Asymptotics for Elliptic Mixed Boundary Problems [Reprint 2021 ed.] 9783112577103, 9783112577097

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Mathematica/ Research Asymptotics for Elliptic Mixed Boundary Problems

S. Rempel B.-W. Schulze Volume 50

AKADEMIE-VERLAG BERLIN

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S. Rempel • B.-W. Schulze

Asymptotics for Elliptic Mixed Boundary Problems

Mathematical Research

Mathematische Forschung

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Karl-Weierstraß-Institut für Mathematik

Band 50 Asymptotics for Elliptic Mixed Boundary Problems by S. Rempel and B.-W. Schulze

Asymptotics for Elliptic M i x e d Boundary Problems Pseudo-Differential a n d Meli in O p e r a t o r s in Spaces with C o n o r m a l Singularity

by Stephan Rempel and Bert-Wolfgang Schulze

A k a d e m i e - V e r l a g Berlin 1 9 8 9

Autorem Dr. Stephan

Rempel

Prof, Dr. Bert-Wolf gang Schulze Karl-Weierstraß-lnstitut der Akademie

für

Mathematik

der Wissenschaften

der DDR,

Die Titel dieser Schriftenreihe toren

werden

ISBN

3-05-500676-3 0138-3019

Erschienen

im Akademie-Verlag

(c) Akademie-Verlag Lizenznummert

Berlin

der Au-

Berlin,Leipziger

202-100/410/89

Gesamtherstellung:

VEB Druckerei

Lektorj Dr. Reinhard

Republic "G. W.

Höppner

1065

Bestellnummeri

Str.3-4,Berlin,DDR-1086

1989

Printed in the German Democratic

05600

vom Originalmanuskript

reproduziert.

ISSN

LSV

Berlin

763 988 o

(2182/50)

Leibniz",Grätenhainichen,DDR-4450

Preface

This monograph

contains a pseudo-differential calculus of elliptic

boundary value problems with and without the transmission property, on manifolds with conical singularities and with edges. The basic idea is to study appropriate algebras of operators with symbolic structures which contain in particular the parametrices of boundary problems for differential

operators.

The analogous approach on closed compact

c"0

manifolds is standard and

leads to the well-known classical pseudo-differential operators (oj>DO's) and to the index The calculus of

theory. TjiDO's

has been developed in the past decades in many

directions, e.g. with more general symbol classes, non-standard conventions, continuity

operator

in distribution s p a c e s / F o u r i e r integral opera-

tors and so on. Another direction is the theory of pseudo-differuntia1 boundary value problems, in particular of mixed and transmission problems, and the pseudo-differential calculus on manifolds w i t h singularities. The corresponding operator and symbolic structures are of a high complexity and far from being in a final transparent state. There are also many interesting open problems in this area. The present exposition is an attempt of a systematic theory for the problems mentioned at the beginning in a unified way, which contains at the same time earlier

results.

The intuitive idea is to establish first a calculus in direction

trans-

versal to the singularities which contains the remaining variables a n d covariables as parameters and then to carry out the

i^DO

action in the

other directions. For boundary value problems this leads first to the boundary symbolic calculus along dary and then to a

1fDO

|R+

being

the normal to the boun-

calculus tangent to the boundary with

valued amplitude functions. In the case of edges the

IR+

operator-

calculus has

5

already boundary value problem-valued amplitude functions but then we apply again a

-yDO

calculus

along the edges.

This corresponds to the local model of an edge as manifold with boundary "J x

|R+x {compact

c"

{edge^ .

In this sense boundary value problems for

tyDO's

are a particular

case

of edge problems. The edge there is the boundary and the local model IR+x {boundary^

. In other words, also for edges in general we have to

expect extra boundary conditions along the edge which satisfy an analogue of the Shapiro-Lopatinskij condition. In parametrices we also get potentials which motivate

matrix valued operators in the sense of

Boutet de M o n v e l ' s algebra or Visik/Eskin"s w o r k . Another supporting pr±ncLple is the conormal asymptotics of solutions as a part of the elliptic regularity. It also determines the structure of the smoothing operators (here called Green operators in analogy to those in Boutet de Monvel's a l g e b r a ) . Our exposition just presents such an approach. This is of course a huge program which made it necessary to respect carefully the hierarchies of symbolic and operator levels. The lowest level is the

IR+

with scalar symbols, based on the Mellin formulation of

calculus

IjiDO's

on

The first two chapters are devoted to this theory. In Chapter 3 the theory of

IJIDO'S

(with and) without the transmission

IR + .

follows

property

with respect to the boundary. Chapter 4 contains first the cone

theory

as an operator — valued analogue of the material of the first two chapters and then the edge

yDO

calculus itself. The various steps

are motivated and further commented in special notes sections that also contain many references. Clearly our approach is embedded into the achievements of the

TfDO

calculus connected with the names of many

authors such as Kohn, Nirenberg, Maslov, Hormander, Visik, Egorov and many others. The idea of symbolic structures was already invented and widely used in the theory of singular integral operators. In particular the content of the Chapters 1,2 may be considered in this

tradition.

Concerning the literature,as far as it is not complete in our biblio-

6

graphy with respect to sources and methods of the analysis,we refer to further papers and textbooks such as A g m o n , Douglis, Nirenberg [A l], Visik, Eskin

[v l]...[v 5]

, [E z]

.Gochberg, Krupnik

[ K 5"] 1 Grubb [G ¿¡, Hormander [H 3 ] , [P 7j, Rempel, Schulze

[.7 l] , Subin

Lions, Magenes

[s 3] , Trêves

A c k n o w l e d g e m e n t : We are grateful to 3 . Leiterer

[G 3], Kondrat'ev

[L 4 ] ,

Plamenev3kij

[ï 2~] .

(Karl-Woierstraß-

Instiute of Mathematics Oerlin, and M . Lorenz (Technical University Karl-Marx-Stadt)

for a number of

hints and technical

improvements,

further to the Editor, in particular O r , Höppner from the A k a d e m i e Verlag, for his patient

cooperation.

7

Contents 1. 1.1. 1.2.

Operators on the half axle 11 The Mellin transform .......11 Spaces of functions with discrete conormal singularity.......16

1.2.1. 1.2.2. 1.3. 1.3.1. 1.3.2. 1.4. 1.4.1. 1.4.2. 1.5. 1.5.1.

The spaces and .16 Tensor products and Green operators.. .33 Mellin operators.... .........45 Definition and basic properties...... .46 Compositions and Fredholm property..... 56 Pseudo-differential operators.. .......61 Mellin expansion and conormal symbols .......61 The action in 68 r, 1 The algebrapC(!R ~ \0) 71 Compositions of Mellin and Green operatore with pseudodifferential operators. ...................71 1.5.2. Compositions of pseudo-differential operators................73 1.5.3. Ellipticity of operatore i n c C f R " " 1 ^ ) 78 1.6. Notes 84

2. 2.1. 2.1.1. 2.1.2. 2.1.3. 2.2. 2.2.1. 2.2.2. 2.2.3. 2.3. 2.3.1. 2.3.2. 2.3.3. 2.3.4. 2.3.5. 2.3.6. 2.4.

Continuous asvmptotics and higher order operators............89 Spaces of functions with continuous conormal singularity.....89 The spaces and 0^.(11?+) 89 Spaces of the type C~(Jl.$A). C*V2,i^*). 115 Other spacee with asymptotics... * 124 Operators with continuous singularity.. ............126 Ths action of opsrators on .........126 Spacss of the t y p e Z ^ i , and the algebra 147 The algebra^(iLdRd) and ellipticity 174 Boundary symbolic calculus... ..178 The Mellin transform of Sobolev spacee............. ...178 Spaces with asymptotics .191 Grsen operators 198 Mellin operatore ...................................208 Pseudo-differential operators of arbitrary orders 218 The algebras cC and 236 Notes 246

3. 3.1. 3.1.1. 3.1.2.

Boundary value problems.....................................252 Function spaces...... ..................252 The spaces (IR?) 253 Subspacss with asymptotics......... ....................262

9

3.2. 3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.3.

Boundary value problems without the transmission property Green operators without boundary symbols...... Green operators with boundary symbols Mellin operatore The class # ^ Ellipticity and Fredholm property..... Notee

4. 4.1. 4.1.1. 4.1.2.

Mixed boundary value problems on manifolds with edqee 320 Function spaces. . .......322 Function spaces on a cone 322 Function spaces on a wedgs .....328

4.2. 4.2.1. 4.2.2. 4.2.3. 4.2.4.

Operators on a cone......... Green operators..... Mellin operatore The algebra e£ on a cone..... The caee of a cone with boundary

4.2.5. 4.3. 4.3.1. 4.3.2. 4.3.3. 4.3.4. 4.3.5. 4.3.6.

Notee Operatore on a wedge.... ?2-Green operatore 92-Mellin operators The class d].C - 0 Ths class : djC T P Ellipticity and Fredholm property Notes

376 379 380 388 393 397 402 406

References.

407

Index

415

10

270 270 280 .....294 301 307 .....313

..331 332 337 .345 365

1.

Operators

on the half

axis

1.1. The Mellin transform In this section we remind of some definitions and identities on the Mellin transform. Since the material is classical, we only give the statements (for proofs cf. [ J !•])• Let IR+ « J t fe IR : t > o ] . The Mellin transform of u£CQ(IR + ) ia defined as

oo u(z) - Mu(z) - ^ u C t J t 2 " 1 ^ , 0

(1)

which is -considered for z £ C or likewise for Re z o 1/2* The Mellin transform (1) is an entire function in z £ C satisfying the estimates I u(z)l ' c n (l + | Z |)" , n

a'R8

for any m £ Z + with a constant c m > 0 and supp u - [ a ~ i ' a ] > a > 1. Proposition. The Mellin transform has an extension to an isomorphism M s L2(|R+)

L 2 (Re z - 1/2),

(2)

00

where |u(t)| 2 dt - jir

5 and

^ Ju(l/2+iy)| 2 dy

(3)

«o M" 1 f(t) -

\ t-i^^Jfil^+iyJdy

.

(4)

"to

For general u € L 2 Q R + ) the Mellin transform u(z) is only defined on Re z > 1/2. If u(z) has an extension to C, say as a meromorphic function, we also consider this extension as the Mellin transform of u. An example is the Euler T(Z)

-

T •o 5

function, tz"1e"tdt.

(5)

0 which is meromorphic with simple poles at z • - j, ] f Z + , oo We also have an extension of M from C0(IR+) to the space r* f

i* { u €• SS'OR + ): u - r + v for some v 6 "£'(|R)J,

11

r + being the restriction to IR+* Then for u » r + v t- < v , r + t z _ 1 >

Mu(z)

(6)

is defined and analytic for Re z > r for some r £ |R and independent of the choice of v with u - r + v . (6) will be denoted as the Mellin transform of u* Let

be the space of all functions f(z)

that

are defined

for Re z > r, for some r € IR, and are analytic there, satisfying an estimate of the form Sc(l+|z|)BaRez

|f(z)|

with constants c,a > 0 , ra 6- Z. If f^'f^

are

in

some half

Re z > const, f^ will be identified with fg* Then

plane

is a commutative

algebra with respect to the addition and multiplication of functions in Re z > const* On the other hand, r

+

i s

a commutative algebra

with the convolution * as product < u m v, «p>

1/2. 2

belongs to L (|Ry ) for all x 0 . Finally let us mention the following simple 8» Lemma« Let f(z) be a holomorphic function in the strip

< Re z ^ f g

and ^|f(x+iy)| 2 dy * a 2 Then for any

for

x < ^

2

.

£.>0 |f(z)|

=

9

113r

,

for

^

+£

< Re z


with arbitrary

8 . &

"t I V l'

t &

18

X Z j-0 k-0

* 0 and N -

(15)

*•}

_

u(t) - 2 . 2 . j-0 k-0

-

6

i .p^jk* 3

109

l0

9kt

°(t)

i1" ° < t ) }

N' » ^

C- L (|R+)

(16)

for all

V

s

Tfej-S.S']

"

and 1 fr Z + . If y

are fixed, we also write

•

Similarly as above w e have the Oo 4 . Proposition« C , is a Fr&chet space with the system of norms TI V fy(u)

»- ( I M * 2

|| t ^ ^ 1

•

8 - (S.S'.Y. 1) fr ¡R+ xlR + x R x Z The assumption Re p^




(17)

].

