A Calculus for Classical Pseudo - Differential Operators with Non -Smooth Symbols Witt, Pseudo-Differential Calculus

Page missing Number 539

240 Math. Nachr. 194 (19911)

the program. Although once the general pseudo - differential calculus is establisl~ncl one has it especially for classical operators, it turns out that as a rule proving resi111,c for non-classical operators is much more involved than for classical ones. For t l w l

non -regular calculus this distinction becomes significant. Therefore, in this paper wtc

develop the pseudo - differential calculus for classical operators first. It permits UR t,o work out in detail the structural elements of the calculus (which are the same as To1 the non-classical calculus) so making the main ideas transparent.

However, in the theory of partial differential equations on several occasions ~ I I V

is forced to leave the range of applicability of classical operator calculus and to i , ih advantage of pseudo - differential technique in its full strength. Therefore it is desiralh to generalize the constructions given below to obtain a calculus for non-classicxl operators in which one is mainly interested in. This is subject of a further paper (tw

Several authors contributed to pseudo - differentid operators with non - regular SYIII

bols under quite different aspects. In order to mention only some topics, the mappiiiy, properties of pseudo - differential operators with non - smooth symbols, the mappiiiy, properties of the adjoint operators and commutator estimates were studied, e. g., in 141, (51, [15], [16], [18] and many other places. Also further elements of pseudo-differentid calculus like compositions and asymptotic expansions were treated, e. g., in (21, [3], 171

An important r61e was played by BONY’S paradifferential calculus, see [3], [17). H ( V P it had been for the first time that a pseudo - differential calculus for operators wiili non -smooth symbols was realized. Moreover, additionally to the complete calculiih a parametrix construction for the elliptic operators was given in full generality. Tlic, remainder terms were characterized by their mapping properties.

In [l], [2], BEALS and REED proposed a calculus for pseudo-differential operatois with non -smooth symbols having coefficients in certain L2 - Sobolev spaces. Thoii exposition is distinguished by the very simple estimates used. In this paper we f i t l l

back on some of the ideas exploited there and develop them further. In the result W I D

obtain a calculus which on the one hand is sufficiently general for applications in noii linear partial differential equations and in which on the other hand the simplicity ol the approach of BEALS and REED is preserved. One main feature is that parametricch to elliptic operators can be constructed within the calculus. Especially this point, is

important in non-linear microlocal analysis and, in [l], [2], it had not been completc.ly realized.

Before we describe the content of the paper in more detail we want to discuss sonic’ of the ideas involved in the constructions. The coefficients of the operators must I ) I , at least continuous. Discontinuities in the coefficients lead with necessity to symbolic levels additionally to the principal one which then, e. g., enter in ellipticity conditionn to provide only one argument. Continuity of coefficients, however, immediately caum the next observation, namely that asymptotic expansions that typically appear i n pseudo - differential theory have to break off after finitely many steps. For instance, iri case of smooth coefficients the symbol c(z, 7) of the composition b ( z , D)u(z , 0) allow3 the asymptotic expansion c(z,q) - &120(l/cr!) (8:b)(zlq) (Dgu)(z ,q) , and variollh differentiations with respect to z arise.

1251).

\Vit.t.. Pseudo - Differential Calculus 24 1

' h e main parts of operators in the calculus of BEALS and REED were of the form

M

j=1

for some M < o, aj E H'(R") for s sufficiently large, and p , E Sm(R" x R"). 'Fhen it is obvious that a general parametrix construction is possible only after an rippropriate completion of expressions of the kind (1.1) is chosen. This immediately loads to tensor products. Our approach is based on the work with completed tensor products on the symbolic level.

In [25], the constructions rely on the weak symbol topology, T, on symbol classes Sm(IR"; E ) introduced in [24], e.g., for E being a FrCchet space. Recall that for the fipace Sy(IR"; E ) we have Sy(IR"; E ) = S?(R")&E. Moreover, a mapping from Sy(Rn; E ) , where E stands for certain coefficient space, into some space of bounded operators between Sobolev spaces realizing concrete pseudo - differential operators is cmtinuous in general only when the latter space carries the strong operator topology. In this paper the considerations are based on the symbol classes ST(R"; E). Here we have Sz(R"; E ) = Sz(IR")&E due to the fact that Sz(Rn) is a nuclear space. For that reason, arguments simplify considerably. So we are now allowed to work with the natural FrCchet topology on the symbol classes, hence deducing boundedness of the operators under consideration in a direct manner. The simplifications give us the possibility to introduce the several elements of the calculus without disturbing about the more difficult topological considerations used in [25] in estimating the remainders.

In the paper we make use of standard notation in the theory of pseudo - differential operators, the reader is referred to classical textbooks dealing with this subject, e. g., [GI, [ll], [21]. In order to be definite, let z E IR" be the space variable, <, q E R" be frequency variables. Introduce the Fourier transform

such that the inverse Fourier transform becomes F;&ii(<) = (2n)-" J e'"CC(<) @. For brevity, in what follows, we forget about the factor (2n)-" and work in Fourier space with a renormalized Lebesgue measure, i.e., d< is the usual Lebesgue measure times ( 2 ~ ) - ~ , and

~c:~ii(<) = e'"< G(<) d<. I ( P P , 5, D,,) -(a = J

Then, with symbols p ( < , 2, q), we associate operators p ( D , 2, D),

(1.2) < - 171 V ) C ( r l ) drl I

where f i (< , <, q) = F Z - , ~ { p ( < , 5,~)). We shall see later on that this definition makes perfectly good sense for our symbol classes under consideration. Intuitively, the operator p ( D , 2, D) is applied to u from the right to the left acting first by differentiation, then by multiplication, and finally by differentiation again. That holds precisely if P(<,z ,v) has product form, i.e., P ( < , ~ , V ) = ~ i ( O a ( z ) p o ( ~ ) , where PO(^), ~ i ( q ) are

242 Math. Nachr. 104 (1998)

certain symbols with constant coefficients, and a(z) is multiplication by some cod ficient. For symbols p(z ,v ) , q((,z) , at least formally, (1.2) becomes the standi~l~~l operator convention in pseudo -differential theory, i. e.,

and q ( 0 , z)u(z) = J e-'(z-v)cq(<, y)u(y) d y e . The plan of the paper is as follows. In Section 2 we discuss classical vector-valuctl

symbols used in the sequel to establish the calculus for classical operators. Then Scia- tion 3 is devoted to the introduction of the several operator classes and the derivatioii of their basic properties. Compositions, adjoints, commutators, and mapping prop erties between Sobolev spaces are treated. Special emphasize is put on the accuratv description of the mapping properties of the remainders. Also in this section BEAM and REED'S estimate is reproduced and the basic technique in estimating the remainders is established. Finally, in Section 4, some further topics are grouped together like the parametrix construction and elliptic regularity, invariance under coordinatc. changes and operators on manifolds, and the equivalence of ellipticity and the Red- holm property on compact manifolds. In the notes at the end we comment on somo additional material.

2. Classical symbols

In this section we discuss several aspects of classical pseudo - differential symbols. Our standpoint is to treat symbols having their coefficients in certain function spaces as vector-valued ones. The classes Sz(R"; E) are defined if E is a Mchet space.

First we are concerned with abstract vector-valued classical symbols. The maiii result, S z ( R " ; E ) = Sz(R")&E, is stated in Proposition 2.4. Then we deal with symbols depending on two covariables which are separately classical in both covariables. In a final subsection, we specify the results previously obtained to symbols having their coefficients in Sobolev spaces. In addition, auxiliary symbol classes arising during subsequent calculations, e. g., symbols depending on three covariables, symbols one coefficient of which belongs to CF(R") and so on, are briefly introduced.

2.1. Abstract vector - valued symbols

In treating classical symbols it is convenient to replace the elliptic symbol (fJr E S;(IR") usually used, where (<) = (1 + 1 ( 1 z ) 1 ' 2 , by another classical symbol which shall again be denoted by (0'. We choose { I-) (0 to be a smoothed noriii function, i.e., (0 is positive on R" satisfying (0 = 2 C, and some constant C > 0. The symbol estimates are not effected by this substitute. Wheii dealing with classical symbols we also need 0 -excision functions $, i. e., functions 1c, E Cm(IR") satisfying $([) = 0 for 5 C1, $(() = 1 for 1" 2 CZ, and sorno constants CI, CZ, where CZ > C1 > 0.

for all ( E R",

Witt , Pseudo - Differential Calculus 243

Let E be a FrCchet space with fundamental semi-norm system { [ I JII}I~N. &call I h t , a fundamental semi - norm system for S"(R"; E ) is given by

for all Q E IN", 1 E IN. S"(R";E) equipped with the resulting locally convex topology is a F'rCchet space. The space S-"(R"; E ) = nmER Sm(R"; E ) of E-valued Hymbols of order --oo is the Schwartz space S(R";E), and we have S-"(R";E) =

If E is a FrCchet space, then it is possible to form asymptotic sums in S"(R"; E ) . S-M (R")& E.

Proposition 2.1. Let E be Q R t c h e t space. Let {mj}jEm c R be a sequence aatisfying m, --t -00 as j + 00. Suppose further that we are given symbols aj E Sm> (R"; E ) , j = 0 ,1 ,2 , . . . . Then there is a symbol Q E S"(R"; E) , m = r n a x j E N mi, such that for every r E IR there en'sts M E IN, M 2 1, such that

u. is uniquely determined modulo S-"(R"; E ) .

Proof . We make the ansatz

j = O

with $J being a 0 -excision function. Then it is routine procedure to check that the reals cJ' > 0 converging to 0 sufficiently fast can be chosen in such a way that, for every M E IN, the sum CFM $(cj<)a, converges absolutely in S"& (R"; E ) , where m'M = m a x j > M mj.

Then, for every M E IN, M 2 1, we have that the symbol

M-1 M-1

j=O j = O j = M

belongs to S'(R"; E ) provided that r 2 ml,, which shows that (2.2) is valid. The uniqueness statement is obvious. 0

In the case that (2.2) holds we also write W

a - C a j . j = O

Now we are in a position to introduce classical E - valued symbols.

Definition 2.2. Let E be a F'rCchet space, m E R. Then the space Sz(R";E) of classical symbols of order m consists of all functions a E Sm(Rn;E) admitting

244 Math. Nachr. 194 (l!)!)M)

asymptotic expansions into homogeneous components, i. e., there are homogerwoiin functions E S("-j)(R" \ 0; E ) , j = 0 ,1 ,2 , . . . , such that

00

holds for an arbitrary 0 -excision function $.

Notice that the homogeneous components of a symbol a E Sz(R"; E ) are uniqucdy determined. The space S(")(R" \ 0; E ) is defined to be the space of all functiorin a E Cm(R" \ {O};E) which are homogeneous of order m, i.e., which satisfy a(X<) : X"a(t) for all < E R" \ {0}, X > 0.

The spacw S(")(IR" \ 0; E ) are topologized by identifying them with CW(S"- ' ;E) , S"-' iw ing the unit sphere in R". In particular, S(")(IRn \ 0) is a nuclear fi6chet spa(*('. Now, once a 0-excision function $ is fixed, we have natural continuous injections

Next a suitable F'rbchet topology for S$(IR";E) is introduced.

S(")(R"; E ) @ S"-'(R"; E ) Lf S"(R"; E ) , (2 .5)

(a(rn1,arn-l) * +(Oa(rn) + am-1

leading, for all M E IN, to continuous mappings

M M-1 @ s ( m - j ) ( R n ; E ) ~ m - M - 1 (Rn; E ) + @ S("-j)(R"; E ) @ S"-M(IR"; E ) j = O j = O

with the mapping in the first M components being the identity. The space Sz(R"; 8) is algebraically a projective limit,

M-1

(2.6) s$(R"; E ) = proj -1im @ s("-~)(R"; E ) @ s"-~(R"; E )

with the limit is extended over M + 00, and Sz(R"; E ) is equipped with the projw tive limit topology. This definition is independent of the choice made on +.

For example, for i k11

M E IN, we have that

j = O

From the representation (2.6) we draw some conclusions.

M-1

(2.7) s;(Rn; E ) = @ s(*-j)(IR"; E ) @ s:-"(R"; E) j = O

holds in a topological sense. In particular, S:-" (R"; E ) is a complemented subspa(:($ in SC;2(Rn; E ) . Similarly, it is seen that S-"(R"; E ) carries the topology induced I)y Sz(Rn; E ) . However, S-"(IR"; E ) is closed but not complemented in S;(R"; E ) .

