=Gx+cfrOK
(9.11)
u^=2f- E°xoKIX+zfrE°x61C I
as in sec.5)
A 2f TE°xx+IX
Here we might be tempted again to look at the observables (5.11)-(5.18), to obtain results on their corresponding first order correction symbol c+ Note that (9.2) with u^ of (9.11) is .
fairly simple now. We will not pursue this further, since it might deviate to far from the subject of this book.
REFERENCES
[Arl] N.Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order; J.Math.Pures Appl.36(1957)235249.
[B1] R.Beals, A general calculus of pseudo-differential operators; Duke Math.J.42 (1975) 1-42. [B2]
,
Characterization of pseudodifferential operators and corapplications; Duke Math.J. 44 (1977) ,45-57 ;
rection, Duke Math.J.46 (1979),215 [B3]
[B4]
.
,
On the boundedness of pseudodifferential operators;
,
Comm.Partial Diff.Equ. 2(10) (1977) 1063-1070. LP and Hoelder estimates for Pseudodifferential ope-
rators; Ann.Inst.Fourier Grenoble 29,3(1979),239-260 [BA ,1 M.Ben-Artzi and A.Devinatz, the limiting absorption principle for partial differential operators; Memoirs AMS vol. 66 (1987) Providence. [Bzl] I.M.Berezanski, Expansions in eigenfunctions of self-adjoint operators; AMS transl. of math. monographs, vol.17, Providence 1968. [BS] A.Boettcher and B. Silbermann, Analysis of Toeplitz operators Akademie-Verlag Berlin and Springer-Verlag Heidelberg 1989. [BdMI] L.Boutet de Monvel, Boundary problems for pseudo-differential operators; Acta Math. 126 (1971) 11-51. [BS1] J.Bruening and R.Seeley, Regular singular asymptotics; Adv. Math. 58 (1985) 133-148.
[BC1] M.Breuer and H.O.Cordes,On Banach algebras with a-symbol; J.Math.Mech. 13 (1964) 313-324 [BC2]
.
, part 2 ; J.Math.Mech.14 (1965) 299-314
[Ca] A.Calderon, Intermediate spaces and interpolation,the complex method; Studia Math.24 (1964) 113-190. [Cal] T.Carleman, Sur la theorie mathematique de l'equation de Schroedinger; Arkiv f. Mat., Astr.,og Fys. 24B N11 (1934). [CV1] A.Calderon and R.Vaillancourt,On the boundedness of pseudo-
differential operators; J.Math.Soc.Japan 23 (1971) 374-378. [CV2]
, A class of bounded pseudodifferential operators;
370
References
371
Proc.Nat.Acad.Sci.USA 69 (1972) 1185-1187. [Chu1]J.Chaillou, Sur les ensembles bornes A de distributions polynomes inversibles dans D'(T) at d'inverse A 1 born4 et sur les hyperfaces T-hyperboliques. Thesis, Univ. of Paris 1969.
[Che1]J.Cheeger, Spectral geometry of spaces with cone-like singularities;Proc.Nat.Acad.Sci. USA (1979) 2103. [Ch] P.Chernoff, Essential selfadjointness of powers of generators of hyperbolic equations ; J.Functional Analysis 12 (1973) 402-414 [CM] R.Coifman and Y.Meyer, Au dela des operateurs pseudodifferentielles; Asterisque 57 (1979) 1-184. .
[cc1] P.Colella and H.O.Cordes,The C*-algebra of the elliptic boundary problem;Rocky Mtn.J.Math. 10 (1980) 217-238 .
[Cd1] E.A.Coddington,The spectral representation of ordinary selfadjoint differential operators;Ann.of Math.60 (1954) 192-211. [CdLv]
and N.Levinson, Theory of ordinary differential equations,McGraw Hill,New York 1955. [C,] H.O.Cordes,Elliptic pseudo-differential operators,an abstract theory.Springer Lecture Notes in Math.Vol.756, Berlin,Heidelberg,New York 1979. [C2]
,
Spectral theory of linear differential operators; London Math. Soc. Lecture Notes vol.76, Cambridge .Univ. Press Cambridge 1987.