1/2 was made for convenience.

2

In this case we have

C L ( R + ) . A more general definition then

follows by posing oo

r

T if , T ° y

I

oo

•

J

ot.oi' with a non-vanishing function g

[ " g

1

(t) -

«j

satisfying

for1
2 . It is our next goal to characterize the in

iiL and C * r fit

in

terms of the Mellln transform and to introduce equivalent systems of norms* To this end w e first evaluate the Mellln transforms of certain particular

functions*

5. Lemma» Set ^P'k(t) p £

Re p


(iii).

i s meromorphic w i t h t h e

C o r o l l a r y 7 . For

l i f r ^ u ^ d t k 1» m^. Thus the map

N

' "

p

¿jk(u)c0

¿1 2 !

is a continuous projection where

aCJ • 1 -

< 1/2,

M^ i ^

fk J

' '

—^ ^

N-N(i.y).

with finite-dimensional rang»,

is the complementary projection. Since im /ti^ S i ^ ^ y

(29) is proved. For proving (30) it is sufficient to consider the case 1 • 1. We first prove that

id'u

for u

l | t

r

V

+ 1

dy . Per definltionem we know that

rf.ul

2

- It^Vtfu-

¿*u)| F

o

L2

for all 1 (r z + , i| • 0, S • 0. Moreover, ^ ' Thus we have

k

. -

P j

o

P

r

k

+

k

o

P

J'

k

-

1

iog k t

+

.

V "ZZ ^^3k* ^.k+i(u>)* 3 , u

N

J

where

8 J

Since

m

"ok"°

N

Then

"

+

Z

-j

-p 3

Z j-0 k-0

l0

9ktSl[tt)(tx

— £ j k ( u ) f c ( E is continuous, the mapping ify ? u — »

-

is continuous, too. Then the continuity of - T ^ " ^ 1 * 1 aC U + T V * 0 6 It remains to consider (31), which is a simple consequence of follows from T ^

1

^ ^ )

t1-(^t)

24

„fr o+lt

(t*u)

.

t

i - M

u >

$

0

J

Later on it will turn out that

ty^

Remember that the bijection u —

t

all

is a Green operator, cf. 1.2.2. ul u gives a definition of

$

with a topology following from the requirement that t*

^fty

for ! ^

—

is an isomorphism« The third statement of Lemma 10 shows that

there is no contradiction if Re p^ *

1/2, Re(Pj+)f) ^

1/2.

11» Remark» Aseume Sy.

j e z+

,

(32)

i.e» (P i .m i ) € ^ = > ( p 1 - J .n^) éf

for all i.J fe 2 + . Then,in the definiPj fk tion of the conormal asymptotics we may replace 16 J by ~ p i3 ' k

"P1 k -t (t) »- t J log t s r .

(33)

{ r + u , u e ^P(IR)]

(34)

12. Remark» Set tf(R+> with the Schwartz space f

0

-

^PÇR)( {(Pj'"•;)) : Pj — j . Wj - 0,

j e 2+ } .

(35)

Then

The conormal 'singularity' of the type

corresponds to the Taylor

expansion of a function which is C°° up to t » 0 , It is then convenient to talk about a Taylor asymptotics (of some order

if we have a

Taylor expansion up to j • Remember that the topology of

^

was derived from a countable system

of scalar products , sr.(u.v) " •

(u.v)

2 Z

L (1R+)

+ (t1-

à

1

*

P (37) u. t1"° ft1 /if v) 2 J Î L*(IR+);

(tj. 8, X) 6 « , cf. Proposition 7. In this way

^

is a countable

normed space with Hilbert norms* If

ïfk*'' is defined as the Hilbert space being the completion of

with respect to the scalar product

y

- lim «

tt^ , we have

tPC°':?

¥

1

.

(38)

25

where lim denotes the projective limit over

nit« .

Now we introduce another countable system of Hilbert norms in Let

^,

^

j f 2, be a sequence of real numbers with the property Re

S-j *

for

P|
=

may be i d e n t i f i e d

i'(of) t f

f t

for of

f o r which

which

order « f

is

Then

31

% For

a

1

u r U2+

.

T

S > 0 we set a

{(Pj^j^?

: Re

Pj< -S+1/2}.

18. Lemma. For any 1 t 2 + there oxists a

induces a bisection

1

^ S ) — f a Proof. Let f £

S ;> 0 such that

T h e n & f for suitable

form

(56)

• o

can be represented in the

^ =

Y

W ^ ) J

£0

jpT«(t)dt ^

c with continuous functions g.(t). Since for 6 > 0 v£>

r i» b: large enough — r :

2 L

:p(8) ^y(S)

(57)

a s

a

f),

cont

is

^

c o n t i n u o u s

nuous

• 0 ^ i 6 1, (57) can be extended to

linear functional. Denote that extension by

Furthermore,

'hus

(l-c^)f is automatically given on H>(f) 0

:= 0

defines an extension of f to

/Hj(tif) + (l-ii)f )

anc

'

'

>

> 's®^

aro

obviously

the identities on the corresponding spaces. This follows immediately by completion of

¡f^ and

) w i t h respect to Sobolev norms of order 1

and extension of the functlonals to the completions which coincide. 0 We have V

V

)

(cf. the proof of Lemma 10) and mappings

which are canonically induced by iT^

and Atj, respectively. In view of

Lemma 18 w e have a bijection (56). Thus the elements in may be denoted by the same letters.

32

and

From = f t ¿y , u t i j

< f , /IT U > 6

s

, we thus obtain the

19. Theorem» For any 1 ff

+

there exists a

S >0

such that for any

I^ we have a unique docoraposit docoraposition f = g + h 1

IM

m

with g f ^ ¿ i , h t

* Π. m = ^

( m j + l ) , N = N(£.y).

j=0 1.2.2. Let y

Tensor products and Green operators be the set of all sequences {(p^.m^) 6- C x Re Pj

Similarly

»--so

as j

«o , Re Pj

: j t-




The Mellin image of 1/2 - S = -

^j

. Ai -P

* a-mCk)

consists of analytic functions in Re z

>

and

oc' (v) = {5l(-S k + iy) 1 v(-S k + iy)| 2 dy - J l O f ^ + i y ^ v ^ + i y ) 1 2 d y } 1 / 2 < k, 1 for vfciTj^,, k,lfc? + . Set

£jU

:»

Then

1/2 is a countable set of norms on

Jy

for k , l t 2 + .

Set

Vi(u>2>1/2-

«.(k)(U)»(X i=0 Then of ^

i3

again a Hilbert norm on J y . Let

with respect to

^k)*

Let

ify^ be the completion

be the Hilbert space of all ho-

lomorphicfunctions f(z) in the strip B k for which \|($ + iy) 3 f(J +iy)| 2 dy

sup "

8

< °°

k < W

for j = 0,1,,..,l(k). From the results of the previous section then easily follows the 1. Proposition, The map u

- M

u |B

+ ¿^u, u £ ^

* £ » £ k + 1/2,

has a unique extension to a topological isomorphism

and

T

= v

&{k) © *m(k) n

* - 2fT

l+z1^

1

'

tpl^*1^

K(z,w) is obviously square integrable. Thus Hilbert-Schmidt and Proposition 2 is proved.

is

•

As a corollary we thus obtain 3. Theorem» The space Proof.

is nuclear.

¿ y is the projective limit of Hilbert spaces

The nucle-

arity holde per definitionem if for any k there is some k' such that » position 2.

is Hilbert-Schmidt. This was just established in Pro0

Theorem 3 reminds

of results on the nuclearity of spaces"of analytic

functions. Let

and consider the spaces ¡Py and iP^

with the systems of

norms (31"^ )oi(.oi and (^ijJgeB' respectively. Here ar(u) 0.

gj

V(0.t)

assertion.

elementary

with ~

S) '

(13), tho

the

a,b&IR

1 W^

11

L*(R+

x |R + )

way

0 R + x IR + ) (25T)

-2

w

1

z

1 '

v(iv,z)

||

2

L

:jj| w

vV ((i vW , z )2 ) J] ' «

L

SrJ1

v(s.t)

< r i * r i >

2

L

v(W|Z)

( r i X r i )

L2(

+

(|R +

x |R+ )

r i

2

2

2

S

2

+ II Z 2 1

2

2

c{ I

z

I ^ 2 " ' l W t

v

(

2

S

)

s

'

t

)

x

r i > i

2

Il 2 « L (IR + x I R + )

J

] j

and » ì

(2JT)

«

v

-1

-

v(s,t) L

i/

1

„ V t '

(1R+ x |R + )

8

v ( w , t )

2

t

V(s.t) l/"(IR+ An a n a l o g o u s

estimate

( V -

for si-

4

SrJ'vts.t) L

44

v(s,t)

x 1R + )

(l« +

X|R+)

L2(|R+

x

|R+)]'

combined with the other estimates finally yields

I

s

«

=

cj t

~

l (t?+ x

|R+)

~

s2(1J

^

~

v(s.t) II "l/^

S

x |R+)

+ Bt2^ ||

The estimates in the proof of Proposition 10

( 2 5 )

^

t

v(s.t)|| "l^C^xrJ

may be modified by using

the inequality

xayb = cj

xsgn

1 a( | a| +£) + yt" b( |a| +

for all x,y€-|R + , a,b€-|R and arbitrary

t) j

£.>0 with a constant c = c(a,b,f)

> 0 . The proof oasily follows from the obvious case when a , b € I R + replacing x(y) by x

—1

(y

—1

by

).

Then (25) can be replaced by estimates where on the right, for instance, we have an exponent of s close to

~ r if - b

~ "r instead of 2(^-6). This has to

be compensated by enlarging the other e x p o n e n t s . In other words,the s (or t strip) needed for estimating the mixed terms in (25) may be close to the original one. The assertion of Proposition 13 is well-known in the special case ^ s for

«= y Q . Then it follows immediately that Proposition 13 is valid Y . 0|

containing only a finite number of points. Using the notion

of infinite sums of topological vector spaces (cf. 2.1.5.) one could derive another proof of Proposition io 1.3. Mellin

from, this .

operators

Now we introduce operators of Mellin type which give a first

expansion

of the algebra of Green operators discussed in 1.2.2. At a first glance the Mellin operators are similar to

Tj.D0's

with Fourier transform

re-

placed by the Mellin transform. Note, however, that they are smoothing outside the origin and are localized modulo Green operators in an arbitrary small neighbourhood of

0. Moreover, no Mellin symbols with

explicit t-dependence w i l l occur.

45

1,3.1» D e f i n i t i o n In for

this f

and b a s i c

s e c t i o n we s h a l l

,0| t y

so-called O °P

where f t

introduce

, which are

The s i m p l e s t

M

properties

closely

Mellin [f]o

operator

: ^

and o p M [ f J

'

of o p e r a t o r s

connected with

~

has

in

, ^

the M e l l i n

the

form (1)

with a cut-off

is- the M e l l i n

)

convolution.

,

m e a n s t h e m u l t i p l i c a t i o n r

a class

function

on I R + ,

convolution

co

opM[f]u(t)

o 5 f(t/t')u(f)df/f

, u£-L2(K+)

.

(2)

0 B e l o w we s h a l l

prove

that

the choice of

ti

does not a f f e c t

modulo Green o p e r a t o r s . More g e n e r a l l y ,

we s h a l l

the

and c o n v e r g i n g

form t ^ " ^ l i o p M [ f j t"'

^

O.

J "0 necessarily

First in

opM[fj] Ojt

3

in C

oo

tion with

is

for

any f ( r f

mappings For

and continuous

and f — >

46

: ^

f

case of

by t h e

f j that

of

sums are

not

function

O

the m u l t i p l i c a t i o n

then tho o p e r a t o r

occurring

by a

func-

, w i t h max(,lepj

, Mf

: Gy

M^ i n d u c e s

*

+ r;eq|J< 1/2,

o n o )

—

M^ o f

multiplication

operators *•

we g e t M

symbols

deal with operators

singularity.