Next we will recognize that S,;l(R") is a nuclear space.

Propos i t ion 2.3. Sz(IR") i s a nuclear fie'chet space.

\YII.I., Pseudo - Differential Calculus 245

1 ' 1 oof . Setting E = S~(Rn), we have to show that for every continuous semi- I I ~ I ' I I I p on E there is a continuous semi-norm q on E , q 2 p , such that the canonical iiinpping 8, + Ep is nuclear. Thereby, Ep denotes the local Banach space to a given n c ~ i n i norm p , i. e., the completion of the space Elkerp normed in a canonical way.

WP start with the representation

M-1

s~(R") = proj-lim @ S("-~)(IR") CB S ~ - ~ ( I R " ) .

I rot , p be a continuous semi-norm on E. We may suppose that, for some M, p is a rontinuous semi - norm living on @:;' S(m-j)(R")@Srn-M(R"). Since in the direct ti i i iri the first M summands are nuclear, we may further suppose that p is a continuous ticmi-norm living on Sm-M(IRn). Assume that, for some r 2 1, only estimates of tlcrivatives up to order r - 1 are involved in the semi -norm p . Then we can choose q I,O be a continuous semi - norm on @;=: S(m-M-j)(R") @ Srn-M-r (R") living on the h s t T summands such that q pulled back to Sz(R") estimates p from above. Again wing the nuclearity of S(m-M-j)(R") we conclude that the mapping Eq --t kp is nuclear. 0

The following proposition is basic in proving continuity between Sobolev spaces of operators arising in the classical calculus. It is interesting that its proof can be I)ased on the weak symbol topology that has been introduced in [24]. Another proof independent of the weak symbol topology shall be given in the notes at the end of the paper.

j = O

Proposition 2.4. Let E be a fie'chet space, m E R. Then

(2.8) . SZ(IR"; E ) = S,;l(IR")&E.

Proof . The assertion (2.8) is implied by the following calculation:

M-1

S~(IR"; E ) = proj -1im @ S(~-~)(IR,"; E ) SY-~(R"; E ) j = O

= proj -1im

= S;(IR")6&

@ s ( ~ - ~ ) ( I W ~ * + R * ) @[s r-l j = O 2" 1 Hereby in the first line we have used the continuity of the embedding

SF(IR";E) L) S"'(R";E) for rn' > rn, in the second and the forth line the nuclearity of the spaces S(rn-3)(R") and Sz(R"), respectively, and in the third line the fact that the injective tensor product is well - behaved under forming projective limits.

0

246 Math. Nachr. 194 ( I !WN)

2.2. Multiple classical symbols

The constructions in the classical calculus prompt to symbols p ( < , z , ~ ) whicli III 1 1

separately classical in both covariables (<,q) E R2". Here we introduce the coiw sponding notions. The spaces Smnm'(Rn x R"; E) were considered by many authoih Recall that a fundamental semi - norm system is given by

for E being a FrCchet space with fundamental semi-norm system {I[ I I 1 } l E ~ .

sums in S ~ * ~ ' ( R " x R"; E). As in Proposition 2.1 one shows that it is possible to form double indexed asympi,ol.lr,

Proposition 2.5. Let E be a Re'chet space. Let {rnj}jEN, {mi}kcN be sequenw?i of reds satisfying m j + -00 as j + 00, mk -+ -00 as k + 00. Suppose further t l i t i l

we are given symbols a j k E SmJ*mL (Rn x R"; E) for j , k = 0 ,1 ,2 , . . . . Then there a,?

a symbol a E Sm,m'(IR" x R";E), where m = maxjENmj, m' = maxkENm;, sudr that for all r , r' E IR there ezist M, M' E N, M 2 1, M' 2 1, such that

M- 1 ,M' - 1

(2.10) a - c a j k E SrqlR" x R"; E ) . j,k=O

a is uniquely detennined modulo S-w*-OO(R" x R"; E).

In the case that (2.10) holds we write

00

u N C ajk. j,k=O

Notice that in (2.10) asymptotic summation can take place first in one index and th(1ir in the other leading to the same result. That means, e. g., if we put

(2.11) m

aj c a j k I k=O

where this asymptotic sum exists in Sm~*m'(lR" x Rn;E) with the result uniqiicm modulo Sm~--m(Rn x R";E), then we can asymptotically sum up the aj's iii

Sm*""(R" x R " ; E ) with the result unique modulo S-OO*m'(Rn x Rn;E), and w(' find m

(2.12) u - C a j . j = O

The definition of the classes S~*"'(R" x IR"; E) is the following one:

Definition 2.6. Let E be a FrCchet space, m, m' E IN. Then the spaw SZ'"'(R" x Rn;E) consists of all functions a E Sm*m'(R" x R";E) for which,

\YII.~., Pseudo - Differential Calculus 247

I'IW j , k = 0 , 1 , 2 , . . . , there are symbols a(m- j ) , (m~-k) E S ( m - j ) , ( m ' - k ) ((R" \ 0) x (IF." \ 0); E ) homogeneous of multi -order (m - j , rn' - k) such that

00

(2 .13) a(<,q) - @(<)'$(q) a ( m - j ) , ( m ) - k ) ( [ , 17) j ,k=O

Iiolds for an arbitrary 0-excision function '$.

I11 (2.13), the homogeneous components a ( m - j ) , ( m ~ - k ) ( < , q ) are uniquely determined. 'I'hc space S(m),(m')((R" \ 0) x (R" \ 0); E ) is defined as the space of all functions (1 E c"((R" \ (0)) x (R" \ (0)); E ) satisfying a ( ~ < , p v ) = ~ ~ ' p ~ a ( [ , q ) for all (, I ] E R" \ { O } , x > 0, p > 0.

In order to topologize s,";'"'(R" x R"; E ) we provide the spaces S(m)*(m')((Rn\O) x (R" \ 0); E ) with FrBchet topologies by identifying them with C" (S"--l x S"-'; E ) . Moreover, for m, rn' E R, M, M' E IN, we have natural continuous injections

M-1 ,MI-1

@ S(m- j ) , (m' -k ) ((IR" \ 0) x (R" \ 0); E ) 0 Sm-M*m'-M'(IR" x R"; E ) j .k=O

L) S"*"! (R" x R"; E ) ,

(a(m),(m))i * . . , a(m-M+l),(m'--M'+l), am-M,m-M')

M - 1 ,MI- 1

~ ( < ) ~ ( ~ ) a ( m - j ) , ( m ~ - k ) + am-M,m'-M' . j .k=O

'I'hen Scys"' (R" x IR"; E ) becomes equipped with the projective limit topology

sty' (R" x R" ; E ) M-1,M'-1

(2.14) = proj-lim @ S(m- j )dm' -k ) ((R" \ 0) x (R" \ 0); E) j , k = O

Sm-M,m'-M' (R" x R";E) .

Proposition 2.7. Let E be a hdche t space, m, m' E IN. Then we have

(2.15) SyqIR" x 1R";E) = S,;l(R";S,"f'(IR";E)).

Proof . The proof of (2.15) is a straightforward, but lengthy exercise in employing the representations of Sz(R"; S,;l'(R"; E ) ) , S:*m'(Rn x R"; E ) as projective limits

0 according to (2.6) and (2.14), respectively.

Especially, from Propositions 2.4 and 2.7 we get

(2.16) S y q R " x R"; E ) = s,"f(R")63,Sy(IR")&E.

248 Math. Nachr. 194 ( I ! ) l ) H )

Notice also that the linearization of the continuous bilinear mapping

S$(IR") x SZ'(R";E) + S,";+"'(IR";E) , (a,a') c) aa'

extends by continuity to a continuous surjective mapping

(2.17) S:'"'(R" x R";E) -+ S,";+"'(R";E) , a( [ ,q ) I-+ a([,q)l,,=t.

2.3. Coefficients in Sobolev spaces

We particularize the results on abstract vector - valued symbol classes to syml)ol classes used subsequently. We will be mainly concerned with symbols having thvii coefficients in L2 -Sobolev spaces, H'(Rn), where s > f. This means, in particulilr., that the coefficients are at least bounded and continuous.

In the case that the coefficient space is F ( R " ) , e.g., F = H " , Hfs,,, C r , we shidl denote Sm(Rn;F(Rn)) = FSm(R" x IR"), Sz(R";F(R")) = FS$(Rn x IR."), S m ~ " a ' ( l R 2 " ; F ( R " ) ) = 3Sm,"'(R" x R2"), etc., where the first set of coordinatth refers to the space variable z and the second set of coordinates to the frequency variables t , r]. As mentioned in the introduction, for symbols p E FSZ(R" x R") w(' adopt two different operator conventions, p ( z , D) and p ( D , z). In the first case t h , symbol is denoted by p ( z , q ) , in the second case by p([ , z ) .

To describe later on the behaviour of operators under compositions, we introduw further symbol classes. For 8 , s' E R, the space H'-8'(R" x R") consists of dl tempered distributions u 6 S' (R2") satisfying

(2.18) ( o s ' ( v ) ' a , d E L2(R" x R").

H8-8'(lRn x R") is the Hilbert space tensor product H'(R")&,HH''(R"). Note that, for u E H8J'(R" x R") we have

.(z) = U(Z,Y) l y=z E H " ( W

if Is'( 5 s, s > f. This is seen by writing 8 ( [ ) = JG([ - q , q ) dq. Let m, m', m" E R, s, s' E R, s > f , s' > 5. Then the symbol class

~ + 8 ' " l m ' > m ' ' ( ~ ~ ~ X R ~ " ) consists of all functions p ( < , z, I , y, q ) E c"(R~"; H~*"(R~")) satisfying

(2.19) sup (€v(d€~s"

([)-""+I"I(C)-"'+1~1 (q)-"+lyl p p p ; u ( < , . , I , . ,q)llH.,.' < 00

This definition is in complete analogy to those in (2.1), (2.9). For symbols in (2.19) we choose the following operator convention:

(2.20) ( P P , 5, D, 5, W u ) - ( t ) = 1% t - I , C I C - 71 7)wl ) 4 4 .

The first under the integral sign refers to the partial Fourier transform of p ( [ , 2, C, y, r ] ) with respect to z, y.

252 Math. Nachr. 194 (II)UR)

ajk(z)

aj(z)

a$(z)

a(z)

for certain s E R.

Ha(R") H'(IR")

H ' - ' (R") H*-l(R")

H'-'(R") H'(R")

H'-2(R") H q R " )

n s > - + 2

2

Next we rise the question what happens if one considers a given operator I I I dtT{'d(R") as an operator with asymptotic expansion shorten by 1. Obviously, H'I~

have m),d(Rn) g d(m),d-l (3.11) A:, cl a , c ~ (R")

for any integer d 2 1. The next example shows that (3.11) is due to a "lack" id

smoothness of coefficients in the remainder term.

Example 3.6. Consider again the linear partial differential operator from (3.!)), Under the above assumptions we also have

(2) ,2(Rn) (3.12) A E d a , c l

The minimal hypotheses on the coefficients under which (3.10), (3.12) are true uritlvi the restriction that coefficients should belong to L2 - Sobolev spaces are shown in 1.1w table:

The operator classes " (R") are defined as follows:

Definition 3.7. Let s, m E R, d E IN, s > f + d, Id - d'l 5 2s - 2d. Tlwii dl (R") denotes the class of all operators P which can be written in the forilk

d-1

p = C P j ( Z , D ) + Pd I

j = O

where p 3 ( z , q) E Ha-JSc~-l(IRn x R") for j = 0 , 1 , . . . , d - 1 and

a-d-max(d.d')

(3.13) Pd E L(Ht+m(R"), H t + d ( R " ) ) t=-a+d-min( d,d')

W j t , t , , I'scudo - Differential Calculus 253

'1'110 property (3.13) means that the scale on which the operators act is shortened by tl' 1, from above if d' > d , from below if d' < d. For d' = d we get A!:!' d l d(R") = Id

,/1:',':,:' "( IR"). The mapping properties in (3.5) are changed into

r - m u ( d . d ' )

( X I 4 ) P E n L ( H ~ + ~ ( I R " ) , P ( W ) ) . t=-s+d-min(d,d')

(4, d, d' Ni)l,o that Id - d'l 5 2s - 2d is required because otherwise the class ds,c, w o i ~ l d be empty.

(R")

With the enlarged operator classes thus defined, the inclusion

(3.15)

(m), d , d' niibstitutes (3.11). Later on we shall see that the classes ds,cl (R") also appear, rb.g., in the composition of operators and consequently are an integral part of the I-illculus.

