Spectral Theory of Linear Differential Operators and Comparison Algebras (London Mathematical Society Lecture Note Series)

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London Mathematical Society Lecture Note Series. 76

Spectral Theory of Linear Differential Operators and Comparison Algebras

H. 0. CORDES University of California, Berkeley

CAMBRIDGE UNIVERSITY PRESS Cambridge London

New York

Melbourne

Sydney

New Rochelle

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521284431

© Cambridge University Press 1987

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1987 Re-issued in this digitally printed version 2007

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data Cordes, H. O. (Heinz Otto), 1925Spectral theory of linear differential operators and comparison algebras. (London Mathematical Society lecture note series ; 76) 1. Differential operators. 2. Linear operators. 1. Title. QA329.4.C67

1986

515.7'242

ISBN 978-0-521-28443-1 paperback

85-47935

P R E F A C E

The main purpose of this volume is to introduce the reader to the concept of comparison algebra, defined as a type of C*-algebra of singular integral operators, generally on a noncompact manifold, generated by an elliptic second order differential expression, and certain classes of multipliers and 'Riesz-operators' As for singular integral operators on In or on a compact manifold

the Fredholm properties of operators in such an algebra are governed by a symbol homomorphism. However, for noncompact manifolds the symbol is of special interest at infinity. In particular the structure of the symbol space over infinity is of interest, and the fact, that the symbol no longer needs to be complex-valued there.

The first attempts of the author to make a systematic presentation of this material happened at Berkeley (1966) and at

Lund (1970/71). Especially the second lecture exists in form of The cases of the Laplace compari(somewhat ragged) notes [CS] .

son algebra of Rn and the half-space were presented in [C1]

.

In the course of laying out theory of comparison algebras we had to develop in details spectral theory of differential operators, as well as many of the basic properties of elliptic second order differential operators. This was done in the first four chapters. Comparison algebras (in L2-spaces and L2-Sobolev spaces) Finally, in chapter X we are discussed in chapters V to IX .

recall the basic facts of theory of Fredholm operators, partly without proofs. The material has been with the author for more than 20 years and has been subject of innumerable discussions with students and

vi

associates. Accordingly it is almost impossible to recall in detail the origin of the various concepts introduced. Especially we are indebted to E. Herman, M. Breuer, E. Luft, M. Taylor, R. McOwen, A. Erkip, D.Williams, H. Sohrab, in chronological, not alphabetical order. We are indebted to S.H.Doong, S.Melo, R.Rainsberger, M.Arse-

novic for help with proof reading. This volume originally was planned under the title 'techniques of pseudodifferential operators', but then split into two parts, with the second yet to appear. We are grateful to the publisher, Cambridge University Press, for cooperation and patience in waiting for the manuscript.

Berkeley, September 1986

Heinz 0. Cordes

TABLE OF CONTENTS

Chapter 1. Abstract spectral theory in Hilbert spaces

1

1.1. Unbounded linear operators on Banach and Hilbert spaces 1.2. Self-adjoint extensions of hermitian operators

1 6

1.3. On the spectral theorem for self-adjoint operators.

12

1.4. Proof of the spectral theorem

17

1.5. A result on powers of positive operators

24

1.6. On HS-chains

29

35 Chapter 2. Spectral theory of differential operators 2.1. Linear differential operators on a subdomain of En 35 2.2. Generalized boundary problems;ordinary

differential expressions

41

2.3. Singular endpoints of a 2r-th order 49 Sturm-Liouville problem 2.4. The spectral theorem for a second order expression 53

Chapter 3. Second order elliptic expressions on manifolds

59

3.1. 2-nd order partial differential expressions on manifolds;Weyl's lemma; Dirichlet operator

60

3.2. Boundary regularity for the Dirichiet realization

67

3.3. Compactness of the resolvent of the Friedrichs extension

71

3.4. A Green's function for H and Hd, and a mean value inequality

78

3.5. Harnack inequality; Dirichlet problem; maximum principle

87

3.6. Change of dependent variable; normal forms; positivity of the Green's function Chapter 4. Essential self-adjointness 4.1.

of the Minimal Operator Essential self-adjointness of powers of H0

93

103

104

viii 4.2. 4.3. 4.4. 4.5.

Essential self-adjointness of H0 Proof of theorem 1.1

109

Proof of Frehse's theorem More criteria for essential self-adjointness

118

111

Chapter 5. C -Comparison algebras 5.1. Comparison operators and comparison algebras 5.2. Differential expressions of order < 2 5.3. Compactness criteria for commutators

123 125 127 132 137

5.4. Comparison algebras with compact commutators 5.5. A discussion of one-dimensional problems 5.6. An expansion for expressions within reach of an algebra C Chapter 6. Minimal comparison algebra and wave front space.

144 150

155

161

6.1. The local invariance of the minimal comparison algebra

162

6.2. The wave front space

169

6.3. Differential expressions within reach of the algebra J0 6.4. The Sobolev estimate for elliptic expressions expressions on a compact 0 Chapter 7. The secondary symbol space

175

181 186

7.1. The symbol space of a general comparison algebra 7.2. The space M\W , and some examples

189

7.3. Stronger conditions and more detail on ffi\W

200

7.4. More structure of 2s , and more on examples

.

193

210

Chapter 8. Comparison algebras with non-compact commutators 8.1. An algebra invariant under a discrete translation group 8.2. A C -algebra on a poly-cylinder. 8.3. Algebra surgery

8.4. Complete Riemannian manifolds with cylindrical ends Chapter 9. Hs-Algebras; higher order operators within reach 9.1. Higher order Sobolev spaces and Hs-comparison algebras

218

220 228 239

246

250

251

9.2. Closer analysis of some of the conditions (1.) and (m.) J

3

258

ix

9.3. Higher order differential expressions

within reach of C or C

s

262

9.4. Symbol calculus in Hs

267

9.5. Local properties of the Sobolev spaces Hs

272

9.6. Sobolev norms of integral order

279

9.7. Examples for higher order theory;

the secondary symbol

283

291 Chapter 10. Fredholm theory in comparison algebras 10.1. Fredholm theory in C C L(H) 292 10.2. Fredholm properties of operators within reach of C 296

10.3. Systems of operators, and operators acting on vector bundles

303

10.4. Discussion of algebras with a two-link ideal chain 309 10.5. Fredholm theory and comparison technique in Sobolev spaces Appendix A. Auxiliary results concerning functions

315

on manifolds Appendix B. Covariant derivatives and curvature

320

Appendix C. Summary of the conditions (xj) used

324

List of symbols used

327

References

328

Index

340

322

TO HILLGIA

CHAPTER 1. ABSTRACT SPECTRAL THEORY IN HILBERT SPACES.

In this chapter we give a short introduction into spectral theory of abstract unbounded operators of a Hilbert space. In sec. 1 we give a discussion of general facts on unbounded operators. In sec.2 we discuss the v.Neumann-Riesz theory of self-adjoint extension of hermitian operators. Sec.3 gives a general discussion of the abstract spectral theorem for unbounded self-adjoint operators. We discuss a proof of the spectral theorem in sec.4. Also, in sec.5 we discuss an extension of a result by Heinz and Loewner useful in the following. Finally an abstract result on Fredholm operators in a certain type of Frechet algebra related to a chain of Hilbert spaces generated by powers of a self-adjoint positive

operator is discussed in sec.6. The typical 'HS-chain' is a chain of L2-Sobolev spaces. The chapter is self-contained and elementary, and only requires some familiarity with general concepts of analysis and functional analysis of bounded linear operators. 1. Unbounded linear operators on Banach and Hilbert spaces.

The term "(unbounded) linear operator" (between Banach spaces X and Y) is commonly used to denote any linear map A:dom A -+

Y

from a dense linear subspace dom A of X to Y. The space dom A C X then is called the domain of A. Here we distinguish between a linear map X + Y and a linear operator: A linear map X -F Y by definition has its domain equal to X .

The term "unbounded linear operator" will be used with the meaning "not necessarily bounded linear operator", so that the bounded linear operators are special unbounded operators. A bounded linear operator, satisfying (1.1)

sup {UAuU/UuI

:

0

9 u E dom A} < - ,

I.1. Unbounded operators

is necessasily continuous, hence admits a unique extension to X in which dom A is dense. This is why we usually assume that a bounded linear operator also is a linear map. The class of all such unbounded linear operators between two given spaces X and Y will be denoted by P(X,V). In particular the class L(X,Y) of all continuous linear maps X -. Y then is a subset of P(X,Y) .

Since unbounded linear operators are not linear maps from X (but only from their individual domain to Y ) their sum and to V product needs the following special interpretation: For A,BEP(X,Y) ,

and C E P(W,X) we define the sum A+B E P(X,Y) and the product AC E P(W,Y) by setting (1.2) dom (A+B) = dom A n dom B

,

(A+B)u = Au+Bu for u E dom(A+B),

and

(1.3) dom AC = {uEdom C

:

CuEdom Al

(AC)u=A(Cu) for u E dom AC

,

where it is assumed that dom(A+B) and dom AC are dense in X (or else, we will say that dom(A+B) or dom AC is not defined ). Also we define cA = with the identity operator 1 E L(X,X) A linear operator A E P(X,V) is uniquely characterized by its graph, defined as the linear subspace .

(1.4)

graph A =

{(u;Au)

:

u e dom Al

of the cartesian product XxY = {(u;v)

:

uEX

,

v(=-Y}

, where (u;v)

denotes the ordered pair. Vice versa, if for any linear subspace T C XxY the set of all first components is dense in X , and if T does not contain elements of the form (O;u) other than (0;0), then a unique unbounded operator A E P(X,Y) is defined by setting (1.5) dom A = {uEX:(u;v)ET for some vEV}

and then we have graph A = T

, Au=v , for u E dom A ,

.

For two linear operators A,B E P(X,Y) we shall say that A extends B (or that B is a restriction of A ) if graph A D graph B. We then will write A D B (or B C A )

.

Notice that the cartesian product XxY of two Banach spaces is

a Banach space again, for example under the norm II (u; v) II = II uIl +II vII Therefore it is meaningful to speak of a closed subspace of XxY An unbounded operator A is defined to be closed if its graph is

2

I.1. Unbounded operators

a closed subspace of Xxy . The class of all closed operators in P(X, Y)

It is clear that a continuous linear

is denoted. by Q(X, Y) .

map A E L( X, Y) is closed, since (uk;Auk) ->(u;v) implies v=lim Auk

, hence (u;v) = (u;Au) E graph A.

= Au

An operator A E P(X,y) is called preclosed if the closure of graph A is a graph again. Then Ac with graph Ac= (graph A)closure is called the closure of A .

In the following we will be mainly interested in unbounded = P( H, H) , where p is an infinite dimensional separable Filbert space with inner product (u,v) and norm

linear operators A E P( H) Ilull

=

{(U, U) }1/2

.

In that case the graph space HxH becomes a Fil-

bert space again, under the inner product and norm

(1.6)

((u;w),(v;z)) _ (u,v) + (w,z)

, I1(u;w)II

=

(I1 ull2+I1w11

2)112

The following facts, regarding adjoint and closure all work with proper amendments. However for general Banach spaces X Y ,

,

we will get restricted to the case X=Y=H in all of the following. For an operator AEp(H) we will say that the (Hilbert space) ,

adjoint A E P(H) exists if the space TA = J(graph A) is a graph and " L" again. Here J:HxH -. HxH denotes the map (u;w) (w;-u) denotes the orthogonal complement in the graph space (with respect A" Then we define the adjoint to the inner product (1.6) E P(H) -

,

.

of A by setting

(1.7)

graph A

TA

Notice that A , if it exists, is closed, since all orthogonal complements are necessarily closed and since J is inverted by -J

It is clear that A C B implies B C A assuming that A and B exist. The proposition, below, translates the definition of the adjoint into a more transparent form. (The proof is left to the reader. )

Proposition 1.1. Assure that A E Q(H) exists, for some AE: P(H) Then dom A* consists precisely of all u E: H for which there exists

such that

an element v E

it

(1.8)

(u,Aw) = (v,w)

,

for all w E dom A.

Moreover the element v thus defined for each u E dom A determined, and we have A..u=v

is uniouely

.

to

Proposition 1.2. An operator A_E

P(H) admits an adjoint A

if and

3

I.l. Unbounded operators

only if it is preclosed. Moreover, then A C 'xx A and the closure A of A equals A

4

k

also admits an adjoint

,

Proof. Let first A E P(H) have an adjoint Ask E Q(H). Then let T =

(graph A)clos* contain the element (O;z) It follows that there exists a sequence wk E dom A with (wk;Awk) - (0;z) Substitute w = wk in (1.8), and conclude that (u,z) = 0 for all u E dom A .

.

since the inner products in (1.8) allow a passing to the limit.

Since dom A

is dense, it follows that z=0, so that no element of the form (O;z) is in T, except (0;0). Also, T J graph A, which implies that the set of first components of elements in T is dense. Therefore T indeed is a graph of some AcE P(H) and A is ,

preclosed.

Vice versa, let A be preclosed, and let again T be the closure of graph A

the set P of all first components of elements 1If(i.e., of all second components of (graph A) L

in TA=J(graph A)

)

is not dense in H then there exists 0 (z,v) = 0 for all (u;v)E(graph A)

9

z E H with ((0;z),(u;v))=

But this implies that (O;z) (graph A)clos =T. However, for preclosedness of A it is required that T does not contain such elements. Thus the set .

A)H

E (graph

must be dense in H. On the other hand if (O;v) E TA = graph A then (1.8) yields (v,w)=0 for all w E dom A , so that v=0 since V

,

dom A is dense. This shows that then indeed A is a well defined operator in P(H), hence in Q(H) q.e.d. All continuous linear maps in L(H) are closed,hence have an adjoint. Moreover L(H) is adjoint invariant, and, for an A E L(H), ,

the above adjoint coincides with the well known Hilbert space adjoint A of the bounded operator A Also, if A E P(H) is (pre-)closed, then yA+B is (pre-)closed, for every B E L(H), 0 # y E M. Then (YA+B)o=YAo+B, (yA+B) * A*+B in the sense of (1.2)

Our main interest, in the following, will focus on selfadjoint unbounded operators. Here an operator A is called selfadjoint if (A exists,and) A =A First of all a self-adjoint operator allows a spectral decomposition as a direct generaliza.

,

tion of the principal axis transformation of a symmetric matrix. Second we will learn about important classes of differential operators which are unbounded self-adjoint operators, and therefore allow such a spectral decomposition. Third, we will show how results on unbounded non-selfadjoint differential operators can be achieved by " comparing" them with certain standard self-adjoint

I.I. Unbounded operators differential operators.

Note that a bounded operator (i.e., a continuous linear map) A is self-adjoint if and only if it satisfies the relation (u,Av) _ (Au,v) for all u,v E dom A ,

(1,9)

(where dom A = H ).A general unbounded operator A satisfying (1.9) needs not to be self-adjoint, because (1.9) just implies that A D A , not that A = A . Such an operator is called hermitian If the closure of a hermitian operator A is self-adjoint then we speak of an essentially self-adjoint operator(i.e., A*=At ).

Note that a hermitian operator A indeed has an adjoint:If ukEdom A Uk+O , Auk--w, then we may substitute u=uk into (1.9), for fixed v, and pass to the limit, resulting in 0 = (w,v), for all v E dom A It follows that w=0

,

so that A is preclosed, and A

*

exists, by

prop. 1.2. Comparing (1.8) and (1.9), it then follows at once that A E P(H) is hermitian if and only if A C AF If A is (essentially) self-adjoint, and B bounded hermitian then yA+B is (essentially) self-adjoint for all 0#y E R

.

More generally, two operators A, B E P(H) will be said to be in adjoint relation if (1.10)

(u,Av) = (Bu,v), for all u E dom B

,

v E dom A

.

The above conclusion, showing that hermitian operators have

adjoints, can be repeated to prove that operators A, B in adjoint relation must have adjoints (both)

and that,moreover, A and B are *

in adjoint relation if and only if A C B

(or if and only if

BCA* ). One of the first major problems occurring in our discussion of differential operators, in later sections, will be the construction of all self-adjoint extensions of a given hermitian ope-

rator. It turns out that not all hermitian operators possess selfadjoint extensions. On the other hand, the problem of characterizing all self-adjoint extensions was solved by v. Neumann [vN11

and F. Riesz [Ri11. We will discuss the v. Neumann-Riesz theory in section 2, together with some other constructions of self-adjoint extensions.

5

1.2. Riesz-v.Neumann extension 2. Self-adjoint extensions of Hermitian operators.

In this section we discuss the v. Neumann-Riesz theory of self-adjoint extensions of hermitian operators.

It is at once clear that a hermitian operator A has a hermisY st

;:'e

sk

is

C A Since a tian closure Ac = A , because A C A implies A self-adjoint extension B = BC of A is necessarily closed, it also .

must be an extension of the closure A Therefore, in looking for self-adjoint extensions of a given hermitian operator A we may .

look for such extensions of the closure, and thus assume that A is closed, without loss of generality. Also if A C B = B then B = B C A extension B of A is a restriction of A ,

A C B = B

( 2 . 1 )

,

so that every self-adjoint as well: We have

,

C A

Proposition 2.1. A hermitian operator A satisfies the identity (2.2) l1(A-a)u02=11(A-Re A)u112+(Im A)2Hu112

,

for all uEdom A

,

AEC

Proof.We have II(A-a)u11 2=((A-u-iv)u, (A-u-iv)u)=((A-p)u,(A-u)u)

where we have written u = Re A + v2(u,u) - 2Re ((A-u)u,ivu) Here the last term vanishes, due to (Au,iu) + (iu,Au) v = Im A ,

.

i((Au,u)-(u,Au)) = 0 , using that A is hermitian, q.e.d. For a closed hermitian operator A it is implied by prop.2.1 Indeed, consider that im (A-A) is closed for every nonreal A E E a sequence uk E dom A such that (A-a)uk -> v It follows that .

.

(A-X)(uk-ul) } 0 (2.3)

,

as k, 1 1 °°

U(A-A)ull

>

But (2.2) implies the inequality

.

I Im AINO

,

u E dom A

.

Substituting u=uk-ul into (2.3) yields Iluk-ulll - 0, since Im A 90, u for some u E H by assumption. Hence uk and (uk;Auk) -. (u;v) ,

in HxH. But graph A is closed since A is closed. Thus it follows that (u;v) E graph A , or, u E dom A , v=Au .

In the following we first consider the special case A = ±i Since we have im (A±i) closed, by the above, we obtain a pair of

orthogonal direct decompositions

1 (2.4)

H = im(A±i) $ V.

,

D..

= (im(A±i))

The two spaces D+=D+(A) are called the defect spaces of the closed

6

1.2. Riesz-v.Neumann extension hermitian operator A indices of A

.

and their dimensions are called the defect

,

We write

def A = (dim(im(A+i))

(2.5)

i ,

dim(im(A-i))

L (v+

JA-) )

Note that the defect spaces D,. are just the eigenspaces of We have the adjoint operator A to the eigenvalues ±i :

D+ _ (im(Ati))1 = ker(A* i)

(2.6)

Indeed, f E D+ ,for example,amounts to 0 = (f,(A+i)u) for all u E dom A

We write this as (f,Au) = (if,u) concluding that f E dom A*

.

with (1.8) X =

±i

(2.7)

,

,

,

u E dom A A*f = if

,

and compare

As another consequence of prop.2.1 we note that (2.2) implies

for

,

=

(IIAuN2+llull2)1/2

n(Ati)ull =

hl(u;Au)H

u E dom A .

,

Note that graph A. as a closed subspace of HxH is a Hilbert space under the norm and inner product of HxH Moreover graph A is in linear 1-1-correspondence with dom A which may be used to trans.

fer that Hilbert space structure of graph A to dom A In other words, dom A is a Hilbert space under the (stronger) norm .

(2.8)

IIuIIA =

(Ilull2+llAull2)1/2

with inner product (2.9)

(u,v)A =

(u,v) + (Au,Av)

,

u,v E dom A

The latter is true for every closed operator B E 0(H)

,

not

only for hermitian operators. In particular we may apply it to the adjoint B=A* of our closed hermitian operator A , obtaining a cor* and inner product (u,v)

responding norm llull

A

,k

,

u,v E dom A

A

In fact, we then get Pull

Pull A ,for u E dom A C dom A* , and A

*

dom A appears as a closed subspace of dom A under graph norm. Note that (2.7) may be interpreted as follows: The two ope,

In fact these rators (A±i) are isometries dom A ; im(A±i) = D. im(A-i)-+im(A+i) isometries are 'onto'. Therefore V =(A+i)(A-i)71 .

:

defines an isometry between the two closed subspaces of H Proposition 2.2. For a closed hermitian operator A we have *

(2.10)

dom A

= dom A ® D +(A) ® D _(A)

7

1.2. Riesz-v.Neumann extension

8

as an orthogonal direct decomposition of the Hilbert space dom A* under its norm and inner product. Proof. We already noticed that dom A is a closed subspace of

* * dom A , under graph norm. The eigenspaces D+ of A are closed subspaces of H , as nulspaces of the closed operators A*'+i. Moreover,

on D+ we have

_ V Hu II ,

nu II *

so that D+ are also closed under -

A

graph norm. For f+ E D+ one confirms that (f+,f ) *= 0, using that - A *

A f+ _ -+if+. Also, for u E dom A

,

(Au,±if+) + (u,f+)

(u,f+)

A 0, so that the three spaces in the decomposition ±i((A±i)u,f+) = *

(2.10) are orthogonal. Suppose f E dom A

satisfies 0

=

(f,u) * A

=(A*f,A*u)+(f,u) for all u E dom A. Comparing this with (1.8) it

is found that A*f E dom At, hence f E dom (A*)2, and (A*)2f+f = One also may write this as (A*+i)(A*-i)f = (A*-i)(A*+i)f = 0 *

*

*

0.

*

In particular, f E dom (A +i)(A -i) = dom (A -i)(A +i), in the sense of (1.3). Now we write f = (A*+i)f/2i - (A*-i)f/2i = f++f noting that (A +i)f+ = 0. This proves that every f orthogonal to and thus is in D+®Ddom A , under graph inner product of A* completes the proof. ,

,

Corollary 2.3. A closed hermitian operator A is self-adjoint if and only if its defect indices vanish (i.e. def A = 0 ). Or, for both, "+" and equivalently, if and only if im (A±i) = H ,

Indeed, A = A

*

:e

implies dom A = dom A , so that (2.10) gives D+ = {0}, hence def A = 0. Vice versa, if def A = 0 , (2.10) gives *

*

*

dom A = dom A , hence A =A since A DA , q.e.d. We now state the v. Neumann-Riesz extension theorem. ,

Theorem 2.4. The closed hermitian extensions B of a given closed hermitian operator A are in 1-1-correspondence with the extensions W W- -> W+ of the isometry V = (A+i)(A-i)-1 im(A-i) + im(A+i) This as an isometry between the closed subspaces W. D im(A±i) :

:

.

correspondence is established by assigning to B D A the operator W = VB = (B+i)(B-i)-1 (which is an isometry extending V between the spaces 0+ = im(B±i) D im(A±i)). Vice versa, given an isometry one must observe that W , as W :W--t W+ extending the isometry V ,

an extension of V, is determined by its restriction W0=WIW°, where W0= W n(im(A-D) = W nD is a subspace of the defect space D and ,

where W0 is just any isometry W0, W0 = W+nD+C V+. Then we have the

1.2. Riesz-v.Neumann extension closed hermitian extension B given by dom B = dom A $

(2.11)

{W0-

E W0}

:

,

where the direct sum again is orthogonal in (.,.) * A

.