~

f t

with Mellin

on t h e m u l t i p l i c a t i o n

1 , Lemma. L e t

Mf

, ]f>0,

a simple special

conormal

by f d e f i n e s

fc

operator

3

, .

we c o n c e n t r a t e

(1). This

, j

the

o

continuous continuous

t>

Jy^

and f - » M f

induces

operators M

f

mappings

^

y

'

continuous

Pr^of:

Let

?

(t)

for

all 1 f

v

i

t

a

f

•

"

. oj = {

(

{

«

O

(t).

Then

" i

,1

obtain z j=0

^

• •^)]

.

x

+

2

logJt

S = 0 we )

, m j )] .

P j

—z

^ Z "i to

= { (

(with

o b v i o u s m e a n i n g of

q..k

subscripts)

¿t

V ,

k=0

=

4

u - ^ T j=0

k=0 p . ,k

^

N

tX "

£

op (if

TJ4f

® J *

Vy. >

)

with

Proof. Per definitionem we have ( t J - ^ Q o p M ( A ) 0

for

\

Then the sequence of conormal

J

symbols

from M + tif.

G r e e n o p e r a t o r s and M e l l i n o p e r a t o r s Yi

»•»0

tf-i «-o (U-60)

operator

arbitrary.

o p e r a t or Ml + A

t

shows t h a t

(cf.

Hence

same c o n c l u s i o n a p p l i e s

a given

uniquely

«s« J "" to + t

J

a compact

than

. > 0, t h e J

^ • For

j **

t

e

a sum o f

norm l e s s

= 0 but ^

(U-K)t

is

J

j ""

t

Lemma 7 ) it

to

is

tj t

J"" iT-i J

and

compact. I f "C -j t J +

=

J

+«LG t h e p r i n c i p a l

determined according

to

an

Proposition

Mellin

10.

Consider

now 3-1 -

. ^ T « * « »

h ^ ) , « ) . * '

k=0 /fti^ b e i n g d e f i n e d

by t h e

finite-dimensional orthonormal with operator we h a v e

last

equality.

space M..

respect

' 1(

J_

v

to the L

)"

be a b a s i s

(for

0R + )

6l3 with finite-dimonsional « i «--Oi, = t

Let

the notation

scalar

range

product.

cf.

For

"¿ijU

yields

being uniquely t

1

0,

instead

then

the

we

and

(AH +4l) =

.

(Atv).

of is

the

sub-

on Re z = 1 / 2 .

G , j t 2 + , are subalgebras of

, the argument principle (cf. (Xl], p. 90)

gives

5. Proposition. Let ot, «.' have the same principal conormal symbol h Q

and itf 1 +

+ OL^Q. a.' fe 1 + ind

+ ° t 6 0

f o r t

for

t > 0 op ( i ; J ) ( S ) u ( t )

- prJjj \ ( t - s ) j " V

* Now we i n s e r t

0

the e

^t-s)u(s)ds.

expansion

-S(t-.,

£

=

^

( t

.

3 ) k

+

V j ( t i l ) i

k-Q where R^ j ( t , s ) ds • - t r

_2

has a z e r o of o r d e r N+l a t

s = t.

Then w i t h r = t / s ,

dr, oo

t ^ (t-s)

;i

" (t-s) u(s)ds 1

k

0

» ^ 1

(t-t/r)k+j"1tr"2u(t/r)dr oo

= tk+J 5

( e - D ^ - V ^ ^ u t t / s ) ! *

1 u e- CqOR + )«

Thus

$

(t-s)k+j-1u(s)ds

0

[ /«kj(8)U(t/8)

I

s

0 f

/ " k j ^

= tk+j

( a - D ^ - V i ^ »

I 0

f o r s > 1. f o r s 6 1.

and we may w r i t e

65

^ (t-Bj^-MsJds

- tk+J

op M (m k )u "kj'

0

(10)

with mkJ(z)

= M/ikj(z)

- S

s ^ ^ ^ ^ s - D ^ j ' ^ s .

1 A classical

integral

formula

„..(*)«

r ( i - z ) r(1+iop+(l+

k —

for

» € C, Re « > 0 .

with

T(i-Z) r>+k)

Hi-2)

=

lc! R l - z + t f + k ) T ( « )

K

,s)

_ )(S)U>

The e x p a n s i o n of P r o p o s i t i o n 4 remains v a l i d k

( t - s ) ( t — s g ( t

i n ÏR+ x IR+ up t o t h e bound-

TO

0

+

\

/

k

( 1 1 )

Rl-z+tf+k)

Set 5°(a)(z)

;- a* g + ( z )

+

a^ g ' ( z ) .

(12) 1

9+(z) 1 à l.

Below we a l s o

G~^(a)

a

I(^)g"(z>}

flu-*)"

B !

^ Î C ^ !

8

t h e c o n o r m a l symbol of a or o p ^ ( a ) ,

where t h e dependence on

S

(13)

of o r d e r - 1 ,

lé

action,

i n t h e e x p a n s i o n of P r o p o s i t i o n 3

out.

5» Theorem. L e t a t S ^ ° ^ ( | R n \ 0 ) N e 2+ there

,

»

Now we d e r i v e t h e M e l l i n e x p a n s i o n of t h e p s e u d o - d i f f e r e n t i a l

i s cancollod

1

write

«"m 1 and c a l l

+

and

i s a k=k(N) , k ( N ) —*• oo

où be a c u t - o f f

function.

a s N —• oo , such

For

every

that

N Oop ( a ) ( £ ) o -

66

+

j-0

''Pg^N^^U')

(14)

with

some k e r n e l b ( a )

Proof.

Let

-

b (a)(t.s.4')

Oj^ be a n o t h e r

Lemma 1 a n d P r o p o s i t i o n

cut-off

f

c V ' ^ O ,

(

N

x

^

o opM(Aj(^.S)g+

- IC^T

+ A~(^,8)g-)

)

o (1-J)(£)d

C^op

mod o p g ( Inserting

the d e c o m p o s i t i o n

fo

N

»op ( a ) ( £ ) o

-

r

N

X

Z°

j-0

k=0

0 P

of

Proposition

A

m(

Î^"

S )

9+

4 we o b t a i n modulo

opQ(

+

N +

b

°PG< N.S>

« V ^ ^ P M ^ ^

(15)

k»0 N ^Tu>opH(A*(£\X)g+

+

+ A~(^,S)g-)

01opG(b).

j-0 The s e c o n d a n d t h e t h i r d °Pr(

®

+ OopM(g~)t

* . J

term on t h e

right

of

(15) are

o^ = OtJopM(g

opM(hI

0

n

Q

n ¿P

o p

obvious M

( X

),

'o

^ ^ ) .

tJ+l 3 ( l - it^ ) . We c a n a p p l y

the decomposition

are of

69

cO op (a)Q, from 1«4»1, Theorem 5, and obtain l ^ P * op ( a ) C 3 ( l - oT^) *

N ¿_

1

* ( 1 - )

f 1 « o p M ( h 1 ( a ) ) t o ( l - if ) + opG b N (a)

1=0

with b N (a)

® *

rO

rO

and suitably generalized Mellin symbols hj^a),

V (1- 0«

and every f C-Qpfi

The vector

is bounded

for

|lm z|>£

for a n y £ >

space is an a l g e b r a with {o\

sided ideal. Denote by CSfft Q T / l ) f in

g

real

as a two-

the vector space of

w i t h coefficients in W f t Q t H

polynomials

and by ( K ^ © ^ )

the sub-

space of h o m o g e n e o u s polynomials of order k. Remember that for op (a)) t

t

M o r e o v e r , for m jt oi M (lR n _ 1 \ 0) , 5- (MV) t W U J l ] , J

(a)

.

• w. 6 ^ (|R n _ 1 \ 0), ol = op^(a) + M

The conormal symbol of order - j of

+

is defined by 6-^(0i) = 6 ^ ( o p

(a)) +

which is an element in

[ a

the subspace of sequences ( > ( h ^ k ^ 2" s j . If

to

(«).

•J e Y

by

of

p.

the d i s c r e t e

as the smallest are

k-0

1

belongs

carried

le

M('fl)

V-ii! N = N(-£),

h)(P>-

in

a s

^ ^ 0 define for

(-4dw

an e l e m e n t

£2

For

(-D

given

j

for

conormal which

constants

«-P.

aw

J

Let

•{ w €• (D

:

w •

p.. ,

j

j

(4)

and r(v) Y The

!=

3T,. y

1/2 - -J }

.

function

belongs to

iS4(C)

for

F(t,w)

:» t ~ w e i ( t )

any f i x e d

t £ IR+. For

4^u(t) and u (:

90

n { Re w i

- u(t)

i s equivalent

- ^

)

F

(5) = 0 we o b v i o u s l y

have

>(t)

to the existence

of some

£

(.

(r($ ) y )

of

the

form

(3)

such

= t ^ ^ u

V

for a l l

U

Define

the

-

that

of

F » ( t )

U2(Rt)

€

(6)

function = ^ t

z - 1

o solution

which i s a fundamental is

1

3

r

Cj.^.l) to/ .

(z,w)

4>(z,w)

- < ^

a moromorphic

F(t,w)dt

(7)

— o.

of

function with

a simplo

polo

at

z=',v.

Instead

( 6 ) we may w r i t e

for

2 (X € o< , H q b e i n g

all

form f ( z )

=

^(z)

+

Ti_,)?

t

•i defined v2(z),

as

the

(8)

s p a c e of a l l

2 L (IR

v ^ fe

IL 2

+)

, supp V j

functions

of

the

bounded,

v2

£

2 Remember 2

L (|Re

that

z = 1/2^).

an e x t e n s i o n tion.

the M e l l i n

to

It

transform

may h a p p e n ,

some d o m a i n

This w i l l

also

u

for

of

every u £ L

instance

D. C (E a s a s i n g l e

be d e n o t e d

(|R + ) b e l o n g s

when u t ¿^y valued

by u a n d c a l l e d

, that

to u

holomurphic

the M e l l i n

has

func-

transform,

again. Now we e x t e n d called

the concept

continuous

of

the d i s c r e t e

asymptotics

a s s o c i a t e d w i t h more g e n e r a l

and a n a l y t i c f u n c t i o n a I s c a r r i e d set

A C

being

t,

A

the union of

disjoint Re w

and a

S

°

by t h e c o m p a c t

- U




' ^

(15)

also h a s an a n a l y t i c e x t e n s i o n to Re z > 1/2 - (S^t-f). S i n c e both £ , a r e c a r r i e d by the compact

set

z >

(15) is e n t i r e » From t h i s w e shall c o n c l u d e

1/2 -

(S+£)j

that

the

function

= £ g

Let Fo(t,w) = t"w £(t) being the c h a r a c t e r i s t i c

function of GO

(16)

(0,1) a n d Z

t ~1Fo(t.w)dt

< p o ( z , w) =

=

.

(17)

o Then •\|)(z,w) =

4>(z,w) -

(z , w )

is e n t i r e s e p a r a t e l y in z a n d w . T h e n the f u n c t i o n > ( z ) < ^ , 4 > > ( 2 ) d«- , 8 denotes the measure on ^ U = 3 induced by the metric of , which differs

with respect to the Cauchy kernel cf>Q by an entire

function. For any U ^ U j

where dff

( A^j) . equiva-

(20)

\

95

cfrj(•) i s a norr.i on $ ' ( Ag^ . since c p ^ ( =

0 implies

= 0

for a l l z t U. Then ' vanishes f o r a l l h °

n

h

h(z)

> " ITI

(22)

dz

s i (|*

sup

|(w.z)|

for a l l z t L5, D = 'BU. Then

B c

=

^ u ( t ) 2 \*

B

su W

PA

I4>(w'z)l2

d 0, B = ^ U. The n u c l s a r i t y of

jC ( Ajjj) f o l l o w s i n the same way as f o r standard spaces of holomorphic f u n c t i o n s . I n f a c t , according to (22) each f u n c t i o n a l £ sented by the holomorphic f u n c t i o n Let U

. B = 9 U , ¿fe ^

96

z

>( )

can be repre-

i n the e x t e r i o r of A

.0

' =

SU P l < < t ,4> > ( z ) l Zf B

•

(24)

6. Lemma« The topologies on !

norms

and

I^B

Proof. The topology of

! B

are

"

St'( Ajgj) defined by

to that defined by {Yq^I

( A^^) defined by the systems of semi-

{tyy}

®quival8nt» is stronger or equal

• On the other hand it is obvious that the

topology is stronger or equal to the {^¡3} topology. Thus the

desired equivalence follows from Lemma 5*

0

Now we show that the convergence of analytic functionals implies the convergence of their potentials

,]> in the space of rapidly decreas-

ing functions on the lines r 7, Lemma. Let

> ^

.

in

{RO

z

iSi" ( A ^ y

-

e

3

and

.

