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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
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W. METZLER & A.J. SIERADSKI (eds)198 The algebraic characterization of geometric 4manifolds, J.A. HILLMAN199 Invariant potential theory in the unit ball of Cn, MANFRED STOLL200 The Grothendieck theory of dessins denfant, L. SCHNEPS (ed)201 Singularities, JEANPAUL BRASSELET (ed)202 The technique of pseudodifferential operators, H.O. CORDES203 Hochschild cohomology, A. SINCLAIR & R. SMITH204 Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT, J. HOWIE (eds)207 Groups of Lie type and their geometries, W.M. KANTOR & L. DI MARTINO (eds)
London Mathematical Society Lecture Note Series. 202
The Technique ofPseudodifferential OperatorsH.O. CordesEmeritus, University of California, Berkeley
AMBRIDGEUNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.orgInformation on this title: www.cambridge.org/9780521378642
© Cambridge University Press 1995
This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.
First published 1995
A catalogue record for this publication is available from the British Library
ISBN 9780521378642 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 1
0.1. The Fourier transform; elementary facts 3
0.2. Fourier analysis for temperate distributions onIn
9
0.3. The PaleyWiener theorem; Fourier transform for
general uE D' 14
0.4. The FourierLaplace method; examples 20
0.5. Abstract solutions and hypoellipticity 30
0.6. Exponentiating a first order linear differential
operator 31
0.7. Solving a nonlinear first order partial differen
tial 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 52
1.0. Introduction 52
I.I. Definition of pdo's 52
1.2. Elementary properties of ipdo's 56
1.3. Hoermander symbols; Weyl pdo's; distribution
kernels 60
1.4. The composition formulas of Beals 64
1.5. The Leibniz' formulas with integral remainder 69
1.6. Calculus of 1pdo's for symbols of Hoermander type 72
1.7. Strictly classical symbols; some lemmata for
application 78
Chapter 2. Elliptic operators and parametrices in Mn 81
2.0. Introduction 81
2.1. Elliptic and mdelliptic Vdo's 82
2.2. Formally hypoelliptic pdo's 84
2.3. Local mdellipticity and local mdhypoellipticity 87
2.4. Formally hypoelliptic differential expressions 91
2.5. The wave front set and its invariance under yxlo's 93
viii Contents
2.6. Systems of ,do's 97
Chapter 3. L2Sobolev theory and applications 99
3.0. Introduction 99
3.1. L2boundedness of zeroorder do's 99
3.2. L2boundedness for the case of 6>0 103
3.3. Weighted Sobolev spaces; Kparametrix and Green
inverse 106
3.4. Existence of a Green inverse 113
3.5. Hscompactness for ftpdo's of negative order 117
Chapter 4. Pseudodifferential operators on manifolds with
conical ends 118
4.0. Introduction 118
4.1. Distributions and temperate distributions on
manifolds 119
4.2. Distributions on Smanifolds; manifolds with
conical ends 123
4.3. Coordinate invariance of pseudodifferential
operators 129
4.4. Pseudodifferential operators on Smanifolds 134
4.5. Order classes and Green inverses on Smanifolds 139
Chapter 5. Elliptic and parabolic problems 144
5.0. Introduction 144
5.1. Elliptic problems in free space; a summary 147
5.2. The elliptic boundary problem 149
5.3. Conversion to an &nproblem of RiemannHilbert
type 154
5.4. Boundary hypoellipticity; asymptotic expansion
mod av 157
5.5. A system of fide's for the Vj of problem 3.4 162
5.6. LopatinskijShapiro conditions; normal solvabi
lity of (2.2). 169
5.7. Hypoellipticity, and the classical parabolic
problem 174
5.8. Spectral and semigroup theory for ado's 179
5.9. Selfadjointness for boundary problems 186
5.10. C*algebras of tpdo's; comparison algebras 189
Chapter 6. Hyperbolic first order systems 196
6.0. Introduction 196
6.1. First order symmetric hyperbolic systems of PDE 196
6.2. First order symmetric hyperbolic systems of
fide's on n. 200
6.3. The evolution operator and its properties 206
Contents ix
6.4. Nth order strictly hyperbolic systems and
symmetrizers. 210
6.5. The particle flow of a single hyperbolic pde 215
6.6. The action of the particle flow on symbols 219
6.7. Propagation of maximal ideals and propagation
of singularities 223
Chapter 7. Hyperbolic differential equations 226
7.0. Introduction 226
7.1. Algebra of hyperbolic polynomials 227
7.2. Hyperbolic polynomials and characteristic surfaces 230
7.3. The hyperbolic Cauchy problem for variable
coefficients 235
7.4. The cone h for a strictly hyperbolic expression
of type e° 238
7.5. Regions of dependence and influence; finite
propagation speed 241
7.6. The local Cauchy problem; hyperbolic problems
on manifolds 244
Chapter 8. Pseudodifferential operators as smooth
operators of L(H) 247
8.0. Introduction 247
8.1. ,do's as smooth operators of L(H0) 248
8.2. The 11DOtheorem 251
8.3. The other half of the gbbOtheorem 257
8.4. Smooth operators; the V algebra property;
'Wdocalculus 261
8.5. The operator classes 'S and 'IVL , and their
symbols 265
8.6 The Frechet algebras y"x0, and the Weinstein
Zelditch class 271
8.7 Polynomials in x and ax with coefficients in 'TX 275
8.8 Characterization of qtX by the Lie algebra 279
Chapter 9. Particle flow and invariant algebra of a semi
strictly hyperbolic system; coordinate invariance
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
Smanifolds 293
9.5. Conjugation with eiKt for a matrixvalued K=k(x,D) 296
x Contents
9.6. A technical discussion of commutator equations 301
9.7. Completion of the proof of theorem 5.4 305
Chapter 10. The invariant algebra of the Dirac equation 310
10.0. Introduction 310
10.1. A refinement of the concept of observable 314
10.2. The invariant algebra and the FoldyWouthuysen
transform 319
10.3. The geometrical optics approach for the Dirac
algebra P 324
10.4. Some identities for the Dirac matrices 329
10.5. The first correction z0 for standard observables 334
10.6. Proof of the FoldyWouthuysen theorem 343
10.7. Nonscalar symbols in diagonal coordinates of 350
10.8. The full symmetrized first correction symbol zS 356
10.9. Some final remarks 367
References 370
Index 380
P R E F A C E
It is generally well known that the FourierLaplace trans
form 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 log
ically consistent theory. Thereafter it became the FourierLaplace
method for solving initialboundary problems for standard PDE. We
recall these facts in some detail in sec's 14 of ch.0.
The technique of pseudodifferential operator extends the
FourierLaplace 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 'wellposedness' (or normal sol
vability) of certain such abstract linear operators. Accordingly
both, the FourierLaplace 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 ope
rator 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 mani
festation with respect to such operator algebras: conjugation with
xii Preface
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 'distribut
ion 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 in
variance of idoalgebras under conjugation with evolution opera
tors, and do not touch the type of oscillatory integral and fur
ther discussions needed to reveal the structure of such evolution
operators as Fourier integral operators. In terms of Quantum mecha
nics 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 197480 (chapters IIV) published as preprint
at U. of Bonn, and (ii) a set of notes on a seminar given in 1984
also at Berkeley (chapters VIIX). 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 theo
ry 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 IIV, pointing to manifolds with coni
cal tips or cylindrical ends, where the 'Fredholmsignificant sym
bol' becomes operatorvalued.
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 Uni
versity 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 pseudodifferential 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 FourierLaplace trans
forms of temperate and nontemperace distributions. In sec.4 we
discuss the FourierLaplace method of solving initialvalue pro
blems 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 hypoellipticity and "boundaryhypoellipticity"
can lead to existence proofs for classical solutions of initial
boundary problems. Sec.6 is concerned with the operator eLt , for
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 mini
introduction 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 (2n)n/2 , ox = xndx1dx2...dxn = xndx .
(b) (x) = (l+Ix12)!/2 , (1+1 12)!,2 , etc.
!
2 0. Introductory discussions
(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 nvector
j J
with components ulx , V"u for the kdimensional array with compoJ
nents ulxi xi .. For a function of (x,i;)=(xl " '.'fin)z '
it is often convenient to write u(a)=aaXU.
(d) A multiindex is an ntuple of integers a=(a1,...an)
We write laI=lall+...+lanl , al=all...an! , ((3)=(R )...( ),
a axa= xl 1...xn n , etc., IIn={all multiindices} .
(e) Some standard spaces: &n = ndimensional 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 complexvalued
functions over a domain or differentiable manifold X (or sometimes
only X=1en): C(X) = continuous functions on X ; CB(X)= bounded con
tinuous 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 LaurentSchwartz 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, distri
bution spaces D', E', SO.
(g) LPspaces: For a measure space X with measure dµ we wri
te Lp(X)=Lp(X,dµ)={measurable functions u(x) with luIp integrable}
for lsp<oo; L00(X)={essentially bounded functions}.
(h) Maps between general spaces: C(X,Y) denotes the conti
nuous maps X>Y . Similar for the other symbols under (f), i.e.,
CB(X,Y) ..... .
(i) Classes of linear operators (X= Banach space) : L(X)
(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(xy)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 limEOa(E)= alE..O
(m) Abbreviations used: ODE (PDE) = ordinary (partial) diff
erential equation (or "expression"). FOLPDE (or folpde)= first or
der linear partial differential equation (or "expression"); 'pdo=
pseudodifferential operator.
(n) Integrals need not be existing (proper or improper) Rie
mann or Lebesgue integrals, unless explicitly stated, but may be
distribution integrals By this term we mean that either (i) the in
tegral may be interpreted as value of a distribution at a testing
functionthe 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 impro
per 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 adjointAT=A*,
in case of a matrix A=((a.k)), respecti
vely. For a symbols a* (or a+) may denote the symbol of
the adjoint 'tpdo of a(x,D) , as specified in each section.
(p) supp u (sing supp u (or s.s.u)) denotes the (singular)
support of the distribution u.
1. The Fourier transform; elementary facts.
Let u E L1(&n) be a complexvalued integrable function.
Then we define the Fourier transform u^= Fu of u by the integral
(1.1) u^(x) = J x E &n ,
with x =x. = an existing Lebesgue integral. Clearly,
(1.2) u^ (x) I s IIu11L1 n9lxlu(x)l=
Note that u^ is uniformly continuous over In : We get
u^ (x)u^ (y) I s 2f¢tI
(1.3)
s NIX111 llullL1 + R1
where the right hand side is <E if N is chosen for <E/4,
4 0. Introductory discussions
and then Ixyi< a/(2NIIuiI ) Moreover, we get u^E CO(&n), i.e.,L
limlxI.00u^(x)=0, a fact, known as the RiemannLebesgue lemma.
To prove the latter, we reduce it to the case of u E CO(&n):00
is known to be dense in LI . By (1.1) we getThe space CO
(1.4) Iu^ (x)v^ (x) I s IIuvilL1 < e/2 , as v E CO , IIuvuIL1<E/2.
Hence limI X I
OOv^ (x)=0 implies I u^ I s I u^ v^ I+ I v^ I< e wheneverx is chosen according to Iv^I < e/2 .
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) IxI2u^(x) =J0 =_f =(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
L1/1X12 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) , or, u"= Fu , where
(1.7) u" (x) = 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 nonintegrable func
tions, 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 Hil
bert 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. )(1.9)
Qglxu^ (x)v^ (x) = _1
Neix
7 j "j qIx.N 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 sNs340 , = 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
(1.10) fQN 41xu^ (x)v^ (x) = j
where cp(t) = (2/n) 1/2 sin t , t340, continuously extended into t=0.
For vE C(Rn), as N , the function will converge
to independent of . Thus one expects the inner integral at
right of (1.10) to converge to j = since
(1.11) P sin t dt/t = n/20
Legalization of this argument will confirm Parseval's rela
tion, 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+
f1OjZbcos(NO)wl0(O)dO).
The latter gives 100 s (11vIILW+ 11v' 11L00) , 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 IIuujIILl  0, 11uuill L2 0,
as is possible. Then, since ujulE CO C L2 , (1.13) with u=v=ujvj
implies Iluj^u1^IIL2=IIuju11IL2 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^ ,(p)=fz pdx for all cpE Cc". This yields f (u^ z)cpdx=0 for
all such cp, hence u '=z (almost everywhere), since CO is dense in
V. Substituting u=v=uj in (1.13), letting j3, 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) =(eixxo_
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:
(1.16) (20)Ixo+b
v(x)dx = JgIxv^(x)eixxo sin 6x
xo b
Assuming only that v , v^ both are in LI , it follows indeed that
(1.17) limn0(26)nfQ v(x)dx =f9lxv^(x)eixxo= (v^ )I (xo ), xoE 2n.
xo ,s(Actually, our proof works for n=1 , xo > 0 only , but can easily
be extended to all xo , and general n . One must replace the deri
vative d/dxo by a mixed derivative an/(axol...axo n). ) Indeed,
0.1. The Fourier transform 7
letting 60 in (1.17) we obtain (1.15), using that sin(6x) /(Sx)
 1 uniformly on compact sets, and boundedly on & , as 60 .
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(xo0))/2
Again for n=1 a limit of (1.16), as exists, if only
(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^) are defined for all u E L1(&n) , and we
have u^ , u" E CO(1Qn) . For u E L1(In)1L2(In) we have Parseval'srelation (1.8) . 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(xY)w(Y) = f41Yv(xY)u(Y)
Indeed,
(1.20) IIwIIL1=JIw(x)Idx s xnfdxfdyIu(xY)IIv(Y)I = KnIIuIIL1IIvIIL1
Prop.1.2, below, clarifies the role of the Fourier transform F for
this Banachalgebra: F provides the Gelfand homomorphism.
Proposition 1.2. For u,v E L' let w = u*v . Then we have
(1.21) w^ ( ) = u^ E &n
Proof. We have
fgixe fpiye
The substitution xy=z , dy=dz thus confirms (1.21), q.e.d.
The importance of the Fourier transform for PDE's hinges on
Proposition 1.3. If u(P) E L1 for all (3s a then
(1.22) 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 , P(D) = a Daja sN ax
where fE L1(&n), Dx =iax , one might attempt to find solutions byj 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(xy)f(y) .
Presently, (1.26) can only have a formal meaning, since nor
mally (1/P)(4 L', or f (4 L' , or u(4 LL, in practical applications.
However, as to be discussed in the sections below, the Four
ier 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).
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
22) For general n obtain the Fourier transform of
a_ ax, 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 pro
blem). 4) Construct a function f(x) E L1(&n) such that f^(4 L1
5)The RiemannLebesgue 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 trans
form satisfying f(x) = O(1/x) , as Jxj is large, and satisfying
(1.27) (f(x+0)+f(x0))/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 , possibly after
changing its value at boundary points.
0.2. Fourier analysis for temperate distributions
2. Fourier analysis for temperate distributions on &n
9
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 conti
nuous if (f,gj).0 whenever (pj0 in D. The latter means that (i) fj
E D, j=1,2,..., (ii) supp cpjE KCC &n , K independent of j, (iii)
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 local
ly 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
( 2 . 2 )(f(a),(P)
= (  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) , p E D .
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 distribut
ion 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 clo
sed set E (intersection of all closed sets E) such that f=0 (such
that f isC00
) in the complement of E .
The concept of Fourier transform can be generalized to distri
butions on &n, with multiple benefit: Some nonL'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.
10 0. Introductory discussions
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
define u^ for general uE D'(&n) , but it no longer will be a dist
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 multiindices a and k=1,2,...,
(2.4) sP(a)(x) = 0((X)k)
 the derivatives of y decay faster than every power(x)k
Note that, equivalently, we could have prescribed that for eve
ry 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) , or is o(1) , or is
CB , or CO , or L2 , 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 bijec
tive linear maps S S , inverting each other.
Proof. Using repeated partial integration and
get fdxe
=iIaI+ISIa hence
(2.7) (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 11
(2.5) , for u E S . Therefore the right hand side is in CO , for
every a,(3 , so that u^E S , again by the equivalence (2.5). Thus
we get u^ E S for all u E S . Similarly for """ . Also, the Fou
rier inversion formula holds for u E S , and the left hand side of
(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 tempera
te distributions is denoted by S'. Clearly, S DD , so that a func
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 multiindex 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)  0Temperate 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 cpj0 in D , then c.>0 in S (by (a)),
hence (v, p) =(u, (pj) 0. Hence vE D' . Furthermore, if u, wE S' haveuID=wID=vE D', then for gE S let (p. be a sequence of (b) above.
Get uwE S' , (uw, ccpj) 0. Hence 0=(uw, (pj) _(vv, cpj) +( uw, (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, cpj0 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 we have y. 0 in S, proving (a).
To prove (b), let X(x)EE
CoO(&n)
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)=1x(x/j), get p.=gc.=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 jco).
Also, sup{Iwj(Y)I}=jIYIsup{w(x):xE 2n}s c. Thus Vj>0 in S, q.e.d.
12 0. Introductory discussions
Note that polynomials, and delta functions S(a)(xa) are ex
amples 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 SIt is sufficient to prove this for F. Again we need an equi
valence like (2.2), now for the property in S' :
Proposition 2.6. Let q E S , j=1..... . Then 'ypj 0 in S ' is
equivalent to each of the following conditions:
(x)kgpj(a) 0 , or Apj(a) a 0 , or (xs(pj)(a) 0
(2.8) for all multiindices a , or k=0,1,2,..., in one (and
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).
Indeed, if pj 0 in S , we have IlxRcpj(a)IILI> 0 , j hence
(xaq)j^ ) (P) IICB_ 0 , implying (pj^  0 , q.e.d.For a given u E S' , observe that u^ , defined by
(2.9) (u^ ,p) = q E S ,
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) (u,cP^) = ix= (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 L1functions
(cf.(1.1), and (1.7)) . We have the Fourier inversion formula
(2.12) (u^ )` = (u` )^ = u , for all u E S'.
Also, for u E S' we have E S' , 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 a(3YxaYu(RY) , xau(1) = EdaRY
(xaY u) (RY) ,
with finite sums and constantscaRY , daRY
b) We have
(2.14) Ixals(x)lal , and (x)ksc (x011 with a constant ck laIsk
k
c) We have
(2.15) (lullLp
s II(x)kllLp
II(x)kull Loo, Isp<oo , k>n/p
d) We have
Ilull scIIU^ II lscll (1+Ixl )n+1u^ II c 7, Ilxau^ II(2.16) L L L lalsn+l L
= C E II (U(a)
)^ II "o s c I IN(a)
II
lalsn+l L lalsn+l L
Problems: 1) Show that the following functionals define distribu
tions 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
rp(x)dxX (for n=1). 2) Obtain the first partials of theIxarl
distributions of pbm.1. 3) Show that distributions f.E DI(I) are
defined by (f+,T)=lime0,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
14 0. Introductory discussions
E D'(2), but eax S', as Re a O. 9) Let Tpol be the class of all
kaE 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' inva
riant. 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 distribu
tions (If necessary, show, they are in S'): a) xa, a E 2+; b)
S (P), E Zn ; c) eiax , aE 2n. 11) Obtain (p.v.1)^, for the disxo + 3F
1tribution of pbm.1. 12) Define a distribution x E S',
using the same kind of 'principalvalue integral' as in pbm 1.
Calculate (p.v.slnh X)^ 13) Obtain the Fourier transform of a
2nperiodic C00(2) function a(x). Hint: Use that a(x) has a uni
2nformly convergent Fourier series a(x)=y00 ameimx, am aeiz°xdx
0
14) Let f(x)=sin xJ . Show that fE S' and evaluate f^.
3. The PaleyWiener 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 (1x)TE D(fz), andsupp (1x)TC supp (1x) is disjoint from supp u, hence (u,(1x)(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 rx=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) com
pactly supported uE D'(fz) coincides with the set of continuous lin
0.3. The PaleyWeiner 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 ypj0 in E means thatTj (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, Tj0
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)=aizx,
is meaningful for all zE Cn, (not only &n), and defines an exten
sion 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) , E > 0
For the directional derivative Vhez of ez at z , we get
(3.5) V. = (ez+eh ez)/E  Ohez 0 in E
Indeed, this only means that aXVE, 0 uniformly on KCC 2n, as rea
16 0. Introductory discussions
dily verified for Continuity of u then implies
(3.6) limE0,Ex0wE _ (u,Vhez) ,
confirming that v(z) is analytic for all z. Formally we then get
(3.7) (u^ ,q)) = rP(U)) = f91 (u,e
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 fR n \K
as K runs through a sequence Kj with LX.=&n,
again, with convergence in E. Again, convergence in E just means
local uniform convergence with all derivatives. One confirms easi
ly the local uniform convergence in the parameter x , since the
function e,(x) = e'x is continuous. Similarly for the xderiva
tives, again continuous in x and . This, and the fact that the
xderivatives 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 deltafunction 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 distri
butions. The result is called the PaleyWiener theorem.
Theorem 3.3. An entire analytic function v(z) over Tn is the Fou
rier 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 PaleyWeiner 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 , and
">" holds in (3.9). Or, I(pja)(x)Is for all lalsj Ix1sj , j=1,
2,... , implying uniform convergence Tja)(x) 0 ,j*oo , a contra
diction, since 1=(u, c).) does not tend to 0.We get u^(z)=(u,Xzez) xz=X(IzI(IxIi)) where xEC00(R)
x(t)=1 , t<? , =0, t>1 , x decreasing. It follows that supp X. C
{IxI511+ } so that (XZez)(a)(x)= O(e'1IImzI+1(z)k). Combining this
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=
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
(3.11) u(x) =
Indeed, this is a matter of Cauchy's integral theorem, applied for
a rectangle in the complex i;jplane 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 com
plex integral over the boundary vanishes. For A the integrals
over Re tj=±A, O<Im i;j<0il tend to zero, in view of (3.8) for k=2
for example. The integration pathes have length 0i and the inte
grand is O((A)2e(1x)IOI). The integral (3.10) may be written as
nfold iterated integral over It. The above proceedure allows the
transfer of the integration from R to the line { j+i0 xjE I}
Next let us estimate (3.11):
(3.12) u(x) = O(e1101xO)
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,
t>0, for u(x)=O(etlxl(11Ixl)), The exponent is <0 as 1xI>1, and
18 0. Introductory discussions
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 ( )=x^ (ei;)+x^ (0)=1, as Moreover, for any cpE S get 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 uCxI2M+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
independent of j, and (ii) mj=Max{Ivj(x)I: x ENn}0,
as
Corollary 3.4. The Fourier transform F: u , u^ establishes a li
near 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
form uj=vj^ we get 1Iuj(a)ii 0, so that indeed in D. Q.E.D.00 _>L
Following [GS] we now define a Fourier transform of a gene
ral 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
(3.13) (f^ ,(p) = (f,(p^) , 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
0.3. The PaleyWeiner theorem 19
ous maps S>S. Also T j0 in Z implies cp.> 0 in S. For uE S' the restriction u1Z determines u and we have ulZE V. Hence get a naturalimbedding 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.
Notice that u^ , for uE D' in general is not a distribution,
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<O and
f=O(ecx) , some c, one commonly defines the Laplace transform by
(3.14) f(r;)= f glxe lxtf(x) , Im t < c0
where the integral exists and defines a holomorphic function in
the complex halfplane Im i; <c (we have modified the standard
definition, by a factor i). The inverse transform then is given by
(3.15) f(x) = +«'+iy plteixif (i;)oo+iy
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) f (t) q), (t) dt = P91xf (x) f eixtg. (t) dt _ (f,g)Im =y Imt=y
The integral dxdIl;I exists absolutely, hence the interchange, by
Fubini's theorem. Also, we get f = fR , 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 ZIm?=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 func
tions (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 (multivalued) function well defined
in a halfplane Im z < y .) 3) For uE D'(ien) with supp uC {xix0}=
20 0. Introductory discussions
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,
show that fE Z' allows definition of a product XfE Z' , setting
(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 FourierLaplace method; examples.
We now will discuss the 'FourierLaplace 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) ,
(4.3) H = axo  A = at  A (the heat operator) ,
(4.4) = axo2  A = at A (the wave operator)
(4.5) +m2 (the KleinGordon 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 FourierLaplace method.
Certain initialboundary problems may be converted into free
space problems: (a) An initialvalue problem for (4.3),(4.4), or
0.4. The FourierLaplace method 21
(4.5) seeks to find solutions u of P(D)u=f in some halfspace,
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 t<O, letting v and g be the exten
ded functions. We will not have P(D)v =g then, but, rather, P(D)v
=g+h , with a distribution h, supp hC {t=o}, since normally v will
jump at t=0. The initial conditions on u often are well posed if
they allow to determine h, making the initialvalue problem equiva
lent to the free space problem P(D)v=g+h , where g+h is given.
(b) Another example: If Au=f (A of (4.1)) is to be solved
in a halfspace under Dirichlet bondary conditions  say, Au=f in
x1>0, 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
halfspace 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 sur
face t=0(x) , x(= &n. Both techniqes may be combined to reduce cer
tain initialboundary 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 initialboundary 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 ex
ists 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) we get, respectively,
(4.6) P(x) =Ix12 , =k2Ix12 , =it+x12 , =Ixl2t2 , =m2+Ix12t2,
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 {t20} whenever supp gC {tao}, so that u=
(e*(g+h))I{t20} will solve the initialvalue 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
For (4.3)(4.5) we will construct such eED,(Rn+1) solving
P(D)e%r2_xn+I6 , supp e C {t20}, using the setup of sec.3, pbm.3:
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 in
verse Laplace transform of an inverse Fourier transform of a sui
table 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) ,
where w(r) E LI()e+,rn1dr). Then the transforms f^ and f, are sphe
rically 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(n1)/2X(r) = Hn/21(r(n1)/2(o(r))(4.7)
r(n1)/2w(r) = Hn/21(r(n1)/2X(r)).
where
(4.8) H (X(r))(p) = fV pr Jv(rp) ?(r) dr , Re v > 20
with the Bessel function J (z) . The second formula (4.7) is validv
if in addition X E L1(&+,rn1dr).
TProof. For an orthogonal nxnmatrix 0 get paxeix
Jxei(OTX)
Tw(IOTxI) = folye Thus f^ has the same
symmetry: with some X(p). We may set
(4.9) f^ I l ; I , 0 , ... , 0)=f RIXeipx w(r)=Kn f rn1w(r)f eirpz, dS0
where the inner integral I is over the unit sphere Izl=1. Evaluate
this inner integral by converting it to an integral on the n1di
mensional ball IXI2 sl, setting z=(z1,?,). We know that dS=
With a contribution from the upper and lower hemisphere where z,=
172 , writing dX=an2dodl, o=IXI, etc., we get
1
I=2 on2 1662dEcos(rp)=tan1 f 6n2cos(rp)0
A substitution a= sin 0 of integration variable yields
(4.10) 1 = 2an1x /2
d9 sinn20 cos(rpcos 0)0
0.4. The FourierLaplace method 23
2 (n1)/2with an1 the area of the nldimensional='((n1)/2),
unit sphere. Using Poisson's formula ([MOS],p.79) we get
(4.11)fn /2
cos(rpcosO) sine28 d9 =2n/22V.(n21)Jn/21(rp)
0
Substituting into (4.10) and I into (4.9) confirm (4.7). No change
ifaix in (4.9) is replaced by elx . Thus lemma 4.1 follows.
Recall also (1.26), now under the aspect of 3,pbm.4. In det
ails, 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 varia
bles 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 initialboundary 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 distri
bution in S'. This is a homogeneous distribution of degree 2
Hence e is homogeneous of degree 2n . It is also sphe
rically symmetric. Conclusion:e(x)=cnIxI2n
, with a constant cn.
Clearly e E Lpol . The constant cn may be evaluated by looking at
(2n)n/2g(0)=(e,Ag)= cnfAW(x)1x12ndx =cnlims0 fdSicpdrr2n
r=e=cnan(2n)T(0) . It follows that
(4.14) en(x) =(n2)w 1xI2n n n 2 nn
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^ one finds that
(4.16) e(x) _jxI .
24 0. Introductory discussions
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
(4.18) e(x) = x1loglxl
is a spherically symmetric L11ocfunction satisfying Ae=b (justevaluate 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 timeindependent 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)=(X1xI2)1E C°n Lpol. An evaluation
of e(x) is possible, using lemma 4.1, as long as ns3 . We get
(4.19) e(x) =IxI1n/2
J Pn/2X Jn/21(Plxl)
For larger n this integral ceases to exist. However, it still
will exist as improper integral in the sense of distributions 
Athat 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/21 may be expressed by trigononmetric functions. For example,
in case n=3 we get J1/2(z) Or,
(4.20) e(x) = VnII J 24 sinpixl0
We may write X = x2 , picking the root x with In x>O . Then
(4.21)
sinpr = 2 KPp sinpr =41 eirpxdP = 2ixlxl Hence0000
e(x) =7 e'xlxl/lxl
(4.21) may be confirmed, noting that e=eiicr/r solves (A+X)e=2n6
For x=k real P(x)=k2lx,2 vanishes at the set Ixl=k, and
is not Ltioc Then look at p.f.(P(1x)). Or else, observe that
(4.22) 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 FourierLaplace method 25
both will solve (A+k2)e=(2n)3/26 . 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/21Kn/2_1(KIxI) , x > 0
with the modified Hankel function Kv(z). Again get (4.24) more dir
ectly, observing that e(x)=y(Ixl) solves (A+X)e=0, hence y(r) solv
es the ODE y"+n1y'x2y=0. Substituting y=rvb, v n1, we obtain
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=
ein14Jt,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 8(t)=(it+a)1 has inverse Laplace transform 04(t) 2neat,
ta0, =0, t<0 , calculating f eateittdt. For e'=F 1(P) get0
(4.25) e' (t) =(2n)1/2et1x1 2
, as ta0 , e' (t)=0 , as t<0 .
Recall that (e  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 aV_2t we thus get
_I X12ta0 =0 t<0(4.27) e(t,x) _() 11
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 uAu = 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
(4.29) Hv = g + 6(t)(9p(x) = h
Thus v=e*h , or,
J9Tye(t,xy)w(y).(4.30) u(t,x)=xn+lftstgltgTye(tt,xy)f(y) + xn+1r
III) Now we look at (4.4), or, the wave equation
(4.31) u = (at  O)u = f
We apply the FourierLaplace method as for (II): The function
(4.32) P = IxI t' 21x7{t+Ixi  tIxI}
has inverse Laplace transform (in t) given by
itjxl_ itlxl sin( x t) ta0(4.33) 2i Ix (e ) = V2n1xI
,
(and zero for t<0). Looking for F;1 of the function (4.33) we can
not 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), roE D(Rn) , r=IxI
using that (cos Ixlt)lt = Ixl sin IxIt . 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,(,)=V2z , for n=3  we get
f°=w(IXI) w(r)=(V_2n)(N(_2
00
)
f '(r) sin r13 cos2ptdp . Or,
0
w(r) = r Psin rp cos ptp
=r fco(sin p(tr)  sin p(t+r)) P
= r{ sgn(tr)  2} . Conclusion:
(4.35) e(t,x) =li atH(tIxI) , 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<O. We are tempted to write atH(tIxI) as
6(tIxI), but then must remember the proper interpretation.
Converting the Cauchy problem for the wave equation,
0.4. The FourierLaplace method
(4.36) ulttAu=p , t>0 , u=p , ult=V at t=0
into a full space problem we get
(4.37) vltt Av = S'(t)Clp(x) + 6(t)W(x) + g = h
where v and g are u and f , extended zero for t<0 . Evaluating
e*h we get the solution of (4.36) in the form
u(t,x) =4iJlxylst
yf(tlxyl,y)(4.38)
27
+ 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)) _
V(f j,((at)j(p)where fj=±sPn°+t or fj=tcPs,1t (with p=lxl) foreven (odd) J. The inverse Fourier integral of fj exist at least as
limAJ lxlsA' we apply lemma 4.1 as above. For odd n we get
(4.39) fj°(t,x)=wj(t,lxl) , wj(t,r)=*r v P7V(rp)ls'ncos pt}
for even n this will be of the form
(4.40) fj°(t,x)=wj(t,lxl) wj(t.r)=trv
J JV(rp){cos pt} dp0
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)JV2natwj(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]21=v, 1=, n=0(mod 4).
We use a WeberSchafheitlin 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= (rzt2)1/2cos(vsin1(r))=4Pj_1(F)'0
with a certain polynomial Pj1(t) of degree j1. Note that the
integral enters (4.41) under the derivative at , so that a=0 for
28 0. Introductory discussions
t>lxl For r<t we get I= rj(t+ txrx)j by the sametrformula. 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<r
whereQ21(2) =
=0(i)1 m(22m1)T2m(1_i2)1m
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/2Q2114P , t>r , =p (t+ t  ) 21 , t<r ,
(4.43)bQ (t) 1(_1)lm( 21 )t 2m+1(i_T2 )1m1 , 1 n!2>0,
21 1 =_0
2m+1 4
while for n=2 (i.e., 1=0) we set Q2i_1=0.
Case (c): n,j odd,
[nn1] n2
=j, n=2j+1, 1='', j=21+1, n=3(mod 4).2
Let I= f00
Jv(rp)cospt =v 2 f Jj112(rp)J112(tp)dp. Use formula 2,
p.50 of [MO] with (a,b)=(r,t), (v,n)=(1+?,1). Get vn=2, I=O, t>r,
(4.44) I=N""1.3.5... (211)xF, t<r2r 2 4 6 2.1 2 2 r 2r
where the hypergeometric function (hence Q21(z))
x1 (11(4.45) 2Fi(1+1;2;z)=11(21+1)z+li =](21+1)(21+3)2r + ...
is a polynomial in z of degree 1=n43. Combining (4.39) and (4.44),
(4.46) e(t,x)=(_i)1+1xr 211x21+1 t __n _
t IQ21(r)H(r ,)} 1n4 , rlxl
Case (d): n odd, j even, j=n41, n=2j+1, 1= , n=i(mod 4) . Now we
need I= f Jv(rp)sinpt r = n2 f Jj_i,,2(rp)Ji/2(tp)dp . Again use0
the formula of case (c), now with Get
(4.47) I=V7(t)3.5.. 21+1 2F, (1+1,11;3;tx )_`/ (t, t<r,2r r 2.4 2t2 2 2 r 211 rand 1=0 for t>r , with a polynomial Q211(z) of degree 211. Hence
(4.48) e(t,x) _ (1)lnr 21at1{Q211(r)H(rt)} n41 , r=IxI
IV) Finally, look at the KleinGordon equation (4.5), n=3 only
0.4. The FourierLaplace method 29
(4.49) (p+mz ) u = utt Au + mz u = f
Equation (4.32) now assumes the form
p = Xz ltz ={t+a.  tip} a(x) m2+ x
FesintVx mx VTP m
For n=3, v=2, j=n211 use (4.7),(4.8) on x(IxI) of (4.51), for
(4.52) Vrw(r) = nVt f
00
J1/2(rp)J1/2(t )p3/2/Vg +W dp0
This integral diverges, just as for m=0. Applying a formula of So
nine (1880) and Gegenbauer (1884) (first on p.104 of [MOS]), for
(a,b)=(t,r) , (µ,v)=(2,2 E) , m=z , s>O , we get
foo
J1/2(rp)J1/2+E (t +p ) pJ mp"I ',(4.53) 0
IF
1Elimn Js1 (MVt2
with existing weak limit in D'(13) . Or,
(4.54) e(t,x) = nm limE>O,E>O{H(tIxI)JEl(m t2 rz)/ tE
Recall, for v=s1 we have
Z/2) E1 z 2(4.55) Jv(z) = r(s)  0 r( +m+E m+
6>0
The second term, at right, goes to J1(z), as E0, uniformly on KM &n , 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 )/ t2rz ,(p) , Ti
2n limeOfrd(t)
(tt x )1E , where r(E)=E(1+c,E+ ...). HenceIxlst
T1=2n lime.,O{f dxdt t+rx (tEr)E =2n limfdtdr{(tr)E}t
(4.56) IxIst
=2n1imEOf1xlstdxdtat{r+t x }(tr)E =2nfH(tr){ }tdtdx
In other words, TI=n(:.a tH(tr),g) . Result:
(4.57) e(t,x) = r tH(tIxl)  J1(m tzxz)
30 0. Introductory discussions
5. Abstract solutions and hypoellipticity.
In sections 1,2,3,4 we have deployed the FourierLaplace
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
P(x)v^=g^, providing an 'inverse' for P(D) in the form spea
king roughly. In later chapters, this type of inverse construction
will be extended to variable coefficients  and more general boun
daries, but the central role of the Fourier transform is maintai
ned, 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 L2Sobolev spaces
 in theory of 1pdo's will have to be studied. Theory of (bounded
and unbounded) Fredholm operators in Hilbert spaces will be cru
cial for elliptic i,do's, as well as compactness properties.
Often existence of a generalized solution to a PDE (a boun
dary 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) = 1lull2 = (u,u) , u,v E H
The Laplace operator A= ax 2 defines a linear map CD(cZ)C'(S)C H7
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 selfadjoint extensions. A distin
guished such extension H , called the Friedrichs extension, may be
constructed using a rather simple formclosing 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 selfadjoint it possesses a spectral measure, as we will not
discuss here. Moreover, it follows that the unbounded operator K=
1H is invertible in the following sense: There exists a continu
ous operator R E L(H), called K 1 = (1H)1, such that R:H> dom H
C H is a bijection inverting the linear map K:dom H > H .
Accordingly, due to existence of the inverse R, it follows
that the linear equation
(5.3) (1H)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 uAu=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 (1L)u=f, for given fE HC L'loc(n)
C Du(st) is a distribution solution of ua(x,D)u =f, since
(u,c)a(x,D)"cp) = (u,g)((Lu) q,ED(R), usingthat (Lu,T)=(u,a(x,D)p), gE D(st)=CO(st)
Question: (a) If, in addition, we have fE C"(c), will u be
smooth  and a classical solution of ua(x,D)u=f? (b) If, in addi
tion, st has a smooth boundary r, and fE COO(clLE') , will also u beC0O(szlf), and will it satisfy boundary conditions?
For a(x,D)=A both questions have a positive answer. In par
ticular, 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 = b3a+ p , aa/a=1 jj=
xJ
with real Co coefficients bj(x) , j=1,...,n, and complexvalued
p()E C, all defined in some domain S2 C &n. By definition eG will
be the solution operator of the initialvalue problem
(6.2) atu=au/at =Gu , xES2 , to 0 , u=u0 , t = 0 ,
i.e., u(x,t)=eGtuo(x), uoEC00
(12), is the unique solution of (6.2).
The single first order PDE atu=Gu becomes an ODE along cer
tain 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 realvalued functions
t,xj of a parameter s. The first equation (6.3) gives ts=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 realvalued 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 non
intersecting smooth curves covering the entire domain R . A solu
tion x(t) with x(0)=x0 will exist in a maximal interval Ost<t0,
where either t0=oo or x (t) E St\K for tE [ tK, t0 ) for every compactset KC Sz with some tK<t0 . 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 , as 0stst,, can be verified from the fact that
x(t) solves (6.4) 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) ifand only if along every solution curve (t,x(t)) of (6.3) the com
posite function u(t,x(t))=y(t) solves the ODE
(6.5) dt = Y(t)q , Y(t)=p(x(t))
Indeed, atj(t) _ b.(x(t)) implies that
(6.6) EbJulx )(t,x(t))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 neigh
bourhood 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 solu
tion 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 max
imal tinterval (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) along0
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 S2S2 is 11by construction, and it may be inverted by solving the reverse
initialvalue problem (follow x(t) through x° for a tinterval of
length t° in negative tdirection). The map vt is Coo by standard
results of dependence of solutions of an ODE on initial values.
We thus obtain a 1parameter family F = {vt : tE &} of dif
feomorphisms vt : St  St , having the group property
(6.7) Vto 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) u(t,x) = ft(povt)(x)dt}0
Indeed our above description of u(t,x) translates to (6.8).
In particular the diffeomorphism vt is inverted by vt, 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 : tE le} of
linear maps etG:C"O(S2)C00(St) , defined by
(6.9) etGuO(x) = u(t,x) , u0 E C'(c) , t e & .
We also have
34
(6.10)
0. Introductory discussions
etG: CO(Sc)  CO(SZ) , t E &
The relation between equations (6.2) and (6.4) remains inva
riant 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
characteristic equations locally given by (6.4) , assuming that
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 diffe
rent 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<t .
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° . Let y (t)=1+d(x° ,x(t)). We knowy(t) is Lipschitz continuous in t and that
(6.12) Idy/dtl s {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(tt0), or
(6.13) x(t) E {d(x,x0) s ec(tto) 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,
locally of the form dµ=Kdx, 0<x(=C00. We still assume that the cha
0.6. Exponentiating a fast order operator 35
racteristic flow F is defined: All solutions of (6.4) extend inde
finitely. Clearly the restriction GO=GICO(st) may be interpreted as
unbounded operator G0 of H with domain dom G0 = CO(n).
Assume G skewselfadjoint: with realvalued bj , p we have
(6.15) G = Ebjaj + 1/2 x I(xbJ)Ij
+ iplx
then G0 with domain CO(n) clearly is skewhermitian: We have00
(6.16) (u,GOu) + (G0u,u) = 0 , u E dom G0 = CO(ft)
If u(t,x) solves (6.2) , for some u0E CO(sz), (6.10) implies that
u(t,.)E D(st), for each fixed t. The y(t)=Ilu(t,.)III EC00(1),
and
(6.17) 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) = vt(x) ,
with a realvalued positive Co°function w(x) independent of u0, by
(6.8), with p=(2x)I(xbj).
. From (6.18) and (6.19) we conclude1
rf
x
(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 skewselfadjoint 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 COOCOO the sense of the norm of H
36 0. Introductory discussions
Then eGt:co>C admits a continuous extension to H, a unitary000
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£,oU(£)Eu =G1u exists in H}
is a skewselfadjoint realization of the folpde G , extending G0.
This realization is unique, since domeGot=CO
is dense in H. G1 is
the closure of G0, and iG0 is essentially selfadjoint.
Finally consider a dissipative expression G (i.e. GO+G*sO).
G must have the form (6.15) with realvalued bj but complex p sat
isfying In p(x)a0, xE S. Hence (6.16) now assumes the form
(6.25) (u,GOu)+(GOu,u) _ Jdµ(Im p)lul2 s 0 , u E C,(a)
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) ,
1 n
where F is a given realvalued 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 applicationwill 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 in
itial condition at some nidimensional submanifold rC tt. Let I' be
given by x=x(s), s=(si,...,sni)E EC &ni, with ax/6s=((axj/asl))
of rank n1. 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 equat
ions resulting for pj=ulxj locally admit precisely one solution,
so that ulx is fully determined along the hypersurface E.
0.7. Solving a first order PDE 37
Thus it is convenient to assume not only u=w(s) but also p=
ulx given for sE E. Of course certain conditions on w(s),p(s) must
express the fact that p=ulx. In other words, we assume given
(7.2) x = x(s) , p = p(s) , u = w(s) , s E E ,
with ax/as of maximal rank n1, while x(s), w(s), p(s) satisfy
(7.3) lp aslx = aslw , s E E , 1 = 1,...,n1
(called the strip condition) and
(7.4) F(x(s),w(s),p(s)) = 0 , s E I .