The proof is almost self-explanatory. It is clear from the above that W = (B+i)(B-i)-1 is an isometric extension of V

for

,

every closed hermitian extension B of A . Vice versa, that W0 determines W

,

as described, follows from the well known fact that

isometries preserve orthogonality. Then, of course, the operator B

,

if it exists,should satisfy

(2.12)

W(B-i)u = (B+i)u

u E dom B = dom A $ ZO

,

,

with a certain subspace Z0 C D+$D-, because dom A C dom B C dom A* and due to (2.10)

.

For u E Z0 let m = Bu-iu

X = Bu + iu = W0

,

It follows that u = (W04)-4))/2i, Bu = (WO4)+4))/2, in agreement with (2.11). Now one simply must verify that the operator B of (2.11)

is closed and hermitian. The closedness follows if we show that * ZO = 4 E WO } is a closed space, under graph norm of A {W0$-4

But we have (2.13)

11W0 _O*2 =

11W04)_$112

+

11W04)+4)112

=

211W04)112 +

2114)112

=

4114)112,

is, in fact an isometry (up to the factor 4). Since W is closed, Z0 also is closed. To verify that B of (2.11) is hermitian is only a calcuwhich shows that the map 4)-W04)-4), taking W0 onto Z0

,

lation; since we know that Bldom A = A is hermitian one must show that (AiYu,v)=(u,A*v) for all u,v E Z0, and for u E dom A, v E 20. Both follow trivially, q.e.d.

Theorem 2.4 has the following important consequence. Corollary 2.5. A closed hermitian operator A admits a self-adjoint extension if and only if def A = (v,v), with v=0,1,2,...,- arbi-

trarily given. In other words, we must have (2.14)

codim im(A+i) = codim im(A-i)

.

Then every self-adjoint extension B of A is obtained by picking an arbitrary isometry W0

:

D_

D+

between the two defect spaces

(2.6), and then defining B with (2.11)

,

and WO , with WO±=D±

The proof is evident.

Although the v.Neumann-Riesz theory completely clarifies

9

1.2. Riesz-v.Neumann extension

the problem of self-adjoint extensions, other criteria are useful, of course, because the construction of isometries between defect spaces is not always practical. In particular not every closed hermitian operator A satisfies the condition (2.14), so that a self-adjoint extension need not always to exist. There are two well known general criteria giving existence of self-adjoint extensions. Shortly, 'real' hermitian operators as well as 'semibounded' hermitian operators always have self-adjoint extensions. The concept of real operator refers to a given involution u

u of the Hilbert space H

.

In most applications we will have

H = L2(X,dp) with some measure space X and measure dp

,

and then

refer to the complex conjugation u(x) - u(x) of the complex-valued function u(x)EH=L2. However, one may think of an abstract space K and an involution map u -r u (U-)_ = u

,

with the properties

(c1u+c2v)

,

= clu +c2v

(2.15)

(u ,v

)

_

(v,u)

,

for u,v E H

,

cjE T

,

j=1,2.

Then a real operator is defined as an operator A E P(H) satisfying (2.16)

(dom A)

= dom A

and

(Au)

= Au , for all u E dom A

Now, if a closed hermitian operator A is real with respect to any such involution of H

,

then one confirms at once that

D+- _ fu-: u E D+} = D-

(2.17)

.

Indeed if f E D+, i.e., (f,(A+i)u)=0 for all u E dom A,then we get ,((A+i)u)_)=(f

0

= ((A+i)u,f)=(f

,'(A-i)u ), hence f E(im(A-i))

using (2.16). This conclusion may be reversed, so that (2.17) follows. Also it is clear that D+ and D+- have the same dimension. We have proven:

Proposition 2.6. A closed hermitian operator A which is real

with

respect to some involution of H has equal defect indices and hence admits a self-adjoint extension.

A hermitian operator A E P(H) is called semi-bounded below, if there exists a real constant c such that

10

1.2. Riesz-v.Neumann extension (2.18)

(Au,u)

> c(u,u)

for all u E dom A .

Similarly one speaks of semi-boundedness above if (2.18) holds

with ">" replaced by "

(u,u)

,

for all u E dom B

,

so that B has the same lower bound 1 as A .

In order to show that B is self-adjoint refer to (1.8) and Now consider the linear functional 1(u) = (g,u), for g as, above and all let (f,Bu)=(g,u), for all uEdom B, and a given f,gEH

u E dom A . We get 11(u) l




B_1(=- L(H)

Pull ,

,

IIB-111

u E dom B , which yields < 1

PB-1fII H" is an isometry between H and H"

H- = im C

,

and (lull

= PCuF`, for all u E H

.

We have C-1, with dom Cr -1H"' C H a self-adjoint operator in P(H),

with lower bound 1. Moreover, the restriction C-11dom A still is essentially self-adjoint.

The proof is left to the reader (cf.also sec.4). 3. On the spectral theorem for self-adjoint operators.

The resolvent of a closed operator A E Q(H) is commonly defined as the inverse R(a) _ (A-a)-1

similarly as for linear maps. More precisely, the resolvent set Rs(A) is defined as the set of all X E M such that (3.1)

im(A-A) = H

,

,

and P(A-A)ull > cllull

,

u E dom A ,

1.3. Spectral theorem, general discussion with a positive constant c

.

It is clear that (3.1) holds if and

only if the linear map A-A between the spaces dom A and H has an inverse (A-A)-1

H -> dom A C H which constitutes a bounded

:

operator of H, with 11(A-A)-lII (c/2)IIuII

,

u E dom A

so that the second condition holds for a neighbourhood of A0 Regarding the first condition we observe that (A-A)u = (1+(a0-A)R(A0))(A-A0)u

(3.3)

,

u E dom A

by a simple calculation. We know that im(A-A0) = H , and boundedness of R(a0). For small JA_a0) the first factor at right of (3.3) is of the form 1+ E takes H onto H A-A01

,

,

11E11

< 1 , hence is invertible in L(H)

,

and

Thus (3.3) shows that im(A-a)=H for all small

.

and Rs(A) is open.

Now we conclude from (3.3) that R(a) = R(A0)(l+(\0-A)R(A0))-1

(3.4)

,

JA-A01

< c

where only bounded operators in L(H) occur. It is evident that the right hand side of (3.4) provides a norm convergent power series expansion of the operator R(a) in powers of (A-A0) for A close to \0

.

,

Therefore it follows that R(A) is an analytic

function from the resolvent set Rs(A) to L(H)

.

Let us now return to self-adjoint operators. Theorem 3.1. For a self-adjoint AE Q(H) we have Rs(A) J cC\

g, i.e.,

Sp(A)C 1, and

(3.5)

IIR(A)fl


c (or Ac (or A A) if B-A is positive. An orthogonal projection (here shortly 'projection') of the Hilbert space H is a bounded hermitian operator P satisfying P2 = P (i.e. an idempotent). One easily verifies Proposition 3.2. The orthogonal projections P of H correspond to the orthogonal direct decompositions of H :

(3.7)

= im P $ ker P

H

For u E H we get u = v + w

,

.

corresponding to (3.7) , where v = Pu

and Pw = 0. For any direct decomposition H = M ® N with M , N orthogonal the assignment u - v defines a bounded hermitian idempotent

operator P such that M = im P , N = ker P Pu=u in M Moreover, for two orthogonal projections P,Q we have P0

.

circle case, respectively, with a notation referring to his special construction. 2.Generalized boundary problems;ordinary differential expressions. In section 1 we derived a direct decomposition (1.15) of the domain of the maximal operator, assuming that the expression (1.1) is strongly hypo-elliptic and self-adjoint. In particular we concluded that all self-adjoint extensions of the (hermitian) minimal

operator L0 are given as closures of certain restrictions of the maximal operator L1, called e.s.a. realizations of L. In turn the e.s.a. realizations are characterized by an isometry W:D- -+ D+ between the defect spaces D+

,

similar as in the v. Neumann-Riesz

theorem (I, thm.2.4).

For a given e.s.a.-realization A of the expression L the condition

uEdom A ', imposed on a function uEdom L1, amounts to a generalized boundary condition, because if u e dom L1 satisfies '

u E dom A , then so does u + v = w , for all v E C0

,

due to C0

C dom A. In other words, the condition u e dom A is not influenced by the behaviour of u away from the boundary, it depends only on the properties of u in some (arbitrarily small) neighbourhood of the boundary of the domain D. In this sense we associate to every e.s.a.-realization of L a (generalized) boundary problem

One will be tempted to ask for classes of e.s.a.-realizations with boundary conditions of the conventional type. For an investigation of this kind involving partial differential operators, and more generally, dissipative and accretive boundary conditions cf. [ CFO]

.

In this and the following section we will get restricted to n=1, i.e.

,

to the case of an ordinary differential operator.

Spectral theory of "ODE's" was completed to a high degree of per-

41

II.2. Boundary problems

,

ODE

fection, starting with the work of H.Weyl [We1]. From the large number of contributions to this subject we mention the work of

Hilb [ H11] , K.Kodaira [ 1 < 0 1 ] E.C.Titchmarsh [Ti1]

,

M.G.Krein [ Kr1] , N.Levinson [Lei] ,

,M.A.Naimark [Ne1], E.A.Coddington [Cd1],and

others (cf. also the monographs [ Ne2] , [Oh] ,

[ Bz1] , [ RS] , [ CdLi] )

We write L =

(2.1)

J jN=0

a.(x)3

3

with extended real numbers a<S

.

a < x
0 , and let E = E(p+6) - E(p) . Write u+6

(y(a)-y(u))dE(a)i

h(u+6) - h(u) = E(f-Y(u)$) -

(4.17)

u

u+6

In the second expression, called J2, we may write

J

u+0 u+6

instead of

1

,

since the integrand vanishes at p

.

This gives

u

u+6 (4.18)IIJ2n2= u+0 y

For the first expression, J1, in (4.17), let w = f - Y(V)* Note that (4.19)

Vu;w) J ()(u;f) - Y(u)(D(u;ip)

by definition of y

=

0

,

Thus, by proposition 4.2 again, write w =

.

(A-u)w, with some w E dom L2. If u is an eigenvalue of A then one may add a, multiple of the corresponding eigenfunction u(x;u) as to obtain a revised w which is orthogonal to u(x,u) is in dom A . It follows that

,

and still

u+6 (4.20)

11J112

= IIE(A-1)0

2

(X-u)2dfE(A)wI12 = 0(62)

=

u+0

Similarly for 6 < 0

,

q.e.d.

Lemma 4.5. We may write (4.13) in the form (4.21)

EA,f = Jo,D(A;f) dE(X)g

,

g = Jo

Here the function g E H is independent of the specific choice of subject to the conditions of lemma 4.3. * ,

Proof. It is clear that we may write (4.13) in the form of

56

11.4. 2-nd order spectral theory (4.21). To show that g is independent of , let i"' be any function with the properties of 'P . Applying lemma 4.4 with f = '_ EA,*' = J6,0(a;*`)/D(X;'P)dE(a)' , hence g" =

we get

dE(A)V`/D(a;V')

JA

This completes the proof.

We will write g = gO

,

since this element g E H

is determined by the choice of the interval A (4.22)

gp, = EA,gA

Introduce a function

g

I + H

:

(4.23) g(a) = lime-++0,u+a+0

,

as A' C A

.

evidently

One finds that

.

by setting as A>0

g[e,u)

,

=-g[a,0)

as 1 < 0.

Then one confirms that gO, = g(u2) - g(ul). For f E CO we thus get b (4.24)

(1) dg(a) , (X) = (D(a;f)

(E(b) - E(a))f = a

b ? 0, with 'P as in (4.10). Also (4.23) implies

for all a 1, not necessarily compact, but paracompact (and connected). In fact, for convenience we assume existence of a countable atlas. On 0 we assume given a positive Co-measure du locally of the ,

form du = Kdx , with K being C' in the local coordiates x. By H we denote the Hilbert space L2(O,dp), with inner product and norm ,

, NO = (u,u)1/2

(u,v) = J2uv du

(1.1)

u,v E H

,

Also we assume given on 0 a second order formally selfadjoint strongly elliptic partial differential expression hikK2 k + q

H = -K-18

(1.2)

x

x]

with Co-coefficients. In particular hjk denotes a symmetric positive definite contravariant tensor with real Cm-coefficients: hjk =

hkj

=

-k0

>.0

,

,

as E 9

0

,

and q denotes a scalar real-valued C -function defined over 0 In the next sections we use the summation convention to always sum from 1 to n over a pair of an upper and a lower index, in a tensor, denoted by the same symbol. Also we will write local not coordinates with superscript indices x=(x1,x2,...,xn) E In ,

subscripts, as for a subdomain 0 C In , according to convention. As in II,1 the expression H induces minimal and maximal differential operators of H domain we get (1.3)

(u,H0v) = J0(hJku

v Ix7

For the minimal operator H0 with

.

Ik + quv)dp = H(u,v)

lx

,

u,v E dom H0

Clearly H0 is hermitian and real, in the sense of I,prop. hjk are real. Accordingly there exist self-ad2.6, since q and joint extensions of H0

.

60

III.l. PDE on manifolds

In much of the following we will be interested in the case where the sesqui-linear form H(u,v) of (1.3) is an inner product, In fact if we assume q > 1 in G , then we get called (u,v)1 .

(1.4)

>

Ilulll

Hull

,

for all u E dom HO

with the norm Ilulll=((u,u)1)1/2

,

,

since the tensor hjk is positive

definite. Condition (1.4) will be crucial in ch.5f, although the condition q > 1 will be weakened later on (cf. prop.5.2).

For an expression H with (1.4) the Friedrichs extension of H0 is well defined as restriction of H* to (dom H*) fl H1, with the completion H1 of dom HO = C3(Q) under the norm Ilulll (cf. the proof of I, thm.2.7 ). In fact, the norm Ilulll of (1.4) precisely coincides with the norm null

of that proof, and the completion H- defi-

ned there will be denoted by H1 here. We know that H1 is naturally imbedded in H In ch.5 we will assume that (1.4) holds, and then denote the .

Friedrichs extension by H again. This will be a self-adjoint operator,satisfying H > 1 We then shall speak of a comparison ope.

rator H

The space H1 will be called the first Sobolev space of the operator H Clearly H-1 E L(H) has a unique self-adjoint .

.

yk=0(-1)k(lk2)(1-H-1)k

positive square root A = H-1/2 = < A < 1

0

(1.5)

, and

We have (cf. I, prop.2.8)

.

H1 = dom A-1

,

(lull

=

IlAupl

,

u E H

,

Hvfl1 =

RA-1vll, v E H1.

That is, A is an isometric isomorphism between the Hilbert spaces Generally the restriction of the self-adjoint operator A-1 to dom H0 still is essentially self-adjoint, and A-1dom H is H and H1

.

dense in H

,

(while the corresponding is not generally true for

H dom H0 = im H0 ). Note that H1 may be identified as the space of all functions u E H such that a sequence um E dom H0 exists which is Cauchy in the sense of H1 and converges to u in H It follows that there exists a unique covariant tensor (called the gradient of u , and .

_

(u

,

even though the components are not proper derivatives)

Ix j)

which is p-measurable and satisfies

S2

h0k(u Ixj - um Ixj )(u Ixk - um Ixk )du

->

0

, m

61

III.1. PDE on manifolds

It is clear that u

.

is the local distribution derivative of u E

Ix]

L2(O,dp) C D'(O) while the above entitles us to speak of the strong L2-derivative. One may introduce L2-norm and inner product of gradients just as for covariant tensors in general by setting (1.6)

hlku

(pu,pv)

Ix3

v

Ixlc

dp

=

Ilpull

,

(pu,pu)1/2

Returning to the general case, where (1.4) needs not to be satisfied, we observe that the expression H is elliptic, since we assumed the tensor

hjk

positive definite

(1.7)

0

,

:

We have

for all E

0

at each point x E 0 It was mentioned in II,1 that ellipticity implies hypo-ellipticity, but this will be discussed in general .

only in FC31. Since we need (strong) hypo-ellipticity (or at least Weyl's lemma) in the following we shall offer a short independent proof of Weyl's lemma here, which is quite similar to the proof of II, lemma 2.2, in the case of an ODE .

It is clear why a proposition like Weyl's lemma is desirable: Thm.l.3 of ch.II carries over literally, with the same proofs, to the present case of an elliptic operator on a manifold. We express this in thm.l.l, below, the proof of which will be left to the reader, (except for our proof of Weyl's lemma, i.e., thm.1.2. ) Theorem 1..1. The defect spaces 4-(H ) are subspaces of ao 4:'a **0 H1= H0 (i.e.,"weak" _ HnC (Q) . Moreover, we have H0= Hl e

,

"strong"), and the direct decomposition (1.8)

dom H1 = ((dom H** )nC'(0)) $ D+(H0) ® D-(H0) ,

which is orthogonal with respect to the inner product of graph H0. The self-adjoint extensions A of H0 precisely are given as the closures of the 'e.s.a.-realizations' A- obtained from an arbitrary isometry W: + D , using formulas II,(1.16) and II,(1.17) Theorem 1.2. If (HO-A)f=g, for any fE dom H0 and gE HnC(2), then we have f E C°10) and Hf-Xf = g , hence f E dom H1 ,

.

,

The proof depends on use of an (E.E.Levy-type) local parametrix of the form as n > 2 e(x,y) _ (p(x,x-y))2-n

,

(1.9)

= log p(x,x-y)

,

as n=2

62

III.l. PDE on manifolds

with p=p(x,z) _ (hjk(x)zizk)1/2 of generality.

(We may assume A=0, without loss

) Note that e(x,y) is defined only

locally, in local coordinates, for x, y E WC 0 , with a chart 0'. We note the proposition, below, which also will be useful later on.

Proposition 1.3. For a function m E

let

v(x) = Je(x,y)$(y)dy

(1.10)

Then we have v E C-(Q') , and (1.11)

J.,

Hv(x) =

y(x,y)$(y)dy

with a positive C'(Q')-function c(x)

,

,

xEQ'

and with a function

y(x,y) of the form (1.12)

y(x,y) = y0(x,Y,x-Y) + 2 i(y3(x,y,x-Y)) x

Here the functions y!(x,y,z) j=0,1,.... n, are C,(Q'xO'xjjn*) In and homogeneous of degree 2-n in the variable z , as n>2 ,

.

the case n=2 we have (1.13)

yi(x,y,z) = h1(x,y,z)log p(x,x-y)

,

and homogeneous of degree 0.

where the h0 are

Proof. We only consider the case n>2

.

The other case n=2 may be

treated similarly. Observe that the function e(x,y) may be written in the general form (1.14)

e(x,y) = f(x,y,x-y)

,

f(x,y,z) E C,(O'xQ'x]Rn*)

where f is homogeneous in the variable z (1.15)

.

,

(Presently we have

=(hjk(x)zjzk)1-n/2

f(x,y,z)

,

independent of y and homogeneous in z of degree 2-n. For any function g(x,y) = f(x,y,x-y) of the form (1.14) we have (1.16)

(axi + a j)g(x,y) = fj(x,y,x-Y)

n

y

where fj again has all the properties of (1.14) homogeneity degree as f

.

, with the same

This follows, because a function a(x-y)

63

III.1. PDE on manifolds is constant in the directions x. yj hence has (2

.+8

x

Now a single derivative 2

.)a=0.

y

may be applied under the inte-

gral sign of (1.10), because the differentiated integrand still is integrable. Applying (1.16), and a partial integration, we get (y)dy +

vlx.(x) =

(1.17)

where gj(x,y) = fj(x,y,x-y) has exactly the properties (1.14) again, (with homogeneity degree 2-n). In particular the partial integration first may be performed on the integral over Ix-yI>e where boundary terms may be explicitly evaluated. These boundary terms tend to zero, as 6+0 , so that no special terms at x=y from the partial integration appear in (1.17).

Accordingly the process may be iterated, it follows that indeed all partial derivatives v(a)(x) exists (1.18)

v(a)(x)

=

for x E Q'

,

, and

I

a2. In case of n=2 one finds that cn = c2 = 27. Since the determinant is C' and >0 this term accounts for the first term in (1.11).

On the other hand another partial integration in the third term of (1.20) will remove the derivative from and give an ,

(x,y)¢(y) instead, confirming (1.12). (In view of

integrand h3 Iv'

(1.16) we again may write

h

l

+k with k(x,y,x-y) (degree

-hlYi=

2-n), etc.) This completes the proof of prop.1.3.

Now the proof of thm.1.2, in essence, is a repetition of the argument used in the proof of II, 1.2.1. Using our parametrix e(x,y) of (1.9) in place of the fundamental solution e(x,y), we derive a local integral equation of the form 11,(2.11) for any pair of functions f,gEH satisfying II,(2.6), with L=H, and with the present Hilbert space H=L2(c)

.

Then, if gEC-, the right hand

side is continuous, using the nature of the kernels. Hence f(x), after correction on a null set, also is continuous. Knowing that f is continuous, one concludes that the right hand side even is C1

,

so that fEC1, etc. Continuing on, one concludes that fEC

Let us get such formula in detail. Let u(x) =

(1.24)

with a local cut-off function X(x)=l near some x0EO', XEC0(Q') c>0 , where ¢ECD(In). and a "regularizing kernel" Assume X=1 in 1"C2'

, 0 open, and let x`EO" , and e so small that

supp 4e(x--.)C 0'. Then we may apply (1.11) for Hu(x) = c(x)me(x--x) +

(1.25)

with c(x) of (1.11), and 6(x,y) of the form (1.12) again. In fact, 6 is a sum of X(x)y(x,y) and terms of the form a(x)e(x,y), (*x,y), with a,b3 E C' (c2') , where a and b3 vanish out-

b3(x)e Ix

65

III.1. PDE on manifolds side supp x

.

All these terms may .be written in the form (1.12).

Substituting u and Hu into the (present) equation II,(2.6), and passing to the limit a+0, we get (1.26)

K(x")C(x")f(x") + JQ,S(x,x")f(x)dux

Jduxg(x)X(x)e(x,x-),

=

valid for x"E2". This relation is of the general type of II,(2.11) However, the kernel 6 requires a somewhat refined proceedure. If g E C_ , then the right hand side is a C--function h(x"), by prop.1.3. The integral at left is at least Hoelder continuous, since we find that the function x" + S(.,x") , with values in H is Hoelder continuous: We get 2/lx"-x"Ie = 0(1)

(1.27)

,

L

for a suitable e>0 and x", x" E Q" , by a calculation. Hence (1.26) shows that f is Hoelder-continuous as well. To show that f is even C1 we write (1.26) as (1.28)

(KCf)(x) = h(x) - JS(y,x)f(y)dvy

and investigate the function (1.29)

e(x) = JS(y,x)f(y)dpy

The difference quotient ve = (9(x+e)-e(x))/e may be written as (1.30)

ye = Jvd(y)f(y)duy

Jvd(y)(f(y)-f(x))duy

=

where My) _ (S(y,x+c )-S(y,x))/e (1.31)

Jvd(y)duY

=

+

f(x)Jvd(y)duy

Now we use (1.12) for

.

My .

J(V6 0K-V6JK ly]

The right hand side limit exists under the integral sign, and

represents a Cm-function of x, since again the derivatives may be taken over to the function K , using (1.16) The other integral in (1.30) has a limit as well, since f is Hoelder conti.

nuous, so that the limit of the integrand is L1 (1.32)

8 0(x) = k(x)f(x) + JSI

.

Thus we get

(y,x)(f(y)-f(x))dpy

,

xj

confirming that fEC'

.

In particular, k(x) is a Cm-function.

Knowing that f is Cl we may integrate by parts in (1.32) (1.33)

(KCf)Ixi = hlx7 - kf +

J

S°(Y,x)flx.(y)dy

,

, for

66

III.1. PDE on manifolds with a function d° of the general for (1.12) again.

It is clear now that this may be iterated, proving that Kcf , hence f is C_ , q.e.d.

2. Boundary regularity, for the Dirichlet realization.

It will be of interest to note the following variant of thm.1.2, applying in the case where g is only continuous, not necessarily smooth.