(25)

P fi A ^ o

Then

in L 2 (x Tp ) ' for all 1 f Z + . Proof; For fixed

^

we find some U 4 Qlj. with Re z / ^

for all z t U .

Then ( f +iy)|2

d

y

Oo

00

^ I ( ^ + i y ) 1 $( X n ( w ) - X(w))( e + i y . » >

-

-to

B -A(w)|2|dw|

= *• 0 as n

d w | 2 dy

» £» .

f tiy)1!2

p + iy.w)| 2 |dw| dy

0

Our next aim is to construct a semi-norm system on 2

under which it

becomes a Fr6chet space. It turns out that the L (|;+)-nortns of tho expressions

(13) are not sufficient

the operators

for a topology on

such that

are continuous. Je need an extra term for tho func-

97

tionals. So let us define on 7L,(u,v) := (u.v)

a system of scalar product

+

2

+ (t-S+V(u-s(u).F>;.

t"S+1l»1(v-

s

Here

(*>|,S,1,B) , where

(v).F>))

^S

"»| t

, S e d , 1 ۥ

and 3 = 1 U for U

The corresponding index set will be denoted by l/2 X . (u,u) 6

T.(u)

L/(ir+)

.

. Set

.

0

Let us define a further system of scalar products on

by the ex-

pressions aj„(u,v)

:=

7

(u.v)

„ L

+

1 (•z1~ u,

z1^)

„

L V . )

(27) •(B)

and associated norms / V

u

)

!=

/

(23)

S e B, where S is the set of all tuples B • (S.p.l.B), J t d ,

p

- 8 : 8«- d ] \J (Z+ \ 0) , 1 (r 2 + , B as a b o v e . Instead of (27) and respectively, we also can consider the system of scalar /j R (u,v) 7

+ (zXu, z 1 " ) „

:= (u,v) „

*

L

2

+ (u.v)

LZ(Tf)

^ )

i

i

products (29)

?

L

(28),

(3)

yp

and the associated norms yu^(u) = (^jg(u.u))'

.

Another convenient semi-norm system is 'yu14R (u)

:= max 1| y sup eiR I u(1/2+iy)| ,

(30)

sup | (p + i y ) 1 u(p + iy)) , s u p | u ( z ) | ] x x J y (: IR z t B B e B. equipped with the system of norms ( 3 T v ) v tf v . i s a /\ o 5 I nuclear Fr&chet space. The topology is equivalent to that defined by

8 . Theorem.

i

the system ( ^ ( ^ g f c B * Other equivalent systems are ( ^ ^ I g f

98

ar|

d

(/Vets* Proof. The simple arguments that

is Fréchet are left to the reader.

Let us check the equivalence of ( jf^. ) and «te have to show that

(^

• (^

))

*

(y^).

( b ^ . C/J B )) is continuous with

continuous inverse, which means that (i)

to any 6 f B there are finitely many /JB(u) = c

"¡f^tf

, k = 1, ... ,kg, with

sup

(31)

with some constant c = c(B , ^ , . . . ) , l kB (ii) to any

t ^

there are finitely many B-,

3, 1 = 1,...,1 oc

—

Jf^(u) S c

sup

with some constant c = c(^ , B

L

*

(r

with

/ " ^ ( u ) , U f. ^ 1 #

...). 0/

(32)

First remark that

L

V2>

and that we may restrict ourselves to those equals one of the

for which

, j ۥ 2 . We have -|z1(u(z) - < >

^U(z)||

(u).4>Xz))|

2

(u).ct»>(z)| L

(rV2-S+

and (31) is a consequence of Lemma 5 and Lemma 7 . Conversely, , .i(S(„

-


w

„

^

^

®o , cf. also Section 2 . 1 . 3 .

In order to prove the equivalence with (yUg) first observe that norms

as

on the functions u (r b^y are finite and induce

which is stronger or equal to that defined by the ^ g * 2

vergence of u in L ( B ) implies the convergence of

the

a topology

Indeed, the con( A j ),

^ j ( u ) in

which follows from | I = I — — 1 1 ^S 2ir i ' 2 cB

\ u(z) h(z) dz ^

sup

| h(z)| | J | u ( z ) | 2 d / j * 2 .

z

Thus

^^

B

is Frfechet with respect to (yUg) and hence both

topologies

coincide. The proof of the equivalence of (y'-'gJgtg

t0

t 1e

'

other systems

follows

in a similar way as that of Lemma 6 and 1.2.1, Lemma 15. 0 It is easily seen that the particular properties w i t h respect to

A

choice of d with the mentioned

does not affect the topology and that a

countable subset of numbers

= 0 with

fj — >

as j — » o o

Similarly, in ( ^ g ) we may replacc the sequonco of tho numoors another one, say ( P^

= (3, k t-^,

9 . Remark. Let A t A

. ^t^ ^^

—*•

+oo

, Aty) = A

by

is

a

as k — _ + oo . >

cf

and tho topology of

1.2.1. coincides with that of ^

*

^

(4)-

. In particular

no immediate characterization of the Mellin

rhan

^

defined in duct ion is closod

Let us point out that the semi-norm systems (yUg) , image

in^^.

or (yu^) give • Out tho above

calculations easily lead to a simple necessary and sufficient tion for u t

^

whore

0 A

closed subspace of

suffices.

condi-

H ^ . Oo

Let A = I be a closed set. Thon a function ^ excision if 0 = ^

100

C (£) is called an A

= 1 and ^ = 0 in some open neighbourhood of A .

10. Proposition. The functions Tj t

are characterized by the proper-

ties (i)

u

(ii)

¿4(C

s

A ) .

oo

\|( ^ + i Y )

f + iy)u( (> +iy)l 2 dy


^

*• ^

z e r o . Then i t

the

• As

suffices

to

d e f i n e d by t h e norms ( 3 8 ) we h a v e

top,.[f]u||^(A) w i t h some c > 0 . T h i s e s t i m a t e

= C

is

M #

obvious

k ( A )

with

. c = sup { | f ( z ) | : Z Bkk V ckk ] I t a l s o i m p l i e s t h e a s s e r t e d c o n t i n u i t y of o p M f . J . 0 oo .le a l s o have an a n a l o g u e o f t h e s p a c e s o f t h e t y p e C , ,cf. T'f

Section

1.2.1,

A'

set

here

of a l l

i n the s e t t i n g closed subsets {z

To any

A*

A'

numbers w i t h

of

analytic

functionals.

¿j

the

/\' C £ w i t h

s 1-z

t

A' j

t

A

t h e r e e x i s t s a system d'

¿^ = 0 ,

Denote by

< ^j + 1'

•

= ( { ' ^

~ ° °

a s

f

of

?

^

non-negative

. d ' r> A " =

Set

19. D e f i n i t i o n . non-neqative

Let

reals.

C°°(IR ) s u c h t h a t

for

112

=

A t

A ,

By C*?

^ ( u

| Re z Z 1/2 + A"

^ A'

every

ar>d

I'}

d, d '

o A'

.

be a s s o c i a t e d

systems

. . (IR ) we d e n o t e t h e s u b s p a c e of a l l

A • A

a '(6V

£V t*

/y

+

S td,~

such

,F>

S' 6 d'

there

exist

elements

that

-

0, s-

T

the f i n i t e

0

qf : £ L

f

is called continuous at

and each closed neighbourhood U of

is a closed set l 0 ,

{ K^.UjJt

we have for every j

f K _ u(y.t) t C*°(int Kj , jfi 1 '*) ) J J being the operator of restriction to int l

L 2 (/R + ).

L 2 (IR + ) - L 2 (IR + X IR ) be a given function and denote

by opG(«v) : L 2 ( R + ) the associated linear continuous op_(w) u(t) - \

¿+

*

L 2 (|R + )

operator

w(t,s) u(s) ds » (w(t,s), Ou (s ) ) -,

L

2 u

L (IR+). Obviously, o p Q ( w ) is a Hilbert-Schmidt operator. Denote by

opQ(L2(R+)

L 2 (|R + )) the space of all those operators and by o p G ( ! T )

the subspace of operators with kernels in T C

L2(R+)„- 3 $ 1 ) . Then (1 + o p g ( b ) ( y ) - 1 + opG(w)(y) for some w t C

Q

)

^

Q

C) It Q S V ^ 1 ) .

Proof: First observe that b(y.t.s)

fr

c ^ X i . i i 2 ® ^ ^ ^„(g^ai?1)

c f . 2 . 1 . 5 . Lemma 3 . In particular b £ C°(Q ) (x^. has a representation b(y.t.s) where c.^ C- C°°(.Q)» f^

oo ^ c i (y) f A ( t ) g i ( s ) , i-0 an d the series converges in the

tensor product topology. Write M " M (y.t.s) = ^ c i (y) f 1 ( t ) g ± (s) i-0 and t^(y.t.s) - b(y.t.s) - b M (y,t.s).

130

r

Then IIfa'(y, . , . ) U

0 as H —»00 uniformly with all deriva-

*o 0 such that MV ° y

(33)

j-0 f

converges in addition *MK

°r

^ V" = ^

a l l

A*-A

w i t h h(A) f A

and that in

*• AM^

(34)

J-0 converges in

( ,

^

^

) for all 77 €- A

with h * ( 7 T ) f A .

If

is

defined in an analogous manner by another choice of c^ and constants c., "¡J^ , then / M - M b Proof; First

is a Green o p e r a t o r .

remember

the representations

» lim

(A).AtA, fixed

in Section 2.1.1., where the norm of ^ ^ ( A ) - k < : R e z < ^ + k j .

refers to a strip S ^ -

Then we have to prove that M V c o n v e r g e s as ope-

rator MV

140

:^Ck(A)

*

3Ck(h(A)>

for

e v e r y k a n d t h e same f o r

MKm

+

W

1

the a d j o i n t ,

uniformly

N

as a f i n i t e

-

anS^,

D

types

k for in

show t h a t

since

^

all

i

is

„ IB

a finite

-

II z

1

-

k}

lines

sum o f

3

, "

n

I *

parallel

( z ) - M [ t

for

l(

z

-* C f

w

j

does not •

'¡f •

e n o u g h we

have

affect

the

and hence i t

singu-

suffices

operators

of

the

(A))»

form

;

(35)

/

of

finite

length, P

finitely

many

lines

axis

M_1{ hjiz+^J

Jf^, c = c^ , h •

c-1f c 1 " "

h(w+p

N + 1,

expressions

to the imaginary

M( U ( c 3 t ) t h^.

j

u] ]

.

Then

»c' ^sw~1+lf6i(s)u(|)

c

(t)

c ^ f f s ^ y ^ s j u i f )

| £ ] d »

,

¿w -

^

J

r i -

and N l a r g e

Thus,it

LZ(B)

^«(Cjt)

abbreviation

nj(z)

zero«

t h e embedding ^ P ^ i A ) ^ ^ ^ ( h

M _ 1 - [ h ( w + ^ )M( i O ( c s ) s u ) ( w ) } c

.

2

smooth c u r v e

M( O ( c s ) e ^ u ) ( w ) -

.