We have again operator classes L?~T",~'d'd'(R") defined to consist of the formal ad-

,joints to operators in d6~~'d1d'(R"). These classes are given as in Definition 3.4 with Iwoperty (3.7) replaced by

#-d+min(d,d')

(3.16) Qd E n C ( H t - d ( R " ) , P"(R")) . t = - s + d + m u ( d , d ' )

3.3. The full operator classes

(m), d , d' The aperator classes dr, cl (R") are preserved under compositions. In verifying that fact, however, we encounter additional operator classes. Furthermore, as we shall see, the parametrix to an elliptic operator in dL:{'d(X), e.g., for X being a closed compact manifold, belongs to the operator class d8, cl (-m)' (X). Thus we are going I,O complete the operator classes introduced so far with respect to these operations.

We start with an analogue to Lemma 3.1 and (3.8):

Lemma 3.8. Let s, rn, m' E IR, s > 5. Then, f o r p ( < , z , q ) E HsSc~'m'(R" x R"), p(D, z, D ) tnduces a continuous operator

(3.17)

f o r all t E IR, (ti 5 s. p ( D , s , D ) : Ht+rn(IR") + P - m ' (*?2& . . *

w r i

Definition 3.9. Let s, m, m', d' E R, d E N, s > + d , Id - d'l 5 2s - 2d. Then dLr",{' (m')' d' d l (IR") denotes the class of all operators P of the form

(3.18)

254 Math. Nachr. 194 (IWN)

where p j ( ( , z , q) E H8-jS2-J'm' (Rn x R") for j = 0 , 1 , . . . , d - 1 and

8-d-rnax{d,d'}

W")) . , (3.19) p d E n L( H t + m (Rn), Ht-m'+d

t=-.+d--min{d,d'}

Similar remarks apply as for the classes d L y { ' d ( R " ) . For example, (3.18) giw" I I

finite asymptotic expansion into operators belonging to d8-j, (m-j) , (m') ,d-j ,d'-j (R") ll)l j = 0 , 1 , . . . , d. For the mapping properties we find

8-max{d,d'}

(3.20) P E n c (Ht+rn(IR"), (IR")) t=-a+d-min{d,d'}

for P E A:,,, m ) * ( m ' ) l d t d ' ( R n ) . The total order is m + m'. The classes for m' = 0 are the same as before:

Proposition 3.10. Let s, m, d' E IR, d E IN, s > f + d , (d - d'l 5 2s - 2d. Tlre71

(m)' (O)' dl '' (R") is obvious. To obtain t,Iw (R") E d,,cl other direction it suffices to deal with d' = d. But then the assertion follows from tlro

I 1

(m), d, d' Proof. The inclusion d,,cl

proof of Proposition 3.15 given below, in the special case m' = 0, T = 0.

Proposition 3.11. Let s , m, d' E R, d E IN, s > 5 + d, Id - d'l 5 2s - 2d. Then

Proof . The relation A::{' (m')* d'd' (IR") = ( D ) m ' d ! ~ ~ m ' ) * (O)' dl dl (R") (D)-"' fol- c1 lows from the definition. Proposition 3.10 then yields the conclusion.

In an analogous manner the classes (m)> (m')' d'd' (R") are defined. An operator I)

belongs to a:, m), cl (m'), 4 d' (IR") if it has the form

(3.23)

where qj E H8-jSrn*"'-j cl (IR" x RZn) for j = 0 , 1 , . . . ,d - 1 and

8-d+min{d,d'} (3.24) Qd E n L(HL+"-d(IRn),H'-m'(Rn)) .

t=-r+d+max{d,d')

Witf,, Pseudo - Differential Calculus 255

\Mi have that BLyi' d' d' (IR") = BS,d (o) ' (m) 'd 'd ' (R") . The adjoint to an operator in

A!;;:- (m' ) , d, d' (R") belongs to B ! ~ / ' c m ) ' d ' d ' ( R " ) .

rtlltkipating explanations given in Subsection 4.1 on symbols. ds-d I H ti Banach space by interpolation. An operator P E dl,cl il(tc!rmined homogeneous symbols p, E H J - j S ( m + m ' - j ) ( T * R " \ 0) for j = 0. . . , d - 1 (txv Definition 4.2) leading to a representation of d ~ ~ { ' ( m ' ) ' d l d ' ( R " ) as a direct sum:

(R" ) (R" )

Wo conclude this subsection by topologizing the operator classes dl, (m), cl ( m ' h d , d'

(m-d), (m'),O,d'-d

(m)? (m')* d * dl (R") has uniquely

d-1

Notice a certain indetermination contained in the composition (3.25) consisting in rillother possible choice of the 0 -excision function 11, in fixing the splitting in (4.3). It is, however, plain that the topology of the locally convex direct sum is independent of that choice. Then d8, ( * ) I cl (m'll d 7 '' (R") becomes equipped with the resulting F k h e t I<opology.

3.4. The basic technique

The resclts leading to the different conclusions concerning the operator calculus will 1 ) ~ derived by applying Taylor's formula to produce the asymptotic expansions and !,hen estimating the remainder terms showing that the remainders obey the right mapping properties. Thereby, in the latter step we encounter expressions of the following kind: For given functions G(<, q), g(<,q) we consider

(3.26) (Th)(<) = 1 W ! V ) d t - q, V M r l ) dq

for h E L2(IR"). We want to find conditions under which T realizes a bounded operator on L2 (IR").

In [2], BEALS and REED made the following simple observation. For the sake of completeness we indicate the proof:

Lemma 3.12. Let G(<,q), g(E,q) be measurable functions on R" x R". Suppose that

or

256 Math. Nachr. 194 (I!J!M)

Then (3.26) defines a bounded operator on L2(IR") satisfyng

(3.27) IITNlt2 5 CCCB l l 4 l t 2 * Proof. We only treat the case when the first of the assumptions is fulfilled. ' I l i v

other proof is similar. For u E L2(R"), one estimates

Lemma 3.13. Let s, t , r E R. Suppose that, for some 6 > 0, min {s, t , s + t - 6) 2 0 and

s , t , s + t - - - 6 2

hold. Then (3.28)

Proof. The proof follows by splitting, for fixed < E IR", the integral in (3.28) i i i h ~

four integrals over the regions {q E R"; I< - q1 5 1, 1771 5 l}, {q E R"; I < - 711 :. 11 1711 I 11, {rl E IR"; It - 771 I 1, Id > 111 and (77 E R"; It - 71 > 1, 1771 > 1 I t

respectively. I I

Now we come to the announced proofs of Lemma 3.1 and Lemma 3.8 so establishill,: the basic technique used later on. Of course, it would suffice only to establish Leiniiiii 3.8, but we use the proof of Lemma 3.1 to familiarize the reader with the techniqiic, used in [2].

P r o o f of Lemma 3.1. We present two different proofs. The first one makes ditcv.1 use of the estimate of BEALS and REED and works only in the case when t 1 0, wherc,;ih the second one additionally uses considerations involving the projective tensor prodiic.1

(a) Assume that t 2 0. Under this assumption we are going to show that the 1,' norm of (<)t+"'&(() yields an upper bound for the L2 -norm of (()'(p(z, D)u)*(<) .

To do so write

Witt, Pseudo- Differential Calculus 257

i l l i d apply Lemma 3.12 with

where the first of its assumptions is fulfilled.

( I E H s ( R " ) , po E Sz(R"), Lemma 3.12 applies to (3.29) with (b) Exemplary we treat the case t < 0. Writing p ( z , q ) = a(z)po(r]) with

with now the second assumption fulfilled, showing that the bilinear mapping

Hd(IR") x S?(R") -+ C(H'+m(Rn),H~(R")), (a,m) H a(z )po (D)

H'(R") €%I S?(R") + L(Ht+m(R"),Hf(R")) 1 a@po - a(z )po (D)

is continuous. Equivalently, the linearization of the last mapping,

is continuous and extends by continuity, in view of Proposition 2.4, to a continuous ttiapping

Thereby, the symbol p ( z , < ) is mapped to the operator p ( z , D ) , since this is true on H"(R")@S~(IR") and the mapping H'S$(R"xR") + S'(R"), wherep I+ p ( z , D)u,

0 Notice that, the first proof also does not work for symbols p ( < , z, 7 ) ) even in the case

H8SZ(R" x R") + C(Ht+rn(IR"), H y m n ) ) .

is continuous for each u E S(R").

that t 2 0. But the second proof does, and we obtain:

P r o o f of Lemma 3.8. We apply again Lemma 3.12. But this time we only provide Ihe functions G(<, q), g(t1 q) showing continuity of the trilinear mapping

S,;l'(R") x H'(R") x S?(R") + L : ( H ' + y R " ) , H f - m ' (R")) 9

(Pl,%PO) * n ( D ) a ( z ) p o ( D ) f

In case t 2 0

in case t < 0

G(t)71) = (< - ( 1 7 ) - 1 q ) s ( < ) - t ( J r m ' P l ( 0 (rl)-mPo(77) 9

where in any case g(<,q) = (<)"&(<), h(q) = ( ~ ) ' + ~ & ( q ) . 13

Finally, we provide a natural companion to Lemma 3.12 which we use in the proof of Proposition 3.20. For that we consider the formal expression

(3.30)

258 Math. Nachr. 194 ( I ! ) I I N )

for given functions G(<, C , q), g(<, C , q).

A possible generalization of Lemma 3.13 is as follows. Let SO, 91, t , r E IR satisfyill/l, min { s o , s1, t , so + s 1 - $ - 6, so + t - $ - 6,s1+ t - $ - 6, so + s1 + t - n - 2 ~ ) 2 o arlt~

(3.32)

n n 2 2

so,s1,t,sO + s1 - - - 6,so + t - - - 6,

for some 6 > 0. Then

(3.33)

A proof follows by noting that for k = min {SO, t , SO + t - f - 6) the quantibp min { 31 , k, SI + k - 5 - 6) is equal to the right-hand side of (3.32). Therefore,

3.5. Change of representation, compositions, and adjoints

After having introduced the several operator classes in previous subsections wv come now to the basic elements of the calculus for classical operator like compositio~rs, adjoints, and commutators.

Before we proceed some words about the logic involved in the proofs that follow We start OUT considerations with Proposition 3.15 in which is stated what happerrh when one representation is changed for another. In it a commutator estimate is hitl- den, and its proof relies on Lemma 3.17. But at the beginning Lemma 3.17 is orrly

\ V ~ L I , Pseudo - Differential Calculus 259

11iii I iitlly proved, what in the following, however, allows us to work in operator classes ,111;: \ , ( 1 1 1 ' ) , d, d' (R") with d' 2 d. Keeping this in mind, in Proposition 3.15 and in all wii1)svquent propositions first only the parts corresponding to d' 2 d are dealt with. I r r particular, at this stage (3.35) in Proposition 3.15 remains unproved. Collecting 1 1 1 1 the facts that can be derived under the assumption d' 2 d we are led in Proposi- I Iorr 3.21 to the formulation of a genuine commutator estimate which appears in two tlilkrent forms. The watershed is Proposition 3.19 for which two different representa- ~,Ic)iis appear for the first time, cf. (3.53). Later on these representations turn out to be rqiiivalent in view of relation (3.35). The commutator estimate allows us to complete I I I P proof of Lemma 3.17 and then, repeating all foregoing arguments, all the other ~"'OOfS.

Therefore, right from the beginning all statements are given in their final formulation. ' 1 ' 1 1 ~ proofs are also given in a way that works for the more general assumptions. Ilcfore Proposition 3.19 it is always mentioned which part is actually proved without liiiving Lemma 3.17 in full, while starting with Proposition 3.19 the two possible clifi'erent formulations are provided yielding the complete statement once (3.35) has Iiocn established.

Proposi t ion 3.15. Let s, m, m', r E IR, d E N, s > f + d, Jm' - T I 5 2s - 2d. Then we have

(R"). (m+m'-r), (r),d,d+m'-r (R") c 4 , c r (3.34) A;;;, (m'h d

In particular, we have

We need two lemmas. The proof of the first one is straightforward.