Corollary 2.1. If g of thm.1.2 is only continuous, then we still get f E C1(0) , and its first derivatives are Hoelder continuous.

Moreover, if g is Hoelder continuous, then also fEC2(0) follows, and f is a classical solution of (H-X)f=g

.

Indeed, it is clear that (1.26) is valid for general f,gEH satisfying (HD-X)f=g . If g is only continuous then the right hand side of (1.26), called V(x) still is C1(0) and p ,

,

IxJ

satisfies a Hoelder condition, by the above arguments, using an estimate like (1.27) for the first derivatives of e(x,x-)

.

Moreover, if g is Hoelder continuous, then the conclusion leading to (1.32) may be applied to integrals representing i

1.

.

It fol-

1 exist and are conti-

lows then that also the derivatives lxj

x

nuous. This shows that, in the first case, f will be C1

,

and

that (1.33) still holds. Also the first derivatives of f then still will be Hoelder continuous. For Hoelder continuous g thus we may iterate once more, getting that also fEC2 , q.e.d. As in 11.2 we next try some boundary application of our Let us investigate the Friedrichs extension in the special case where we have H of the minimal operator H0 only regular boundary points. Similarly as in the 1-dimensional case parametrix e(x,y)

.

of 11,2 it may occur that the manifold 0 is a subdomain with smooth boundary 30 of another manifold R^

,

and that the triple

{0,du,H} extends to a triple {0^,dp^,H^} on 0^ satisfying our general assumptions. (That is, 30 is an (n-l)-dimensional Co-submanifold of 0^

,

and we have du = dp^I0 , H = H-JQ .) In this

case we will say that 0 has the regular boundary 30

.

Moreover,

if even OU30 is a compact subset of Q^ then we will speak of an expression H (or a triple) with regular boundary, or we will ,

67

111.2. Boundary regularity

say that all boundary points of c are regular. In the case of a triple with regular boundary one defines the Dirichlet operator Hd J HO by setting (2.1)

dom Hd= {uEC'(SZUM): u=O on 8O}, Hdu=Hu , for uEdom Hd

Specifically the condition "u=0 at x(=-22" is referred to as the

Dirichlet (boundary) condition. Theorem 2.2. The Dirichlet operator Hd

,

for an expression,H with

regular boundary, is an e.s.a.-realization of H , and its closure ** H = Hd is the Friedrichs extension of the minimal operator HO Proof. First we note that Hd indeed is a restriction of the FrieFor clearly we have HdCHd* drichs extension H of HO (i.e., .

Hd is hermitian), since a partial integration confirms that (2.2)

(u,Hdv) _ (u,v)1 = (Hdu,v)

,

for all u,v E dom Hd

(The boundary terms vanish, since u and v vanish at 2D .) Also, HOCHd , trivially, hence Hd*CHO* so that we get HdCH O* For any u E dom Hd it is easy to construct a sequence ujE dom HO with nu-u101-0 , as j- , since only the first derivatives of u must be modified, while the 1-st Sobolev norm 11.111 contains only first derivatives as well. In details, for a boundary chart c', ,

thought of as an open set in &+ _

{xn>0} ,

X(xn)EC,(J)

let

,

X?0

X=0 near xn=0 , X=l for xn>1 , and let X5(xn) = X(Jxn) uj(x) = u(x)Xj(xn)

.

Then u-uj = u(1-X].) has support in O<xn 0 hjk as in (1.2). Such an expression may be brought into the K , ,

form (1.2) (3.2)

:

One has

L = -K-1a xi

K82h3k3 k + q + q0 x

,

q0 = - a (h0kKs

)

1xi

ix

k

71

111.3. Compactness of the resolvent Indeed, we get (u,Lv) u v E dom HO , (u,qv) + (4(Bu),V(Bv)) where we write V(Bu) = $Vu + u48 , and apply a partial integration ,

,

to the term (BVB,uVv+vVu) = (BOB,V(uv)), from the second inner product, to free the term (uv) from its gradient. For more detail, cf. the derivation of formula (6.6), below. If we assume that (u,Lu) > (u,u)

(3.3)

u E

,

then the facts discussed for H in section 1 apply to L as well.

We get a well defined Friedrichs extension L of the minimal operator LO , and its inverse positive square root L-112 is a bounded The space L-1/2C0(St) is dense in H self-adjoint operator of H and L112 is essentially self-adjoint in dom HO .

Let us adopt the convention to write

(3.4)

limx

y(x) = Y0

(in S2

,

)

,

where y(x) is a function over 0 , (with values in a space X) , and if for every neighbourhood N of YO a compact set K C S2 YO E X We can be found such that y(x) E N whenever x E 0 is outside K Then condiare not excluding the case of a compact manifold ) ,

.

.

tion (3.4) is void: It is true for every function y(x) , limit y0° and space X whatsoever. In particular condition (3.5), below, is generally true whenever S2 is compact.

Theorem 3.1. Let the expression (3.1) satisfy (3.3) and

limxq(x) = - ,

(3.5) Then L 112

(in S2

)

.

is a compact operator of H = L2(c,dp)

The proof is rather technical,and will be prepared by a series of propositions. Let w,X E C'(l) be such that (3.6)

0<w,Xl.

For 0 < A < - define the two functions (3.7)

XA(x) = w(q(x)-A)

,

uA(x)

= X(q(x)-A)

It follows from (3.5) that aA has compact support.In fact, we have (3.8)

q < A+ 1

in supp aA

q > A 9

in supp

uA

72

111.3. Compactness of the resolvent

Proposition 3.2. For every A > 0 there exists a constant cA with

(3.9)

(u,Lu) > 7 IIUAull2 + IIV(AABu)112 - cAIIXABull

u E C0(52)

,

Proof. We get (u,Lu) = J9A 11A2(hjk(Bu)

+

gIuI2)dp

Ik +

IXJ

x

(Bu)Ik + gIul2)dp x

Ix]

The second term at right is bounded below by AUPAuO2 first term we use (3.2), with B replaced by XA

,

.

For the

to obtain, with

a=XA , u=VA ,

JaA2h3k(Bu)

J82

kdu = Uy(AA6u)112+

(Bu) Ix7

K(Khjka

Ix

klul2dp. IxJ

Ix

Here we estimate the last term at right by cA'

Hull

llABull

cA'

,

= supxES2 IS(Khjkx

Note that Hull

=

II(A2+P2)uU
AUP U112 + 114(aABu)U2 +


0 there exists finitely many functions zj E C_ (S2)

,

j

= 1,...,M , M = M(B)

u E C0'(S2)

,

(zj,u) =

0

(u,Lu) > Bllull 2

,

such that

1,...,M

111.3. Compactness of the resolvent

75

Proof. Apply proposition 3.3 , with A = 2B , and assume without loss of generality that each of the compact sets supp wj relates to a subset of the cube Q under the coordinates of the corresponding chart Q.

.

In that compact set we also get a positive bound

below for the positive definite matrix function ((h3k(x))). Thus AO(AAwju)112

f

> (B + dA)IIAAw.u112

AAwj u sinalix dx =

0

,

I a I

< N.

.

Since only finitely many indices j are involved (i.e., those with supp wj n supp aA 1 0 (3.18) translates into (3.15), and (3.16) follows by combining (3.10) and (3.17), q.e.d. ,

Proof of theorem 3.1. Setting u =

L-1/2w

in (3.15)

,

(3.16) we get

(3.19) IIL-1/2 w112 q

,

((h" 3k)) > ((h 3k))

setting (.,.)Q" =

1

in c"

.

..du

,

we get

111.3. Compactness of the resolvent (u,L-u),,- > (u,Lu) > B(u,u) = B(u,u)0-

(3.23)

,

whenever (zj,u) = 0 , as a consequence of prop.3.5. Here we were extending u to 0 by setting it zero outside QNotice that the .

conditions (zj,u)=0 may be written as (z-i,u),- = 0 , with z-i _ ziIsi` E HIn other words, we conclude that prop.3.5 is valid also for L- in the subdomain Sl- C n, except that now the functions .

no longer are C'(c-), but only in C'(Q)1H`

z`3 .

.

This weakened

condition still allows its use for the argument of the proof of thm.3.1 , because we still get (z-j,L--1/2w) _ (L--1/2z-j,w) q.e.d. E Hwhere L--1/2z-.

,

We now return to comparison triples. Let triples {c,du,H}, and {c^,dp^,H^} be given, both satisfying the basic assumptions of sec.l. We write {2,du,H} {Q,dp,H}

)

if

(i) 2 is an (open) subdomain of Q^

,

(ii) du = dµ^Ic

(3.24)

(iii) ((hjk(x))) > ((h^Ok(x))), q(x) > q^(x), for all x E Q.

In this case we will say that the triple {Q^,du^,H^} sub-extends the triple {P,dp,H}

.

The main point for introducing this more complicated notion is the fact that many such singular problems display a very different behaviour near a point x0 E 2Q C 2^ than at infinity (of 0^

)

.

q -

For example, as shown by cor.3.6, we do not have to require L-1/2 near such a point x0 to get compactness of This will still become more evident in V,3, where we inves,

tigate compactness of commutators of the generators of a comparison algebra C Here it may become important that a proper change of depen.

dent and independent variable may be necessary, before Q^ can be defined (cf. the examples in V,4 and V,5). Theorem 3.7. For a comparison triple {c1,du,H} let there exist a

triple {0^,dp^,H^} (c> {c,dp,H} satisfying (3.25)

q^(x) > 0 for all x e 2^\K

where K C Y^ is compact

.

76

111.3. Compactness of the resolvent i.e., with

Then, for every function a E C(Q) with a = a2 = 0(q^)

(3.26)

the operator aA =

,

and

aH-1/2

0

,

(in 2-

is compact in H = L2(Q,dp).

Proof. Assume first a has a real extension 00

.

But then

77

111.3. Compactness of the resolvent (3.28)

IlBull

Or, IBIS EI

< e

,


1 , and

81

111.4. Greens function

has a well defined bounded hermitian inverse H-1 to (4.18) we conclude that

H-I

E L(H)

.

Applying

H-1v(y)=Jcx(y)v(x)dp =Jg (x,y)(K(x)/K(y))v(x)dx, vEC-(In).

(4.19)

Then the hermitian symmetry of H-1 implies that (4.20)

g (x,y)K(x) = g°(y,x)K(y) , x,yE In , x#y

We have proven the following result in the special case of a triple {In,dp,H) , coinciding with the Laplace triple outside a compact set. In case of a general manifold 0 we relate the

integral operator to the global measure du=Kdx again. Also we denote the Green's kernel by g(x,y), not g°(x,y) ignoring the ,

earlier meaning of g(x,y)

.

Theorem 4.4. For any triple {c2,du,H} satisfying (1.4) the Frie-

drichs extension has the form of an integral operator H-1u(x)

(4.21)

=

Jg(x,y)u(y)duy

,

u E H = L2(,,n)

with the 'Green's function' g(x,y) (E C-(HxQ\{x=y}) and with g(x,.), g(.,x) E C(H,H). Also, g(x,y) = g(y,x), x,y E H, and, in

local coordinates, on a chart n', g has an expansion of the form g(x,y) =

(4.22)

R7)e(x,y) + Ek=1gk(x,y)

,

x,yEH',

where e(x,y) denotes the parametrix of (1.9) and X(x) denotes an even cutoff function (X(-x)=X(x) X?0). X(=-C0 , X=1 near 0 ,

,

,

Also, gk(x,y) = 0((x-y)k-n+2), k=l,...,n-3, for all n>0 gn-1 E C(H'xH') , while gk , k=1,...,n-2, have Furthermore, the first support contained in a ball Ix-yl0 for k=n-1 l-X1EC0_(H) , the funca cutoff function X1EC'(H) 1=0 near x0 .

,

tion *(y) = X1(y)g(x0,y) is contained in C.fldom H

For a proof of thm.4.4 in the general case observe that the function g(x,y) is uniquely determined by the relation u(x) = Jg(x,y)(Hu)(y)dpy

(4.23)

For xEH'

,

a chart of H

,

,

for all u E dom H

let 8 and H' be as initially in this

82

111.4. Greens function

83

section. Let g'(x,y) be the Greens function of (the Friedrichs extension of) H' in In

,

and let the cutoff function X have

support in 0' and be = 1 in B

Then we also get

.

u(x) = Jg'(x,y)(H(Xu))(y)duy

(4.24)

, uE C'(Q)

Subtracting (4.23) and (4.24) we get

(4.25)J(g(x,y)-X(y)g'(x,y))(Hu)(y)duy=J([X,H]g'(x,y))u(y)duy,uEC-0.

In particular we note that the commutator [H,X] acts on the y-variable, and is a first order expression with support in 1'\B so that the function [X,H]g'(x,y) = qx(y) is C-(c') tion of y only)

.

,

(as a func-

,

Setting px(y) = g(x,y) - X(y)g'(x,y)

,

(4.25)

assumes the form (px,Hu) _ (qx,u)

(4.26)

,

for all u E C0'(S2)

We even request that (4.26) holds for all u E dom H

.

This deter-

mines a Cm-function p E dom H such that Hpx=qx in the sense of x ordinary differentiation. Then we define g(x,y)=px(y)+X(y)g'(x,y).

At least for all uE dom H with HuEC' we also get uEC_(G) , and the right hand side of (4.25) may be written as (g'(x,.)[H,X]u) so that (4.23) follows.

For the last statement observe that (i,Hv) _ (gx0,H(X1v)) + ([H,X1]gx0,v)

(4.27)

,

vGCCOfldom H

E HnC"

with gx (y) = g(x0,y) , where z = [H,X1]gx

,

,

since the

0

0

commutator [H,X1] vanishes near x0

.

Note that the first term at

right of (4.27) equals GH(X1v)(x0) = 0. Hence the selfadjointness and essential self-adjointness of its restriction to as stated. Therefore the proof C'fldom H implies that Edom HnC' of H

,

of thm.4.4 is complete.

Next we will investigate the Greens function g(x,y) near its singularity. Given some fixed x0Ec let us ask for the surface (4.28)

g(x0,y)=6(x0)n2-n},

Cx001={yEc:

(or g=dlog n, as n=2)

with 6 = 6(x0) = cnVET-x T , as in (4.22) , where the constant n 0

,

111.4. Greens function

is assumed sufficiently large. More precisely, if g in (4.28) is replaced by g0 , then the surface (4.28) will be an ellipsoid, in close local coordinates. For g one expects a smooth surface C x0,n

to that ellipsoid, and with normal close to the normal of the ellipsoid, as n is small. We leave open the possibility that other points y exist in the set (4.28). These will be ignored, i.e., the statement "y sufficiently close to x0 should be added in (4.28). Let us assume without loss of generality that, at the point x0, we have hjk(x0)=6jk=0, as jtl, =1, as j=1, and that x0=0 so that the equation of (4.28) assumes the form if

g(x0,y)=dn2-n

,

(4.29)

IYI2-n + Y(Y) = .2-n

where we have (4.30)

0(ly15/2-n)

Y(y) =

o(IY13/2-n)

, VY(Y) =

.

For n=2 the first term in (4.29) must be replaced by -loglyl, and the second term is continuous with derivatives satisfying (4.30). We again focus on the case n>2, with n=2 to be treated similarly. It follows that for small IyI the first term at left of equation (4.29) is large as compared to the second term. Along each ray ty0 , with 1y01=l , 0 A0 = J(p0)

This completes the proof of prop.5.4. I

Proposition 5.5. Any u E H1 \

{0} which minimizes the functional

89

111.5. Harnack inequality

J(u) is in dom Hd ,and is an eigen function of Hd to the minimal eigenvalue a0

.

Proof. For a given function u with the assumed properties and any w E dom H

let 0(t) = J(u+tw)

ble function,since u+tw E Hl \

This is a well defined differentia{0} for small t

Using the fact

that u minimizes J we conclude that (5.10)

dJ/dt(0) = 2 Re ((u,w)/(u,u)' - (u,w)'/(u,u)'2) =

Or, replacing w by we

u

with suitable real u

,

0

it is found that

'Re' in the last equation (5.10) may be omitted. It follows that I

(5.11)

(u,w)1 =(u,Hw)' = A(u,w)'

A = J(u) ,for all w e dom H.

This implies that A is an eigenvalue of H hence of Hd ,by prop.5.3. It is clear,from prop.5.4 that A is the smallest eigenvalue of Hd

,

q.e.d.

Proposition 5.6. If u is a real-valued eigenfunction to the smallest eigenvalue a0 of Hd ,then either u - 0 or u 9 0 in all of 0'. Proof. In view of prop.5.2 it is sufficient to show that u cannot assume both positive and negative values. If the latter were true then the function v(x) = Max {u(x),0) vanishes identically in some open subset of 0' but does not vanish on all of 0' We claim that v is in Hl ,and that it minimizes J(u) This,if proven, implies that v E dom Hd ,and Lv = a0v a contradiction because v then will have to vanish identically,in view of prop.5.2. We require the following special case of Sard's theorem

(cf.

[ Mll] , [ GPl]

.

)

Proposition 5.7. For any real-valued C'(0)-function f the set of all critical values (i.e. values n E I such that n=f(x) at some xEO with Of(x)=0) has Lebesgue measure 0 on the real line I .

Using prop.5.7 we conclude the existence of a decreasing sequence {ej} ,ej + 0

such that the level sets u(x) = ej do not contain a critical point for each j Indeed the nulset of all .

critical values of u(x) cannot contain any interval (O,e) , e>0 It follows from the implicit function theorem that the sets = {x E 0': u(x) = e

ail;

are finite unions of smooth compact

J

n-l-dimensional submanifolds of ill. In particular, ;Q

naO'=o, J

since u=09ej on aO'

.

(In case of infinitely many components there

90

III.5. Harnack inequality would have to be a limit point in the interior of the compact set O'U8O' , which would make E. a critical value.) It follows 1

J

1

that the set &IE U8S2E , with S2E J

J

is a finite dis-

_ J

joint union of compact manifolds with boundary. Accordingly we may integrate by parts in these sets to obtain JonhJku (5.12)

u IxJ

kdu=-JOn(u-e1)K-1(KhJku

,

Or, setting S2+

kdu

)I

IxJ

I x

x

I S2"

=

S2

E1

, we get

= US2E 1

(5.13)

(h Jku

(v,v)1 =

k + gIul2)du = X0(v,v)'
0

.

Let u E C2(O) be a solution of

the differential inequality

(5.14)

Hu < 0

, x E S2

Then, if u assumes a positive maximum at some point of 0, or if q(x)=0 , we have u = const.

Proof. Assume that uM=u(x0)>u(x) for all xEO, and some x0ES2. Let um >

0

.

Since we have q>0 , and since Vu = 0 for x=x0

,

it

then follows from (5.14) that (5.15)

hjkulxjxk > 0 at x = x0

.

At a maximum we also have the 'Hessian matrix' ((u

k)) = M IxJx

negative semi-definite,as well known. On the other hand, it is no loss of generality to assume that hJk(x0) = Sjk , by virtue of a suitable linear coordinate transform. Then (5.15) amounts to the condition that the trace of the matrix M is nonnegative. One thus concludes that the Hessian matrix must be zero at x0

.

A contradiction results if we even have Auu>O at x0 , since then it is implied that the trace of a negative semi-definite matrix is positive.

91

111.5. Harnack inequality

A proof of the maximum principle results by proving that a suitable modification of u indeed results in a function w with Hw>O at its maximum. The construction, below, has been adopted from Protter-Weinberger [PW1] p61f. ,

Suppose we have u(xl)O in KUBKi, using ellipticity of H ar )- 1 Finally define w = u + ez , with Confirm ,

.

0<e1 .

compact, we always can arrange for q-=l

In the following we often no longer require q > 1, but only that (1.4) holds. Proposition 6.2. Let (cl,du,H) satisfy all assumptions of early sec.

1, especially (1.4), i.e.,(6.10), below, but not necessarily q>l. (u,Hu) > (u,u)

(6.10)

, u E C_(Q)

Then, if 0 is noncompact, there exists a positive C-(0)-function y, solving the differential equation HY = y in all of 0 .

Using y in a transformation of dependent variable u = yu-,

as described, one arrives at an equivalent triple of the form {0,du-,l-&u-} with

-= K--18

(6.11) u

K-h3k xl

k

, du = K-dx

2

,

K- = Y K

x

in local coordinates.

On the other hand, if 0 is compact then the equation Hu=au,

for some real constant A > 1 admits a positive solution y which may be used as above to obtain an equivalent triple with q-=A > 1.

This proposition is an immediate consequence of the facts derived on transformations of dependent variable,as soon as we establish existence of a positive solution y of the equation HY=y.

Such a solution will be constructed below. In fact it will become

95

111.6. Normal forms; positivity of g

evident that the condition H > 1 is necessary and sufficient for existence of a positive solution. The result, below is required for prop.6.2,and perhaps of independent interest. It was shown by the author for an ODE and a compact interval in [CEdI, p.65 (in 1957) (also [CS],IV,1.9.),

and for elliptic operators of varying generality by Allegretto [Al1l, [Al21, [Al31, Piepenbrink [Ppl], [Pp2], Moss-Piepenbrink

[MP1l, Agmon [Agl]. We are indebted to M.Meier for the suggestion to use the Harnack principle, which independently also was used by Agmon (Agl] for the same result. ,

Theorem 6.3. Given a positive measure dp and differential expression H as in (1.2) on a manifold Q, with real symmetric positive definite hjk and real-valued q , all C` , but not necessarily q > 1 ,

.

(a) If D is noncompact, and the expression H-1 is positive, i.e., (6.12)

j(hJku lx

Iu lx k

+ (q-1)juj2)dp

> 0

,

for all u E Cp(D)

then there exists a positive solution u = y E C-(D) of the partial differential equation Hu = u .

(b) If D is compact then the same is true under the additional assumption that (6.12a)

inf {(Hu,u)/(u,u)

:

u E C0-(S2)

,

u 4 0) = 1

(c) Vice versa, regardless whether 0 is compact or not, if a posi-

tive C'-solution of Hu = u , defined over all of 0 , can be found then H-1 is positive (i.e. (6.12) holds. If 0 is compact then also (6.12a) follows. ,

Remark. Notice that a positive solution y of Hy = y may be used for a transformation of dependent variable, as in section 1, to take the triple (0t,dp,H) to an equivalent one {St,dp-,H-), with q=Hy/y=1, using (6.6). Since the new triple satisfies q"'=1>l, we trivially get (1.4), coinciding with (6.12). Moreover, if 0 is compact, then a positive solution of Hy = y trivially represents an eigenfunction of H to the eigenvalue 1. Since we know that (6.12) holds we then conclude that (6.12b)

1=(Hy,y)/(y,y) < (Hu,u)/(u,u), for all u E C0_(S2)=C"(0).

This evidently implies (6.12a) . Accordingly we already have pro-

96

111.6. Normal forms; positivity of g ven (c), using the above transformation of dependent variable.

Proposition 6.4. On a manifold n satisfying our general assumptions there exists a C'(O)-function 4) taking positive values on 0 such that limx_,°°4)(x)

_

°D

,

in the sense of sec.3 (or app.A).

Proof. If Q is compact then the function 0 = 1 will satisfy all with

assumptions. If Q is non-compact then set 4)(x) _

the partition of unity of app.A.

Q.E.D.