AVij

LZ(D 2 )l 2

J

B being a piecewise

for

N + 1 . Thus

n (z)|

B

Set

-

c o n v e r g e s a s a sum o f

r

n j

fixed

may be c o m p o s e d w i t h

r 1

or h a l f

2 !

j-N+1 o p e r a t o r s has o r d e r

For k

| Re z > j

AM^)

then i t

|

"

>

-(j-Vj ) < 5 -

to

MC(1)

MKy

j-0 zero order

sum o f

to deal with

Qo

^

1

larity

Write

) , MV(o)

suffices

for A t A .

c

"

*

^

j

|

St-w

i

h(w+|)

c1_m

)u(|)

dw j d t

vievi of the concrete definition of the scales ^^(A)^

$k(/\*A.°)

io" N = N(k ) sufficiently large wo get that both : ^k(A) — *

#k(A+A°)

and op G (b N ) : ^ k ( / \ ) — ^

^ k ( A + A°)

are con; iiiucus. Thus «op

(a)U

t

k

(

A

+A°)

ii continuous. Mow we have op-j, (a)

a

^op^ia)^

+ 'oa op

+

(1-W) op^(a)to

( e ) ( l - c o ) + (1-co) o p ^ ( a ) ( l - i ^ ) .

The continuity of ( l - w ) op

—>

+/

i? obvious as well ar thai of bouor^e of

Theorem :\ qnd the continuity of

ifL i A o • Further it /\

let

ci' be another cut-off function with o = co'co. Then (1-co) op^(a)ai = (1- W

) op^(a)o

+ (oi - CO' ) o p ^ ( a ) w .

For ( —

cû" ; op^(a)co

op^(a)co

we may apply 1.4.1. Lemma l,wnich shows that (1- to') op^(a)co

:

>

/\°

we argue a s in the beginning,whereas for (1- cO* J

continuous, t o o . The continuity of (43) may be

proved by obvious estimates of the involved semi-norms. 0 Note that

A°

B

A ( Y 0 ) . cf. 1.2.1. (36), 2 . 1 . 1 . (4), and that w e also

have a continuous operator

2 3 . Proposition. Let aj. é o p G C P r

j t 2 + , a (r S ^ ° ^ ( l R n \ 0 ) . Then, for every fixed o p ^ a M ^ g .

6 6

, T~J T

T, A t A

° P

G

( %

A

£

° ®

r

» P, T - j A =A,

f 0 (44)

^ > '

opG(fr(x)r^A+Ao) .

(45)

Proof : We show (44), then (4b) follows by duality. In view of Proposition 2 and Theorem 8 it suffices to know that °p^(a) $

*

«fop^a)* t

S £ (

l 2

o r

+

) - A

¿£(L 2 (IR + ),

o )

)

(46)

/ .

(47)

(46) is obvious because of Cj^t ¿£(L 2 (IR + ), ifp ) and Proposition 2 2 . (47) is obvious , too, since op^(a)*f

S£(L 2 (IR + ), L 2 ( R + ) ) and d ^ f

¿£(L 2 (|R + ).

).0 Note that we have even op^(a)cj. € o p G ( and similarly for

Oj,op^(a)*

Let MK tie as in Theorem 16 and a t (38) by

TkZ^) U ij ( ^Z 1 J+k-1 of

C^ia)

for

s j tz"l J

«s V 1 . Assume

that V1(rZd.

The

proparties

llm z.\ — » show that then f x ( z ) f ty?) j. W i t h

''l'^l'lt 2 ' Then we have the

146

0 ) . Replacing m^ in formula

—k ffM (a) we get.a function f^(z) which is holomorphic outside

constan,:s

following

as

above, we can pass to some operator

(39).

24. Theorem. Under the

described

op^(a) w i t h some G r e e n o p e r a t o r op^(a)

as the l e f t

Proof: Applying

c o n d i t i o n s we h a v e

- /W + h

o and, hence, in the Intersection. Since

9

X l ^ ' V c t j=o

for all N £ N —»oo

g.

2

for all j t 2 + . we have

^

$

&

^

)

• This follows, since the scalar product in

is induced by the diagonal of

H

) © ( -i&H^ )0

J *

and the only point is to ensure the convergence for

in ( £ )@

152

)• Then, for an orthogonal base (9j)j

we have convergence oo > (h.g,,). 5«o

( h , 9

respectively.

^Ih^-)' (5"

anc

' t ' 1 i 8 induces at the same time the above

)-scalar product.

0

X

A s an example let A

2

.A

A

€ A

compatibility condition that Fr&chot

(which doe9 not necessarily A

+A

2fe

A

satisfy the

)• Then we have the sum of

space3 :PAi*

C

1

^

A -

A

1

+ A

2

,

2

L (|R + ).

This notation will be justified when we show that the space not depend on the decomposition of A (cf. Proposition 4). O n the ^

does

into the sum of elements in

spaces w e have a n additional

from the representation as projective limit of Hilbert

,x

^

A

structure

spaces

j

Then

Since the embeddings 0

!

j

^J-l'

w h i c h are induced

1

"

1,Z

'

the inclusion ^fj C L 2 (|R + ), satisfy

by

el|

.

e

2

|

we get e

j

+

e

j

s

+

*i

^ j

~ -

ttj-1

+

for all J * Z

and

The same is true of arbitrary finite sums

P

y

We shall need also A

infinite sums of of compact sets

j

i-1 . • One reason is that each A ^ A

is an infinite sum

A1

Aj•

Consider a n arbitrary sequence ( A ^ ) ^ ^ ^ sup | Re z : z t A * } Then w e can represent each j there exists i Q

*• - ««

^ 1. as ¡P x. « lim A A •*— » i Q ( j ) such that °

• A* t A

satisfying

a s i — * «> . % i n .J c»

such a w a y that for i . # for all i > i 0 * k-1

J

153

In fact,

t h e norm d e f i n i n g

for

f i x e d s t r i p c ( j ) < Re zz < | for large i .

fixed j

t a k e s c a r e o n l y about a

t o w h i c h no s i n g u l a r i t i e s a r e

contributed

Therefore,

Z *} - Z a}

i-1

i=l

and we s e t

oo

Note t h a t

^

check t h a t

i-1 for

Definition

2).

(

I ^ •

i»l ^

^ i s a nuclear oo ^

A

A "

i-1

Ai

Z

•

i-1

/x

FrSchet

space, a g a i n . I t

«L Aj_ c o m p a c t , ^

$ A.

J\

i-1

E v e r y A & A ( c f . t h e d e f i n i t i o n of A i n 2 . 1 . 3 . ) o» ~ ^ » A j • where A j € A compact and

i s easy

= ¿R

A

(cf.

to 2.1.1.

can be r e p r e s e n t e d

as

J-l sup ^ Re z

: z t A ^

—

-

oo

as j —>

oo

.

&•£^ •

Then we s e t

J-l

4 . P r o p o s i t i o n . The F r & c h e t s p a c e Ji does not depend on t h e c h o i c e of oo ^ the decomposition ^ » ^ A j w i t h t h e above p r o p e r t i e s , i . e . i f A " J"

1

\

i s a n o t h e r such d e c o m p o s i t i o n , t h e n £ j=l

k-1

and t h e t o p o l o g i e s a r e e q u i v a l e n t . rn P r o o f : Denote by A ^ a n o t h e r c o v e r i n g of A common r e f i n e m e n t

of

Aj

and

A£,

i.e.

by compact s e t s w h i c h i s a oo ^ ^ ^ = A ^ compact and 1=1

f o r each j

there

isanl(j)

isanl'(k)

such t h a t

a s 1 —>oo

and i f y

obtain continuous

154

such t h a t

a"* A j ^ . ^ j C. C+ rx , lf-L(j). The main

£(u) £

( A j). which is

possible according to the following 5. Lemma. Let K.K^.K^ C C be compact, simply connected sets. K =

i in

A

t o show t h i s we may r e s t r i c t

contrast

ourselves

the

the

the s i n g u l a r i t y

of

conditions

quantities

functionals

about

no a n a l o g u e o f

c o n n e c t e d components

the

singularity

analytic

the s i t u a t i o n

are

that

convex.

sequence

a

the sequence of

which g e n e r a l i z e s

1

reader. if

in

functions general,

left

U are

function

associated

A

is

^f-iS^i'(A^)

coefficients

-spaces,

quantities«

only with

the

More

such

preci-

bounded

t o t h e unbounded o n e s . to

in

without

In

order

bounded

components. 9.

Proposition.

dense

in ^

Proof: Let

Let

tffe C 0 ( R + ) ,

= 1 near

bounded c o m p o n e n t s . Then

l,and

a,,

each u

t ^ c t ' ) ! !

^ • 27T

set

«

^

? c ( z )

(«fcUf(Z)

be w i t h o u t

t*

we

have

\

«fc(z-V»)u(w)dw n - i j

ri

T

compact this

is part

compact

of

occurring

a s t r i p 77

part

it

In

cp(t

),

< jy i t

with

0 < c »

?(f)

i p c ( z - w ) u ( w ) d w (:

r

t h e semi-norms of

Re z

coincides

=

^ t , ^ . !

^ 1

a contour

[u(K»J - U(z)] dw

r

| ] .

admits a decomposition u

where u M f

I

/

+

= ^

, vfc 7

^

, and

hence

} ) t

for all k t-Z + and any

/

tV*

L 2 (IR + )

f, t- 0, The decomposition of u follows oy arguments

of solving a Cousin problem, similarly as in che proof oT Lemma 5. Then v^(t) = where

£

f M'(eii)'

of u . In contrast

^

to the

unique, since in general

160

co(t)

+

r y O

,

represents an asymptotic

theory here the choice of ¿^ ft

quantity

is not

Now we are in a position to extend the calculus of 2.1.4« to the spaces ^

,A

» It is clear that the abstract description of Green o p e r a -

tors a p p l i e s , we denote by

the class of all op^ib) L

\ \ 2

acting on L ( R + ) or

2

N

M

i

l2(r

+

}

®x A

]

.

The equality -

^

®

r

established in 1.2.2. Proposition 13 reflects a special property of the iPp

spaces. The proof is based on the fact that tor each fixed weight is only a finite-dimensional modification of

spaces, A ۥ A

, we have

ifQ. For the

/ © r . nore precisely tho proper

inclusion

«V

*

- V

r V

r

A

r

Assume that

^ , (xL ^ „ = ^ (x) „ as sets tor A r J "A1 ^ A^ A A ^^ the topologies have to coincide and each norm on

$ 1 ® A In other words,for given piece-wise smooth curves Pj

j , f"^ .

Fig. 3

„ Ij

,,

_ 'if1' 2

) such that \ )

(r I 5 G

C C \

Iu(zi)v(z2)p

0Utsi
/ ( C j )

side

\ C

K

1

( l + f t l

-

1

of

give3 2

Z

the

}

1

2

•

inequality

the (

1 +

^

X

in

desired

I^|)

question

contradiction I d z J I

2 I < Ì

we g e t

the

esti-

\ — > oa

for

dz2|

Ti

v

L

^

M

*

(

i

.

i

)

!

2

)

^

!

K

I

~2

^

J

v

^

M

l

^

5"

|dw2| *

x

2

J

ft

-

t

f

M

C

M

^

I

w

J

I

^

2

10.

Définit ion.

ing

properties

(i)

Zc

(ii)

| z & Z

(iii)

for

is

:

i

any

obvious

Définit ion. 2 7T

9

is

of

all

closed

d Z ,

Z

1 x

- Z

Denote

Proposition

1 Oli)

all

Z tZ.

12.

Proposition.

are

equivalent.

proof of

arbitrary

is

(Z)

Qy

piecewise

smooth

curve.

Z f Z

can Z

by

^

be w r i t t e n ,

tha

X

ZCC

)

2

!

with

!

^

K

i

the

follow-

i

=

Ç2(rlR defined

in

the

of

In

spaces.

For In

the

nuclear

=

each

definition

Frêchet

tyf)

Z (rZ

a

+ Cjff}

is

2.4.1.

2

for

Z

correct.

since

U-,

tha

Qtil^)

= O f ) ^ independent

and of

In

.

for

space

wo h a v e

particular

is

space,

= lim

associated

in

1.2.

space

'.Vrita

2.1.3.was

i

=

the

of

1.2.

inductive with

the

view

each

limit Z f Z .

topologies

choice

of

the

Z.

based

on

Proposition Z ^ Z ,

a

that

as

form

Z

shows

is

Fréchet

decomposition

all

points

• Z2.