[C3]
,
On the technique of comparison algebra for elliptic boundary problems on noncompact manifolds. Proceedings Symposia Pure Math. 51(1990),part 1, 113-130.
[C4]
,
On some C*-algebras and Frechet-*-algebras of pseudodifferential operators; Proc.Sympos.P.Appl. Math.43 (1985) 79-104.
[CC]
,
On compactness of commutators of multiplications and convolutions,and boundedness of pseudodifferential operators; J.Functional Analysis,18 (1975) 115-131.
[CD]
, A pseudo-algebra of observables for the Dirac
[CE]
, A version of Egorov's theorem for systems of hyperbolic pseudo-differential equations;J.of Functional Analysis 48 (1982) 285-300.
equation; Manuscripta Math.45 (1983) 77-105.
References
372
, A pseudodifferential Foldy-Wouthuysen transform;
[CF]
Comm.in Partial Differential Equations,8(13) (1983) 1475-1485.
[CG]
,
[CI]
,
On maximal first order partial differential operators; Amer.J.Math.82 (1960) 63-91. On geometrical optics;lecture notes,Berkeley,1982 The algebra of singular integral operators in Rn;
,
J.Math.Mech.14 (1965) 1007-1032. On pseudodifferential operators and smoothness of
[CFo]
,
[CL]
special Lie-group representations;Manuscripta math. 28 (1979) 51-69 .
, A global parametrix for pseudodifferential
[CP]
operators over in ;Preprint SFB 72 (Bonn) (1976) (available as preprint from the author). , An algebra of singular integral operators with
[CT]
two symbol homomorphisms;Bulletin AMS ,75 (1969) 37-42 .
, Ueber die eindeutige Bestimmtheit der Loesungen
[CU]
elliptischer Differentialgleichungen durch Anfangsvorgaben. Nachr. Akad. Wiss. Goettingen Math.-Phys. Kl.IIa, No.11, 239-258 (1956). [CPo]
,
On the two-fold symbol chain of a C*-algebra of singular integral operators on a polycylinder;
[CDg1]
Revista Mat. Iberoamericana 2 (1986) 215-234. and S.H.Doong, The Laplace comparison algebra of
,
a space with conical and cylindrical ends; Pro-
[CEr]
ceedings, Topics in pseudodifferential operators; Oberwolfach 1986; Springer LN, vol. 1256. ,and A.Erkip, The N-th order elliptic boundary problem for non-compact boundaries;Rocky Mtn.J.of Math. 10 (1980) 7-24 .
[CHe1]
,and E.Herman , Gelfand theory of pseudo-differential operators; American J.of Math. 90 (1968)681-717.
[CHe2] [CMt]
Singular integral operators on a half-line; Proceedings Nat. Acad. Sci. (1966) 1668-1673. , and R. McOwen, The C*-algebra of a singular elliptic
problem on a noncompact Riemannian manifold; Math. Zeitschrift 153 (1977) 101-106. Remarks on singular elliptic theory for
[CM2]
complete Riemannian manifolds; Pacific J.Math. 70 (1977) 133-141 .
[CMe1]
,
and S.Melo, An algebra of singular integral opera-
References
373
tors with kernels of bounded oscillation and application to periodic band spectra; J. Diff. Equ. 75 (1988) 216-238.
[CH] R.Courant and D.Hilbert,Methods of Mathematical Physics, I: Interscience Pub. ltd. New York 1955; II: Interscience Pub. New York 1966.
[Dd1] J.Dieudonne, Foundations of modern analysis,Acad.Press New York 1964 [Dj1] D.Z.Djokovic, An elementary proof of the Baker-Campbell-Haus.
dorff-Dynkin formula; Math.Z.143 (1975) 209-211. [Do1] H.Donnelly, Stability theorems for the continuous spectrum of a negatively curved manifold; Transactions AMS 264 (1981) 431-450. [DS1,2,3] N.Dunford and J.Schwarz, Linear operators, Vol.I,II,III,
Wiley Interscience,New York 1958,1963,1971. [Dx1] Dixmier, Les C*-algebres et leurs representations; GauthierVillars, Paris 1964. [Dx2] Sur une de E.Heinz, Math.Ann.125 (1952) 75-78.