Lemma 3.16. Let d E N. Then there ezist symbols xj(f,ql() E S!;d-J'0(lR3n) such that

(3.36) d

(t)d = C Xj(C1 71, t - 71) - j = O

Proof . Choose a partition of unity { $ ~ k } ; = ~ on R", i.e., I,!Jo(t) + EL==, I , !Jk( t ) = 1 for ( E R", where $0 has compact support and, for k = 1,. . . , n, $k E SEl(R"), and It1 2 C for some C > 0, l t k l 2 &It1 on suppI,!Jk. Then (t)d = $0(0( t )~ + c:=, $ k ( t ) ( O d , $o( t ) ( t )d E S--(R"), while, for k = 1,. . . ,n ,

d

$k(t)(t)d = $i(~tkd = Cxjt(tiq,t -71)

with certain $i(<) E SE,(R") and xjk((,q,() E S:id-jv0(IR3") using the binomial 0

The main step in the proof of Proposition 3.15 consists in establishing the next lemma. The particulars in its proof are carried out in detail making use of the tech-

j= l

formula for (," = ( q k + (& - ~ k ) ) ~ . The conclusion follows.

260 Math. Nachr. 194 ( 1 OM)

niques developed in Subsection 3.4. Later in similar proofs we shall confine ours(!Ivm to certain steps.

Lemma 3.17. L e t s , m, m', r E R, d E IN, s > q + d , - 2 s + 2 d 5 m ' - r 5 2 s - r l Let further p ( < , z, 77) E HaScT*m' (R" x ELZn). Then we hove

(R") if d < m ' - r 5 2 s - d , m+m'-r), (r) ,d,m'-r 4, C l

- 4 8 , C l

(3.37) p ( D , z, D) E A b ~ ~ m ' - r ) ' ( r ) ' d (R") if 0 < m' - r 5 d ,

(R") if -2s + 2d 5 rn' - r 5 0 (m+m' -r ) , (r), d, d+m'-r

B e g i n of p roof . The relation (3.37) is derived from

(R") (m+m'-r), (r) ,d,m'-r (3.38) P(Dlz,D) E -48 , c l

which is first shown to hold for d 5 m' - r 5 2 s - d and which then turns out to I w true for (d - m' + rI 5 2s - 2d.

Let p E H8Smim' (R" x R'"), d 5 m' - r 5 2s - d. We set p ( < , z, 7) = (<)rq(<, x,7/), where q(( , z , ~ ) E H*Sz*m'-r (R" x R'"). Then

(P(D9I m)-(t)

= J(0Wl t - TI, rl)&(rl) drl

Hence i t remains to prove that the remainder R given by the third line in (3.39) is :I bounded operator from Ht+m+m'-r(R") to H1-r+d(R") for each k

t E [-s + d - min {d,m' - r } , s - d - max{d,m' - r } ] .

According to our general procedure we prove it when the symbol p ( < , 2, 77) E H8Sc~1m'(Rn x R") has product form, i. e., q(<, z, 77) = q1(<) u(z) q o ( r / ) with a E H8(R"), qo E Sz(Rn), q1 E S2'-r(Rn). In that case the defining formiilii for the remainder R becomes

\Yit.tt, Pseudo - Differential Calculus 261

Now the expression Ji (1 - O ) d - l (a?ql)(q + 6’(( - 7)) d0 can be rewritten as k(<, q) . ( ( ) m ’ - r - d + kl (t, d(v)m’-r-d , where ko, kl E Lm(lR” x R”). Because we have 1 1 1 ’ - T - d 2 0, this is seen by writing

where we haveset k ( [ , q ) = ~ ~ ( I - B ) d - ’ ( B E P q 1 ) ( 7 7 + O ( t - 7 ) ) ) d B , and x ( M ) ( < , q ) is the vliaracteristic function of a set M

In this situation Lemma 3.12 applies. For the first summand, the first assumption i l l Lemma 3.12 is fulfilled in case t E [-m’ + r , s - d - m‘ + T ] if we set

IR~*.

h(q) = (q)t+m+m’-r C(77) 1

!7(<9 4 = (oa-d(m.4Yt) I

(,c):+m’ -r

G(t177) = ko(t1s) (t - q ) s - d ( q ) t + m ’ - t (rl)-mQo(77) 9

whereas the second assumption is fulfilled in case t E [--9 + d - m’ + r , -m’ + r] if we S C t

C(77) P

dtld = ( t )E-d(m4-(t) 7

t+m+m’-r h(V) = (59

(V)-t-rn’+r

G(tl11) = kO(t177) (< - qp-d(t)-t-rn‘+r (77)-mQ0(77) ’

This shows that this part of the remainder R defines a bounded operator acting from /lL+nl+m‘-r(IRn) to Ht-r+d(lRn) for any t E (-s + d - m’ + r,s - d - m’ + r ] . In a similar fashion we argue for the second summand. This time the first assumption in Lemma 3.12 is fulfilled in case t E [ -d,s - 2 4 if we set

whereas the second assumption is fulfilled in case t E [-s, -4 if we set

G ( d I

dt, 77) = ( t ) E - d ( m ) - ( t ) >

t+m+m’ -r h(rl) = ( r l )

262 Math. Nachr. 194 ( 1 l ) l ) N )

Thus the second part of R defines a bounded operator acting from Ht+m+m'-"( 111 " ) to HL-r+d(Rn) for any t E (-s,s - 2 d ] . All in all we have obtained that

8-d-max{d,m'-r)

RE n L (Ht+m+m'-r (R"), Ht-'+d(R")) -a+d-min{d,m' -r)

as required, in the special case that the symbol p ( < , z, q ) has product form.

shown that the trilinear mapping Since Lemma 3.12 also provides us with corresponding estimates, we have actmllv

is continuous, with R given by (3.5). Hence, according to the properties of the p i x i

jective tensor product, the linearization of (3.41) is continuous as mapping

and may be extended by continuity to a continuous mapping

(3.42) Has,;l,m'(Rn R2n) + dir:m'-r-d).(r).O,m' -r-d(R").

Thereby, it is seen that the symbol p ( < , z, q ) E H8SZ'm' (R" x R2") is mapped to tJii-

operator R given by the third line in (3.39), since the mapping

is continuous for each u E S(IR"). Thus (3.38) follows. To conclude the proof, for a symbol p ( < , z,q) E HaSc;l*m' (R" x R2") we write

d

P(<, 2,771 = (q)dq(t, 2 , ~ ) = C xj(<, 77, t - q)q( t , 2977) 1

j=O

where q E H8Sc;(-d*m' (R" x R Z n ) and, for j = 0,1,. . . , d , xj E S:io'd-j(R3n) accoi~cl- ing to Lemma 3.16. Now the operator with symbol xj(<, q, <-q)q(<, 2, q ) can be rewri1,- ten as an operator with symbol hj(<, z, q ) E H8-jSZ-d'm'+d-' (R" x R") , whc:rv hj(<jz,q) = F;f.(xj(<iqi~)6(<iCiq)}. Thus, by (3.38)) the operator P ( D , ~ , D ) has been represented as the sum of operators in d~~~$-r - ' ) ' ( r ) ' d' d+m'-r-j (rn." ) for j = 0,1,. . . , d if 0 _< m' - r 5 2s - 2d. That is, in this case we have seen that

(3.43)

Wi1.t.. Pseudo - Differential Calculus 263

Nol.ice that if (3.38) is shown for Id - rn' + r( 5 2s - 2 4 then we obtain (3.43) for (711 , ' - T I 5 2s - 2d, and vice versa.

Observe that the choices of g(1, q), G((,q) made in the proof of Lemma 3.17 can be

(3.37) follows from a discussion of the cases resulting from (3.38), (3.43).

iccorded, e. g., for the first case in the form

dt - rl , d = (t - r7)a-d(m)-(< - r7) 1

liming it open whether t + m' - T 2 0 holds or not and whether in the assumptions of Lemma 3.12 integration of the square of the modulus of g(< - q, q) takes place with respect to 1 and 77, respectively.

B e g i n o f p r o o f of P r o p o s i t i o n 3.15. We show (3.34) in case 0 5 rn' - r 5 2s - 2d. As we shall see, (3.34) follows in general for Irn' - r ( 5 2s - 2d when (3.43) I i a s been established on its full range (m' - r( 5 29 - 2d.

k t P E Aa,c i (m) ' (m' ) 'd (Rn) . According to Definition 3.9 write P = xj<dPj + P d

with Pj = p , ( D , z , D ) , pj E Ha-JSc~-J'm'(R" x R") for j = 0 ,1 , . . . ,d - 1, I'd E A a - d (m-d)'(m')'o(IR"). Fkom (3.43) we obtain

(R") (m+m' - j - r ) , (r) , d- j, d+m' - j-r

Pj E d a - j , ci

for j = 0,. . . , d - 1, so it remains deal with Pd. We show that independently of the sign of rn' - r d ~ ~ ~ d ) ' ( m ' ) * o t - m ' + r (R") =

(Rn) holds, i. e., (m+m'-r-d), (r) ,O,m'-r '8-d

a-d-max{d,d-m'+r) n c ( p + m ( R " ) , H t - m ' + d (R")) t=-a+d-min{d,d-m'+r)

a-d-max{d,d+m'-r)

- - n ~ ( ( ~ t + m + m ' - r (R"), H'-r+d(R") ) . t=-r+d-min{d,d+m'-r)

To do so we have to verify s-d-max{d,d-m'+r} = s-d+m'-r-max{d,d+m'-r}, -s + d - min{d,d - rn' + T } = -s + d + rn' - r - min{d,d + rn' - r}. The first relation is implied by max{d,d - m' + r } = 2d - rn' + r - min{d, d - rn' + r} = 2d - rn' + r + m a { -dl -d + m' - r } = -rn' + r + max{d, d + rn' - r } , and the second one follows in an analogous manner. Therefore,

(R") (m+m'-r-d), (r),O, m'-r (R") E d a - d

( m - 4 , (m'), 0 p d E A a - d

which proves the first part of the proposition.

part follows from Now anticipating the situation when (3.43) has been completely proved, the second

264 Math. Nachr. 104 (l!J\tH)

by what that has been already proved, and

by the same argument. I 1

In the same manner as for (3.37) it can be proved that for s, m, m', r E R, d E IN, s > f + d , -2s + 2d 5 r - m' 5 2s - d we have

(R") if d < r - m ' < 2 s - d , (m+m'-r), (r),d,r-m' B*, cl

(3.44) p(D,s, D) E Bj:Tm'-r)l(r)vd (R" 1 if 0 < r - m' 5 d , if -2s + 2d 5 r - m' 5 0 , (R") (m+m'-r), (r), d , d+r-m' i 4, cl

provided that p ( ( , z , q ) E H8S:'m'(Rn x R2"). Notice that so far we have only 1.111~ first two lines of this relation when following the lines of the proof of Lemma 3.17, Note that (3.44) is derived from

m") (m+m'-r),(r),d,r-m' (3.45) P(D, 21 D) E q c l

which is first obtained ford 5 r-m' 5 2s-d and which holds for Id+m'-rl 5 2s-2tl In the next result it is asserted that B-classes constitute merely another represcii.

tation for A - classes.

Proposition 3.18. Let s, rn, m', d' E R, d E IN, s > + d, Id + m' - rl 5 2s - 2d Then we have (3.46) & m), (m'),d ( ~ n ) Bj;;m'-r), (r),&r-m'

In particular, we have (IR") *

(3.47)

Proof. We show (3.46) if d 5 r -m' 5 2 s - d . The general case Id+rn'-r( 5 2s-2rl follows when (3.37) and (3.44) have been completely established.

Let, P = Cj<d Pj + Pd, where Pj = pj(D,z,D), p j E H (R" x R'") for j = 0,1,. . . , d - 1, Pd E ds-d (m-d)'(m')'o(lR,"). For j = O , l , ..., d - 1, we havi,

(IR") by (3.44). Furthermore, it is readily seen thar Pj E B s - j , c l

a- j~ rn - j~m' cl

(m+m'-r), (r-j) , d - j , r-m'-j

(R" 1 (m-d) , (m') , 0, r-m' - d (m+m'-r), (r-d), 0, r-m'-d A#--d (R") = B 8 - d

r - m' 5 2s - d. (m), (m'), d , r--m' (R"), we infei (R") c d,&l

(m+m'-r), ( r ) , d Since we analogously obtain B#, cl

The second part of Proposition 3.18 can equivalently be formulated as

(3.47). c1

(3.48)

W it,t., Pseudo - Differential Calculus

wl i id i is seen by setting r = m' + d'. Recall that the classes I?,,,,

265

(m), (m'), d , d'(Rn) have

I , c v r i designed to incorporate the formal adjoints to operators in dL$/'(m)'d' " (R"). 'I'lii~s as a further corollary to Proposition 3.18 we obtain that P E d ~ ~ { ' ( m ' ) l d ' d ' (R") ill,,,lies po ridjoint representation.