Proposition 6.5. If n is noncompact then there exists an infinite sequence

521C 522C ... C 52

, U SZj

=

52

,

52j

0

,

#

0

,

(6.13)

0i = U1 nil , S2j1r1 njm =

19m

,

where each 0j1 is a nonvoid open subdomain of n with compact closure and nonvoid smooth non-self-intersecting boundary (i.e., 8S2 j1

is an n-l-dimensional submanifold of Q ). Every sum in (6.13) is finite,and every inclusion in (6.13) is proper. Moreover, each nilclos is a proper subset of some unique SZj+l,m Proof. Using proposition 5.7 on the function 4) of proposition 6.4 we conclude the existence of a strictly increasing sequence nj + 1 < nl < n2 < ... < nj < ... such that the "level set" ,

an3 = {x E 0 = 0

:

4)(x)

= nil does not contain any point x with Of(x)

Indeed the set of all critical values cannot contain any

interval (n,°°) (6.14)

so that a sequence {nj) must exist. Then define SZj

=

{x E 0

:

4)(x)

< nj}

,

so that an. is the set defined above, containing no critical points. By the implicit function theorem we conclude that an

indeed is a compact n-l-dimensional manifold,not necessarily conis compact, by con-

nected. Each SZj is nonvoid and each Q.U3S2

struction of 4) Thus each connected component is the interior of a manifold with compact closure and smooth (compact boundary). There can be at most finitely many components. (Otherwise a subsequence of components must converge to a point x0E852., near which 8n. while it is supis homeomorphic to a connected open piece posed to contain an infinity of nonconnected boundaries of components.

) Clearly each component 0 j1 has a nonvoid boundary, since

97

111.6. Normal forms; positivity of g we assume 2 connected. Clearly we get 37

C

0j+l

, which implies

the remaining statements, q.e.d.

Let us observe next,that prop.5.6 establishes theorem 6.3 for the case of a compact manifold Q

.

Proposition 6.6. Let 2 be noncompact,and let

be defined as in 2j

(6.13) Then if X3 is the smallest eigenvalue of Hd in Qj the sequence {a0} is strictly decreasing and 0 lim 0X > 1

Remark: The operator Hd for Qj is defined as the direct sum of the corresponding operators for the components Qjl .

Proof. Notice that the minimal eigenvalue v of Hd is minimal eigen value for at least one of the connected components Pjl. But Sljl C S1 j+1'm ,

for some m , by prop.6.5, and this is a proper inclusion.

If i * 0 is a corresponding (positve) eigen function then define (6.15)

w = i in S1 jl

,

=

0 elsewehere in 0j+l,m

Then the conclusion of prop.5.5 may be repeated to show that w E H1 - for 0j+l, and J(u) = v, so that the minimal eigenvalue m, v' of Hd in Qj+1 cannot be larger than v If v=v' then w will have to be an eigen function of Hd in Qj+l,m which is impossible since w as defined in (6.15) vanishes in some open set, by prop.5.2, thus will have to vanish identically. This .

amounts to a contradiction unless v' > v Thus we indeed get a strictly decreasing sequence,as j increases.Evidently there is .

the lower bound 1 as well ,q.e.d.

For any j and some wjE C'M

,wj > 0 we now solve the

Dirichlet problem (6.16)

u E C-(Q j)

, uI92j = wj

, Hu = u in 2.U32j

.

This is possible,since 1 is less than the smallest eigenvalue of Hd for 0j (due to prop.6.6 and prop.5.3). The solution will be called u. Proposition 6.7. The function uj is positive in all of QjUaRj. This proposition follows exactly as prop.5.6

.

Proof of theorem 6.3. To complete the proof in the noncompact case let us now consider the sequence {vj = ajuj :j=1,2,...} the positive reals aj chosen to normalize vj according to (6.17)

sup {vj(x)

:

x E Q1 1 = 1

,

j

= 1,2,...

with

98

111.6. Normal forms; positivity of g Using the Harnack inequality (i.e., thm.5.1) we get sup {vj(x)

x E 03} < C inf {vj(x)

:

x E S23

:

}

(6.18)

< C inf {vj(x)

x E 01} < C sup {vj(x):x E 01) = C

:

where the constant C is independent of j

= 4,5,...

j

,

.

Now (6.18) at once implies a corresponding estimate (6.19)

J dp(glvjl2+hkly.

< C'

,

C' independent of j

Indeed, with a cutoff function X E C'(923)

,

X=l in 02

2

(6.20)

1 k vj lx lx

_ (Xv,A v)0

(Xv,(q-l)v)0 3

=-Jhkl(Xv) 3

,

kv lx

we get 1dp

lx

where we set v=vj, for a moment. In the right hand side write (Xv)lxk = Xvlxk + vXlxk , and, for the integral with the second

term, integrate by parts again, using that vvlx1 = (v2)Ix1/2

.

It follows that J52 2

(6.21)

hkly

lxk vI xldp
l/C in S21, since sup v. = 1. Accordingly v=0

-

is impossible.

A similar argument may be carried out for each 0k , instead of 921

Using the Cantor diagonal scheme one arrives at a sub-

99

111.6. Normal forms; positivity of g sequence of vj converging in every nj

.

The limit v cannot vanish

identically anywhere and will be the desired positive solution of Hv = v

.

This completes the proof of thm.6.3.

Finally, in this section we will show that (1.4) also implies positivity of the Greens function of the Friedrichs extension H .

Theorem 6.8. Assume that condition (1.4) holds. then the Greens of the minimal function g(x,y) of the Friedrichs extension H ,

operator H0

,

as constructed in thm.4.4, is positive for all

, x9y

x,yESI

Proof. First we note that it is sufficient to consider the case of q=const.>1, by virtue of thm.6.3, since we have (1.4). Indeed, the Greens function of the transformed operator y-1Hy=His given by (6.22)

g-(x,y) _ (y(y)/y(x))g(x,y)

,

which is positive if and only if g is positive. Notice that we certainly have g(x,y)>0 as x and y are sufficiently close together, by virtue of the expansion (4.22) Indeed, the first term in (4.22) is positive and is large in com.

parison to the other terms. Accordingly the positivity of g follows from the maximum principle (thm.5.8), whenever 2 is compact.

If G is noncompact, then we apply prop.6.5, constructing an increasing sequence Q . with smooth boundary, and let H

the Dirichlet operator of D.

d,j

be

Let gj(x,y) , x,yEGj be the Greens

.

function of Hd,j. From thm.4.4 we know that X(.)gj(x0,.)Edom Hd,j, as x0EDj, so that cor.2.2 gives gj(x0,.) EC'(PjUe3j\{x0)) , and gj(x0,y)=0 as yEaDj . Accordingly we again may use the maximum and x#y , and principle to conclude that gj(x,y)>0 as x,yENj ,

then Harnack's inequality to show that even gj>0

.

Now we conclude the same for g by showing a type of weak convergence gj-*g . Define the sequence of operators Gj`= GjXG

, with 7

of Qj , where it is understood

the characteristic function XN 3

that the function Gj-u , defined in G. is to be extended zero out-

100

111.6. Normal forms; positivity of g

101

Let vj=Gj-u-Gu. Note that vjE Hl, and that even Ilvjhl0 let the set ER =

{xES2

:

for some given x0. For some

0

a(x) 0

0 < p < R introduce the

,

continuous, piecewise linear function n = np'E on with This function is Lipschitz n(t)=0 on [p,°) , n(t)=l on (--,p-e] continuous, hence the composition fi(x) = n(a(x)) defines a Lipand we get V = n'Va , almost schitz continuous function over S2 .

,

everywhere. For u,v E CO(Q) we get Cv absolutely continuous, and is locally bounded. Thus Green's formula may be applied for (DUu vC + hjku

0 = JE

(2.3)

Ix.(cv)

R

I

x

k)du

It follows that

(2.4)

(Auu v + hjku

JE

JE

In'IIValIVulIvldu R

R

Note that (2.4) holds for all p

,

0 < p < R , and fixed e >

We may integrate (2.4) dp from 0 to R

,

0

observing that the only

term at right depending on p is the term n'

=

dnpeE/dt(a(x))

Hence we get R

(2.5)

J

0

As e

-*

x

x

R

converges to the characteristic function

, boundedly, and almost everywhere, while JRIn'Idp converges

of Ep

to 1

0 the function

v +h3kul jvI k) <JE duIVoIIVulIvl dpIn'I

,

as easily calculated. Therefore (2.2) follows, q.e.d.

109

IV.2. Limit point case for H

We now can prove thm.1.1 for m= 1 , as follows. Suppose a function a exists with the properties of the thm. since a-as x-, all the sets ER of lemma 2.1, formed with this function a, are compact. Now the operator H0 is >1

,

(if necessary we may

replace H by H+c0, using rem.1.3). Thus it is sufficient to prove that im H0 is dense (I, prop.2.7). Or, we may show that (f,H0u)=0 for all u E dom H0= C0 implies u=0. In view of Weyls lemma (i.e., III, prop.l.2) any such f must be C'(O), and then must solve Hf=O. Thus we are finished if we can prove the lemma, below. Lemma 2.2. Under the assumptions of thm.l.l

,

for m=1

,

if

f E H12W(O) solves Hf=O , then we get f=0 Proof. Define the functions $(p) , I(p) , A(p) , by setting .

VR) =

(2.6)

Note that

JRIIflValdg dp , *(R) = 0 p

JRIVfII2 dp, 0 P1. Now we go back to (2.6), and conclude that

< c2angfll2 =

c3an

c4/X-ran/2

(2.8), for A < (2.10)

,

A'

>

since IIfU .


pl

with a positive constant c5 . Now it is well known (and easily

110

IV.3. Proof of theorem 1.1

derived) that the differential inequality (2.10) cannot have a positive solution in an interval of infinite length, whenever n0. The latter proves impossible, because then a3/kr3/2 = (X2/X.)3/2 , while we showed above J

J

J

that it tends to 0 as c-0

.

J

It follows that

, -3/2(R)

(3.17)

R

aj dr


0 and nondecreasing, and I

j=l,...,m, X ceer , as r>p, with c >0, implying the statement. ,

a(r)>a(p)ee(r-p)

Corollary 3.4. We have either K-0 or r K-T(P) - K_T(r) > c`(r-P)

(3.31)

where the right hand side tends to - , as r-

,

as r>P

,

, while the left

hand side remains constant. This results in a contradiction so that we must have K = 0 for all R. This, in turn, implies that f =

0

.

This completes the proof of thm.l.l. 4. Proof of Frehse's theorem.

In this section we finally want to discuss a proof of thm. 1.9. As in the discussion of thm.l.l we only must show that every C'-solution of the differential equation Hu=O in H=L2(In) vanishes identically, under the assumptions of the thm., such as (1.9). Here we may assume that us is real-valued, since the coefficients of H are real-valued. Let us assume q?0 again, without loss of generality.

In the following we consider open balls Br with center x0 kept fixed for a while, with radius r. For uEC(1n) solving Hu=O, and EEC_(Br) we have (4.1)

fB

0

r

In (4.1) we introduce $ = X2IulPsgn(u), with sgn u = ±1 and 0, as u>0 , u2. Proposition 4.2. If n>2, and q>O , then every C(I)-solution u

of Hu=O satisfies the estimate (4.18) with C2 of the form (4.19) where C5 depends only on n , while the center and radius of the ball BR are completely arbitrary. Exactly the same proof works in the case of n=2, with the following modification. Hoelder's inequality implies that (4.22)

Ilullp

< V(2-p)/2pnull2

, with V = 1B dx = nR2 R

and then apply prop.4.1 with p = r

Here we set (4.23)

(tR2)(2-p)/2pllVCll2

11CI2p/(2-p)
0

.

However, for n=2, a simpler proceedure, not using the above technique, still will give the same result as for general n, as will not be discussed here. Proof of theorem 1.9. In view of thm.1.10 we may assume that n>2

,

so that prop.4.2 holds. Let us first focus on assumption

(1.10). For R=r/2 , with the r>0 of (1.10) we conclude that

sup{1u(x)I:ixl=n} < KJ n_ryl , where yl is a positive constant. Proof. One may repeat the proof of thm.1.5 in the following amenand define ded form. Let T(x)=dist(x,3U) ,

(5.1)

O(x) = n-llog(E/T)

for suitable constants E,n>0

as T(X)<E

,

.

,

=

0

,

as T(X)>E

,

Then use this redefined function

o(x) as in the proofs of lemma 2.1 and lemma 2.2, to get essential self-adjointness of H' .

In particular the 'spheres' ER again are defined by the inequality o(x)O (1.2)

(u,Hu) > (u,u)

,

and that HO > 1

,

,

i.e.

for all u E CD(D)

Then we introduce the first Sobolev norm and inner product by (1.3)

IxjvIk + quv)dp = (u,v)1 , u,v E dom H

(u,H0v) = JD(hJku

x

The Friedrichs extension H is a well defined self-adjoint Clearly H-1 E L(H) has a selfrealization of H We get H > 1 .

.

adjoint positive square root A = and 0 {c,du,H} (which may be {52,du,H} itself) , and assume here in addihjk h"jk tion to III,(3.24) that the tensors and coincide on S2 , and that q" >,l outside a compact set K C 52" (we may achieve K=0 by (1.2) and III,prop.6.2, changing variablEs).In particular the norm of a tensor is unambiguously defined, since the two metric hjk and h"jk coincide. Generally (w) will be required for tensors {O,du,H} , but not for {52",du",H"}

.

In view of IV,thm.1.5 it is

naturalt expect q to tend to - near 30 C 0" whenever (w) holds. Accordingly thm.3.1, lemma 3.2 and lemma 3.3 often will have the best use for 0 = 02" , although they are formulated for general O C 0"

(Indeed the condition Va=o0(ge/2) is weaker than Va"=

oQ"(gne/2) near a point x0 E 30

, whenever q -* - as x -r x0 , since

a" and q" must stay continuous near such a point.

)

On the other

hand, thm.3.4 and the lemmata following it strongly depend on condition (3.13), below, which cannot be satisfied for q near a point x0 E 30 C 0" unless q stays bounded there (which is impossible if (w) is to hold). (This reflection assumes 0 to look like ) Note that near such

a manifold with smooth boundary near x0

.

point x0 the conditions (3.16) still allow some degeneration of the coefficients bi , p of a differential expression D , but weaker than near infinity (of 0")

.

The sequence of results, below, discusses conditions on the generating sets A# and D# implying compactness of commutators in the comparison algebra C(A#,D#). Thm.3.l deals with commutators of the form [a,DA], while thm.3.4 looks at [DA,FA], a E A#, D,F E D#.

We use the nctation oO(.)

,

etc., of app.A.

Theorem 3.1. Let aEC_(O) satisfy the condition Va=o0"(q"£/2)(in 01)

for some 0<s0

.

In other

words, we must require that (4.13)

n > B"'/(B"+n/2)

.

Notice that the right hand side of (4.13) is 0 there exists Sk >1 such that (sk )

0

0

holds for every S<Sk 0

For a concrete example we now set q"=1, and then, with A#, D# as above, introduce the classes A#=A# , and

(4.15)

D#={DEV#: (x)-Sb7EA#, P=O((x)a'), p

j=o((x)-a+a)) 1x

The symbol spaces for some such algebras have indeed been . We shall discuss this in VII,thm.4.8.

worked out by Sohrab [S3]

Example (D): This example, and example (E), below, only point to special cases of thm.4.1 where the various conditions can be translated into a simpler (or different) form. None of the algebras have been investigated in generality, regarding their symbol or symbol space. Let 52 be an open subdomain of In with compact connected

smooth boundary 30 , where 2S2 is an n-l-dimensional smooth submanifold of In Let H = q - A , with the Euclidean Laplace operator A , above. Here the real-valued Co-potential q>l must satisfy (4.3), above, with the Euclidean distance. Set S2" = I dV = du" .

,

= dx, H^ = 1-A, to get all conditions satisfied, including (4.1). In other words, we have 0 either the interior or the exterior of a compact subdomain of In Now the class A# consists of all bounded C'(c2)-functions a c such that, for suitable e , 0<e nk(x - a)-2 near a, or q(x) > nk(x - S)-2, near B

,

respectively. (Both for 2 finite ends). Specifically (w)=(s1)

holds for q with (5.12), for nl = 12 , and (s) _ (s,) holds for (5.13) (x-a) 2/q(x)=o(1)

,

as x+a

,

or, (x-a) 2/q(x)=o(1)

.

as x-5,

respectively. (Both conditions, if a and $ both are finite, but if only one end is finite then the infinite end never requires a special condition,except the semi-boundedness of the operator). From (5.11) and (1.7) we know that the operator FA , with A = H-1/2 is bounded whenever, for some y>0, y E C-(I) ,

b = 0(1)

(5.14)

c + bY'/y

,

= O(/q--ywTy)

.

This condition may be simplified by introducing the function (5.15)

6

= Y'/Y ,

6'

+ 62 = Y"/Y

,

Y = Y(x0) exp(JX 6(t)dt) 0

Proposition 5.3. The operator A = (FA)** is in L(H) whenever there

exists a real-valued function 6 E C'() such that (5.16)

b = 0(1)

,

c + b6

= O(V`q-77 67)

The proof is immediate. Note that change of coordinates does not give an improvement of conditions for b. To evaluate the second condition (5.16) we assume that the operator bound, hence the 0(.) -constant in (5.16) is 1 , as always can be achieved by multiplication with a positive constant. Then (5.16)2 is equivalent to the Riccati type differential inequality (5.17)

6' + 26 Re(cb) + (l+Ibl2)62 < q -

Icl2

.

V.S. One dimensional problems For example, consider the case of b-0 , where (5.17) yields

0 < icl2 < q - 6' - 62

(5.18)

.

For 6 = 0 (5.18) gives the trivial condition IcI A more general 6 will improve this only if (5.19)

6'

over the interval I

+

62


0.

X

Accordingly, while the possible improvement of q may be large at some specific point, it will become small in the average, over a finite interval (x,x+h) only if q+0

,

as x-*0

.

,

as x-*

, and can be significant

Since we get q_l in suitable coordinates,

it follows that q, in the average cannot be too different from 1 Accordingly the improvement of the boundedness estimates by coordinate changes may be of little use, if the interval is (--,+°) On the other hand, if the right end B of the interval I=S2 is finite, then one may think of possibly significant changes of

q by change of coordinates.

153

V.5. One dimensional problems Example 5.4. (5.23)

The choice 6(x) = (B-x)A

y(x) = exp(-(B-x) X+1/(X+1))

,

0O then write A2R(A)=1+AR(A)

,

158

and LN+l = (ad H)N+1L

for a moment. For u E im(1-AH0) we get, A2R(X)u = (1+XR(X))LN+lu (6.7)

= LN+l(1+R(A))u

+ X(R(a)LN+lu-LN+lR(X)u) = LN+lA2R(X)u+AA2R(A)LN+2A2R(A)u

,

and again we may extend this to H. . Substituting (6.7) into (6.5) yields (6.5) for N+1 q.e.d. ,

Proposition 6.6. In the notation of lemma 3.3 we have (6.8)

Jr A2JR]+1(A),s/2da

=

2rti(-l)j+l(s,2)As

with binomial coefficients norm convergence of L(H)

,

s>0, j=0,1,...,

Here the integral converges in

as an improper Riemann integral. Proof. Integrate by parts in formula (3.6), using that Rk+l(a) d/dXRk(A) = k ,

.

Proposition 6.7. With the notations of prop.6.5 we have (6.9)

AsA

(-1)3(s/2+j-1)

=

((ad H)OL)As+2j+N + RM

0

where RM = i/27r JrA2R(a)((ad H)M+lL)A2M+2+NRM+1(X)Xs/2+MdA

(6.10)

Proof. Multiply (6.5) by i/(2n)as/2 , and integrate over r Note that the integral may be interchanged with the operators (ad H)3L = Lj , which are preclosed. Then use (6.8) to evaluate some of the integrals.

Theorem 6.8. For all (real or complex) s with -2M-2

>

SM s(a) = AM+1/2+s/2-eAM+l-sRN+l()

,

0<e0

,

2 and SM s()A 2e-M-1

>

as s

bounded and analytic on the (A,s)-set specified.

We should emphasize that, by writing (6.11) we intend to ASAA-s

imply that the (product of unbounded) operator(s) is bounded, and that its continuous extension to H should be taken in its place.

Remark 6.9. It is an immediate consequence of thm.6.7 that the remainder SM

s is a holomorphic function of s (with values in L(H) ), defined in the strip -2M-2 < Re s < M+1 and that even ,

AE-M-1 , the operators SM's AS-M-1, for 0<s<M+l for and SMM's S A -2M-2+e<s<e are in L(H) . Moreover, it should be observed ,

,

that the two factors S1(A) and S2

(A) both are functions of A hence commute with every function of A The proof of thm.6.8 is a matter of analytic continuation

of the identities (6.9), (6.10) from the positive real s-axis onto the s-strip specified. First we get (6.11) for 0<s<M+1, by Next we note that all multiplying (6.9) from the right by A-S terms at right are entire functions of s except the remainder. .

On the other hand, the remainder may formally be written, as specified in (6.12), where the function Si are analytic. Also, boundedness of the functions S!

, near their singularity a=0 is

a consequence of estimate (3.9). Thus one concludes that indeed the remainder integral admits an analytic extension into the strip -2M-2< Re s < M+l Finally, we at least have the complexvalued function ((ASAA-S - A)u,u) _ 4(s) analytic in the strip, for every fixed uEH. . One thus concludes that (6.11),(6.12) .

holds in the same inner product sense. This, in turn, implies for s in the H-boundedness of the pre-closed operator ASAA-s

strip. The remaining estimates follow as part of the above proof, or are trivial consequences.

Remark 6.3'. Note that the above definition of Sobolev space is meaningful not only for integers NO but also for general s>0 .

159

V.6. Expressions within reach

Similarly, for s1 (1.1)

.

q°

In this representation define = X(nkT)-2h1kTlxjTlxk ,

T

= jj=12-3ejwj

with our partition {w.} of app.A, where e.}0, as j}-, and 1 3 3 < (Max{l, sup{IVwj(x)I 0 < £j x E nj}}) Also, nk denotes the positive real number specified in IV, thm.l.l, and X denotes :

.

162

VI.l. Local invariance any C-(Q)-function with 0<X Mi

, with 1

}

.

We shall prove that u defines a homeomorphism onto Mj. Assuming this to be true let us observe that the proper transformation

law is valid,since they will inherit the co-variant tensor property of the 8

.

x]

To verify that u is 1-1 we recall that the maximal ideal mEffi.

is characterized by its corresponding homomorphism J -' M

,

given

VI.2. The wave front space

by A - aA(m) , A E J J. Let m, m' be such that i(m) = t (ml) = x, and as above. Note that this implies a4)(m)=a4)(m'), and aA (m)=aA (m') J

for all

= 1

E C0-(S2j), regardless whether 4)(x)

J

or not. Since a

,

is a homomorphism, and since the m and Aj listed form a set of generators of J., modulo K , it follows that aA(m) = aA(m') for ,

all A E,J Thus the homomorphisms corresponding to m and m' are identical, which means that m = m' Thus the map u is injective. .

Finally we focus on surjectivity of u

above every x GO

.

It is clear that,

there will be at least one point (x,l;) E u(ffij). j

For the map i maps onto I

so that some mE M with x=t(m) can be found. For this m we then get the l;j by our above construction. To show that every (x,n)

,

, for general n E

construct an automorphism of Jj

n is an image, we

, leaving K invariant, which

carries (x,F) into (x,n) , in form of a coordinate transform. Let M=((m.k)) be a constant real nxn-matrix with M>;=n, M*J2M=J2,

where J=((hJk(x)))1/2 (i.e., JMJ-1 is orthogonal). Such a matrix exists for every i;,n E gn and one even may choose det M = 1 ,

,

so that JMJ-1 is a rotation, and can be smoothly deformed into the identity matrix. Denoting x=x0 , for a moment we design a diffeo-

morphism e:Q.- 52. as follows: Let R(T), 0x/(x) taking In onto { x 0. A calculation shows that (2.14)

IV=0 Aj *A

1

198

VII.2. Secondary symbol space so that 0< no

(1'Yv=1In.12)112 .

Thus we have the following.

Proposition 2.7. The space M\W is homeomorphic to a compact subset of the set fn={,EBn+l.