[Dv1] A.Devinatz, Essential self-adjointness of Schroedinger operators; J.Functional Analysis 25 (1977) 51-69. [Du1] R.Duduchava, on integral equations of convolutions with discontinuous coefficients; Math. Nachr. 79 (1977) 75-98. [Du2] , An application of singular integral equations to some problems of elasticity; Integral Equ.. and Operator theory 5 (1982) 475-489. [Du3]
,
On multi-dimensional singular integral operators,I: The half-space case; J.Operator theory 11 (1984) II: The case of a compact manifold; J. 41-76 Operator theory 11 (1984) 199-214. ;
[Du41
,
Integral equations with fixed singularities; Teubner Texte zur Math.,Teubner,Leizig,1979.
[Dn1] J.Dunau
Fonctions d'un operateur elliptique sur une variete
compacte; J.Math.pures et appl. 56 (1977) 367-391 [Dy1] A.Dynin, Multivariable Wiener-Hopf operators; I:Integ.eq.and operator theory 9 (1986), 937-556. [Es.] G.I.Eskin, Boundary-value problems for elliptic pseudodiffe-
rential operators; AMS translations vol.52, 1981. [Eg1]Yu.V.Egorov, The canonical transformations of pseudodifferential operators; Uspehi Mat Nauk 24:5 (1969)235-236. [ES]
,
and M.A.Shubin, (ed's) Partial differential equations IV, microlocal analysis and hyperbolic equations;
374
References
Springer-Verlag, Berlin New York 1993.
[Erl] A.Erkip, The elliptic boundary problem on the half-space; Comm. PDE 4(5) (1979) 537-554. [ES]
,and E.Schrohe, Normal solvability of elliptic boundary problems on asymptotically flat manifolds; J. Functional Analysis 109 (1992) 22-51.
[Esl] G.Eskin, Boundary problems for elliptic pseudo-differential equations; AMS Transl. Math. Monogr. 52 Providence 1981 (Russian edition 1973). [Fa1] H.O.Fattorini, The Cauchy problem; Encyclopedia of Math. and its applications, Vol.18, Addison-Wesley 1983. [Fr1] K.O.Friedrichs, Symmetric positive linear differential equations; Commun.P.Appl.Math.11 (1958) 333-418. Spectraltheorie halbbeschraenkter Operatoren Math.Ann.109 (1934) 465-487 ,685-713 and
[Fr2]
110 (1935) 777-779 Pseudodifferential operators; lecture notes 1968, Courant Institute Math. Sci., NYU, New York. and P.D.Lax, On symmtrizable differential opera.
[Fr3]
,
[FL]
,
tors; Proc. Sympos. Pure Math. Chicago (1967) AMS 128-137.
[Fhl] I.Fredholm, Sur une classe d'equations fonctionelles; Acta Math. 27 (1903) 365-390. [Gal] L.Garding,Linear hyperbolic partial differential equations with constant coefficients;Acta Math.85 (1-62) 1950 [Gei] D. Geller, Analytic pseudodifferential operators for the
Heisenberg group and local solvability; Math. Notes 37 Princeton Univ. Press, Princeton 1990 [GT] D.Gilbarg,and N.Trudinger, Elliptic partial differential equations of second order; 2-nd edition; Springer New York 1983. [GS] I.Gelfand and G.E.Silov, Generalized functions; Vol.1,Acad. Press, New York 1964.