Abr"c;+d')'(m-d'),d, d' (R"). We call a representation in B-classes an

Next the behaviour under compositions is established.

Proposition 3.19. Let s, mo, mb, ml, m: E R, d E IN, s > f + d , Irnb + mlI 5 2 s - 2d. Then we have

(mo),(mb),d (R"). (mo+mb+ml), (m;), d , d+mb+mi (R") c d8,d (ml) , (m;) ,d (R") . d,,d (3.49) d8,d

Ilr (3.49), the composition on the left-hand side i s understood in the opposite da- wction, i. e., it is meant that P E d ~ ~ ) ' ( m b ) ' d ( R " ) , Q E da+, (mi) , (m;) td(Rn) imply

(R"). (mo+mb+ml), (mi ) , d,d+mb+mi [t? p E A8.d

To prove Proposition 3.19 we need the following lemma:

Lemma 3.20. Let s , m, m', m'' E R, d E IN, s > + d , -2s + 2d 5 m' 5 2s - d . Then, for p E H8*8SZ9m'~m" (R2" x R'"), we have

(R") if d 5 m ' S 2 s - d , (m+m'), (m"), d, m'

(m+m'), (m"),d

(m+m'),(m"),d,d+m'

4, C l

(R" 1 if 0 5 m' 5 d , (3.50) P ( D , z, D, 2, D) E d8, c,

(R") if -2s + 2d 5 rn' 5 0 . I An, cl

Proof . We give here the proof in case 0 5 m' 5 2s - d . The general case -2s+2d 5

We first show (3.50) in the cases m' = d and m' = 0. Let m' 5 2s - d follows when (3.37) has been completely established.

~ s , s s c ; l , d , m " (R2" x R3") ,

where we aSSume that P ( ~ ~ ~ , < ~ Y , v ) = ~2(t)al(2)pl(C)ao(y)po(rl), ao, a1 E H8(R"), po E S~(R"), p1 E S,~,(R."), p2 E s~"(R"). We write

( A D , z, Dl z, +)^(t)

266

and obtain

Math. Nachr. 194 (JOlJN)

with k ( C , q ) = s(l - O)d-'(8rp,)(q + B((c - 7 ) ) d being bounded yielding t l ~ i ~ t

R E d s - d (m)3(m")'o(IR"). This shows (3.50) for m' = d. Next let p E Hs~sSc~'O'm' ' (R2" x R3"). Then (3.50) follows by writing

d

ti x, I , Y I 77) = ( v ) d q ( t i x , ( 7 Y, 77) = 1 xj(77, C, 7 - C) q(<, 2, C, Y, 7) j = O

with Finally we treat (3.50) simultaneously in the cases 0 5 m' 5 2s - d aiitl

-2s + 2d 5 m' 5 2 s - d . We again assume that p ( [ , x , C,g, 7) is given in protl-

E Hs.sScY-d~o~m" (R'" x R3") and x j E S:;d-Jt0(IR3n).

uct form, i. e., p ( t , 2, 5, Y, 7 7 ) = PI (t, 2, C ) PO(<, Y, 77) with PO E HSS:lm' (an x R'") pl E HaS~;m"(IRnxIR2") . By Lemma3.17we have~o(D,x ,D)Ed!~~m') ' (o) 'd 'd' ( , ' I ) , where d' = m' if d _< m' 5 2s - d , d' = d if 0 5 m' 5 d , and d' = d + m' i f -2s+2d 5 m' 5 0, respectively. Thus we write po(D, x, D) = x;zi p o j ( D , x, D)+PO,,,

(IR"), and f i i i t l where p ~ j E H s - j S c ~ + m ' - j l o (R" X R'"), Pod E d8-d pl ( D , z, D) poj (D, x, D) E di::z'-j)' (m") 'd-J (IR") for j = 0,1,. . . , d - 1, whilv

0

( m + m ' - d ) , ( O ) , O , d ' - d

p1 (D, %, D) Pod E AL:d+m'-d), ( m " ) , o * d ' - d (IR") by direct calculation.

Note that in the proof of Lemma 3.20 we have obtained as a byproduct

(R") (m+m'), (m"), d , m' (3.52) P ( D , x, D, 2, D) E As, cl

\ V i t , t , - Pseudo - Differential Calculus 267

w l i i d ~ so far has been seen to hold for d _< m' 5 2s - d and which then turns out to l ) l * [,rue for Id - m'( 5 2s - 2d.

Now we are prepared to give the proof of Proposition 3.19.

I' I o o f of Proposition 3.19. We prove (3.49) in the form

(ml),(mi),d (R") A::)- (mb),d (R") .A8, el

d(mo+mb+mi). (mi), d, d+mb+mi (R") if O < r n b + r n l < 2 s - ' 2 d ,

(R") if - 2 s + 2 d 5 r n b + m l 5 0 . g { 8 v c '

(3.53) ma), (mb+ml+m:), d, d-mb-mi 4, C l

Izirst let d 5 rnb + r n l 5 2s - d. €+om (3.52) it follows that

(R" 1 (mo+mb+mi), (m;),d,mb+mi (3.54) dD, 2, D)P(D,X, D) E d8,

j<d k < d

where Pj = pj(D,x, D), p j E Ha- jS~'mb- ' (R" x R2") for j = O , l , . . . , d - 1, (lR" x R2") for k = 0, 1, . . . , d - 1. Then, for [Jk = qk(D,x,D), Qk E H8-kSc,

I = j + k < d, we obtain

ml - k,m;

Qk pj d(mo+mb+ni-l),(m;),d-l,mb+ml-l 6 4 , cl (R") 3

whereas, for j + k 2 d, it is checked that

(R"). (mo+mb+ml-d),(rn~).O,mb+mi-d Qk p 3 E d s - d

Now, if 0 _< mb + ml 5 2s - 2d, Proposition 3.18 yields that

(R") (ml ) , (m; 1, d (R") . A,, C l (mo-d), (mb+d),d

(mo+mb+ml), (m; 1, d , d+mb+ml

(R") = B8,d

c 4 C l

(rnl), ( m i ) , d (R*) .A8, c l A;;)3 (mb)? d

(R") *

For -2s + 2d 5 mb + ml 5 0 we argue in an analogous way just changing the r6les 0

Next we come to a discussion about commutators. Although it is possible to describe commutators in the calculus in general the formulas arising for remainders are

played by the A - and the f3 - classes. The proof is furnished.

268 Math. Nachr. 194 ( I !WN)

complicated. Therefore, we confine ourselves to the special case when one of thv iqi

erators involved has smooth coefficients and postpone the general ease to [25]. 'I' l i i+

special case is sufficient, e. g., in treating semi-linear partial differential equatitiiirc Furthermore, it meets our needs in proving (3.37) exactly.

The results on commutators could be achieved by considerations similar to t l i c MI!

above, but we prefer to take advantage of the elements of the calculus developrtl M I

far. In doing so, we have to anticipate two points of our later discussion: The firsf, O I I ~ '

is that to construct globally parametrices to elliptic operators within the calculus wv adjoin the pseudo - differential calculus of classical operators having their coefficitvic t4

in Cy(R") to our calculus, i.e., we work in the operator classes

(3.55)

Here L:+m' (R") is the space of classical pseudo -differential operators of order mi 111'

defined on R" with uniform symbol estimates in the space variables. Then it is ('ilsy

to see that our non-smooth calculus obeys the ideal property in the calculus givi~ii by (3.55), e.g., we have

The second point to anticipate is that operators have a symbolic structure. That poiiil will be discussed in detail in Subsection 4.1.

Proposition 3.21. Let s, m, m', mo E R, d E R, s > 5, d 1 1, lmol 5 2s - 'Ld

Then, for A E d8,c, (m), (m'),d (R"), P E Lr;O(R"), we have

(R") . (m+mo - 1 ), (m'), d - 1 , d+mo - 1 (3.56) [ P , A ] = P A - A P E d , - l , c l

Proof. We prove (3.56) in the form

(R") if 0 5 mo 5 2 s - 2 d ,

d(rn-l),(rn'+mo),d-l,d-rno-l a-1,cl (R") if - 2 s + 2 d 5 m o < O

(m+mo- l ) , (m'), d-1 , d+mo-1 4 - 1 , el

It holds that A P E A!I",:mo)'(m')'d(R") and P A E A!:{*(m'+mo)'d(R"). By Prol~i

sition 3-15, db:{'(m'+mo), d ( R n ) ~b:;mo)~ (m'),d,d+mo (R") if 0 5 mo 5 2s - 211,

(R") if -2s + 2d 5 rno 5 0. RII while thermore, both operators A P and P A have the same principal symbol. Therefoils,

In the end we complete the proof of Lemma 3.17. In the course of its proof (3.3h) shall also be verified. Then, as explained in the beginning of this subsection, we equilll~ obtain the remaining cases in Proposition 3.18, Proposition 3.19, and Proposition 3.2 I , as well as in Lemma 3.20.

(m), (m'+mo), d, d-mo (R") G &l (m+mo), (m'), d

(3.57) holds. I1

E n d of p r o o f of P r o p o s i t i o n 3.15 a n d L e m m a 3.17. By induction on d wi'

show that, for a symbol p E H8ScT1m' (R" x R'"), we have

(3.58)

\ V i t , t , , Pseudo - Differential Calculus 269

I f -2s + 2d 5 m' - T 5 0. This is relation (3.43), hence we obtain (3.38) and (3.37). A R shown at the end of the proof of Proposition 3.15, this gives us (3.35), for operators with asymptotic expansions of length at most d.

For the beginning of induction, d = 0, there is nothing to do. Thus let d 2 1 and hiippose that the proof for d - 1 has been supplied. By the usual tensor product rirgument, we may assume that p ( < , 2, r ] ) in (3.58) has product form, i. e., p ( < , 2, r ] ) = ([)"ql(<)a(z)qo(q), where a E Ha(R"), qo E Sg(R"), q1 E SCn;'-'(R"). Then we find

P(Dl2l D) = (Wa(z)ql(D)qo(D) + (D)' [Ql(D), .(.)I Qo(D)

(R") (m-I), (m'), d-1. d-m'+r-l (R") + 4 - 1 , C l ( m + m ' - r ) , ( r ) , d

(m+m'-r), ( r ) , d E d*,c1

= d8,d (R") 1

m+m'-r-I), ( r ) ,d - l , d+m'-r-1 (R") + A6-1,,, where in the second line the already proved part of (3.56) is used and in the third line I Ire. inductive hypothesis that (3.35) holds for operators with asymptotic expansions of length at most d - 1. We get (3.58), and this completes the inductive proof. 0

4. Further elements

In this section, several further elements of the classical operator calculus are developed. In Subsection 4.1, principal symbols and subordinated homogeneous compo- iicnts for complete symbols are introduced. The results obtained there are used in the 1)iirarnetrix construction for elliptic operators which is performed in Subsection 4.2. I?lliptic regularity and the Fredholm property for operators on closed compact mani- lidds are dealt with in Subsection 4.4. Before, in Subsection 4.3, we discuss coordinate iiivariance and operators on manifolds.

1.1. Symbolic s t ruc tu re

Next we become acquainted with the symbolic structure of operators in d~~~~(m')7d'd'(Rn). For those operators we have principal symbols as well as com- I'lete symbols, the latter circumstance is due to the fact that we are working on R".

Lemma 4.1. Let s, m, m', d' E R, d E N, s > + d, d 2 1, Id - d ' l s 2s - 2d. Let At, 5 1 11) E H 8 S C , m'm'(Rn x R'"). Then p ( D , 2, D) E dLr~~i'(m')'d-l'd'-l (IR") if and

only if p ( t , z, q) E H q - l s m ' (IR" x R2").