DEnxEn , where

(2.15)

It is clear that

Iv_olnvl2=1

,

no>o)

Bn is topologically equivalent to the ball

En (of infinite radius). In [CHe1] is was seen that

MW = aBnxBn = aBnxBn

(2.16)

.

From V,4 we know that the algebra s0has compact commutators, as a subalgebra of the algebra obtained in example (A) there. Similarly, for the algebra N of V,4, example (A) we get ffi\ID = aMA # X En c

(2.17)

.

Example (B) of V,4. Let us use the function algebra A# and xEO=M3k=gjxTk in

folpde-class D# introduced in V,4,(B). Write

the form x=(x',x") , with x'EB3 , x"ETk . A function a(x',x")EA# must also have its x"-derivarives decaying, as x(i.e., as Ix'I

- ). This implies that functions in A# must be nearly constant on the k-tori x'=const., Ix'I large. In VIII,2 we will consider a larger class A# which, however, will involve non-compact commutators for the corresponding algebra C. The compactification MA c

pjk

#

will look similar to the Stone-Cech compactification of M3k

One may use exactly the same basis Dv , v=0,...,n

as defined in

(2.12) (with D. , v>j, now acting on a periodic function). Again the Aj=D.A are self-adjoint, and A0>0 - . Again we get (2.14). 13

Accordingly it follows again that MW C aMA #xIn

.

However, in the

c

present case, we do not have equality. Rather, the secondary symi.e., to the characbol space MW)int is homeomorphic to B3xZk ter group of the group Mj,k (cf. thm.3.1 of [CS],ch.5 ) We postpone the discussion of other examples of V,4. Here let us first discuss an example of different type. Bn Example (H): Let 0 be an open subdomain of , with smooth smooth boundary, compact or not. Assume the boundary ao to be a smooth n-l-dimensional submanifold of En. We will focus on two ,

.

199

VII.2. Secondary symbol space special case only: either QUaO is compact, or 0 _ l+={xeRn:xl>0}, referred to as "cases (a) and (b)" respectively. Consider the Euclidean Laplace-triple,in either case, with the classes A# and V# of restrictions to Q of the functions (folpde's) of As and Vs of example (A), above. Let C be the coresponding comparison algebra. Again we may use the orthonormal base Dv of (2.12),for every xEaMA#

.

and MA#=B+=(g+)clos

(We have MA#=cU2c2 in case (a)

with the closure of In in En .) The dimension is 6=n+l everywhere. However, an investigation shows that the operators Aj no longer are self-adjoint, although we still have (2.14). In particular, focusing on case (b), the operator Al , involving the derivative normal to the boundary, is not even self-adjoint modulo E

. Accordingly the set n(t-1(x0))

, for a point x0 with x0=0

only is a subset of the complex unit sphere 1v=O1nv12=1, nEWn+1. For a precise description of the space M cf.[C11,V,thm.10.3. The description there is in a slightly different representation, to be translated into the present form. Over points of the finite boundary of 0 one only gets the (still well-defined) cosphere bundle, and, in addition certain 1-dimensional 'filaments'. Over points lxl=00, xl=0, one obtains the ball JEJ0 , and again some 1-dimensional filaments). For other points

Ixl=° we of course get the same as for D=Yn

,

A# , VS again.

In [C1],V we also investigate the commutator ideal E of case (b) in detail, and thus obtain a complete Fredholm theory for the algebra C. We have not analyzed the space ffi\T in case (a)

, but

expect a very similar result.

Note that the last example pertains to the case of a genuine boundary problem, where cdn.(w) is never true. In [C1] and [CE] we also show that the elliptic boundary problem with 'LopatinskyShapiro'-condition is solvable with our algebra C 3. Stronger conditions and more detail on ffi\W

.

It should be realized that our assumptions on A# and V# as well as on the triple {O,du,H} are minimal, so far. In sec.2 we have studied some examples which allowed a more complete description of the space ffi\W

.

We now first will look at conditions to

200

VII.3. Stronger conditions allow a description of the sets i-1(x0) C I\W as compact subsets of a real upper hemisphere ffl

1, just as in the examples of sec.2,

although only locally.

the following observa-

For the Rn-related examples of sec. 2 tion may be useful.

Remark 3.1. Suppose near a point x0E3M # the manifold Q 'looks A neighbourhood Nx like Rn,, in the following sense: x0 has a 0

relative to M 4 , such that = U is a chart. In particular, QnNxO A assume that, if U is considered a subset of In , then either x0 is a point of a smooth boundary piece of U or JxoJ=- , and ,xl for all xENx \R . Suppose then that, in the coordinates of U , 0

the coefficients hjk(x)

, q(x) extend continuously onto Nx

, and 0

still define a positive definite (n+l)x(n+l)-matrix, at all points of N x0 Suppose also that D# contains folpdes D v such that .

(3.1)

D0 =

1

, Dv = -i3 v

,

v=l,...,n , for xEU .

x

Then we have dx=n+l in Nx

, and it is possible to use the folp-

0

de's (3.1) as a base of Sx, although not necessarily orthonormal. Accordingly cor.2.4 applies in the entire Nx

,with n defined 0

by (3.1). For a generator A =a(x), or =DA have the value of aA(m) at some mEt-1(x0)

(3.2)

oa(m) = a(x0) , aDA(m) = b3(x0)nj

or =(DA)* of C we then , x0EaM #

,

given by

' a(DA)*(m) =

b3(xO)nj.

for D=InbvDb3Dj+p, summation convention used from 1 to n,only for non-greek indices.

In thm.3.2, below, we will show that, in this case, we get (3.3)

hJk(x0)njnk + q(x0)1no12 = 1 , n0>0, for nj = nj(m)

.

Therefore we may write (3.2) in the form (3.4) aa(m)=a(x0),

aDA(m)=(b3(x0)Cj+p(x0))/(hjk(x0)Z* jCk+q(x0))1/2

201

VII.3. Stronger conditions

using the coordinates j=nj/n0 , with a E E ]En

.

Here the vector _ is not always finite. It is assumed to be a point in the directional compactification In of Cn , defined similarly as

n above. If, in addition, it can be shown that all nj are real, then we have E,n (but we should ,

recall example (H) of sec.2,

of a comparison algebra on a manifold with boundary, where we do not get all nj real). In this representation, the space N x$n appears as a part x

0

of the compactification of the cotangent space T*c2 of 0 , generated by the bounded continuous functions (3.4) over 0 . Moreover,

the representations of t-1(x0) C M of thm.2.2 are obtained by a change to 'projective coordinates'. In the following we will seek to recover these features. While we return to the general case, assuming only (a0),(a1),(d0), and (m1),(m2), (and not (w)) we impose (some or all of) the following additional conditions. Condition (m3): For each x0EM # and open neighbourhood N of x0 A there exists a function a E A# with 00 C again maps into

,

so that

HS-1

To show the surjectivity, let us introduce 'projective coordinates', for (x,E)ET S2 , with x near the point x0 , by setting

j=nj/no , n0>0 . We get (3.28)

(bjnj+pno)/(hjknjnk+gn02)1/2

Here we introduce n0=r(x)/q(x)

,

nj=hjk(x)ck(x) , with the (real)

coefficients of some folpde F=-icla

j+r, into the right hand side. x

1/ Then it will assume the form {F,D}/{F,F}2

.

For j=nj(x)/Jr(x)J

208

VII.3. Stronger conditions

209

we thus have ±{F,D}/{F,F}1/2

(3.29)

In (3.29) set D=Dv . Also assume Dv orthonormal in some neighbourhood of x0 , and let {F,F}(x)=l , near x0 , without loss of gene-

rality. It follows that (3.30)

TD A(x,C) _ ±{F,Dv}

where the same sign holds for all v

,

,

since the sign is determined

as the sign of the coefficient r(x) If F runs through all unit vectors of the 6-dimensional space, at some x , then the 6-tuple .

({F,D

v=l,...,d

v

runs through the entire unit sphere. Thus, for

any point XES6-1 we have either X or -X assumed by (TD A(x,&)) The same property then must hold for the values gy(p)

,

.

as p runs

H6-1 But we already know that (p) E Thus it 0 (H6-1)int follows that all points of must be s assumed. Since in Ha-1 is closed, we then conclude that im = i.e., is sur-

through it-1(x

)

.

.

,

jective.

It then follows that the map p = v(m) (3.31)

,

for mE1-1(x0)

is well-defined. Let

We get

.

Ta(p)=a(x0)=aa(m)

,

TD

A(p)=Cv(p)=nv(m)=oD A(m)

U

so that indeed formal symbol and symbol coincide.

On the other hand, if (m6)x

is false, i.e., 6x 0} ,

.

But if 6x dim Sx 0

= ax 0

,

0

for all x-'Q, and every neighbourhood of x0 contains such points.

Repeating the above conclusion we find that the vector (TD V

now assumes either C or -C, for every

CEBa-1

.

The same again is

true for the vectors c(p), as p runs through w-1(x0) Thus, in B+6-1, this case, the map also goes onto and we again may define .

the map v(m)

.

Thus we have the same statement, and thm.3.6 is

VII.4. Structure of ffi\W; Examples

210

established.

4. More structure of M.

,

and more on examples.

Remark 4.1. Before we attempt application of the results of sec.3 let us point out that thm.3.2, cor.3.5, and thm.3.6 all have 'local' generalizations, in the following sense: Under cdn's (m3)x

only, for some given point x0

and (m6)x 0

0

E 3M # , we still have the statement of thm.3.2 true for that A point, regardless of all other points, assuming the other (mj), of course. Under (m3)x alone we still get cor.3.5, at x0 .

0

Again, under (ml')x '

0

(m3)x

,

with the other (mj) required for

0

thm.3.6 we get the homeomorphism of thm.3.6 at least between

C1(x0) and r-1(x0) , although perhaps not between ffi and 3F SZ Here, by (m3)x we mean that the function a of (m3) exists 0

for that x0, and its neighbourhoods. By cdn.(mi')x

we mean that 0

(ml') holds 'near x0', i.e. {D,D} and function in A# near x0

.

each coincide with some

Similarly (mi)x

, which may be substi0

tuted for (ml)

,

above.

We shall refer to these local generalizations as thm.3.2x

,

0

etc. Note that there also is a thm.2.2

x0

Let us point out again, that, speaking in terms of thm.3.6, examples have been given which show that the set i-1(t ) , for some x0E3M # , may range from the minimum (the set Wclo n -1(x0) A Tr

to the maximum (all of ff-1(x0))

.

In particular, example (B) of

V,4 shows that cases between these two extremes occur: For the Laplace comparison algebras of the polycylinders G3k we may get more than Wclo:h n-1(x0) but not all of n-1(x0) , rather, only some sub-surfaces of the hemisphere Ha-1 are contained in I

.

As a partial answer to the question for the set i-l(x0) we prove the result, below. Theorem 4.2. Under the assumptions of thm.3.6x , if there exists 0

VII.4. Structure of 81\W; Examples a function aEA# with a(x0)90 , and aD.0AE E , where D0 denotes the

folpde assumed in thm.3.6, with {D0,D0}={DO,DO}0#0 we have

t-1(x0)

(4.1)

at x0 , then

,

t-1(x0)f'$Tclos

=

Proof. Let mEHts = ffi\Wclos = ffi\,s , and let 2(m)=x0. Since aDDAEE

we conclude that 0 = aaD A(m) = a(x0)OD A(m), or aD A(m)=0, since 0

A

d

by assumption. Therefore the statement is an immediate consequence of cor.4.3, below, q.e.d.

Corollary 4.3. Under the assumptions of thm.3.6x , with the 0

operator D=D0 of thm.3.6, we have (4.2)

t-1(x0)nIDclos={mE1-1(x0):aD A(m)=0}={mE1-1(x0):TD A(m)=0} 0

.0

Proof. First notice that TD A(m)=0 for points of Wclos, Indeed, 0

we know that, for an (x,F)EW we have

so that the formal

symbol assumes the form (4.3)

T

DA

(x,E) = b0E?/(hOkE0E0)1/2

,

3 k

1

for

1

01=1

With the technique of the proof of thm.3.6 we express this as TDA(x,C) _ ({D,F}1/({F,F}1)1/2)(x)

(4.4)

,

where F is a suitable folpde. (Note that the sign of the 0-order term no longer is essential.

)

Since we have {DO,D0}1(x0)=0 , we

conclude from (4.4), that TD A(p) =

0

,

for all

p E Tr(x0)

whenever p E Wclos, which proves the above. Now, again in the terminology of thm.3.6, we must show that the restriction of the map p to the set {mEt-1(x0):r) 6(m)=0}

maps onto the sphere (or ball) ns=0 However, for xE12 near x0 the values of the vector (aD A(m)) , mE1-(W), are exactly the .

v

values of (TD

(({Dv,F}/({F,F}1)1/2)(x))

,

taken over all

folpdes F, defined near x. From this one concludes that also in the limit x-x0 every vector of the sphere or ball is assumed. This completes the proof. Remark 4.4. Let us emphasize again, that thm.3.6, and cor.4.3

211

212

VII.4. Structure of ffi\W ; Examples

require conditions (m4) and (m5), which imply that, for a basis (mod A#) of self-adjoint folpdes DIV the corresponding generators DvA are self-adjoint mod E . If these conditions are violated,

as in the case of a comparison algebra associated to a boundary of section 2), then N no longer must problem, (cf. example (H)

be a subset of the space 3P 0 In [C1] and [CC1] we discussed such an example, where it turns out that certain additional .

'filaments' of N occur, which cannot be found from the values of the formal symbol. Remark 4.5. As another feature, illuminating the role of the let us observe that, formal symbol and the compactification 3&*sl ,

in all of the results of sec's 2, 3, 4, involving the sesqui-linD,FED# , we may use the form ear form {D,F} {D,F}^(x) of any ,

triple {O^,dp^,H^} (c> {O,dp,H} , defined as (4.5)

{D,D}^ = h^jkFibk+IPI2/q^

,

and by polarization for {D,F}^ . Actually, one may use just any similar sesqui-linear form, positive definite over each space Mn+1 of n+l-tuples (Vu(x),u(x)), at every x E 0 , as long as the estimate V,(1.6) remains satisfied with {D,D}^ instead of {D,D}. It is clear then that we get corresponding conditions

(mj=1,6,7, as well as (ml')^, and the corresponding local etc. (Note that cdn's (mj), j=2,3,4,5, do not conditions (mj)^x depend on the form {D,F} ). Here it is essential that (ml implies that {D,D}^ {S1,dp,H} in the sense of q^. We have q > q^ V,3. For formal reasons it is convenient, however, to impose ,

the following. Condition (ql)

:

We have q(x) = q^(x) in N^\N , with some neigh-

bourhood N of 30 To satisfy V,(3.13) and (w) we again require (as in V,4) that .

q(x) > Y1(dist(x,3c))-2 , and V(log q^) = o(1) (in R^)

(4.7)

Now it turns out that the classes A'

,

Do of V,4 in general

will not satisfy conditions (m.), j=1,2. To discuss an example only let us work with the form {D,F}^ of the above h^jk and q^ and the following more restricted classes called A# and Do AC is the class of all bounded functions a E C'(c) such that Va = o,^(q^E/2)

(4.8)

,

for some e, 0<E1 on I

.

In order to reach the generality of [S2] all con-

ditions imposed below must be translated back to the general case.

In order to have only one point of M # at each end of the interval we select A#=algebra finitely generated by C_(I) and s(x)=x/(x) , and D#=span {DO=/q, D1=-i3x}(mod A#). We want E=K(H), hence we require V,(3.13) and V,(3.15), V,(3.16), with q=q^. A

calculation shows that this requires V,(3.13) (4.13)

q'

= o(q)

,

i.e.,

215

VII.4. Structure of M\Q1; Examples

as only additional condition. (Actually, we could allow the larger function class A#:A# of V,4, and still would get E=K under (4.13) only, but this would give us many points in aM # over ±o'.) A Note that (4.13) will not allow'potentials of exponential growth, while potentials of arbitrary polynomial growth often are acceptable. In particular, the potentials q(x)=l+x2 and q(x)=1+x2 +x4

of the harmonic and the anharmonic oscillator in quantum mechanics satisfy (4.13).

All conditions (m.), (ml') hold, as well as the assumption of thm.3.6 (with

We will show that

(4.14)

IN

=

3P*(R)

,

where the form {D,D} of hll=1 and q=q^ is used. We trivially have W= Ix{-oo} u Ix{-} c1. For the secondary symbol space Sohrab looks at the operator T = observing that the range of p2(m), in the set 1-1x0 , for either one of the infinite points x0=±. with a cut-off

coincides with the essential spectrum of function X=l near x0 , = 0 near -x0 .

Formally we get T = L-1, with a self-adjoint realization L associated to the expression (in R = (--,o) ) (4.15)

32q-1/2+1

L

= -

3xq-1 ax +1-q-1/2(q-1/2)

Transforming L onto Sturm-Liouville normal form again we get

x _ (4.16)

L` _ -a2 + l+q"/(4q2) -5q'2/(16g3)

y

,

q

-cc0} . If the manifold 0 has a regular boundary, onto which the expression H, and as well as A# and V# may be extended, meeting similar ,

measure du

conditions as for interior points, then commutators are no longer compact. However, both quotients, CIE as well as E/K(H) , turn out to be function algebras. The maximal ideal space 1 of C/E looks similar to the spaces encountered for complete manifolds like except that it has certain one dimensional 'filaments' over the boundary of 9

n

,

.

On the other hand we get (0.1)

E/K(H)

__

C(N,K(h))

,

with the Hilbert space h=L2(R+), and the symbol space ID of a well If the E-symbol investigated Laplace comparison algebra over In .

of an operator AEL(H) does not vanish, then A may be inverted mod E , which corresponds to the first step of the common approach of solving a boundary problem: That of using a fundamental solution (or parametrix) of the differential equation to solve the differen tial equation, though not yet the boundary condition. This then must be followed in common theory by solving another system of equations- for example a singular integral equation over the boundary, in case one uses the well known boundary layer approach. The latter corresponds to the inversion mod K(H) of a (system) 1+E , with E E E .

Presently we do not study such manifolds with boundary, but rather turn to a pair of different examples. First (sec.l) we consider an algebra over I , where some of the generators are periodic functions. (It was seen in V,5 that a commutator [ e1X,DA]

is

VIII.O. Introduction not compact)

.

It is found that again a 2-link chain results. Now

we get

E/K(H) E C(N,K(t2))

(0.2)

with the Hilbert space t2= L2(Z)

,

i

S1 U S1

=

and two disjoint copies S1+ of

,

the circle S1

In direct relation to the above special importance of L2(Z) this algebra is invariant under translations by 2kir . Again there

is a tensor decomposition of H involved, together with a certain unitary map of H. Results are related to those discussed in Gohberg-Krupnik [GK), ch.l1, (cf. also Sarason [Sail

,

and [CMel1

where we expand on the present discussion).

Next, in sec.2, we look at a product manifold O=1n'>B, where B is a given compact manifold. The simplest example would be the circular cylinder XxSl , with the circle S1 We studied a compaXxSI rison algebra on in example (B) of sec.4, using an algebra .

A# of bounded Cm-functions having all derivatives tending to 0, at ±That including the derivatives in the 'S1-variables' algebra had a compact commutator. In sec.2 we admit more general .

,

generators, and obtain noncompact commutators. Again a 2-link ideal chain is found. Again the second quotient is of a similar form: E/K(H) __

(0.3)

C(ffi,K(h))

,

where I is the symbol space of a certain algebra of singular integral operators over in In case of n=l we get I = R U I , .

In each case we have h= L2(B). a disjoint union of 2 copies of I Again a tensor decomposition of H is involved, after a unitary .

transformation. The ideal E is identified as a certain algebra of singular integral operators with compact operator valued symbol. In section 4 the corresponding is done for a manifold with finitely many cylindrical ends. To make this discussion possible we introduce the technique of 'algebra surgery' in sec.3.

This simply means that we cut out the portion of a comparison algebra over some manifold 01

,

corresponding to an open subset

U C 0 , and compare it with the corresponding portion of another

algebra C2 belonging to another manifolds 02 , but with the same subset U

.

Of course algebra surgery also may be performed if com-

mutators are compact.

It should be noticed that sec.3 only discusses the basic

219

VIII.O. Introduction

method of surgery, at the example of a 'compact cut', - i.e. the boundary of the cut-out portion is assumed to be compact (not the cut-out portion itself). It is not hard to remove this assumption, at the expense of additional conditions near the infinite parts of the cut. We find that these additional conditions are too complicated to state, except for very specific geometrical models. For a discussion of surgery on a manifold with polycylindrical

ends we refer to a[ CDgl]

,

sec.5.

In all three examples considered here the symbol space of C (i.e. the maximal ideal space of CIE) is a compactification of the wave front space. In other words, the secondary symbol space is void, fon each of these examples. The theory of sec.4. has an application to a problem on In, studied first by Nirenberg-Walker[ NW1J, then by Cantor [Ctl],

Lockhart [Lk] and McOwen [ M1] (cf. also [ LM] ). This involves differential operators with coefficients constant at infinity and homogeneous symbol in certain weighted Sobolev spaces, where derivatives of different orders have different weights. Details of this application will be discussed elsewhere. We note that there are numerous other approaches to singular boundary problems on manifolds with cylindrical or conical ends (cf. Agmon-Nirenberg [AN1]

,

Bruening-Seeley [BS1,2] '

Melrose-Mendoza [ MM]

,

Schulze [ Schu1]

). Also let us point to

a direct extension of our present theory discussed in [CFb] and It turns out that the E-symbol can be extended to the

[ CMe1]

.

entire algebra C One thus obtains Fredholm criteria without involving a chain of two inversions. .

1. An algebra invariant under a discrete translation group. Let us consider the Laplace comparison algebra C on I of example (G) of V,5 with generators V,(5.27). We have discussed the commutator ideal E in V,prop.5.5. Now we want to obtain more details about the ideal chain C D E D K(H) and the quotients CIE and E/K(H). First we study the symbol of the algebra C .

Proposition 1.1.

The compactification M A

#

of the function class

A# described by V,(5.27), as a point set, is given by collapsing in the compact each circle {(x0,y):yESl}, for a fixed x0EI infinite cylinder [-o,o]xSl , into a single point. The points of ,

220

VIII.l. Periodic coefficients the two circular 'caps'

1(-co,y):y ES1} and {(co,y):y 61} correspond

to the maximal ideals {a(x): limk}-.a(2sy+2kir)=0} (and with the limit k}+oo, respectively, for integers k) of the algebra C A#

The topology is the weak topology induced by CA#. In particular, the relative topology induced on the two caps is equivalent to the Euclidean topology of the circle S1 . Every neighbourhood of one of the points 6 of the circle caps contains an open neighbourhood of the point 0+2km, for all sufficiently large (sufficiently small) integers k corresponding to the cap at respectively. ,

The proof is left as an exercise. Theorem 1.2. The symbol space ffi of the algebra C generated by A#

andD# of

(5.27) is homeomorphic to a compact subset of M #x[-co,+wl. Using the homeomorphism as an identification we have A ,

(1.1)

ffi

That is,

ffi

= M

A is the closure of the wave front space in the above

product. The symbols of the generators are given by QA(x,E)=a(E)

(1.2)

,

cDA(x,E)=s(E)

, ca(x,E)=a(x),

where the functions A , s and a must be continuously extended. Proof. We use the argument of Herman's Lemma again (cf.[C1],IV,2): ,

The algebra C has the two commutative subalgebras C0 = C # and A C# = algebra span of the operators S=s(D) and A=a(D). These alge-

bras have the maximal ideal spaces M #, and (the interval) A respectively. Clearly both of them together generate the algebra C.

be the duals of the injections C0

Let r0:M-M # and iT #:M A

iC and C# + C

.