A (2n+1)tuple of functions (7.2) satisfying (7.3) is called an
((n1)dimensional) strip: The plane in (x,u)space through the
point (x(s),u(s)) with normal vector p(s) is tangential to the
nlsurface (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) exten
ding a given nidimensional 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) satis
fying (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 characte
ristic 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 &, p0E 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
38 0. Introductory discussions
(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) de
fines a nondegenerate Cc'curve in xspace ien.
We claim that (x(t),u(t),p(t)) also may be interpreted as a
1dimensional strip in (x,u)space, along which F is constant.
That is, x=x(t) , u=u(t) defines a Co curve E in (x,u)space; at
(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)(xx(t)) = P(t)(xx(t))
defines a plane tangent to EE at (x(t),u(t)). Indeed the vector
(x (t),u clearly lies in the plane (7.9). We get
(7.10)d/dt(F(x(t),u(t),p(t)) = Flxx + Fluu + Flpp
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 1dimensio
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 nldimensional integral strip
(7.2). For each (x(s),w(s),p(s)), sE E, we obtain the unique char
acteristic integral strip through (x,w,p). That is we construct
(7.12) x(s,t) = x(s1,...,sn1,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) , s E E
The functions (7.12) exist only for (s,t) E no , with
(7.14) no = {(s,t) : s E E , Itl < t1(s)}
with a suitable function ti(s) > 0 , 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 xspace
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
(7.15) u = w,X , i.e., u(x) = w(s(x),t(x)) .
Indeed for x e l' we have t = 0 , 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
(7.16) ulxjaslxj = asl =p151
ulxjatxj = atw =i
pjatxj , (s,t) E 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 .
In particular we again get t=0 , 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
(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
initial conditions as our constructed x(s,t) , w(s,t) , p(s,t)
so that we must have x=y , K=w , p=q , for all sE I , ltl<tl(s)
Since K = uoy = u,x = w , we then find that u = w,X coincides with
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 nldimensional C"sur
face in In with local parametric representation, rank as/ax = n1.
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 nldimensional surface r={xE c:ip(x)=V(x0)}
is given, where VIx#O , and F(x,p(x),Vlx(x))=O on r. Then, if x=
x(s) represents (some part of) r, we find that w(s)=p(x(s)), p(s)=
Vlx(x(s)) defines an nldimensional integral strip of F. Suppose
this strip is admissible. Then thm.7.1 yields a solution cp(x), de
fined near r, such that p(x)=p(x), and l (x)=yrlx(x) on r. It fol
lows that r also is given in the form T = c , at least locally.
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) L = a(x,D) = a(x)DaI aTsK
a
is the CauchyKowalewska 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 realvalued 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 analo
gous to the real Co case, and will not be discussed in detail.
0.8. Characteristics 41
Accordingly we assume that L of (8.1) has complex C" coeffi
cients, but that the coefficients of the principal part polynomial
(8.2) a
laaa 0
=Nare realvalued 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
(8.3) aN(x,Cplx) = 0 ,
with the gradient Tlx of T. Clearly (8.3) is a single first order
PDE for the unknown (realvalued) 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
(8.4) x =
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 homogen
eous 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 sys
tem 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
(x(s),c)(x° with x(O)=x° , (0)=9Plx(x°where 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 ex
pression (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 of
(8.3), defined in B={Ixx°l<a} such that Tf1B = {xE B: cp(x)=c}
for some constants c, a>0 .
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
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 expres
sion L of (8.1) if r is given by cp(x)=0 with g(x)=0 , cplx(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 thataNI
(x°,c)lx(x°))#0, and then sol
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 in
tegral 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 expres
sion 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 initialvalue problem of (8.1): For a given (hyper) sur
face 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 N1
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 theo
rem (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 tan
gential to r, let us start prescribing arbitrarily the functions
u=1p0 , Dvu = 1U1 , ... , DNu = 1IrN1 , on r , where
(8.6)
0.8. Characteristics 43
This will determine all u(a) : lals N1, on r, as tangential deri
vatives of the Vj: For some local parametric representation x=x(k)
'xn1) 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
(8.7) pDNu + Fpaaa j = f
Now if q E C1 withTlx
yd 0 is constant on F we find that
(8.8)
ax7= (TIxj/DvT)Dv + Tj , j = 1 , ... n ,
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) p =(Dvq))N
I
N aa(x)cplxa(x)a
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 calcu
lation of all Dvu on F. Clearly this allows construction of a uni
que solution of (8.5) by its Taylor series, assuming that an analy
tic solution exists. The recursion may be used for estimates pro
ving convergence of the Taylor series and existence of an analytic
solution, if the coefficients are analytic. This sketches a proof
of the CauchyKowalewska 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 (of (4.1)) : The characteristic equation
(8.10) cpIx2 = Lj=1 'pIx2 = 0
has only constant realvalued 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 complexvalued analytic functions.
There exist solutions of (8.10) in the n complex variables x1,...,
xn, defined in a domain of T . Setting such q constant will given
44 0. Introductory discussions
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=atA, 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 indepen
dent 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)=tt0 we have cpjx (1,0), hence a2(glx)=a2101110=0
c) 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) 2 2j=1 JIx, = 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 noncharacteristic surfaces (cf.sec.4 and ch.7.). Again the
KleinGordon operator (4.5) has the same principal part and charac
teristics as the wave operator, and a similar Cauchy problem.
d) L is an elliptic operator, with Coocoefficients. That is, we
have as xE cz, E n, O. Again the characteristic equa
tion (8.3) implies cplx=0 assuming T realvalued. Real characteri
stics 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 hyper
bolic 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 ach
ieved by a linear transformation of independent variables. Then
the characteristic equation decomposes into the N equations
(8.15) TIxn = tj(roIx1,...,(' Ixn1) ,
with t =the real solutions of
0 for _In the presence of multiple roots we cannot expect that the
tj depend smoothly on 1If smoothness can be arrangedthen 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 Nth 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 pro
blem 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<rl , in G then is a path of NxNmatrices together
with its derivative V'=di,/dt. For i,(0)=e=unit of G , and gE G ,
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 neigh
bourhood 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 .
46 0. Introductory discussions
In these coordinates the tangent space at e=y(O) (i.e., the
corresponding Lie algebra A) consists of all a=(a3) E &v, repre
senting the matrix Aa = 2aJylu (0). We get 11correspondencesj
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)=jajygluJ
(0) . a(g) defines a global folpde La on G  i.e.,
a(yg(u))=La(yg(u)), with La, in coordinates y(u), near e, given by
( 9 . 1 ) La D Y.A30 (u,O))Y(u)a/aul, where 9(u.v)=Y(Y(u)Y(v)),1 j j
similarly for general g with yg . Again A **IL a:aE Av} , so that
aE A appears in 4 forms, as a vvector (a3), a matrix Aa , or a
folpde La on G , or again as a(g)=2ajygluj (0)=gAg .
For aE A define a smooth curve V(t) in G as solution of
(9.2) 3 = a(p(t)) , 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 i
parameter 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 1parameter 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'
Va(t), Ostsr's , for fixed aE A. This shows that, indeed, g=
Va1(E1)Va
2 (E2) .... VaM(Em) ,
for finite M and sjss.
0.9. Lie groups and Lie algebras 47
Corollary 9.3. We have
A t(9.3) Va(t) = e a = A,,3/j! , Aa = Va.(0)_EalY luj(0)
AThe subgroup Ge is generated by all e at, aE A, O<tse>O
Completing the structure of A, we define a bracket operation:
Consider the commutators [Aa,Ab]= AaAbAbAa and [LaLb]=LaLb LbLa
of the matrices and (matrixvalued) first order expressions, resp.
Get y(u)y(v)=y(9(u,v)), near uv=0, with 0(u,v) of (9.1). Differen
tiating this conclude that [y lu (0),Ylu (O)]= 2jcjlYlu (0) , with1 r r
xjl = 9rlu v (0,0)9rluv (0,0) . Accordingly,
7 1 1 j
(9.4) [AaAb]=Ac , c = (cr)=( is 1ajbl)
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(Pjgllujgjplluj)au1
At u=O get pj=aJe qj=b3e , 0(O,v)=v , hence 0 (0,0)=Olk Thus1Ivk
{q+}lu.(0)={ij }Ylu.(0)+E{bkPllvku,(0,0). We get
(9.6) j(pjgllugjplluj) _ 7,(ajblbjal)Ylu=b1AaalAb
with our above Aa, using the symmetry dlluvk=01lvjuk, at u=v=O.
Accordingly, [La,Lb]Y(0)=I(b1Aaa1Ab)Ylu1AaAb 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 **Lc
, under our 11correspondence.
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, skewsymmetry, 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) commuta
tors [A,B]=ABBA, A,BE A. Assume dim A = v . Prop.9.1 and cor.9.2
48 0. Introductory discussions
then suggest a direction of approach for generation of a corres
ponding Lie group G  a Lie subgroup of GL(in) : G should contain
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 on
ly 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 1parameter group, unless A and B commute. For general A, B we
will prove the BakerCampbellHausdorff formula: For small Iti ,
(9.8) Vt = eC(t) , 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+eAtBeAt )Vt ,
where the coefficient A+eAtBeAt E A is a convergent power series.
Proof. (9.9) follows trivially by differentiation. Also, the
family Bt = eAtBeAt satisfies the differential equation
(9.10) Bt' = adA Bt , 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) = eAteBt1 = (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
0.9. Lie groups and Lie algebras 49
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 j1) 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 . Note that the
matrix coefficients cj1 of C(X) = log X = > 1(1)iXJ/j are con
vergent power series in the N2 complex variables xjk , where
X=((xjk))j,k=1,...,NIwhenever IIX(t)II<1. The latter holds for
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 compu
tation. For the general case we will show that C(t) solves an ODE
((9.18), below) which leads to a recursion for the Cj.
We calculate that (eC)'e C = = 1=0 N(!k,)l(Cm),Ck
( 1)nm 1 (Cm),Cnm = I II(n)(Cm),Cnm
(nm)'m. 1 namn m
xn nm n m nm m I k mk1X00 1I_Oh C'C Thus=ten=1 n!=1(1) (m)(C )'C Here, (C )'=
k=0(_1)n m1(m+1)CkC,Cnk1the inner sum equals Tn
= min=01
7t1
1
=0CkC'Cnk1=k(1)nml(m+n
1) _CkC'Cnk1Zk
, where Zk =
nk1= (_1)n1k(n1), by induction. We get T ==0 7 k n
4_01 CkC'Cnkl(1)n1k(nk1 ) _ (ad C)nIC' whence
(9.17) ' 1n!(ad C)nIC' = (eC)'eC = Vt'Vt 1 = A + eCBeC
50 0. Introductory discussions
adusing (9.9) and eC=eAteBt=Vt. With (9.12) we get eCBeC= e C B
The left hand side of (9.17) may be written as f(adC)C'
with the power series f(ey)= 1n1/n! =(ex1)/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 tkadck)A + h( tkadC
k)B , h(x)=g(x)ex
The coefficient of the kth power of t , at right, is a finite
linear combination of a finite number of applications of adC toj
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. Let All... ,A, be a basis of A (as a linear space),
and let A(u)= "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
(9.19) eA(u)eA(v) = eA(f(u'v)) ,juI,IvI<s0
.
Proof. As in the proof of lemma 9.5 we conclude that eA(u)eA(v) _
eC(u,v)where C(u,v) is a convergent power series in u and v as
IIA(u+v)ll<log 2 . Lemma 9.5 then implies that C(u,v)E A for suffi
ciently small Iuj,IvI. Hence we may write C(u,v)=A(f(u,v)), where
again fk(u,v) are convergent power series. It then is evident that
fk(u,v) = uk+vk + higher powers. Q.E.D.
From the given linear subspace A of L(EN) we now define a
group G  G(A) as the collection of finite products of matrices
eA , with A E A , where the group operation is matrix multiplica
tion. Clearly G contains I = e0 . We intend to show that G is a
(vdimensional) Liegroup. Moreover, the tangent space of G at
0.9. Lie groups and Lie algebras 51
its unit element I equals the Liealgebra A . And, vice versa,
if we depart from a general Liesubgroup 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 neighbour
hood 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<s0
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, Then
eA(u0)=eA(v0)Z , Z=GG'1E G , for some u0 , v0 E BE . Using cor.
0 09.6 we get Z = eA(f(uv)) , 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 v
v(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)eA(v(u))for u close to uO.
Or, YG() = YG,(v(u)) for small luu0l 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 de
rivatives of curves in G starting at I) coincides with A .
Thus, indeed, we established a 11correspondence between the
connected Lie subgroups of GL(2N) and the Lie subalgebras of L(&N),
Chapter 1. CALCULUS OF PSEUDODIFFERENTIAL OPERATORS
0. Introduction.
In this chapter we deal with the details of pseudodifferen
tial 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 re
minder 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', referen
ced 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 dis
cussions has our sympathy. For other approaches to the same sub
ject cf. ch.7 where the 4do's of certain symbol classes are ident
ified as operators on H=Lz(&n), smooth under action of certain Lie
subgroups of U(H). Or else, cf. [ C1] and [C2], where regular andsingular elliptic boundary problems are approached with tools of
C*algebras, avoiding entirely the adocalculus.
The calculus presented generalizes formal calculus of diffe
rential operators. We get a collection of Frechet algebras contai
ning differential operators, with formulas for product and adjoint
like Leibniz formulas, containing generalized inverses (socalled
Green inverses) of their elliptic and hypoelliptic 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 pseudodifferential operator generated from a 'symbol'
E C00 (1e3n), with ten , we will write a(M1,Mr,D) to indicatean order of operation: Multiplication corresponding to the x(y)
variable is carried out to the left (to the right) of the differen
tiation D of the avariable, hence "M " (hence "Mn') "(cf. cor.2.3).1
Introduce the space ST of all C'(&3n) satisfying
( 1 . 1 ) 0(()x(k).((x,Y))X(1)IaI+I1Isk, IYIs1
with nondecreasing x(k),X(l) , k,1=0,1,..., such that
(1.2) j) = limj.(?(j)  j) = 0°
Here, and in all of the following, we use the abbreviation
(z) = (1 + IzI2)1/2 , z E 1em
(particularly ((x,y))=(1 + x2 + y2)1/2) Another useful shorthand:
(1.4) 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 y u(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 impro
per 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 u(y) exists as descri
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 (xY) = (xY)
m=0,1,2,..., with the Laplace operator Ay = 32 . A partial inteJ 7
gration for fdy is applied over a finite ball. The boundary terms
54 1. Calculus of pseudodifferential operators
die out, as the ball tends to In, by (1.1) and uE S. This yields
(1.7) 2m 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
(1.8) ei (xY)(Da a)(x,Y,t)u(R)(Y), 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 ).
A substitution y = (x) y , dy = (x) ndy brings 'f 11 in (1 .0) to
(1.11) (x) ,(0)+nffJ(Y)1(0)(1+(x)2lyl2)q/2s c(x)X(O)+n,
c=rr1J(Y)%(0)q
Therefore
(1.12) (0)+n
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
(1.14) x(2m)  2m < nIal
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)+n0
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,
1.1. Definition 55
estimates u(a) by IxIq, as the only way, u enters (1.12).
Let us prepare a few other simple lemmata for thm.1.1.
Lemma 1.3. For a E ST we get aaSY=DaDYDYaE ST and ST, for
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 calcula
ted 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 differentia
bility 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)
fDO
X= iaxj
where we set cp [a,t] , the characteristic function of [a,t]. Theinner 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 multiindex y we have
fdjdy u(y)(1.19)
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(xY)a(x,Y,U)}=fda jfdyu(Y)eit(xY)a(x,Y,)
56 1. Calculus of pseudodifferential operators
3) The right hand side of (2) is zero, because boundary inte
grals 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
(1.22) 6+s=a, a+t=(3.
Using Dxel (xY) = in (1.22) we may integrate by
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
We also applied lemma 1.5 and another partial integration Note
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
(1.24) fdAccordingly 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 cO(x)n+k(1)lllulik
,
)x(2m+IPI)2m
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) IIAuII10
s 0OIIulik , u E S
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
gral interchanges. Get L1(&2n), vE S, by lemma 1.2.
Hence (v,Au)
foJoyfpix ... = f oyfRi f41x ... _ (u, A" v) ,
for A = a(M1,Mr,D) , with a 3fold 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:
(2.3)*
A = a(Mr,Ml,D)
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) (Bu,c)) = (u,A (p) u E S' , (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 SS or, S'S'. Thm.2.1 gives AC B.
Generally we will not distinguish in notation between the ope
rators 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 complexvalued functions under
pointwise addition and multiplication, with (real) involution aa;
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) (Da'DR'DY'a) (Da"DR11DYlob) , a'+a"=a , l'+0"=(3 , Y'+Y"Y .x y i x yIn (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) , o(j)and estimate the products of (2.5) by
t(j)
(2.6)(x) IaI+ IRIa(la' I+IR' I )a( ICE" I+IR")( (x,y)) I? It( IY' I)t( IY"I )
Then define al(j)=Min {a(1)+a(jl)} , t'(j)=Min {t(1)+t(jl)},lsj le. j
We get the estimates (1.1) for c with x'=ja', X'=jt' instead of
58 1. Calculus of pseudodifferential operators
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) ba(x)ca(y),aasNwhere ba , ca E COO(&n) satisfy the estimates
(2.8) A a(x) = 0((x) P) , Dc(x) = D((x)p) x e &n ,
with a constant p , independent of x , a , R . Then we get
(2.9) A = a(Ml,Mr,D) = ba(M)Daca(M)
a differential operator.
Proof. The Fourier transform u^ = Fu and its inverse u" = F lu
(2.10) u" () = J91x u°(x) = JF1 u E S
satisfy (c.f. 0,(1.22)).
(2.11) FDjF1 = 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
(2.12) (Au)(x)= =fdye
for u E S , with the Fourier transform (2.10).
Proof. The first formula is clear, using (2.10). Also,
(2.13) (Bu)(x) _f4eix
,f
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
(2.14) c EST , and c(Mi,Mr,D) = a(Mi,D)b(Mr,D)
Proof. From Theorem 2.2 we conclude that c EST. Using (2.12),
A(Bu)(x)=ff
u(y)
= (c(Mi,Mr,D)u)(x) , u E S , q.e.d.
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)=
for u, v E S . We have seen that f9Ixfpig fyiy=f9I JJTx9Iy .Note that a
temperate distribution k E S'(&2n) is defined by
(2.16) (k,w) = J JoxPzy w E S(&2n)
By the techniques of sec.1 we get
(2.17) (k,w) = J<) 2mfP1xPlyei (xY) (1Ox)m(aw)
For large in this is in the span of
(2.18) (x,Y) , IPI+IYIs2m,
where (1.1),(1.2) give existence of the
inner
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
(2.19) (v,Au) _ (k,v(gi) , u,v E S .
Formally, with distribution integrals, we may write
(2.20) (Au)(x) = fk(x,y)u(y)piy , (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,xY) = a'(x,Y,a)Ia=xY
in the following precise sense: The inverse Fourier transform a
of with respect to its third (its 2n+1st...3nth) argu
ment is a distribution in S'(13n) which may be written as a con
tinuous family pa =a3'(x,y,(xy)+a) of distributions inS,(R2n),
(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+xy) , AE D(13n)
with a distribution integral in (2.22) defining p0EC(&n,S,(R2n))
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 .
60 1. Calculus of pseudodifferential operators
Proof. Clearly a family p0E C(&n,SI(it2n)) is defined by setting
(2.24) (P0,V) = )'W(x,y) , PE S(&2n)
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) EC0(ln)
We get fdo(p6,ypa)=
,(P)
confirming (2.23), where we have used a linear substitution of in
tegration variable, and some Fubinitype interchanges. Q.E.D.
The distribution kernel's singular support will be of inter
est in the following. In sec.3 we introduce restricted symbol clas
ses with the property that the kernel k=ka for such a symbol a has
singular support contained in the 'diagonal' {(x,y)E1e2n:x=y}.
For general a e ST this is not true: Consider
E ST. Then sing supp ka coincides with the set {(x,y):IxyI=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) ,
with unique symbols bE ST and b1E ST depending on x, only.
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 represen
tation A=a(M1,D) for all y,do's (also written as A=a(x,D)), follo
wing 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) resul
ting 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) DaaDyDya = o(()m,+S(IaH+IRI)p' I?I(x)m2P2IaI(y)m3P3IRI)
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
(3.2) for a given m=(mt m2 ,m3) , p=(pi p2 ,p3) , 6 , where only thefirst condition (3.3) will be required, and no longer S<p3. Thus,
Ssm'p,5 in general will not be a subset of SS.
Let 4h and 1phm,p,S denote the classes of a(x,U)E SS or SSm,p5S'
(i.e., a is independent of y). Clearly,for yah and yadm,p,S, the con
stants m3, p3 are redundant: Any m3 a0 and any P3 E= 1 may be chosen
since aaa=0. We then will set m3 =0, p3 =1 , or omit m3 , p3 , writingm=(ml,m2) and p=(pl,p2) as 2component 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 ine
qualities 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).
62 1. Calculus of pseudodifferential operators
Accordingly we shall refer to symbols in the classes
SP,6 , SS , q* , SSm'P'b , Vh,,p,b as Hoermander type symbols
From (3.2) it follows that
(3.5) Im, I+Im: I)
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,
(3.6) SS CST , SSm,p,o CST , as 6 < 1 .
Lemma 3.1. Let a E SSm'P'b , b E SSm,P1'S', then we get a + b ESSm 'p '6 , 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 sym
bols 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,s with
(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 with m , p , b of (3.8) such that
(3.9) 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
(3.10) C* = a((M1+Mr)/2,D)
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 diago
nal {(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) out
side 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
IxYIF ((Y)(1laXaYkISCapYE'
as yE K , IxYIZE))
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 from00 2n[C,]), let wE CO(I ), w=0 near e. Then supp w has positive distan
ce from O, and IxyI mE C00(supp w). By (1.6), with a partial inte
gration, 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(xY)A a(x,Y,U)/IxYI2m
give the derivatives of k(x,y) (k is independent of m; the inte
64 1. Calculus of pseudodifferential operators
that sing supp k C 0 . Very similar arguments will show that
y alaIayklscaRYE, as x E K , jxylze , etc. Q.E.D.
4. The composition formulas of Beals.
Proposition 4.1. For aE C0*(&2n) , bE C0OO(&3n) we have the formulas
(4.1) a(Mr,D) = p(M1,D) , b(M1,Mr,D) = q(M1,D)
with p,q E S(&2n) given by
(4.2) iyrla(xY,tj), 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,xY)u(Y)
with '"y" indicating the inverse Fourier transform with respect to
the 2nd (set of)variable(s) of by integral interchange.
Write ay(y,xy)=aa(x(xy),xy)=c(x,xy), i.e., c(x,z)=a'(xz,z)=
fg eil'za(xz,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)ell;z
E S(>Q2n), and get c(x,z)=py(x,z), c(x,xy)=pr(x,xy). Accordingly,
(4.4) a(Mr,D)u(x) = J41yp (x,xY)u(Y) = JR (xY) u(Y)
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
Then C=AB=c(M1,D), A*=a*(M1,D) with c(x,i;), given by
)eiyq(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(yz) u(z)
1.4. Beals composition formulas 65
We may write a 4nfold integral, the integrand being L1(R2n). Use
(4.8)
and continue (4.7) as follows.
f(4.9) = e1T1(xz) u(z) aDenoting 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)
achieve the transition M1 **Mr
, and the calculation of the symbol
of operator product and adjoint. We will show that the formulas re
main 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 Weylrepresentation 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 4nfold integral:
Proposition 4.3. Let a,b E C"(&n) , and c E C0OO(I3n). We have
(4.10) C = c(M1,Mr,D) = P(Mw,D) 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 (xY+z/4, rl+t/4) b (xYz/4, nt/4) ai ('1ztY) .
Proof. For (4.10)i depart from (4.3), writing
c' (w+z/2,wz/2,z) , with w=(x+y)/2, z=xy. Defining p(w,l;)=d'(w,l;)
=JFizd(w,z)e 12t, get Cu(x)=fslyp (x ,xY)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
(4.12)Qu(x) = (ABu)(x) = fr(x,z)u(z)ylz ,
r(x,z) =
Again write r(x,y)=s(w,v)=r(w+v/2,wv/2), w=(x+z)/2,v=xz,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 Had
amard's finite part (cf. also [Ci],II).This will be a distribution
integral using partial integration, based on identities like (1.6)
(4.13) aiY'"
=(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
)Nein
(4.14)f
= aN(z,t)=(lOZ)Na(z,t)
Using the other identity (4.13) we get, similarly,
(4.15) f 9jYPTqe 1(Y) 2M(l0I)M(('il)
Here the expression (10I)M((,1)2NaN(xy,trl)) is in the span of
(4.16) 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) are0((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.},
also satisfying (1.1), ( 1. 2 ) . Thus (xy) X=0((x) k+(y) X); 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
(4.17) fdsdaeisw(s,a) ,
with an integral over &2m, and wE If W(= Cm(12s) satisfies
(4.18) asaaw(s,Q) = 0((s)X(IaI)(a)%(jPj)) , 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 abo
ve 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)
(4.19)=fdsdaeisa(a)2M(1Os)M((s)2N(1_A(3)Nw(s,a){
=f dsdaeisa(s)2M(1A0)M((a)2N(l_A5)Nw(s,a))
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
(4.21) limjIN,M(wj) = IN,M(w) , 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.,
(4.22) ?(2N)  2N < n , 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) = jIaIX(a)(s/j) . On the other hand, the inte
gral 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> 1iyyl
The proof is a calculation: Just use (4.19), with M,N suitable
68 1. Calculus of pseudodifferential operators
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:
N(4.23) 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)(xy,
r0 ,J91y911le11aXayaOb(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 ( (xy) ( rl)%(2N)) ,
in the first case, and inthe 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 desi
red 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.
4.2 , and a,b,c of prop.4.3 are chosen in ST1, ST ; STI , STI
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)
1.5. Leibniz formulas with integral remainder 69
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
p, c, a q again satisfy the strongerthat the composed symbols
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 qTrW =qT=OPVt, with transition formulas between representations of
AE W given by thm.4.7. Moreover, LI' is an algebra under operator
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 condit
ions 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 differen
tial 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 dif
ferentiation 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
70 1. Calculus of pseudodifferential operators
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 inte
gral remainder. There will be a finite sum, as in (5.1), but also
a remainder term, as in Taylor's formula. Unless the symbols be
long to SSmp,b, 6<pi, however, the formulas may be of little va
lue, since the remainder does not decay. The formulas will be cal
led Leibniz formulas with integral remainder.
We recall Taylor's formula (with integral remainder):
(5.2) I OsNa(O)(x,U)(T1)O/O! + PN ,
1
pN (N+1) I (r1)0/0!P0,N ' PO,N =IOI=N+1 0
Substituting (5.2) into (4.2), (4.6), (4.11) one obtains the fol
lowing 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) , cl(Ml,D) = c(Ml,Mr,D) = cw(MH,D) ,
(5.3) 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):
(5.4)arl(x,) = Lj=0(iaxa)Ja(x,)/7! + (N+1)PN
1
PN(x,) = di(1i)NIN!p.f. 91YF1Tleiyr!(iaX a N+1a(xY, Tj);0
(5.5)cl(x,) =J=II/j!{(iay a )N+1c(x,Y, )}x=y + (N+1)PN ,
1
PN(x,U) = dt(1s)NINIp.f.f plyAeiyr!(iay a N+1c(x,xY, T1);0
cw(x, )=`7=01/j! (ia ay)Jc(x+Y/2,xY/2,)j + (N+1)pN(5.6)
J
1
PN = di (1i)/N! p.f.fP1YAe_iy1(idya N+1c(x+y/2,xY/2,il);
0
p1(x,5)= 01/j! (ia ay)1(a(x,)b(Y,i1)jx=Y,=1 +(N+1)pN(5.7)
1.5. Leibniz formulas with integral remainder 71
(5.7) 1 _pN (1t)N/N! di p. f. ly"(iay
0
pw(x, )= =0, ayaX a,l)J(a(x, b(Y,Y1) Ix=Y, +pN,(5.8)
1 N 21(zrlyt) i N+1M t) dt e (2(asaKataa>> a(s.a)b(t,K),pN0
M=N+1, s=xy, a finite part integral.
(5.9) a (N+I)pN ,
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
(5.10) iy = ay(1 )" = (2x)n/2ayO(Y)
in the distribution sense. A partial integration must be used with
(5.11) (ax ay)1/1! = I aXay/8e 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 ei"1(1)e/O!
For a,b E C0*(&2n) this is easily transformed into
(5.13)
,=(i) 10 91Y9lT1 ay(e
=(i)I0I/O!JP1YPlrl
with N sufficiently large, but with the npartial integrations
reversed to the status M=O. This still gives a well convergent
integral, as long as a,b are Co . Recall that bN = (1Ax)N b
Here the iiintegral may be evaluated. It gives the Fourier
transform ((Y1)2N)^=((,q)2N)' =EN, the unique fundamental solutionin S' of (1A)N. An expression for EN in terms of modified Hankel
functions is known (cf. (0.24) for N=1; the method works for gene
72 1. Calculus of pseudodifferential operators
ral s>0.) EN decays exponentially, as lxIgoo. Without
Hankel functions one confirms easily that EN equals a function
in S, for 1xIa1. (c.f. [C1],II). Hence (5.13) is continued as
(5.14)=(i)'°1/91
(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
fp.f.f91y51rl aiY110
(5.15) _ 91Y91r1 aiyrl (1z)Ndia(e)(x,
0
1(li)NdiP1YP1r1
0
as a matter of Fubini type integral exchanges. Write (rl)°e iyr) _
(i)IOIaye1Y11 , and carry out a partial integration, for
n r
0
(5.16) iai [l IT f _,.... ,a%
0
completing the discussion of (5.7), for a,b E COOO
(0)(xY,) ,
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 cor
responding asymptotic expansions of arl' cl' cw' p1' pw' a*The key is an estimate of the remainders pN of (5.4)(5.9).
Theorem 6.1. Let a E E  =0 S, b E y>hm, 1, 61 , c E SSm",p","Pwith the 3 sets of paramaters satisfying (3.3). Then the remain
ders pN of (5.4) through (5.9) satisfy the following estimates:
1.6. Calculus of pseudodifferential operators 73
For (5.4):
oPPPl(
)m.(N+1)(p.b) (x)mz(N+l)pz )
For (5.5): =of( )m."(N+1)(p'"b") (x)m2"+m3"(N+1)p3°`.
For (5.6): o(()m. "(N+1)(P' "b") X)M2 "+m3 "(N+1)p)
p=Min{p2 ",p3 J"} .
For (5.7): 0f(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)(pbe'),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' P ),b ' for (5.5),(6.1)
pN as for (5.5), with p3" replaced by p" , for (5.6) ,
PN E 4*m+m'(N+1)(pb'e'),p^,b^ ,for (5.7) and the same with 6'
replaced by b^,for (5.8) .
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 sup
ply estimates, uniformly in s, of the integrand di only. For exam
ple, to estimate PN of (5.4), it suffices to consider
p.f.fdydrl ah 1a(6)(xY,xT1)
(6.2)
=fdydTl e
again with aN (1AX)N a, for sufficiently large M,N. Use (3.2) for
(6.3) =olf dy(xY) mz pz 1111 (Y) 2M fdll( iTl) m. p. 19 1+6 101(,n) 2N)
.
Here we also employed the estimate
(6.4) ((Tl)R)(a) = o((Tl)RIa
which is easily derived. Then apply the well known inequality
74 1. Calculus of pseudodifferential operators
(6.5) (x) s(xY)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) (xY, TI
Then the above estimate may be repeated for
(6.7) PN(a) = 0(()m,(N+1)(P'b)P IaI+OIl1(x)m2(N+1)p:PZ IPI),
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'
cl, cw, pl, pw, a* of thm.5.1 are in yah , provided that, for
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 0m,p,S Cl,cwEyhm ,P,s
with m =(m, ,m2+m3) , P =(PI ,P2 ) , p2 =Min(p2 ,P3) , P1 , Pw EApm+m',p^,S^ with p^ , 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
while b1 may be obtained by writingc(Mr,Ml,D)=(c(Ml,Mr,D)*
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(xy), 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. Similar
ly 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 mul
tiplication, 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.
1.6. Calculus of pseudodifferential operators 75
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 repres
entation A=a(Ml,D), and then will use the more conventional nota
tion 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 , and a sequence of symbols aj, j=0,
1,2,..., ajE yt j , m+ =(mi,m2), mk.oo, as joo, k=1,2, we willm ,p,S
say that a has the asymptotic expansion =0 aj , written as
(6.8) a =0
aj (mod Vh_.) , (or simply "(mod S)"
if both orders of the symbol pN=ae=0 aj tend to oo, as N (that
is,more precisely, if rNE h N , with µN=(µN,µ2), whereµ ,P,5
j , 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) a (mod 0_,O)
as aE 0M OP 16, 6<p3, and similarly for the other 5 formulas.
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 mi00,
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 , and where mJ=(mi,m2) , with
(6.10) mk>mk>mk> ... , as jaoo k=1,2
There exists a symbol a Eh m = m0 such thatm,p,S
(6.11) sN = a  4=0 aj E yh N N=0,1,2,... .
M ,p,b
Proof. Indicating first the idea of the proof, let
(6.12) s 1/2 , o )=l , as jxI+1fl z 1 .
76 1. Calculus of pseudodifferential operators
Assume wE C°°(1e2n) , and let x=1wE CD . Assume Os w,x sl in &2n
For an increasing seqence {tj} , tjE I j , to be
determined later, define
(6.13) w(x/tj,/tj)
Conclude that w.=0, as /2. Since t.>oo the sum in (6.13)
is finite near Thus is well defined. We get
(6.14) a  flaj = Lj=oxjaj + > N wjajJ=O
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}
with 9S =md m 9'Z =mz lZP . By choosing tj sufficientlylarge the supremum in (6.15) can be made less than 23/cj . Usingthis in (6.15) , and then substituting (6.15) into (6.14) we get
(6.16) a  Lj=pai = 0(()mi (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 =a`zX j=0a. , and,I J
(6.17) aXo sN = 0
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.) insteadm ,p,S
ofw a. with c., mN1, mz1 replaced by cj,a,,ml1+Sjajpiipi
13 3m2 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 2j, 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 I1+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
1.6. Calculus of pseudodifferential operators 77
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 bjkE 0 m_jIl,p,6 again
form a sequence of symbols with the assumptions of lemma 6.4. Then
(6.19)7=0 0 bjk = =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 yahmkµ,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 =71
b 1 . Note thatj+lsN j
bN  b'N is a finite sum of symbols of order mNµ hence itself
is of that order. Moreover, b'  b'N is of order mNµ , while bbN
=b  4 4=0bj + =0(bj  Vk=0bjk) again is of that order. Hence
bb' _ (bbN) + (bN_b'N) + (b'Nb') is of order mNµ , for all
N = 0,1,2,... . It follows that bb' 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 differentiatedmj,p,s
term by term . That is, (6.8) implies
(6.20) a(P)=Oaj(a) (mod S
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)enl 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(C)={uE COO(C) : aX u(x, , . E
S(kn_l),j=0,1,..,uniformlyin x}. Show that a tpdo A=a(x,D)E OpVt0
with periodic (2n) in x, may be regarded as a map S(C)
COO(C), given by Au(x)= _ooel3xl
x° )uj
2nwith u j(x° )= 2a
faljx'u(x,,x° )dx, and the Fourier transform
0
""" with respect to x° . 2) The operator A=a(x,D) of pbm.1 may be
regarded as a 4d o on &n1 with operator valued symbol, acting on
78 1. Calculus of pseudodifferential operators
functions on Rn1 with values in C'°(S'): For1n1 define
A(x° ,t") :COO(S' ) ,C00(S' ) by A(x" t" )u(9)=eljea(O,x" , j , where
2nuj= e1 u(9)dO. Then write v(x")=u(.,x") , and Av(x")
f oweix"t" )v ( "),for VE S(Rn1,COO(S' )) S(C). Show that,
for a symbol aE 9pto (and even for aE STS ) this operator A mapsS(2n1,C"(S'))
+ S(IIn1,C,(S')), and continuously so when we equip
S(C)=S(1n1,C°°(S')) with the Frechet topology of all norms
II(x")ku(a)Il 0 , k=0,1,..., aE IIn. (Recall that must be 2nL
periodic in xi . ) 3) Investigate linear operators C00(S' as
B=A(x" 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 and2nperiodic (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 Hil
2nbert 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 DX 2)1/2 and B2 =Dx Bi inthe 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 , UCr = U m , '_O, = n "m ,
setting p1 = p2 = 1 , 6 = 0 . 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
1.7. Strictly classical symbols 79
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 seminorms, respectively,
(7.3) sup {ja(")(x, )fi(x)m2 x, E&n} , a,l EIIn
and
(7.4) sup a(P(a))(x,
)I(x)mZ+IaI( a,p EIIn
Similar sets of seminorms 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 seminorm 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) , a E A , b E B}
is bounded inm+m and the sets A* ={a*} of symbols of adjoints
and P = {p E i1cm : p(MrD) = a(M1,D) a E A } are bounded in wcm
Similarly , if ,c is replaced by it , in all the above.
Lemma 7.2. Let a E pcm , b e iycm, . Then we have , with e
a(M1,D)b(M1,D)  (ab)(M1,D) E OpyrCm+m'e'
(7.6) a*(M1,D)  a(M1,D) E Opcme
a(M1,D)  a(Mr,D) E Opicme
Also, if A = a(M1,D) , B = b(M1,D) then
(7.7) [ A, B] = AB  BA 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)
form bounded sets of Vcm+m'a, V. me' 'c me, Vcm+m respectively.
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 thatrespectively, for the 4 relations in (7.6), and (7.7),
(i ) {a0} bounded in tpCme, , and {b°} bounded in 1 cme. ,
80 1. Calculus of pseudodifferential operators
(ii) {a(8)} bounded in 4cme
(iii) {a(9)} bounded incme
(iv) {a(e)} bounded inVcme'
, and {a(0)} bounded in Vcme2 ,
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 11correspondence 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 boundedness of sets follow by just going again over the proofs leading to
thm.6.1, keeping in mind the boundedness of sets in Frechet spa
ces. 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, regar
ding (5.5), as well as the corresponding constants for the deriva
tives 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) m2E '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
OpWhp6
: They contain (generalized) inverses of some of their ope
rators  called (md)(hypo)elliptic. For historical reason such
an inverse is called parametrix or Green inverse. Speaking algebra
ically, we deal with inverses modulo some 2sided ideal, either
the class', of integral operators with kernel in S(12n) (Kparame
metrix), or the operators of & with finite rank (Green inverse).
Integral operators with kernel in S(&2n) are compact opera
tors of L2(&n), as well as in every Sobolev space (cf.III,5). Thus
a Kparametrix (Green inverse) of A will be an inverse mod K(H)
(a Fredholm inverse) as well, under proper assumptions, giving nor
mal solvability of the equation Au=f (cf. [C,],App.Al).
For Oplhm pS
'elliptic' usually will be 'mdelliptic', amoun
ting to ellipticity 'at and at jxI=co'i.e., "d" and "m"
ellipticity. This is parallel to discussions in algebras of singu
lar integral operators ([Ci],[Cz]): For operators of order 0 'md
elliptic' simply means that the symbol is #0 at 1x1+1V_o .
An mdelliptic operator of order m will have a Kparametrix
of order m , and existence of a Kparametrix of order m is in
fact necessary and sufficient for mdellipticity of AE Op'hm,p,S.
However, it is possible for a nonmdelliptic operator to have a
Kparametrix of order >m. One such class of operators  the for
mally andhypoelliptic operators  is studied in sec.2.
The ellipticity concept can be localized, in the xvariable
as well as in the 5variable, and 'local parametrices' may be con
structed under various assumptions. In particular, (formal hypo)
ellipticity (for all ) over an open set 92 C Rnimplies hypoellip
ticity 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 discus
sion 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) mdellipticity of a matrix
of pdo's, together with certain partial inverses.
81
82 2. Elliptic operators and parametrices
1. Elliptic and mdelliptic pdo's.
Historically a parametrix is an integral kernel with a sin
gularity 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 Kparametrix of
A if we have ABlE Opyho. , BA1E Op%ph,O. Clearly then A and B are
Kparametrices of each other. We speak of a left (right) Kparame
trix B of A if only the second (first) condition holds.
A symbol aE Phm,p,6C 1ph will be called mdelliptic (of order
m) if (i) for all sufficiently large and (ii)
the function b=1/a, well defined for large equals some
symbol in Vhm,p,S' for large jxI+1g1. Then the yxio A=a(x,D) also
will be called mdelliptic. We will see that mdellipticity is a
property of the operator, insofar as, for an mdelliptic A=a(M1,D)
=b(Mr,D)=c(Mw,D) all 3 symbols a,b,c will be mdelliptic.
Proposition 1.1. A symbol a E 1phm,p,8 C Vh is mdelliptic
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 mdelliptic we get b=1/a for large and bE
m,p,S' hence (x)m` ), using I,(3.2). This implies(1.1). Vice versa, if a symbol satisfies (1.1) then
a(x, )
C w1, as w0, 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) {[IIi (a(,CE3j)/a)]/a : 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 mdelliptic ; (ii) c is mdelliptic ; (iii)
is mdelliptic (of order m ). (iv) A is mdelliptic.
2.1. Elliptic and mdelliptic operators 83
This follows from calculus of do's: The symbol a may be ex
pressed 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 mdelliptic if and only if A* is.
(b) The product AB of two mdelliptic operators AE Opwhm,P,a,
B E OpVhm,,P1b , again is mdelliptic (of order m+m')
Again this follows from calculus of ,do's.
Recall that a differential operator A =IaJSN
ac`Da is called
elliptic at x0 if its principal symbol
(1.3) a aNaa
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 mdelliptic (of order (N,m2)) if and only if
(1.4) a p(x)m" , for all x , and all 1
and also, with suitable constants p, c>O,
(1.5) Ia(x,l;)I a for all x,g with lxi a c .
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),
using that the terms a l < N , all are N1( x) m,) .
By (1.5) mdelliptic differential operators are elliptic in
&n. (x)mA even is 'uniformly elliptic' in ien. (1.4), in effect,
is another ellipticity, with variables x, reversed.
Theorem 1.6. An operator AE OptNhm.P.b , admits a Kparametrix in
Optph of order m if and only if A is mdelliptic.
Proof. If A and B are Kparametrices of each other then the rela
tion AB=1 and the uniqueness of the left representation implies 1=
modulo terms of order Ee, e=(1,1), E>O. Thus it fol
lows that jabi a1/2, as ac, for large c. Accordingly, jal
a2 b ap( )m' (x)m' , I x I + I I ac, p>O, using that b is of order mhence, m'). Thus A is mdelliptic (order m). Vice
versa, let A=a(M1,D) be mdelliptic. Using a recursion and I,lemma
84 2. Elliptic operators and parametrices
6.4 we will construct a Kparametrix for A.