[G11] I.M.Glazman, Direct methods of qualitative spectral analysis of singular differential operators; translated by Israel Prog. f. scientific transl. Davey, New York 1965. [Gb] I.Gohberg, On the theory of multi-dimensional singular inte-
gral operators; Soviet Math. 1 (1960) 960-963. [GK] I.Gohberg and N.Krupnik, Einfuehrung in die Theorie der eindimensionalen singulaeren Integraloperatoren;
References
375
Birkhaeuser, Basel 1979 (Russian ed. 1973). [G1] B.Gramsch, Relative inversion in der Stoerungstheorie von Operatoren and i-Algebren; Math. Ann. 269 (1984) 27-71. [G2]
, Meromorphie in der Theorie der Fredholm Operatoren
mit Anwendungen auf elliptische Differentialoperatoren; Math. Ann. 188 (1970) 97-112. [Gr1] G.Grubb, Problemes aux limites pseudo-differentiels depen-
[Gr2]
dent d'un parametre; C.R.Acad.Sci.Paris 292 (1981) 581-583. Functional calculus of pseudidifferential boundary problems; Birkhaeuser 1986; Boston Basel Stuttgart
,
[GLS] A.Grossmann,G.Loupias,and E.Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space; Ann.Inst. Fourier,Grenoble 18 (1968) 343-368.
[Hdi] J.Hadamard, Lectures on Cauchy's problem, Dover pbl. inc. New York 19 [Hg1] S.Helgason, Differential geometry, Lie groups, and symmetric spaces; Acad. Press, New York 1978. [Hbl] D.Hilbert, Integralgleichungen; Chelsea, New York 1953
[HP] E.Hille,and R.S.Phillips, Functional analysis and semi-groups Amer.Math.Soc.Coll.Publ. Providence, 1957. [Hr1] L.Hoermander,Linear partial differential operators;Springer, Berlin, Heidelberg, NewYork ,1963.
[Hr2] L.Hoermander, Pseudo-differential operators and hypoelliptic equations; Proceedings symposia in pure and appl. Math. Vol 10 (1966) 138-183. The analysis of linear partial differential operators; Springer New York I: distribution theory and Fourier analysis, 1983; II: diff. operators with constant coefficients, 1983; III: pseudodifferential operators, 1985; IV, Fourier inteoperators, 1985. [IZ] C.Itzykson, J.B.Zuber, Quantum field theory; McGraw-Hill, New York 1980. [Ka1] T.Kato, Perturbation theory for linear operators; Springer New York 1966. [Hr3]
,
[Ka2]
,
[Ka3]
,
Perturbation theory for nullity, deficiency and other quantities of linear operators; Journal d'Analyse Math. 6 (1958) 261-322. Notes on some inequalities for linear operators;
References
376
[Ka4]
,
[Ka5]
,
Math.Ann.125 (1952) 208-212. Boundedness of some pseudo-differential operators; Osaka J. Math. 13 (1976) 1-9.
Linear evolution equations of hyperbolic type; J. Fac. Sci., U. of Tokyo, Vol. 17 (1970) 241-258; part II; J. Math. Soc. Japan 25 (1973) 648-666.
[Kel] J.L.Kelley, General topology; Van Nostrand, Princeton 1955. [KG,] W.Klingenberg, D.Grommol, and W.Meyer, Riemannsche Geometrie im Grossen; Springer Lecture Notes Math. Vol. 55 New York 1968. [KN] J.Kohn and L.Nirenberg, An algebra of pseudo-differential ope-
rators; Comm. Pure Appl.Math. 18 (1965) 269-305 [Krl] M.G.Krein,On hermitian operators with direction functionals; Sbornik Praz. Inst. Mat. Akad. Nauk Ukr. SSR 10 (1948) 83-105. [Kg,] H.Kumano-go, Pseudodifferential operators; MIT-Press 1982. [LMg1] J.Lions and E.Magenes, Non-homogeneous boundary value problems and applications; I, II, Springer, New York 1972. [Lvl] E.E.Levi, I problemi dei valori al contorno per le equazioni lineare totalemente ellitiche alle derivate parziali; Rome 1909. [Lwl] K.Loewner, Ueber monotone Matrixfunktionen; Math.Zeitschr. 38 (1934) 177-216. [MO] W.Magnus, F.Oberhettinger, Formeln and Saetze fuer die speziellen Funktionen der Mathematischen Physik; 2-nd edit. Springer-Verlag, Berlin, Heidelberg, 1948 [MOS] W.Magnus, F.Oberhettinger and R.P.Soni, Formulas and theorems for the special functions of Mathematical Physics; Springer-Verlag, New York, 1966. [MNP] W.G.Mazja, S.A.Nasarow, B.A.Plamenewski, Asymptotissche Theorie elliptischer Randwertaufgaben in singulaer gestoerten Gebieten. Akademie Verlag, Berlin 1991. [Mr] J.Marschall, Pseudodifferential operators with coefficients in Sobolev spaces; Transactions AMS 307 (1988) 335-361.