1

for

2 + Z* = Z.

12 b e l o w

nuclear

a

set

compact

of

every

Proposition

proof

sat

is

the

that

1 Z ,

The

tho

Re z =

y ,

11.

over

¡

= Z

Z

are

by Z

Denote

Definition It

•

?

i

tha 6 are

decomposition repeated with

= l,....Pi

Z

: =

i=l

principle. obvious Z

1

we

The

arguments

modifications. have

of

tho

For

163

z -

Qtflz±

.

i-1 Now consider Mellin operators with symbols in omit the assumption that Z some

fcC.

Re

Z

d

2 e Z. First we i"* |.

but only suppose that

f) Z • (3 for

• 0 . In a second step the latter condition will also

be dropped* The second step is based on the first one by applying the decomposition principle in the sense of Definition 11

and Proposition

12. Let h ( r Q Z

t- Z.

Pj 0

Z - 0, Re fl - 0 . Fix some

¿T-*

( A +

z

}

A^A.

Let Z 4 Z be arbitrary and h (r ^ ^ 0 + Z1'1 .

h±fl

r j O T

4

z 1 - 0 - j». r

"

Z1'1 f

"here Z x

2

H T " S i z1'1 - 9

Z

.

5

i hen «"If-, Vi lfi tot J op h (T J h.)t J ci = O t

S0 ° o p M ( T °h -1 op (i ii

t according to the translation ro

Let hj * hj

o)t

S,

l0

+

' fj. , ;t c2> mod j*i

principle.

+ h^ ^ be trie corresponding decomposition of h^. Then

Q

¿ q

h

j,o

c~>

q

(v,

- \ h , « V" > k,o

j=l

k=l

p

h, ^ - V " k,l >

h. „ o,l

k«l

ana Doth sides ot the latter equation are holomorpnic near Tnus ^-Mi-

op M (T S o f

cOop'^W^iO

and

A/vf

and

is the formal L 2 adjoint, then ) - T~*

«""'V)*".

Now let us calculate compositions* First assume h P j fl Z »

^

n

fl W • fi and consider M. - ci o p ^ h ) o

ot 6-

Re U = 0 . Set for 3

X't

t

tOopH(n)0.

X ۥ (E, t > 0

»|z f t ¡ § - £
0 sufficiently s m a l l . Then Z = Z sition argument yields h » h

Thus W - iO o p ^ - ^ T - S i j

. h 2 ) i 0 mod ^ . Q

.

T h i s p r o p o s i t i o n shows t h a t t h e c o m p o s i t i o n l a w f o r c o n o r m a l s y m b o l s from 1 . 3 . 2 . r e m a i n s v a l i d i n t h e p r e s e n t more g e n e r a l

168

case.

The f i n a l sists

of

c l a s s of M e l l i n o p e r a t o r s w i t h conormal symbols in infinite

—•+«>

for j

sums o f M e l l i n o p e r a t o r s o f •

l e m s we o n l y n e e d assume t h a t

(cf. OC^ •

t h e numbers

opj^ , a r e r e a l and

] (• 2 + ,

In the applications

t o mixed

prob-

F o r t h e same r e a s o n s a s i n 1 . 3 . 1 .

we

in the d e f i n i t i o n

of

satisfy

, j

A s shown i n t h e p r o o f

of

-

tf^

> oo

Proposition

g i v s n Z-' t- Z ,

j (• 2 + ,

as j —

f

o

r

1 4 we may r e s t r i c t

o f Z G- Z i n t o a sum o f

decompositions

con-

c o n o r m a l o r d e r « j , R e of^

j f ^ j , which are i n v o l v e d

—+

Tf^

arbitrary

j,

1.3.1.).

Qff}2

two s u b s s t 3

where Z °

each i ourselves

: Z » Z°

satisfies

.

+ zi

to

For

i ) Z° = 0 ,

we

con-

s' s i d e r an a r b i t r a r y j tf ^ l l l j - 1

rea

j

s e q u e n c e h^ (- ( f { ) j .

-"- n o n - n e g a t i v e "

Y j . i — •

Choose two sequences {

^

numbers s u c h t h a t J - Tfj i °

V i . i — " *

•«o , i a

3

a s

•

1

1.2,

0,1.

-y. F o r g i v e n d e c o m p o s i t i o n s Z ^ » 2? P o t

j

,J

- ZJ

« fl we

1 1

obtai

* Z^'1,

0PM(T*3.°

O P M ^ ' defines a continuous

j-

operators singularity

>

J.°

j ( 1

1

)t

Yi

i

f

T " * ' ( A + (Z-*

4 »oo J ^opj^hjjco

for

j —

,

, 0

)

• ^ •

i « 0,1,

(z

3 , 1

),*

only a f i n i t e

i n t h e sequence c o n t r i b u t e t o the 1 1 — t y p e i n e a c h f i x e d s t r i p j • c < (!o z < j . i h e

( CO o p ^ O V j j a ) *

define

H

0 z ^ a y , + h . « » h . .and a.i J'

T

1 h.

operator

Vp A Since

ft

,i

continuous

-

op

+ co t

opM(r

M

(/j'° J'

1

h

J > 0

h

j ( 1

)

*

0«

A

A

number

• of

resulting adjoints

tJ t

e

+ J '

1

^

operators

169

tf Since

^

—

-P

> oo

as j —-> oe , i = 0,1 , the resulting singularity in

1

1

each strip j - c < R 0 z < j i a tors ( 60 o p ^ ( h j ) c à ) *

affected only by a finite number of opera-

in the sequence« The following result can be

proved in the same way as 1.3.1. Proposition 15. 19. Proposition. Let h..é ," P -12 0 J Ofift H . Z t Z"" rary sequence. If op'^Cn^) is defined with

° 9> 3 *

an

+

o as above, opj^

A

arbitOP^'

one can choose constants c^ > 0 such that

M < -

£0(c^t) o p ^ t h j )

«(c^t)

J»o converges in

, $

,

,

,

n

) for all & i à

t

and rn converges in

(J

*

= ^ ^(c^Mopj^ih..)) 3=o f (z

}

for all A t A . Another choice of 20. Definition. Denote by tha

„

§Q A

j

¿C- 2 (1R + ))

GÙ(c..t)

form AN, +

M

+

tà G

t

( ij.o)

+

(z

}

< SJfl)i

and c^ preserves

"e

, where

s

Pace ç

)

M V modulo ^ q «

operators in an

° AW is a Meilin opera-

tor of a form which occurs in Proposition 17. The function S ¡ ^ ( W * ép(z)

s- h j (z)

is called conormal symbol of order - j . Note that the sequence of conormal symbols is uniquely determined by the operation and does not depend on

In

fact, the conclusions

of 1.3.1. Theorem 13 can be applied with obvious modifications* A s in 1.3.1. Corollary 16 we have the exact sequence

170

where

denotes the i n d u c t i v e

- P .

limit

over a l l t y f ) 2 o ,

Z°tz,

["*, f ) Z °

'

2

I n t h e same way a s 1 . 3 . 2 . 2 and P r o p o s i t i o n

P r o p o s i t i o n 3 we g e t

Proposition

16

2 1 . Theorem. % M

from 2 . 2 . 1 .

*

algebra,

i s a two-sided i d e a l

in

a n d

k+l-j Now we d e f i n e an a l g e b r a the parameter Q

(R ) d

^(R^)

i| 6- IR6* ( c f .

we d e n o t e t h e s e t

of

families

of o p e r a t o r s d e p e n d i n g

the analogous d e f i n i t i o n s of a l l

families opQ(b)(i|)

b ( t . s . - y ) (r C ° ° ( I R d . ^

(• IR C| , I *n | h/2 X u(Xt).

Choose a f u n c t i o n

-

» c o n s t . , where t h e c o n s t a n t

£ €- c " ^ ^ ) ,

may depend on

£ ? 0, f

/•(t) ^ and

-1

^opG(b)(^)

1

?

< I t

for

t >

,

for

t > 1

2

,

set

C j > 0 some c o n s t a n t D e n o t e by Cjj^. . . .

d e p e n d i n g on j

t h e s p a c e of homogeneous p o l y n o m i a l s o f ) with

coefficients

ded i n t h e f o l l o w i n g

way.

22. Proposition.

h^ 6 QV}

sequence,

(2+,

Let

in

T f l ^ . Then P r o p o s i t i o n

, Z^ t Z ,

j

P ^ 0 Z ° = fS, and d e f i n e o p ^ ( h ^ ) ( i j

be a n

order k

in

19 may be e x t e n -

arbitrary

) a s a b o v e . Then

there

2 are constants

c^ •> 0 s u c h

that

eo

171

))

converges in (ZJ,0)W

^(Z

3 , 1

)^

for

a 1 1

A

J

*^ '

X

^ u n i f o r m l y in '»|é-KCClR d . and 00

J

"°

^

for

converges in

+

^

a 1 1

ù 0 , w i t h the Hilbert

s X structures induced by 3[ ' (|R+) and

H

s

(|R+), respectively.

c < d < «> : = t3-1[0.d] + H8 [ c ,

s.^R«

endowed with the norm of the sum.

In view of

178

the set

,

space

*8'*[c.d] =

Hs[c.d] =

*9'*[0.d]nH8[c..]

we get an equivalent definition for another choice of c,d, in other words

is independent of c,d. Remark that C ^ (1R+) is dense for a l l s.tfirlR. Moreover, < p & s

in

(|R+ )cj£ s

(|R+ ) for every

ij)tc" ( R + ) . This follows easily from the corresponding for

property

3

2C '*(IR + ). 5f?s'^[0.d]

Since

a Hilbert

, Hs[c,«o"]are Hilbert spaces, we also can

introduce

space structure in the sum.

It w i l l be fixed then once and for all. So J D O action on IR. For S = 1 we simply w r i t e Remember

op(l+).

that

o p d + S ) ( S ) : h£(R+)

H^"Re8(R+)

for a l l rtIR, s f t, S 6 IR , and o p ( l ~ s ) = o p ( l ® ) - 1 . H e r e H£(IR + ) = { u f e H r ( l R )

: supp u £ R + } .

179

A s u s u a l l y L 2 (|R + ) a n d H°(IR + ) w i l l

: L 2 (IR + ) — ^

op(l®)(g) The notation o p ^

Moreover, r e c a l l

= r+op e+,

r+

the

the o p e r a t o r of e x t e n s i o n

by

zero.

fixed p é ID a n d

stants c ^ , c 2 > 0 such

for

(1)

function

4.4).

For e v e r y

Re p P

cjimzl

e

i.e.

the f o l l o w i n g a s y m p t o t i c b e h a v i o u r of the V

1 , Section

2. Lemma.

tolR+,

+

identified,

h;3(R+).

w i l l be u s e d as u s u a l l y o p ^

restriction operator

(cf.T

be c a n o n i c a l l y

cf R+l

there are

con-

that

, n, < \ J Ç ^ £ l \
c , w h e r e c is so l a r g e that t h e r e a r e

p o l e s of T ( z + p )

in

S j S Re z < b 2 ,

no

|lm z | 2 c. CO

D e f i n e the b i n o m i a l Then we have

the

coefficients

the

ot-k.

, but

dense

is

then i t

obvious, follows

since in

in

Thus u>jtX o p M ( T - A h ) = ca{tX

t"X

-

opM(h)}q>

opM(T"Xh1)t-X-

opM(h1)]cJ

-185

c a n be c a l c u l a t e d 5.

by t h e same a r g u m e n t s a s

P r o p o s i t i o n . caop°(l°| ) ( S ) c ^ I 1

continuous

Proof;

i n the o p e r a t o r

on t h e

converges Let opJ(l;)(S)

The

in

9

for

in general.

. Since

for

=0.

converges C * (IR + )

is

From

Û

' * " r(lR+ ))

for a l l

s.^eiR.-^f

+ (l-c^)op

+ 2+.