First, let b1 be a symbol such that ab1=1 for large
by definition of mdellipticity. If B1=b1(M1,D) then AB11=e1(x,D)
where e1 is of
order 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 lpdocalculus for AB2. Together,
we get A(B1+B2)=1+E1E1+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 Bf=1Bj is of order (N+1)a, by definition.
Also A(1Bjj is again of order No. Thus AB1 is of or
der No, for every N = 1,2,.... It follows that AB1 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 Opph0), q.e.d.
Finally, in this section, we show that Op4ho,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,xY) =
For aE ihoo S(le2n) (1.7) is a well defined integral. The Fourier
transform and b(x,y) b(x,xy) are isomorphisms S S , q.e.d.
2. Formally hypoelliptic yxlo's.
In sec.1 we found that a ipdo A of order m has a Kparametrix
of order m if and only if it is mdelliptic. It is possible, how
ever, for A E C Wh to have a Kparametrix in OpWhm 1P'6 P'S
with mj amj without being mdelliptic.
Let aE iphm,P,S
C Wh. We shall say that a (or A=a(x,D)) is
2.2 Formally hypoelliptic operators 85
formally andhypoelliptic if for we have (i)
and (ii) a=O((l;) (x)m2 ), with some m'=(m1 ml ), and (iii) forall a,(3 the functions a((cc)/a satisfy the estimates
(2.1) a(a)/a = of( )P'
In this case we shall call m'=(m1 m' ) an inverse order of a
Lemma 2.1. Let a E m,PSC ph . The symbol a is formally andhypo
elliptic if and only if (i) a and a(c)/a are (ii)
the functions of (i), for all a,(3, extend to symbols in 0m',p,S
and Vh , respectively, where, (3ma P,S
(2.2) ma',p =
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 in the00
span of the products
(2.3)=1(a( )/a) , 3=163=a+k=103=(3+x , r=1,2,...
Again formal mdhypoellipticity 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 mdhypoelliptic if and only if b is
Proof. From formula I,(5.4) we conclude that
(2.4) j(i)191/9!
a is formally mdhypoelliptic then the asymptotic sum
(2.5) c = '01/e! a(e)/ae
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 mdelliptic 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)/aE 1ih_(N(P'S),Npz),P,S
86 2. Elliptic operators and parametrices
This may be multiplied by the symbol a , using lemma 2.1, for
(2.7) ca a(9) EmNp+NSe',p,S
N=0,1,2,....
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) 8(i)I0I/0l a(6+a) (mod
Then, repeating the above conclusion we see that
(2.9) b/a , a/b E 0 0,P.o b(P)/a E'iph
Again, the functions (2.9) must be modified on some KCC I2ninde
pendent 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 «,(3m ,p,S
so that b is formally mdhypoelliptic. Vice versa, if b is formal
ly mdhypoelliptic then a may be obtained as asymptotic expansion
in terms of the derivatives of b. The above conclusion may be re
peated to show that also a is formally mdhypoelliptic, q.e.d.
Proposition 2.4. (a) AE OplphMIP16C Op1ph is formally mdhypoellip
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 mdhypoelliptic then so is AB.
(c) The formally mdhypoelliptic symbols of (b) form a group un
der pointwise multiplication, for each p,S. (d) An mdelliptic sym
bol of 1phmIP.SC 0 is formally mdhypoelliptic, inverse order m.
Theorem 2.5. A formally mdhypoelliptic A=a(M1,D)E Oplphm,p,S ad
mits a Kparametrix B=b(M1,D)E Opom"p'S, for suitable m'am.
Proof. As in the proof of thm.1.6 we focus on construction of a
right Kparametrix. First set b1=a, and B1=b1(x,D). By
assumption b1E 1phm,,PS, and the product formula yields.
e
(2.10) AB11=E1=e1(M1,D) , e1= Ie ,0(_i) 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
VhNp+NSe',p,S' I0I=N. Accordingly el E y't2_p+Se',p,S has orderless 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 mdellipticity 87
(2.11) AB2+E1=e2(M1,D) , e2 = 10 f0a(0)(el/a)(0) (mod h,0)
A term a(e)(el/a)(e) for 101 = N is in the span of
(2.12) { (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 mdellipticity, and local mdhypoellipticity.
Recall that (formal) md(hypo) ellipticity of a symbol aE
y,h , is a condition for large If a is modified in a set
KCC R2n these conditions are not disturbed.
A parallel, though slightly different concept of mdellipti
city introduced in [Cy l],for the operators of some C subalge
bras of L(L2(&n)). Similar algebras are studied in [C2] for gene
ral 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
is only defined at the boundary ffi=d(1QnxPn)=Pnxpn\R2n of PnxPn.
Then AE A is called mdelliptic if its symbol is #0 on M. mdellip
ticity is necessary and sufficient for AE A to be invertible modu
lo operators of finite rank. The algebra A is generated by the spe
cial pdo's a(M) and b(D), where a,bE COO(R2n) are bounded and have
a(a)=o(1), bO)=o(1), as lxloo, 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 mdellipticity. Within A we speak of md
ellipticity 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 withinCB(R2n).
The maximal ideal spacePP,6
of this function algebra is a compac
tification of R2n  again the smallest to which all functions of
Vh0,P,6 extend continuously. We set ffiP,b=dPP s=PP,b\R2n again.
Proposition 3.1. A symbol aEm,P,6
is mdelliptic if and only if
(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,6The proof follows from (1.1) and prop.1.4, and mdellipticity
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,pb,
assuming pj>O, 0<6<p, (cf. V,10, for pj=1, 6=0).
Following the above lead, a symbol aEm,p,b will be called
mdelliptic 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 mdelliptic in MC ffip'b
if there exists a neighbourhood N of M in Bp,b with (1.1) in
Kfl it2n for every KCC N. A symbol aE 0m,p,b is called formally md
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 defini
tion of local mdellipticity, 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 StoneCech 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 p,b generate a subalge
bra B of Ab,
for p sl, ba0. The maximal ideal space of B is a
compactification of &2n again it equals BnxBn, with the directio
nal compactification Bn of Rn (cf. sec.5 or IV,1, below). Bn con
sists 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 homeo
morphism 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=L1N' where N' is a neighbour
hood 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,6needs not to extend to C(I0), but for symbols of differential ope
rators and their parametrices we have that property in W below:
I° = a (BnxBn) = IQnxaBn U aBnxaBn U aBnxln
the first term at right denoting xE ien, E aBn}=w. The set
WC ffi° will be called the wave front space, in view of an applica
2.3. Local mdellipticity 89
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 con
ventions for differential operators. For more general MC W another
notation is in common use: Such sets are called noncharacteristic
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)
ellipticity in some aol , 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 cor
responding lpdo A=a(x,D). We mainly will consider the case MC W, no
ting 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 mdhypoelliptic with res
pect to q if for xI+I xj , some fl>0, we have (i) ay60 in supp cp
(ii) TE 1hv+m',p,6 '(iii) the restriction of a(R)/a to the set
supp T , extends to a symbol in VhR
,
m ,P,S
with ma'P of (2.2). If m'=m we will call a mdelliptic with res
pet to T. Clearly a (formally) md(hypo) elliptic symbol is just
a (formally) md(hypo)elliptic symbol with respect to T=1.
Theorem 3.3. Let Tr= lphv P16 , and let aE lPhm,p6be (formally) md
(hypo)elliptic with respect to T. Then we have ejE hvm',p,S
j=1,2, where m'=m in the elliptic case such that A=a(x,D), Ej=
ej(x,D), (D1=cp(x,D)=y(M1,D), (D.2
=cp(Mr,D) satisfy the relations
(3.2) AE1 = (D1 + K1 , E2A = 02 + K2 , K1,K2 E Op'tihoo .
Proof. 'Elliptic' just means m'=m, hence needs no special conside
ration. We look at E1 only, taking adjoints for E2. With E2=B we
once more go through the construction of the right parametrix of
thm.1.6 (or 2.5). Set b0=a, getting b0E v+m',p,S' by (ii). Write
(3.3) AB0=(1+c0(x,D) , c0=J J11
90 2. Elliptic operators and parametrices
where the asymptotic sum c0E hm+v+m'p+6e',p,6' The next correc
tion will be b1=c0/a. Dividing (3.3) by a we get bl as an asymp
totic sum of terms in the span of
(3.4) (a(0)/a)(a)(9) 101 Z1
It follows that bl E 0v+m'p+Se'Iand that
(3.5) AB1=c0(x,D)+C1(x,D) , c1= j(i)101/0!a(0)(c0/a)(9)
This then leads into the recursion bN =cN1/a , BN = bN(x,D)
(3.6) ABN =cN1(x,D)+cN(x,D) , CN 1.f1(0101/0!a(0)bN(0)
N=1,2,... . We show that, successively, bN, cN are of proper order
with B. well defined, while the orders of C tend toI N
As instrument in this proof we use the lemma below.
Proposition 3.4. Each symbol bN ,N=1,2,... , is an asymptotic sum
of terms in the span of the expressions
(3.7) nj=!(x((33/a) : LjOal=L40PJ=101
where 101 a N , for bN .
Clearly prop.3.4 yields bNE 0v+m'N(pSe')' hence cN abN+l
E v+m+m'N(pSe') 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) , for 101=N. Each term (3.9)
is of the form (3.7) , for 191Z N+1 , q.e.d.
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_ Kwith KCC NC BnxEn , as in def.3.2. Then the symbol a is (formally)
2.4. Formally hypoelliptic expressions 91
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
function where xE C (&n ), 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 mdellipticity on a cylinder in the setting
of 1,6, pbm's 14. Require the operator family
C00(S') invertible for i xA I + I ° I large; also, in the norm (III II of
L2(S' ), that ')II= O((x^)mzPz MIpt IaI+olII), a,(3E IIn, as in I,(3.2)(3.3). Discuss construction of a Kparametrix
B with 1AB, 1BAE4V (!fit as in 11,0, but with L(L2 (S' ))valued kernels). 2) Show that the abovetf(with operatorvalued kernel) con
tains noncompact operators of L2(S'x&n1), but that KE Xwith ker
nel in K(L2(S')) is compact. 3) For mutual Kparametrices A,B, as
in pbm.1, a K(L2(&n1))valued symbol may be defined for each, AB=
I+K and BA=i+Kz, K.E. If and only if these symbols are inverti
ble the operators A and B are Fredholm operators of L2(S'x&n1)
4) The generators [Cz ],VIII,(2.2) of the C *algebra C there i.e.,
on our polycylinder C=S'xRn1, the multiplications by a(x)E COO(e)
(2nperiodic in xi), and the operators A , ax A , axzA, ..., A =
(1A)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 hypoelliptic differential expressions.
Let a be a domain of &n, and let
(4.1) L = a(x,D) =IaTsN
aa(x)D'
be a differential expression with Co coefficients : aa E C'(S2). We
shall call L (or its symbol )formally hypoelliptic if for
every compact set K C c there exist constants Mi, i, p, such that
(i) a(x,l;) 30 0 , as x E K , ICI Z 11 ,
(4.2) (ii) as x E K , ICI a 'n ,
(iii) a(() /a as x E K , ICI a 1 .
(ii) is superfluous if L is locally of order >0  i.e., L is
92 2. Elliptic operators and parametrices
of order Nx near x, with 0<NxsN, aa(x)#0 for some laI=Nx. Then
NXP) , 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 sp usually may be assumed negative :
For a constant coefficient L=a(D) of order >0 formal hypoel
lipticity is equivalent to hypoellipticity in the sense of the de
finition in ch.0, sec.5. This was shown by Hoermander [Hr1],p.101.
On the other hand there exist hypoelliptic expressions
with variable coefficients which are not formally hypoelliptic.
Here we use our parametrices of sec.3 for the following:
Theorem 4.1. A formally hypoelliptic differential expression is
hypoelliptic. If M, of (ii) can be chosen <0 (as true for L of
locally positive order) then LX is hypoelliptic for all XE M .
Proof. The second statement follows from the first, because for
M,<0 we get
liml l uniformly on KCC
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 apoint 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) a(x0+(xx0)x(Ixx0I) , ) , for x, E in
(4.4) a0E PcN.O C Vh(N,0),(P,1),O ,
and that a0 is formally dhypoelliptic in the latter space, with
inverse order M=(M1,0). By prop.3.6 a0 is formally mdhypoelliptic
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 Opiho,. Clearly
a=a0, as xE B(x° ,6), hence cpa = cpa0, for all assuming theleft hand side zero whenever a is undefined. Accordingly
(4.5) cpf = cpLu = Lcpu + [cp,L]u = L0((pu) + [cp,L0]'yru
with another cutoff p, supp V C B(x,26), =1 in supp [cp,L0]0
2.5. The wavefront set 93
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
the complement of sing supp h . Hence x° sing supp Eh , using
thm.5.4, below. Applying the operator E to (4.5) we get
(4.6) epu+ (1x)(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=1xE CO(&n)C S, thus w(D)cpu f 9Tyw' (xy) ((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 hypoellipticity in [C,],II,IV.
Hypoellipticity of L?. , especially of L.ti is required to make
the Carleman alternative work, as well as in general the construc
tion 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 "cosphere bundle", i.e. the bundle of unit
spheres in the cotangent space, with respect to a suitable smooth
Riemannian metric. It is clear then that the unit vector of a
pair with x E =n just indicates the direction tx , as t   ,
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 cutoff functions called
cp(x), and respectively. For x0E sz a cutoff cp is a C func
tion, cpa0, y=1 near x°. For a "direction" o0E a>sn given by °E &nwith 1°1=1 a cutoff 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
equals some homogeneous function of degree 0 for large One
where x=0 near 0,may think of a C(I)function
s=1 near oo , µ=1 near l;°We define: (x0, 0)E WF(u)c (i.e., is not in WF(u))
if there exist cp , ,p , such that, with the Fourier transform 11^61 ,
(5.2) E S
Note that Tu admits a natural extension to I , zero outside S2, a
distribution in S' , hence has a Fourier transform in S'.
Proposition 5.1. We have WF(u)c if and only if there existcutoffs cp,1p, near x and resp., such that the ipdo C=1p(D)g(x)E OpipcO'OO takes u into S ip(D)cp(x)u E S .
The proof is evident,since C=F11pFg, and because the inverse
Fourier transform F1 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) isthe projection of WF(u) to the first component Sz of W=S2xa18n
(5.3) s.s.u = {x: there exists F; such that (x,l;)E WF(u)} .
Indeed, if xE (s.s.u)c, then a cutoff 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 (P_
be a cutoff near x with cpp =cp for all j (i.e. T0=1 on supp
(p ), then ((pu)^1pj E S, by prop. 5.3. Hence (cp u) (Jipj)E S. But
it is possible to arrange for wY, 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 1w/(w+x)=x/(x+w)E Co, and
((p u)^E C. Hence cp uE S, uECoo
near x, or xE (s.s.u)c, q.e.d.
We have used the proposition below, also useful later on:
2.5. The wavefront set 95
Proposition 5.3. If we have ((pu)^tpE S , with a pair of cutoffs
q, 1p and a distribution u, and if qi cp=cp, for a cutoff qi , then wealso get (qf u)^ V E S .Proof. We know that ((pu)^,p E S is equivalent to p(D)q>(x)u E Sand similarly for qi instead of T . Write
(5.4) '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= X01i9lp(0) (0)(x) (mod S)>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). Q.E.D.
Theorem 5.4. Let u E ES' , and let A = a(x,D) be a ido with symbol
in some 1phm,p,s , with the usual inequalities 0sO<p1, 0<p2. Then
(5.6) WF(Au) C WF(u) , 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 differentia
tion 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 cutoff func
tions <p0, W1' ..., (V0' V1' ) such that q<p0=p0' T0T1=T1'
Tj(pj+1=,pj+1 ''' (and 'VOV0' VOy'1V1 " Prop.5.3. implies
(5.8) Cju E S , j = 0,1, ... , where Cj = 1pj(D)gpj(x) .
Proposition 5.5. We have
(5.9) C0C =CC0 =C0 (mod O(oo)), CjCj+1= Cj+1Cj= Cj+1 (mod O(0°)).
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
the last term, called V for a moment, is in O(oo) . But we have
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)) . Similarly for CiCj+1 . On the other hand,
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) , le (mod S)
which again implies W* E O(oo) since all terms vanish, q.e.d.
Using prop.5.5, we may write
CNAu = CN1CNAu (mod S) = CN1ACNu + CN1[CN,A]u (mod S)
(5.11) =CN2CNI[CN,A]u (mod S) = CN2[CN1'[CN,A]]u (mod S)
_ ... = [C,[C0,[C1..... [CN1,[CN,A]]] ...]]u (mod S) .
Here the N+2fold commutator has multiplication (differentiation)
order  (ml(N+2)(plO)) . 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 cutoffs cp" , 1p" with cp" qpk=f" "Vk=V" , forall 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 mdhypoelliptic with res
pect to a symbol q' . Let uE S', and let Au=f, and cp(x,D). Then
(5.13) WF(f) C WF(u) , and WF(4i) C WF(f) .
Morever, if q) (i.e., if A is formally mdhypoelliptic),then
(5.14) 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 mat
rixvalued 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 vxµmatrices to indicate that the entries
are in the corresponding symbol class. The operator A of (6.1)
will act on spaces of µvectorvalued functions or distributions.
All formulas of the pdocalculus of ch.I remain valid, except
those involving a commutator, such as I,(7.5).
For mdellipticity we focus on square (µxµ) matrices of Idols
We call A=a(x,D)E Opyahm,p,b (and its symbol a) mdelliptic if
(6.2) E011m,P,6
is mdelliptic. A symbol aE SS (and its operator A) will be called
mdelliptic if is mdelliptic.
Lemma 6.1. A squarematrixvalued symbol aE 1hm,p,b is mdelliptic
if and only if for with some T>0 , the matrix
a squarematrixvalued symbol bE
Vhmp,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 Vhm p,b ,resp., we have (for some 1>0)
(6.3) 1 = ((6j1))j11=1,.,v' for all rj
then we will call a right mdelliptic and b left mdelliptic .
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.
Theorem 6.2. Let a vxµsymbol aE m,p,bbe (right) (left) andel
liptic, and let bE yhm,p,b be a (right) (left) mdinverse symbol.
There exists a (right) (left) Kparametrix E=e(x,D), eE 4hm,p,6'
of the pdo A=a(x,D) such that
(6.4) E'U (m.p.+6,m2 p2 ),p,6 '
98 2. Elliptic operators and parametrices
Vice versa, if a (right) (left) parametrix of A of order m
exists then A (and a) will be (right) (left) mdelliptic.
We prove thm.6.2 like thm.1.6, constructing a right or left
Kparametrix. If both exist they coincide mod O(o3), as usual.
Many 'hypoelliptic' arguments work as in the scalar case.
For matrixvalued symbols there is another interesting case
to consider: Assume the matrix of constant rank for large
and that, moreover, aE yahmp,, allows a decomposition
(6.5) a = a0+a1 , a1E 0m',p16 , m'<m ,
where a0 allows a 'partially mdinverse symbol' b0E 11h,,,,b , in
the sense that is a gxvmatrix with
(6.6) for large
where is a vxvprojection 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 1p0 vanishes on im a0, i.e,
(1p0)a0=0. Introduce Then get g0q02=
b0a0b0a0b0a0=b0a0bOp0a0=b 0(1p0)a0=0, so that also is a
projection. Evidently rank g0(x,t)=rank p0(x,t), im p0 = im b0 .
Generally, if b=b0+b1, where b1E yhm',p,S, with m'<m , we
will call the symbols a and b partially mdinverse to each other.
Theorem 6.3. Assume that aE and bE Vhm,p,S are partially
mdinverse symbols of each other (where p>0 , pl>S). Let p=ab ,
q=ba , and let A=a(x,D) , B=b(x,D) , P=p(x,D) , Q=q(x,D) . Then
(6.7) AB  P = r(x,D) , BA  P = s(x,D) ,
where the symbols are vxvmatrixvalued and µ.xµ
matrixvalued, respectively, and are in`Vh_Ee,p,S ,
for some e>0
The proof is calculus of ,do's. We cannot expect better para
metrices making r or s of order  , with improved B, etc.
Matrixsymbols also occur in VI,IX,X, under different aspects
Chapter 3. L2SOBOLEV THEORY AND APPLICATIONS.
0. Introduction.
In this chapter we consider ipdo's as linear operators of
a class of weighted L2Sobolev spaces over in . We specialize
on L2spaces and neglect LPtheory, because 'ydo's in general are
not continuous operators on LPSobolev spaces, for p#2 . To be
more precise, general L2boundedness theorems for xdo's are true
for A=a(x,D)E OpiPh0,p,6 , assuming pa0, 0s6!.pi, 6341, but corres
ponding LPboundedness statements are false, except for p'=1.
There is an extensive theory in LPspaces of Sobolev and other
types (cf. Beals [B 41, CoifmanMeyer [ CM] , Marshall [ Mri ] , Muramatu [ Mml] , Nagase [Ng, ], Yamazaki [Ymi ]).
In sec's 1, 2 we prove the L2boundedness theorem, for 6=0,
and 0<6s1 , respectively. This result often is quoted as Calderon
Vaillancourt theorem. In sec.3 we look at weighted L2Sobolev
norms. Our class of spaces Rs=HS1,s: is left invariant by the Fou
rier transform, just as many of our ydoclasses. A y,do of order
m=(mi,m2) is a bounded map Hsa Hsm , for every s. For every m E
12 an order class O(m) is introduced  the operators SS extending
to operators in L(Hs,Hsm ) for all s . 0(m) is a Frechet space
under the norms of L(Hs Hsm) ; 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) el
liptic Ado's. Such an operator admits a Green inverse the equiva
lent 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 nega
tive order (m,<0, mz<0) are compact operators HsCHs, for all s.
1. L2boundedness of zeroorder 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).
More precisely: The restriction AJH maps and belongs to L(H).
99
100 3. L2Sobolev theory
This result is quoted as the CalderonVaillancourt 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 {Ixjl<r,j=l,...,n} choose OsgE CO(Q2), T>0 in Q312. Define
cpa(x)=cp(xa), aE 2n, and Wa a(ncpa)1/2. Clearly then
(3E2
(1.1) Va E CD({xE &n:lxaI<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 unita
ry we may show Ba FAaF1E L(H). But for aE ipt0, supp aC {Ixlsp}
FAFlu(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 in
terchanges. The integral operator at right of (1.3) is L2bounded
by Schur's lemma We get the operator bound
(1.4) sup f,Jdtla``(tq,t)I : ,T1 Elen}
In particular,
( ''1) 2Nav( l1, ) = J x a(x, ) (lAx)Neix( Tl)
=J1xaN(x, )eix( q) , with aN = (1Ax)Na
which implies
(1.6) D((rl)2N IIaNIi
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,)
3.1. L2boundedness of zeroorder operators 101
Taking the Fourier transformed operator A^ for A=a(M1,Mr,D) again,
with a(x,y,l;) vanishing for (x,y) outside KCC &2n , we get
(1.9) A^u=FAFlu(T1)= flixilJFTKely'{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 = (1Ax)(1Ay)Na
Note that (1.11) implies
(1.12) f IIaNNIILOO
0((K_,,) 2N)
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 L1norm has to be estimated by the corresponding Lc norm. Also,
2N(K) 2N= f (T1x)) 2N, and f ((t) (2st) )2Ndt
ItIfIsl + ItIJIsl = I1+I2 , where 12
c(s)2N, c=f(t)2Ndt
For the integral I1 the inequality implies 12stI22IslItI2lsI
so that the integrand is bounded by (s)2N(t)2N , i.e., we get
the same estimate as for 12
. This shows that indeed (1.12) holds.
Next we transform the operatorAa(3
, as follows.
(1.13)Aapu(x) = f91 JR1Y((1A )
M eii;(xY)u(Y)
=am (M1,Mr,D)u(x) , am )Map CEP
where the partial integration is justified as in sec.l.
Proposition 1.2. We have supp aap contained in the set
(1.14) Ixals2n , IYPI12n}
Moreover,
(1.15) Ilaa(3NNIILo 5 c(aR)2M max :Ia152M, ItI52N}
102 3. L2Sobolev theory
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) IlAapllL2 s c (a_I)2M[ [a]]2N,2M ' 2N > n ,
with
(1.17) [[a]]k,l = max {Ila(a)IIL : 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= I S(ua,AaRvt) I S cl
aIIualIlIvnII(aR)2Ma
s c2{ Iluall2 ;Ilvpll2} 1/2 = c2llull llvll
with 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 12boundedness 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 SS0000 0 '
(i.e., m=0, p=0, 6=0).
The Aa, are of this form anyway. Instead of (1.16) , (1.17) we get
(1.20)
with
IIAaPIIL2 s c (ap)2M [[[a]]] 2N,2N,2M
(1.21) [[[a]]]j,k,1 = maxLOO
:IaIsj,ItIsk,IkIsl} .
Thus we have
Corollary 1.4. All operators A=a(M1,Mr,D) with a E SS0,0,0
are
L2(len)bounded. Moreover, we have
(1.22) IIAIIL2 s c[[[a]]]N,N,N ' N = [n/2]+1 .
3.2. L2boundedness for positive delta 103
2. L2boundedness for the case of O>0 .
Note that thm 1.1 gives L2boundedness 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 L2boundedness
for general m=0, and pjz0, plab, OsS<1. It was first proven in
[CV 21. Other proofs may be found in [ Ka2] , [ CM] . We discuss aproof after a scheme in [B3] which again is relatively short.
Theorem 2.1. A pdo A=a(x,D) with symbol a Eiih06e',6'0<6<1
induces a bounded operator of H=L2(&n) .
Proof. We use cor.1.4 and the following decomposition. For a par
tition wj, j=0,±1,±2,..., like Va but for n=1, i.e.,
(2.1) wj(t) = w(tj) , supp w C {It Is2} , yjwj2 = 1 , wja 0
set xj=wj(log Clearly xj has support near i.e,
2j. Since we get xj=0, as j<1. The partition of
unity Exj2()=1 has supports in concentric spherical shells:
(2.2) supp xj C {2(32)/b sq) s 2(3+2)/s} .
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
((2EIj1I)) is 12bounded, and prove an estimate of the form
(2.4) IIAj1IIL2 s c2 EIjlI , 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"=FAj1F1 instead of Aj1 again. Here
Ajl"=Pjl=pji(M1,MrD) with Get pj1=O unless
(2.5) 272s(x)6s2j+2 , 212s(Y)O 21+2 .
Repeating once more the partial integration of (1.13) we may write
Pjl=pjl(Ml.Mr,D) . pj1=(xY)2M((lO )M pjl)(x.Y. )
(2.6) _ where e(x,Y)=(Y)26/(xY)2
and bM(x, ) = q)21 (1Ox)Ma(x,U)
104 3. L2Sobolev theory
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 indepen
dent of 1 , for each fixed M=0,1,2..... .
Applying the Fourier transform we may prove IIbM(x.D)xl(D)II
s c instead. Note also that E O,6e',6 ,so that
E 00'6e' .b .
Proposition 2.3. The family gl(x,T)=c1(21x,210 is bounded in Vt0
Indeed,
(2.7)g1((3)(x, )=
21(lalIRI)ci(R)(21x.210
= 2l(lal1pl)0((210 s(IPIlal))
with 0(.) independent of 1. Also, x1(21 )=0 except if (210 6 dif
fers 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 L2bound for
gl(x,D)=Q1 . Note that Q1=VibM(x,D)xi(D)Vl1, with unitary V1:u(x)
2In/2u(2lx), 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 IjlIZk0 with suitable fixed k0.
Proposition 2.4. For Ijllak0=5S , and s= 6(16) we have
(2.8) e(x,y) s c 2 EI3lI, as (x,y) belong to the set (2.5)
with c independent of j,l.
For the proof we use the estimate
(2.9) (xY) 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 2iI31IZ32, i.e., lili
a5S=k0 the right hand side of (2.10) is bounded by
c41/4TIjsc41(13)=c4hlljlI, as j>l, and by c4l/41l=C4(r11)lI7ll,
as j<l. Hence we get the statement with s=212=2(16)/6 .
Regarding the proof of thm.2.1 note that Pji has distribution
kernel OM(x,y)k(x,y), with the kernel k of The
latter symbol defines a uniformly bounded operator family, while,
3.2. L2boundedness 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 ( Y)  3 (xY) k I a lk+j=1R1
Indeed, for IaI+It1=1, writing aaayv=j for a moment, we get a(3=
(log O)ias=26 (log( y) )iap2 (log(xy) )kR in the span of P and SSlap
with T=( y) , S=(xy) . For any " , t , and I E + I rl =1 we get ( t3 )ts 11 =E,P+rj + (p4ap)(R where we may set R=S,T,O. An induction argu
ment gives 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+ ,3
31+76 3 , with multiindices aj, (3 j , y j , 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 theabove products are O(SIaIk T1), where k+l=I(3I, implying (2.11).
Now we again recall the above unitary map V1. Look at (2.6),
for M=1 , describing a symbol of Pjl , rewritten as
(2.12) p 1(x,y, ) =
The operator Wj1=V11Pj1V1 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
with respect to the norm (1.22), while a[[[w.,]]]N,N,Nsc2 E13ll,
so that cor.1.4 gives (2.3). Indeed, axayaywjl is in the span of
(2.14) 1U(v) , x+µ=a , X+v=P
where (x,y)=e(21x,21Y), wj(x)=X(21x),,21x)
Note that aaaRE/E=21(Ia'1+1p1)(aaaI0/O)(21x,21y). For boundedness
in supp wjlxwe show that 21/(21y) yand 21(21x21 y) are bounded, by(2.11). In supp wjl get 21/(21y)sc21/21/6sc. Also, 21/(21x21y)s
c21/12j/621/61, as near (2.10). With similar argument (and 1=1/6)
we get sc2l(1r1) for 1>j, andsc2113,
as 1<j, so sc in both cases
In other words, the first factor of (2.14) is bounded. We al
ready know the last factor bounded, since riE Vt0. For the second
106 3. L2Sobolev theory
we get 6Xw=2IIL I1O((21x)IRI)s121R1(1hj), bounded for 1>j, but not
for 1<j. The third factor is bounded by c2 61j1I. Together we get
the estimates for (2.4), via cor.1.4 , but only for 1>j .
For j<l observe that prop.2.2 does not involve the support
restriction (21x)6, 2j, while prop.2.3 remains intact if that res
triction is replaced by (21x)s215 a jxjs /4541 . This means
that we may replace the factor wj(x) in wjl by wj(x)0(x), using a
cutoff z=0 for jxI2:26, =1 near 0. For j<<l we get wj=wj6. Repla
cing wj in (2.13) by wj6 we get a new wj1 with also the third fac
tor of (2.14) bounded for j<l. The old wjl will now be wj(x)wjl
giving the yido wj(M)Wjl , i.e., the correct old Wjl. The estimates
after (2.11) also remain intact with the new xrestriction.
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) IIAllsc IPI)IILOO
: IaIs2,I,Is[Z]+1}L
with the L2operator norm IIAII .
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; Kparametrix and Greeninverse.
Define the spaces Hs Hs1 ,s2 ' for s=(si ,s2 ), as the classes
(3.1) 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 , or S' > S' , with inverse
(3.3) s1= nS = n_s(Mr.,D) = (D)sl (M)s2 E Op4c_s
In particular 11s is mdelliptic of order s , and,
(3.4) ns1  r1s E OPVCse
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' .
3.3. Weighted Sobolev spaces 107
We refer to the spaces Hs as (weighted L2) Sobolev spaces
over &n. In the special case s2=0 we obtain the ordinary (unweigh
ted) L2Sobolev 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) Ilulls2 = JR1 Iu^2s
with the Fourier transform u^=FuE LIloc(&n), where f exists as a
Lebesgue integral. In particular,
(3.7) He = {u E S' : u^E L2(&n,(,)2sdU)}
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 halfline (,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
(3.9)
(3.10)
llullk = {a1sk
llu(a) ll2} 1/2l
This follows from (3.6) :
llullk = Jdfslu^ l2(1+7)2k = [aJfd2alu" () 12l afsk
(k) Ilu(a) ll2ask awith the multinomial coefficients (k) = k!/(a1(klal)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) HullLOO
s csllulls , cs = ( .J(S)2sd )1/2 ,
and
(3.12) Iu(x)u(Y)lsss(lxYl)Ilulls , 6s=(2Jsin2(1t/2)( )2sd)1/2
108 3. L2Sobolev theory
Moreover, if even s >n/2+j , for some integer j >0 , then we
have a compact imbedding HsC03(&n) , defined as the space of all
uE CO such that also u(a)EE CO , for all lags j , and inequalities
of the form (3.11), (3.12) are valid for all u(a) , lalsj .
Sobolev's inequality ( 3 . 1 1 ) follows by showing that the Fou
rier transform u^ is LI for u E Hs, s>n/2 (a matter of Schwarz' in
equality). (3.12) is derived by estimating the Fourier transform
(cf.[C1],III ). The compactness of the imbedding is a consequence
of the Ascoli theorem: Equicontinuity of Hsbounded 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 Hslal , and
(3.13) Ilu(a)IlsIaI 5 llulls
This is a consequence of the inequality
(3.14) Ilu(a)lls IaI=JIu^ JlU.
using that 5
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 Hs are mutually adjoint under the pairing
(3.16) (u,v) = (Asu,Asv) = fuvdx , uE Hs , vE Hs
defined by the second expression, involving the inner product of
H=L2, using that As:H5. H and As:Hsa. H define isometries.
Returning to weighted Sobolev spaces H5H5I,sz
of (3.1) we
note analogously the following properties of Hs
(i) We have
(3.17) Hs = {uE S' : II cull<co} = {uE S' : II s*ull<o0} ,
with the operator s*=(D) s' (M) 52 =ns(Mr,D). Moreover, the norms
(3.2) and (3.18) below are equivalent.
(3.18) llulls = Il s*ull = II(D) S' (M)S' U11
This follows by calculus of ado's : ss*I and s* s1 be
long to OpWcOC Op'pt0, They are L2bounded, by thm.1.1, so that
3.3. Weighted Sobolev spaces 109
(3.19) csllulls s Ilulls s csllulls
(ii) We have
(3.20) 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/CS. tz Ilulls. tz s ct/cs. ,tz Ilullt .
using (3.2) and (3.19). or, Ilulls=ll sull=llLt(rc1ns)ull=llrc1 sullt .
where Kst I11 E OpyPcst is a bounded (compact) operator of Ht,due to 17tKst1 '= $sr2 St(sltrlssl )= stLst, where LstE OpipcOCL(H), r,=( x) (D) E K(H), as rj<0 (cf. thm.5.1).
(iii) For s1=ka0 , k an integer, we have
(3.22) Hs = {uE S' : u(a) EL2(&n,(x)2szdx),
115k}
and a norm equivalent to Ilulls is given by
llI(x)szu(a)l
(3.23)la>sk
For an integer sz=j20 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. , kz ,
(3.25) %+k = { uE He : xau (0 ) E Hs , as l a l sk. , l l l skz }
and Ilulls+k is equivalent to
(3.26) 2 llxau(R)ll'II lsk. , Ialskz
(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 Hs are mutually adjoint, under the pairing
(3.28) (u,v) = ( Su,II sv) = ,fuvdx , uE Hs , vE Hs
where (I'su,II sv) (with (.,.) of H) defines (u,v), uE He, vE Hs.
110 3. L2Sobolev theory
Note that n =rI , 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 Hs and (.,.) of (3.28).
This follows because S :HS H and lI s:Hs.* H are isometrics.
(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 HsC S', with (u,(P)=(u,g), gE 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 umu may be generated, for uE Hs by setting
um=xm*((Omu), X m nx(mx), wm=w(m), with suitable cutoff's x,wmThe spaces Hs will play a central role. They allow a descrip
tion of topologies on pdoalgebras in terms of L2norms.
Theorem 3.1. For a symbol a E tptm the corresponding yxlo A=a(x,D)
is a continuous operator H. Hsm , for every s E &2 . Similarly,
for a E0m,Se',o ,O<6<1 , A is continuous from Hs to Hsm
The proof is simple, due to our preparations: Since 1Is is an
isometry H5'H, one must show that smA $1 is L2bounded. By I,thm.
4.8 this is an operator in Op,t0, due to smE Op1csmC Opy)tsm
sIE Oppc 5C Opy,ts. Hence the desired Lzboundedness follows from
thm.1.1. Similarly for aE using thm.2.1 instead.
We now define a type of order classes: An operator A:SS,
with the property of thm.3.1 (i.e., A admits continuous extensions
As:Hs>Hsm, for all sE 8:2, and given mE &2 ) 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=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
(3.31) O(m)O(m') C O(m+m') .
The class 0(0) has the natural locally convex topology, in
duced 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 seminorms is given by
(3.32) { IIAIIk : kE Z2} .
Moreover, 0(0) , 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
coincides with the class yad,, , and, again, with the class ' of
all integral operators with kernel in S(&2n)
Proof. We have seen in II, lemma 1.7 that ih00 is the class of all
integral operators with rapidly decreasing kernel. Also, as a con
sequence of thm.3.1, get Op h =Opit C O(co). For the converse
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 kxE H, for xE >en, by the FrechetRiesz theorem. Similarly,
(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 It follows that
(3.36) Au(x) = fk(x,y)u(y)dy , k(x,y) = kx(y) .
But the same conclusion also applies to all AaPa,s, . It follows
by integration of (3.36) for large a, 3, a', I' that k must have
derivatives of all orders, for x, y , and that
112 3. L2Sobolev theory
(3.37) ±ifxaYa'a 4'k(x,Y)u(Y)dY
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 Kparametrix of a y'do of II,1. We now
recognize that a Kparametrix 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 opera
tors A,BE O(oo) will be called Green inverses of each other if
(3.38) AB1 , BA1 E F ,
where F C O(oo) denotes the class of operators with finite rank.
Notice that F is a 2sided ideal of the algebras O(oo), 0(0).
Clearly a Green inverse is a special kind of Kparametrix. In sec.
4 we will show that an operator admitting a Kparametrix 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:%'Hms,B:Hms'Hm continuous, for every mE &2 , so
that again A and B are Fredholm inverses of each other, as maps
Hm HMS . Hence then also A is Fredholm as a map Hm msFor general r=s+t>0 we may regard A as an unbounded operator
from Hm to Hm+t with domain dom A = Hm+r.Then B:Hm+t Hm is
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 SS , S' >S' , Hm*Hm5 (or dom A = Hm+rC Hm Ht )
Problems. 1) For the t,do's on the cylinder of I,6II,3,pbms, if
IIA(a) (x° IIL2 (S' )=0(1) , all a,(3, show L2 (S' x&n1 )boundedness ofA. 2) Define weighted L2Sobolev spaces on the cylinder, using the
2:tnorms dxi fdx° (x°) 252 I (1A)s' uI 2=IIuIIs, with properly defined po
0
wer of 1A=1aX,A6. 3) Discuss H5boundedness of y,do's as in (1).
3.4. Existence of a Green inverse 113
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 Kparametrix. 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 complexvalued symbol but even
for a matrixvalued pdo A =((a.k(x,D)))j,k=l,...,v,although we
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
orthocomplement 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 proper
ties 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 :Hms> Hs is the opera
tor 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 SS, %_sHS. Then AE O(m) amounts to the condition
(4.2) smA S1 GL(M) , for all sE &2
For the adjoint A* this translates into
(4.3) sl*A*lsm = h1 sA*rcls E L(H) , s E H2
Or, replacing s by ms, rIsmAs
*rf lE L(H), i.e., A*E O(m). It fol
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 adjointA*
of I,(2.3).
Theorem 4.1. Let AE O(m) have a Kparametrix BE O(m'). Then the
114 3. L2Sobolev theory
operator A , as a map A:S' > S' has the following properties.
(i) ker A C S , ker A* C S;
(ii) dim ker A < ao , dim ker A* < oo ;
(iii) in A = If E S' : (f,T) = 0 for all T E ker A* }
(iv) For fE He with (f,q)=O for cpE ker A* all solutions u
of Au=f are contained in Hsm,. There exists ps>0 such that
(4.4)IIff1s=IIAuIIskpsllullsm,, whenever uE Hsm (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 Kuj
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)<oo , implying dim A <oo . Similarly for A* , proving (ii).
For (iii) we first observe that, trivially, every f E S' ,
f=Av , satisfies (f,g) _ (Av,y) = (v,A*T)=0 , for all g E ker A*
In order to show the converse we will first establish (iv) and
show that a solution u E Hsm, exists for fE Hs, satisfying (4.4).
To clarify the relation with the inner product of H we assu
me 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 IIuIIsm,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 Kparametrices 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 HH(0, while IIsAsl
E O(r) has Kparametrix 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 = 0Thus 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,
(4.5) IIAuII a pllull for all uE Hr don A with u 1 ker A* .
Suppose this inequality is false. Then there exists ujE Hr ,
uj 1 ker A , IIujII=1 , with IIAujII o , j . But we get
(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 jK'ul11_0, j,l . Thus,ujE Hr, uju (in H), 11u11=1, Auf. 0. Hence uE don Ac , with the clo
sure Ac of the unbounded operator with domain Hr, as discussed.
But we also have IIAu jlls+r_' 0, Iluujlls 0 for ss0, s+rs0, where nowA: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 uju 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
(4.7) (u,Av) =(A*u,v)
for all u,v E dom A = domA*
 Hr ,
by definition of A*. In other words, A and A* are in adjoint rela
tion 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 Hr . Then A*f=0 wth A*:HH_r . OruE 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
the operator A:HHr. Indeed, ujE Hr don A, u j*u (in M), Au j Acu(in H) implies ujE H uju (in H) , Auj> Acu (in Hr), hence
Acu = Au , since A:H , Hr 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 Kparametrix B. If BEO(m'), then also B0E O(m').
Also, B0 may be chosen in such a way that (4.1) is valid.
Proof. A Green inverse is a Kparametrix. Thus we must construct
116 3. L2Sobolev theory
a Green inverse whenever a Kparametrix exists. Applying thm.4.1
we define an operator Bo by setting B u=0 for uE ker A* , and B =0 * 0 0(AI(ker AO)1)1
in in A =(ker A )1 . Every u65' is in some Hs, and
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 =(kerA*
and AB0=0 for uE ker A* . In other words, (4.1) is valid.
On the other hand, (4.4) implies
(4.8) IIA lulls_m, s
P
IIuIIs for all uE H. , u 1 ker A*PS
for the abstract inverse A 1 of A:{uE Hs+m: u I ker Al > Hs
Actually, we get (4.8) for Ac1 with the closure Ac of the unboun
ded operator A:dom A =Hs+mr=H
sml+Hm , and Ac1 is defined in all
of {uE H : u I kerA*}.
By our definition of B (4.8) implies
* 1s ) is= JIBOVIIs_m,SpsIIvII csllulls, where u=v+w, wE (ker AIIBOuIIs_m
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) , v=uw , hence IIv1Is
s IIuIIs+ JIlrojIlsllrojII_slluIIs , 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 ope
rator 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 Kparametrix constructed there.