[MO] R.McOwen, Fredholm theory of partial differential equations
[M1] R.McOwen
on complete Riemannian manifolds; Ph.D. Thesis, Berkeley 1978. Fredholm theory of partial differential equations on complete Riemannian manifolds;Pacific J.Math.
References
377
87 (1980) 169-185
[M2] R.McOwen
.
On elliptic operators in In ;Comm. Partial Diff. equations, 5(9) (1980) 913-933.
[Me,] R.Melrose, Transformation of boundary problems, Acta Math. 147 (149-236). [MM1] R.Melrose and G.Mendoza,Elliptic boundary problems on spaces
with conic points; Journees des equations differentielles, St. Jean-de-Monts, 1981. [MM2]
Elliptic operators of totally characteristic type;
MSRI-preprint, Berkeley, 1982. [Mml] T. Muramatu, estimate for the norm of pseudo-differential operators by means of Besov spaces; Conf. on pseudodifferential operators; Springer L.N., Vol 1256 330-349; Springer-Verlag, New York [Ng1]
M. Nagase, On sufficient conditions for Pseudodifferential operators to be Lp-bounded; Conf. on Pseudodifferential operators, Oberwolfach; Springer L.N.,vol
1256 (1987)350-359; Springer-Verlag; New York. [Pal] K.Payne, Smooth tame Frechet algebras and Lie groups of pseudodifferential operators; Comm. Pure Appl. Math. 44 (1991) 309-337. [P1] B.A.Plamenewski, Algebras of pseudodifferential operators; Kluwer Acad. Pub. Dordrecht, Boston, London, 1989. [Pr] S. Proessdorf, Einige Klassen singulaerer Gleichngen; Birk-
haeuser Verlag, Basel, Stuttgart 1974 [Rbi] V.S.Rabinovic, On the algebra generated by pseudodif£erential operators on &n, operators of multiplication by almost periodic functions, and shift operators. Soviet Math. Dokl. 25(1982) 498-501.
[RT] J.Rauch and M.Taylor, Decay of solutions to non-dissipative hyperbolic systems on compact manifolds; Comm. Pure Appl. Math. 28 (1975) 501-523. [R1] C.E.Rickart, General theory of Banach algebras; v.Nostrand Princeton 1960.
[Rfl] M.A.Rieffel, Deformation quantization for actions of Rd Memoirs AMS 106
,
No 506 1993
.
[RN] F.Riesz and B.Sz.-Nagy, Functional analysis; Frederic Ungar, New York 1955. [Rol] D.Robert, Proprietes spectrales d'operateurs pseudo-diffe-
rentiels; Comm. PDE 3(9) (1978) 755-826.
[Schrl] E.Schrohe, The symbol of an algebra of pseudo-differential operators; Pacific J. Math. 125 (1985) 211-224.
References
378 [Schr2]
,
Potenzen elliptischer Pseudodifferentialoperatoren; Thesis, Mainz 1986.
[Schr3]
,
Spaces of weighted symbols and weighted Sobolev spaces on manifolds; Conf. on pseudodifferential operators, Springer LNM 1256, 360-377, New York.
Complex powers on noncompact manifolds with singularities; Math. Annalen 281 (1988) 393-409. [Schu2] B.-W.Schulze, Mellin expansions of pseudo-differential operators and conormal asymptotics of solutions; [Schr4]
,
[Schu2]
,
Procedings, Oberwolfach conference on Topics in pseudodifferential operators 1986; Springer LNM vol.1256 (1987) 378-401. Pseudodifferential operators on manifolds with singularities; North-Holland 1991, Amsterdam.