Then

(1* ) (S ) ( l - i ^ )

op ( i * - ) ( 8 ) e c°°(ir + xir + ) of o p ^ ( l " ) ( S )

in classes with weights

t

: #

S , Î

(R

+

)

Ûf~

is

unique

(cf. S e c t i o n 2 . 3 . 3

induces a continuous

modulo

below).

operator

r

+

N e x t we want t o e s t a b l i s h H S ( I R + ) = r + H s ( I R ) , and supp u bounded, i s

184

the a s s e r t i o n

£ . Moreover, the d e f i n i t i o n

S^Re

' " r ( I R + ))

f u n c t i o n s , cd^tù = id^.U^ti^ = l ù ^ .

-

s.yeiR,

u(zK

r

3 obviously

which

op*(l")(S)

(i)

s _ r

: -C3op* 0 with constants c, a depending on u.

This is an immediate generalization of 1.1. Theorem 5. It that the limit of the integrals in (ii)

for i - » 0 is also

A n a n a l o g o u s characterization can be given for =

with bounded support, y i f , ^ ( v i n T i 2 ~ of

(i), that

u

poles, and in (ii)

finite.

ue$s,3(lR+)

+ ^

"here we assume,

(IR+)

instead

3

is meromorphic in Re z > ^ - s =

follows

with the corresponding

Note that for s < 0 the latter

restrictions are not relevant and for s > 0 , only a finite number of poles has to be counted. Let s e IR and %(s) Denote by g

: = max [ j ۥ 2 +

: j -c |s| - | ]• .

8 f 0, the linear span of the derivatives of orders

M,(s) of the Dirac-measure at the origin

linear span of the functions 8. Theorem.

H S (IR + ) =

Let

{t-'io(t)

and

0 j 8 , s 2 0, the

: j = 0,...,

M.(s)] .

aelR, then J^S,S(IR+)

for s > O,

t^- 3 , 8 (IR + ) + S 5 8

for s i

3 £ S ' S ( P + ) + iT® ,8,3. *

)

O, s f | mod

for 3 > 0, s * |

mod 2

for s i 0.

The proof w i l l be given in several steps. First show the 9. Proposition, ll/e have for ^H®(|R + )

2

following.

a > 0 S

(IR + ).

with equivalence of the topologies, U a cut-off

function.

185

Proof : First w e show that there is a continuous 3

ciH®(IR+

S

' (IR + ). In view of the isomorphism

S

u e H Q ( l R + ) there is a unique v f L

2

embedding (1) for every

—g (IR+) with u = op(l + )v. Let supp u

be bounded , tC>u=u , and choose a sequence { v j } 2

Vj — • v

—

in L. (1R+). Then u^. = o p ( l +

tion 3 we know that car+oP(i;s)

Vj

(IR+ ) with

8

8

) v^ —*• u in H Q (l£ + ). From

Proposi-

= « o p ° ( i ; s ) vj

for all j and that the right-hand side converges in

8

limit u'. Thus u ' = r + u and hence C 0 H 8 ( R + ) |

for a l l e fetR.

Ct0^8,8(R+)

()R + ) to some

In particular for s i 0 w e get riop^i;8)

v = w °p°(i;a) v

(6)

for all ve-L 2 (IR + ). Now let u . . £ H 8 ( R + ) , u^ —*• u in H 8 (IR + ). Then = op(l8) u j - » v

Vj

in L 2 (IR + ) and

From (6) applied to Vj

it

Oop°(i;8)

Vj

u • in

' S

follows u'=u, u^ —»• u in

S

(R+).

(/R + ).

For the reverse direction we apply Proposition 5 and obtain the continuity

tOcOi

3

où. For

QR + ) w e have

cdu =. cd r + o p ( l ~ 8 ) o p ( l 8 ) e+c»u = ti> r + o p ( l ~ 8 ) e + +

(1-ttj) r + o p ( l 8 ) e+6)u

W r + o p ( l ~ 8 ) e+(Jj o p ( l 8 ) e+£^u

By Proposition 5

it

follows

ci r + o p ( l ~ 8 ) e * ^

op(l8) e+«u

= Or+op(i;S) e+«t op8(l8)«u Now choose a sequence 8

ft '

8

of

. functions

(IR+ ). Then f

186

.

J

: a

opjt1')«»

,Qvj — •

u,

u(

in

L2(iR+).

Thus

u':*

ci r + o p ( i ; 8 )

e+fj

u 1 t H® ( K + ) .

Moreover, uj

: - U>r*

converges 0,

(7)

H®(R+)

=• X

3

'

®(IR+)

for

80,

Proposition. 3

Proof:

(ir+)

Let

us

We h a v e

for

s>0,

first

that

Qu =

ti> = tOcOi* Then,in

op^(l~8) i^op

^

Moreover,

(IR+)©rS],

u t H (IR+),

c^u = u .

: L2(IR+) — * H S ( I R + )

s

(l~ )v.

Let

the

Remark

Propositions =

10 a n d

that

t O o p ° ( l ~ ) c ~ 0

in

c ^ o p ^ l ^ 8 ) ( 1 - i) cOj^v .

c O o p G ( b ) cO.v

s

inclusion.

cut-off

13 we

+

of

vf-L2(IR+)

we f i n d a

(IR+) w i t h v = l i m v^

u o p ° ( l ~ s ) c^v..

14 shows

Then, i n v i r t u e

be a n o t h e r oe

Choose a sequence Vj t CQ

v i e w of

j.

S

inclusion

s Let

0> o p ^ ( l ~ s ) U ^ j for a l l

1 a ^ mod 2 4>H (IR + ) =

s *

show t h e c o n t i n u o u s

1 s £ ^ mod 2 .

isomorphism

continuous

+

c - i H S ( I R + ) C_» ( ¿ { t f S ' s>0,

to a

(IR+). }

= {(

>tJ:

Re

r(S)^

= { ( P j . - ; , ) ^ :

Re

= {(P

R«

P j

l

..

(

3

i

+

the

operations

by

KS)>

Obviously, v

z < §

P j

- & ] .

< J - S 3

.

P ^ f - S ] . Pj =

+

f o r

e a c h

a n d

v

For \ t

w o

|(Pj • >

+

4(1R+)' w

•

where ={(0u + (l-CO)v

: u(^(t!t),vtHS(K+)J

.

Then # f 8 C R + ) = ¿¿t{r)y J ^ f V

:

+

-

CR+) •

^("V

¿(F(y)y). +

=

(10)

- ¿ ( r i ^ y )•

It is clear that

^ / ( « V TP

T-^

(5)

'

X f- IR. Thus,these spaces could be transformed

to some fixed weight.

For notational convenience we also write

Remark that spaces

can be written as a projective limit of Hilbert j

The choice of such a scale is not

and follows from corresponding

scales for

and

references below the scalar products and norms belonging will be denoted by

194

canonical For

to

(IR+ ) ^^ ^

(...)

s

and

I.I

,

(7)

respectively. The notation tf^ will also be used for the 3C - spaces,

The description in the Mellin image will always be given for

(Ju,

whereas the influence of M(l-c«i)u will be neglected at all. This is i necessary here, since M(l- c«^)u is holomorphic only in Rs z ^ 2 1 (e.g. when u ( L ) but Mt^u is of interest in Re z < ^ • where may be negative, and then we would have an overlapping

strip.

Let us also introduce the following small modification of our spaces (5)

that only contain an asymptotic information in the open half 1 space Re z > ^ . First observe that

for 8 > s', ^tt'

with continuous embedding.

Then we can define the projective limit tf-*-0**

: = 11.

)•

¥'< vf 11 0

Note that

= { Urn One may ask whether the spaces

ft8'*"!

+

with discrete singularity typos are

invariant under certain natural operations such as multiplication by functions in

or coordinate difisomorphisms. In this connection

it is natural to introduce the condition that ao j=0

T

c v

.

Observe that the Taylor singularity

(o)

=

|(-j.c)j j t-2

is

°f that

type. 3. Proposition. The operator of multiplication by induces a continuous operator

(=

fo

195

aj = ^

and t h e same w i t h o u t

^ . More g e n e r a l l y , f o r

i f fc

j=o

*

JT^/CCjRe z < o ]

and

( 2 2 ) and 1 . 2 . 3 . Proof:

For

we have an a n a l o g o u s

Lemma

have

and

if:

= H*

for

— s > -

Theorem 8 (remember

Moreover, result

tj> : ^

follows

applies

for

case a r e

u

left

is

by t a k i n g

to the

: IR + —•>• R +

constant

(cf.1.2.2.

g (t)

equals

obvious,

t h e sum,

cp(

that

t h e case w i t h o u t

Now l e t

if

= H9 for

o 2.3.1.

with

1).

e f t ^ ( I R + ) we f i r s t

=

result

in

particular

Moreover, l e t

that

induces a

to(0)

ot(r) = r

difisomorphism

= 0 ) and

for

let

r > c

with

some

c > 0.

Proposition.

L e t «x> ba a s m e n t i o n e d ,

then

n,

induces

continuous

operators otf

oo at = 1

T"-'v

•

8

s

• if £

U3*

> yi y •

R

ar| d

t h e same w i t h o u t

^ . Moreover, u*"

operator

*

">t Proof.

Let

represented %(r)A

follows

t

=

—

.

(10)

o t ( r ) and X ( - C . T h e n t h e s i n g u l a r

= ( I ! 1 j=0

c

isarbitrarily

from t h e T a y l o r

= a^

^

i n new c o o r d i n a t e s

U > « r ) )

where N £

196

(9)

'

j=0

induces a continuous

W(r)

ML*.

—

•

+ a2r2 +

J

t*

ui { t )

by

rX+J

+ r

N +

L(r)) (-Mr)

1

given,(^(r)

= ci(%(r))

e x p a n s i o n of «0 a t

+ a ^ r "

function

- 1

+

a

N

(r)

r N

r = 0

,

,

(11) cfNfcc"(IR+).

This

is

= a r(l 1 where a 1 f 0 , a ^ C W

-

.

+

(IR + )

J

X

(r)

é- C (1R+ ) .

( i XX

+

( l

! a r ax

singular

of

. . .

+

of

, x.*'g^~ 9 (r)

= r^~s + o ( r ^ " 9 )

î?o0R+)

g

shows t h a t proved.

r-s

also J i 3 ' ^

(r)r

N

) ,

respect

=

of

to

logarithmic

\

yields

terms.

the

Then

(10)

{ u fc ¿f(IR) : supp u c iR+

f u n c t i o n s under

sflR. for

N

r V

(11) w i t h

2 . 3 . 1 . Theorem 8 and t h e i n v a r i a n c e for a l l

g_

b

of

the s i n g u l a r

*" : ^ ¿ S ' S

B

+

functions with

from t h e i n v a r i a n c e

and t h e b e h a v i o u r

rN)

• bx r

Differentiation

t r a n s f o r m a t i o n of follows

r

r +...+ *£L

and

r * 1

= aJ b N

a1

u* . I n view

the Sobolev

As proved

of

s p a c e s we g e t

above

r n e a r 0 . Then t h e

isomorphism

„

8

is

invariant

under

. Taking the sum,(9)

is

0

5. D e f i n i t i o n . for which

A

c

D e n o t e by

{Re

z < |

, ) f t IR,

-tf] . S e t

the set

A =

Then we h a v e t h e n o t i o n of s p a c e s

^

W X

with

of a l l

elements

^ t

Z

A (if)continuous

asymptotics,

namely ^(IR+) A 6 A (if) • and i t

: = tX

iTlA(|R+)

. The r i g h t - h a n d s i d e

c a n be p r o v e d a g a i n t h a t

This leads

to the

it

is

is

then

well-defined

independent

of

X

.

spaces

and "A " + ' A t A

. I n

spaces of

the

c a s e of

o" + '

^

continuous s i n g u l a r i t y

t y p e s we a l s o

admit

sort

197

0 so s m a l l

that

z

p^ 4

{l

-Pjli£j

well

defined

for

define

=

\ ^jk^

0 i k S m. ,

l S j i l

by V

"

:

= 2¥T I

hjk(z)Mu(z)dz. J

Then

is

on 2.