Proof. Indeed, that Green inverse is a special Kparametrix, and
we observed earlier that, as inverse modulo the ideal O(oo), a
Kparametrix is unique up to an additive term in O(c) .
We conclude this section with the remark that a Green inver
se 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 opera
tors 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. Hscompactness 117
with uj=,1nf a liWu(p)dp , as in I,6,pbm.3, is L2 (S' )bounded if weassume that =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*sub
algebra 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=ajaj_1 , and some p>O , for all (some?) 1 and k.
5_HScompactness 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 0pOm,p,sp,>0, pz>5, is E K(HS,Ht) whenever mj<sjtj , j=1,2.
Proof. 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
(5.1) pE(x) =(x)E (x) = cE'vlxl(En)/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(xy) 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 , are compact, from H to
H . Second, we have lIQEQElI  0 , j>oo , by Schur's lemma, so that
QEE= K(H), since K(H) is closed under uniform operator convergence.
With Lzcompactness of QE we get the statement at once: Com
pactness 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: HsHt is compact if these inequalities hold. Q.E.D.
Chapter 4. PSEUDODIFFERENTIAL OPERATORS
ON MANIFOLDS WITH CONICAL ENDS
0. Introduction.
In the present chapter we will focus on pseudodifferential
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) = Fla(M)F is meaningfull only for func
tions or distributions defined on ten , the kind of ydo's we intro
duced has a natural environment on a type of differentiable mani
fold, 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 'Smanifolds' 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 mani
fold 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, cutoff's,
partitions, as well as admissible coordinate transforms, all
designed to give S(c) similar properties than S(8n) . In particu
lar a Riemannian metric is introduced, making n a space with coni
cal ends. In sec.3 we prove invariance of pseudodifferential ope
rators 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 mdelliptic and formally
mdhypoelliptic operators, defined by their local symbols.Results
of ch.'s II and III regarding Kparametrix, Sobolev spaces, order
classes, Green inverse, etc., all generalize almost literally.
Similar results, on coordinate invariance as well as andel
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
tial operators we point to [CDg1] , [C2] where similar C*algebras
of singular integral operators on L2(a) are considered, with cor
responding 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 C00manifold a of dimension n (always assumed para
compact even with a countable atlas) the space D=D(a)=C00(n) is
well defined, and the convergence cpj0 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
(1 . 1) (u,(Pj) > 0 whenever y j  0 in D(n) .
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 , O<K(= Coo , on a paracompact manifold n , and then define
(1.2) (f,P)=f 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 topo
logized 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
120 4. Operators on manifolds with conical ends
we follow our habit of [Cz], and always assume given a pair {sl,dµ}
of a manifold st and positive Cmeasure 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 expres
sion 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 com
pactly 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 lar
ge lxi. In 11,5, looking at wave front sets, we introduced a type
V of directional cutoff and regarded f belonging to S in a coni
cal 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
structure on sl, allowing only special transforms as
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 SGstructures
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) , x = s1(y) = y/ 1 y , = t(x)
Introduce [ y] = 1 y Z , then,
4.1. Distributions and temperate distributions 121
(1.6) Y = x/( x) , x = Y/[ Y] , [y] = 1 /( x)
Proposition 1.1. The map
(1.7) u(x)  v(Y) = u(s1(Y)) = (uas1)(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.
Remark 1.2. A function fE COCO(B1) has a natural extension to in

f=0 there. The extension v is CQ(&n), with supp vC Bi=(B1)clos.
Vice versa, for vE COO(e) with supp wE BC we have wIB1E
Proof. If vE CO"(B1) then v and all its derivatives vanish of in
finite order at lyl=1, by Taylor's formula. Thus we get
(1.8) v(a)(y) = 0(((11y12)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
(1.9) v(R)( )II ay
X
k(xjk/(x)) , IRlslal , lYk = «
by induction. Get aX(xj/(x))=O((x)IY I) , easily confirmed. Eachterm (1.9), hence ua is 0((x)k) , for all k, i.e., uE S
Vice versa, let u E =S and v(y) = u(y/[y]). Note that
(Y/[Y])2=1+Iy12/(1IY12)=1/(1IY12)=[Y]2, hence (Y/[Y])=[Y]1
(1.10) u(a)(Y/[Y]) = O((Y/[Y])k) = O([Y]k), for all k .
For a=0 it follows that v(y) E CO(B1) . For arbitrary a one again
uses the chain rule to express v(a)(y) as a linear combination of
(1.11) u(P)(Y/[Y])nay j(Y1 /[Y]), IPIslal, j=a ,
similarly as (1.9). Here we get aY(yl/IYI) = O([y]21Y11), andy
(1.10) still implies all terms (1.11) to vanish, hence v(a) =0
at IYI=1 . Thus v E COOO(B1) follows and prop.1.1 is proven.
The above suggests the concept of an Smanifold as the inte
rior 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).
122 4. Operators on manifolds with conical ends
aSt0 is called the infinite boundary of a; its points at infinity.
We will insist, in the following, that a0 be a compact C
manifold 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 Car
tesian products with Ij=I or I.=le+=[0,oo) , depending on j.
Notice that 2 xln=R2n 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 aI 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 Sma
nifold 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, : I1Isk} , k=0,1,2,...L (B.)
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) 71 O J1(1i)Nv(0)(y°+i(yy° ))diIOI=N+1 0
For O#yE Bi let y° = y' , so that I (yy° )6 I s yy° j16 I s [y]K. Then
(1.16) v(y)= Max{IIv(0)II :I8I=N}.O([y]N+1) , N=0,1,2,... .L (Bi )
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.... .
4.2. Distributions on Smanifolds 123
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) , whereyE D(In), V(y)=q(x). We get an injective map S'(&n)>D'(BI). Actu
ally, v defines a functional on the subspace CO"O(B1)C C00 (Bc1)
extending to w:COO(B?)T , by the HahnBanach theorem. More precis
ely,v:C000
(B1)'T is bounded with respect to some Ilvil k Then Hahn
Banach, applied for the Bspace Ck(B7) gives a continuous functio
nal 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 ofC00(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 Smanifolds.
2. Distributions on Smanifolds; manifolds with conical ends.
We return to the discussion of Smanifolds 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°  of (U°) C Bi
with a homeomorphism of between the open sets U° C sz° and oi' (U°)Then an Sadmissible chart of sz is of the form
(2.2) w:U  w(U) C &n , where U=U°flsz , w= s1.(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 Sadmissible charts.
An interior chart UCC Sz of the noncompact manifold Sz is S
admissible, since we may define U°=U, of =so (o. A general chart w:U
.w(U)C ien is Sadmissible if and only if the map s,w:U B1 extends
to a diffeomorphism of :U°  V° between open subsets where UC U° CSz° , sow(U)C V° C B1 , and where U and so w(U) are dense in U° , V1,
124 4. Operators on manifolds with conical ends
respectively, and U°n sz=U. For a given Sadmissible UC sz the chart
U° C fe is given as U° = U U (Uen ac° ), with the closure U0 in n' ofU . 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'
{ lyy° l<e , I y < 1 } , with jy° j =1 , y° image of p°, in B1coordinatesof sz°. With s1 this goes to NE={j y°kss}. After a rotation as
sume y°_(1,0,...,0). The neighbourhood system NE={(1{x2+ ss2}
is equivalent to ME={1 sa, se}, where x=(x,,x ). ME is the(x) I
intersection of the solid hyperboloids
(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 cutoff cone
(2.5) CC yo ,T1={xE In: IxI>l IXI
Y° 1s11 } , 11>0
where we may choose r1==1(E) 40 as e0 . Vi Ice versa, any set Na contains a CCy°,,, , for a smaller 1>0. This describes the neighbour
hoods of our infinite points. The sets (2.5) form a base at p° .
An Sadmissible coordinate transform w,XI:x(U n V)>w(U n v),
for Sadmissible charts w:U In , x:V' In , must be of the form
( 2 . 6 ) x=w,X1=s1°x° , s , with x° = w ° °X° I :X° (U° n V ° )  of (U° n V°)
a diffeomorphism between open subsets of B1For investigation of maps of this form we focus on the Jaco
bian matrix P(x)=ay)/ax=((ay,/axl))j,1=1,...,n of a global coordina
te transform q=t°rp°s, t=sI 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)(Is(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(Is(x)s(x)T)
with 111=arp/as(s(x)) . Here A.(x)=y = 1 , hence
(2.8) P(x) _(1Si )1/2(I+t('V(s))t('U(s))T)11J(s)(IssT)
4.2. Distributions on Smanifolds 125
Taking determinants we get
(2.9) JT(x) = Idet P(x)I = II(s )3/2Idet'(s)I
using that det(I+t(x)t(x)T)=1+t(x)2=11x2. Therefore is (ex1 2
tends to) a smooth positive function on B1. Moreover, 11ss=
1 distance(s,I) E C'(Bc) (with t=aB?) follows, ause1+ 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 ntuple of funtions V. , where dist(V(s(v)),r)=
(v). Clearly V1=0 as v1(s)=0. Thus ''1/v1E C , and it isclear that this quotient is >0 , since yr is a diffeomorphism.)
This result may be extended, as follows.
Proposition 2.1. The function Q(s)=Pos1=P(t)=(acp/ax)("), 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)(1ssT) 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 , we may write
(2.10) I ssT = (1Is12)s°s°T + L1 sjsjT
hence
(2.11) W(s)(I ssT) = (1Is12)(w(s)s° )s°T + _i(tY(s)s. JT
Applying the matrix t(V(s))t(V(s))T=1s
V(s)?Q(s)T to the termsV 7
at right of (2.11), the first term gives aCC'(B1)matrix,
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 1V s also is Coo near IsI=1 . Q.E.D.
There exists a finite Sadmissible atlas. Indeed we may choo
se a finite atlas of the compact c° and construct the cor
responding charts {U3. ,w.} of (2.2). Moreover, for a partition of
unity P,9=1 subordinated to (with supp XOC UO) a correspon3 J 7 J 1
ding partition subordinated to {u wj} is defined by %j=(X9j )osj, 3
Notice that, in admissible coordinates, we have SSO,(1,1,1),0'
i.e.,%a)=0((x)Ial) for all a, (T extended 0 outside a(U) in3 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 Sadmissible partition of unity, and cutoff function,
respectively. (Often we drop "Sadmissible" for "admissible").
With an admissible partition {Xj} and finite atlas {Uj,wj}
we define a Riemannian metric on st by setting
(2.12) ds2 = 71j%jdx2 ,
with the Euclidian metric dx2 of wj(Uj)C ten, in each summand.
From prop.2.1 it is evident that an admissible coordinate
change cp:Rn 10 converts the Euclidean metric dx2 of Rn into a
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  gjkos1 extends as a Cc'function to the infi
nite boundary of the chart, and the matrix is >0 there.
Proposition 2.2 For each infinite boundary point p° and admissi
ble chart p°E in there exists a cutoffcone CC Y,
,)C w(U)
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
the sense that c , xx° l s d (p, p° ) s c xx° 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 , andthe 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 Sstructure' or
'Smanifold', 'Sspace', in the following sense: An Smanifold be
comes 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 Smanifold 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 Smanifolds 127
face measure dµ=dS (=det((g]k))1/2)dx) of the conical metric ds2=
= jjkdxjdxk as for our distributions and Sobolev spaces
In particular we introduce the Hilbert space H=L2(SZ,dµ) .
As a side remark it may be mentioned that the quasiEuclidean
metric ds2 has a corresponding BeltramiLaplace operator A . Using
the comparison triple {Sz,1A,dµ} in the sense of [C2],V one may
generate L2comparison algebras with complexvalued symbol, con
taining 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 dis
tance 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 (thinkof 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.)Such an end will be called conical if (a suitable) Sk' is isometric
to an end of an Smanifold, 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 Sstructure, making 91 an Smanifold is in
troduced by declaring certain charts, coordinates, coordinate chan
ges etc. as admissible. Admissible charts, atlantes, cutoff's,
partitions of unity, all are obtained from charts,..., of nO,
using the map s(x)= : InB1 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 Smanifold sk possesses a distinguished type of Rieman
nian metric ds , called conical metric. In admissible coordinates2
128 4. Operators on manifolds with conical ends
ds2 has properties (i)(iv) listed in prop.2.2. A manifold with
conical ends is defined as an Smanifold 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))nldv with a positive Co measure
dv on the compact S2° , and (d (p)) = 1 +d (p )1e) We have
(2.13) S(S2)= {uE C00(St): (Xu)ow lE S(len) for all w,X} ,
and
(2.14) S'(S2) = {uE D'(S2): (?u). (x lE S'(ien) for all w,X} ,
where, in each case, w:Uw(U)C In and X denote an admissible chart
and subordinated cutoff 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
(2.16) (u,) = J u(pdµ , TE S(n)92
g) Define L(S2) as the space of differential expressionsL: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'multi
pliers (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°)
it may be generated by all seminorms II(d(p))kLull00
where k=0,1,..,L
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
PE &n and m=((mjk))j,k=1,..,n is invertible, mjkE &
That is, if Au=a(M1,Mr,D)u=f, where u,f E S(ien) , a E ST , and
u(mx'+p)=v(x') , f(mx'+p)=g(x') , then we get
(3.1)
where ImI=Idet ml, for a moment. With =m t=(m 1)t, we get
(3.2) g(x')=
ff
Joxlfol t 'MY')
Or, g=b(M1,Mr,D)v, where a(mx+p,my+p,m generally is
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 sub
collection of OpST. We only admit coordinate transforms T:1n1n
with the property that sog0s1= V extends to a diffeomorphism
B +.B1 . Clearly then the diffeomorphism T:2njkn extends to a hom
eomorphism 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 homeomor
phisms EnH En preserving this manifold structure.
Given such g:1n2n 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 in
tegral signs, where co(z)=0 for lzIse, w(z)=1 for jzj22e>0. Writing
g=g1+g2 , 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
(3.4) f2(x)=JRiJRye1(xY)w(s(6(x))s(6(Y)))a(x,Y,)u(Y)
130 4. Operators on manifolds with conical ends
With the inverse function t= l=s.0.sl we have s.O=l s , hence
(3.5) f2(x)=fo1
We use the identity The inte
grand vanishes near x=y. An Nfold partial integration gives
f2(x)=fO1 foy.il(x))
(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)IaplxyI , with some p>0 . Hence
the integrand in (3.6) vanishes forp
sp . In other words, we may assume that Is(x)s(y)Iap .
Observe that sl(x)=t(x)= x , where [x]=V . We get
(3.7) T(x) = ((tilxl)) =l 3((6il+titl))
=(I+t)(t)
implying T(x) to be real symmetric, T(x)a , xE B1. Notice that1XI
t(x)t(y) = fdr UTx))) ={ fdtT(x+t(yx)}(yx) = S(yx),
where SaMin{ th , [ y] 1. We get 1 =(t (x)) , hence
(3.8)It(x)t(y)Ia(Min{(t(x)),(t(y))})Ixyl, x,yE B1 , or,
Ix Y1Min{(x),(Y)} s(x)1s(Y)I ,
x,y E&n
Thus we may assume that
(3.9) IxYI2N 5 (g)2NMax{X2N(),x2N(Y)}
, fi(t)= I
Assuming aE SS., P'6 , 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)
fk2(x,Y)= IxYI
where N is arbitrary (sufficiently large), and where fd exists,
since the iorder of aN is m12Np1 < n , 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(P
(y) = k2IT(x,y) under the coordinate
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,xy).
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 zderivatives, 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,xY)=O((x)m:(Y)m') , as x,yE &n , IxYIae>0
For the factor k3 in front of the integral in (3.10)2 we get
(3.13) aaayk3(x,y) = O(?.21(xy) 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)(xy)11
using the well known inequality. Accordingly,
(3.15) aXayk3(x,y) = O(X1(x)X1(xy)), 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(os1 , 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) = foI (Y)v(Y)
where X=X(s(x)s(y))=0 for js(x)s(y)ja2E . Write
1
(3.17g(x)(P(Y)= 0 di{atcP(Y+i (xY))}= M(x,Y)(xY)
where M(x,y) = dx(aq)/ax)(y+i(xy))0
If x and y are close then clearly M(x,y)zt aye/ax which is an inver
tible matrix. We have (T(x)cp(y))= Then an inte
132 4. Operators on manifolds with conical ends
gral substitution Mt(x,y) , for fixed x,y, close together, may
be used to convert (3.16) into the form of a ipdointegral 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(xy)) with Q(s)=0
a(p/ax( s ) E C'(B1) (by prop.2.1). The curve {s(y+t(xy)):t E &}=T
is a halfellipse with center 0 and vertices atX_Y
and 7
z= vector to the point of the line {y+t(xy)} closest to
easily seen. The arc {s(y+t(xy)):Ostsl} connects s(y) and s(x) on
r. It is contained in the ball Is2(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) , with
a detM(x,y) a(y(x),T(Y),M 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(1,1,1),0 More precisely, we have
(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
targument of a , or anywhere else. In the first case we get a
factor 0 x1 1p,(( ) in addition to the already existing ones.
In the other cases we get a factor 0((x) p: (l;) S ) or 0((x) (All this for an xderivative).
It follows that a E SSm,p,, with 6°= Max{S,1pi}. 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<cs(q(x))/(x)sC are easily derived. Summarizing:
Theorem 3.3. Let a E SSm,p,6 , 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 thats,(P,s 1
(with s(x) (X = ) , a map extends to a diffeoVTI T7
morphism B?H B1 of the closed unit ball B1. Then the linear opera
tor 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,P,
66 =max{6,1p.}. Ups
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.  i.e., 6<p, and
1p. <p. ap. >2. We therefore have
Corollary 3.4. Let aE tphm1 p s with pz >O , p. >2 , S<p. . Then thecoordinate 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 , 1p. } . The new symbol, up to terms inm'<m , is given by a with a" of (3.18).
Problems. 1) For n=1, consider the transform tx=q(t)=et, a diffeo
morphism & .&+=(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. Exp
ress M and M like Fourier integrals. Show that M is the Fourier
Plancherel 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 R2*=R2\{0}as
Riemann space with metric ds2=y,(Sjk+ni nk)dxi dxk , 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<t<oo, Os9s2n} with me
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 par
ticular, 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 C00(Z). Try fora concept of Kparametrix, for mdelliptic operators to be defined
especially with an operatorvalued symbol at the conical tip.
134 4. Operators on manifolds with conical ends
4. Pseudodifferential operators on Smanifolds.
We start our discussion with a generalization of I, thm.3.3.
Proposition 4.1. For the special Smanifold S2=1Qn let 0,?. be admis
sible cutoff functions such that0°=00s1
and 7v°=X s1 , extended
to B1, have disjoint supports. Then, for any AE Op1Uhm we have 0A?.
E Opph00 , an integral operator with kernel in S(12n).
Proof. By I, thm.3.3 the distribution kernel k(x,y) of A has singu
lar support at x=y only, thus the kernel K(x,y)=k(x,y)0(x)X(y) of
0A?, is C'(3k2n). 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 Ixkklxyl. 1y1"Ixy1 , as xE supp 0 , yE supp k ,
IxI , lye >> 1 , 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
terior of S2° , a compact manifold with boundary, as in sec.2. We
will use admissible cutoff functions now, defined as 0=01os, with
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 cutoff 0 and the chart U
will be expressed by writing 0 ® U . The same notation will be
used for another relation: Writing
(4.1) 0 (> O X , or , 7v m 0indicates that 0=0°,s , X=X°,s , where 00 , X° are cutoff'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 cutoff function 6:* U we
may construct an infinite sequence 0=0 0<10 01 ® 02 ® ...<XJ
U.
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 cutoff function X and chart w:UTen , with 7`GQ U, there exists A,=a,(x,D)E Optphm p S , KXE r with as x w(U), and
(4.3) A(Xu) = KXu + (AX((Xu)0w I))°w , uE S(S2)
where we assume m,p,S independent of k and w:U>&n .
4.4. Operators on Smanifolds 135
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, 1pisO<p.. The class of
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, 1p.sO<p., enabling local cal
culus and coordinate change. Write LSO.,p1s=U LSm,p,s , U LSD LSOO
= LS. We shall see: =LSooflLSm. The special case p=e=(1,1), O=0
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 identifi
cation of the points of U with those of w(U)  i.e., as the iden
tity map, which may be omitted in writing. Then (4.3) reads
(4.3')
Note, we also get
(4.3")
A(Xu) = AX(a.u) .
A(au) = Kku + k° AX(Xu) ,
with any other admissible cutoff ? o , %< 0 ko Q U . Indeed, we get
X< 0U9 with the admissible chart Uo={Xo=1}into U =wI Uo . By defin
ition 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<pj, 1pi2.6<p.
If we have both XQ U, 0<) U, for two cutoff's X,O , then
A),O=a),(x,D)XO+KXO=a0(x,D)XO+KOX, showing that (aka0)(x,D)OXE(f .If X(p)*O, O(p)*O , for some pE ft° , then prop.4.1 implies
existence of an admissible cutoff x with x=1 near p such that
(4.4) S(12n)
Clearly (4.4) expresses a uniqueness property of the local symbol.
Note that LSm,p,5(&n)=OpVhm,p,s , assuming p.>O, 1p.sb<p.
First let A=a(x,D)E Opiph. For X.ZJ?o Q U , w=idI U , construct%QXo<CU. Then (1Xo)A?E(`',by prop.4.1, hence (4.3"). Similarly
for a general w extending to an admissible coordinate transformNn .,in
. One then must use cor.3.4. For general admissible w:U&n
first cover UC 2n by a finite collection U Uj such that w IUj is
approximately linear, hence wlUj extends to an admissible map In
ea&n . Then, with an admissible partition 1=Eµj , p,j<EoUj , let
%j=%Rj . Get AJ%j=ak (x,D)k j+KX , and sum over j, for (3.3). For7 7
136 4. Operators on manifolds with conical ends
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
(4.5) Au=2 AXju=y, XA), (Xju)+Ku , uE S , KE LS00 aj<F)k I<E)Uj
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 cutoff's 0=0°os , ?=1,.°os, if
(supp 8° )fl(supp x, )=Jd, then OA), E LS00 , for every AE LSm .
Proof. Construct a finite admissible atlas and partition of unity
Eoj=1 , %.<10 U. , wj:Uj_wj(Uj)C In such that for each j either
Ujfl supp or U9fl supp 0° = jd . With this partition write
OA?. 29(A), 29(0A), A +K )k
The sum may be extended over j with Oflsupp o, 0, else the fac7
for 0X vanishes. In the leftover terms we get sinceJ
then k jk = 0 , so that 6AX = E0Kj). E LS00, 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 cutoff with supp(.Aj)C Ufl Uj . The
coordinate transform w o : w . (Ufl u w(Ufl u.) is of the formx=wow 1 = s1x° os with a7diffeomorphism x° :w.( U° fl u .) * of (U° fl u .)
7 0 J J 7We 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 . To be
precise, in the formulation of (4.2) we have
(4.7) (A1((X), ju) "wjl))owj = (Aj°((AAju)oW ))ow
with properly extended diffeomorphism wowjl , 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), withan admissible cutoff 0j, supp 0 j C U. By prop.4.2 the integral
4.4. Operators on Smanifolds 137
operator Kj has a kernel of the form v1(x)u2(y)kj(x,y), with admis
sible cutoff'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)fP= I(3I) for all a,3As 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 opera
tor 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 corres
ponding facts in thm.4.4. Proofs are straight extensions, and will
not be discussed in detail.
Theorem 4.4. The classes LSmP.S
for form algebras for
each given p?0, 1pi <S<pj . More generally, we have
(4.8) LSm,P,S.LSm',p,S C LSm+m',P,S,LCm.LCm,C LCm+m'
.
One finds that LS,Ois an ideal of LS0 , LSOO, LSmPS , m=0,0
.
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=
9
i6
a )bx(e) ax=
6
Kl;9
{xax}(9) (mod h,O)
where "f=g (mod (Z) near p" means (fg)xE(; with a cutoff 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
138 4. Operators on manifolds with conical ends
of each other if AB1 , BA1 E LS00 . 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. , so that BJA11=A..+Kj
Write Aµ Aj . Notice we also have .X2A . = X, X. + K j , sinceE Op,phco , by prop. 1 . 1 . As a consequence, ? BjXjA =
.B .X .A .k3 + .1B .??A(1X3 ) _ c +K , using prop. 4.3 with A=T,?333 J J J J J J J j j JX=1X.. 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
Kparametrix of the operator A E LS . Involving adjoints to possi
bly convert right into left parametrices, we have proven:
Theorem 4.5. Let AE LSm . For an admissible partition X>,.j
=1 , ?.i 61 ®Uj with admissible cutoffs ej and charts w:Uj.&n ,
let the local operators A. of (4.3') admit left (right) Kparamj
etrix in 4*m',P,B , (with inverse order m') with respect to Xj ,
for all J. Then A admits a left (right) Kparametrix of order m'
A pdo A E LSm,p,6 C LS is said to be mdelliptic (of order
m) if for every admissible cutoff 6 ®U , w:U > ien an admissible
chart, there exists Oo (E>6 , Bo o` U , such that the local symbol
a0, is mdelliptic (of order m) with respect to 0 .
Theorem 4.6. An mdelliptic 'do AE LSmp S C LSm admits a parame
trix B E LSm,P,6 . Vice versa, if AE LSa,P'6 C LSm admits a Kpa
rametrix in LSm,P,b then it is mdelliptic.
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'am and some admissible finite partition 1=EXj , %j<0ej®U. , with admissible ej , wj:U.in , the local operator A0 of
J
(3.3) is formally mdhypoelliptic (of inverse order m') with res
pect to %j. Then A admits a Kparametrix 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 perio
dic functions. 2) Consider y,do's A with operatorvalued symbol
A(x,i), where for fixed x, , is a pdo in LCo(B), for a
4.5. Order classes and Green inverses 139
smooth compact manifold B. As in 1,6, pbm's 14 and 11,3, pbm's 1
4 such operator should act on S(C) , C=Bx&n , properly defined. We
should get a t,docalculus, a Kparametrix construction, and, gene
rally, results analogous to those in the problems mentioned, inclu
ding 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 cylindri
cal ends both, as well as conical tips: In a subdomain sic contai
ning 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 operatorvalued symbol at a cylindri
cal 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 cutoff's (o,x with support
in a chart UCC st , we have wAx a ipdo on &n . 5) For a distribution
uE S'(ft) , 12 a manifold with conical ends, define the concepts of
WF(u)  the wave front set  and ZF(u), looking at ZF (Xiu) of
11,6 and VI,7, with an admissible 1=1k , xi <s)Uj , 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 Smanifolds.
On an Smanifold sZ introduce the weighted Sobolev norms
(5.1) HAS = {lllxiulls} 1/2 uE S' (st) , s=(s ,s2 )E &2
with l=? 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 (L2Sobolev)
L (Rspace Hs=Hs(s2) consists of all uE S'(c) with finite norms Ilull s
Generally, when writing hulls we imply h1uh1s<oo , i.e., uE Hs .
Again we regard the maps wj as identifications, hence kiu
as functions on &n (extended 0 outside wi (Ui)). Notice that
*(5.2) hlxiuIhs = s Sk.u) , as uE Hs
140 4. Operators on manifolds with conical ends
with the pairing III,(3.28), or, with the inner product of L2(in),
as uE H2s, where Qj=Xjrlss
E Oppc2s . Hence we get
(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
a ido , P2sE LS2s,e,0 = LC2s , as follows by comparing (5.3) and
(4.6). Moreover, P2s is mdelliptic of order 2s, as confirmed by
looking at the local symbol: At each point it is a finite sum of
transforms of the nonvanishing symbols of Qjkj, all of them local
ly mdelliptic and nonnegative, so that they cannot cancel each
other, looking at formula (3.18) . It follows that P2sP2t and
P2tP2s both are mdelliptic of order 2s2t, 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.,
(5.5)S(cz)=
SHS(n), 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' : kiu E Hs(in) , j=1,...,n} .
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 opera
tor 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,
(5.7)IIPsu112 = 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:HsH is
4.5. Order classes and Green inverses 141
continuous for every s. As mdelliptic operator Ps admits a Kpara
metrix Q_S, a i,do in LC_s. If uE H, Psu=O, then 0=Q_sPsu=uKsu, Ks
E LS_00, =::,, u=KsuE S =* (u,Psu)=7'11 s/2Xju112=0, ju=0 u=0, showing
that Ps:HSH is 11. In H let fE H be 1 to im Ps. Then (f,PS(q)=0,
cpE S 0=(f,PSQsp)=(f,(1LS)i) , i E S , where LSE LS_Q. Hence
(fLSp,i)=0, iE S f=LspE S b0=(f,Psf) =7' IIns/2kjfII2 f=0 . It
5H5 defines anfollows that P5:H5H has dense image P5. Then Ps1:P'
unbounded operator from H to Hs with dense domain.
Proposition 5.1. For WE 12 we have Ts=PsP5E LC0 . TS is bounded
in H=L (S,dµ) and has a bounded inverse Ts :HH. Moreover,1E
LC0 is a yido as well, and thus is a special Kparametrix of T5.
Proof. We already noted that TsE LC0 is mdelliptic, hence has a
Kparametrix 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:HH. Indeed TSu=PSP_SU=O yields uE S, us
ing the Kparametrix RS, then PSUE S=>PSu=0 u=0, so TS is 11.
Similarly, using Rs, show that (f,Tsp)=0 for cpE S implies f=0, hen
ce im TS is dense. Existence of the (L2bounded) Kparametrix 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
have R5T5=1L5, with K5, LSE This implies RS=
T5 TB KS=Ts L5Ts . Or, TS =RS+Ts Ks, and T5 =Rs+L5T5 . Substi
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 bel
ongs 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
inverse. Indeed, T5=PsP_S implies Ps1= 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 IIPsulk=2Iln 5(XjP5u)II2=
7,11n s( KkA`ks/2s11 /2?.ku)112 The point is that, for fixed j, all
142 4. Operators on manifolds with conical ends
terms r1,2Ps/2?.ku must be taken to the coordinates of the jth
chart, using thm.3.3. and only within a neighbourhood of supp XjC
Uj. They will give operators of order s , insuring L2boundedness
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 . There
exists a y'do Q_SE LS_S , acting as an inverse i.e.,
(5.8) PSQ_S = Q_SPS = 1 .
We have Q_S:H>HS continuous, i.e., PS
: HS« H and Q_S:R HHS define
isomorphisms between Hs and H . With constants cs , cs>0 we have
(5.9) c5IIPsuII s (lulls s csllPsull , u E RS
Using the isomorphism PS we can prove HSboundedness of ipdo's:
Theorem 5.3. A i,do AE LSm satisfies
(5.10) llAulls_ms cSllulls , for all u E HS , cs independent of u.
Proof. We have llAulls_ms cfl(Ps_mAQ_s)Psufl , where PS_mAP5 E LSO is
L2bounded, 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'Hsm. Then, clearly, LSmC O(m) for all in.
We again define O(±ao) by III,(3.30), and get LS_00=0(oo). The con
cept of Kparametrix 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
Kparametrix BE O(oo) with 1AB, 1SAE O(oo) of finite rank.
We now can repeat every line of argument of 111,4, showing
4.5. Order classes and Green inverses 143
that a Green inverse  and even a special Green inverse, with all
properties III,(4.1), exists if and only if a Kparametrix 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 admis
sible vector bundles (the latter denoting the restriction to St of
a C"ovector bundle over ? , where evidently Hsnorms and Hsspa
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 BeltramiLaplace 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 cal
culus 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 Kparametrix exists: mdellipticity 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 re
gard to elliptic and parabolic problems. There we applied the Fou
rierLaplace method to freespace problems of elliptic equations,
and to evolutionary halfspace problems of the (parabolic) heat
equation, all with constant coefficients. We covered Dirichlet
and Neumann problems in a halfspace, for elliptic equations.
With the tools developed in I, II, III, IV we now can give a
similar "FourierLaplace 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 summa
rized (in more general form) in sec.1, below. If Sz is compact we
need ellipticity, else mdellipticity of the operator. Not only
(md) elliptic operators on a complexvalued function but even
maps between crosssections of vector bundles are considered.
Note that there is a different approach  a functional analy
sis approach  to these theorems, not using ado's at all. Elliptic
theory, in its beginnings, was developed for 2nd order equations.
Such 2nd order theory is of dominating importance for many physi
cal 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):uau , aE C(Zn) (cf.II,3) and the ope
rators D.(1A)1/2=5., defining S.=s.(D)=FIs.(M)F as Fourier mull 3_1/2 J 7 7
tiplier or else (1A) as inverse positive square root of the
unique selfadjoint realization of 1AaO, A=a ([Cl],III,IV).7
Similarly, in case of a general Riemannian manifold with co
nical and cylindrical ends, and conical tips (cf. IV,4,5, and the
problems of IV,4) we may generate such C*algebra fromD(1A)1/2
with the BeltramiLaplace operator A of n under the metric discus
144
5.0. Introduction 145
sed for such manifolds, with DE D# , and a(M) . aE A# , with sui
table 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 ellip
tic but also to hyperbolic theory (of ch's VI and VII):Conjugation
with the evolution operator eiLt of a first order hyperbolic equa
tion defines an automorphism of the above algebra A . The dual of
this automorphism will be a Hamiltonean flow in symbol space deter
mining propagation of singularities, as in Egorov's theorem.
In sec.2 we start focusing on the general elliptic boundary
problem for smooth boundaries under LopatinskiShapiro type boun
dary 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 Diri
chlet or Neumann problem over a halfspace into a freespace pro
blem. Similarly, for a general boundary problem, here we will ex
tend all functions from MX into all in , and the elliptic operator to an mdelliptic operator on in (sec.3). The boundary problem
will become a 'RiemannHilbert type problem' involving (genuine)
distributions. We get a type of distribution best classified as
'multilayer potentials'. In potential theory one uses a single
or multilayer ansatz, using the same kind of distribution for sol
ving a boundary problem. Here we find that every solution necessa
rily 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 distribut
ion u solving Lu=f for a hypoelliptic 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 assump
tions on L (i.e., hypoellipticity, and that r is noncharacteri
stic 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 'Lop
atinskiShapiro type'. Of course we use the "multilayer 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 Smanifolds should
involve no new ideas, under proper assumptions on a and r .
The multilayer 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. Act
ually, for existence and uniqueness considerations we need two dif
ferent such elliptic systems, one n/2xn/2system, the other nxn .
In sec.7 we turn to the parabolic initialboundary 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/atL is hypoelliptic. For
existence of the semigroup we assume standard results, such as the
HilleYosida theorem. Boundaryhypoellipticity (sec.4) will be
usefull to secure C00solutions instead of distribution solutions.
An efficient use of results such as the HilleYosida theorem
in sec.7, will require investigation of R(X)=(LX.)1 of an ellip
tic operator, as will spectral theory of elliptic PDO's. Thus, in
sec's 89, we look at spectral theory and R(?.) , for elliptic L.
We only consider the cases ain , s2 = compact manifold , n =
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) exte
rior boundary problems, (iv) RiemannHilbert type problems are
fairly evident, but are left for the reader to deploy.
Existence of a Green inverse for operators defined with boun
dary conditions is discussed as auxiliary result (thm.8.3), exten
ding results of sec.6. Mainly we focus on compactness of the resol
vent, under proper assumptions. Results on selfadjointness and dis
sipativity of differential operators (making thm's 7.3, 7.5 appli
cable) are discussed under various assumptions (thm.8.5 for n with
out boundary, thm.9.1 for LYE 1n with boundary). In all cases pro
per deployment of the results is left to functional analysis  in
sofar as the spectral theorem , the HilleYosida theorem, or exi
stence of an orthonormal base of eigenfunctions for selfadjoint
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.61
6<p2, of IV,4, with similar proof, not discussed here.
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
Assertion:(1) The
differentlia_lN
equation
(1.3) Au=f , f e C00(n) ,
is normally solvable, for uE C°°(s2) . 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
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 LSN = LS(N,0),(1,0),0 , the latter as in IV,4, with
(1.5) GA=1P , AP=IQ ,
where P is a projection onto ker A, annihilating some (arbitrarily
chosen) complement of ker A , and Q a projection onto an (arbitra
rily chosen) complement of im A annihilating im A .
Proof. The compact manifold a is a special case of Smanifold stu
died 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
thus write LSm,,p,6 1
and LCm in case of p,=1 , b=0
148 5. Elliptic and parabolic problems
Also, since there is no infinity, mdelliptic means the same
as 'elliptic'  i.e., cdn.(1.2) above. Thus the existence of a
Kparametrix in LSN 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:IY , 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 roxromatrixvalued, while (1.2) is replaced by
(1.6) det 0 , xE U ,=1 .
Then the assertions of thm.1.1 hold again, as follows:
(1) The equation
(1.7) Au=f , fE Y ,
is normally solvable , as u E I  that is, dim ker A <oo
codim im A <oo .
(2) There exists a special Green inverse G , locally an
roxromatrix of pdo's in LC_N , 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 
X(n) still containing all ofC000
(n) ; (ii) the coefficients as of
(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 (mellipticity).
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:XX , 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 a belongs toasN
5.2. The elliptic boundary problem
VC(NIO) .) Finally, instead of (1.2) we ask for
(1.8) x as rl , with some c,rl>O
149
That is, A must be adelliptic 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 Sadmis
sible vector bundles of the same dimension ro, with components in
Hoo, x , 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).
Theorem 1.3. Given an mdelliptic differential operator A:X>Y
with the spaces %, Y of Co sections on two Sadmissible 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=ar , and let A be mdelliptic of order (N,K) . That is,
in the coordinates of an Sadmissible chart UC St, we have (1.1),
(1.9) aa(P)(x) = 0((x)xIPI) , for all P ,
and (with the ro xro matrix norm 1 . 1 , and some c,rl>0)
(1.10) Ia(x, )I=I a asasNThen the differential equation
(1.11) Au=f , fE Y ,
is normally solvable, for uE %  i.e., dim ker A <oo , codim im A
< oo . There exists a special Green inverse GE LS (N,x) such that
(1.5) holds, with projections P,Q as described.
Problems. Consider the paraboloid P:xi=x:2+x32 and the one and
twoshell hyperboloids B+:xI2x:2x32=t1 in &3 . Let A be the Bel
tramiLaplace operator of the Riemannian metric induced in P and
B+ by the Euclidean metric of &3. 1) Show that the natural Sstruc
ture of 23 discussed in IV,1 also induces an Sstructure in B. :An
atlas of admissible charts may be obtained by restricting admissi
ble charts of 1R3 to B. .) With this Sstructure, 1A is mdellip
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 boun
dary problem on compact domains with smooth boundary.
150 5. Elliptic and parabolic problems
Consider a compact subdomain SEC 1n with boundary r=anC &n
where r = U rj is a finite disjoint union of compact smooth n1
dimensional submanifolds of &n . Let A=a(x,D) , Bj=bj(x,D), j=1,..
..M, be differential operators of orders N, Nj, with smooth coef
ficients, defined near rtlf and near r , respectively. Assume com
plex (globally defined) coefficents, for the moment,  i.e.,
A = aa(x)D" , B. = ba(x)Da'(2.1) IaIN J
where aaE C°O(N) , baE C"(Nj) , ttU'CC N , Ft N j, N,N1C in
For given functions fE C0O(c&f) , PjE C°'(r) one seeks to finda 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 2nd order equa
tion 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
5.2. The elliptic boundary problem 151
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 exam
ple 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 nxnmatrix, 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 mdelliptic expression on In :
A=a(x,D) , a={cIN} , where c= cas E tc
(2.5)
1aIsNa N,m2
is mdelliptic 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 mdelliptic
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 Bs(N,m2) %.. Accordingly we tend to think of similar spaces Boo , %00 , also for nonempty 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 LopatinskiShapiro condit
ions  will be introduced below. Then we will prove normal solvabi
lity 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 LopatinskijShapiro 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 quest
ion parallel to hypoellipticity, as in II,thm.4.1. For open SIC In
if uE D' (n) has CuE CO(n) then uE C00(12), by hypoellipticity 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 notdefined. 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 hypoellipticity.
As already mentioned, a boundary problem (2.2), in its ab
stract 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 f0 on s or TjO, j=1,...,M , on F. That is, either
(2.2') Au=O , xE Slid' , Bju=Tj , xE 1, , j=1 , ... ,M .
Or,
(2.2") Au=f , xE Slid' , B ju=0 , 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 suchv can be assigned to every f, such that v=Xf becomesa 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=uv. Then
w solves (2.2') with ypj replaced by pjBjv. Since (2.2') is normal
ly solvable we get a solution w if only cpjBjXf satisfies finally
many linear conditions, translating into finitely many linear
conditions for f,cpj . Clearly also the null space of (2.2) coinci
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=uv
again. Then w satisfies (2.2") with f replaced by fAv. A solution
w exists if fAv 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 thatthe equations Bv=c)translate into a system of differential equa
tions for (vO,...,vN1) , 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 mdelliptic hence has a special
Green inverse G (III,thm.4.2): CG=1P, P a finite dimensional pro
jection 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, biorthogonal to {aj}.
Try to select Rj such that (3j=O near s F. We only must construct
a set {y j } of linear independent functions yJ.E S such that yJ.=0 inB. ={ I x I s,q} , for some sufficiently large rl , and that no linear combination 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 LStheory 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 inde
pendent in X,1=&n\BI : The matrix ((JX aja1dx))=Z is nonsingular.I
For a suitable cutoff 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=gDj(aj,g)=f for x near
szld'. Thus a v solving Cv=f near nLr exists: Set v=w in N(szLf)Moreover, the above construction has given a right inverse of A
154 5. Elliptic and parabolic problems
as required, and cdn. S always holds, under above general assump
tions, 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 condit
ion  the fundamental solution cnlxy,2n (or c2log1xyl) 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 Green0
00inverse G of C let z=Gf', f' an extension of fE C(nLf) to CO(In).
Then u =uzlnLr transforms (2.2) to the following problem:
(2.8) Av=y , xE Snl,r, B,P=1l j=(,jBjz , 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 Inproblem of RiemannHilbert 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 "zeroextension to ln"
by z" , so that z" =z in c&f, z" =0 in In\S2\r. Thus g=f" , v=u .
We have Cvg=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 Cvg=
h (interpreting the derivatives of C as distribution derivatives),
(3.2) Cvg = 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)(xa) (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 RiemannHilbert problem 155
where av denotes the kth 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 uni
que, so that (t,v) are useful coordinates. Then ak simply denotes
the kth 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)cpdxS2
The right hand side may be integrated by parts, for
(3.4) (Cu ,q) = ffcpdx+
jaj+1PjSN1
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)=(Cvg,<p) _IaI+1P +j+ksN1 T
We still may integraterate byy g parts along r using that COO(r)
Accordingly, all tderivatives 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+ksN1 T
where the P1 have smooth coefficients.
Note that (3.6) amounts to
(3.7) h = =Ub k ,T , V PNlkjavu , 6.,T of (3.3)kkehence it gives the desired decomposition of h explicitly.