( D ) m ' ( q o ( z , D ) + Q'), where qo E H8S,;l(Rn x R") and Q' E dLr<ii'd-l'd'-l (R")l

(IR"). Writing p ( D , 2, D) = (m-I), (m' ), d- I , d ' - 1 Proof . First assumep(D,s ,D) E d8-l,cl

we are reduced to the case that m' = 0 and p ( = qo) E H8S,;l(Rn x R"). By assumption, there exists some t E [-s + d - min{d,d'},s - max{d,d'} - 11

~ u c h that p(zlD) E L ( H t + m ( R " ) , H t + l ( R " ) ) . Now suppose po(z,t) f 0, where IJO denotes the principal part of p . Choose some (zo,&,) E R" x (R" \ 0) such that p~(zo,Q) # 0. Further choose u E H t ( R " ) , u 4 H ~ ~ l ( ~ o , ( ~ ) , where the latter means u is not in Ht+' (R") microlocally at (ZO, t o ) . Then, for 4 E Cr(R"),

270 Math. Nachr. 194 ( I !Uh)

x E S:l(R") with 9 supported in a small neighbourhood of 20, ~ ( z o ) # 0, a i d \ supported in a small conic neighbourhood of (0, Ix([)l 2 c for [ E R" in SOIIIV

smaller conic neighbourhood of 6, 2 C, we find a symbol q E HaScTm(R" x I I i " ) such that p ( z , () q(z, <) = ~(z )x (<) . By Proposition 3.19 we have p ( z , D) q ( 0 , .is)

rO(z, D ) +R', where rO(t, () = p ( z , () q(z, <) and R' E da-l,;l (R"). This i r n ~ i l i i v i

#(z )x (D)u = p ( z , D ) q ( D , z ) u - R ' u E Ht+'(Rn), which contradictsu 4 H ~ l ( z o , ( o ) Hence, po(z,<) f 0, i.e., p E HaS2-'(R" x R").

I I

(-1) ( 0 ) d - 1

The reverse direction is straightforward.

Definition 4.2. Let s, m, m', d'ER, d € l N , s> f + d, d 2 1 , Id- d'l<2s - 2d. 1 , ~ t

further PEd!r",),'(m')'d'd'(R"). Then ( p o , p l , . . . , p d - - l ) , where for j = 0,1,. . . , d - I , p j E Ha-j S(m+m'-i) (R" x (R" \ 0)) is called the complete synbol of P if

holds, with 1c, being an arbitrary 0-excision function. po E HaS(m+m')(R" x (R*\O)) is called the principal symbol of P.

Each operator P E dt:{' ( m ' ) * d l d ' (R") has a complete symbol found by arranging t,Jw homogeneous components of the symbols appearing in Definition 3.2 of the operatot (D)-mf p (0)"' A;;Tm' ) ,dy d' (R"). As a consequence of Lemma 4.1 we obtaitr uniqueness of the components of the complete symbol.

For Q E Ba,cl ( m ) ' ( m ' ) ' d l d ' ( R " ) , define thecompletesymbolasd-tuple ( q o , q 1 , . . . , ~ d - ~ ) ,

where qj E H'-JS(m+m'-j)(R" x (R" \ 0)) for j = 0,. . . , d - 1, such that

holds. qo E HsS(m+m')(R" x (R" x 0)) is called the principal symbol of Q. Principal symbols behave in the same manner as they do in the case of pseudo

differential operators with C r -coefficients. We list some of their properties in Propo- sition 4.3. Proofs follow by examining the asymptotic expansions provided in the proofs given in Subsection 3.5.

Proposi t ion 4.3. The principal symbol of a n operator is independent of the reg). (R"), i. e., I / resentation chosen for this operator in one of the classes ds,cl

does not change altering the representation in accordance with Proposition 3.15. I / even does not change passing to the adjoint representation according to Pmpositiori 3.18. Under compositions of operators principal symbols are multiplied. The principnl symbol of the adjoint to an operator i s the complex conjugate of the principal symbol of that operator. The principal symbol of a commutator is the Poisson bracket of tlw principal s p b o l s of the operators involved t imes i-' .

(m), (m'), d , d'

Wit,t, Pseudo- Differential Calculus 271

Proof . We must be careful in compositions, since the proof of Lemma 3.20 has been rii.t,her implicit. But checking the single steps, the proof can also be accomplished in that, case. 0

The subordinated components of the complete symbol depend on the chosen repre- wntation. They also depend on the elliptic operator, (D), the powers of which enter irito the reduction to the standard operator classes. We shall provide, in [25], some formulas demonstrating how subordinated components alter when the representation is changed. The corresponding formulas for the classical situation are produced by c~illecting homogeneous components in the right way.

In the sequel let the symbol m be an abbreviation for the data set (m, m', d, d'). Let further the set (m - j , m', d - j , d' - j) be abbreviated by m - j.

As a conclusion to Lemma 4.1 we obtain the short exact principal symbol sequence:

Propos i t ion 4.4. Let s , m, m', d' E R, d E IN, s > +d, d 2 1, Id-d'l 5 2s-2d. Then we have a short exact split sequence

(4.3) 0 + dr--ltc,(R") --+ drcl(R") 4 HsS(m+m')(IRn x (R" \O)) + 0

with the natural injection and u being the principal symbol mapping.

Using subordinated symbols, for j = 0,1,. . . , d - 1, we obtain short exact split sequences:

j (4.4) 0 + d~-;<-l~cl(R") -+ dFCl(Rn) + @H8S(m+m'-k) (R" x (R" \O)) -) 0 .

k=O

Notice that for j = d- 1 the sequence (4.4) is a reformulation of (3.25) thus completing the discussion around the topologization of A,, cl (R?. (m), (m'), d, d'

4.2. The parametrix construct ion

Now we come to the parametrix construction for elliptic operators within the calculus. To start with record that a(.) + 0 holds as 1x1 + 00 for a E H'(R"), s > %. For that reason it is impossible to obtain uniform ellipticity estimates for operators in d~r",)l'(m')rd(R"). Moreover, the identity Id = op(1) does not belong to the calculus.

To get round this problem, we adjoin the classical pseudo-differential operator calculus, as mentioned in Subsection 3.5. Thus we work in the operator classes

(m) 9 (m') , d, F- (4.5) L;+m'(IR") + A,,, -.

Here L:+m' (R") is the space of classical pseudo - differential operators of order m+n' which are defined on R" with uniform symbol estimates in the space variables, i.e., the coefficients are taken from CF(R"). Recall that the non-smooth calculus has the ideal property in the larger calculus given by (4.5), e. g., we have

272 Math. Nachr. 194 (100H)

Notice that on C" -manifolds, X , to be considered in Subsection 4.3, in which ~ ~ I , N I I

the coefficients are from local Sobolev spaces, H,B,,(X), we have

Definition 4.5. Let s , r n , r n ' , d ' ~ R , ~ E I N , s > $ + d , d 2 1 , I d - d ' J 5 2 s - 2 d . T ~ I I an operator A E L~+m'(IR") + d(m)'(m')*d'd' I , cl (IR") is called elliptic if it has an ellip1.h principal symbol ao(z,() E C,OOS(m+m')(R" x (IR"\O)) + H'S(m+m')(IR" x (R"\O)), i. e., ao(z, () satisfies the estimate

(4.7) laO(z1OI 2 wlm+m' for all (z,() E R" x (IR" \ 0 ) , and certain constant 6 > 0.

Definition 4.6. Let s, m, m', d' E R, d IN, s > $ + d, d 2 1, Id - d'I i

2s - 2d. Let an operator A E L:+"'(Rn) + d8,cl (m)'(m')vd'd'(IR") be given. TIII~II

P E L;"-"'(IR") + da,,[ (IR") is called a paramet* to A if i t satisfics (-m' ), (-m), d , d'

(R") 1 P A - Id E ( - m ) v Ot d'-d (4.8)

(R") * (-m'-d), (m'),O, d'-d (4.9) AP-Id E d a - d

(R") is immediate from (4.8), (4.9) if i t Uniqueness modulo da-d

Preliminary to Proposition 4.8 we establish a lemma providing the formal Neumaim

(-m'-d), ( - m ) , O , d ' - d

parametrix exists.

series argument required in the parametrix construction:

Lemma 4.7. Let s, m E R, d E IN, s > f + d, 3d 5 2s + 1, d 2 1. hrther assuirw that C E L,'(IR") + d ~ ~ ~ ~ { ' ( - m ) ' d - l (IR"). Then

(4.10) (Id - C) (Id + C + C2 + . . * + Cd-' ) = Id

(R"). (m-d). (-m), 0 i s satisfied modulo da-d

Proof. In view of Proposition 3.19 we obtain by induction

(4.11)

for j = 0 , 1 , . . . , d. Then the assertion follows from

and

(Id - C) (Id + C + C2 + . . . + Cd-') = Id - Cd n

\Yit.t., Pseudo - Differential Calculus 273

Notice that in the proof of Lemma 4.7 we get another interpretation for d' addition- rilly appearing in Definition 3.9. Now i t yields that the expansion given by (4.11) is risymptotic.

Proposi t ion 4.8. Let s, m, m', d' E IR, d E IN, s > f + d , 3d 5 2s + 1, d 2 1, Id - d'l 5 2s - 2d. Then, for an opemtor A E J~Z+~ ' ( IR") + d,,=, (m)*(m'),d?d'(Rfl), the /idlowing conditions are equivalent:

(a) A i s elliptic in the sense of Definition 4.5. (b) There ezists a pammetriz P E L;"-"'(R") + d~~~') ' ( -m)~d'd' (Rn) to A .

Proof . It suffices to deal with A E Lz+m'(IR") + dL:{*(m')3d (R"), since

Suppose that A is elliptic. Let

ao(z, €) E C~S(m+m')(IR" x (R" \ 0)) + H*S("+m')(IR" x (R" \ 0)) be the principal symbol of A . It is clear that, under condition (4.7), po(z,<) = o.o(z, <)-l belongs to CpS(-"-"')(IR" x (R"\O))+H'S(-m-m')(R" x (R"\O)). Thus iising the short exact sequence (4.3) (and its analogue for classical pseudodifferential operators with coefficients in Cp(R")) we find an operator PI E L;"-"'(IR") + d~~~''~'-m''d(R") having po(z, () as its principal symbol. Then by the composition rules for principal symbols and the short exact sequence (4.3) again we conclude that

Therefore, by Lemma 4.7, P = (Id + C + Cz + + Cd-') F'1 is a left parametrix to A . In a similar manner, a right parametrix to A can be constructed. The argument is completed by noting that a left parametrix and a right parametrix if they exist are

The reverse direction follows from the composition rules for principal symbols. I7

We conclude this subsection by remarking that the parametrix construction in Proposition 4.8 can also be achieved either only locally, or on a closed compact manifold as discussed in Subsection 4.4, or right form the beginning in larger classes of operators taking their coefficients in Hioc -spaces. But the element of adjoining the smooth calculus to the non - smooth one is familiar in the analysis of non - linear partial differential equations; thus we have decided to do the constructions in the indicated way.

equal modulo d!~~'-d)'(-"')'o (R").

4.3. Coordina te changes and operators on manifolds

In this subsection we discuss the invariance of operator classes dL~~t(m')' ' I d ' (IR" 1 under coordinate changes and introduce the classes da, (m)t(m')>d!d'(X), c, X beingaCM- manifold.

274 Math. Nachr. 194 ( I ! l I l N )

Our first goal is to explain the action of the pseudo-differential operator p(D, : I ,? / I )

for p E H;b,Sz'm' ( X x IR2"), s > f, and X being an open set in R". The exprcssioii p ( D , x, D) will in general not be defined. This corresponds to compositions ill ~,Iicl

C'-theory in which one of two operators has to be properly supported. HcIv \\'I'

have to make a similar assumption. We discuss only a rather special case, wlIic4, however, suffices for the applications we have in mind.

Notice that, for 4 E Cr(R"), we have 4(x)p((,x,q) E H'S2'm'(R" x R.2") I I p E H/ocs;*m' (x x IR2").

Lemma 4.9. Let s , m , m ' ~ R , s > p. Let f u r t h e r p ( ( , x , ~ ) ~ H ~ o c S ~ ~ m ' ( I R " x 113.") (a) Suppose that the Fourier transform F'2p{p((,z,~)} is compactly support iir / I

uniformly in (x, q). Then p ( D , x, D ) induces a continuous operator

(4.12)

for all t E R, (ti 5 s. (b) Suppose that the Fourier transform F,&,, { p ( ( , z, Q)} i s compactly support in p'

uniformly in (<, x). Then p ( D , x, D ) induces a continuous operator

p(D,x,D) : Ht+rn(IR") + H;,-,m'(R")

(4.13) p ( D , x , D ) : H,t,+,mP(R(IR") 3 Hi-' (R")

for all t E R, It1 5 s .

Proof. (a) Suppose that supp(F;' {p((,z,q)}) E K x IR" x R" for some compac*t set K c R". Then, for 4, @ E Cr(R -rI ) satisfying (supp$-supp4')nK = 0, we haw

4(x) (4'(Y) P ( t l Y I Q ) ) (Dl x, 0) = 0 .