200

obviously or

Example.

as

on e v e r y Let

y

=

desired. space ¡ ^

Note where Q

t h a t 4- i s f J ^ 1

_1.

. Then

also

4-jo

u

= (jt) d " ( ^ I t - o

is a trace operator of positive type on

• J=°

^

¡P. (IR ) in the sense of To +

boundary symbols in Boutet de Monvel's algebra. Thus the above 4extends the notion of such trace operators for the general conormal asymptot ics. 3. Definition.

An operator

£•({)-«(, q(Q. P) is called a Green operator

of discrete singularity type if and ft in

for

"£(»() ® ^

ojQ has a kernel in ^

certain

t y

©T J

Denote by

O^ g (Q.P) the class of all Green operators with given 0=(X,0|), (

U^.^OjVary with

)

iVe also write

C L g ( X . V = ct G ((X.O), (j.O)) . It can easily be verified that this definition is correct, i.e. independent of the representation of oj_ as a matrix

(3).

It is obvious that ^t(l)

implies

-¿g(Q,0).

oj2

fc(l')

'and

^

the

-¿G(o_,P) same

without

C1)-

All essential results hold both for left Green and the proper Green operators. So,from time to time we give the formulations only for one case, where the other one is then left to the reader. It is convenient also to use Green operators that have finite smoothness both with respect to the interior and conormal orders. To thi9 end we first observe that every ej£(1)-«¿q(Q> p ) induces continuous operators $

+ ify _^3-()f-\)

ft** for all k

^

IR+ , j ( 2 + , where

+

k,A+k-0

( 4 )

+k

(IR+).

(5)

tJ| depends on ^ .

Denote by

201

(l)-^G(Q.P)(~k) the set all

of a l l

3t2+

as i n

(1)

(6)

for which

(4),

(5) are continuous

(l)-^G(Q,P)(-k>

(7)

n (1)-^ (0,P)(-k) k tz+ ^

(3)

for

.

Then (l)-^G(Q.P)(~k_1)

C

and (1)-^

°

(0,P)

=

Moreover, € (l)-^G(Q.Q)("k) implies d,

G

g2

&

( 1 )

( 0 , P ) d e n o t e s

which &

-^G

, oj 2 £ ( l ) - ^

( Q , P ) (

"

k )

the subclass

has a k e r n e l i n

G

(o,P)("

k )

' of a l l

oj. 6 ( 1

¿(op©

+

( 0 , P ) '

G

'

f o r

for /l0li

SOma

TP

and

^ G (0,P)

HS-^"X)+I w h e r e the inductive

(1)-jLG(Q,P)(-k)(«0).

= 11m

limit

is taken over a l l

«Of

and in the

same w a y (1)-«(, G (Q.P) =

lim_(l)-^G(O,P)(«0).

Now we can proceed in a similar w a y for the class w i t h o u t we have to introduce the subspace -

£

:

f

o

10 ) ¿G(Q,P)

= lim

and so on. For the are left to the

¿„(Q,P)(-k).

"^.-classes we can do the same. The obvious details

reader.

2.3.4. M e l l l n operators Let us now extend the calculus of Mellin oparators of Section 1.3 with respect to arbitrary Remember that ^

conormal

orders,

denotes the set of all countable subsets £ of 1. ¡f - k i X ,S i $,

ca1t~tM'kopJi(h)6a2 - % 6

210

h

^^

t-r+kopJ(h) «2

«¿Gii-r'i)(-k)

+

«¿ r is-r-» , ) C " k )

Then

t^i • i=»1.2.

for arbitrary cut-off functions

rso

fO

The flat term vanishes w h e n 4. Definition. A n

operator (2)

is said to have a (left-) Mellin expansion near t=0 with the conormal symbols e^-^et) if for certain

tf

= a J (r Qtfl^. , J

t R ,£ -j5 ^

f _u (O^Ot- t r g

< $ , j t 2 + , and every N £ Z +

tfj •? H taopM(aj)

arbitrary cut-off

(3)

functions.

We talk about a Mellin expansion if (3) holds without Note that the definition implies 5. Remark. symbols

"¡f0= ^

and hence

(1) on the

aQ t

right.

•

Let ft, have a left-Mellin expansion. Then the conormal (OT,) aro uniquely determined, j 6 2 + , They are also unique

for j=0,...,N-l

if the expansion only holds with fixed N. This is a

consequence of 1.3.1. Theorem 11. 6. Proposition. Let OLbe an operator satisfying 00 induces continuous

operators

01 : # J ( R + > — for every s t IR,

Definition 4. Then

a ^ O V

11p (y) with sorae Ojt y

(4) depending on ^ and SI.

If 6i is an operator satisfying Definition 4 we can define an extension to

rijfj'p

for arbitrary 'pt'p

by octfy , Then ot defines continuous

operators

and also continuous operators in the sense of

(4).

211

7. Definition« and denote

Let

by o i

tf

M + G

s p a c e of

all (5)

T

which

a-l^f

L

Z 1 f N o p j , ' (a ) 1 C 5 2 t 4 , ( Q , P ) ( - k > J j ==o0 "

for c e r t a i n If

, oj (•? (£-|. ^ -~ j J i£ U j a£ If 'ft •. iO ^^ -. OO gg aa rr bb ii tt rr aa rr yy ccuutt--ooffff

"

OL s a t i s f i e s

class

(6) w i t h (

(l)-lM+G(Q,P) -

i_l^G(U,P)("k) -^¿h+g(Q.P)(_I

"0

Ve

( Q

'

P ) (

"

k )

'

'.Ve have by definition

w h e r e the elements If w e replace o^ by

act on

by OlJTy .

the assertion of Theorem 10 remains v a l i d .

Moreover, w e have an immediate analogue of Proposition 11, A s for operators with discrete asymptotics for every

. P) (""k)

we can recover the conormal symbols, and then r j

1

(8)

is exact. For

216

JQm+q

we

have an immediate analogue of Theorem 9. The precise

formulation

is

obvious

and

left

In

( Q . P ) we h a v e a n a t u r a l

defined

by a s i m i l a r

Now l e t

u s make two

both 15.

for

procedure remarks

the d i s c r e t e

Remark.

only

X

exception

that

provided

fits

a discrete

R e m a r k . The c a l c u l u s

can

be g e n e r a l i z e d

to

respectively. by t h e

flat

section

subspaces

Note

of

the

specific 2],[l

also

transform

this

y

versions

every

that

the

sequence

conormal

in

of

the d i s c r e t e

every

^

of

Green o p e r a t o r s could but

We s h a l l

in

and

then has

to

between

our a p p l i c a t i o n s

deal

every J

1 + G (¡f-^-5")

spaces

distinguish

an

aQ.

and

M+G(if-f =

k+p=j

Repeating the conclusions in the proof of 1.4,1. Proposition 3 we get (with the notations introduced in the beginning of 2.3.3.) 1. proposition. Kor each N £ Z + , N>|t+1 and such that for each

S>0

there exists

^

N

X , 0 < X < N - p - 1 , X 4 ^ mod 2,

where P Q ( X ) = ( X . r ( \ ) ^ ) , such that for all u ۥ C Q

(R + )

N-l «¿op (a)u = ^ { A ^ ( S ) o p M ( g + ) + A - ( S ) o p M ( g - ) ] o p J=0

218

(l^-^S.xJJu+^ (2)

ju.

Proof«

Set

V s ^

= 5>oitt

V s

** •

( T )

where N-1 aN/S(t)

= a(-c) - ^ { A ] ( S ) 6

+

(X)

+ A-(8)e-(T)}

l^C«.*).

j=0 Since

|aN

we have

-1 + op^(0 ) = opM(g~)

I n v i e w of with

c

s

k e r n e l 6)b N j ( t - s ) .

(2)

Then,for

bN

6 H^P""1^).

g

f o l l o w s , where X under

g

is

the

operator

consideration

00 = |Jb K

i

cl u 1

P

0

( X )

N i

|

o

^P

0

(X)

L

Tho mapping p r o p e r t y

for

$

follows

cj N g h a s

b^ j ( s - t ) o i a s k e r n e l .

Note t h a t

t h e mapping p r o p e r t i e s

3N.S

j(t-s)u(s)ds

6

2. Definition.

The

t h e same w a y ,

since

0 of

* * 1

f o r

in

$ imply

mod

* * *

0

that N-r-1'

function

: = { a y i z ^ l

+

«¡g-(z+r)j

^ ¿ ¡ f J — j .

(3)

H I t 2+1 is

called

conormal order

t h e c o n o r m a l symbol of

, where

2.3.4. Definition

( 1+j » 0 ) } j ^ In 6 "

case 1

(a)

J«. 6 2 + 1

1).

>L,

to

the

= { (l-j-|«,0) ,(l+j ,0)j

ot>f

+

J

F o r -M-^ Z + we h a v e

,

X ot, f= { ( l - j - p . O ) ,

. a^ = a^

fe

belonging

(*•-!•

Note that (cf.

atSc^(|R)

for

1 = 0,1,...,L-1,

• where = { (1+j ,0) ] + can be r e p l a c e d by

^2

+ since

for

we h a v e 1 < L . I f , in p.. 1

i 8

addition, 8

Poly~

219

nomial then. If p. t 2 M.-1

ffj^ (a)

and

a^ ^ a^

for some

1, the conormal symbol z = 1,2,..., 1-J4.

has poles of second order at the points

Our next objective is to derive the Mellin expansion for higher order IpPO's on IR+. First we consider the case when the weight y

of the

desired domain of definition satisfies ^

>

max ( - f.p).

(4)

Then we know from 2.2.1. that there are continuous extensions of

op for all

— •

ae-r-t-r

s£IR. The Mellin operators

. We get from

.

) . is the Mellin

(2) a finite sum of terms of

the form A~(8)Skeao(Y1(g±)tlt+J+k'r

opM(Ak>._

k=0

'

For given

1t2+

all terms with

term with conormal order pendent of letting

j+k = 1

. >

contribute to the Mellin

-(tf-p+l). Since the left-hand side is inde-

S j we may simplify the calculation of the Mellin

8 — 0 ,

i.e. we take into account

symbols

only the absolute terms of

the polynomials in ^ . We have

AÎ and only

k=0

< > ° = aÎ

occurs

, i.e. j=l. In view of

« f ^ r ^ ô op^gV*1-!*) =

T

r - v

we get ( ° P ^ ( a ) H 2 ) = l a Î 9 + ( z T ) • a^g-(z T )J

^ ( a

It remains to find the singularity type of the Green

) (z).

remainder

terms.

Clearly tJ-rf

(tg)

k

oP;

(

hkiJ

¿ r f - s . ^ - r " ' * - ^ )

k=N and ùop^g1) e

for each

^opM(g

X S f -f+j+N, i

Jt^TJ" k=N

and

\ 4 -g mod 2,

r!

(tS)kopJ(hkfj

therefore

¿CK. 8 , t f . DC^-iJ^). '

0

setting

221

N-l j=0

k=N

w e gat

g1 t

Consider

. ^"(x')

now

N-l

N^l

j=0

k=0 N-l

- cit-r y ^ o p j t a Mr t a ) ) « * 1=0

,

where by d e f i n i t i o n of

a l l M e l l i n s y m b o l s of order >

v a n i s h . C o m m u t i n g on the right

+

op M (g"~) and

+

^ w e get in v i e w

of 1 . 3 . 1 . P r o p o s i t i o n 8 G r e e n o p e r a t o r s w i t h image in The a r i s i n g M e l l i n o p e r a t o r s of c o n o r m a l order of o r d e r

$ p - N h a v e image in

jw. - N

(j + k—

)^

> p-N d i s a p p e a r and

in the second a r g u m e n t . For

t _ X

N o t e tha't for

X'2 0,

5u( j ) = 0, j = 1,2,... ,

°PM(hk,j-r)^

terms

g N £ w e proved that it is a g a i n

the other c o n t r i b u t i o n s w e use that for u £ C ^ (IR+) such that

those

.

F i n a l l y , we e s t i m a t e the s i n g u l a r i t y type of the G r e e n r e m a i n d e r

V

For

f j mod Z a n d

+

= °PM(T"Xhk,j-r)

t C^" (IR+),