,6,TRemark 3.2: For a distribution TEE DO(in) of the form T=Yj
with SN I, of the form (3.3) and a finite sum, the distributionsJ,
Sv T E D'(T) are uniquely determined (as long as we define av byJ,
means of the above coordinates (v,t)). Indeed, for any given inte
156 5. Elliptic and parabolic problems
ger ka0, and cpE D(1'), a function P°E D(kn) may be constructed such
that jk on 1' , j=0,1,... . Then clearly
(3.8) ('!'k,(p) (T,q°)
showing that indeed the ,yk , 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))
(3.9) Cu = f" + h , supp h E T .
Thus Au=f in fz, and h coincides with the distribution (3.7).
Indeed, we trivially have Cu =f in the open set fz . Since u,
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,fEC00
(nLr) , the PDE Au=f
is exactly equivalent to (3.9) with h of (3.7). For a boundary pro
blem (2.2) we must solve Au=f, hence (3.9), for given f, where uE
C"(fzlf)' VO' V1' ... VN1 E C"(F) are to be determined.
The right hand side of (3.9) belongs to S'(2n) . Therefore
(3.9), for given y,j , may be solved with the methods of ch.II (or
V,thm.1.1). The mdelliptic operator C admits a Green inverse G .
We get CG=1P 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 + ,jwj , tyjE C" (r)Vi
with X jE C, and a basis wj of ker C. For arbitrarily given f, +yo,.
'VN1 one will not expect such v to vanish outside fall'  i.e.,
v is not a zeroextension 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" "VN1E 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 hypoellipticity 157
other two terms involve the operator GE OpVcN,m2 , of different
iation order N< 0. G sends Hs to Hs+(N,m2),as we know. From II,
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,xy)
g=symb(G) is equal to some function in S(&n), for fixed y and lar
ge lxi (and uniformly so for yE RCL 1n). We have proven:
Proposition 3.5. v of (3.11) is C'(kn\r) . Also, v is equal to a
function in S(In) for sufficiently large lxJ .
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.
The vj will be called multilayer potentials (induced by G) . This
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
Green inverse is natural. The two distributions vo , v, are known
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 CN1 (Rn) . Moreover the Nth derivatives w(a) , JaI=N , 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), respectively2) For 'tpE C°°(r) , and j=O,...,N2 the multilayer potential
vj=GSA is a function in CN2j(Nn). For all derivatives (a) the
restrictions to n and &n\(nu) extend to functions in CO0(nLr) andin C00(&n\ft) , respectively, j=0,...,N1. (That is, all derivatives
of order s N1j 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 compa
ring the particular asymptotic expansions of f" and fi,.
4. Boundary hypoellipticity; 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 notat
ions of sec's 2 and 3 focus on PDE's of the form (3.9), i.e.,
(4.1) Cv=f" +2_0Sj
where vE D'(1n). We will mainly be interested in the case p=N1
158 5. Elliptic and parabolic problems
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 Llloc(&n), as will be assumed henceforth. Define the spaces
(4.2) Z ,=j wE D' (&n) : supp wC nil , wE C" (n) } , Y %szm l loc(&n
)
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 .
Since r is compact N contains some set NE=IExr , IE={jvjsE} , E>O.
Here the neighbourhood NE of r may be regarded as a subset of the
cylinder n'= &xr as well. With a cutoff function x , supp x C IE,
x=1 near 0 , and w=xv , equation (4.1) takes the form
(4.3) 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<v<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 regar
ded 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=oCNj(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 vsE, 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 , i.e.,
(4.5) cO(0,t) # 0 (on all of r) ,
is sufficient for our present discussion. We assume C hypoellip
tic (not necessarily elliptic) apart from (4.5). By assumption f
is C"(nld') . It admits a Taylor expansion in the variable v :
(4.6)
q 7
f(v,r) _ =0(ayf(O,t))j! +Jrq(v,t)V q
rq(v,t) _ (aq+lf(O.t))(vq dx , q=0,1,... , vaO0
For the extended function f" we get an expansion valid in a 2si
ded 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<O .
5.4. Boundary hypoellipticity 159
For N=0,1,2,...,
(4.8) f (v,t) _ oavf(O,t)xj(v) + rq (v,t ) vE T , Bilks
The xj(v) are homogeneous of degree j in v . (4.8) may be
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
(4.9) f (v,t°) _ =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) ).
We have r25_, av(p(O,t° )xj=rq E Cq(St`) by well known pro
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 , pjsn , and allowing certain nonhomogeneous terms
for integers pjkO . Here it is practical to extend in a somewhat
simpler way: For j=1,2,... define the distributions
(4.10) X_j = avc0 = av1b
Observe that then, for k=0,1,...,
aYXi  Xj_k , jE Z , vkX (k+ )IXk+j jZ0
(4.11)
vkX_j= (_1)k _lk i Xk_j , j>k , =0 , jsk
All xj , also for j<0 , are homogeneous distributions of degree J.
We shall say that a distribution T E %., allows an extended
Taylor expansion (mod av) at v=0, (abbreviated "ETE") written as
(4.12) T = 0pX? yj (mod av) at v=0
where pE Z , and yjE DI(T) , j=p,p+1..... , if for every M=0,1,...
there exists NO=NO(M) such that yjE CM, near v=0, as Na
N0. The largest such p will be called the order of the expansion.
Actually we always will require that yjE C°O(T) , as ja0 , while
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 0extensions 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 func
tions yj(t)E C°°(r) such that
(4.14) g='.0 xj®yj (mod av) at v=0
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
CN({vZO})function, for every N , hence is C00({vz0}) , q.e.d.
Proposition 4.2. 1) An extended Taylor expansion may be differen
tiated 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 determi
ned 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 hypoellipticity, as well as prop.3.6
now are settled by proving the following.
Theorem 4.4. Let C of (4.4) be hypoelliptic 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 hypoellipticity 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) CNj(v)avuO = xp+j®CNj(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=uu0. 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(w1u1)=C(uu0u1) has an expansion of order p+2.
By iteration we get u0,u1,..., such that C(uu0...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+j1+ "' + 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(uvq)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
162 5. Elliptic and parabolic problems
ble cutoff'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(uv) has an ETE with all terms vanishing.
This means that C(uv)EC00(c'). By hypoellipticity of C this imp
lies uvE 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=
f"+` j_16v assuming that f,'j belong to suitable Sobolev spaces.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 pro
blem. 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 LopatinskijShapiro 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 multilayer 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
is representable (mod ker C) as a sum v= i of multilayeri
potentials, with certain viE Coo(r) . That is, u is the continuous
extension of v l Sa to Said' , 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 , with certain y,iE C"(Sa) , hencei
(5.1) v = e4_oGbj + 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)
(5.2) ",J=IBkGOI +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.
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,...,N1J
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 modi
fication, systems of pdo's. Fixing some y,i, (2.2') alone is conver
ted 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
164 5. Elliptic and parabolic problems
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 combina
tion of multilayer potentials with respect to the half space &+
with boundary y given by {x,=0} = &n1 . 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
ajx=(v,t )E &n:v>0} ,I'=&n1={x=(v,t )E &n:v=0} , 1E CO (r)
(5.3)
v =
G is a special Green inverse of CE OpVCN,ma , an mdelliptic
differential operator of order N . For our boundary conditions
(2.2) we also need derivatives Dav , IaIsN1 , 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 6family:
Choose any family 6E(v)E= E>0 , with as E0 , weakly
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(5.5)
limE0TE
TE= DaGD1(SE (v) P=DUGDiE Opwc(IaI+k)el m
Clearly 6E0 1V E CD(1n) C S(e), hence (with x=(v,t)=(x,,x°))
(5.6) TE(xI ,x°)= JtfoKeiT(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 Fubinitype in
tegral exchanges, due to 66^E S, p of polynomial growth.
Introducing the nldimensional symbol (with parameter m)
5.5. A system of operators
(5.7)PE
(XI ,x" J9iveix, v be. (v)p(Xi ix" 'V'V
we get TE= PE (x, ,x° ,Dxs )1. We get 6E^ S^ =, as E0, 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
RE=p(x, x° D X, )O E , for fixed x° implying the convergence(5.8) for each fixed in S'(&). Formally we thus expect
(5.9) ikD v(v,t) = Yo (v,t.D,C)1U , PO (v,t. ' ) =
with the inverse Fourier transform ,3' for the 3rd argument.Let us analyze the special form of
Proposition 5.3. (and its derivatives for x, of arbitrary
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 Nj+Ial+k in 1;. Specifically,
e.=(1,...,0)(5.11)
rj(x,U)/cN+1(x,U) , rjE Wcjmje+(Ial+k)e,
where the rj are polynomials in of degree sNjj+Ial+k with coef
ficients in VcO'jm j , and where denotes the principal
part polynomial of (i.e. the homogeneous part of degree N).
Asymptotic convergence mod ICI at 1_oo means that
(5.12) 0( as RZRo (q),
uniformly for xE KCC In , all K , and with all derivatives.
Proof. G is the operator B constructed in II,thm.1.6, up to an ad
ditional term in 0p5(1Q2n). In the notations used there,
b = bj (mod pc.)  where b0=c (large IxI+
IL;I ) 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
166 5. Elliptic and parabolic problems
for large again. At the moment we have x varying over a
bounded set, so, 'large a'large ICI ' . For our present
notation we use g=symb(G) and replace bj by gj in (5.13).
Proposition 5.4. We have (for xE KEE In , ICI large
(5.14) gjE span{iII(c(aj))/c) : 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) : 7,I aj I =j 113 j I =k . Such a term is a symbol in Vcmke
m=(N,m2), e=(1,1). Asymptotic convergence of Igj , in our case,
means that 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 By reordering the expansion we get
(5.15) g =L1,=0Pj/e7+1
(mod ICI) at p0=1 , PjENj_j,Nmz_j
where pj are polynomials in of degree s Njj. Write c=Lj=0cj,
with homogeneous cj of degree j. The mdellipticity 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, °)111 +...+cN(x, °) ISIN}1
cN(x, IF
where the yj are homogeneous of degree 0 . Uniform convergence of
the series 1+y1III+... in (5.17) for large ICI and xE K is evid
ent, using the geometric series +y 1y+y2+... . Again, a study
of the coefficients yj, from expanding yj=(C ...CNshows that
(5.18) c = c Oqk/ck degree gkskNkN
The series converges uniformly for xE K and large ICI with all
(term by term) derivatives. A fortiori we have asymptotic conver
gence mod Ii;I. Similar expansions result for all powers cI . 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
(5.19) P=Y 1 )101_
In other words, the symbol of P=DaGDX is a finite combination of
symbols g (x,U) . Since we know that (5.12) may be differen
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 parti
cular 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 trans
form "I" term by term, we must deal with the distribution integral
(5.20) Nr2_npJ f dxeivxpj(v,T,K, ° )
where pj = rj/cN+1 is a homogeneous rational function of degree
k+IalNj in the variables For fixed the pj are
rational functions of the single variable K . Their poles, in the
complex xplane, are independent of j  just the roots of the
denominators cN+1 i.e., of the polynomial equation
(5.21) 0 .
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 semicircle in the upper or lower halfplane: For v>0 (v<0) the
function eiync=eiVRex.eVImKwill decay as Im K > oo (Im x > 00)
Let the roots of (5.21) be called x1 . Then it is clear that
(5.22)f
K, as v>0,
 mx <0 ResK as v<0.1 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
168 5. Elliptic and parabolic problems
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(ien1)
where is smooth in for all and satisfies
(5.25) n±(T'b° ) = j=Onj'+(T,S°) (mod I ° I ) at °=oo
with
(5.26)III, >0 Resx pj(0't'x' °)
1 1
n7'(t'°) _ Y'Imx <0 Resx pj(0't,x,°)1 1
for TEE K= ten1 , all K , uniformly in t , and for partial deriva
tives of all orders. In particular, the nj'*(t,°) are smooth in
as 1;°340 , and positive homogeneous of degree N+Ial+kj+1
(5.27) Resxlpj(0't'x'V) = JIxx11=E pj)dx
with small a>O .
Proof. Note that the roots of (5.21) are continuous
in 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 Nj+Ial+k we
find that the pj'±(t,t°) are homogeneous of degree Nj+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 represen
ted by an integral over a countour containing all roots in the
given halfplane, where the countour may be locally kept constant.
As for the asymptotic convergence in (5.25), the remainder
Pq = p2gp . =0 ( 1 1 _ R ) as qZq° (R) , and I x I R={ I ° 1, +xz } R/2
hence pq (p2gpjyp = O(fdKltlR) = O(jt°jR+1) . Similarly we may
argue for partial derivatives.
Finally, regarding smoothness of we look at the ex
5.6. LopatinskijShapiro conditions 169
pansion (5.15) (which, in fact, is mod S(R2n), not only mod ICI).
That is, the remainders are O(()) not only 0(1 1_R) , so that
the "a"remainders are arbitrarily smooth. Regarding(pj/cj+1)0,
The same residue calculus may be applied  but remember, the poly
nomial may have real roots, in a bounded set IKI51 . But
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
countour  aC00function. 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. LopatinskiiShapiro 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
(6.1) Bk bk(t,Dx)= k ba(t)Dxa , k=1,...,M , NksN1
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) k=0glk(t,Dt)Vk = W1B1W 1=1,...,M
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 aresmooth in and have asymptotic expansions
(6.3) glk = 1=O(mod I ) at °
k kglkj=(i) ba(t)nJ'+'I sN
Here the nj,t,k'a are given as sums of residues, of the pJ,=pj,k,a
in the upper and lower halfplane, 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
(6.4) glk0 mx1>OResxl(ix)k( Ia sNl ba(t)(x. °)a)/cN(O.t.x. °),
simplifying to
(6.5) fy dx(ix) kbl(t,(x ixi °
+
170 5. Elliptic and parabolic problems
with a positively oriented countour y+=Y+ in Im x >0 , sur
rounding all poles there. Similarly we get
(6.6) glkj =fy+dx6(ix)kIIalsN1 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 sNjj+Ial+k. Thus the polynomials in the nume
rator of (6.6) are of degree s Njj+k+N1 . Arguing as for prop.5.5
we find that the glkj are (finite) sums of homogeneous functions
with largest degree Nj+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 trans
lates into a system of pseudodifferential equations
(6.7) =OGlkk = p1B1w , 1=1,...,M
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
(6.8) glk(t, °) _ 7j Oglkj(mod 1) at 1;°=oo
where glkj are homogeneous of degree N+N1+1j+k in ° , and
smooth in t . Moreover, the highest order terms are given by
(6.9) glk0(':' ° ) = fy+°+x n)
+ N
Here bl, 1(t, °)= 1bla(T)l;a is the principal part polynomial
N
of '1( ) 1 °) In (6.9) we used the notation n=(1,0,..,0)
=e,= interior unit normal. is the above countour.
For existence of a solution of (5.2') we must find density func
tions V0 .. VN1 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,...,N1 , and interior
by exterior derivatives. We get the following:
Proposition 6.2. The system (5.2)1 is equivalent to a system of
pseudodifferential equations of the form
(6.10) `k=OHlkPk = D1 w , 1=0,...,N1
5.6. LopatinskijShapiro conditions 171
withtpdo's Hlk on I' . Locally, in the boundary coordinates of sec.
6, the symbol hlk of Hlk allows a homogeneous asymptotic expansion
(6.11) hlk(i, ) _ Ylj 0 hlkj (mod °) at °=o
where hlkj is homogeneous of degree 1+1Nj+k. Specifically,
(6.12) (i)kf7 dK k,1=0,..,N1.
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 LS: (The LopatinskiShapiro conditions) (i) In the
boundary coordinates of sec.3, for every &n1 , the equation
(6.13) 0 , KE C
admits exactly k=N/2 roots 1=1,...,N/2, in the upper
halfplane Im K >0 , counting multiplicities.
(ii) For every 0A°E In1 the polynomials
(6.14) b1, 1(t,l;°+Kn) , 1=1....N/2N
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 LopatinskiShapiri conditions, is normally solvable.
Proof. The point is that, under cdn.LS, (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.LS, the polynomial c+(K) is
of (precise) degree M=N/2. By the Euclidean algorithm we may write
(6.15) (31(K)=b1,N1(t,°+Kn) = a1(K)cN(K)+pl(x) degree pl<M
with uniquely determined polynomials pl(K) of degree <M . cdn.(ii)
of LS means that the M polynomials p1,...,pM are linearly inde
pendent. Looking at (6.9) we find that
172 5. Elliptic and parabolic problems
(6.16) glkO  fy+dK(1K)kpl(K)/cN
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,..,M1 invertible.
Focus on the matrix ((glkO))1=I,..,M,k=O,..1M1=Z . To show that
Z is invertible we must show that
(6.18) Y=((JY dK KJ+l/cN(O,t
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 , such that
Jy dK 6(K)?(K)/cN(K) =0 for every polynomial X(x) of degree <M.
However, 9(x)/cN(x) must have poles inside y+, since
ahas poles
N
of total order M there, while s is only of degree <M , hence can
not cancel all poles. If x0 is a pole of O/cN
then set T,=II(Kxl),
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 MxMsection
(6.19) ((g for R=0,1,..,M,1k0 1=1,..,M,k=R,...,R+M1 ,
is nonsingular , for all T 1;° 340. Also, note that (b) followsfrom a similar argument: just note that c(K)=Zmx <0(Kxl) 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) Ik=0glkOzk 0 1=1,..,M and "k=0hlkOzk 0 1=0,..,M1
Repeating the above argument, this means existence of a polynomial
X(K) of degree <N (with coefficients (i)kzk ) such that
(6.21) J dKX` 0=0 , j=0,..,M1 , and J dx KJ=O , j=O,..,M1y+ 0N y cN
Thus it follows that X/cN is regular inside both curves y+ . Since
a nonzero polynomial of degree <N cannot cancel all the poles of
5.6. LopatinskijShapiro conditions 173
1/cN it follows that %mo, i.e., z0=...=zN1=0 . Note we only used
half of the matrix hlkO'and have shown that the square matrix
(6.22) (g1kO " ..'gMkO'hOkO " ..'hM1k,O)k=o,..,N1
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 ellipti
city 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(I0)w ,
with (10) 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,...,N1 .
lAlso, multiply the 1th equation (6.7) by AN1N, and the 1th
equation of (6.10) by AN11. 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 or
der 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 cplBlw satis
fy 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 LopatinskiShapiro condi
tions working for an elliptic system of R equations in R unknowns,
174 5. Elliptic and parabolic problems
under proper boundary conditions, just as (5.2) . Then generalize
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 LS in a form using matrices of
complex integrals. 2) Same as (1) for an even order PDE on a comp
pact 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 zeroextensions (=0 in a, this time).
5) Same as (1), for a RiemannHilbert 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 clo
sure 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 $(lnl)  or only S'(ln1) ? . Also, of course,
we then must require if , f E S,(ln) .
7. Hypoellipticity, 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] ). onespeaks of an evolutionary problem. The abstract differential equa
tion (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):XX 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
initialvalue 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 conven
tional rules hold, it is confirmed at once by formal differentia
tion that u(t) of (7.2) satisfies (7.1).
The family {U(T,t) : t2T} is called the (family of) evolu
tion 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 U(T,t)=eL(tT). The collection
{U(t)=U(O,t)=eLt : tao} , then has the properties of a semigroup:
(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(tT) .
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 16: 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) , we define
(7.4)Y = dom L = {uE X: Bju=O , j=1,...,M} ,
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 condit
ions are needed. First we assume that at A(t) (as a PDE in (t,x))
is hypoelliptic. 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 hypoellipticity or by cor.4.3, because it proves a distribut
176 5. Elliptic and parabolic problems
ion solution of the PDE atuA(t)u=f (under proper boundary cdn's.
There are well developed abstract tools available, for this
2pivot approach: For example we have a detailed abstract semi
group theory (cf. [HP] , [ Yo 1 ] , [ Ka 1 ] , [DS,]). Also, an abstracttheory of evolution equations [Ka5]. For simplicity we stay with
semigroups  i.e., L(t) of (7.1) is independent of t.
Definition 7.1. A strongly continuous semigroup of a Hilbert or
Banach space I is defined as a 1parameter family {U(t):tZO} of
U(t)E L(I) which is strongly continuous in t (i.e., limttoU(t)u=
U(to)u in I , for every uE I , toZO), and satisfies (7.4).
The HilleYosidaPhillips theorem [DS1],p.624, states that a semi
group 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
(7.5)Au=dU(t)u/dt(O) = limt..0,t>0((U(t)U(0))/t) ,
for uE dom A = {all WE X for which the limit exists} .
If AE L(I) is a bounded operator, then we have
(7.6) U(t) = Lj=O(At)3/j! = exp(At)
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 oc
curring as infinitesimal generators of a strongly continuous U(t):
Theorem 7.2. (HilleYosidaPhillips) The class of infinitesimal
generators of C0semigroups coincides with the class of closed ope
rators A such that (i) the resolvent R(X)=(AX)l exists for a
half line {X>Xo}, (ii) for a constant M>O independent of we get
(7.7) IIR(X)JIJ s M(XXo )j , a,>ko , j = 1,2,... .
Then we have
(7.8) U(t) = limj>Oo(1tA)j = t>0
and, vice versa,
(7.9) R()) f dtektU(t)dt , Re X >ko0
For TE dom A we have u(t)=U(t)cpE C1([0,00),I) .
5.7. The classical parabolic problem 177
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 halfplane.
Looking at (7.1) we will focus on the following two cases:
Case (F) : L=A=a(x,D) with an mdelliptic 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 LopatinskiShapirotype 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 containsC000
(n), dense in H, hence is
dense). Thus the closure L** exists. We may look at the problem
(7.1') u(t)E CI([0,oo),H) , au/at = L**u , ta0 , u(0) = tp E Y .
Similarly, in case (F) , we introduce the operator L by
(7.10) dom L = CW(a) , 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 (7.1') with this L**
Now, suppose the closed operator L** of H satisfies the con
ditions of thm.7.2. Then indeed a solution of (7.1') is given in
the form u(t) = U(t)q , U(t) = exp(L**t) . Indeed, (7.3) implies
(7.11) au/at(t) = limh,0,h>0 U h)1 u(t) = L**u
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) atu = L
**u+f , t20 , 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/atAu=f in (0,oo)x11 . Indeed,
(7.13) fu(app/atA"V)dxdt= f (atuL**u)4dxdt=(f,tp) , y,E C0((0,00)xst).
178 5. Elliptic and parabolic problems
Thus, if fE C'([O,oo)x12) and atA is hypoelliptic, get u(t)=u(t,.),
where u(t,x)EC00
((0,o)xn): u is a classical solution of (7.1).
In fact, we can put to work our boundary hypoellipticity
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<O , satisfies
(7.14) (at A)v=6(t)0 u(0)+f = 6(t)O cp + f ,
in the sense of distributions on $x12 . 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
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
hypoelliptic. 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 solut
ion 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 noncompact. Of course, the condition at x= that
u(t)EE C1([O,00),H) , H=L2(St) , insures uniqueness as well.
Now let us look at case (B). If the semigroup 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
hypoelliptic, and (ii) A is noncharacteristic along r :
Theorem 7.5. In case (B), assume that L** is infinitesimal genera
tor of a C°semigroup. Assume (i) and (ii) above. Then (1.1) ad
mits 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 16, give control on cdn.(7.6) for L** above: Clearly (i),
(ii) of thm.7.2 only involve (L**X)I
= R(?) , for L**. Elliptic
theory will supply a Fredholm inverse, even a (special) Green in
verse for L? : In case (F) we need mdellipticity of AX, for
5.8. Spectral and semigroup theory 179
case (B) we need A elliptic and cdn's LS for Bj (cf. thm.8.3).
Still, looking at a variety of examples one finds that
mdellipticity of A% alone in case (F) or ellipticity and cdn.
LS for case (B) though it might be helpful, is not enough.
The insight of the present section is the fact that (i) hypo
ellipticity of C=at A and (ii) the noncharacteristic 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** 1vent (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 semigroup U(t)=exp(L**t) as a para
bolic 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 selfadjoint 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 mdelliptic of order
(2,0), for a.>0. This requires ajkE 1)c0 , i.e.,
(7.16) ajk(a)(x) = 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 selfadjoint (admits a
spectral decomposition) and semibounded 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 dif
ferential 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 com
pact 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 L2Sobolev 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. uAu as uE dom Lo =C00(a) )For aCC &n with smooth r we assume L the closure (in H) of an ope
rator of the form (7.4) (that operator is called L' here). Equat
ion 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
LopatinskiShapiro, respectively. For a noncompact St (with conical
ends) we assume A mdelliptic (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:(LX)lE L(H)J. Control of Sp(L) (and of spec
tral 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 com
pact 2) not only do we have A mdelliptic, but we even have AX
mdelliptic, for every XE M . Thus a Green inverse Gk of AX=AX
exists for every XE T . Moreover, Gk may be chosen such that
(8.1) (AX)GX=1PkerA.,GX(AX)=lPkerA.*
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(LX0)=0 , im(LX0) 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) = (LX)1 is a compact operator of H , for every XE Res(L)
Proof. First we note that ker(LXo)=ker(AXo)={u6S'( °°:(A%o )u=0}.
Indeed let LX=LX. 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 hypoelliptic we conclude that uE Coo. Moreover, from III
thm.4.1 we know that uE S, since A% is mdelliptic. Vice versa,
every uE S with AXu=O belongs to ker LX. By our assumption it fol
5.8. Spectral and semigroup theory 181
lows that ker Axo={0}. Hence PkerA =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 (LXo )I is bounded.Its domain is H (since it is closed). Moreover, R(Xo)=(LX
)1=Gxo
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 ana
lysis (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 el
liptic and (A,B1,...,BM) satisfy cdn. LS, 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)). The term Green
inverse should be redefined for operators acting on a bounded
domain of &n : We will take this in the sense of the well known
generalized Greens function, defined as integral kernel of a spe
cial Fredholm inverse of an operator of the form (7.4).
Theorem 8.3. Under the general assumptions of sec.2 let A be ellip
tic. Assume that (A,BI,...,BM) satisfies cdn.LS. Then there
exists an integral operator G. : C°O(s LF) ' dom L' ,
(8.3) Gxu(x) = fa
gk(x,y)u(y)dy , u(= C'(fl )
such that
(8.4) GXLXu=uPXu , uE dom L1, LkGkv=vQ%v , vE C00(slld')
where P., QX are projections of finite rank onto ker L' and a com
plement 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 CX, C as in (2.5), as in sec.3. Calling that ker
nel h (x,y), for a moment, we have gXhXEC00(S1xSa). Moreover,
GX
: C (Slid') dom L' extends to a compact operator H+H.
Proof. We may assume X=0, since A and AX=AX have the same princi
pal part, hence AX is elliptic ((AX,Bj) is LS) if and only if A
((A,B.)) is. Given a uE dom L' solving L'u=f  i.e., a solution003
uE C (Said') of (2.2"). Let C be the mdelliptic expression of sec.3
182 5. Elliptic and parabolic problems
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
VN/2= " 'VN1=0 , then solve the system (with interior derivatives)
(8.5) `j/U1 BkGO = Bkf" k=1,...,M=N/2j
With the procedure of section 6  using the ipdo's A defined near
(6.23) we set Vj=A iwj , and multiply the kth equation (8.5) by
N1NA 1
. One obtains an elliptic system of order 0 for the wj
N1N(8.6) Hw = (A B1Gf")1=1,...,N/2
N1N1With 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
u = Gf" + /21 GSA , xE n , where(8.7) 4=0 Vi
N 1N'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 : Let xE a. For fE C0(n),i.e
f=0 near r, get f " E C00(&n) , a'Gf" _ JOa g(x,y)f (y)dy, with the
distribution kernel g of G. The integral exists as Lebesgue (or im
proper Riemann) integral. a g is C" except at x=y, where it has a
singularity of order NnIal. The A3 may be written as jth order
differential operators with 0order yido's as coefficients  for
N1Nexample A=A1P1, Ps of IV,(5.3). Hence Kk1A 1B1Gf =27kaaaGf
with 0order ydo's Jka , and a sum over laIsN1 . For xE T , yE
supp f the kernels ax (x,y) are C00(T) . Hence we get
k N1N1(8.8) A Kk1A B1Gf"(x) = Jvk1(x,Y)f(Y)dy I vk1E C (tuts)
Combine this with
f(8.9) GO (x) =(1)J aj
r vYg(x,Y)W.(Y)dSY
to conclude that the second term in (8.7) is as described.
5.8. Spectral and semigroup theory 183
An operator Ku(x) f k(x,y)u(y)dy withk(x,y)=O(IxyIEn)
is bounded as a map L2 (St) L2 (I'), whenever E>2, by a variant ofSchur's lemma (We leave the little exercise to the reader). Conclu
NINsion: 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/21<N1, since our sum
in (8.7) only extends to these cases. For JalsN/2 look at aaGSkk=
jl.dSyaXavg(x,y)V(y) where the kernel also has a singularity of or
der NnkIalaln. These operators are of the form adjoint to tho
se L2 (SZ) L2 (I') used earlier. Thus they are bounded as well, bythe same Schurtype 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 LC1).
The kernel of G has the corresponding properties, hence the two li
nes 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<oo, by standard elliptic theory applied to H. We
still get (8.7) if only the right hand side of (8.6) is 1 X. This
amounts to f 1 vj, j=1,...,R, with v1E COO(c&f). With the orthogo
nal projection Q in H onto span{vj} introduce F=Z(1Q), Z the ope
rator of (8.7). Confirm that L'F=1Q, while F of course still is
compact in H and has a kernel just like Z. In particular F is a
right Fredholm inverse of V. But we saw in sec.7 that L':YX is
Fredholm. The space ker L' consists of C°° functions. We get a left
Fredholm inverse W of L' such that WL'=1PkerL" Then get WL'F=W
WQ=FFPkerL " hence F is an Hbounded Fredholm inverse of V. The
special Fredholm inverse satisfying (8.4) differs from F only by
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(1Px)uUI=IIGxLx(1Px)uIIs cIIL'(1P,)uII
uE dom L' . Taking closure it follows that
(8.10) IILull a cliull , for all u 1 ker L' , uE dom L
184 5. Elliptic and parabolic problems
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 isclosed hence im Lko=R, R(X)=Lk1 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 expres
sed by direct reference to the differential expression A : 'Her
mitian' 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 expres
sion A will be called selfadjoint (dissipative) , respectively.
For a boundary problem L' we also need the above conditions
for the expression A . But, in addition, a postulate on the boun
dary conditions results: Certain selfadjoint 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 selfadjoint (dissipative) and the boundary conditions are
selfadjoint (dissipative) with respect to A .
For the general case of an LSsystem (A,B the discussion
with selfadjoint (dissipative) expression A the discussion of
general selfadjoint (dissipative) boundary conditions is quite
complicated, and will not be attempted. A simple example will be
the case of the Laplace operator L=0 : Selfadjoint LSconditions
with real coefficients all will be of the form
(8.13) bo avu+b, u=0 , xE T ,
with realvalued bo, b, , bo 2 + b 2 =1We 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
5.8. Spectral and semigroup theory 185
cal ends. Let A be mdelliptic of order (N,mz) with N>0 , m2a0 .
Assertion:1) If A is selfadjoint then L is not only hermitian but
even a selfadjoint operator of R=L2(f): It admits a unique ortho
gonal 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 eigen
functions {cpj}j=1,2,... , with (pE S(c) .
2) If A is dissipative then L is infinitesimal generator of
a semigroup 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 solu
tion u with uE H . But we know that for mdelliptic A+X every
such u belongs to S . Note that AA , for X=tEi (?.=E) indeed is
mdelliptic for small s>0 due to m:a0 , and since we get
real (or Re for selfadjoint (dissipative) A .
On the other hand, we clearly get SC dom L , hence (L+1,,)u=O . Thus
(8.15)0 = Im((L±Ei)u,u) = e1IuJI2 u = 0 , for A selfadjoint,
0 = Re((La)u,u) sEIIu1I2 u = 0 , for A dissipative.
This argument shows that indeed im(LtEi) is dense in H for A self
adjoint (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 selfadjointness
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) _ (LE) E=L(H) , and, IIR(E)IIsE1 . This implies
(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
latter for m2>0 only  from thm.8.1 again, with Xo=a . Q.E.D.
186 5. Elliptic and parabolic problems
9. Selfadiointness 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 selfad
joint 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 , consider the sesquilinear form
(9.1) Q(u,v) = f dx(AuvuAv)dx , u,vE C"O(r&f)
In coordinates (v,t) (sec.2) for u,vE CO'(nUP) let uA=(aju)1=0..N1
v°=(ajv)j=O,...,N1' 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 NxNmatrix Q=((gjk(t,at)))j,k=O...N1 of differential
expressions Qjk gjk(t,at) in t . Here Qjk 0 for j+kaN, and Qjk is
of order N1jk for j+ksN1. Specifically the 'antidiagonal
terms' Qjk=gjk(t) : j+k=N1 , 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) Q(u,v) , u,v E CO0(c f) , hence
(9.3) (Qu° vA )I. = (u° QvA for all uA vA E C°°(r) .
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 bounda
ry 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. Selfadjointness for boundary problems 187
Clearly (9.5) may be described by 'adjoint boundary conditions'
(9.6) Sjf° = 0 , 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. LS .
Let the differential expression Q of (9.2) be of order 0 (i.e.,
an invertible NxNmatrix ofCOC
functions on r) , and let the
boundary operators Bj only contain normal derivatives. Assume
L of sec.8 hermitian. Then L is selfadjoint  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
(9.7) (f,(L'z)u)=O , for fixed zE C , all uE dom L' .
Selfadjointness 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 (Az)f=0 , xE 1t , in the sense of distribu
tions. Since A is hypoelliptic conclude fE C00(f2). Let f and C be
as in sec.3. We get (Cz)f = h , supp hC r . The distribution h
must be of the form h=f=OS , y, jE D' (r) , S of (3.3).j j
In fact, we claim that R=N1. 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(Az)(vNu)=0 for all uECOO(fLf). In turn, this yields
(9.8) (f,(Cz)(vN(P)) = f of (CZ) (vNcp)dx=O for all cpE D'(&n),
with C and the "zeroextension" f of f (=0 outside f&f) defined
as in sec.3. In other words, we get vN(Cz)f=vNh=O in D1 (,n)A calculation shows that (,N6 j,,P) =('4', av (vN(P) r) r =0 as jsNIbut =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
188 5. Elliptic and parabolic problems
(9.9) j=p ,(p) _ (Qfg°)r , q ED'J
as a careful comparison of (3.4)  (3.7) with the above shows.
We were assuming that the matrix Q contains no differentia
tions 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,...,N1 also for our above f E H : We get
(9.10) (Cz)f h = =O6`J
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,...,N1 such that, locally, near r , in the coordinates (v,s),
jED'(r)( 9 . 1 1 ) ( cz)(f"j
3 = 0j=a
Clearly these fj are uniquely determined. A close examination
shows that then f°=(fj)j=0,...,N1 , may be substituted in (9.9),
above, under our present assumptions. In other words, we get
(9.12) ((Cz)f,(p) = (Qf°,q ) r , (pED'(1Rn) .
Introducing some complex conjugates we conclude that
(9.13) 0 = ((Cz)f ,(p)=(f,(Az)gPIcj) (Qf° q>E D(1n)
Or, in other words, equation (9.8) reduces to
(9.14) (Qf ,u°)=0 for all u° with bjTu° = 0
Or, recalling selfadjointness of the boundary conditions, one
concludes that we must have the boundary conditions Bjf=O satis
fied for f in the following generalized form: We have
(9.15) Bjf = bjTf° = 0 , j=1,...,M=N/2 .
We also have (Az)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=njy, must satisfy a homogeneous elliptic system
of Wo's. Since ellipticity implies hypoellipticity, the Vj are
C00(ct) after all. Hence also the fj are C" , and f E C" (c&F) follows. In other words we get fE dom L'. Then, however, we must have
(9.16) 0=Im(f,(Lz)f) = (Im z)J1f112 f=0, if In z #0. Q.E.D.
5.10. C*algebras 189
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 def
ine a C*subalgebra (called As) of L(HS). Commutators in Optpc0
are in Opyice , 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 opera
tors 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) subalg
ebra 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 A °= A K(Rs s /
s) is a commutative C*algebra.
Let us focus on A, = AO. . By the GelfandNaimark 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 ano
ther 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 asso
ciated 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
190 5. Elliptic and parabolic problems
symbol aA(m) , defined on It , does never vanish.
As in [ C, ] , [C: ] we call II the symbol space of the algebra AOAsking about the nature of I and the homomorphism E:AC(D1) we susp
ect a relation between 'algebra symbol' aA and '1pdosymbol' 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 tpdotheory ([CHe1], thm.36 the
algebra C is called T there  (cf. also the algebra called 4s in
[Ci],p.135/136,pbm's 14). Clearly the 4dosymbols 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 idosymbols (10.2) of the generators (10.1) of C
extend to continuous functions over the compact space
(10.3) BnxBn = n} .
Moreover, by the StoneWeierstrass theorem, the si(x),sj( ) genera
to 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 pdosymbols:
(10.5) as,(M)=si(x), as BnxBn ,
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 pdosymbol
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)
by stating that AE C°C is adelliptic (of order 0). By II,
thm.1.6 this is necessary and sufficient for existence of a Kpara
metrix (hence also a Greeninverse) in Op4co .
5.10. C*algebras 191
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
products of sj(x) and The sj(M) and sj(D) do not all com
mute, but their commutators are in OpVce , hence their pdosym
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 existen
ce 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 mdelliptic of order 0 (cf. [Ci],p.149).]
Let us now state a result for the algebra A. The symbol class
VCo forms a subalgebra of CB(RnxIn), the set of all bounded conti
nuous functions on Rnx1n. The closure (i,co)c (under supnorm of
CB(RnxIn is a * n n(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 spa
ce of all maximal ideals mxo o= for given
(x°,1;°)E Rnx1n. Or, P is a compactification of Rnxln. It is known
that a 11corresponce between compactifications of R2nand C*sub
algebras of CB(22n) containing 1 and C0(12n) exists: Each compacti
fication defines a subalgebra  the algebra of its continuous func
tions. Each subalgebra defines a compactification  its maximal
ideal space. This relation is order preserving. The largest  cal
led StoneCech compactification  belongs to CB(Rn) itself. The
smallest is the 1point compactification  it belongs to I+CO(12n).
Our P will be 'larger than' EnxEn , since pCo contains the
jgenerators s(x) , sj() of C(EnxEn). The map P>EnxEn is given as
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 . (cf.II,3).
Moreover, for A=a(x,D)E Op14)Co the symbol aA equals the restriction
to I of the continuous extension of a(x,l;) to P , for all sE P2
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 Op1ce 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.
192 5. Elliptic and parabolic problems
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 , E 1n , where Ixjl+I jlo0 . Either we have
IxJI or or both. We may assume
I 3Is11 are mutually disjoint. Pick pE with
II(vIIL2=1, and let uj=c)j° , u=cp` , go, "= 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
(10.9)IlbAujll = IIbJF ti IIbfol
First consider the case where Ix3I remains bounded. In the
compactification Enx)gn of It2n will converge to
where I° I =oo , 1x0 I <00 . We have the estimates s
eclxx°I+clx°xil+Ej
where the second and third term at right tend to zero, as jo0
For e>0 pick a ball Ixx°I< a/c=ry and conclude that
(10.10) aEE j , as Ixx° Is rj , Ej0 as j*oo
Accordingly, choosing bE CO({Ixx°Isi}) , we get lla(x,gj)bullz
(aE)Ilbujll  Ej . Next, to control "u" , 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 (aE)llbujll  Ej , Ej 0 , as j.00 .
Here Ilbu jll=llbull#0. As j co get IIA+KIIzaE . This for all E>0, hencefor E=0. With "inf" over K get (10.8) so far only if xj is bounded
In the leftover case Ixiloo we follow the same argument, but
5.10. C*algebras 193
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 , as 0 elsewhere. We get u(x)=2S1x x , uj=eix ju.
Now we have
al s
sc/(xj) I xxj I + E j ,
hence (10.10) now holds with I xx° I srl replaced by I xxj I srl(xj)We now will pick b=b. depending on j , according to b.=1 in
Ixxjls(xj) bjE C0"0({Ixx3l<Tl(x3)}) , IbjIs1 , insuring that
(10.13) c(xj)1/2 s Ilbjujll s C(xj) 1/2 , j=1,2.... .
A choice is bj(x)= b((xxj)/(xj)), bE Co(IxI<q), b=1, IxIs1/2
It is fairly evident that now it suffices to show that
(10.14) Ilbj(x)f O(Ej(xj)1/2)
and that
(10.15) II[bj,A]ujIl = o(Ej(xj)1/2) , Ej = 0(1) , as joo .
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 KRepeating the arguments of (10.12) one gets
IIAvjllz(aE)IlvjllEj,
hence II(A+K)vjIIZ (aE)IlvjllEj , again proving (10.8).
Let us offer some details: For (10.14) look at
1f
J
1
J1.1 o(.) as Ixx31si1(x3) , jioo1
implying (10.14). For (10.15): We get [bj,A] Z (bj,a)(x,D) with
the Poisson bracket (bj,a)=bjlajx
bjlxa,, = P(x,l;) .
Clearly IIP(x,D)ujll=llbjlx"
plral,(x,l;3+rj)11=o(Ej(l;j)) as desired.I
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(xy)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 ob
serve that, for cpE C0(1), we have (v.,cp) = 0 for large j , since
supp v C{ I xxj I s i(xj) } where 11<1 may be assumed so that I x I a
I xj I ri(xi) oo , as jo . since Ilv Its C and Co is dense in H weget I (v j,(P) I s I (v j,fl
I+ CIIroVII <a with suitable y,E Co . Q.E.D.
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 trlox with the canonical injectionWe 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 mdelliptic
(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 haveS
1 Ails = A + s1 [ A, s]Fredholm too, for all sE &2 since s1[A, s] E O(1)C K(H) .
Here Ill = ns (x, D) = (x) $2 (D) as in 111,(3.1). Recall thats:Hs H is an isometry between Hs and H. The above means that not
only is A:HH Fredholm but also A:Hss
. For clarity let us deno
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) = (ns(x,D)u,ns(x,D)v)Lz , uE Hs , vE Hs ,
not with respect to the usual Hilbert space pairing (u,v)S . Thus
we have AS:Hs Hs , in fact, As=a(Mr,D) with a symbol indepen
dent of s . This explains why dim ker AS increases with increasing
sj.) The fact that ker* ASC Hs, dim ker A. =const then implies that
ker As C S , and ker As C S as well. Both spaces are independent
5.10. C*algebras 195
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=1Pker A*'BA1Pker A ' also
defines a Green inverse of A. Thus, if A:H H is Fredholm, it fol
lows that a=symb(A)E 1pco is mdelliptic, 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 inA 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
the set "ola1 ) . But a *homomorphism u:XY between C*algebras
always defines an isometry u°:(X/ker u)*Y (cf.[Ci],p.323,thm.7.17)
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+(1X) = Z has cZ=Xt * +(1X)#0 onA 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 11:z 1/rI . Usingj J j
(10.8) we get an inverse mod K(H) , called Wj with JlW111s 2, and
71a Fredholm inverse Vj with JIV1Ils 1 . This implies that Z is Fred
holm as well (cf. [C,], corollary on p.267).