Then u:ffi + MA#x[-co,+co]

,

defined as u=TrOxTr#

defines a continuous map onto a compact subset of the space at right. This map must be 1-1, hence a homeomorphism, because 7T 0(m) _ 70(m') and n#(m)

=

7#(m')

,

for m,m'E ffi implies that the

homomorphisms hm:C+M and hm,:C-M corresponding to m and m' coincide on C0 and

C#, hence on C

.

Then hm=hm, implies m=m'.

Now N must contain the set $x{-o,+oo}

,

identified as the

221

VIII.l. Periodic coefficients

wave front space, by VII,thm.l.5. Indeed, the map n0 is identical

with i of the proof of VI,thm.2.2 over interior points of O. The cosphere at each x0E I contains exactly two points which must 70-1(x0) Since N is closed, it must agree with the two points of i.e., the space (1.1). On the other contain the closure of W , hand, N cannot contain any (x0,E0) in the product set having 0 Then f(D)EE finite. Indeed, let 0ECO(I) be such that Using the associate dual map one finds that the and also E C On at m=(x0,E0) symbol of A=O(D) assumes the value the other hand aA=O since AEE (by V,prop.5.5), a contradiction. This shows that N coincides with the set (1.1). Then formula (1.2) is a consequence of the fact that aa(m) = a(s0(m)), aECO and af(D)(m) = (x#(m)) , O(D)EC , by virtue of the properties of the associate dual map (cf.[C1],AII,5). Q.E.D. Next we examine the quotient E/K(H). In fact it will be possible to obtain a much more precise control of the ideal E. First, in that respect, it is useful to look at the C -subalgebra F of E with generators .

.

(1.3)

j=-Naj(D)eiTx E E , ajECO(l)

E

, N=0,1,25...

.

Note that the equation (1+E)u=f , for such an E, is equivalent to the linear 2N-th order finite differences equation (for the inverse Fourier transforms u' and f') (1.4)

u' (x) +

j=-N

a.(-x)u'(x+j) = f' (W )

,

x E I

J

Here (1.2) relates only the values of the vectors (1.5)

u'=(u )=(u'(x-j)) i=O,±l i

for each fixed x

,

O<xl, (i.e., some noncompact part of 0 is identified with a subdomain of a polycylinder) we get a noncompact boundary, however. Such type of

algebra surgery is attempted in [CDgll. First of all it is clear that the wave front space T of C is just the cosphere bundle of 0 , and the restrictions of the

VIII.4. Cylindrical ends symbols of generators to W are the functions a(x)

(4.5)

,

bJ(W )j

aEA#

,

D=bJ3 j+p E D# x

,

for a multiplication operator a , and for DA , respectively.

The important question to be answered is about the i.e., we ask for points of BI

secondary symbol space M\01

,

over infinite points of the compactification M # . At each end A the open subdomain 0, of the manifold S2 is identified with an open subdomain Uj

Bjxl I =(0,oo) = of the cylinder 0. Thus we have the configuration of thm.3.1 , assuming that

the generating sets A# A#

,

D# on S2

3

J

.

,

,

D are matched with corresponding sets In that respect let us choose A# and D# in accor,

#

3

3

# J

dance with (2.2) and the proposed sets A D following (2.2), chosing n'=l and B=BJ . , j=1,.... N This and (3.1) describe the ,

.

D# uniquely, since in any part of 0 away from its ends we just get all Cm-functions (folpde's). sets A

,

Next we look at the commutator ideals E. Laplace comparison algebras C.

of the These

=

j=l,...,N

=

were studied in sec.2. In fact, thm.2.6 gives a precise desciption of the Ft-conjugated ideal E.^ of E. , where Ft denotes the Fourier transform (2.3) in the t-direction of 0

.

.

J

Remark 4.1. The ideal E. contains the operator A. 3

(l-A.)-1/2

=

1

0

where 4j = D2 + Ax3j denotes, the Laplace operator on Stj , with

the Laplace operator A

on B x,j

.

Indeed, we get

J

Aj^ = F.t1AjFt = (1+T2-Ax,j)_1/2

(T)-1

Also the operator valued function S(T) _

(1+T2_A

(4.6)

-+

0

x,j )

lTI takes values

in K(hx'j) because the manifold B. is compact. hx'j = L2(B.) Thus we indeed get Aj^E CO(l,hx'j) C Ej^ or, A.E E. ,

,

The above remark shows that VII,thm.4.7 may be applied to each of the cylinder algebras Cj , showing that the secondary symbol space is empty. (In particular we have MA #

= Bas

I

easily seen. One thus confirms easily that all assumptions of VII, thm.4.7 hold.) In view of remark 4.8 we have the following result. Theorem 4.2. The secondary symbol space of the Laplace comparison algebra C(A#,D#) is void. The symbol space Iffi of this algebra coincides with the compactification of the wave front space 1

247

VIII.4. Cylindrical ends

under the symbols (4.5) of the generators Next we look at the commutator ideal E(A#,D#) Here we may apply thm.3.4, which will link E with E. . Again 0 and have .

the common subdomain Uj=2j=BjXE+ Using thm.2.4 we get a complete description of the commutator ideal E. Then we may gene.

.

rate the comparison algebra - here called C. - of the subdomain U.C. SZwith the classes (3.3) and call its commutator ideal E.-. J

J

J

Note that Cj" and Ej- also live on the manifold 0 , where they again are generated by the common subdomain Uj=cj of 0 from restrictions of the classes A# and D# defined on 0 For each j=1,...,N choose a cut-off function 1(T), depending .

on t only, and meeting the requirements of thm.3.4. Then, we get E C'(N)

For any EEE we have 0E ,

E K(H)

, by

VII,thm.1.3, and [Pj,E]EKr(), by V,thm.3.1 and ECC. Accordingly (4.7)

E =

jj=1pjEipj (mod K(H))

y* EP +

Using thm.3.4 this implies the proposition, below. Proposition 4.3. The quotient algebra E/K(H) allows the direct decomposition (4.8)

E/K(H) = E1-/(H1

® ...® EN"/(HN')

On the other hand, we now look at E

,

HL2(O.)

as a subalgebra of Ej,

again using thm.3.4 and remark 3.9: Theorem 4.4. We have, with !z (4.9)

.

K(hj)

E3.°/K(H.-) = C(Ekx'j)

,

hj=L2(Bj)

Ii = E

,

j=1,...,N.

Proof. We apply thm.3.4 and conclude that, for EEE ., we have J

(4.10)

E + K(Hj) =

K + K(Hj)

*jEj*j + K(Hi)

,

with some EEEj, and H=L2(0). Since K(Hj-)CK(Hj) with the imbedding of thm.3.4, we get a map u:Ej-/K(Hj")} Ej/K(Hj) , which is an isomorphism, by virtue of (3.17). On the other hand, thm.2.7 implies that E./K(Hj) = C(E+,kx'j)®C(E-,kx'j). Also, from (2.31) we conclude that the symbols of operators of the form 4) jEj vanish identically on E . Again, checking on the symbols of all

248

VIII.4. Cylindrical ends

the generators, as they are listed in thm.2.7, one finds that there are enough symbols of this form to generate the entire C(tk ), by the Stone Weierstrass theorem. We still introduce x,j

the notation Ij = E + _ I for each j=1,...,N, and find (4.9) established, and our proof is complete. ,

We now summarize the results about the commutator ideal E below. Remaining proofs are standard and will be left to the reader.

Theorem 4.5. For the quotient algebra E/K(H) we have (4.11)

EIK(H) = C(&l,fzxej) a) C(E2,hx,j) ® ... ® C(EN,kx,N)

I

where each I. is homeomorphic to I , and where fzx'j are the compact ideals of the spaces L2(B.)=h3.. Moreover, E (mod K(H)) is generated by (4.12)

:

$jax,jlj*j, *jDx,jAj,pj

with the classes

and

Dx'jED#j, j=1,...,N,

ax'jE

of of (2.2), for B=B.

.

In (4.12)

may be chosen arbitrafor arbitrarily large R.

the support of the cut-off functions rily far out - i.e. within t>R The generators (4.12) have symbol ,

(4.13)

a

D x,jA2x,j T?(T), x,jAx,] T.(T), J J

TEE =1 ,=0 on E J

,

k'

k#j

with (4.14)

A

x,j

= (1-A x,j )-1/2

T.(T) = (1+T2A2 i

)-1/2

x,j

Thm.4.5 has an application to a singular elliptic boundary problem on In considered by Nirenberg and Walker [NW], which we shall not discuss in detail.

249

CHAPTER 9. Hs-ALGEBRAS; HIGHER ORDER OPERATORS WITHIN REACH.

In the present chapter we attend to two different, but related problems. First, we propose to also study comparison algebras in L(Hs) , where Hs is an L2-Sobolev space on 0 of order s. Here the spaces Hs always are understood as spaces of the HS-chain induced by the (self-adjoint) Friedrichs extension H of HO5 for a a given triple {O,dp,H} (cf.I,6). In that respect we must assume cdn.(s) in order to be in agreement with the customary definition of Sobolev spaces.

Second, we will focus on more general N-th order expressions within reach of a given comparison algebra C (cf. V. def.6.2). Every compactly supported expression already is within reach of the minimal comparison algebra, as we know from V,6. However, fm a

general algebra C on a noncompact 0 the general expression L no longer is within reach. Thus we will ask for criteria to decide whether a given L is within reach. The relation between these two tasks is discussed in sec.l below. There we also discuss the organization of the present chapter. In particular we point out that some of the theorems have to be 'recycled', in the following sense. The first application only applies to the minimal comparison algebra J in L(H) 0 (or in L(Hs) ). That brings certain higher order differential expressions 'within reach' of JO , hence of larger algebras, so that, in turn, the same theorem now may be applied again, to get larger classes within reach, etc. Unfortunately this makes it necessary to allow as assumptions, in many theorems, a variety of special cases, so that the theorems look complicated.

IX.l. Sobolev comparison algebras 1. Higher order Sobolev spaces, and Hs-comparison algebras.

In this section, and in the following we assume cdn. (s): All operators Hm are essentially self-adjoint, for m=1,2,...

.

Then it becomes possible to work with general Sobolev spaces Hs as already introduced in V,rem.6.3 (and rem.6.3'). We recall the definition:

For s > 0 we define Hs = dom A-s (u,v)s = (A-Su,A-S V)

(1.1)

,

(lulls

=

,

and

(u,u)1/2 , u,v E Hs

For s < 0 we define (lulls and (u,v) s by (1.1) for all u,v

E H = dom

A_s

and then Hs as the completion of H under ilulls The spaces H. and H_. are defined as the intersection and .

the union of all Sobolev spaces: H. = n{H

(1.2)

:

sell

,

H__ = V[Hs

:

s E 11

.

In particular H. is considered a Frechet space under the s E 1) The space H_. is provided

topology of the class {Il.lls

:

.

with the inductive limit topology. For details cf. [C1],III,(l.23) (but this will not be required in the sequel). Notice that the collection (H:Isl0 For s=l and uEC0 the Sobolev norm Ilull1 coincides with a (weighted)

L2-norm of the (n+l)-vector-valued function (u;Vu), by V,(1.3). Correspondingly, every norm Ilullk , k=2,3,..., may be expressed as a weighted L2-norm of the 'jet' (u(a):Ial[n/2]+k+l s

Accordingly Hs consists of smoother functions as s grows larger,

and H.C C(). For sn/2 , and H C Ck(Q), s

as s>n/2 + k

.

In particular, for any triple over D2, we have

C' (D) C H

C C0(Q)

Note that we are not attempting an investigation of the spaces Hs at infinity. For example, for S!=Xn it is known that the

functions of H. have all derivatives vanishing at infinity. More generally one could ask how Hs will be influenced by a change of hjk the metric tensor or the potential q. Discussions of this type will not be made here.

Assume now that we are given a function algebra A! and a Lie-algebra D! satisfying (m5'), and (1.), j=1,2,5, as well as (a0),(al)(or (al') for D# only, D!=D#+{al}), (d0), (or also, we admit the case of the minimal classes).

Under these assumptions we will show that the operators

(1.9)

a ,

DA

, AD ,

for a

A!

, DED!

belong to L(Hs) (cf. thm.4.2). Accordingly, for s E B , the Banach algebra Cs = C5(A!,D!) obtained as norm closure in L(H5) of

the finitely generated algebra C0 of the operators (1.9) will be called the Hs-comparison algebra of the triple {S2,du,H} , and the generating sets A! and D! Similarly we introduce the algebra C. as closure of C0 in 0(0) For a precise definition of the opera.

.

j+p in terms of distribution

tcrs (1.9) one may think of D=bJB x

derivatives. Or else, one also may think of the closures (in Hs) of the unbounded operators ajHs, D0,(AIHs), (AIHs)D0, respectively.

For technical reasons we introduce a second type of finitely generated algebra, called CO , obtained from the generators (1.10)

a

,

Clearly we get C0 C

DAj

,

AJD

,

for a E A!

,

D E D!

,

j=1,2,....

while C = Co whenever (m5') holds,

since then we have A E CO, hence DA3= (DA)Aj-'E CO, and similarly A2D E CO. If (m5') does not hold, then (1.9) and (1.10) still generate the same Banach-subalgebra of L(H), (and of L(Hs) or 0(1) as well, as will be seen later on). Indeed, from (al') we get existence of a function X E A! such that XD=DX=D, for any given D E D!

,

and with OX of compact support. Then we get DA=(DA)(AX) 2

257

IX.l. Sobolev comparison algebras

+D[x,A2], where the last term is in KO), by V,lemma 3.2. By (11) so that the first term is contained in CO Also, C D K(H), by V,lemma 1,1, thus it follows that DA2 E C. A similar conclusion implies DAB E C, j=3,4,... , and AIDE C, j=2,3,... In thm.4.2 we also will show that, under above conditions, the algebra Cs is a C -subalgebra of L(Hs), while the ideal chains C and Cs are in agreement, and the symbol spaces remain the same. In fact, under suitable additional conditions, such as (14), even we have x E D!

,

.

.

the symbols of the generators remain independent of s. 2. Closer analysis of some of the conditions (1.) and (m.). In this section we assume (a0),(d0),(al)(or (al')), and more cdn's as stated. In thm.2.1 we show that (m6) implies (14), assuming some other (mj) and (lj)

. Actually, we show that (14) and

D=b03x.+p E D# implies p E D#, and D0=b33

(**)

D#

.

imply a 'global Parseval relation' of the form

{D,F} = JN,=lava{D,Dv}{D,,F} for all D,F E D#

(2.1)

with a symmetric positive ((a,X)), avXElk, and a system D1,..,DNED# while (2.1), even with avxEA , and without (**), but with inver-

tible ((avx)) implies (14). Thus (m6)x

gives a local (2.1),i.e., 0

H = 16,X-1DvaVADX

(2.2)

DvED# ,

,

near x0,

from which a global (14) may be pieced together, if (m6) holds. Theorem 2.1. Assume cdn's (m1),(lj),j=1,2,3. Then (m6)x

implies 0

(2.2), while (m6) implies (14), and even (with certain DEED#) 6=6(x)=n+l

H =

(2.3)

with a partition {xj:j=l,...,N} of MA#, xjE A#, au,=a,u=ai.E A#. Vice versa, (14) and (**) imply (2.1) with certain DvED#, avxE I Proof. Assume (m6)x

for x0E 3M #. With some orthonormal base {D.} A ((a.l(W))) invertible for xE N=Nx , where 0

we get

of Sx 0

(2.4)

0

ajl(x) = {Dj,D1}(x)E A# , aj1(x0) = Sjl

258

IX.2. The conditions (1)

259

Using (13) on det((a J.l)) we get X((ava(x)))-1=X((avX(x))), with ava E A# , with a cut-off function X E A#, in accordance functions

with (m3), for x0 and N

We introduce ((bvx)) _ ((av,A))1/2, and

.

find that Dv=lXbvXDX , v=l,...,5, is an orthonormal base at every xE N (Perhaps D )"'E D#, but still D v"E S ). Use Parseval's relation x for the DJ J. Define the local matrix H"= ((hv,X-))v

n by

a=0

hvA =hvX for v, a>1, =q-1 for v= A=0, =0 for v=0, a>l, and v>1, u=0, for a moment. Also let D,-= Jbva j+b0

, and then define the matrix

x

B=((b)) v=0,...,n,a=1,. .6 X

,

with'v=row-index, u=column-index.

The above is valid in local coordinates, near points of Q. dense in M

Observe that then Parseval's relation can be written as

A#

(2.5)

b*H-b

=

b(H-BB*H"')b

,

for all b E !):d

This relation may be polarized, and then yields

,

xE NO

H"'-1=BB4 .

Or,

(2.6)

which is valid for all x E N

,

u E Md

.

If we substitute u0=u(x),

.(x), for some u E C' (O) , and multiply by X(x) , we get

uj= u IX

1b_1IIXDv""uII2

J0dpX2(hJku

(2.7)

u IxJ

k+gIul2)

=

,

u E C .

Ix

Now we get Dv = XcvXDX- , with the matrix ((avX))1/2

.

((cva))=((bvX))-

As a consequence the right hand side of (2.7) may be

written in the form (2.8)

Lsa=iJ0dpX2avXDvuDXu

Finally, assuming (m6) U(m6)x, use (13) to construct a partition of unity {X1,...,XN} on the compact space M # , where we A require that XjE A#

,

that (2.7) with (2.8) Dv = Dvj (2.9)

.

j=1,...,N , while JX? = 1 on M # A

,

and such

hold for every Xj with suitable

Taking a sum over j we then get

(u,u)1 = LN=l1 v,a=1(XjavADvju,DXju) for all u E CQ(O)

IX.2. The conditions (1)

Here we finally may polarize and change notation i.e., the first half of the theorem.

for (2.3),

Vice versa, let (14) and (**) hold. Accordingly we can write (2.10)

H = Iv=lavDvFu , av E A#

,

D')

,

Fv E V#

where all coefficients may be assumed real-valued, since H has real coefficients. Using (11),(12),(m1) one may rewrite this as (2.11)

IN

v=1D

H =

Fv

with changed, but still real-valued DV,FV Let Gay , X=1,...,R, be a basis of span {D,,, D,,, F,,, F :v=1,...,N}, with the principal parts DV , FV of Dv Fv We may express Dv and Fv as linear combinations of the G. and (2.11) takes the form .

,

.

,

H = iR,X=lavXGv*GX

(2.12)

' a,,=

a,X= a.X E R

Now use that H is self-adjoint, so that H = 1/2(H+Hh) (2.13)

H =

.

We get

IR

X=lavXGv*GX

where again the avX have been changed. Now the matrix ((avX)) is real and symmetric. Returning to our old notation we have proven Proposition 2.2. If condition (14) is valid, then we have (2.14)

H = IR,X=lavADv*DX , with avX=aX=aXv E A# , Dv E V#

where all functions and folpde's are real. Now we write (2.15)

Dv = bj3xj+b

,

B =

n,v=1,...,N

where B will be considered an (n+l)xN-matrix , i.e., j indicates the rows and V the columns. It may be assumed that DV, v=1,...,M, have bR=0 , while D

v=M+1,...,N, are of order zero. Now let

denote the sum (2.14) taken only over v,X with at least one >M While g' formally is of first order, it really is of order zero,

0 0* _ (DVV 0+D 0* )b v 0

0 0 due to bXDv + (bXDv)

0 0 + [bX,Dv], where the right hand

side is sum of products of two zero-order terms in V#. We thus may

260

IX.2. The conditions (1)

261

repeat the above proceedure once more on 1', to arrive at a new matrix ((aV), )) with a.VX=O as v<M , X>M or V>M

,

X<M

.

Comparing

coefficients in (2.14) we then find that, with H- of (2.5) we have H--1 = BAB*

(2.16)

Then (2.16) implies H_=(H-B)A(H-B)* which translates into {D,F} = IN

(2.17)

X=lava{D,DV}{DX,F}

for all D,F E D#

,

s

This completes the proof of thm.2.1. Next we will look at condition (m5)

.

This condition is a

consequence of (11), if (a1) holds. On the other hand, under (al') only, prop.2.3, below, shows that (m5) follows for the algebra C of the 'truncated classes' AU,O' DU,0 of VIII,3. (We, of course, then must work with a potential q going to - at 2U as was used ,

in VIII,prop.3.3.)

Proposition 2.3. Suppose D# contains a sequence of functions (i.e. zero order expressions) X. j=1,2,..., such that ,

0 <Xj l, and A-tS

M,-s

At E E , where we know that

From cor.3.4 we get LjAN+2j E E. Again we use LjE 2N+j,0 V,(6.11) on Lj , and for -t instead of s, to express W. by .

Lj+kAN+2j+2k

and another remainder. The first terms are in E as posted. The remainder is an L(H)-convergent integral with inte,

grand in E

.

Thus WtE E

,

and L.AM+2!E Et

.

The A-powers may be

taken into the remainder S and will give an integral V,(6.12) M+N+l M,-s, with ALM+1A replaced by V. above. We just showed that VtE E (we have Vt=Wt, for j=M+1). Thus also Sm (4.4) for A = LAM

.

s

E Et

,

and we get

Similarly for the other generators B = ANL.

The same follows trivially for AEC.0, or for A finitely generated N by LAN, A L , with LEAN and the proof is complete. ,

In thm.4.3, below, we now summarize the consequences of thm.

270

IX.4. Symbols over HS-spaces

271

4.2. These facts will be pursued in closer detail in X,4, where it will be seen that the Fredholm theory of an Hs-comparison algebra is similar, or even the same, as that in the corresponding H-comparison algebra. The assumptions are the same as for thm.4.2. Theorem 4.3. The quotient algebras Cs/E5

,

s e I

,

all are isome-

trically isomorphic, and also the quotient algebras Es/K(H5), SEEM,

all are isometrically isomorphic. In each case a *-isomorphism Cs/Es } Ct/Et and a *-isomorphism Es/K(H5) - Et/K(Ht) both are induced by the natural isometry A - ArAA-r , r=t-s Accordingly, the maximal ideal spaces Ms and ffit of Cs/Es and .

Ct/Et are homeomorphic under the associate dual map of the above isometric isomorphism, and, if E/K(H) is a function algebra

CU,K(h)), as in the examples of VII,1,2, and 4, then also Es/K(Hs) = C(Es3K(h))

(4.8)

,

where all spaces Is are homeomorphic to E = E0

.

Moreover, under condition (l5')(P.), if the homeomorphisms with M _ MO then the operators in the algebra finitely generated by CO and LAN, ANL , for used to identify all the spaces ffi

,

LEQN, N=0,1,..., have their Cs-symbol independent of s The proof of theorem 4.3 is evident. We will continue this

discussion in X,4. Note that a dependence of the Cs-symbol of a (finitely generated) operator A on s is possible, if only (15)(P00)

, not (15') holds. We shall not discuss such problems

here.

Finally let us shortly look at the (Frechet-)algebra C_ already mentioned. Note that the algebra C.0 C 0(0) also has a closure C, in the natural topology of 0(0) i.e., the locally ,

convex topology of all operator norms

(4.9)

IlAlls

= sup{llAulls: Hull sO, with a weighted L2-norm of the 'k-jet' of u First let the assumptions of thm.1.2 hold: We are given we will focus on equivalence of Ilufk

(5.1)

(O,dp,H)

,

,

{O,dv,K)

on the same manifold 0, coinciding outside some compact set. Only the minimal sets A# and D# are needed in this discussion. We assum m me condition (s) for the first triple. The sets Am , Dm , with the first triple, then qualify for all results of III,3,V,3, V,6 and will give us a minimal algebra of expressions- explicitly known as the set of all C'(O)-differential expressions of compact support, with LN consisting of all such expressions of order 1 we conclude that K3

Therefore in K3 is dense in H

.

is essentially self-adjoint, by I, thm.3.l, q.e.d. Denoting the Sobolev norms of the triples (5.1) by 11.0 and I1.11 s K

(5.6)

H =

Iluus

where 11.1

ss H

for a moment, we note that

,

IIH-s/2uII

,

Iluns

is the norm of L2(S2,dp)

=

K .