[Schw1] L.Schwartz, Theorie des distributions; Hermann, Paris 1966 [Sel] R.T.Seeley, Topics in pseudo-differential operators, CIME Conference at Stresa 1968. [Se2] Integro-differential operators on vector bundles; Transactions AMS 117 (1965) 167-204 .
[Shl] M.A.Shubin, Pseudodifferential operators and spectral theory Springer-Verlag 1987, Berlin Heidelberg New York. [S,] H.Sohrab, The C*-algebra of the n-dimensional harmonic [S2]
,
oscillator; Manuscripta.Math.34 (1981),45-70 C*-algebras of singular integral operators on the line related to singular Sturm-Liouville problem; Manuscripta Math. 41 (1983) 109-138. Pseudodifferential C*-algebras related to Schroedinger operators with radially symmetric poten-
[S3]
[S4]
[S5]
tial; Quarterly J.Math.Oxford 37 (1986) 105-115. , A class of Pseudodifferential C*-algebras; Quart.J. Math.Oxford (2), 40 (1989) 119-131. ,
Spatially nontemperate pseudodifferential operators sphere extensions, Fredholm theory; to appear in Rocky Mtn. J. Math.
[Sml]A.Sommerfeld, Atombau and Spektrallinien; Vieweg,Braunschweig 1957
.
[Spi] F.O.Speck, General Wiener-Hopf factorization methods; Pitman Adv. Publ. Prog. 119 (1985). [Svl] L. Svensson, Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part; Arkiv f. Mat. 8 (1969) 145-162. [Ti ] M.Taylor , Pseudo-differential operators; Springer Lecture
379
References
Notes in Math. Vol. 416, New York 1974 [T12]
.
Pseudodifferential operators;Princeton Univ.Press Princeton 1981
,
.
[Tl3]
,
Noncommutative harmonic analysis; Math. Surveys
and Monographs 22 (1986) AMS, Providence. [T14] Pseudodifferential operators and nonlinear PDE; Birkhaeuser, Boston, Basel, Berlin, 1991. [Un1] A.Unterberger, Quantification relativists; Bull.Soc.Math. France 119 (1991) Memoire No 44-45. [Tr1] F.Treves, Introduction to pseudodifferential operators and Fourier integral operators; Plenum Press,New York ,
and London 1980 An algebra of pseudodifferential operators and the asymptotics of quantum mechanics; J. Functio.
[Vr] A.Voros,
nal Analysis 29 (1978) 104-132.
[Wm1] J.Weidmann, Linear Operators in Hilbert space. Springer-Verlag, New York 1980 [W1] A.Weinstein, A symbol class for some Schroedinger equations on In ; Amer.J.Math.107 (1985) 1-21. [WZ]
,
and S.Zelditch, Singularities of solutions of some
Schroedinger equations on In, Bull.AMS 6(1982)449-452 [We1] H.Weyl, Ueber gewoehnliche Differentialgleichungen mit Singularitaeten, and die zugehoerigen Entwicklungen willkuerlicher Funktionen; Math.Ann.68 (1910) 220-269. [Yo1]
K.Yosida, Functional analysis; fourth edition; SpringerVerlag; New York 1974.
[Z1] S.Zelditch, Reconstruction of singularities for solutions of Schroedinger equations;Ph.D.Thesis,Univ.of Calif. Berkeley (1981).