But if A were compact, so is XA*A compact. Hence 1X then is
a Fredholm operator. The latter is impossible: Then we get a Fred
holm inverse BE A and CBz(1X)=1 on all of at =*OrB(po )(1x(po ))
=1 , 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
s, using the isometry 5:H5> H . Also,
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 vxvmatrix of symbols k=((kjl)), kjlE 1Ce, for fixed t.
In essence 'hyperbolic' means that either the matrix is
selfadjoint, or at least has real eigenvalues, both mod loco .
In sec.1 we discuss a symmetric hyperbolic first order syst
em 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 LZSobolev spaces of ch.III.
In sec.3 we will look at properties of the evolution opera
tor 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 hyper
bolic equation, proving existence of a characteristic flow. Actual
ly 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 Nth order (systems or) equa
tions will be reduced to the case of ist order systems, as alrea
dy 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
6.1. Systems of PDE's 197
single linear first order PDE of the form au/at=Lu , L=2 bjaxj +p ,
with realvalued bj(x) , essentially is equivalent to an ODE along
a set of curves in (x,t)space. The curves were called the charac
teristic 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 xvariable  cf.[CH],II, ch.V, for example. We
regard it as a principal merit of yidocalculus that it will resto
re 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,...,nJ
is a first order vxvmatrix 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+a02aJ
jxj
Clearly it also has bounded coefficients.
We regard (1.1) as an ODE in t with dependent variable u ta
king 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)
198 6. Hyperbolic first order systems
By V,thm.7.2 the operator L will generate a strongly conti
nuous group {E(t) = eLt = eiKt :t E 1e} such that
(1.5) dom L = {u E EH : d/dt(E(t)u)lt=0 exists in H } ,
and that u(t) = E(t)u , for u E dom L , satisfies
(1.6) u(t)E C1(R) , du/dt  Lu = 0 , t E le , u(O) = u ,
provided that we can verify the assumptions for all real IXIay .
The tderivative 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) , tE & , (Eu)(0)=p
An integration then solves (1.1) : For f(t)E C(I,H) , TE H, we get
t(1.9) u(t) = E(t)T + E(tt)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 L2solution 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) (Xy)(u,u) s Re((L0+%)u,u) , u E dom L0
Taking closure we get
6.1. Systems of PDE's 199
(1.12) (kY) IIull2 s IIull II (L+X)ull , u E dom L
which implies that (L+X) is one to one and has closed range and
(1.13) II(L+? )ull 2 IXY I IIull , u E dom L , 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,
we get exactly the same properties , and the estimate (1.13) also
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 11 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)1IISIY .
This implies the
assumptions of the HilleYosida 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.,
(1.16) fE H , (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
be constructed such that fk + f , Lfk  g , (in H) , as k ' o .
Introduce the convolution operator JE , E>0 , by setting
(1.17) JEU(x) = JwPE(xY)u(Y)dy , gE(x) = E nT((xY)/E) ,
with some cp E C"(&n) , supp p C {Ixls 1} , q);0 , jpdx = 1 (P even.
It is clear that fE=JEf+f (in H) as E>0 , for every fE H . Indeed,
for continuous f with compact support one finds
IfEfl(x)=lfroE(xY)(f(x)f(Y))dylSMax{If(x)f(Y)ItlxYIsE}+0, E+0,
while supp (ff) stays within a fixed compact set, implying that
f,)f in H . On the other hand, JE=V£1JlVE with the unitary (dila
tion) operator VEU(x)=En/2u(Ex) , shows that IIJEII=c is independent
of E . Also C0(In) is dense in H
For fEH satisfying (1.16) let hFVFfE=tPEJEf, with VF=cPE^^(Ex), as in (1.17). Then h. )f in H holds as well. Also, hEE C0
*
200 6. Hyperbolic first order systems
for every E>O . To prove lemma 2.1 we must show that also LhF0g .
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 tendsJ
to zero uniformly on &n. Hence [L,'tp2]fE*0, and we must show LfE+O.
Let us write a3(x) = ((aJ (x) )) and introduce the vector (P,k
(y)kl E,x
((PE,x(y) )j=l,...,m = (bkj(PE(Yx)) For the kth component
of LfE we get (summing over all double indices)
(LfE)k(x) _ J(akl(x)axj+a01(x))rPE(Yx)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 esta
blished 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) areO(En1)
, 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, remo
ving the derivative from qE(xy) implies that Tf+0 in H , as O.
Combining the two facts we conclude that Tf0 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 sys
tem of pseudodifferential equations over &n , of the general form
(2.1) au/at + iK(t)u = f(t) , t1 5 t 5 t2 , u(0) = qp .
For convenience we assume a compact interval 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 val
ues in a space HSO CV of all u=(u1,...,uv) , ujE Hs (of III,(3.1))
For simplicity we write Hs instead of HSO Cv . We assume K(t)E
Opipce , e=(1,1), introducing the class VcV of vxvmatrixvalued
symbols with entries in ipcm. By III,thm.3.1 get K(t)E L(Ht'Hte)
for all tE 22. In general we do not get KE L(HS) but may interpre
te 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) lal E C(I,CB(&2' ,iv)) , a, PE IIn
Here 1v denotes the algebra of complex vxvmatrices,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 nmultiindices 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) andkk*
E C(I,i,cO) using the Frechet topology of tpcm .
Clearly (2.4) implies that the skewsymmetric part 1/2(K(t)
K*(t)) of K(t) is of order zero hence bounded in every HS . This
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. Under the assumptions (2.3), (2.4) the Cauchy problem
(2.1) admits a unique solution u E C(I,Hs) f1C1(I,Hse) , for each
cp E HS , and f E C(I,HS) , where s E II2 is arbitrary.
Remark. As a 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 , atu are continuous in every HS
202 6. Hyperbolic first order systems
Proof. The key of the proof is the socalled 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
ArzelaAscoli theorem, will provide the desired solution of (2.1).
As in section 1 we choose a regularizer (or mollifier)
(2.5) 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) = EnV(x/e) . Observe that yr E S . We introduce
(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 hermi
tian symmetric in the Hilbert space H = H0 .
b) The family KE(t) defines a bounded function [0,1] x I
L(Hs ,Hse) which is norm continuous in (0,1] x I , and strongly
continuous in [0,1] x I . We have K0(t) = K(t) , t E I .
Proposition 2.3. (Energy estimate). Let u(t)E C(I,HD) f C1(I,Hse)
be a solution of (2.1) 6 for given s , s , y , f(t). There exists
a constant c independent of s,s,q and f such that
(2.7) Ilu(t)IIs s IlaPllse°Itl + e°ltlteoillf(z
sgn t)Ils ds0
with the norm II.II 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
(2.8) IIuE(t)Ilss c, , 0 s E s 1 , t E I,
6.2. Systems of operators on Rn 203
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)Itse+ Ilf(t)Its sc.
We also were using prop.2.2. Similarly,
(2.10) II(M)(D)uE(t)Ils s cslluE(t)lls+e s c
all with constants independent of t and s . In (2.10) we used thatthe 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 Ar
zelaAscoli 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)usj
(t)
(2.11)= f(t)  iK(t)u(t) in S and every Hs uniformly over I
Since us and its derivative uE converge uniformly the limitj j
function u(t) is in CI(I,HS) for every s and in CI(I,S) Also
uE a u and u solves the Cauchy problem (2.1)o = (2.1)
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 solu
tion of (2.1) with cpj and fj, it follows that ujul solves (2.1)
for cpjcpl and fjfl. Thus the energy inequality (2.7) implies
(2.13) Iluj  u11lss c{Ilavj cpllls+ MaxtE Illfj f11ls}  0 , j,l °°
204 6. Hyperbolic first order systems
Thus {uj} is a Cauchy sequence in C(I,H5), and u =lim ujE C(I,H5).
Similarly u(= CI(I,HSe) , 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 {LeV (EM) : 05s51} , and {JE=FLEF}, are
bounded in L(H0) . Both have strong limit 1 , as E0 , in L(H0)
(Indeed the family of functions IV (ex) : 05651 } is bounded,
and (Ex) = yr(0) = fgIx p(x) = 1 , pointwise, for all x .)
2) The family {K(t) : 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 jE]/E } , and {[ s,LE]/E}, where 0<e5l,
and s = (M) S2 (D) s' , with fixed s E H2, are bounded in OpPcse
Indeed we may apply I, lemma 7.2 onto(ED)](D)s' (M)s2 = [(M)S2 Ir (ED)](M) s2
noting that ((x)s') (0) EO, s2 1' while { (p (Ex) (0)/E : 0sss1} is
bounded in Wsi1.0' as 101=1. For the latter one confirms that
(2.14) a )s cEl (1+E )1=O((
with 1 = 101 , for any (o E S . Similarly for LE5) 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
E=Vice, :0< E s 1} is bounded in Vce, for 101= 1.
7) The family {[JE,LE] : 0 < e s 1} is bounded in Opipce
(Proof similar to the above.)
Proposition 2.4. We have with the inner product (u,v) s of Hs
(2.15) I(U,K Oses1, tE I, sE &2, uE S,
where the constant 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 =((KK*)v,LJV) +(Kv,[J,L]v)+([K,J]v,Lv)
(2.16)+(v,[K,L]Jv)+([I],LKJ]u,v)+(v,[LKJ,II]u where [II,LKJ] =
= [ 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
(2.18)dµ(t)/dt = µ (t) = (u (t),u(t))8 + (u(t),u (t))s
= i((K9u,u)s  (u,K9u)s) + 2 Re(f,u) s
Thus (2.15) implies
(2.19)I 1A (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)ect  (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,Hse) 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:
(2.21)(t) = (u ,JLu)5+ (u,JLu )5=i((K9u,JLu)5(u,JLK9u)5)
+ (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,
(2.22)(Ku,J6L6u), (u,JSL6Ku)5 = ((KK*)v,Mv) + (Kv,[11,M]u)
+([II,K]u,Mv) +(v,[K,M]v)+(v,[MK,f]u) + ([I1,K]u,[II,M]u) ,
206 6. Hyperbolic first order systems
by a calculation.As 60 the right hand side of (2.22) converges to
(2.23) ((KK*)v,v) + ([II,K]u,v) + (v,[K,II]u) = p(t) ,
uniformly for t E =I . Indeed,the second,fourth and sixth term,at
right,tend to zero,using (2) and (4) ,(6) , (5) and (4) ,respecti
vely. The remaining three terms of (2.22) converge to to the terms
of (2.23) in the order of listing,using (1) and (3) ,(1) and (5),
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 p'6* both converge uniformly it follows that the limit
µ(t) is C1(I) 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 , the assignment
cp  u(t) defines a linear operator U(t): He 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 deri
vative U (t) = dU(t)/dt exists in strong operator convergence of
L(HS,HSe) and is C(I,L(HS,HSe)) under the assumptions stated.
Similarly, under the assumptions of thm.1.1, the evolution opera
tor 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'3S', 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 ,
6.3. The evolution operator 207
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 It0 = [tlt0,t2t0], 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,Hse), solving (3.3). Thus get a more general
evolution operator U(t,t): mapping Hs'H5, sE &z , satisfying
(3.5) atU(t0,t) + iK(t)U(t0,t) = 0 , t E I , U(t0,t0) = 1 .
If K(t) is independent of t we get U(t0,t) = U(tt0) = eiK(tto).
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. Regar
ding 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.,
(3.7) (x)1+Ial{ 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
(3.8) wE + iKE(t+t0)we+ iKE (t+t0)vE = 0 , WS(0) = 0
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
d/dt(IlwElls) = O(lIwEII$ + Ilvalls+e) ,using (3.7). This implies
(3.9) IlwEll2 s ecltIIlvell22+e 5 e2CItIIIVP112+e
with c independent of E,t,t0 , and cp . The family {vE(t0,t):0<E51}
is equicontinuous; 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, Ek0, vE converging to v(to,t)E C(I,S),k
208 6. Hyperbolic first order systems
in every Hs. Also atVF k=iKEk(t)v£
k(t) converges in S, hence atvE
C(I,S). Moreover, the function z5(t)=(v(tO+S,t)v(t0,t))/S solves
(3.10) z5*(t)+iK(t+t0)zO(t)=i(AK/O)v6 , z5(0)=0 ,
with v (t)=v(t0+S,t) , AK=K(t+t0+S)K(t+t0). For the difference
p=zSzS, , S , S' >0, with A'K=AK, for 6=6' , we get
(3.11) p +iK(t+t0)p=i(AK/S (v Sv")+(AK/SA'K/S')v
', p(0)=0 ,
t+to +Swhere AK/S=S1 K (x)dx is a bounded family in L(H,,H,_ )
t+tosince K'(t), tE I, is bounded in OpWce. Moreover, AK/6K'(t+to)0
in L(Hs'HSe), 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 p0
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
(3.12) w + iK(t+t0)w +iK (t+t0)v = 0 , w(0) = 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 11vv100s * 0 , 11v V1 "se > 0 , as in the
proof of thm.2.1, with vi the solution for Tj . Using (3.12) we
also get flwwillse  0 , for wi = at0Vi . Consequently,
(3.13) v(t0,t) E C(I,Hs) (1C1(I,Hse)
Similarly u(t0,t)=v(t0,t0+t) , the solution of(3.3), satisfies
(3.14) u(t0,t) E C(IxI,HS) f C1(IxI,Hse) .
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 SS, and HsHs, sE ie2. Its first partials atU , atU exist in
strong operator convergence of L(Hs,Hse), sE &2. They are E O(e).
(ii) U(t,t) is bounded and strongly continuous over IxI , in
L(HS) , s E H2. atU and a
tU both are bounded and strongly conti
nuous in L(Hs,Hse) s E H2, t0,t E I
6.3. The evolution operator 209
(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
(3.17) V(T,t) = U 1*(t,t) = U*(t,t) , t,TE I ,
is the evolution operator of the adjoint equation's Cauchy problem
(3.18) V'+ iK*(t)v = g , 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(3.19) u(t) = U(T,t)tp + U(x,t)f(x)dic
T
Next we notice that higher T,tderivatives of U will exist
if higher derivatives of K(t) are assumed. For example, let
(3.20) symb(K) = k E C,(I,4ce) .
Then it is clear that K E C*(HsHse
) for all s. Accordingly it
follows from the differential equations (3.16) that atU and a
TU
have first order partials for t and T in L(HsHs2e) in strong
operator convergence. Thus the second order partials of U for t,T
exist and are strongly continuous in L(HsHs2e)' Using this in
(3.16) again we find that the third order partials of U exist in
L(HsHs3e),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 opera
tor convergence of S and S'. We have ataiUE O((j+l)e). That deriva
tive 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, con
tinuity of atU,ayU in strong convergence of L(HS'HSe)' (for C1),
and atazUE O(j+l)e°),corresponding existence, continuity (for c0o).
Proof by reinspection of (3.16), under the new assumptions.
4. Nth 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
Nth order hyperbolic system (4.1), below, with a detailed theory
to be worked out in ch. VII . With I=[ ti , tz ] , t. E I, let
(4.1)Lu=1=0(i)iAN_j(t)(at)ju=f(t), t(= I, dlu/dt1=W1, t=t0, Osl<N
where Ak(ttt)=ak(t,x,Dx) are kth order differential operators in x
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) A 1=Ae,l=nee=ne° (x,D)=(x)(D), a°=e, =(x), °=Z.
Note that Ae,E Opyc_e,. With the new dependent variables
(4.3) uj = AjDtu , j = 0,1,...,N1
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, g=(0'...,0,iAlf)T, "((p0,...,(iA)N1(PN_1)T'
and the square matrix of yido's
0 1 0 . . . . 00 0 1 . . . . 0
(4.6) K = A 1 . . . . . . .
0 NO ... 0 N 1 IAN,A AN_1A'' A, ... , 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
6.4. Nth order strictly hyperbolic systems 211
one unknown function then (4.4) is an NxNsystem. In general, if
(4.1) is an vxvsystem, then (4.4) will be an NvxNvsystem. 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 con
ditions 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) ECoo(I,1jea)
,j=I,...,N
Or, as a possibly stronger condition we require
(4.8) aa, j(t,x) E C'(I,Vc(0,(N_j)fz )) , 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 )= N_j(Dx)Dt. If L is hyperbolic then the roots
of PN(X, ) 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 vxvmatrix 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, ,f2We then will speak of a strictly hyperbolic system of type f=eA .
An Nth order vxvsystem (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 oneparameter family
{R(t)=r(t,Mw,D):tE I} of zeroorder 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 mdelliptic, uniformly over I, and that
212 6. Hyperbolic first order systems
(4.10) (RK)*(t)(RK)(t)=K*(t)R(t)R(t)K(t) E C'(I,OptpcO) .
Here "*" denotes the Hadjoint. 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
R(t) , defined for every bounded positive selfadjoint Operator
H36 , is a pdo of order 0 again,together with its inverse S1(t) .
Moreover, S, S1are mdelliptic, and we get S , S1E
Coo(I,Optpc0).
Proof. Write S(t)=Af dX\/X(R(t)X)1 as a complex curve integralr
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 Kparametrix of II,thm.1.6, mod
O(oo). In particular, r may be laid such that R(t)a, is mdellip
tic for tE I, ?E r. The terms of the resulting asymptotic expan
sion of the symbol of the integrand are analytic in X and smooth
in t. The 0order term yields s0(t,x,U)=2nf r t x% ,
using a
harmless interchange of integrals. Lower order terms contain
higher powers of rX in the integrand, but are well defined sym
bols 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)S1(t) +iS (t)S1(t) .
Clearly K"E COO(I,Opyicf) , by prop.4.1. Moreover, we confirm that
6.4. Nth order strictly hyperbolic systems 213
(4.14) K*(t)K (t)=Sl(K*RRK)Sl_i(SlS'+S'S_l)E C'(I,Oppc0)
using calculus of tpdo's. For a =e all assumptions of thm.2.1, thm.
3 . 1 , cor.3.2, cor.3.3 now hold for K. We have S, S I O(0), they
map HsHs, and SS, 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.
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) satis
fies atuE Hs_je, (not only E Hsje), 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 fol
lows 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
(4.16) 2Tt1>0, j=1,...,v1, f2 IxI+ft krl0.
with a constant rt1 > 0 , independent of t,x,l; . 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
(4.17) rO+r_e
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,hence r0k0 X jpjTpj , using that pjpl=pjoj1 . Also,
with pTplT=pTSj1 , and k0=q,1p1T , we get 0r0=Xjpj Tpj=r0k0 , for
214 6. Hyperbolic first order systems
tE I, If we can show that tpco then we
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
teristic equation ®(? =0, with ©, a poly
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
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 tderivatives, one finds by successively differentiating
(4.18) recursively that t,j(P)= O(xJaJe'JPJez(x,U)) , so that
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 esti
mates, thus give symbols in C°°(I,tvc0) equal to the µj, for large
IxI+lgl. For a°=e' use a cutoff x(t) , =1 in ItIa2rI, =0 near
111n} , and use µjx instead of µj . Similarly for aA=ez
For the pj we have a complex integral representation
(4.19) i/2xJµj (=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
p jE C0o(I,Vcyo) , after modification outside J f2 Jx J+f. I t I x2rI} _ E11 .
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 Cm(I ,'grc0) , the eigenvalues R j of k0 aresymbols in C°°(I,'Wa0) , 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 re to be determined
For e°#e add (1x) with or x(x) to r0 at right of (4.17), res
ectivel Observe that R *p y. O=2(r0(t,x,D)+r0(t,x,D) ) has the proper
form, by calculus of pdo's , while R0 is a bounded selfadjoint
operator of H by III, thm.1.1. Evidently R0 is mdelliptic of
order 0 , and so is ROy , 0sys1/n , because, for zE &n , we get
(4.20) z r0z = 1ipjzl2 anlIzl2
Thus we conclude that the selfadjoint operator RO(t)y is
Fredholm, for y<l/n , hence can only have discrete spectrum there,
by 11,4. Moreover, the eigenfunctions of this operator are in S,
hence its spectral projections are operators in O(ao). Thus we may
define R(t)=R0(t)(1Py(t)), for example, where Py(t) is the ortho
gonal projection in H onto the span of all eigenvectors to eigen
values s y < 1/n . Here y must be selected locally, for tE AC I
such that none of the eigenvalues ever equals y . Then P(t) E
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 , containing t , and a complexvalued
symbol kEC00(I,Pce)
. The conditions (2.3), (2.4) then amount to
(5.2) k1 realvalued, 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 stu
died 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 evo
lution operator U(t,t) of (5.1) constitutes an order preserving
automorphism of the algebra Op4v of pseudodifferential operators
with symbol in e = yic00 , as defined in 1,7, for all t,% E= I. In
other words, for A = a(M1,D) E OpVcr , rE 12 , we have
(5.3) At ,.t = U(t,t)AU(t,t) = at,t(Mi,D) ,
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 allo
wing continuous extension of symbols aE VcO. We trivially have
(5.4) U(t,t)KSU(t,t) = Ks , KS=K(HS)
From thm.5.1 we conclude that
(5.5) U(t,t)ASU(t,t) = As
Accordingly there is an induced automorphism of the commutative
C*algebra AS/KS a C(Me,O). It is well known that every such auto
morphism 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
(5.6) v' = vi't , 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 aat,t of the symbol of A=a(x,D) un
der 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 217
(5.10) V1mt,t 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 compac
tification 1e,0 amounts to the statement that
(5.12) a E lc0 aovt't E Vc0
(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 yice .
For simplicity of notation assume t=0 , and write
U(t) = U(0,t) . At = AO,t , at = a0,t , vt = v0't , v't = V10't
From thm.3.1 we know that U(t) hence At admits a strong
tderivative in L(SsHse
) Using (3.16) we find that
(5.14) 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 'pdocalculus. From I,(5.7) we conclude that
(5.15) i[K(t),A] = (k,a) (Ml,D) + Rt , Rt E Opyice ,
with the Poisson bracket
(5.16) (k,a) I j=1(k(e)a(8)  a(8)k(0)) .
Assuming ( 5 . 1 3 ) , and in addition that rt exists and rt E la_e
(5.17) at (x, ) =
a symbol of order e . It is practical to replace k(t) by
the realvalued 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 =kIN t I, x0=x , 0
modulo terms of order e again.
We try to satisfy (5.19) precisely, not only modulo e, obser
ving 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 realvalued 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
obtained in sec.6. Hence for a,1E IIn, aXa t(x,) EC00(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 equat
ion (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 realvalued k1E pce the flow vs't defi
ned 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 supnorm 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,OMe,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 at.t=aovt,,L we get
(5.21) atat,t(x, ) + (ki,at'z) = 0
as follows by using (5.20) in
(5.22)
)atxt,t +a,, (xt't )at t.tFor a E Vcm let us use the operator
(5.23) Pt't= U(t,t)At't U(t.t) . Ate.,=at,t (x,D) , at,,=acv t',t
Clearly At't EC00(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'Hsme)) 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(me). Also Vt is strongly
continuous in t as a map HsHsm+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'HSm+e)
T
This confirms that at least (5.3) holds modulo an additive term in
0(re). To get more precision note that Vt=U(t,t)WtU(t,t), with
Wt E OpVcme , 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(r2e), as in the integral of (5.26), etc. Thus, for all N,
(5.27) U(t,t)AU(t,t) = at,t(M1'D) + "l=1Tt t + XN
with Tt E Op"cr_Ne XN of order r(N+1)e Then we get
00l Tt't + X00(5.28) U(t,t)AU(t,t) = at't(M1,D) + Yj
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 vxvsystems 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.
220 6. Hyperbolic first order systems
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
(6.1) cP=cP(x,)=VP(x,,t,i)=tt(x.)
Writing k1=k , for simplicity, we then have (5.19) in the form
(6.2) f'=kl (f,g) , cp'=kix(f,g) , f=x , T= at tr .
Proposition 6.1. The functions f , cp satisfy the aprioriestimates
(6.3) 0<CS(f) /(x) sC , 0<cs( (P) /( ) sC ,
for t,t E I , &n , with constants c,C independent of
Proof. One derives the following inequalities for (f)' , ((P)' :
(6.4)(f)' = f.f'/(f) = kl .f/(f) = O((f)) ,
((p) * = cp cp * /(cp) =k I x . w/(Cp) = 0((q)) )
Integrating (6.4), under the initial conditions (f)=(x) , ((p)_( )
at t=v we get
(6.5) log((f)/(x)) = O(1) , log(((P)/(l;)) = O(1) .
Since (f) , (q) a 1 this indeed give (6.3), q.e.d.
Note that prop.6.1 gives the apriori estimates required for
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)1IUI()IaI) ,
a C°function consider the composition
(6.7) b(f(x,5),g(x,t))
The derivative c(R) is in the span of the terms
(6.8)
6.6. The action of the particle flow 221
(PXvb)(f. ))njri f03 nlsl
where Iajl+If3 I21 , Ia'lI+IP'lIa1 , Yaj+2q'l=a , 2Pj+D "=p
t
and where the tensors V V b and vectors f(a) , are to bex (PIcontracted in arbitrary order of indices, f with T with 0
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 f(a) + k (a) f(a)=x(a)
(6.9) (3) Ix (1) (0) (1) , as t=t
(P(P) klxxf(R)  klx(P(3) ,
,(P)_(P)
Here we have written klxf=4Ixjfj , butjfj , etc.
Multiplying the first equation (6.9) by (cp)/(f) we get, as matrix,
(6.10)
(p)/(f)f(R)I
(a)(P k l X'( f) k l xi; I I a, I
The 2nx2nmatrix in (6.10), called P1 , 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 nxncorner. There we find the addi
tional 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(IttI)=O(1). Calcu
lating 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+I1Isr1, 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
klgx
(6.13) p* = Pp + q , p(t) = 0 , q = (qZ )
222 6. Hyperbolic first order systems
with the above matrix P , and with q' , q2 in the span of
(6.14) (0) 111=1 (R4)
and of
(6.15) (Qs+IQs'k(f,T))n3=1 f(3) 1Z1=1 T(e)
respectively. In (6.14) and (6.15) we assume that s+s'sr, and
1sIajI+I3jI<r, 1sla'll+IP'lI<r, jaj+Ya'l=a, 2Pj+2P'l=P. Again con
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 re
peated 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 Ip12.
= O(IpI2+IgI2) 1 IP(t)I2=0 . This may be integrated
for completing the proof for j=1=0.
Next we look at t and tderivatives of f,q . First it is
evident for the vector of (6.13) that also p'=O((x)IPI(t)1Iai)
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 con
tains only terms involving the 0th and 1th tderivatives of
f(a and cp already known to satisfy (6.6) Using this, one
finds that also p''=0((x)IRI(1;)1IaI) In fact, the proceduremay 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 tderivatives. 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'
y(a)=pz. It follows at once that (6.6) holds for 1=0, j=0,1,2,...,
for reason of symmetry. But mixed t,tderivatives may also be est
imated in this way: For example, writing p'=Qp+r, we get p''=Qp'+
6.7. Propagation of maximal ideals 223
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,tderivatives, 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) . For example,
(6.17) 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 oftt
A t=U(t,t)AU(t ,t) is given as
(7.1)('A = 'A°vtttt
for every AE A. Similar for AE As , the Hsclosure 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 hyp
erbolic 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
(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) 4k0 () , as in 11,5 , we must have A mdelliptic 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 such that EiA=q(Mr,D)+
Ki , Ki E O(oo) , Et E Op4c0. It follows that (tk, ((px° u)" )" =E, AuK, uES , 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 thereexist cutoffs 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° , while does not belong tochar CB . Hence we also get ":" , q.e.d.
Thm.7.1 first prompts us to define an Awave 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}
with the wave front space W(A)=i1W
= b1(&nxa1n) of A , using the
surjective map L:Me,O  I of 11,3 . Clearly we get
(7.6) LWF(u,A) C WF(u) .
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,
(7.8) ZF(u)J WF(u) U WF(u^) .
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)) , ZF(u(t),A)=v,t(ZF((p,A))
Similarly , for f(t)#0 , 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 JZF(f(K)A)), K between t and t
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.
6.7. Propagation of maximal ideals 225
Corollary 7.3. Let k0(t,x,l)=symb(K0(t)) of (4.9) allow an asympto
tic expansion k0 Dc0j (mod ICI) at 1;=, as in V,(5.10). Assume kj
E Co°. for tE I, 1 1;I=1, xE KG: &n, all K. Then we also get formulas
(7.9) for WF(u) instead of WF(u,A) .
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 (ra
ther 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 commut
ators in K(LZ(&n))? (2) For the L(L2(in))norm closure A of an al
gebra 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 impor
tance 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 FourierLaplace method of ch.0,4: Let h=(1,0,...,0).
Taking the Fourier transform in (x2,...,xn) and the Laplace trans
form in xl will convert P(D)u=f into P(t+ith)u^=f^, where the ima
ginary part t of the t1variable must be sto, for some to, in ac
cordance with the rules of the Laplace transform of ch.O,(3.14).
Thus (ii) insures that we can write as tsto. Spec
ifically, for f=6/xn, the inverse transform (P( +ith))"=e defines
a fundamental solution of P(D). An analysis shows that (i),(ii) in
sure 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 {xh>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 halfspace 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 hyperbolici
ty (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 timelike vectors. In sec.2 we
226
7.1. Algebra of hyperbolic polynomials 227
consider strictly hyperbolic (differential) polynomials. The class
shyp(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). "shyp(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 shyp(h)
at x=x° if a(x°,D)E shyp(h). For an a(x,D), strictly hyperbolic
over a domain SIC Rn the vector may depend on x. If is can be cho
sen 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 Nth order
vxvsystem, looking like an ODE in t with coefficients PDE's in x.
We prove existence and uniqueness for a relativistic and a nonrel
ativistic normally strictly hyperbolic Cauchy problem.
In sec.4 we simplify the assumptions, essentially by linking
strictly hyperbolic differential operators with the strictly hyp
erbolic i,do's of ch.VI. In sec.5 we discuss the 'hyperbolic featu
res'  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 hy
perbolic 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 coef
ficients, except perhaps for a common complex factor # 0 .
Proof. We have Thus must equal
the product of all the (real) roots of hence must be
real, for all real . This implies that 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
(1.1) P(T) = P(l+ith) = PN(ih){tN +
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
const. Accordingly, Re tk = Re p1  I Re tj s Re p1 (N1)t0j7q
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) , forall t . Suppose q(t)=0 , for some real t . There must be some com
plex 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 ta a t0 . Since
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 indepen
dent 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 <N , will be hyperbolic, if (and only if), with constant c
(1.2) 7,IR(a)( )I2 s c 1PN(a)(,)12,
(=ina a
We will discuss this in sec.2.
A polynomial P (or a differential expression P(D) ) will be
called strictly hyperbolic (or Petrovskyhyperbolic) (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 shyp(h) instead of hyp(h)
for strictly hyperbolic polynomials or expressions. Note that
7.1. Algebra of hyperbolic polynomials 229
"shyp(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
"shyp(h)" implies "hyp(h)". Thus PNE shyp(h) has the property
that all P=PN+R , deg R <N , are hyp(h) .
For a polynomial PE hyp(h) , O#hE mn , we introduce h=h(P,h)
as the set of all TIE In such that the polynomial p(X)=PN(rl+?h) has
only negative real roots. Prop.1.1 and prop.l.4 imply that the
roots are real. The highest coefficient P (h) is constant, hence
X changes continuously with q . Thus hC I is an open set. For rl=h
we get p(X)=PN(h)(1+X)N, with all roots ?=1. Hence hE h , and h
is nonvoid. 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)<O. In terms of special rela
tivity, for the special hyperbolic expression D2+...+DnDi , this
cone h plays the role of the cone of timelike 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(tt j) , and all t are< 0 . Hence PN (rl) = PN (h) n(c j ) # 0 . We already know that h isopen. 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
point of h would be in the interior of m. Let TIE h , and =X.rl+µh,
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.
Proposition 1.7. For q G h we have as Re t < t0,
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 continu
ously, since the highest coefficient of the polynomial is indepen
dent of t . By our above remark none of the roots may cross the
230 7. Hyperbolic differential equations
imaginary aaxis 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
0 . We know that all roots of PN(h+1x11) = 0 arenegative real, since 11 E Eh . Consider the family of polynomials
rt (µ) (h+1111)) = PN(11)RN + ... . Clearly we getlim ,oo rt (14) = PN(h+µr1) . Accordingly rr (ti) = 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 , Z; E &n .
We know that PN(1l+ah)#0 , due to 11+EhE h , since h is starshaped
and a cone. It follows that P E hyp(11+eh) , e > 0 . But h is open,
hence contains 11E = 11Eh , for small a>O . Applying the above for
wla instead of 11 one concludes that P E hyp(11) , for every rl E hFinally, to prove convexity of h , let rl1 112E h. It follows
that P E hyp(1l3) , 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 starshaped 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' a2 a2 ...a2xl x2 xn
This operator is hyperbolic and strictly hyperbolic with respect
to the "timelike" direction h=(1,0,...,0) . Indeed, we get
I
I21;12. hence
I. 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 timelike
vectors is given by as follows from prop.1.6.
2. Hyperbolic polynomials and characteristic surfaces.
In this section we shall investigate the relation between
"hyp(h)" and "shyp". It turns out that the set shyp(h) consists
precisely of all PE hyp(h) with simple characteristics. Here the
concept of characteristic (surface) must be understood in the sen
se of ch.0,8. There we already defined the concepts of simple and
7.2. Hyperbolic polynomials 231
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 expres
sion 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 low
est 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 x0E 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 realroot 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 prin
cipal 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 characteri
stics. Thus an elliptic operator always is of principal type.
Proposition 2.2. An operator P(D)E shyp(h) is of principal type.
Proof. Let PE shyp(h), and let for &n. We know that
PN(h)#0 , so that =?.h is impossible. Hence is linearly indepen
dent of h, and PE shyp(h) implies that has a simple
root at t=0. Hence the planes normal to 1; are simple characteri
stics. It follows that all characteristics are simple, q.e.d.
Theorem 2.3. We have hyp(h)C shyp(h). For a polynomial PE hyp(h)
232 7. Hyperbolic differential equations
we have PE shyp(h) if and only if all real characteristics of
P(D) are simple. In other words, shyp(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 po
lynomial R of degree <N satisfy (1.2). Then also PN+RE hyp(h).
Moreover, if P=PNE shyp(h) then PN+RE hyp(h) for every R, deg R
< N , regardless of (1.2). In particular, shyp(h)C hyp(h).
Proof. Observe that PN+RE shyp(h) implies PNE shyp(h), so that
the second statement implies PN+RE hyp(h), reducing the third
statement to the second. To reduce the second statement to the
first, observe that PNE shyp(h) must be of principal type, by
prop.2.2. Conclude that PN is homogeneous of degree N1,
while for all ICI=1. It follows that, with some co>O,
(2.1) aIPNa)( )I2Z IPNI(
)I2. for all E In
However (2.1) implies that (1.2) holds for all R, deg R <N
To prove the first statement it is convenient to write (1.2)
as R< J PN, with the order relation 11<0"" of [Hr1]. Or, F®G means
(2.2) 7, IF(a)(,)I2 s c Ia 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
(2.3) F ( ) = { lIF(a)(,)I2}1/2 s c(T1) F ( + ) ,
for all E In , rl E En , with a constant c(1) independent of .
Accordingly, F®G implies F1. J G,q, for the (real or complex)
translations F,q( )=F( ..... . 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 , j = 0,...,N
Thus indeed, we get Fj <V G , j = 0,...,N
7.2. Hyperbolic polynomials 233
Let PN E hyp(h). From thm.1.5 we conclude that h+Re i; is
timelike for sufficiently small ICI . Accordingly,
as I1Is 2Vnb >0 , E ten. Using Cauchys estimate laltimes, 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)lal1max
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.
This gives (p(1)=T(0)r)j=1((tj1)/s with the roots ti , ltjlz2
of q . Hence the Max at the right hand side of (2.5) is estimated
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
(2.6) c independent of and a
If P<MOPN then we have seen that Pj<u PN , hence
<ZJ by translation invariance. This and (2.6) implies
cIPN(t+ih)12, t E ,n , j=O,...,N1
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 jNILet R = PO+P1+...+PN1 , then we get
Therefore P = PN + R satisfies the estimate
(2.7) IP(t+ith)l Z (1cltl1)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 shyp(h) .
Proof. It is clear that PN must be hyp(h) , so that 0
has only real roots, and PN(h)#O . If PN is not strictly hyperbo
lic 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 sN1, can be hyperbolic.
We may assume to =0. Then the polynomialhas degree Nk s N2 .
Now we will show: If, under above assumptions, P=PN+R is
hyp(h) then the polynomial p also must have degree
sN2. For a general R of degree N1 or less, this can not be true.
234 7. Hyperbolic differential equations
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
(2.8) xr(x,a) = 0 , r(x,o) =
with a polynomial r(x,o) in x, deg r sN1. At x=0 we get the equa
tion PN(i;+ah)=0 , which, as we know, has the kfold 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 K0 , has an expansion I amxm/1Ms
Let s be choosen such that as is the smallest coefficient #0
Clearly szO , since the series vanishes for x=0 . If 0<s<1 , then
ta(1/i) sast 1s/1 =I asI It I Yexp{ i(arg as+y arg t )} , where 0<y =1s/l<1 . For real t we may set arg t =vn , with an integer v . Here v
may be choosen such that arg as+ vyn is not a multiple of n, since
I must be i2. Then we get It Im aj(l/t)I as Itl . However,
we have Imtaj(I/t)) so that Imtoj(1/s)remain bounded by definition of hyp(h). Thus we get a contradic
tion 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
that f(0) = PN(h)4=1(_? (t)) = O(ItINk) . Hence indeed
p (T) = P(tl;+h) must have degree s N2 , q.e.d.
Corollary 2.7. If P E shyp(h) , then also PE shyp(rl) , for every
timelike vector Tl E h .
Proof. The cone h of timelike 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 N1 or less, because each such P is hyp(h) ,
hence also hyp(q), using thm.1.5. Therefore P E shyp(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] .
7.3. The hyperbolic Cauchy problem
3. The hyperbolic Cauchy problem for variable coefficients.
235
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
x°, if the polynomial in is shyp(h). If this holds for
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 shyp(h)=shyp(h,st). For a
general set M we say LE shyp(h,M) if LE shyp(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 "shyp"
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,mTry), where m T=(m 1)T. Choosing
h°=mTh=(1,0,...,0) will make the transformed expression a"(y,Dy)
shyp(h°) . In fact, for a more general transform x=m(y) the con
stant 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 written in the form
(3.1) L = ja jkDjDk + lower order terms , ajk = akj .
The principal part polynomial is given by We get
(3.2) with
ajk a common complex fac
tor). Cdn. "shyp(h)" then means that
tor). "shyp(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 shypoperators are
of the 'classical' form: The principal symbol matrix ((ajk)) has
one positive and nl negative eigenvalues.
236 7. Hyperbolic differential equations
We now assume h=h°=(1,0) , OE Rn, in Euclidean space Rn+l =
{(t,x): tE &, xE Rn}, and an expression L=a(t,x,D) of the form
(3.3) L=11+k:.N
aa'k(t'x)DaxDt=4=OANk(t)Dt' ANk IaIsN_kaa,kDx
with principal part
(3.4) LN aa.k(tx)DxDt4=OBNk(t)Dt, BN_g aa.kDk.IaI+k=N IaI=Nk
Let sz=IxRn, I=[ t] ,t2], and aa,kE COO(R) , for all a,k. For complexvalued aa,k the expression (3.3) is shyp(h0) if and only if
(3.5) a0,0(t,x)'0, k=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 complexvalued coeffi
cients. However, the results, below, are valid for the case of a
vxvsystem as well. From now on we assume that aa,k(t,x)E Lv, with
the algebra Lv of all complex vxvmatrices. A vxvmatrixvalued
expression is called shyp(h°) if the determinant of its symbol is
shyp(h°). For L of (3.5) this means that a0'0(t,x)=identity, and
(3.6) p(i)=det( 7 has Nv real distinct rootsIaI+k=N
for all (t,x) E Rn+1 , 0#
E Rn
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
"shyp(h°)"  while at x=w the conditions do not match.
We focus on the local "nonrelativistic" Cauchy problem
(3.7) Lu = f , (t,x)E S , atu as t = t0 , j
for given f (defined in Sz) and Tj (defined for t=t0E I), assumed
COfunctions. The function (distribution) u is to be found.
The term 'nonrelativistic' expresses that the initial condi
tions 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 'spacetime
transform' i.e., a transformation of independent variable. Recall
the cone h = ht,x of timelike vectors. h is convex, h°E H, and L
7.3. The hyperbolic Cauchy problem 237
of (3.3) is shyp(rl) at (t,x), for every 1E Ht,x (cor.2.7).
A smooth hypersurface I C St will be called spacelike if its
surface normal is a timelike vector, at each (t,x)E S . Since h
is convex and WE h, a spacelike 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 forlarge 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,...,N1.
Here av denotes the normal derivative to S, av defined as in V,3.
We choose the normal pointing to increasing t. Assume f, WjE Co00
given, defined on Si and E , respectively.
The relativistic Cauchy problem may be reduced to the nonre
lativistic 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
(3.9) t' = t'(t,x) , x' = x , (t,x) E SZ=Ix&n
inverted by
(3.10) t = t'(t,x') , x = x' , (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 veri
fies 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 trans
forms 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') , and with aN(xa,ta) = aN(t,x,t,t) . In the Jacobian
matrix J the row index is j , and the column index is 1 . One cal
culates 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 realdistinct roots t' if L is shyp(h) for h of (3.12), all (t',x').
Let t'(t,x)=t8(x) , so that t(t',x')=t'+O(x'), and get h=
=(1,V8(x')) independent of t'. Clearly h is normal to I , as (t,x)
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 shyp(h°), and nonrelati
vistic. 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 spacelike, 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 shyp(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
solution u(t,x)E C0O(I,Hs) , for all sE k2, of the nonrelativistic
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) "shyp(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'=t8(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 spacelike 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 spacelike, but even that
its normal is contained in the cone hX fl(ht,x: (t,x)E M}, with the
"slab" M={z,t,StO(x)st:t2}. Then there exists a unique solution
u of the relativistic Cauchy problem (3.8), defined for (t,x)E M00
and each f, E CO , such that the function u(t0(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 shyp(h°,K), h°=(1,O), for every K(= Ixxn, when
ever 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 ope
rator 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 N1
equal 0 or 1, as in VI,(4.6), but the elements of the last row are
(4.1)(ne°(x,U))NjaN_j(t.x,U) , j=0,...,N1
with aNj 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<...<RN for all
(t,x,t) E IXZe,0 with It was pointed out before that the
functions then must be continuous in this compact set.