IIy-1K s/2ufl ,

as uEC'(c)

For a proof of thm.1.2 it is

sufficient to show that both norms (5.6) are equivalent, for every s E I .

Proposition 5.2. For a given sEI the two norms (5.6) are equiva-

IX.5. Local properties

lent if and only if the operator (H-s/2 bounded (in L(H)).

Ks/2)'3d

and its inverse are

Indeed, C. is dense in both Sobolev spaces, by lemma 1.1, (since we have (s) for H and K), we get equivalence if and only if

cl'ull ls H

(5.7)


v

,

Ilvll=l

.

Since

Js is preclosed, this gives a contradiction. Similarly for JSl This completes the proof of thm.1.2. We also have made progress in the general proof of VI,thm. Vt-1 was 1.6, insofar as the boundedness of the operators Vt ,

shown for all tel. To confirm the statements about the cosets Ut Vt+K(H), we must extend VI,prop.l.8. As mentioned above, the nota-

274

IX.5. Local properties

tion of sec.4 now is in order: all operators in 0(m) are considered as operators H_-H_ , but also are identified with their continuous extensions to any Hs, as maps Hs;Hs-m. Specifically we have H S, Ks E 0(2s) .

Proposition 5.3. For 0 < s,t,a,T E I we have (5.10)

Ho[Hs,K-t]HT E K(H)

, whenever o+T < s+t+1/2

Proof. With R(X)=(H+a)-1, S(p)=(K+11)-1 as in the proof of VI,prop.

1.8 we set up a formula like VI,(1.17) again, but now we will use identity VI, (6.8) to express H-s and K-t by integrals over higher powers of the resolvents. Instead of the commutator formula VI,(1.18) we need formula (5.11), below. Proposition 5.4. Under cdn.(s) we have (5.11)

[Rm(A),Sl(P)] = Rm(a)S1(U)L(X,P)S1(P)Rm(a)

for all X,p >

0

, and all l,m=1,2,..., where L(A,p)=L1 m(A,p)

denotes the polynomial (with rPq=2(m+l-p-q)+3 (5.12)

)

=lp=llq=lLpgaP-l11 q-1,

L(X,p) =[(H+X)m,(K+11)1]

LPgELr

,0

Pq

where we again use the classes A!=Am , D! = Dm

The proof is a calculation, noting that R(a) S(p) E 0(-2). One has Lpq = cpq[Hm-p+lKl-q+l] , using the binomial theorem, and ,

the fact that [HP,Kq]=0, as p=0 or q=0. Also, a straight-forward extension of VI,prop.1.9 is required. We indeed get LpgE Lr

,0.

P9

Now we prove prop.5.3. Using VI,(1.6) for H-s and K t write (5.13)

Ho[H-s,K-t]HT = lP=1Fq=l JdA JdPIpq(a,u) 0

0

with Ipq=cpgm+p-s-211 1+q-t-2 HoRm(a)S1(11)LpgSl(P)Rm(A)HT,

(5.14)

For given s,t,a,T >0

,

cpoER.

if 1 and m are chosen sufficiently large,

then the integrand will be norm continuous and

0((Xp)-1-e)

,

e>0,

so that the improper Riemann integrals exist in L(H) . Also we get Ipq(X,P) E K(H) so that (5.13) is a compact operator. ,

Indeed, the existence of the integral at a=0 or P=0 just requires s<m , tl. Write

275

IX.5. Local properties AaRmHoK6-aS1PY)(Ko_6LpgKT-6)(UYS1K6-THTRmaa)

Ipq=cpq(Au)-l-6( (5.15)

=cpgJpq , Jpq=A.B.C

,

with the three parentheses called A, B, C , where a=(m+p-s-l+e)/2

(5.16)

,

y=(l+q-t-l+e)/2

while $ must be chosen such that B is bounded. From m>s

,

1>t

we also get 00 . Again, we may replace ¢u at

For if that is true for all such ,w then with - = 1 and replace u by ¢u in for the the inequality achieved. Also we may replace u by H2Nu right of (5.20) by u

.

we use it for 4- instead of

,

equivalent relation (we used cdn.(s))

yuull2 , u E C'(S22)

(5.21)

Again in (5.21) we may replace u by u with some function i with V = 1 , and then get (5.22), below, equivalent to (5.20).

yllull2

(5.22)

,

u E C1(S22)

But this is an immediate consequence of V,(6.9) and (6.10). = 0, and also, There((ad H1)1w) = 0

For we may assume

i

.

fore, if the expansion V,(6.9) is applied to A=w, with N=O, s=t, and the adjoint is right-multiplied by , then the remainder is the only non-vanishing term of the expansion. By choosing M large the reminder term can be made L2-bounded, even compensating the H1N yQ,Pu112 q.e.d. power

.

Hence we get IIwA1H1N1ull1
O we have XA2X)Ak-r

(5.23)

E K(H)

,

],k=1,2

Proof. This is a matter of the following identity, where we write Ry(a)=(Ai2-a)-1. We have, with Mi=(ad Hk)jX XA12R1(a)X

-

XA22R2(A)X

,

for N,N'=0,1,...,

=

(5.24)

(-l)NXN+N'AI. 2NRiN(a)(MNAl2Rl()MNt- MNA22R2()MN,)A12N'R1Nt(M) For N=1, N'=0 relation (5.24) coincides with VII,(3.11). For more general N,N' it follows by induction, using that (1-AHD)-1 = Tj

MNT1-1 - TJ-1MN

(5.25)

so that AMN+l =

,

,

MN(1-XH1)

-

i.e.,

2 AAi 2Ri(a)MN+lA12R1(A) = Ai R (a)MN - MNA12R1(a). i

Relation (5.25) serves for an induction proof of (5.24). On the other hand, (5.23) follows from (5.24) with the same resolvent integral procedure used over and over, q.e.d. Finally, to complete the proof of prop.5.5, we write (5.26)

XA24u + (XAt1 x -

XA2t

where the first term is

and the second term as well,

due to (5.23), q.e.d.

We now may approach the proof of thm.1.3. The second statement is easily reduced to the first. If DU is smooth, then the diffeomorphism identifying the two subdomains U. of Qj to constitute the open subdomain U may be extended beyond 3U such that ,

01 and Q2 contain the commom subdomain W , with compact boundary, while UL U C W . Perhaps then the triples are different over W\U. However, we then may use thm.l.2 to remedy this, at least in a subdomain of W , still satisfying the conditions. Then (1.6) holds for U as a subset of W. ,

To prove (1.6), let V C U be an open subset, with VclosC U. Since 3U is compact, there is a compact neighbourhood N of 3U relative to UUBU such that U\NnV=O Then we may select a func,

.

tion

meeting the conditions of prop.5.7, with q=l outside N

,

278

IX.5. Local properties so

particularly

=l on a neighbourhood of Vclos

applied to the 0-order operator find that

E C-(St

.

By V, prop.6.7,

with [H,q,] E Dm , we

is a bounded operator of L(H1.)

, hence qE L(Hs).

Now let u E HS. There exists a sequence ukE C0(S21) such that

We have u E Hs as well, and ctuk4 u in HS. However, since u may be interpreted as the product of the distribution u and the C"-function , and since =l near C U V . Also, the distribution u has support in U since supp Accordingly, we may interpret $u as a distribution on S22 as well, and also we get cu k- u in D' (02) , i.e., in weak convergence of Also, by (5.18), the sequence uk is distributions over S22 a Cauchy sequence in H2 , hence converges in H2 to some vEH22 Since this also implies distribution convergence (on St 2), we find that v=4u E H2 This shows that every restriction ujV of some s u E H1 may also be obtained as a restriction to V of v=mu E H2 uk-r u in Hs

.

we get uIV =

.

Since the assumptions are symmetric, this implies the statement of thm.1.3.

6. Sobolev norms of integral order. In this section, we look at Sobolev norms Ilullk for the case Clearly, with the inner product (.,.) of H , of an integer k>0 .

Iluuk2

= (A-ku,A-ku) = (u,Hku)

u E C0'(S2)

,

expressing Ilullk in terms of derivatives of u. In tin, where H has

constant coefficients, one may integrate by parts for (6.2)

Ilull

k

2

=

a

llu(a)02

,

I

with positive (multinomial coefficients) ya (cf.[C1],III,(2.11)). In order to derive a similar formula under more general conditions, let us assume a triple {St,dc,q-0} in Sturm-Liouville

normal form, i.e., du = dS = Fdx , h= (det((hjk)))-1

.

(From

III,prop.6.1 we know that this is no loss of generality, but it should be kept in mind that cdn (q3), below, refers to this form.) We assume {S2^,dp^,H^}={P,du,H}

, and q>l on 0 . Moreover, we will use the Riemannian co-variant derivatives induced by the metric tensor hjk (cf.app.B.).

Under these general assumptions we impose the following. Condition (q2): The Riemannian curvature tensor R (cf.(B.10))

279

IX.6. Integral order is bounded on n

,

and its co-variant derivatives of

all orders also are bounded over 0 Condition (q3)

.

The potential q satisfies the estimates 4kq = k O(ql+k/2) , k=1,2..... x E S2, where V q denotes the k-th order co-variant derivative (tensor) of q. :

,

Our problem formally suggests introduction of the norms {IqN/2u12+Iq(N-1)/2Vu

(u)N = (6.3)

N =

I92(

u) 2du = IIgN/2u112

+

.. .

,

+ IIVNuII2

with the formal covariant tensor 4k of the k-th covariant deriIIt" vative, and with Itl for t=(t. ), defined by 11...ik ,

,

ItI2= h11]1...h1kOk t.l

(6.4)

1

IItu2=

..1

3

l"

ItI2du

Theorem 6.1. Under cdn.s (q2), (q3) the Sobolev norm 11-11N

,

on

the dense space C' C HN, is equivalent to the norm N. That is, clN < IIuIIN < c2N

(6.5)

where c1

for all u E C0_(S2)

,

c2 are positive constants independent of u

,

.

Proof. The second estimate (6.5) essentially is formal: For N=2m, an even number, one writes (6.1) in the form Hull N2= IIHmu112.

In order to simplify the formalism introduce, for a moment, the (n+l)x(n+l)-matrix ((gvu)) 9jk=hJk , g00=q n goj=gj0=0 and its inverse .((gvu)) 0 Use summation conven,

,

.

tion from 0 to n (over greek indices only, one up, one down). Also define a0k=-hjk, j,k90, a00=q, a03=a00=0, j90 , and, with formal tensor notation, (greek indices from 0 to n), [u]k the 'tensor' with components uv V u=ulv = VV ...V u, with 70u=u ' ., v V k 1 k 1 ,

as V #

0

.

Recall that the metric tensor hjk acts like a constant; it has all its covariant derivatives zero. Suppose, for a moment, that also the co-variant derivatives of q vanish (i.e.,q=const.). Then we may write umvm

u1vl ( 6 . 6 )

H

...a

u

u1v1...umvm

This is just the contraction of the tensors (a)x(a)x...x(a)

280

IX.6. Integral order with m factors (a)= (avu)

and [u) N . Thus it is a matter of

,

Schwarz' inequality that IHmul2 {S2,dp,H}). Then the

symbol is a restriction of the formal symbol of VII,(3.25). For a compact commutator algebra C the operator D of (3.11) then is Fredholm if (though not necessarily only if) the formal symbol se

TDA is invertible at each point of 88 0

Finally we will use our above theory to also consider operators acting on crossections of a vector bundle over Q. We have avoided vector bundles, so far, because our C -algebra approach to symbol and symbol space is developed best for complex-valued symbols, while its extension to vector bundles is a simple abstract matter, to be discussed now.

Let V be a vector bundle over Q with fiber an inner product space Vx of dimension m, with norm IuIx

,

u E Vx

.

We only con-

sider (C'-)vector bundles of the form V=VA#IS2 = 0-1(Q), where VA#

is a (continuous) vector bundle over the compactification M # of A We assume 0 induced by A# (cf.VII,l), with bundle projection 0 .

the euclidean metric 1.1x of the fiber defined for xC M # nuous in x

,

,

conti-

and C' over Q.

For the above we assume given a comparison triple on 2 , and generating classes A# D# satisfying (a0),(a1),(d0) , and then ,

will introduce operators acting on HV = L2(Q,V,dp), the space of crossections u of V satisfying Ilull2

=

J0Iu(x)l2dp


O

limt;-Li(t)=L (eo), limt-.(a kLi)(t)=O

L=

Remark 4.4. Recall, that, at each end, we have a 'chart'

U=

(4.7)

{(x,t): xE BJ .

, O0 there exists a compact set K C 0 such that lf(x)/g(x)l

f=o(g)

,

< c for all x E 0-\K .) We shall write f=O(g)

(without "(in 0`)"

,

,

and

etc.) if no confusion can arise.

Lemma A.1. Let f,g be as above, and let g be continuous over U If f = o0(g)

,

then there exists a positive C-(U)-function ip

f = 0(*) (in R') , and V = o0(y) Proof. Let (x) = f(x)/g(x) , so that we have q(x) bounded over Consider U` and limx ¢(x) = 0 extended to 0 by setting fi(x) = 0 outside 0, then the limit still is zero. With our partition w. define nj = x E supp wj } Observe that nj>0, and 0 For there exists a compact set Ke C 0 such that Ifl<e outside K6 for every e>O Only finitely many of the sets 0. = supp wj can have points in common with Ke Else there exists X. E U nK e with a limit point xE 0 Ke Due to localy finite covesuch that

.

.

.

.

.

.

.

3k

ik

rage x0 has a neighbourhood N contained in only finitely many U. so cannot be a limit point of the x.

.

Hence, for e>0 there

k

exists NE such that all supp wj

,

Accordingly, nj =

j>Ne are completely outside Ke. supp wj} < e

.

Now we just define the function

where

the sum is locally finite, hence represents a C'(0-function. Since y is only continuous, the function $ thus defined is not yet C_(U)

but is at least continuous. It also satisfies the other

conditions: For any x E U we get = p(x)/g(x)

.

This implies Jf(x)I

(x)I =

Jnjwj(x)

=

p(x)

,

i.e.,

320

App.A. Functions on manifolds

. Also we note that limx4Injwj(x) =

f = 0(p) (in SZ-)

= o0(g)

.

Indeed, let KN =

:

j>N+l}

-

0

1j=N+lnjwj(x) < -

as N

,

so that

,

Clearly KN is compact,

S21U....US2N

while x E S2\KN implies that sup{nj

0

To fully establish the lemma we

.

now must make a C"-correction of 4 which does not disturb the other conditions already established. This is accomplished in lemma A.2, below. Lemma A.2. Let f,g E C(Q)

C%1)-functions y

,

,

g >

0

.

Then there exist positive

such that

6

f(x) < y(x) < f(x) + 6(x) , x E S2

(A.1)

,

6

= 0Q(g)

.

x E .jclos} Clearly ej> 0 , since Proof. Let ej = 2-JMin{g(x) UJclos are compact, and g(x) > 0 Using the coordinate transform QjclosC of Uj we may regard S2jC Uj as subsets of in Let wj > 0 .

:

.

.

w. E C0(S2j)

unity clos 0

.

'

jj=lwj(x) = 1 in a

,

i.e., {wj} is a partition of

Using regularizing techniques and the compactness of such that

it is possible to find fjE C0'(S2

wj(x)f(x) < f.(x) < wj(x)f(x) + ej , x E S2j

(A.2)

For each j let Xj(x) E CD(Uj)

,

0<Xj {c,du,H) Cf. (m6)" .

(mi)x (VII,3): There exists a DOE D#, such that {D,D}(x)90, but {D,D}1(x)=0

.

(m7) (VII,3): Means cdn.(m7)x for all x E M A

#

(m7)x" (VII,4): Means cdn. (m7)x ,but with {D,D} replaced by {D,D}"

,

as n (m6)"

(m7)" (VII,4): See (mi)x"

.

(V,6): (Same as 'H-compatible, V,def.6.4). (pl) ( I X , 3 ) : [H,L] E PN+l for all L E PN , N=0,1,... (p)

.

325

App.C. Summary of cdns(x) (p2) (IX,3): Every L E pN is within reach of L(H)

.

(ql) (VII,4): We have q=q^ in some neighbourhood of 3o (q2) (IX,6): The Riemann curvature tensor and all its coriant derivatives are bounded over 0 (q3) (IX,6): The potential q satisfies Vkq=O(ql+k/2), k=1,.. .

(s) (V,1): For all m=1,2,... the minimal operator (Hm)0 H0m of the m-th power HM is essentially self-adjoint. = (sj) (V,1): (where j=1,2,...,oo) (H3)0= H0J is essentially self-adjoint.

(w) (V,1): The minimal operator H0 is essentially selfadjoint.

326

List of symbols used

AA 14

E

ACB, BDA 2

e(x,y)

A# 126,129

E(A) 15

AC

213

graph A

A# c En

144

mn

199

198

Hd

78

219,236,248 62/63

2

B(x,q) 86

Hs, H. 30,157,160,251 H°° , Hm 303

B

i

blj

blb

,

322f

78

10 129

,

76

K(H)

24

cdn.(x3.) 324f

L(H)

2

CB

,

CB(X), CB(X,X) 229

ffi

CO

,

131

CO(X), C0(X,X) 230

ffis

C

129

M #

CO

129

C

257

S C

271

C0

257

C A#

187

, fp

A

0(g)

,

P(X,Y)

2(X,V)

213

Rs(A)

D#

144

dom A D

x

dx

2

Sx

,

Q(X) 2

13

194

T 0 , T 0x 161 161

194

w

194

(x.) 324f

126

P(X) 2

12

Sp(A) 86

{D,F} 193 E

,

187,206 E*0 115

D c

o0(g) 320

115

D# 126,129

d(x,y)

189

187

( x)

146

References

[AN1] S.Agmon and L.Nirenberg, Properties of solutions of ordinary differential equations in Banach space; Commun. Pure Appl. Math. 16 (1963) 121-239.

[Al1] W.Allegretto, On the equivalence of two types of oscillations for elliptic operators; Pac.J.Math. 55(1974) 319-328. [Al2]

,

Spectral estimates and oscillation of singular differential operators; Proc.Amer.Math.Soc.73(1979)51.

, Positive solutions and spectral properties of

[Al3]

second order elliptic operators;Pac.J.Math.92(1981) 15-25.

[APS] M.Atiyah, V.Patodi, and I.Singer, Spectral assymmetry and Riemannian geometry; Math. Proc. Phil. Cambridge Phil. Soc. 77 (1975) 43-69. [AS] M.Atiyah and I.Singer, The index of elliptic operators; I: Ann. of Math. 87 (1968) 484-530; III: Ann. of Math. 87 (1968) 546-604; IV: Ann. of Math. 92 (1970) 119138; V: Ann of Math. 92 (1970) 139-149, [BMSW] B.Barnes, G.Murphy, R.Smyth, T.West, Riesz- and Fredholm theory in Banach algebras; Pitman, Boston 1982. [BDT] P.Baum, R.Douglas, M.Taylor, Cycles and relative cycles in analytic K-homology; to appear.

[B1] R.Beals, A general calculus of pseudo-differential operators; Duke Math.J.42 (1975) 1-42. [B2]

,

Characterization of pseudodifferential operators and applications; Duke Math.J. 44 (1977) ,45-57 cor;

rection, Duke Math.J.46 (1979),215 [B31

,

.

On the boundedness of pseudodifferential.operators;

Comm.Partial Diff.Equ. 2(10) (1977) 1063-1070. [Bz11 I.M.Berezanski, Expansions in eigenfunctions of self-adjoint

operators; AMS transl. of math. monographs, vol.17, Providence 1968. [BW1] B.Booss and K Wojciechowski, Desuspension of splitting

elliptic symbols I,II; to appear. [BdM1] L.Boutet de Monvel, Boundary problems for pseudo-differential operators; Acta Math. 126 (1971) 11-51. [BS1] J.Bruening and R.Seeley, Regular singular asymptotics; Adv. [BS2]

Math. 58 (1985) 133-148. The resolvent expansion for second order regular ,

References singular operators; Preprint Augsburg 1985. [BS3]

, An index theorem for first order regular singular operators; Preprint Augsburg 1985.

[BC1] M.Breuer and H.O.Cordes,On Banach algebras with a-symbol; J.Math.Mech. 13 (1964) 313-324 .

[BC2] part 2 ; J.Math.Mech.14 (1965) 299-314 [Ca] A.Calderon, Intermediate spaces and interpolation,the complex ,

method; Studia Math.24 (1964) 113-190. [Ca1] T.Carleman, Sur la theorie mathematique de 1'equation de

Schroedinger; Arkiv f. Mat., Astr.,og Fys. 24B Nil (1934).

[Ct1] M.Cantor, Some problems of global analysis on asymptotically simple manifolds; Compos. Math. 38 (1979) 3-35. [CV1] A.Calderon and R.Vaillancourt,On the boundedness of pseudodifferential operators; J.Math.Soc.Japan 23 (1971) 374-378. [CV2]

, A class of bounded pseudodifferential operators; Proc.Nat.Acad.Sci.USA 69 (1972) 1185-1187.

[Che1]J.Cheeger, Spectral geometry of spaces with cone-like singularities;Proc.Nat.Acad.Sci. USA (1979) 2103. , M.Gromov, and M.Taylor, Finite propagation speed, ker[CGT] nel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds; J. Diff. Geom. 17 (1982) 15-53. [Ch] P.Chernoff, Essential selfadjointness of powers of generators of hyperbolic equations 12 (1973) 402-414

;

J.Functional Analysis

.

[CBC] Y.Choquet-Bruhat,and D.Christadolou, Elliptic systems in Ns 6-spaces on manifolds, elliptic at infinity; Acta Math.146 (1981) 129-150. [Cb1,2 1

L.Coburn, The C -algebra generated by an isometry, I:Bull. Amer.Math.Soc.73 (1967) 722-726; II: Transactions Amer.Math.Soc.137 (1969) 211-217.

[CL1]

L.Coburn and A.Lebow, Algebraic theory of Fredfholm opera-

tors; J.Math.Mech.15,577-584. [CM] R.Coifman and Y.Meyer, Au dela des operateurs pseudodifferentielles; Asterisque 57 (1979) 1-184.

[CC1] P.Colella and H.O.Cordes,The C -algebra of the elliptic boundary problem;Rocky Mtn.J.Math. 10 (1980) 217-238

.

329

References

[Cd1] E.A.Coddington,The spectral representation of ordinary selfadjoint differential operators;Ann.of Math.60 [CdL1]

(1954) 192-211. and N.Levinson, Theory of ordinary differential

equations,McGraw Hill,New York 1955.

[C1] H.O.Cordes,Elliptic pseudo-differential operators,an abstract theory.Springer Lecture Notes in Math.Vol.756, Berlin,Heidelberg,New York 1979. [C3]

,Techniques in pseudodifferential operators;mono-

[Cu]

,

[CQ]

, A matrix inequality, Proc.Amer.Math.Soc.ll (1960)

[CC]

,

graph,to appear.

On some C -algebras and Frechet-*-algebras of pseudodifferential operators; Proc.Sympos.P.Appl. Math.43 (1985) 79-104. 206-210.

[CD]

[CE]

On compactness of commutators of multiplications and convolutions,and boundedness of pseudodiffe-

rential operators; J.Functional Analysis,l8 (1975) 115-131. , A pseudo-algebra of observables for the Dirac equation; Manuscripta Math.45 (1983) 77-105. , A version of Egorov's theorem for systems of

hyperbolic pseudo-differential equations;J.of Functional Analysis 48 (1982) 285-300. [CEd]

, Ueber die Eigenwertdichte von Sturm-Liouville Problemen am unteren Rande des Spectrums; Math. Nachr.l9(1958), 64-71.