Let u E H'+rn(IRn) for t E IR, It1 5 s. In order to define p(D,z,D)u, we defiiio p ( D , x, D)u on K' for any compact set K' c IR". To this end, choose functions 4, 4' E CF(IR") such that 4 = 1 on K', 4' = 1 in a neighbourhood of K - supp4 aritl set

P ( D , 5, D)u = dJ(x) ($'(PI PKl Y, v)) (Dl x,

on K'. It is seen that this definition is independent of the functions 4, 4' with tlw above properties, consistent on intersections of compact sets K' and agrees with thv usual definition in the case that p E H'Sz*m' (IR" x R2").

R" x IR" x K for some compavt set K C R". Then, for 4, 4' E CF(IR") satisfying (supp4 - supp4') n K = 0, wv have

(b) Now suppose that supp (Fv2p, { p ( ( , x, q)})

(P(& x, Q) 4(4) (D, 5, D) 4 ' (4 = 0 * Let u E H~O+mmp(IRn) for t E R, It1 5 s. We choose functions 4, 4' E Cr(R") suclr that 4' = 1 on suppu, 4 = 1 in a neighbourhood of K + supp4' and set

P(D, x , m = (P(& 21 Q) 4 W ) (D, 2, D)d'(x)u.

\VII,I,, Pseudo - Differential Calculus 275

I 1 is seen that this definition is independent of the functions 4, 4' with the iII)ow properties and agrees with the usual definition in the case that / ' C / P s y q R n x R2n) . 0

Now let p ( ( , x , q ) E H&72'm'(X x R") with X being an open set in R". As I I cwollary to Lemma 4.9 we obtain that we can give a meaning to the expression I ) ( 11, x, D) as a continuous operator

(,1.14) P(D,Z,D) : H::g(x) + H;im'(X)

lor all t E R, It1 5 s, if one of the assumptions of Lemma 4.9 is satisfied. Notice that for symbols p ( ( , x , Q) E HsS2'm' (R" x R") we can always assume that

o i i c or both assumptions of Lemma 4.9 are satisfied. To see this, choose a 0-excision function 11, and consider, e. g.,

( 4 . 1 5 ) FP+& - @(P))F;: ,{P(l ,x ,Q)}} E H"scY1m'(~" x R2").

'I'his symbol fulfills assumption (a), while the symbol FP+ { $ ( p ) F<d,{p((, x,~)}} I(3ads to an operator which belongs to ,C(H-s+m(R"), H ' ( R " ) ) for any t E R.

We need a technical lemma:

Lemma 4.10. Let s, m, m', d' E R , dEIN, s> f + d , Id - d ' l 5 2 s - 2d. Let further pEd::)I~(m'),d,d' (R") and suppose that, for the kernel K of P, we have that K ( z , p) = O f o r x , y E I R n , Iz-vJ-<6, andsome6>0. ThenPEda-d rn-d), ( r n ' ) , O , d ' - d

Proof. We may reduce to the case m' = 0 and P E dL:{'d'd'(R"). Then it is

(R") if d 2 1. (m - 1 ) , d- 1, d' - 1 sufficieflt to prove that P 6 ds-l,cl

KO be the kernel of po(z , D) and K' be the kernel of P'. Then Ko(z, y) = Lo(z, z - p), where Lo(z,p) = F;~,,{po(z, 7)). In particular, $ ~ ( p ) Lo(z,p) E H'(Rn)6?,S(Rn) for an arbitrary 0-excision function +. Choosing $ E Cm(R") in such a way that $ ( p ) = 1 for lpl 2 6 we arrive at a decomposition (1 - $(z - y))Ko(z, p) + $(z - y)Ko(z, p) + K ' ( s , y ) for the kernel K of P , where (1 - $(z - y))Ko(z,y) and $(z - p)Ko(z,y) + K'(z,y) give rise to operators in d6r,t\'d-1'd'-1(R"), since (1 - $(z - p))Ko(z,y)

is supported in 1x - y( 5 6. Hence, we have P E d8-l,cl

is accomplished by the next lemma:

Wri teP = p ~ ( s , D ) + P ' w i t h p o E H " S ~ ( R " x R " ) , P' E d(m-l)*d-l'd'-l 0-1, el (R"). Let

(R"). 0 (m-l ) ,d-1 ,d ' -1

The transition to the operator classes mentioned in the end of the previous subsection

Lemma 4.11. Let s, m, m', d' E IR, d E I N , s > f , Id - d'I 5 2s - 2d. Let further, for some open subset X C R", P : C r ( X ) + V'(X) be a linear continuous operator such that for any 4, 4' E Cr(X) the operator S(R") 3 u t) 4'P(#u) E S'(Rn) belongs to dB,C, (m)' (m')' d 7 d' (R"). Then there are symbols

p j ( I , s,q) E H ~ ~ ~ ~ s ~ - J ~ ~ ' ( X x R 2 " ) for j = O , l , . . . , d - I

276 Math. Nachr. 194 ( I ! ) B H )

satisfying one of the conditions of Lemma 4.9 and an operator

s-d-rnax{ d,d' }

p d E n L: ( ecp (X 1 1 H;Lm'+d (XI) t=-r+d-rnin{d,d'}

such that d-1

j=O

In the notation of the operators S(R") 3 u t+ (b'P(c#m) E S'(R") in Lemma 4.1 I we have omitted the extensions to R" and the restrictions to X , respectively.

P r 0 of of Lemma 4.11. Choose a partition { ( b k } k e N of unity on x and choow fUnCtiOnS $k E cr(x) Such that (bk$k = (bk for all k E ]N and {supp$Jk}kEn\l iS I1

locally finite cover of X. Then, for u E Cp(X), we have

k=O k=O k=O

(R") for all k, 1 E IN. ( m - d ) , (m ' ) ,O,d ' -d By Lemma 4.10, we have (bi ( 1 - $ k ) P q j k E dr-d

Moreover, for u E H ~ , + , ~ ~ ( P ( x ) , the sum C z o ( l - $ k ) P ( ( b k u ) is finite. Consequently,

s-d-rnax(d d ' } Thus, ( 1 - $ k ) P ( b k E nt=-s+d-min{d ,d ' ) ~(~'+m(x),H,k;" '+d(X)) for dl k E IN.

BY assumption, the operators $ k P ( b k belong to d~~{"rn" 'd ' s (R") for all k E N .

(R"). We can arrange thai. j = o , 1 , . . . , d - l , k E ~ , a n d P d k E d , - d

the p j k ( [ , 2, q)'s satisfy, e. g., the first assumption of Lemma 4.9 and , for fixed j , their supports in z form a locally finite cover of X uniformly in ([, q). For the kernels of tlio P d k ' S , we may assume that their supports, both in z and y, form locally finite coveI.s of X. Then, on X ,

Write $ k P d k = C ~ ~ ~ p j k ( D , Z , D ) + P j k , where p j k E H'sZ-J'm'(Rn x RZn) for' (m-d), (m ' ) ,O ,d ' -d

satisfies the first assumption of Lemma 4.9. Thus, for j = 0,1 , . . . , d- 1, these symbols give rise to operators p j ( D , z l D ) : H;o+,",(x) -i H:~~ ' (x ) for every t E IR, It1 5 s . Eventually, the sum CCp=, P d k exist, and

e-d-max{d ,d ' }

F P d k E n k=O t=--r+d-min{d.d'}

\VII I I'srudo - Differential Calculus 277

' 1 ' 1 1 1 ~ pioof is finished. 0

Nvx 1. we are concerned with the invariance of d,, (m)'(m')'d' 'IR") under global changes 111' (soordinates of R". Let n : R" -+ Rn be a diffeomorphism of R". In the sequel, WI' dmll always assume that there are constants c1, c2 > 0 and c, > 0 for (Y E IN" siic.11 that

( , I . 18) c1 5 1 de td (z ) ( 5 c2

i l l l t l

(4.19) I % w ) l I ca Irolt l for all ZER". Recall that, for functions uES(R") and distributions vES'(R"), I I I P pull -back K*U is defined by K * U ( Z ) = u(n(z)) and the push- forward n8z) by

(LV, 4) = (v, n.4 I det I C ' ~ ) , 4 E S(R") .

I ly (4.18), (4.19), we have n* : S(R") + S(R") and K. : S'(R") + S'(lRn). Fur- I.lic:rmore, n* = (n-')* holds on functions, since distributions are transformed as I densities.

For P : S(R") + S'(R"), we define the operator P, : S(R") + S'(R") by

Proposition 4.12. Let s , m, m', d' E R, d E IN, 8 > f + d, Id - d'l 5 2s - 2d. Let further IC : Rn + R" be a diffeomorphism satisfying (4.18), (4.19). Then, for

(R"), we have (m),(m'),d,d' 1' E 4 cl

(4.21) ' E 4 c l (R"), (m),(m'),d,C

Proof . It suffices to show that the conclusion holds for the operator classes ~ i ; ! l ~ , ~ ' (R"), since

ilrld the behaviour of classical pseudo - differential operators with coefficients from C,y(IR") under coordinate changes is known.

The mapping K* : S(R") + S(R") extends by continuity to an isomorphism K.* : HL(R") -+ H t ( R " ) for every t E R. Thus A E d~~,lf;d)'o'd'-d(IR") implies

(R"). Therefore, we may assume that P = p ( z , D ) holds for some symbol p ( z , q ) E H6Sz(R" x IR"). Then, for the symbol p , of the operator P, = p R ( z , D), we claim that

(4.22) p,(~(z), q) = e - iK(2 )qp( z , D)e'+)V .

This is immediate for symbols in product form, i.e., p ( z , q ) = a(z)po(q) with a E H'(R"), po E S~(R"), from the. corresponding result in the COO-situation

m-d),O, d ' - d E d!-d

278 Math. Nachr. 194 (199H)

(see, e.g., (11, Theorem 18.1.17)). In that manner we obtain a continuous bili~~oiri mapping

H'(R") Sz(IR") --$ H'Sz(IR" IR") 1 (aim) a ( K - ' ( 2 ) ) ( P o ) K ( V )

showing that pK(z, r ) ) defined by (4.22) indeed belongs to H"Sz(R" x R") for p ( z , 7) (

H'Sz(R" x R") by invoking the usual tensor product argument. That p,(z,r/) ih

actually the symbol of P, now follows from the fact that it is true on H"(R")@Sz(lR") and that the mapping S(R") + S'(R"), u --t p , ( z ,D)u , is continuous for eadi ZL E S(R"). L I

In the proof of Proposition 4.12 we have shown that the principal symbols of P arul P, are interrelated by

i. e., as usual, the principal symbol behaves like a function defined on T'IR" \ 0.

that In the situation considered in (4.22) we get as in the case of coefficients in Cr(IR.")

in H"S"(R" x R") holds, where p z ( g ) = K ( V ) - K ( Z ) - n'(z)(y - 2). Recall thiit the Do U (e'j'=(V)q ) I y=z 's are polynomials in r ) of degree less than or equal to 14 /2 wit11

coefficients in Cr(R"). One obtains, from (4.24), formulas for the components of lower order of the complete symbol of P, by replacing p in (4.24) by the complet,ci symbol of P and ordering terms with respect to homogeneity.

We go now over to the operator classes d!r",), '(m')'d'd'(X). Fkom now on, let A' denote a C" -manifold.

Definit ion 4.13. Let s, rn, rn', d' E IR, d E N, s > 5 + d , Id - d'l 2s - 2d. Theii d~~{'(m')*d'd'(X) denotes the class of all operators P : Cr(X) + D ' ( X ) such that. for every chart (Y, K ) and arbitrary 4, 4' E CF(Y)

(4.25)

holds.

Note that, by Proposition 4.12, it is enough to ask (4.25) only for a collection of charts ( Y k , ~ k ) such that the Yk x Yk's cover X x X. Further note that, for X = m.", the operator classes ds, (m)' cl (m') 'd' d l (X) introduced in Definition 4.13 are different from

the classes d8, (m)' cl (m')' d' '' (R") considered earlier in that respect that the behaviour of coefficients is now not restricted as 121 + 00.

which are immediate from our foregoing considerations. Below we summarize some of the properties of the operator classes d ~ ~ { ' ( m ' ) s d * d ' (x')

\Yit.t., Pseudo - Differential Calculus 279

I’roposition 4.14. Let s, m, m’, T , mo, mb, ml, mi, d’ E R, d E IN, s > f + d , I t / .- (1‘1 5 2s - 2d, Im’ - T I 5 2s - 2d, Imb + mlI 5 2s - 2d. Then the following prvlierties are valid:

( 1 1 ) W e have

s-max{d,d’)

(XI c n L:(H:,+,”,(x)l H;im‘(X)) 1 ( , 1 .2~ ) dbr”,’,.(”’)t d,d‘

t=-s+d-min{d,d’)

c ir i r l the remainder classes are completely characterized by that property, i. e . ,

8-d-max{d,d’)

(XI = n L:(H:,+,~~(x), H;T’+~(x)) . (m-d), (m’).O,d‘-d (ei.27) ds-d t=-8+d-min{d,d’}

I r r (4.26), for properly supported operators, one can replace either Hi:,’(X) b y ll/Lm(X) or H;im’(X) by H;,-,”d(X).