Under the present special conditions, the matrix k0 and its eigen
values even extend to continuous functions over {Ixffi°xZn: W1.1,
with in of 11,3, and the compactification RA allowing continuous
extension of all functions a(x)E ip0 . Here we assume that
(4.2) aa, j(t,x)E (Nj)(e°e') as
Accordingly, for every jt°j=1 , the limits limpj(t,x,p°)Rj(t,x,«x°) exist, and define Nv realvalued 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( 7vltjbl (t,x, ° )) = 0
j+1 =N
where bN 7 are the principal parts of7 Ian=Nj 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 = 0j+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)
is strictly hyperbolic of type e° , where a°=e or e°=e' . Then
240 7. Hyperbolic differential equations
the differential expression L is shyp(h°,K) for every KC: Ixin .
Moreover, we even have L E shyp(h°,Ix&n) , uniformly, insofar as
the minimum difference between two roots of (3.6) is bounded below
by c° >0 for all 1 ° 1=1 , as e°=e' , and by c° (x) , as e° =e. Furthermore, 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
not influence the above derivation, since the limit will
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 timelike 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) lµj(t,x,°°ll)
, j=1,...,Nv)
where u(x)=neae, (x,i;)=1 as a°=e' , =(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 hlx1={tZC°(x)1 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 differen
ces bounded below by co, just as in the statement of prop.4.1.
Proposition 4.4. Under above assumptions the system VI,(4.4) is
strictly hyperbolic of type e' .
Indeed, we will just make the same decomposition into k0 and
x0 , as above. The negative eigenvalues ij=pj then again satisfy
(4.6) 1 (
k+l=N
with aNj of (4.1) or (3.3). We may wite this as
(4.7) p(t;t,x,y)+ 7, Tk ( )llalaa,Nl(t,x)( )a=0k+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 loc
ally 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 shyp(h°)" is suffi
cient  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 condit
ions of thm.3.1 are satisfied for all compact intervals I=[ti,tz].
Consider the adjoint L* of our expression (3.3), defined as
(5.1) L* = I Dxa'Dt a,k(t,x)
Ial+ksN
Proposition 5.1. We have L E shyp(h0) if and only if L* is
shyp(h0) . Moreover, the cones ht'x of timelike directions for L
andL*
coincide, at each (t,x) , and a surface I is spacelike
for L if and only if it is spacelike for L* .
For L with complexvalued coefficients we know that "shyp",
the cones ht'x , and "spacelike" are completely determined by the
principal part of L. Cor.1.3 implies that the principal parts of L
andL*
differ only by a nonvanishing complex factor y(t,x): We get
realvalued, for some g(t,x), hence the prin
cipal part of L* evaluates as aN gb=(g/g)aN. Since "shyp" implies
"hyp" this proves the proposition for complexvalued expressions
L. For general matrixvalued expressions we still have aN (aN)T ,
and det aN = (g/g)det((aN)T) follows because det aN is a hyperbo
lic 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 piecewise smooth, and consists of
two parts, E1 and E2, both space like surfaces, with equations
t=91(x), j=1,2.
242 7. Hyperbolic differential equations
The surfaces E1 intersect
at positive angle in some
smooth compact nldimen
sional surface. Moreover,
we assume 81 extended over
all of In, and that the
extended surfaces t=01(x)
(5.2)
Fig. 5.1
t=t, satisfy the assumptions of
thm.3.2. Assume the com
pact region ito defined by
rte= {(t,x) :xE &n, 81(x)sts82(x)} .
Lemma 5.2. Suppose a solution u of the relativistic or nonrelati
vistic 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, rl9rtoThen 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
(5.3) f u fdtdx = fuTL*vdtdx = 0RO
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 spacelike surface satisfying the assump
tions 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 spacelike 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
5.2 may be applied. First one notices
that, for each fixed t the function
u(t,.) has compact support: For exam
ple, one may work with a surface E°
carrying a spherical bulge, such as
t=to +Max{ x 2 2a.x ,b}=& (x), with 0<b<la12, smoothened at the rim, where
b must be chosen sufficiently close
to 1a12 . Using cor.4.3 one finds that
such a satisfies the assumptions of Fig. 5.2
243
the lemma, provided that the maximum slope is sufficiently small.
This does not limit the height such a bulge can 'climb'. The
latter works if a°=e' , while for a°=e a somewhat more complicated
lense must be chosen. Once we have shown that u(t,x) has compact
support for all t, we may examine the solution in successive slabs
tjlststj , where tj=ej , for small e>0 . The surface t=tj outside
E° , t=tj1 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 argu
ment 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 shyp(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 jsll>c° with O<c°, Co independent of Assume E to be a spacelike
surface satisfying the assumptions of thm.3.2. Then the relativis
tic Cauchy problem, for the surface E, and arbitrary smooth data
f,Tj, where yj are compactly supported while (supp f)MxlnO Rn+1
for all I( 1, admits a unique solution uEC0'
(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 discus
sion 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(Nj)e2 as jajsNj .
244 7. Hyperbolic differential equations
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 , asIn 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 FourierLaplace method in ch.0,4. Note that use of
ydo's has been eliminated, except in the proofs. For a Cauchy pro
blem 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 spacelike
surfaces through (t°,x°). This envelope will have the normal cone
of the cone hto , x° as a tangent cone, at (t° , x°) . Hence, at thatpoint (t°,x°) , it may be decomposed into a forward and backward
part. The regions RD and RI also may be described by the bicharac
teristics 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 aIxB3 ,
with BY =BY (0)={xE In: I X <Y } , I=[ t1 , t2 ] , we have a differential expression of the form (3.3) defined. Assume that LE shyp(h°,S) ,
and let the coefficients aa,j(t,x) be COO(n) . Assume a spacelikeintsurface EC n . 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,..N1,
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) , with the polyno
mial a (t,x) %k , and with a function 6(p)E=a,k
7.6. The local Cauchy problem 245
C00((0,00)) satisfying i(p)=p as pE [0,1] , 0s9(p)<2 in (0,m) , 0(p)
=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
independent of x as jxja2  we have L° = a(t,0,Dt,Dx) there. In
particular, L° is uniformly shyp(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
setting Tj=0 , f=0 outside IxBi . In this way we obtain a Cauchy
problem satisfying all assumptions of thm.3.2. The solution exists
and its restriction to IxBi will supply a solution of the local
problem.
Now, for a general set n=Ixn° , where n° CC Rn 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 , and let it be shyp(h°,Ixn) . Construct a finite covering of
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) Lu=f , near I atu=4Pj at I j=o,...,N1 ,
where I C Ixn is a spacelike 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 pro
blem 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 pro
blem 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 shyp(h°,Ixft )
there, and a spacelike 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 "shyp(h)", "shyp(h,n°)"
are meaningful for differential expressions defined on a differen
tiable manifold n. Since we have coordinate invariance we may int
erpret h as a vector of the cotangent space T*ax° , at a point x°,
and define L to be shyp(h) if this is so in any set of coordina
246 7. Hyperbolic differential equations
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 shyp(h) _
shyp(h,nz°) if LE shyp(h(x)) , at each xE n' . We then will have
a cone hxC T*nx of cotangent vectors ar x  it is the cone h of
the differential polynomial of L at x, in any set of local coordi
nates; note that the polynomial is now defined on T*nx, and that
its principal part is defined as well. It determines hx as a sub
set of T*n . A spacelike surface EC n will be defined as an n1
dimensional submanifold with normal at xE E contained in hx. Here
we will assume that L is shyp(h,n ) with some domain sE C n , and
that XC n° c n . Then again we may formulate a hyperbolic Cauch
problem: For compactly supported fE CO 0(n) and (fjEC0(E),
N1 (where L is of order N) find a solution uEC (n') of
(6.2) Lu=f , xE n' , ah(x)u=Wj , xE E , j=O,I,...,N1
where n' C :re is some (sufficiently small) open neighbourhood of E,
and with the jfold derivative ah(x)u (for t, at t=0) of the func
tion 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 vxvsystems, in the above
Correspondingly we may allow an expression L mapping between vdi
mensional complex vector bundles on n , in thm.6.2, and then spe
cify 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 spe
cial 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) , with a quadratic form
of rank n and signature n2 .
Chapter 8. PSEUDODIFFERENTIAL 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 Fourier
Laplace 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 exam
ple, the algebraOpCB00
(12n), with CB'(12n)=pto the class of
EC00
(3k2n) , bounded with all derivatives, coincides with the class
of operators AE L(H) which are both "translation smooth" and "gau
ge smooth". That is, with the family {Tz:zE Rn} of translation ope
rators TZu(x)=u(xz), and the family {M:tE ien} of "gauge multipli
cations" Mu(x)=eitxu(x), we get TZATZE C(&n,L(H)) and M,AMtE00 nC (& ,L(H)). Notice that both Tz and Mt are groups of unitary ope
rators on H, representations of the Liegroup In in U(H).
In terms of ch.I,Oppt0=OpCB00
(12n), a subalgebra of L(H) by
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 rota
tion 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 par
ticular, we will work with symbol classes 1tm, Vsm, and V1m exclu
sively, 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 'IDOtheorem' des
cribing the new characterization of Opipt0. In sec.4 and 5 we dis
cuss 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
248 8. Smooth operators of L(H)
properties of the new symbols. Sec.8 deals with corresponding Lie
algebras and BealsDunaucriteria.
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
(1.1) Tzu(x) = u(xz) , u EH .
For each character we define the gauge operator Mt=eitm by
Mtu(x) = eitxu(x) .
Note that Tz and M are unitary operators of H . Moreover,
(1.3) Tz+z' = TzTz, , Mt+V = M t M V , for all z,z',i;,t' ,
which expresses that z  Tz and  Mt each define a unitary rep
resentation of the group ien on the Hilbert space H .
We want to emphasize: The differentiable manifold In with abo
ve 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
Ixx°lss with x°(?%')=n ) with small e one confirms that
(1.4) IITzTZIII a , IIMMII 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.
8.1. Smooth operators of L(HO) 249
The pseudodifferential operators within the algebra L(H)
will be defined as those operators A E L(H) for which both fami
liesAz,O and AO,. are not only norm continuous, but even are00
C
(1n,L(H))
, 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
(1.7) Au(x) = (2n)nfd
where a complexvalued 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 existen
ce 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'
TzMtTzu(x) = eit(x+z)u(x) = e1ztMtu(x)
Or,
(1.9) TMTZMle1z
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) .
250 8. Smooth operators of L(H)
Clearly GH with the manifold structure of gh is a Lie group.
Our above condition describing 'WT is equivalent to the following:
For AE L(H) let AZ't = EZ,t,W IAEZ,t,(P . (Note that AZ't
is independent of cp .) Then we have
(1.13) Az,t E C00(gh, L(H)) .
Indeed, (1.13) trivially implies the condition of (1.6).
Vice versa, note that (1.9) implies
(1.14)AZ,t = TzAO,tTz = MtAZ,OMt
If (1.6) holds then use that
Az.t,ii =Ao,t,ii
, Az,,tii = IIAz'0 Az,'011
showing that Az, is continuous in z for fixed t and continuous in
t for fixed z , and either continuity is uniform in the other
variable. This clearly implies norm continuity ofAz,t
in both
variables. Similarly (1.6) and (1.14) imply existence of
(1.15)aaAZ,t= Tza 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 deri
vatives 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 cutoff 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
p(R)E Hk, for all a,(3, with the Sobolev space HkHk,0 of 111,3. By
Sobolev's lemma we thus get p(a)E Ck(I2n), for all k. Looking at p
in dependence of u, v we get II p (a) ii , 11p(p)11 c ii u li ii vii , with normsin Hy and H, resp., and with c undependent of u,v . Following this
estimate through the above argument  Parseval, Sobolev, etc.,...,
we find that also iip(a) ii k s ciiuii iivli , for each This impliesC
(%Az,t)(R) E L(H) , and even ii(XAz,t)(Pa)iis c . Then one concludes
8.2. The pseudodifferential operator theorem 251
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) . More
over, it is well known that u^ is a rapidly decreasing function.
That is u^(t) = for every k=0,1,2, ... , and the
same is true for the derivatives u^(a) . Since a(x,l;)E CBO'(12n)
is bounded, and lelt(xy)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 =((xy)+y)a is a finite linear combination of terms
(xY)OYY , 3+y=a , while
Again the inner integral equals (2n)n/2eixt(xYu)^, hence is a func
tion 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 inve
stigated in 1,1, under weaker conditions on the symbol
2. The 'MOtheorem.
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 transla
tion operators, and the 'gauge transforms' u > eitxu (or, equiva
lently, ghsmooth). Here we were using the standard representation
252 8. Smooth operators of L(H)
GH of the Heisenberg group gh, i.e. GH={Ez,.,(P gh} , with
gh = 2nxlnxl1 ,
$1 = 2/(2nZ) , where
(2.1)H , (Ez,(Pu)(x)=el(T+1'x)u(xz)
Especially,all translations Tz:u(x)  u(xz),for a constant vector
z, and all "gauge transforms" u(x)  ei(T+tx)u(x) for constants
gE It , t E In belong to this class.
For A E L(H), a continuous operator on H , we are interes
ted in the function Az,t : GT > L(H) obtained by conjugating A
with the operatorsEz,.,T
1(2.2)
Az,t =Ez
,t,TAEz,t,g
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 EZ1tAEz,t ,
EZ,t=
and henceforth regard Az,r, as a function 12n > L(H)
We restate the essence of our discussion in sec.1 as a theo
rem, 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 pseudo
differential 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, ) aXa a(x, ) of
all orders bounded over 12n . Then we have
(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) defined by
(2.6) IIAII = sup { IIAufl/IIuhl : 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
8.2. The pseudodifferential operator theorem 253
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)
using that
E =Z ,t
(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 commuta
tors 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 of
(1.7) extends to a continuous operator H H
Proof. It turns out that we need only boundedness of 'a few' deri
vatives 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) = et tm1/(m1)! , as ta0 , = 0 , as t<0 .
One checks easily on the following facts:
Proposition 2.3. We have ymE Cm2(2)fl C"(2\{0}). The derivative
ym (m1) still is piecewise continuous with a single jump of magni
tude 1 at t=0 . We have ym(j)(t) = O(eJtI/2) , as t#O . Moreover,
(2.10) (at+1)mym = 6 ,
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 ).
For k=0,1,2,... let CBk(&n) be the space ofCk(Hn)
with bounded derivatives up to order k including. Also, define
(2.11) x, Elk .
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 'pseudodifferential 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
u E CO(&). In fact, Qu(x) = y2(x) Fly2Fu(x) , so that
(2.13) Q = Y2(x)Y2(D)= x1y2(x) ((l 1
+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 kernel xi222)
y2(x)/(1+i(xy))
E L(&  i.e., Q is of Schmidt class. Moreover, we may write
(2.14) Q=V*U , V=(1ix)1(1ax)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(xY)' µx(Y)°xi(1+ix)(1+ax)((1+i(xY))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 Schmidtnorms [[[U]]] =' x ' L
2 , [[[V]]]= IIv(y)ItL
2
Proof. Use (2.12) to write
(2.17)
f dzdtb(z,tj91l q(xz,t)eixu. () ,
8.2. The pseudodifferential operator theorem 255
the integral interchange being legitimate, for uE Cp, hence u^E S.
The inner integral may be written as q(xz,Dl;) , using (1.7).
For an operator A=a(x,D), aE lt0 (general n) one finds that
(2.18) AZ'Ou(x) =iz = 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) =
ffl a(x,D+t)u(x)
Combining (2.18) and (2.19) we get
(2.20) Az,t = a(x+z,D+t)
Clearly (2.20) holds for a=q as well. Thus (2.17) may be written
(2.21) Au(x) = fdzdl;b(z,l;)Qz,tu(x) .
This leads into the following estimate for the inner product (.,.)
I(Au,v)I= IIbII
LJI(Qz,tu,v)Idzdt,
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(Yz)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} , u,v E Co1
with constants S,T. Here we let v=v run through a sequence in CD
with LZlimit Au/S2, concluding that IIAuIIs STIIuII , uE CD . Noting
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)
We get (2.12) with b(x,l;)=rlj=1{(1+ax )2 (1+a Also, the7 7
U and V are tensor products of the 1dimensional operators, their
Schmidt norms are products of 1dimensional Schmidt norms.
Corollary 2.6. In the general case of n dimensions, we have
(2.23) IIAII 2 n s cnsup{ Irf'1{(1+ax )2(1+a )2}a(x,U) I len},L(R) 3= j j
where
(2.24) I2ffdxdyl(1+ix)(1+a){ +ixx}I2}1/2
+ix x( 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
f1
(x+t%1ej,V 11(a(x+1e7, )  dt alxj3 0
(using (2.20) where z=t=O, without loss of generality). Therefore,
(2.26) (Vq.ja)(R)(x') = O dt
But the uniform boundedness of all derivatives of a(x,t) implies
that each derivative a(R)(x, ) is uniformly continuous over 12n
J1
(2.27) a(x,g) = h Xa(x+th,t) = lhlO(1)0
for example,where 0(1) is bounded for all x,h, E 11 Similarly,
(2.28)a(R)(x,
) = O(Ihl+lll)
Thus the integral at right of (2.26) equals
(2.29) 00111)
with 0(.) independent of Thus (2.23) implies
(2.30) lim1,0 1lV1.jAz.t
alz(x+z,D+t)II = 0
7
Similarly for all partial derivatives, by induction.
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
formula for in terms of A, first in the case A=Op(A)=a(x,D)
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 ) 2a(x,i;) ,
writing as product of 2 Schmidt kernels:
(3.2) p(x,U)4dl1 {(1+a,TF+f 77
In fact, both kernels
(3.3) Y 2
and
(3.4) u(x,rl) = (1+a1){(1if1)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 Fubinitype 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 trans
form 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*) ,
258 8. Smooth operators of L(H)
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
(3.10) Bz, = (1+az)2(1+aO 2Az,t E 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 definedmap TILT> 1Ut0.'To show it to be an inverse of 0 we prove that SA=Oimplies A=O. Indeed, then S is 11, and S(OS1)=SOSS=(SO1)S= 0,
hence OS=1 as well, so that S and 0 are inverses; S maps WT onto
Tyto 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
(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=E1 BEZ,t ,(i.e.(2.2) holds for B)Z,tWe 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) =
with 1P3=1{Y2(xj)Y2(j)}
All earlier arguments of sec.3 may be repeated: With the 1dimen
sional (3.3),(3.4) define Un, Vn on L2(1n) with kernels
(3.16) un(x,T1)= j) , E L2(12n)
8.3. The other half of the theorem 259
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
(3.18) a(z,1) = trace((Bz,tWn)Un*)
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 11 . 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) Jdzdl;ix =
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) =K 0 for all z,l;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 ndimensional (3.2) assumes the form
(3.22)
while (3.21) may be written as
(3.23) 0
using that
(3.24)*EZ'tF = e
iztFEt,z
Multiply (3.23) by T of (3.20) and integrate: Jdj may be pul
led out; its inte rand is L1(,2n)g . 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
260
f
8. Smooth operators of L(H)r
fdzdtp(z,l;)eizteitxe
(3.26) =
Jdzdt(p(z,t)eix4q(zx,tt)

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 (realvalued) 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 BB j=
Q1B + PjBQj , Qj= (1  Pj) . Denote the factor of at left
in (3.27) by X(B) , for a moment. By Schwarz' inequality we get
(3.28) IX(PjBQj)ls [[[u]]]'[[[(1Rj)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 [[[(1R1)v]]}2 = Lj=nJlxj(Y)12dy 0.
For the boundedness we use that IX(PjBQj)Is [[[U]]}[[[V]]]. Simi
larly 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
8.4. Smooth operators 261
z=t=0, by (2.2). Using that XZ'tE C(&2n,L(H)) conclude that
(3.30) 0 = f0(1+a )X etdt = X + f0X (1a )etdt = X t tej,0  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; pdocalculus.
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=
Tz1ATz of (1.5) is C"(&nL(H)) , and gauge smooth if the other
family A0' = M4IAMB. 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 pseudodifferential
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 alge
braT is *invariant  it trivially contains its adjoints, since
A* = A * , A* = A * , and IIB II; 0 implies IIB*II=IIB II  0.z,0 z,0 0,t 0, 3 * 3 3The *algebra 11UT no longer is a C subalgebra of L(H), but
it carries a Frechet topology induced by the natural norms of CB00:
(4.1) 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 Az,
=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,..
262 8. Smooth operators of L(H)
If A = Op(a) , then
(4.3) aZa Az.tOp(a(a))z,t , a(a)(z,t)=trace(p(azat)aya Az,tWU*),
by (2.20) and (3.8) . 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) IIalIks IIAIIks cilPalik, P=rIJ((1+azi)(1+at7))2.
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,01E Coo, and may calculate the derivatives in the
usual way. For example, azj(A Iz,O)=Az,O1(azjAz,O)Az,O1
, ... .
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, nota
bly regarding the Fredholm property of its operators, a 1V*algebra
behaves like a C*algebra.
We notice that the resolvent R(a.)= (AX)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 , assu
ming that A E 407 . In fact, the complex derivative dR(X)/dX _
R(X)2 belongs to qGT as well. Moreover, for the difference quo
tient Q = I(R(X+h)R(X)) we getQz
= h{(Az,t>vh)1(Az,t%)1}.One finds easily that limhOaZaQz, exists, for each z,t, uni
formly as z,t E 2n. Accordingly, the resolvent is complex differ
entiable within the Frechet*algebra 1IGT, defined on Res(A).
Commonly, for a (complexvalued) 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 inte
rior, one defines q(A) by the 'Dunford integral'
(4.5) p(A) =2n
existing in 11GT since its integrand is continuous. We have proven:
8.4. Smooth operators 263
Proposition 4.4. For A E WT and a function cp with above proper
ties 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) c(x,t) = 1Y1dydq
The integrand clearly is not L1(R2n) , and the integral was defi
ned 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))
(4.9) a 1C2f (xy,t11)e iyndydl
where again the integral is a finite part.
Different formulas are known for differential operators
(4.10) A = a(x,D) = IasNaa(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)
whence
(4.13)
(Dx.u)" , (DXu)" (U) = eau" (U) ,J
Au(x) = a(x,D)u(x) = xnfdfdyei(xy)a(x,)u(y)
264 8. Smooth operators of L(H)
A differential operator A of (4.10) never belongs to i1GT,
except, perhaps, if as 0 for a#0 . For differential operators
A=a(x,D), B=b(x,D) with symbols we have composition
and adjoint formulas given by the Leibniz' formula of differentia
tion, 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 1a(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 TaylorLeibniztype 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 TaylorLeibniz 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 wor
king with yrhMIP16 , where both pj>0 , and 0s <p, .
A calculus of ido's will be possible again for operators AE
L(H) which also are rotation smooth and dilation smooth, in addit
ion to the translation and gauge smoothness already discussed.
Here we refer to the groups On of rotations o:&n &n and
of dilations a:1Rn_&n, with an orthogonal nxnmatrix o (of determi
nant 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) 00*AO0 E C'(OnL(H)) , 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 i1GSconsists of pseudodifferential operators with symbol having all
derivatives bounded. We will see that the symbols of 4GS have
derivatives decaying at , and stronger so, as the order of dif
ferentiation increases.
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 nxn
matrix (in addition to translation and gauge smoothness). This de
fines yet another class LIGLC'WTC'WT of pdo's, studied in sec.5.
8.5. Operator classes 265
5. The operator classes 'TVS and WL , and their symbols.
As already observed, the classes 1IGS D WL of sec.4 are con
tained in 1IGT, thus consist of ipdo's (1.7) with symbol in CB00(12n)
The additional smoothness of Az, 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(xz) and gauge transforms u(x)eiz
of sec.1 generate a larger Liesubgroup of u(H) , we will write as
(5.1) GL = (g,z,1,(P) E gl}
where
(5.2)Tg,z,;,tu(x) = (det g) 1/2 ei(l;x+(P)u(gx+z)
, u E H ,
gl = {(g,z,1,(p): g E GL(In) , z,i;Ein
,(PE S1}
,
The group operation in GL (or gi) is best decribed by introducingthe 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
(5.5) (g 1,Xog 1).
(5.6)
For a pair of maps g(x)=gx+z, X(x)=1x+q intoduce the matrix
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 calcula
tion. Thus (g,?.) M defines an isomorphism between GL and the
Liesubgroup of the linear group GL (&n+2) consisting of all matri
ces (5.6). Actually we must calculate mod 2n in the variable cp ,
i.e., work with the quotient of the matrix group modulo the sub
group of all (g,z,t,cp)=(1,0,0,2kn), to obtain an isomorphism.
Speaking in terms of Liegroups we find that (5.6) describes
266 8. Smooth operators of L(H)
a representation of the Liegroup 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 rot
ation o and 0<oE I . Define gs={(a,w,z,t,(p):(ao,z,t,c))E g1}, where
0>0, 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 sub
groups of the unitary group U(H) , where GX , X=T,H,S,L, all are
unitary representations of Liegroups 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 Liealgebra (called ax) is described by the dis
cussion 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 Liegroup 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) = RgIARg , 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) ,
gusing the abbreviation g = g1, gt = (g1)t . Accordingly,
(5.9) Aug(x) = xn(det g
Another transformation of integration variable yields
Z(g)u(x) =Aug(gx) =(det g)xn elxg a(g t )d(5.10)
t
With another substitution of integration variable we get:
Proposition 5.1. For A = a(x,D) E WT the operator Z(g,A)=RgARgis in t1GT again. We have
(5.11) a(gIx,gt )
Note that the symbol (5.11) indeed belongs to lpt0=CB°
Now we can evaluate linear group smoothness; it amounts to
8.5. Operator classes 267
(5.12) Z(g) E C'(GL(len),L(H))
with derivatives existing in norm convergence. Clearly (5.12)
implies the existence of the partial derivatives
a "nn(5.13) (a
gil
)ll...(agnn ) Z(g111 ...gnn) at gjl=6jl
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) ag(g1)
_glhpggI ,
hpq= ((6pi6gj))i,j=l,...,nPq
hence
(5.15)
we get
agjl (gix)mlg=1
= (hjlx)m = 6jmxl
agj1 (hlj)m = slmmj
agjla(g
(5.16) = 4=1(alxm(x,U)6jmxl+ a,
m(x,U)slm
j)
Our attention is directed to the folpde's
(5.17) ejl = jalxlaxj , 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 para
meter) are of polynomial growth in (x,g), uniformly in the parame
ter. We will omit the function uE S in (5.19), derive a formal re
268 8. Smooth operators of L(H)
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 nxnmatrix) k we get
(5.21)DhgZ(g,A) = {E{Z(g+Ehg)Z(g)}}IE.
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) _
(Ej1b)(x,D), by (5.18). The left hand side is
{£{Z(g+Et)Z(g)}}IEa 0 = tpgagpgZ(g,A) , tpq (hg) p4 sjpglq
Together we have (summing over indices occurring twice)
(5.22) 6jpglgagPq
Z(g,A) = glgagJq Z(g,A) = (Ejlb)(x.D)
where
(Ej1b)(x,U) =
(g Ix.gtU)gpjp p
= {gr jglp'lraTpa(Y.1)  glrgpjYraypa(Y.TI)} IY=g' x,rt1=gtt
Summarizing we get
(5.23) (Ejlb)(x, ) = grjglp(Erpa)(glx,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 argu
ment 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 . More
over, 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
contained in CB"O (&2n) . Then we have GL = Opyi10 .
For WS we state a similar result, using the folpde's
(5.26)1ljl EjlElj= j)+(xjaxlxlax,) , j,1=1,...,n ,
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 ar
bitrary 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) [ap0'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 withtop
and Ep0 pulled
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 combina
tions 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 oncompact regions of GL(&n). This implies that A=a(x,D), aE tps0,
must be GL(&n)smooth. For example we get
t(5.29) Z(g(t))u  Z(g(0))u = dt(d z (g(i)))u , u E S
0dC
for a smooth curve g(t). Here the integrand I(i) satisfies III(t)II
sclluII, with L2norms, and c independent of u and t. First conclude
270 8. Smooth operators of L(H)
from (5.29) that Z(g) is continuous in g. Similarly all derivativ
es (5.13) are continuous as maps GL(&n)>L(H). Knowing this we omit
uE S in (5.29) and divide by t. As t0 get existence of the part
ials in norm convergence. Iterating we confirm GL(&n)smoothness.
Vice versa, if A EEL(H) is GLsmooth, in addition to trans
lation 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 skewsym
metric s . Thus, the derivatives (5.13) must be replaced by
a a(5.31) ((at)00(as )12... )Z(tes)
, at t=1 , s=0 , sT=s12
where only derivatives for sjl with j<l occur. For g=tes we get
(5.32)tatZ(tes) _ ljlgjlagjlz(g) _ ;jlpgjlgpj(Epla)(g x,gtD)
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(ases)Pgag
((g))jl jl pq
For s=0 we get
(5.34) as jeels=0 = hjl hlj , hpq = ((s 8 ))i NAP
vq µ,v=1,...,n
Accordingly, using (5.18),
(5.35) asj1 Z(es)I5=O = (EjiEij)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
tx,ot)
8.6. The Frechet algebras
= 0jr0lp(Tlrpa)(o x,ot5)
Accordingly, we have the following substitute for (5.24):
271
(5.38) as.1Z(eso,A)1s=0 = f,p=lojr 1pZ(o,Arp) , Axp=rlrpa(x,D) .
We need the CampbellHausdorff formula of ch.0,9, quickly re
derived as follows here: For sE N(so), a neighbourhood of so write
(5.39) es = e7 (s,so )es°, a. = log(eseso ) = _yn=1 (1eseso )m .
Clearly ?(s,so), is well defined (and holomorphic in the coeffi
cients s jl , so jl ), as long as 1eses0 1 < 1 . Since a=e8es0 ist
orthogonal we get e =a_X
, hence Xt = log(ek) _ % , so that X
is skew symmetric (and realvalued, for realvalued s,sO) . For
fixed so the function %(s,sO) has a local inverse at so . We get
(5.40) asj1Z(es) I s=so = <v(axlLvlasjl)(so ,so )aXµv
Z(e%e$0 ) IX=0
Or, with holomorphic functions xjlRV(s), j<l, t<v ,
(5.41) as.1Z(es)=;,<vKjlµv(s)2r,pltr 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) ((as)a12....)Z(es)
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 WeinsteinZelditch 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,
of restrictions of symbols aE ix0 to Qx&n={xE are subsets
of the Hoermander class S0,1,0(Q) (cf. [Hrz]). That is, we have
( 6 . 1 ) a(Pa) = 0(( )  I a I ) , as xE Q CC Q
272 8. Smooth operators of L(H)
Proof. It suffices to look at i,s0, since V10C iNs0. Then we get
(6.2) xE Q, aEl,s0.
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 = jko0a  Dl?jla =
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 witha(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 , ap0 = np0a = a i xp ' a0q = ll0ga= a I q '
apq = Ylpga , p , q = 1,... , n .
Then there exist symbols y 1 j (=yp 1 2 , y 2 ) E 2 .1 such thatpq e e pq e
(6.8) a 13 n 2jI x q=0 Y pq apq ' a l i = fp, q=Oy pq apq
Here we recall the symbol class Vam , at = (ml,m:) , defined by
(6.9) {a E C°D(12n) : O(( )m1ICI I(x)M2 ICI) }
For symmetry reason it suffices to discuss only one of the iden
tities (6.8). For the second one write (6.2), more explicitly, as
(6.10) apax a=a00+ ?pap0 , ( ptl_ q 1 1 )a=apq+(xpag0 xgap0)
p p p q pThen (6.3) may be replaced by
(6.11)
ppap0)+
pp(apj+xpaj0xIap0)
by the same derivation. Dividing (6.11) by we obtain
a formula of type (6.8) for alb , noting that the quotients7
8.6. The Frechet algebras 273
xp/() 2 , q/( ) 2 , 2 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
(6.12) IRI s j s IaI , 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 intro
duced 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 inter
changed. We also get an xdecay of derivatives on compact asets.
The inclusions of (6.13) are proper. For example, let n=1 and
(6.14) x ,
where x(t), w(t) E COO(&), x=1 for tal, =0 near 0, w=1 near t=1, =0
in It11 a 1/2 . Clearly all derivatives of a , for x and t , are
bounded. Moreover, a and all its derivatives vanish, as X is out
side the interval (1/2,2) i.e., unless we have xs 2T , and s 2x
both. In other words, we have (x) s and s 2(x) in supp aAccordingly 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 talkxalx = 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
274 8. Smooth operators of L(H)
the countable class of seminorms
(6.16) Ile pNgN ..... eplg1 allLoo =IIaII nK 79=(p1,...' N), K=(g1.... ,gN)I
with epq of (6.1) , for p,q a 1 , and
(6.17) ep0 = 1p0 = ax p ' EOq = 110q = a q , (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 seminorms
(6.18) IIVaa(x,D)II = IIIIaIIIIa ,
with the L(H)operator norm II'II , and VaA denoting the partial de
rivatives of Ag,z,t
, or A6101z,
t , or Az,t , at the unit of the
group. Here a will be a multiindex of the dimension of the group.
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 aJ J
with some selection ePjgj , corresponding to the multiindex 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 theejl
include the expressions (6.15).
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 with1 pjgj
the corresponding combination VU of ag , az and a. on Agizi. atpq p q
(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 seminorms 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 make0(or WX) Frechet algebras.
8.7. Polynomials in x and Sx 275
The adjoint invariance and inverseclosedproperty follow from
(6.22) A*g,z,S= (Agrzr0 * I
(A 1)g,z,l; _ (Agrz101 .
If A is differentiable in L(H) then so are the right hand sigrzrt
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
,x of all polynomials in the 2n variables x, with coefficients
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) y)x {a E C00(1 2n) : (x) a(x, ) E px0 }
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 theVx 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) nm(x,U) = Q ) , m = (ml rm2) E &2
we have the estimates (7.4) for all products 1"1=1I1pJqJ of3
(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 µvreplaced by ERv
Proof. Consider x=s only. If aE Vsm , then anl=bE VsO. Hence
(7.5) llpga = (1lpgb)nm + (b(npgnm)/nm)nm
Note that npgnm/nm E yicO. Thus Y1pga=O(nm). Moreover, (7.5) implies
that11
pgaE Vs m, since ,cOC s0, and since Vs0 is an algebra. Hence
we may iterate, for (7.4). Vice versa, let (7.4) hold for a. Then
(7.6) 1lpgb _ nmllpga + b(iipgnm)/nm) = 0(1) ,
using that r1 pgnm/nm 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,
(7.7) E'Wxm+r(e'e2),as dal s r s 1PI , r = integer.
Proof. First let JaJ+JPJ=1. Prop.7.1 shows that 1xm is left invar
iant 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 inr(e'e2)
r=t1,
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
Zn,msm' with Zn,O=Zn , may be introduced, replacing O(.) in
(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 or
der 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,x0In 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
(7.11) [xj,A]=a3(x,D)=ial,j(x,D) ,
satisfies stronger estimates of WeinsteinZelditch type
(7.12)O(((x)/() )1) , IRI1 s 1 s lal
O(((x)/( ))1) l s 1 s lal+1
Similarly for higher order commutators and higher derivatives.
Hence yw is an algebra under a°b of prop.7.4, since in a pro
duct 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
vxm°1m'C m+m' , i.e., 1Vx is a graded algebra, under 16" .
Next we look at calculus of pdo,s, for the algebra OpVs, si
milar as in 1,6 for 0M,P,o We have yixmC y,tm, hence may use the
LeibnizTaylor formulas I,(5.7) , I,(5.9) for product and adjoint
in Opts. The question remains whether (in what sense) the remain
ders 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] , K = k(x,D) , A = a(x,D) ,
where aE yixm, kE Cm 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= 4=0
I
j=j
(i)j/8! (k(e)a(8))(x,D) + RN
(7.14) AK = 4=0l
(i)J/e! (a(8)k(0))(x,D) + RNe j[K,A] = Lj=1(i)j/j! (k,a)j(x,D) + RN
where k(8)a(0) ,and a(8)k(0) , and the 'iterated Poisson brackets'
(7.15) (k,a)j =lef
,(e)(k(e)a()k(0)a(e)) , (8) = j!/e!
belong to Vxm+m're'r'e2' for all r,r'=0,1,2,..., r+r'=j, while
R m+m'+re'+r'e2 ,for all r,r'=0,1,..., with r+r'=N+lNE
Proof. Note that k(e)E 1lcm'je' ' a(A)E m+r(e' e2) ,r=0,...,jas lel=j, by prop.7.2. Their products are E 1Vsm+n'(jr)e're2'
proving the first statement. In the remainder of I,(5.27), i.e.,
_ N+1 1
(7.16) PN (N+1)IAlN+1(01)
P 0,N' Pe,N0
(1t) Ndx IO,N(x,lj,t)I
278 8. Smooth operators of L(H)
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 = (1Ox)Ma, and the usual formal partial integration.In 1,5, by a somewhat difficult interchange of limits, we pro
ved (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 analo
gously. We need an estimate for I. They are in the span of int
egrals (7.18), with replaced by
(7.19) rq) a , 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' N1Ia, xY)m12r( )mi+r,
where la"I s r s N+1+I3"I . Here we can estimaterr
J dy(Y) 2M(x_y) M'2 r
= 0((x) m' Z r)(7.21)
JdTI(1I)2M(zTl)m, N1Ia' I = O((t)mI N1Ia, I )
using I,(6.5). Note (7.21) holds uniformly in t, 0sts1. The estim
ates remain true if. a'=a"=13'=13"=0. Hence,
(7.22)(a)
O( , : ) , r+r = N+1.22) (P) nm+m,rer e
for all multiindices a,3, giving the corresponding for the symbol
rN of RN . These estimates are weaker than the stated ones. But,
(7.23) rN = 71 (i)IOI/61k())a(6) + rRN+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'ezje' by (7.22). Thus,
' 1(7.24) 1 =111pjgjr2j E Vtm+m' re' r' e2 , as N' s j
8.8. Characterization by Lie algebra 279
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)Anms(x,D) : A E WX }
Theorem 7.6. The space Op*xm is characterized as set of all AE
O(m) such that for all T=Tg,z.,.W
of (5.2) the operator T1AT bel1
ongs to the subgroup GX of GL ,
AT
AT defining a C *map gx O(m)
in the Frechet topology of O(m), as discussed in 111,3. Moreover,
a sufficient condition for AE OpVxm is that (i) AE L(Hs'Hsm)
(ii) T1AT isC00(gx,L(HS, Hsm) 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, 910 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 NxNmatrices, 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
Ae , for AE A , by the CampbellHausdorff formula.
Now focus on the 'representation' A > EZ,t,q)
of our Lie group gh of (1.11),(1.12) on the «dimensional space
L(H) . Actually, two representations are involved: We represent
gh (=&nx&nxSl) by (z, ,g) EZ.t, ei((P+K)TZ=ei(T+Mi;)eizD on H (as
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 repre
sentation as the linear span (called AT=AH) of the folpde's
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 ix tIs 3 } , {elt} , {e j }, respectively. These groups consist of
unitary operators of H, thus their generators are skewselfadjoint
each having an orthogonal spectral measure.
Notice that the precise domains of the operators (8.1)
are not identical. Linear combinations no longer are skewselfad
joint. However, as easily seen, the selfadjoint 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 skewself
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 CampbellHaus
dorf 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 an
other representation of the Lie algebra AM , as the (also unboun
ded) operators on L(H) , formally given as linear combinations of
(8.3) ad axj
, ad ixj , j=1,...n, with (ad X)A = [X,A] = XAAX .
(8.4)
Indeed, let then, formally,
dt(eiLtAeiLt) _ 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 take7
A to a bounded operator. In turn A Az' is Co. if and only if AE
OpVto , by the 'tpdotheorem. 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= xjAAxj trivially is a map S'S'. Similarly
8.8. Characterization by Lie algebra 281
(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 areS S'. Thus the statement is meaningful. The proof will be a veri
fication 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 un
itary 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 (un
bounded) directional derivatives of T(g,X)=T at e=(1,0,0,0)
of gl, in directions allowed by the subgroup. In this way we get
AL and AS as the real linear spans of (respectively)
(8.6) xlax + bjl/2 , ax , ixl , i , j,l = 1,...,n ,7 J
and
(8.7) 4=1xkaxk +n/2 x1ax JxJ
,az1 ,j<l,ax ,ixl,i ,j,1=1,...,n .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 CampbellHausdorff formula, of the form
(8.8)
and where
exP (iDY
)T9,z° .V , , where g=eY , z° =ei (y ) z'V
(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 XH :
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:SS' 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 CampbellHaus
dorff formula (0,lemma 9.5) offered in 0,9, with pbm. (1) above?
Chapter 9. PARTICLE FLOWS AND INVARIANT ALGEBRA
OF A SEMISTRICTLY 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 trans
forms. 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 AEL(H) with At eLtAeLtE C_(R,L(H)) for every LEAX In new coordi
nates 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
atv=L"v. Thus "AE 'IGX" means A tetL AetL E C"(&,L(H)), assuming
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 eMtAeMt for certain M  the transforms of LE AX
back to the old coordinates. For coordinate invariance we must
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 skew
symmetric, 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 hyp
erbolic equation atu=Mu , of the form studied in ch.VI.
We always will require ME Opyce M+M E OpVcO so that theexistence theorems of ch.VI are applicable. For a detailed discus
sion 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=eMtAeMtE CO'(&,L(H)).
We prove this for a rather general class of hyperbolic t'ide's
t plyce, KK E Under proper assumptions on Ka u=iKu with KE O
we get eiKt SeiKt=fix, as for Oplyco in VI,5. In fact, Optpx =iKt iKt iKt iKt
me Opyucme and even a Ae E 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 vxvmatrix of pdo's. To include the Dirac equation (ch.10) we
require the system atu+iKu=O to be semistrictly hyperbolic of
type e° , a generalization of a concept of VI,4: The eigenvalues
of (the essential part of) k=symb(K) must be real, of constant
multiplicity, but need not be distinct (sec.5).
It is no longer true that 1W%v (or Opxv) is invariant undermconjugation with aiKt. Rather eiKtAeiKt remains a 4 do in Optxvmonly if A belongs to a certain subalgebra P=P(K) of Opwxv. The exi
stence of this invariant algebra P gives raise to physical specula
tions 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
that its symbol matrix a=symb(A) commutes with k=symb(K) , modulo
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 relativi
stic motion of the particle under the potentials imposed.
For 1dimensional 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. Flowinvariance of VI..
In this section we assume a scalar realvalued symbol k=k1E
284 9. Particle flow and invariant algebra
yce. Let 'fit be the flow of VI,(5.18), for this symbol k. In VI,6
we proved that vt:142n12n satisfies at=aovtE yicm for aE 141cm . Herewe 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 tat + (k,at) = 0 , t E & , a0 = a ,
with the Poisson bracket (.,.)_(.,.)1 of VI,(5.16). Note that
(1.1) is a Cauchy problem for a first order PDE in Our con
ditions on k will insure that at solving (1.1) gives a symbol atE
11 , for all tE & , a E 1 .