[CF]

, A pseudodifferential Foldy-Wouthuysen transform; Comm.in Partial Differential Equations, 8(13) (1983) 1475-1485.

[CFo]

,

[CG]

,

[CI]

,

[CL]

,

On maximal first order partial differential operators; Amer.J.Math.82 (1960) 63-91. On geometrical optics;lecture notes,Berkeley,1982 The algebra of singular integral operators in in; J.Math.Mech.14 (1965) 1007-1032.

On pseudodifferential operators and smoothness of special Lie-group representations;Manuscripta math. 28 (1979) 51-69

[CP]

.

, A global parametrix for pseudodifferential operators over In ;Preprint SFB 72 (Bonn) (1976)

330

References

(available as preprint from the author). [CS]

, Banach algebras,singular integral operators and

partial differential equations;Lecture Notes Lund,1971.(Available from the author). [CTJ

, An algebra of singular integral operators with two symbol homomorphisms;Bulletin AMS ,75 (1969) 37-42

[CWS]

,

.

Selfadjointness of powers of elliptic operators on noncompact manifolds;Math.Ann. 195 (1972) 257-272

[CPo]

,

On the two-fold symbol chain of a C -algebra of singular integral operators on a polycylinder; Revista Mat.Iberoamericana 1986, to appear.

[CDgl]

, and S.H.Doong, The Laplace comparison algebra of

a space with conical and cylindrical ends; To appear; Proceedings, Topics in pseudodifferential operators; Oberwolfach 1986; Springer LN.

[CEr] _,and A.Erkip, The N-th order elliptic boundary problem for non-compact boundaries;Rocky Mtn.J.of Math. 10 (1980) 7-24 .

[CHel]

,and E.Herman , Gelfand theory of pseudo-differential

operators; American J.of Math. 90 (1968)681-717. [CHe2]

[CLa]

[CM1]

Singular integral operators on a half-line; Proceedings Nat. Acad. Sci. (1966) 1668-1673. , and J.Labrousse, The invariance of the index in the metric space of closed operators;J.Math.Mech. 15 (1963) 693-720. and R.McOwen The C -algebra of a singular elliptic ,

problem on a non-compact Riemannian manifold; Math.Zeitschrift 153 (1977) 101-116 .

[CM2J

Remarks on singular elliptic theory for complete Riemannian manifolds; Pacific J.Math. 70 (1977) 133-141 , and S.Melo, An algebra of singular integral opera.

[CMe1]

tors with coefficients of bounded osecillation; To appear. [CSch]

and E.Schrohe, On the symbol homomorphism of a certain Frechet algebra of singular integral operators; Integral equ. and operator theory 120-133.

(1985)

331

References

[CW] _,and D.Williams, An algebra of pseudo-differential operators with non-smooth symbol ;Pacific J.Math. 78 (1978) 279-290

.

[CH] R.Courant and D.Hilbert,Methods of mathematical physics,I: Interscience Pub. ltd. New York 1955; II: Interscience Pub. New York 1966.

[Dd1] J.Dieudonne, Foundations of modern analysis,Acad.Press New York 1964

.

[Do1] H.Donnelly, Stability theorems for the continuous spectrum of a negatively curved manifold; Transactions AMS 264 (1981) 431-450. [Dgl] R.Douglas, Banach algebra techniques in operator theory; Academic Press, New York 1972. [Dg2]

,

Banach algebra techniques in the theory of Toeplitz operators, Expository Lec. CBMS Regional Conf. held at U. of Georgia, June 1972, AMS, Providence, 1973.

[Dg3]

,

C -algebra extensions and K-homology; Princeton Univ.Press and Tokyo Univ.Press, Princeton 1980.

[DG1] J.Duistermat and V.Guillemin, The spectrum of positive ellip tic operators and periodic bicharacteristics; [DS1,2,3 1

Inventiones Math. 29 (1975) 39-79. N.Dunford and J.Schwarz, Linear operators, Vol.I,II,III,

Wiley Interscience,New York 1958,1963,1971. [Dx1] Dixmier, Les C-algebres et leurs representations; GauthierVillars, Paris 1964. Sur une inegalite de E.Heinz, Math.Ann.l25 (1952)

[Dx2]

75-78.

[Dv1] A.Devinatz, Essential self-adjointness of Schroedinger operators; J.Functional Analysis 25 (1977) 51-69. [Du1] R.Duduchava, On integral equations of convolutions with dis[Du2]

continuous coefficients; Math. Nachr. 79 (1977) 75-98. , An application of singular integral equations to some problems of elasticity; Integral Equ. and Operator theory 5 (1982) 475-489.

(Du3]

,

On multi-dimensional singular integral operators,I:

The half-space case; J.Operator theory 11 (1984) 41-76

II: The case of a compact manifold; J. Operator theory 11 (1984) 199-214. [Du4]

,

;

Integral equations with fixed singularities; Teub-

332

References

ner Texte zur Math.,Teubner,Leizig,1979. [D11 J.Dunau ,Fonctions d'un operateur elliptique sur une variete compacte; J.Math.pures et appl. 56 (1977) 367-391

[Dyl] Dynin, Multivariable Wiener-Hopf operators; I: Integ.eq. and operator theory 9 (1986), 937-556; II: To appear. [Ei11 L.P.Eisenhart, Riemannian geometry; Princeton 1960. [Er1] A.Erkip, The elliptic boundary problem on the half-space; Comm. PDE 4(5) (1979) 537-554. [Er2]

, Normal solvability of boundary value problems in

half-space; to appear. [Es1] G.Eskin, Boundary problems for elliptic pseudo-differen-

tial equations; AMS Transl. Math. Monogr. 52 Providence 1981 (Russian edition 1973). [Fe I] C.Fefferman, The multiplier problem for the ball; Ann. Math. 94 (1971) 330-336.

[F11 J.Frehse,Essential selfadjointness of singular elliptic operators;Bol.Soc.Bras.Mat. 8.2(1977),87-107 [Fr1] K.O.Friedrichs, Symmetric positive linear differential equations; Commun.P.Appl.Math.11 (1958) 333-418. [Fr21

Spectraltheorie halbbeschraenkter Operatoren Math.Ann.109 (1934) 465-487 ,685-713 and 110 (1935) 777-779

.

[FL] K.O.Friedrichs and P.D.Lax,Proc.Sypos.Pure Math.Chicago,1966.

[Fh11 I.Fredholm, Sur une classe d'equations fonctionelles; Acta Math.27 (1903) 365-390. [Ga1] L.Garding,Linear hyperbolic partial differential equations with constant coefficients;Acta Math.85 (1-62) 1950 [Gf1] M.P.Gaffney, A special Stoke's theorem for complete Riemannian manifolds; Annals of Math. 60 (1954) 140-145 [GT1] D.Gilbarg,and N.Trudinger, Elliptic partial differential

equations of second order; 2-nd edition; Springer New York 1983. [GS] I.Gelfand and G.E.Silov, Generalized functions; Vol.1,Acad. Press, New York 1964.

[Gl1] I.M.Glazman, Direct methods of qualitative spectral analysis of singular differential operators; translated by Israel Prog. f. scientific transl. Davey, New York 1965.

[Gb] I.Gohberg, On the theory of multi-dimensional singular inte-

333

References

gral operators; Soviet Math. 1 (1960) 960-963. [GK] I.Gohberg and N.Krupnik, Einfuehrung in die Theorie der eindimensionalen singulaeren Integraloperatoren; Birkhaeuser, Basel 1979 (Russian ed. 1973).

[G1] B.Gramsch, Relative inversion in der Stoerungstheorie von Operatoren and ip-Algebren; Math. Ann. 269 (1984) 27-71. [G2J

, Meromorphie in der Theorie der Fredholm Operatoren mit Anwendungen auf elliptische Differentialopera-

toren; Math. Ann. 188 (1970) 97-112. [Gr1] G.Grubb, Problemes aux limites pseudo-differentiels depen-

dent d'un parametre; C.R.Acad.Sci.Paris 292 (1981) 581-583.

[Guil] V.Guillemin, A new proof of Weyl's formula on the asymptotic distribution of eigenvalues; Adv. Math. 55 (1985) 131-160. [GP1] V.Guillemin and A.Pollack, Differential topology; Prentice Hall, 1974. [GK] I.Gohberg and N.Krupnik,Einfuehrung in die Theorie der ein-

dimensionalen singulaeren Integrale; Birkhaeuser, Basel,Boston,Stuttgart 1979. [GLS] A.Grossmann,G.Loupias,and E.Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space; Ann.Inst. Fourier,Grenoble 18 (1968) 343-368.

[Hgl] S.Helgason, Differential geometry, Lie groups, and symmetric spaces; Acad. Press, New York 1978.

[Hz1] E.Heinz, Beitraege zur Stoerungstheorie der Spektralzerlegung; Math.Ann.123 (1951) 415-438.

[H1] E.Herman, The symbol of the algebra of singular integral operators; J.Math.Mech. 15 (1966) 147-156 .

[H11] E.Hilb, Ueber Integraldarstellungen willkuerlicher Funktionen; Math.Ann. 66 (1908) 1-66. [HP] E.Hille,and R.S.Phillips, Functional analysis and semi-groups

Amer.Math.Soc.Coll.Publ. Providence, 1957. [Hz1] F.Hirzebruch, Topological methods in algebraic geometry; Springer, Grundlehren, Vol. 131, 1966, Berlin Heidelberg, New York.

[Ho1] L.Hoermander,Linear partial differential operators;Springer, Berlin, Heidelberg, NewYork 1963.

334

References

[Ho] L.Hoermander, Pseudo-differential operators and hypoelliptic equations; Proceedings Symposia in pure and appl. Math. Vol 10 (1966) 138-183. [Ho3]

, The analysis of linear partial differential operators; Springer New York I: distribution theory and Fourier analysis, 1983; II: diff. operators pseudo-

with constant coefficients, 1983; III:

differential operators, 1985; IV, Fourier inteoperators, 1985. [I1] R.Illner, On algebras of pseudodifferential operators in LP (,n); Comm.Part.Diff.Equ. 2 (1977) 133-141. [Kpl] I.Kaplanski, The structure of certain operator algebras; Transactions Amer.Math.Soc.70 (1951) 219-255. [K1] T.Kato, Perturbation theory for linear operators; Springer New York 1966. [K2]

, Perturbation theory for nullity, deficiency and other quantities of linear operators; Journal

[K3,

, Notes on some inequalities for linear operators;

d'Analyse Math. 6 (1958) 261-322. Math.Ann.125 (1952) 208-212. [Ke1] J.L.Kelley, General topology; Van Nostrand, Princeton 1955.

[KG1] W.Klingenberg, D.Grommol, and W.Meyer, Riemannsche Geometrie in Grossen; Springer Lecture Notes Math. Vol. 55 New York 1968. [Ko1] K.Kodaira, On ordinary differential equations of any even

order and the corresponding eigenfunction expansions; Amer.J.Math. 72 (1950) 502-544. [Kr1] M.G.Krein,On hermitian operators with direction functionals; Sbornik Praz. Inst. Mat. Akad. Nauk Ukr. SSR 10 (1948) 83-105. [Ku1] H.Kumano-go, Pseudodifferential operators; MIT-Press 1982.

[Le1] N.Levinson, The expansion theorem for singular self-adjoint linear differential operators; Ann.o.f Math.ser.2 59 (1954) 300-315 [Lk] R.Lockhart, Fredholm properties of a class of elliptic ope.

rators on noncompact manifolds; Duke Math.J. 48 (1981) 289-312. [LMgl] J.Lions and E.Magenes, Non-homogeneous boundary value problems and applications; I, II, Springer,

335

References New York 1972.

[LM] R.Lockhart and R.McOwen, Elliptic differential operators on noncompact manifolds; Acta Math.151 (1984) 123-234.

[Lw1] K.Loewner, Ueber monotone Matrixfunktionen; Math.Zeitschr. 38 (1934) 177-216. [MaM] R.Mazzeo and R.Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature; preprint Dept.Math.MIT, 1986.

[M0] R.McOwen, Fredholm theory of partial differential equations on complete Riemannian manifolds; Ph.D. Thesis, [M1] R.McOwen

Berkeley 1978. Fredholm theory of partial differential equations

on complete Riemannian manifolds;Pacific J.Math. 87 (1980) 169-185 .

[M2] R.McOwen

On elliptic operators in In ;Comm. Partial Diff.

equations, 5(9) (1980) 913-933. [MS1] E.Meister and F.O.Speck, The Moore-Penrose inverse of Wiener

Hopf operators on the half-axis and the quarter plane; J. Integral equations 9 (1985) 45-61. [Me1] R.Melrose, Transformation of boundary problems, Acta Math. 147 (149-236).

[MM1] R.Melrose and G.Mendoza,Elliptic boundary problems on spaces with conic points; Journees des equations differentielles, St. Jean-de-Monts, 1981. , Elliptic operators of totally characteristic type;

[MM2]

MSRI-preprint, Berkeley, 1982.

[Mi1] J.Milnor, Topology from a different viewpoint; Univ.of Virginia press, 1965. [Ms1] J.Moser,

On Harnack's theorem for elliptic differential equations; Comm.P.Appl.Math.l4 (1961) 577-591.

[MP1] W.Moss and J.Piepenbrinck, Positive solutions of elliptic equations; Pac.J.Math.55(1978) 219-226. [NSz1] B.v.Sz.Nagy, Spektraldarstellung linearer Transformationen des Hilbertschen Raumes; Springer, New York 1967. [Nb1,2 1

G.Neubauer, Index abgeschlossener Operatoren in Banachraeumen I: Math.Ann.160 (1965) 93-130; II:Math.

Ann. 162 (1965) 92-119. [vN1] J.v.Neumann,Allgemeine Eigenwerttheorie Hermitescher Funk-

336

References

tionaloperatoren; Math.Ann.102 (1929) 49-131. [Ne1] M.A.Neumark, On the defect index of linear differential operators; Doklady Akad.Nauk SSSR 82 (1952) 517-520. [Ne2]

,

Lineare Differentialoperatoren; Akademie Verlag Berlin 1960.

[Nt1] F.Noether, Ueber eine Klasse singulaerer Integralgleichungen; Math. Ann. 82 (1921) 42-63. [NW1] L.Nirenberg, and H.F.Walker, The null spaces of elliptic partial differential operators in Rn J. Math. analysis and Appl. 42 (1973) 271-301. ;

[Ppl] J.Piepenbrink, Nonoscillatory elliptic equations; J.Diff.Eq. 15(1974) 541-550. [Pp2]

, A conjecture of Glazman; J.Diff.Eq.24(1977) 173-176

[P1] S.C.Power, Fredholm theory of piecewise continuous Fourier integral operators on Hilbert space; J. Operator theory 7 (1982) 52-60

.

[PW1] M.Protter, and H.Weinberger, Maximum principles in differential equations; Prentice Hall, Englewood Cliffs 1967. [RS] M.Reed and B.Simon, Methods of modern mathematical physics; Academic Press, New York. I: Functional analysis,

1975; II: self-adjointness and Fourier analysis,

1975; III: Scattering theory, 1978; IV: Spectral theory, 1978; V: Perturbation theory, 1979 [Re1] F.Rellich, Halbbeschraenkte Differentialoperatoren hoeherer Ordnung; Proceedings International Kongr. Math. Amsterdam 1954 , vol.3, 243-250. [RS1] S.Rempel and B.W.Schulze, Parametrices and boundary symbo-

lic calculus for elliptic boundary problems without the transmission property; Math.Nachr. 105 (1982) 45-149. [R1] C.E.Rickart, General theory of Banach algebras; v.Nostrand Princeton 1960. [Ril] F.Riesz, Ueber die linearen Transformationen des komplexen

Hilbertschen Raumes;Acta Sci.Math.Szeged 5 (1930) 23-54.

[RN] F.Riesz and B.Sz.-Nagy, Functional analysis; Frederic Ungar, New York 1955.

[Rol] D.Robert, Proprietes spectrales d'operateurs pseudo-differentiels; Comm. PDE 3(9) (1978) 755-826.

337

References

[Sa1] D.Sarason, Toeplitz-operators with semi-almost periodic symbols; Duke Math.J. 44 (1977) 357-364.

[Schu1] W.Schulze, Ellipticity and continuous conormal asymptotics on manifolds with conical singularities; to appear; Math. Nachrichten. , Mellin expansions of pseudo-differential opera-

[Schu2]

tors and conormal asymptotics of solutions; Proceedings, Oberwolfach conference on Topics in pseudodifferential operators 1986; to appear. [Sk1] S.Sakai, C -algebras and W -algebras; Springer, Berlin New York 1971. [Ro1l W.Roelcke, Ueber Laplaceoperatoren auf Riemannschen Mannigfaltigkeiten mit diskontintuierlichen Gruppen; Math.Nachr.21 (1960) 132-149

.

[Schr1] E.Schrohe, The symbol of an algebra of pseudo-differential operators; to appear. , Potenzen elliptischer Pseudodifferentialoperato-

[Schr2]

ren; Thesis, Mainz 1986.

[Se1] R.T.Seeley, Topics in pseudo-differential operators, CIME Conference at Stresa 1968.

Integro-differential operators on vector bundles; Transactions AMS 117 (1965) 167-204 Ca [S1] H.Sohrab, The -algebra of the n-dimensional harmonic [Se2]

.

oscillator; Manuscripta.Math.34 (1981),45-70

.

C -algebras of singular integral operators on the

[S21

line related to singular Sturm-Liouville problem; Manuscripta Math. 41 (1983) 109-138. [S3]

,

Pseudodifferential C -algebras related to Schroe-

dinger operators with radially symmetric potential; Quarterly J.Math.Oxford 37 (1986) 105-115. [So] A.Sommerfeld, Atombau and Spektrallinien; Vieweg,Braunschweig 1957

[Sp1] F.O.Speck, General Wiener-Hopf-factorization methods; Pitman London 1985. [St1] M.H.Stone, Linear transformations in Hilbert space; Amer.

Math. Soc. Coll. Publ. New York 1932

.

[T1] M.Taylor , Pseudo-differential operators; Springer Lecture Notes in Math. Vol. 416, New York 1974 .

[T2]

, Pseudodifferential operators;Princeton Univ.Press, Princeton 1981

.

338

References

[Ti1] E.C.Titchmarsh, Eigenfunction excpansions associated with second order differential equations;part I,2nded. Clarendon Oxford 1062; part II, ibid. 1958. [Tr1] F.Treves,

Introduction to pseudodifferential operators and Fourier integral operators; Plenum Press,New York and London 1980

[Up1] H.Upmeier, Toeplitz,C -algebras on bounded symmetric domains; Ann. of Math.(2) 119 (1984) 549-576. [W1] A.Weinstein, A symbol class for some Schroedinger equations on

In

; Amer.J.Math.107 (1985) 1-21.

[We1] H.Weyl, Ueber gewoehnliche Differentialgleichungen mit Singularitaeten, and die zugehoerigen Entwicklungen willkuerlicher Funktionen; Math.Ann.68 (1910) 220-269. [We2]

, The method of orthogonal projection in potential theory; Duke Math.J. 7(1940) 411-444.

[Wd1] H.Widom, Asymptotic expansions for pseudodifferential operators on bounded domains; Springer Lecture Notes No 1152, 1985. [Wi1] E.Wienholtz, Halbbeschraenkte partielle Differentialoperatoren vom elliptischen Typus; Math.Ann.135 (1958) 50-80.

[Z1] S.Zelditch, Reconstruction of singularities for solutions of Schroedinger equations;Ph.D.Thesis,Univ.of Calif. Berkeley (1981).

339

Index

condition (xj),x=a,d,... 324

A-bounded 294 A-compact 294

covariant derivatives 134f, 322 curvature (Riemannian) 279f, 323

adjoint operator 3 adjoint relation 5

defect index 7

algebra surgery 239

defect spaces 6,38 definite case 41

boundary, regular 67 boundary condition 41,68,315 ,Dirichlet 53,68 ,Neumann 315

Dirichlet condition 53,68 operator 60,68 problem 87

realization 59

boundary space 44 bounded operator 1

domain (of an

Carleman alternatives 41

E-symbol 218f

unbounded operator) 2

change of dependent variable 93f

for periodic

Christoffel symbols 134,322 closed operator 2

for polycylinder

closing 39

closure of an operator 2

coefficients 227f algebra 235f elliptic differential expression 37

co-sphere bundle 170 covariant derivative 134,322

equivalent triples 93

commutator of differential expressions 133f

essential self-adjointness 123 of H0 103, 111

, compactness 137f

commutator ideal E 126,131 commuting unbounded operators 17 compactness criteria 71f of commutators 144

essentially self-adjoint 5

of H0m

105f,lll

expresssion (differential) 36

within reach 127,155,262 extension (of an operator) 2

comparison operator 61,127f comparison algebra 127f,157 , minimal 129,161f

4 (property) 293

comparison triple 128

formal adjoint 36 Fredholm domain 302

complete spaces,

with cylindrical ends 246

folpde 126,129

index 33,293

341

operator 33,180,293 inverse 180,295

maximum principle 91 minimal operator 37 minimal comparison algebra 129

distinguished 34,180 ,

multiplication operator 134

special 295

Frehse's theorem 118f

noncompact commutators,

Friedrichs extension 11,71f

algebras with 218f normal forms 93,94

fundamental solution 42,78

Sturm-Liouville 95,151 graph (of an unbounded operator) 2 Green inverse 161,181,318

order classes 30,267,318 ordinary diff.expression 37

, distinguished 181

Green's function 78f

periodic coefficients 154f,221 poly-cylinder algebra 228f

H-compatible expresssion 157,160

positive operator 14

Hs-comparison algebra 257,267f,315f

positivity of the Green's function 100,130

Harnack inequality 87f Heinz, lemma of 24

preclosed operator 3

hermitian operator 5 HS-chain 29

principal part 36 symbol type 266 symbol space 193

hull 191

projection 14

hypo-elliptic expression 37

positive square root 12

isometry, betw.Hs-spaces 31,268

reach, expression

indefinite case 41 irreducible C -algebra 130 K-invariant under H-conjugation 33,181 Landau symbols 320 Laplace comparison operator 93

within 127,155f,186,262f algebra within 266 general 266 real operator 10

realization,of a differential expression 35,37 regular endpoint 45 regularity, boundary 67f

limit circle case 41,53

Rellich's criterion 75,77,179

limit point case 41,53

resolvent 12

Loewner, lemma of 24

compactness 71f expansion of

maximal null space 47

of H-compat. expr.158

maximal operator 37

formula for As 139,163

342

set 12

square root of an operator 12

Riesz-v.Neumann extens.thm.8

strongly hypo-elliptic 38 Sturm-Liouville problem 51,53

second order express.on 0 132 secondary symbol space 193 rel.to formal symbol 207 local charac.195,211,214

subextending triple 76 summation convention 60 surgery, algebra 239f symbol of an operator 191,137

for Schroedinger opera-

of an algebra 191

tors on In

function 170

215f

self-adjoint operator 4 semi-bounded operator 10 separated boundary cdn's 48 singular endpoint 45,49 Sobolev space 61,157,159,251f first 128 on compact Q 157 on noncompact 0 251f

of integral order 279f local 272 Sobolev estimate 181f,184 imbedding 182,217,251 ,

comparison

algebra in 253,257,315 spectral density matrix 52 spectral family 15

spectral theorem 12f spectrum 13 essential 265

formal algebra 187,206 symbol space ffi

131,189,191,145

, principal 193

, secondary 193

systems of differential expressions 303f tensor 132,322 tensor products,

topological 226,229,303 two-link ideal chains 218f,309 unbounded operator 1 Vector bundles, comparison algebras on 303f

wave front space 161f,169f Weyl's lemma 39,42,60