INDEX
commutativity, spectral
accretive 197
theoretical 340
ad joint
,distribution
compactification 87
57
compactness
,Hilbert space 57, 201 diff. expression 120 adjoint relation 115 algebra, invariant 298f of the Dirac equation
of neg. order pdo's comparison algebra 189 comparison triple 127 complete spaces
with cylindrical ends 138 cone of time-like vectors 229 conical 120
310f, 319f
Baker-Campbell-Hausdorff formula 48 Beals composition formulas 64f Bessel function 24 equation 25 integrals 27 bicharacteristic strip 42 boundary condition Dirichlet 150 Lopatinski-Shapiro 169f
coordinate invariance 129f, 283, 293f transform, S-admissible 124 cut-off function,
S-admissible 126 Dirac equation 313
Dirichlet problem 145 exterior-interior 151 dissipative operator 184, 197
150 Neumann boundary space Zx 186
distribution 9 temperate 11, 118, 123f
boundedness of pdo's in A2 99f Bracket operation 47
Egorov theorem 145 elliptic 83
calculus of 'ado's 72f
Cauchy problem, local 244f nonrelativistic 227, 236 relativistic 227, 237 characteristic equation 41 set 224
md- 82
formally (md-) hypo- 85 local (md-) 87f md- with repect to 89 end conical 127
strip 38, 41
cylindrical 127
surface 41, 230
evolution operator 175, 206f
simple, multiple 42 complex 44
finite propagation speed 241
,non- 89
commutator equation 301
flow 33
380
Index
flow,
381
,
characteristic 33, 196 particle 196, 215f, 220f, Foldy-Wouthuysen transform 319f, 323 theorem 343
Fourier integral 3f inversion formula 5 -Laplace method 20f series 14 transform 1,3f for temperate distributions 12 for general distr. 14f Fredholm inverse 113 special 113 fundamental solution 154 of heat operator 25
boundary 145, 152, 158f
differential expression 91f, 174f formally md- 87f infinitesimal generator 176,197 initial conditions 36 value problem 42 for heat equ. 25 inverse, partial md-
98
Jacobi identity 284 K-parametrix 81
Kirchhoff formula 27 Klein-Gordon equation 29 Laplace transform 19 Leibniz' formula
geometrical optics approach of the Dirac algebra 324f Green inverse 112, 113f, 139f, Green's function 181
with integral remainder 70f Lie algebra 45f Lie group 45f
Lopatinski-Shapiro conditions 145, 151, 169, 180
Hamilton flow 145 system 218 Hankel function 25 transform 22 Heisenberg group 249 representation 313 Helmholz equation 24 hermitian operator 184 Hille-Yosida-Phillips theorem 146, 176 hyperbolic 226f normally 227 polynomial 227 semi-strictly 283 strictly 45, 196, 210, 228 of type f 212 symmetric 196
hypergeometric function 28 hypo-elliptic
L-S-cdn. see Lopatinski-Shapiro maximal ideal space 87 propagation of 223f minimal operator 180, 198 multiplication operator 52 natural classes of pdo's 247 Neumann problem 145 normally solvable 147,161f,189 observable 314f
basic 341, 355, 366 corrected 336 spin 340
standard dynamic 336 order classes 110, 139f Paley-Wiener theorem 14
382
Index
parametrix 81 K- 82 local 89 with respect to 89 Parseval's relation 5 principal symbol space 89
Sobolev estimate 107 imbedding 107 Sonine-Gegenbauer formula 29 Spectral theory 179f spectrum, essential 338 strip 37 characteristic 38f condition 37
type 42, 231 pseudodifferential operator 53
initial 37
integral 37
4 do-theorem 251f, 257f
structure, S- 127
realization,of a differential expression 180 region of dependence and and influence 241f resolvent 262 compact 180 set 180
symbol
classical 78 local 118
of a 4do 53, 249 of a C -algebra 87, 189 classes ST 53 SS 61 STi
60
Vh
61
S-manifold 121
i'c
78
SG-structure 120 Schmidt norm 254, 260, 339 Schroedinger representation 312 secondary symbol space 89 self-adjoint operator 185, 186f semigroup 175, 179f singularities, propagation of 223f smooth operator 247, 261f translation 261 gauge 261 rotation 264, 318 dilation 264, 318 Sobolev norm
Vt
69
V1 'yes
269, 275 269, 275
LS
135
Riemann Hilbert problem 145,154
, weighted 139
Sobolev space 106f , unweighted 107f , weighted 106, 108f
LC 135 Heroander type 62 symbol space 87, 189 principal 89 secondary 89 symmetrizer 210f, 212 systems of ydo's 97f I
wave front set 93f space 88
wave operator 230 Weinstein-Zelditch class 273 Weyl representation 61