Assuming WE V10 we know that lk q a is bounded for finiteij
such products. We apply epq (or a finite product) to (1.1), for a
system of equations in a pqt=e pqat , a pgrst=ers epqat , etc. Then we
will derive apriori estimates showing that apgt, ... remain boun
ded in t,x,l . As a preparation we note the following.
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"Ix Epgjx a,,
with the functions
(1.3) Epq =  as p,q 2Ep0
= 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) + D,pgrsarst =0
'p,q=0,...,n, (p,q)#0.r
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
9.1. Flow invariance 285
teristic curves of the common principal part of all equations,
That is, along the particle flow vt , above, we get a linear homo
geneous 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 .
Since the coefficient matrix ((kpgrs)) is bounded in all variables
x, E 1n , t E It , it follows that apqt are bounded in x, , over
compact tintervals (cf. the detailed discussion in VI,6). We get
(1.10) y(t) _ 2 1a (t)12 IaIscy , y(0) = E 1a I2= 0(1)pp pg pq Pq
c independent of t,x,g, which may be integrated for y=O(ect). Thus
(1.11) at, axPat, a
qat, Epgat pa qat
xgaxPat' p,q=l,.... n,
are bounded in every compact tinterval, as In .
Clearly the procedure may be iterated, under proper condit
ions 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 , implying epcpgrsE
VcO . For the fourth term we get
r
%parsersapgt=P'parsapgrst
using (1.6) again. With this, and the relations (1.5), (1.7) get
(1.13) atapgrst+(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
particle flow with coefficients bounded in Again it follows
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 tderivatives are bounded as well. We proved, for m=0 :
Theorem 1.2. Assume that the realvalued symbol kE Vice satisfies
condition (1.6). Then at aovtE COO(2,i10), for every aE 1lm, mE V.
286 9. Particle flow and invariant algebra
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
Xpgrspa ' etc., are bounded as before. However, the initial values
are Thus, the resulting estimates now will be
(1.14) atapgt' atapgrst ' ... , = O(nm'vt(x,))
However, in VI,6 we proved that the right hand side is
This completes the proof of thm.1.2 for general at.
The class of all symbols kE e satisfying (1.6) will be den
oted VXne. Note prop.1.3, below, describing a subset VX eof
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
Then V), eC V%ne The (realvalued) functions of VXe
form a Liealge
bra over M (k) under the Poisson bracket product (kl,k2), contai
ning the a Jl of (1.3), and we have epq: VX e)XeThe 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) x00x + kd + km ,
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
Theorem 2.1. Let the realvalued symbol be in ryane, and let
a E ilsm Then, with the characteristic flow vt = of (0.2)
and at = aevt , we have atE C00(1k,4m).
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,
9.2. Invariance under particle flows 287
similarly than the ejl in sec.1, such that
(2.4) 1jla = (rljl,a) , j,1=0,...,n , r110 = x1 , '1O1
Proposition 2.2. The (real) linear span of the functions 171, as
well as the class yPe , are (real) Liealgebras 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 &, Then
(2.5) (nra) = Pnr2e( 7' 71P (11pga) + 1100(1100a))p<q
The proof is a calculation
nr a) = P( x) p2p ja l  P( ) P(x)P2D[ja
I, j(2.6)
2 z= Pnr2e pq(xpxgalxq  gxpal p)
xpgq(xpalxxgalx + al,  gal, ) + 7, xg g(xpalx tpal,
pq q p q P pq p pwhich implies (2.5).
For a more general class of functions in "ewe introduce
(2.7) tjl = Ijlne , 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 multiindexnotation, writing
a(2.10) O(t) _(a,a = 'Yan_Iale rla a = '1=0tj1 J1 , YaE T
288 9. Particle flow and invariant algebra
with an (n+l)2dimensional multiindex a = (ajl) . The next obser
vation is that every application r1N rl b = b" again is of theJ_
1 pjgjform (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, rl0gb0 E "e2 . One obtains
(2.11) loon Re= µ(nµe2e2 nµe2e') fljlnµe = 0 ' 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(lv)e (rljlla)).
For j,l s 1 the first term vanishes ; for j=1=0 this term is in
ice, + 4e2 , hence will go together with bd+bm . We must focus on
_ Pq(2.13) n(1v)a nj11a
= (1v)a 4q=1a'pglaE (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 isof 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 (1v)e 11a,a) = a) + n(1v)e zpgyla Epgrlpga .
The last term indeed is a combination of the gpga, coefficients
E tpc0. For the first term apply (2.5), with r=(1v)e. Q.E.D.
Define Vanes nmeyiane, and yianm'=nmevane' , for mRe. Thenyianm consists precisely of all cE 1pcm with c " =rl lpjglc satisfying
(2.15) (C ..) = jl 1jl . Y jl E me
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 . Similarly
for (cd)" , which gives the algebra property.
Proposition 2.5. Every continuous root of
9.3. Congugation with eiKt 289
(2.17) 0(X) = L=09 XJ ON = 1 , 8j E po 0 ' j=0,...,N1,
may be extended to a symbol XE yxt0 , with all ThlPjgj
XE V= O, pro
vided that for large IxI+ItI all roots have constant multiplicity
and their mutual distance is bounded away from zero.
Proof. For a root with constant multiplicity p we can write
(2.18) (2xip)1f i0'(t)/0(t) ,
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'(tApq
Substituting into (2.18) one obtains
(2.21) ( X, ) _ jl1jl , Xjl = (2xip)1 f tdt (Y 3 1l0ely jl)/02
with path of (2.18). The path is kept constant during differentia
tion, the integral being locally independent. For t on the path,
(2.22) Ie(t) I = Iri(X jt) 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.
Since 0 and 0' are Vc0 , but Yj1E IPc_e , we get kjl=O(xe).
The formulas for Xj1 may be differentiated under the integral sign
for Xj1E tpCe. Also we get expansions for rhlp q X : Apply rjpq toj
J
(2.18), using estimates as above. Q.E.D.
Corollary 2.6. Let k E tVaxpe . Then we have
(2.23) (k, a) EVsm+(µ1)e for all a E Vsm , m E &2
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
290 9. Particle flow and invariant algebra
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 SS (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 sim
ple nontrivial case. Slightly weaker conditions on k will be nee
ded in sec.4, to prove coordinate invariance (cf. cor.3.4). A res
ult 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
complexvalued xE0 , and a realvalued koEa,Then,
for every A=a(x,D)E '1m , me &2 ,we have AteiKtAeiKt E Op1lm
tE I. Moreover, we get Ata(t,x,D) with
(3.1) z E 11e, n yle2
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 pdocalculus of VIII,7. Let
(3.2) P(t) = eiKtat(x,D)e iKt
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 atE yurm
and at(x,D) is meaningful. Note
that this is the only direct reference to x=s or x=1 . We get
(3.3) dP(t)/dt = eiKt(i[K,At]  (k,at) (x,D))e iKt ,
where the operators are S > S . By VI,thm.3.1 at(eiKt) exists in
L(Hs,Hse) , and by sec.1 (or sec.2) atat(k0,at) exists in yucm
Hence dP/dt exists in L(HS'Hsme), for all s, and (3.3) is valid,
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+ere'r'e2' r+r'=j, and the remainder RN has symbol in Vxm+ere' 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).EVxmre' r' e2 ' r+r' = j1 , hence RNE Op Xmrei r' e2 1 r+r' =N . For x=1,j=1 get 1=(Egpa) =cgpaE yrxm, while (kd,a) , (km,a) belongto Xm+e' re' r' e2 ' and Vxm+e2 re' r' e2 l resp. With r=1 (r=0) getall terms E Vsm, as stated. Similarly for j=1, x=s. For j>l the
9.3. Congugation with eiKt 291
first sum (1.15) gives 0, since Epglxx epgl qpglxxq pgI 0. The
other terms in (1.15) go as for j=1. To summarize:
Proposition 3.2. Under the assumptions of thm.3.1 we have
3 3(3.5) cA 4=1(i)7/j! (k,a)j+rN , rNE mre'_r,e2 ,r+r'=N,
where the jth term belongs to Vx mre,_r,e2, r+r'=j1.
Applying prop.3.2 for N=1 we get
(3.6) icA (k,at) nvxm_e2At
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 cal
eculus of ydo's between y'c and ix that y is continuous as a map
Hime' (or 4w m_e2 ). Actually, we even get it COO: One knows fromsec.1 or 2 that at is C" , and the asymptotic expansion (3.4) (or
LeibnizTaylorexpansion (3.5)) may be differentiated arbitrarily
for t. Then prop.3.2 may be applied to the differentiated expan
sion. Hence we even get y E C00(I,Vtme/2 ). Again this gives
(3.7) P(t) = A + JtdieiKty(t,x,D)eiKt
0
Or,
(3.8) eiKtAeiKt= at(x,D)  ftdieiK(tt)Y(t,x,D)eiK(tt)
0
This is a decomposition like (3.1), but the reminder (second
term) is not in iyidoform. It is norm continuous in L(HS'Hsmf)' f
=e', e2, e/2. Also,d't(e_iKtAeiKt)
exists in L(HsHsm), 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
(3.9) (t,x,D) f dx'eiK(x' x)W(x,,t)eiK(x x0
with a ydo W = w(x',t,x,D) obtained by substituting y for a into(3.7) . Clearly it follows that wE C00(12,1Vxmre' r'e2) , r+r'=2using (3.6) . Also, yx E C00(1e2,'wxm_f), f=e' , e2 .
With (3.9), for x=tt, in the integrand of (3.8) we get
(3.10) eiKtAeiKt = at(x,D) + zit(x,D) + W1t'
where Wit E L(HslHs_m+re'+r'e2),while the symbol 1;1t is in
C0D(H,1xmf) , f=e' e2 . An Nfold iteration then yields
(3.11) aiKtAeiKt = at(x,D) + 4=1zjt(x,D) + WN ,
292 9. Particle flow and invariant algebra
with symbols zjE C"(&,"mre,r , e2), r+r'=j , and with a remainder
in O(mre'r'e2) , r+r'=N+1 . Moreover, the remainder will have
all derivatives for t of order up to N+1 existing in norm conver
gence of every L(Hs'Hsre'r'e2)' since it is an N+1fold iterated
integral over a bounded, strongly continuous integrand.
Finally we take the asymptotic sum
(3.12) zt = Lj=lzjt (mod Vt00)
It exists, because tjtE yetje/2 follows as above. Then eiKtAeiKt
 at(x,D)zt(x,D)=Vt clearly is of order . Its tderivatives of
all orders exist and are of order  again. Thus Vt vt(x,D), vtE
Cl2n(1Q,S(& )). Also all tderivatives of the asymptotic sum z
iKtt iKtexist in 4wmf, f=e', e2. Writing zt+vt we get a Ae
=at(x,D)+z(t,x,D), which amounts to (3.1). In particular, we have
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) , and a function cE C" (B?), with theclosed unit ball B?C 1n , as in IV. In other words, let
(3.13) k= (x) q +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 [c0coefficients.
Proof. It suffices to focus on the case
A calculation shows that epq:w7,,Key Vx1ce . Thus we only must show
that (c, .) =2' a pq . But (al, . ) =aa 26a =aa ja eq pq q xq lxl q1 xq lxl ql
+Y,alx xlax . Here alx E pco, so look at sax yalx xlax =(x)laax1 q 1 q 1 q q
(s()+xl(sla(s)((x)a(s))Ix1
)axq. Finally, ((x)a(s(x)))lx1=(sl(y)M
+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.
9.4. Coordinate and gauge invariance 293
Looking at prop.3.3 and the proof of thm.3.1: In the case x=
1 we clearly get at well defined again, and E E CC00(&,1V1m) . 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=jl
for again. First
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 =20(s(x))1xleglaj j
+Y,O(s(x))lxixlaxga J . We get a(s(x))lxl=yalm(s)(slm slsm)/(x)E
1Vce2 , hence the first term is in Vlme, n vime: 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 Smanifolds.
We now will look into coordinate invariance of the two alge
bras 1I.L and Y14S and more generally, Op1V1, Opys. Given a globaltransform x=q(y) with a diffeomorphism sp:&n In
, inverted by 1V:
&n In :We have sp(x(x))=x(W(x))=x, xE &n. Assume (the components
of) p(x) , x(x) are Coo(&n). The Jacobian matrix ((a(P,/axl(x))) _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) , (Txu)(x) = u(X(x)) ,
for functions u over In . Assume 0<c, s ldet J(x)Is c: , with con
stants, to insure that TT TX : H > H
If a folpde
(4.2) L = Yaj(x)axJ + a0(x) , aj E C00(In) ,j=0,1,...,n
294 9. Particle flow and invariant algebra
is given, then we get a transformed folpde L" = TgLTx . We get
L =  j(x)ax +a"0(x) , a"0 = T,,a0(4.3) j
a j(x) = Y,(axj/ail)((p(x))al(T(x))
The exponentiations eLt and eL t are related by
(4.4) eL" t = TtPeLtTX
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=q1, with L=ixq ax xgax +2Spq, in case ofWL.P p
For WS we use L=ixq ax , xpax
xgax , and Ex3 +Z . We mustpq p J
verify that the transformed expressions satisfy the assumptions.
as outlined in sec.0. We consider Sadmissible transforms, as in
IV,3, only: With s(x)=5, s:I B1, of IV,(1.5), assume that
(4.5) tP = slo'ylos x=c 1 = slo eos
with diffeomorphisms p:B1 u.Bl , 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=AIIMEE Optpx0 ='WX. Then TWBTXE tIGX
follows. Also, T BT =(T AT )(T P_ T ), where the second factor
belongs to Oppc mC OpVx_XAlso T P T is inverted by T P
MT E
OpVxm. Hence T(P ATx = (TgBTx)(TTPMTx)Ex
For thm.4.1 below we must focus on the expressions M=Mpq
(4.6) MOq ixq(x) . Mp0= Irojlx (x(x))axJ p j
Mpq 2xq(X)Tjlxp(x(x))axj + ZOpq p,q=1.... n
in case of x=L, and MOq , Mp0 , I 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 osIE C"(B?) , and x=sIoOos , hence Tjlx (X(x))P p
l )o6)os(x) =a.
(s(x)) , with=TjIx
o(s_0 Oos)(x) =(((Pjlx os 1pP
9.4. Coordinate and gauge invariance
ajp=(cPjjx os1)oOE C°'(Bcl)p
295
On the other hand, (x)1cp(x)= yr(s(x)){(x) 1 V(s(x)) 2}1 ,
where the second factor may be written as 1,IS, }1/2
E C"(B?) ,y(s)2
1
just as in IV,(2.8),(2.9). This shows that Mpq ikpq(x,D)+x0, where
(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
(4.9) p(x)=gx+w(x) , x(x)=g1(x)+v(x) , gE GL (&n) , w,vE "o
Then (4.8) is valid  i.e., TT and TX
leave OpyJlm invariant. More
over, if in addition we have goo , 0<aE & , o orthogonal, then
also WS and OpVsm are invariant under T(P and TX
.
Next let us turn to gauge transforms. A calculation gives
(4.10) L= eiµ(x)Leiµ(x) = L + i
Pp(x)µlxp(x)
Applying our arguments to the generators ixq, ax , xgax +P p
26pgLpq again we now must prove eMtAEMtE C00
&,L(A), for AE t1Gx,
where M=MpgLpq+Rpq , tOq 0 , tLp0=iR I xp , µpq I A I XI xp.
Theorem 4.3. Let the 'gauge function' µ(x) be realvalued and let
a (x) E yice2 . Then Op1Jixm, mE &2 , X=S ,I, are invariant under
(4.11) 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 tothe 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).
296 9. Particle flow and invariant algebra
Repeating the discussions of IV,4 with W1m instead of gene
rates classes called LLm of 'natural pdo's' on an Smanifold 11 .
On the other hand, we do not expect invariance of OpWsm un
der general Sadmissible 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 matrixvalued K=k(x,D) .
In this section we take up the question of extending thm.3.1
to the case of a vxvmatrix 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
yes , from now on, but note that most arguments will also work for
'Pl. Generally we assume kj1E Vae (cf. sec.2). Instead of requiring
(5.1) atu ik(x,D)u = 0 , u=q at t=0 ,
to be strictly hyperbolic of type a°, as in VI,4, we will accomo
date 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) semistrictly hyperbolic (of type e°) if
(i) We have kea=neako, KE Wceae,
with entries of the vxvmatrix 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 vxvmatrix is diagonalizable for all fzIxl+
f1 I Zrl0, sufficiently large q0. All eigenvalues Xjof kea are real, and of constant multiplicity vj (independent ofx,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=keanea have the
same spectral projections, and the eigenvalues of k0 are given
by lAj = kjnea . We may arrange for
(5.3) vl+...vP=v, f2 lxI+fi ZtIO
As our third condition, we assume that
9.5. Conjugation for a matrixvalued K 297
(iii) 260 , f2JxI+f1Ifl 2110, j=l,...,pl
with some positive constant 60 independent of x, .
We only will admit the weight functions ne, mentioned, al
though 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 semistrictly 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 top
ology of L(Hs,Hsje,). For every sE &2, and j=0,1,2,..., we get
(5.4)at(eiKt) = KjeiKt E 0(jf)
(5.5) r_e(x,) , r_eE 1Va_e
using the spectral projections to of
f2 xl+f1 I 22rI0 , extended to symbols (with coeffivients) in Vco.The discussion follows the proof of VI,thm.4.4, with small amend
ments, left to the reader. The point is formula VI,(4.19), i.e.,
(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 argument 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
conjugation A*AteiKtAeiKt . Letm
(5.8) f ()J ' (0 0)
298 9. Particle flow and invariant algebra
Then clearly k E °e , and a E But a calculation shows that
\(5.9) e1KAe1K =( 0 0) ,
Clearly P = p(D) , p( ) = e21(x)
with
E Vt0
p = 21(D)
but p 0 Vs0
On the other hand, as a simple consequence of thm.3.1 we getiKt iKte As E OpVsm if and are diagonal matrices for
large IxI+IgI. If only a and k commute for large IxI+l I we can
expect (3.8) with an error term Zt (the integral at right) of low
er order, not necessarily a y,do (cf.also [T13]).
Thm.5.4, below, shows that even the error Zt becomes a 4do
E Op sme° ' e° =e` , e2 , if only a suitable 'correction' of lowerorder 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 semistrictly hy
perbolic 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
(5.10)e At = eiKtAeiKt E C(&,OpjsV) , in case of a°=e.
For e° =e' , or e° e2 we replace (5.10) a by
(5.10)e° atAt E C (H,OP1S jj = 0,1..... .
Then we define, for a°= e , e' , or e2 ,
(5.11) P = PK =U
PKKmEV
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(eiKtAeiKt)It=o
)
Especially, if AE Pm, BE Pm , , then AB E Pm+m' Moreover, if AE Pm,
then (ad K)A E Pm(ee ,)' 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)(ee°)
Theorem 5.4. Let (5.1) be semistrictly hyperbolic of type a°.
(1) For each vxvmatrixvalued 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 matrixvalued K 299
is an operator in Pm . That is, in particular,
At = eitK AsitK E OpVsm, t E & , At = at(x,D)
(5.13)
atat E C'(L,'V)S _7(ef)) , 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
aja of ((ad. )iA) . In particular, ajz Vsv
t t iK t t t mj(ef)e(3) If, for any q with (5.12), the symbols z1, z2 both sat
isfy (1) then z1z2 satisfies (5.13) for me , instead of m .
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 commutatordifferential equation in sec.6.
The particle flows of the eigenvalues . (x,t) will be impor
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 Pme . Thus, indeed, z3 allows a decomposition (5.13) for me.
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
(5.15) At +i[K,At] = 0 , t E & , AO = A .
(We first get (5.15), applied to uE S , due to differentiability
of a*iKt in L(Hs
H se) (thm.5.1). Then, since all operators invol
ved are ydo's, we get (5.15) as an equation for Ado's.) Using cal
culus 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)=albbixbl alx , (a,b)j= Ie=j8'(a(e)b(e)b(e)a(0))
300 9. Particle flow and invariant algebra
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 "mrei r' eZ 'r+r'=j1, under our assumptions on k and at. Thus (5.16) implies
(5.18) [kf,at] = rt E 1Vsm .
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.33atp.
+ j7lpjrtpl/(kjkl) = qt + zt
(Just note that at Ppjatpl , and that pj[k0,at]Pl =(1jX1)pjatpl
leading to pjatpl = pjrtpl/(XjX1) ,as j#l , and the solvability
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 jT,)lE 1Vce , (possibly after a correction forsmall f:IxI+f,ll;I), using (iii) above. Moreover, since pj , Xj are
independent of t , we may differentiate (5.19) for a decomposition
of the symbol of Bit , Bj=(ad iK)1AE Op1Vsmj(ee,) using
(5.10). This proves (2) for e°=e.
Next consider e° =e' , noting that the case e° =e2 handlessimilarly, with x and reversed. Again get (5.16), where now wehave kf=kdE 1Vice, , however. Also, atE 1Vs e: , by (5.10)e' Hence,
(5.21) (k,at) j E 1Vsm+e' re' r'e2 r,r'20 , 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) ofour e'semistrict hyperbolicity, we get (5.18), with rtE
But we now also have Xi E 1Vice, , instead of Xi E 1Vice . Thereforethe second term at right in (5.9) still is a symbol in 1Vsmeagain, 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 yisme 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 'Wsme, fl yisvme2
(In particular we know from (2), already proven, that ztE Vsme
Also, the only additional term of (5.16) to be taken into (5.22)
for a relation mod Vsme will be i/2(k,q )2. We will not use this
term, at the expense of a weaker (5.21), resulting in slower asym
ptotic 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
(5.23) [kf,zt] = [K,qt] + i(qt + (kf,gt)) = cpt
Assuming that(Pt
is known, for a moment, (5.23) implies
(5.24) zt = Llplztpl + i j7, pj (qt+(kfgt)+i[K,gt])pl/(),jkl);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
(6.1) [b,w] = d ,
for general vxvsquare 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 dimen
sion independent of x, . As used before (sec.5) the eigenvalues
A,
3
may be arranged as C"(c)functions: a.. is a simple root of
q(p1)(A,)=0, with q(A)=det and the multiplicity p of
X. Hence aq(p1)(Ai)#0. The implicit function theorem yields )vj
E C°D , locally. The real distinct eigenvalues may be ordered by
size, giving globally defined C"(a)functions. For X=Xj ,
302 9. Particle flow and invariant algebra
(6.2) p = i/2nJ (b(x,t) 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
(tdepedent) 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) areC"matrix functions, and (.,.)=(.,.)1 is the Poisson bracket.
(6.3) d = g + (b,g) + [d^ ,g] + d" , g = atg .
For given X , p let T1(x,i;),...,q) p(x,l;) be
a biorthogonal 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) )Consider the local matrices Y=((Yjl)), S^=((6jl)), b"=((% 1 )) of
g, pd^p, pdp, (all leaving S invariant): In detail, yy(t,x,l;),
6^ (t,x,i) , S" (t,x,l) are defined by
(6.4) pgp Y lp j 1 , Pd"j,l j
P= 6j1Pjl Pd P sj1
with Pjku = cpj('y u) , u E CV , Vk = , 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 . , of the form
(6.5) p(gp)'PIS+[v,g0] +d"0=0, d"0 pdpIS ,
where
(6.6) n , of = at + %Itax  xIxat , v = (pd^P + P(PItbIx)P)IS
Moreover, with the matrices y , 6A, 6" of Y0 , d^
0=pd^ p I S ,60
with
9.6. Commutator equations 303
(6.8) a=((e jl)) , O jl=Vi'(p1+
l , 11111=d/ds.
The restrictions g0 , d^0 ,
d"0clearly are linear operators
of S . Prop.6.1 implies that, under above assumptions, cdn. pdp=O
involves only g0, not the restrictions of g to the other eigenspa
ces, hence a 'decoupled' system of p2 DE,s results for each eigen
space 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 con
sider 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) PkgrPk = PkgrPk = 0 ' PkgrlxPk = 0 , k#r .
We have g = 29k ' b = DrPr , hence
(6.10)Pk
Pk = T 1 + T2 '
where
(6.11) T 1 = Pk( M g) pk = Pk
using (6.9). Also, as j#ky6r#j , using gjlxPk = gjPklx .....
(6.12) Pk(Pr,gj)Pk = 0 .
Similarly, as j # k = r , using pkpkl =
(6.13) Pk(Pk'gj)Pk = Pkl gj(1Pk)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(1Pk)gklxPk  Pkgkl(iPk)Pklx
= PkIpkIxgk  gkpklPkIx = LPkpklPklxpk'gk]
Accordingly, using (6.9) again,
(6.15) T2 = Dr 1PkPkIPrlxPk'gk1 = LPkPklblxPk'gk]'
304 9. Particle flow and invariant algebra
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)) _  (('U'j*(Pr))
with respect to any biorthogonal base {(Pj} , {ipj} of Sk, where Yk
is the matrix of gk . Accordingly, (6.5) 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 sol
ution satisfying y=y0, at t=0, defined near each for small
t. Let it be assumed that a = N2n ,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 given value at t=0.
Note that d/ds just is the derivative along the Hamiltonian flow
(1.3) of the function X(x,l;) (instead of k0 ).This flow is well
defined for all t if we assume that X E y)Qe, , for example (of.
prop.2.5 ). We then get existence and uniqueness of
for all (t,x,l;) E I2n+1
Theorem 6.2. Let the assumptions of prop.6.1 hold for f21xI+fII
>rjo , and let bE yiae, , d^ E C'(1t, e), d E C"(1,Vsm+e° e)' Fore°=e assume in addition that the coefficients of b are E Wne. For
e°e even atd^E CO(&, c(1+1)(e°e))' atdoE C00(&'y'sm+(1+1)(e°e))00
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 1 = 0,1,2..... .
Proof.(Assume 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+gjpjpjd pj,
This may be interpreted as another set of ODE's for the v2 coeffi
cients of gj. There is a unique solution of (6.18) under the init2n+1ial conditions of (6.17), defined in N. This must be the same
9.7. Proof of theorem 5.4 305
gj , also solving (6.5).
We have p;=(? j'Pj) E 'cfe' C'(l.'iVcf_e)Therefore (6.18) may be interpreted as a vzxvzsystem of first or
der 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 v2component vector with entries
wile cIE C00 (&,'1cv2e,_e) and the v2vector cj has entries in
C (R,1Vsm+eAe)' 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) Idlgj1m2/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
(6.22) IgJ Im S ril(e) , s E R ,
with locally bounded ril(s) . Then (6.19) implies
(6.23) Iatgk
Ime+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 1 withJp p
coefficcients in ps0 , following the proof of thm.1.2. This will
give estimates for all expressions Id/dsI=lri 1 gI2m, completing7pp 7the 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
306 9. Particle flow and invariant algebra
(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,
belongs to C00(L psm). We also get as solution of (5.22),
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 = eiKtq(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+...In particular, we have used (6.22) and [kf,gt] = 0 . Note thatrt E J=1,2 , and z0 E 'Wsm_e
We now may solve the linear inhomogeneous differential equa
tion (7.2): It follows that d/dt(eiKtWteiKt)=eiKtRte iKt , hence,
eiKtW aiKt = Q +eiKt iKt t iKt iKit (S2t + Zt)e = Z0 Oe Rte di
which implies
(7.3) aiKt(Q+ZO)eiKt = Qt+Zt  foeiKTRt_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 denotingour old rt formed with zt . Here we write rt = rt + rt , with
rt = (K,zt) i/2(k,zt)2 + "' 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 ztE C"(l1S _e) to sol
ve the compatibility conditions pjrtpj=0. Then (5.19), with the
9.7. Proof of theorem 5.4 307
first sum set 0, gives a solution of (7.4) of the form
(7.5) zt= PjirtPl/(%jXl) E C00(R,Wsme' e)(C°°(!,Wsm_ez _e) .
After fixing the symbol ztE C°°(R,Wsm_ej) we also have a well defi
ned r2E CO0(R,Wsme, j'ez) , j+j'2 , using prop.5.3.
Set rt[kf,zt1+rt in (7.3), and 'integrate by parts': Write
] i(k,zt)+(i)2/2(k,zt) + ...),(7.6) rt = rt i([k,zt2 2
with rt=rt+i ([ x, zt] +i(k, zt) +...) EE CO0(R,VPSmje' _j' ez j+j' =2. Get
fteiKTRtT aKt dt toe iKtR _t eiKt dr +if toiKz [ K, Zt_t ]eiRr
eiKtZ2eiKt _ Z2 +f toiKtR2 eiKtd0 t 0 tt
where we introduced rrt Itzt instead of the old rt. Substitute in
(7.3) (we now have Zt+Z instead Zt but Z0=0 by construction). Get
(7.8) eiKt(Q+ZO+ZO)eiKt = Qt+Zt+Zt+Zt  feiKTRt_telKtdt
In particular, we have rt E
Cm(R,Wsmje'j'Jezz)
, j+j'=2 , and
zt E Cm(R,VSm_ je, _ j ' ez) 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 sui
table CO0(R,Wsmje'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
je' j'ez , j+j'=2 , since it must solve (6.5). The rt will be of
this order, with j+ j' =3 , and the zt, solving [ ke, , zt] =rt^ , will beof order mje'j'eze , 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 mje'j'ez, j+j'=3. Thus get
(7.9) aiKt(Q+ZO+...+ZN)eiKt=(Qt+Zt+...+Zt) f triteiKERN_teiKT
where generally
(7.10) zt , rt E COO(R,'WSm_ je' _j I G2 ) ,j+j'=1 ,1=0,1,2.... .
308 9. Particle flow and invariant algebra
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) aiKt(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 me
but only will be of order mej , 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 pro
pagation' 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) .First, zt of (5.23) again is in C°D(&,1sm_e) , but for a different
reason: The right hand side of (5.22) now is in Vsm_e2, but divi
sion by gets this into Vsm_e again. qt of (5.24) has qtE
)sv_e2 . Moreover, atztE C0'(&,'U)svmle2 _e)' and, atgtE C00(N,Wsmlee )'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 decompo
sition a = a0 = q0 + z0 at t=0 and then set
(7.12) qt = Lj=1 gjtpj,
gjt:Sj > Sj , Sj = im pj
9.7. Proof of theorem 5.4 309
Here the linear operators qjt of the jth eigenspace Sj of ks, are
given as unique solutions of the first order system of PDE (7.13),
with covariant derivatives in changing with x and ,
(7.13) Pj((gjtPj)It kjI (gjtpj)Ix+?jIX
satisfying the initial condition
(7.14) qjt = gtlSj , at t = 0
and vj :Sj  Sj are defined by
(7.15) vj = (iPjfPj  PjPIkflxPj)ISj
In "local coordinates"that is if (p1,...,cpPi
, and Pj is a
local biorthogonal basis of the eigenspace S (x,g) , defined and
C" in some open set a of if Kjt = is
the matrix of qjt, with respect to that basis, then, for 1,
(7.13) and (7.14) are equivalent to
(7.16) Kt it + `7I Kt Ix [O 1 ] = 0 , KO = Kl ,
where 0j is the sum of the matrix of vj and (((Xj1Vr*) s))
For the qt thus obtained one next writes
(7.17) zt=z0+i jI p.(qt+(kf,gt)+i[K,gt])Pl/(XjXl) , [kf,zt]=0.
Here 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 = 0with the differential expression of the "jth particle flow"
(7.19) CX j = at + 7,.jl
ax  kj lxat .
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, 'Wsme' "me' n vsmez ' 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
as a small charged magnetic field Wsphere of mass m and
total charge e spinning B=curl A
about an axis through
its center of gravity, i electric field
and moving along some
orbit x(t)=(xi x2 x3 ) (t) . E=grad VIn the sense of /
classical Physics the 0/particle experiences
forces from the fields I Fig.1
E and B determining
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 3vectors, also in the following)
(0.1) tx.
cJ.
= E + C(x'xB)
as equation of motion for the particle, while j(t) satisfies
(0.2) t M 1x'2/c2 B"xj = 0
where
(0.3) B" = B + (1/ec)(1+ 1x'2/c2 )1x'xE
denotes the magnetic field 'seen by the particle' (even if B=O,
310
10.0. Introduction 311
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 nonzero combi
ned force on the particle. We have reason to first ignore this
"SternGerlacheffect", 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 4vector
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) selfadjoint 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 selfadjoint 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 probabi
lity (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
312 10. The invariant algebra of a Dirac equation
>`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 expan
sion (0.4) for "Tj shows. However, measuring another observable
B will transfer the state into an eigenvector of B, hence a follo
wing 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 measu
red in any order, always producing the same result.
The first momentum coordinate pl is given as (the unique
selfadjoint realization of) the differential operator ir<ax
Clearly thus the observables xl and pl do not commute, we have
(0.6) xiplpixl = 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 measure
ment of H must produce the same result at any time. The represen
tion 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 differentialequationinitialvalueproblem
(0.7) radii/dt + i Hy, = 0 , 0st<ao , p(0) = Vo .
10.0. Introduction 313
Or, equivalently, by the Heisenberg representation: Physical
states are constant in t, but observables change, according to
(0.8) A  At = eitHAeitH .
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:
(0.9) H = V(x) + mc2Il + c ..1aj(iaa/axj  2Aj(x)) .
Here V(x) and Aj(x) are the potentials as above, m,e,c,a are mass,
charge, speed of light and Planck constant. Also R , all a2, a3
are certain constant 4x4matrices. We are used to work with
(0.10) «j = i(_aoj) j=1,2,3oIJ
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 4x4matrices
(0.12) lij = fa oii' pj = ro3a, , j=1,2,3
j j)
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 3vector a = (x1'(y 2'°3) ,
and arbitrary formal vectors a,b, components commuting with a
(a) a(aa) = a + iaxa , (aa)a = a  iaxa
(0.15) (b) ((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 4x4system, in the sense of VI,2. Moreover,
we shall see that it is semistrictly hyperbolic of type e' , un
der proper assumptions on V and Aj . Accordingly our results of
IX,5 apply. We are getting an invariant algebra of ydos.
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 theo
ry, but may, at the contrary, prove to be one of its major points.
It then follows easily that the differential expression H
is formally selfadjoint, and, moreover, that the minimal operator
(with domain C0(&3)) admits precisely one selfadjoint realization
In other words, there is a unique selfadjoint 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 stran
ge 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 FoldyWouthuysen
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 59 we focus on the
'correction symbol' first for 'standard observables' then in gene
ral, discussing some finer details.
1. A refinement of the concept of observable.
To motivate an application of the theory of ydo's, let us
focus on the concept of observable introduced in sec.0 . So far,
any bounded or unbounded selfadjoint operator of H = L2(H3T4
(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 x1coordinate, we mention
ned) or differential operators  such as H above, or the momentum
observable, given as pl = iha/ax , etc. . We will meet other1
"standard observables" later on, and mention that multiplication
by a (proper) matrix may occur as well, such as the spin observa
10.1. The concept of an observable 315
ble, commonly defined as multiplication by pj , j=1,2,3, with pj
of (0.12) .
on the other hand, in many respects the admission of all
unbounded selfadjoint operators as observables appears inconve
niently 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 4vectors. We might just as
well assume that H  L2(&3) (with complexvalued 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 trans
lations. In more detail, it is well known that the operator family
Ah = ThATh , 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) like a(x)
= xl/Ixl1 = sgn(xl) .
If we ask for differentiability in the parameter h the situa
tion gets worse: A derivative ahj
Ahu as limit of a difference quo
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 normCw : 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 trans
lations. (Although of course this may be an idealization just as
the concept of derivative  interpreting velocity as time deriva
tive amounts to a strong simplification, but the builtin 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
316 10. The invariant algebra of a Dirac equation
defines a unitary operator of H 'diagonalizing' the momentum obser
vable (i.e., pj _ iax is transformed into the multiplicationj
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:HH is a unitary operator then we get exactly the
same probabilities and expectation values if we replace i, and A by
yf = U, and A" = UAU* , respectively. In sec.0 the position obser
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 smooth
ness condition" (i.e.,cdn.(u)) also in the momentumdiagonal form.
A simple calculation shows that this amounts to the following
gaugesmoothness condition (y): The operator family Ah=eihxAeihx
is normCoo in the parameter h (i.e., Ah E C(&3,L(H)) ).
[Indeed, (FAF*)h = ThFAF*Th = FAhF* , as easily checked]
Remark 1.1. Both conditions (z) and (y) (i.e. translation and gau
ge smoothness together) amount to the condition of normsmoothness
of conjugation within the Heisenberg group HG : The map g > A(g)=
g*Ag isC00(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 pseudodifferential
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 settingr
(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
straightforward 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,toThe significance of the last remarks become clear if we
10.1. The concept of an observable 317
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 L2bounded pseudodifferential operators precisely coinci
des 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}
whereA2.t=
e_itxeizDAeizDeitx, and where P(az,at) is
any differential polynomial (constant coefficient differential
expression) in dz
=a1az
and atj with the property that it admitsj j
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
potentials inverting (1Az)n(1_A )n).
This result was discussed in [CL]. Actually, it coincides
with VIII, thm.2.1, of course, except for some variations in the
symbol formula.
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 selfadioint
,do AE Opyito automatically is a bounded observable satisfying the
translation smoothness and gaugesmoothness condition.
Notice that we are lead into a set of axioms selecting bet
ter 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 stan
dard 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).
318 10. The invariant algebra of a Dirac equation
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 3x3matrix
o define the unitary operator S0
E L(H) by u(x)  u(ox)=S0U(x)
Then the family A(o) = S*AS0 is normCoo 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 normC00 over R+
Note that cdn's (p) and (S) are Fourier invariant: They mean
the same, if imposed on the positiondiagonalized 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 Liesub
group 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=xjxj
qjl=( A , la +(xi axlx1ax,) , 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 multiindices 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.
10.2. The invariant algebra 319
Next we introduce the class ips of all polynomials
(1.8) aa,p E 4s0 .
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 , handlerather well (cf. VIII,7), and we also may write
(1.10) A
with different coefficient symbols as . A LeibnizTaylor formula
as in 1,5 holds in OpVt, with the class 'tVt of polynomials with co
efficients 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 calcu
lus of i4ido's, (although degenerating for cf. VIII,7).
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, andthat a characterization of tINm by conditions like our above
'smoothnesses' (t),(y),(p),(6) is possible with the (polynomially
weighted) L2Sobolev 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'HSm)
with the normCoo properties (u) , (y) , (p) , (6) , all with
respect to the operator norm of L(HSHsm) instead of L(H)
The proof will be omitted.
2. The invariant algebra and the FoldyWouthuysen transform.
As stated above, an observable will be a (selfadjoint) ope
rator of uN , from now on. But it will be seen later on that the
320 10. The invariant algebra of a Dirac equation
algebra ills is not in general invariant under time translation.
Assuming states constant and observables to change by conjug
ation with aiHt, we need At=eiHtAeiHt 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 L2bounded A we require that
(2.1) (x)3a3At E C'(l,L(H)) , j=1,2,....,
and, moreover, (2.1) is not only required for At , but also for
every gderivative 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(HsHsm
) , as in thm.1.5.
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 sym
bol 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 = eiHtAeiHt = at(x,D) , with atE s
and (with e2= (0,1) ) we have
(2.2) at at E PSmjet 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 Vsm still satisfying (2.2) .
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 onm
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 aiHt . It will be cal
led the invariant algebra of the Dirac equation (or, more general
ly 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) = eitH,F is given as the solution of
the Dirac equation (0.7) with V(0)=4i' . Explicitly we are given
an initialvalue 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
10.2. The invariant algebra 321
principal part has constant coefficients. The existence and uni
queness 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 multiindices 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 eigenva
lues of are =tI I. For X960 they both are double, hence
have constant multiplicity.
The hermitian matrixfunction has the diagonalization
(2.6) +(x, )=1(x, ) =
with a unitary 4x4matrix and eigenvalues
(2.7) 11=12=X+ , 13=14=X , X+=V(x)tcfr , where fr m2c2+n2
for all x,1; E &n. ('+' denotes the adjoint 4x4matrix, 'a*' the
Hilbert space adjoint's symbol  a(x,D)*=a*(x,D), in all of ch.X.)
In 1,7 we defined the class of symbols ypcm, m=(ml,m2)E &2, by
(2.8) "'cm = {aE CO: a(T) = O(( )m1 IOI(x)m2 ITI),6,'rE Z+
with ( ) = 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 2fold eigen
values Xt have the twodimensional orthogonal spectral projections
(2.9) a0=me/(m2c2 + a2)1/2. t= /(m2c2+n2)l/2
A calculation confirms that X. as well as the entries of are
in yc0 . Let V1, yr2 be the first two columns of p+ , let V3, yi4 be
l/2the last two columns of p , and let Tj = V Pj/(1+a0) . Then
322 10. The invariant algebra of a Dirac equation
(2.10) cp(x,t) =
is the desired unitary matrix satisfying (2.5) and again has coef
ficients in ta0, by a calculation. We write yicm for the class of
all rxrmatrices 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 Egorovtype result, saying that cdn (t) (or (t')) holds for
every A E ills (cf. VI,6). For a semistrictly hyperbolic system of
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 semistrictly 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=iaxj
, angular
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 complex
valued and 4x4matrixvalued symbols: Let Vtm , VSm , etc. be the
4x4matrices 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 Hamil
tonian H as well as all polynomials in H. In fact, it is clear
that every AE LIS commuting with H  in the sense that eiHtAAeiHt
=0 , using the operator product of O(oo) , satisfies At A=const.
hence will belong to P . For example, the resolvent R(A)=(HA)
can be shown to belong to 1ISe, , as A is nonreal: Clearly the
symbol h(x,l;)A =g(XA)q)+ is mdelliptic 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 selfadjointness of H  it follows that R(A) must be
that special Green inverse. Hence R(A)E 11Se, , implying that
10.2. The invariant algebra 323
R(.t)E P_e' . On the other hand, the operator eiHt does not belong
to W , hence not to P either . The operator eiHtx , for O#XE C00
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
4 4(2.11) a = q + z , z E y)sme, q E s [h(x,i;),q(x,1;)] = 0
the latter for all x,t E R3
(2) Vice versa, if a matrix qE yes of symbols commutes
with the symbol h(x,l;) , at least for all x,T; ER3 satisfying
xl+Z y > 0 , then there exists a symbol z E ys _e such that
A = a(x,D) = q(x,D) + z(x,D) E Pm .
(3) Suppose A1
, A2 E Pm (which must have a decomposi
tion (2.11) both) have the same q, but different z (=z1,z2), resp.
Then b=z1z2 allows a decomposition (2.11) with m replaced by me.
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 oper
ator of H and a pseudodifferential 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 cor
responding unitary operator is called FoldyWouthuysen transform.
Theorem 2.2. There exists a unitary operator X:HH 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
(2.13) au/at+ i(A + T)v = 0 .
Here r is a (4x4matrix of) ,do(s) in O(oo), while A vanishes in
its upper right and lower left 2x2corners. Moreover, we have
(2.14) symb(A) = diag (mod "_e2 ) , e2 = (0,1)
with diag( ) denoting the diagonal matrix with entries listed.
324 10. The invariant algebra of a Dirac equation
A proof of thm.2.2 will be discussed in sec.6, below.
Thm.2.1 and thm.2.2 are