(b) We have

(XI , (m+m’-r),(r),d,d+m‘-r

(X) c (4.28) (m’)*

( c ) The adjoint to an operator in d~~{’(m’)’d’ d‘(X) belongs to d!r”:,+d’)v(m-d’)ld’d’

(d) For the composition of P E ds,c, (molt (mb),d (X)l Q E Aqcl (ml) ’ (m~)*d(X) , where one (S).

(ij these operators is properly supported, we have

(4.29)

(m), (m‘), d , d‘

(S) we introduce the space H~o,S(m+m‘)(T*X\O) as the space of all functions p(zl() on T*X \ 0 which are homogeneous of order m + m’ in the fibres and which, in any c:hart on X , belong to HfbcS(m+m’)(R” x (R”\O)). It is plain that an operator P E ,4~r”,{’(m’)’d’d’(X) has a uniquely defined principal symbol ~ p ( z , <) E H/ocS(m)(T*X\O). A s in (4.3), we obtain a short exact split sequence, i.e.,

(4.30) 0 + dy--ltc,(X) -+ dzc , (X) 4 H’ 1 oc S(m+m’)(T*X\O) -+ 0 ,

where m = ( m , m ’ , d , d ‘ ) . An operator A E d8,c, (m)’ (m’)’ d l ” (X) is called elliptic if its principal symbol U A (2, () never vanishes on T*X \ 0. In that case, using a partition of unity, a parametrix P E d8,c, (-m’)’(-m)’d’d’(X) to A can be constructed, i. e., a properly supported operator P satisfying

In order to define the homogeneous principal symbol for operators in da, el

Conversely, from the existence of a parametrix we conclude on the ellipticity of A. Again, natural F’rCchet topologies on the operator classes ds, cl

duced also using a partition of unity. (m)9(m’) ,d*d’(x) are intro-

280 Math. Nachr. 194 ( l ! ) l ) N )

4.4. Elliptic regularity, parametrices, and the F'redholm property

In this subsection we quote results when X is a closed compact manifold. ' l 'I i i*t i the spaces H;oc(X), H:omp(X) become replaced by H t ( X ) . Note that, for t' > t., t,Iw embedding H"(X) cs H ' ( X ) is compact. The spaces H,"ocS(m)(T*X \ 0) arc IIOH'

denoted by H'S(m)(T*X \ 0). In case X is compact, ellipticity is equivalent to the F'redholm property:

Proposition 4.15. Let s, m, m', d' E R, d E IN, s > f + d, 3d 5 2s + 1, d 2 I , Id - d'l 5 2s - 2d. Let X be a closed compact C" -manifold. Then, for an opefur/rji A E dL:{* (m')* d , d' ( X ) , the following conditions are equivalent:

(a) A is elliptic. (b) The operator A : Ht+"(X) + H L - m ' ( X ) i s h d h o l m for some (and then j ~ i

all) t E (-s + d - min{d,d'},s - max{d,d'}). In that case, there exists a parametriz P E d L ~ ~ ' ) v ( - m ) ' d ' d ' ( X ) to A . M o r e o w ,

elliptic regularity holds, i . e. , u E H-a+m+d-min{did') ( X ) , Au E Ht-"' (X) for S O ~ M

t E [ - s + d - min{ d, d'} , s - max { d , d'}] implies that u E Hf+m ( X ) .

Proof. Suppose that A : H"+"(X) + HL-m' ( X ) is a Fkedholm operator for I I

certain t E [-s + d - min{d, d'},s - max{d,d'}]. By order reduction and some otlli~i manipulations we may assume that m = m' = 0, d = d', t = 0, and A E d r k t ( X ) .

We use a device from [19] to recover the principal symbol QA(z,<) of A: Givcbii (so,<o) E T*R" \ 0, there exists a family of unitary operators Rx, X > 0, on LZ(IR") such that, for u E L2(R") , Rxu + 0 weakly in LZ(R") as X -i 00 and

(4.32) R, ' (A+K)Rxu + U A ( Z O , < O ) U in L2(Rn) as X + 00

for A E drkt(R") and any compact operator K on L2(IR"). A family of operators RA, X > 0, obeying these properties is given by

(4.33) R ~ u ( ~ ) = xnI4 eixzto U ( X ' / ~ ( Z - z0)).

For details, see [19, Section 2.3.4). Using a partition of unity, we find, for fixed (20, (0) E T'X \ 0, a family of isomot

phisms RA, X > 0, on L 2 ( X ) such that RA, R,' are uniformly bounded in norm f o i

X > 0 independently of (zo,<c,) E T ' X \ 0, further, for u E L 2 ( X ) , RAU + 0 weakl~~ in L 2 ( X ) as X + 00 and

R ~ ' ( A + K ) R ~ ~ + O A ( S ~ , < ~ ) ' L L in L 2 ( X ) as X + 00

for A E d F L f ( X ) and any compact operator K on L z ( X ) . Then, if A E drb,d is 4 1

Fkedholm operator on L 2 ( X ) and P E L ( L 2 ( X ) ) is a Fkedholm parametrix to A, i. ( I . ,

P A - Id, A P - Id are compact operators, we get, for u E L 2 ( X ) and K = P A - ] ( I ,

Wlc I , I'seudo - Differential Calculus 281

wlicw~ 11 11 is the norm on L2(X) and C > 0 is some generic constant. Now, if X tends I I I PO, we find, for each u E L 2 ( X ) , u # 0,

( $ 1 M ) 0 < lbll 5 C l ~ A ( Z O i < O ) l 1b11 I

WII 1 1 C' > 0 being independent of (20, ( 0 ) E T'X \ 0, yielding the ellipticity of A . Ckmversely, if A E A!"' (m')' '"' (X) is elliptic, then a parametrix exists. The exis-

0 I (*ii(v of a parametrix implies elliptic regularity and the F'redholm property.

b. Notes and remarks

We conclude with further notes on topics discussed in previous sections. Our ex- 1)osition in this paper is based on the classical stock of pseudo-differential calculus. I;oI references concerning the theory when the operators have smooth coefficients, the iviitler is referred to standard textbooks, e. g., [6], [Il l , [21]. The calculus for pseudo- ilifierential operators with non - smooth coefficients presented in this paper has been riiinounced in [23, Chapter 4.21.

We add references which are close to the subject treated in the body of the paper. Ihrther references may be found in papers cited in the bibliography.

Section 2. The main attribute avoiding the complications in the treatment of non - rlitssical operators in [25] is the nuclearity of Sz(IR"). This property was recognized I)y MANTLIK (Dortmund). He observed that under radial compactification of R" and iippropriate renormalization Sz(R") is transformed into the space Cm(B"). Here B" is the closed unit ball in R", and expansions of symbols in Sz(R") into homogeneous c,omponents correspond to Taylor expansions near the boundary 81".

The topologization of the symbol spaces Sz(R") is taken from [20] and represents ill a closed form the construction given there (cf. (2.6)). The proof announced for Proposition 2.4 that does not rely on the result Sp(IR";E) = S ~ ( I R " ) & E is briefly is follows: We have that Sz(lR") @ E is algebraically a subspace of S z ( R " ; E ) . The induced topology is that of the injective tensor product. Sz(IR";E) becomes a subspace of L ( S z ( R " ) ' , E ) via the mapping a e (Q I+ (@,u)). Here Sz(R")' is the strong dual to Sz(IR"). Note that functionals Q E Sz(R")' can be applied to symbols ( 1 E Sz(R"; E ) yielding elements in E. It is seen that Sz(IR"; E ) carries the topology induced from C (Sz(lR")', E ) when the latter is equipped with the topology of uniform convergence on all bounded subsets of Sz(R")'. We further have L ( S z ( R " ) ' , E ) = Sz (R" )&E (see, e. g., [13]), since Sz(R") is a nuclear F'rCchet space. Therefore,

and Sz(IR"; E ) = Sz(R")&E. The symbol classes Sc;llm'(IRn x El") were introduced by HIRSCHMANN in connection

with a pseudo - differential calculus on IR" for operators having symbols the coefficients of which satisfy certain exit conditions at infinity. From (2.11), (2.12) and the same with the rdles of j, k interchanged it is seen that the definition given in (91 agrees with that one used above.

282 Math. Nachr. 194 (l!NJM)

Section 3. The structural aspects of the pseudo - differential calculus which Iiavci been considered follow the general framework of a pseudo-differential theory. III particular, we had to take care in two respects: only finite asymptotic expansioiih are allowed in the calculus, and the components in these asymptotic expansions J ~ I V

generally of the form p(D, z , D ) . The latter requirement causes that so-called spectd conditions otherwise to impose on the symbols are avoided.

The symbols considered in this paper are infinitely differentiable in the covarial)loH Sometimes one is interested in symbols satisfying weaker differentiability conditiorih Here we did not go into this question, and no attempts were made to obtain opt,iiiiril results. In [25], after the non-classical operator calculus will have been establislwil, we will come back to that topic and will make several comments on it.

It is natural to provide the classes d6T",i"""' d'd'(R") with suitable F'rCchet topolo gies. For example, using these F'rCchet topologies one confirms oneself that thc iii clusions stated in Propositions 3.10, 3.15, 3.18 and 3.19 hold in a topological SCIISI ' .

The operator classes A, (m)'(m')'d'd'(R") to be introduced in [25] will turn out to 1 1 1 3

Banach spaces mainly due to the fact that only finitely many derivatives with respwl to the covariables are needed in estimates. In any case we have

and the embedding is continuous.

Section 4. We compare the pseudodifferential calculus for operators with iioii

smooth coefficients developed in this paper to other possible alternatives. In our ciil.

culus, the operators have coefficients in L2 - Sobolev spaces H'(R"), while, rouglily speaking, in BONY'S paradifferential calculus the coefficients are taken from Holdci Zygmund spaces. One advantage in our approach manifests in the simpler estimaf,cin used not built on Littlewood-Paley decompositions. Moreover, e.g., in view of i l l ) .

plications to quasilinear hyperbolic equations it is desirable to demand coefficient5 in L2 - Sobolev spaces, since solutions are looked for in energy spaces. There aicb, however, applications in nonlinear partial differential equations which oblique one 1 I I

leave the range of applicability of L2 -theory. In such instances it would be bet,t,oi to have a calculus at one's disposal in which the coefficients are permitted in moio general function spaces, e. g., in Besov - or Bessel- potential spaces. In a future pi per, we shall lead into such pseudo - differential calculi along the lines stressed in this paper. In the global parametrix construction in Subsection 4.2 we had to adjoin f.Ii(, operator classes Lz+m' (R") to our calculus to obtain uniform ellipticity estimatw since functions in H6(R") vanish at infinity. This setting-up and also some troublia in formulating the commutator results would be prevented when working from t J i c t

beginning with coefficients, e. g., in Holder - Zygmund spaces. A further subject not touched upon in this paper concerns microlocal analysis, biil

it has been already used in embryo in the proof of Lemma 4.1. Questions related 1.o microlocal analysis and similar questions are also intended to publication in a futui o

paper. Here we only note that the expected effect that proofs simplify considerably i r i

dealing with classical instead of non - classical operators again occurs.

\V I I I , I'scwtlo ~~ Differential Calculus 283

A vk iiowledgements

'/'h rrisearch was supported by the Deutsche Forschungsgemeinschaft. I thank MICHAEL I ) I I I , ; I I I , ; I ~ (TU Bergakademie Fkiberg) for pointing out an error i n an earlier version of the J I I I I I J / O J ~ Lemma 3.17.

I ~ I S A L S , M. , and REED, M.: Propagation of Singularities for Hyperbolic Pseudodifferential Og wiitors with Nonsmooth Coefficients, Comm. Pure Appl. Math. 36 (1982), 169- 184

I j ISALS, M., and REED, M.: Microlocal Regularity Theorems for Nonsmooth Pseudodifferential Operators and Applications to Nonlinear Problems, "lkans. Amer. Math. SOC. 286 (1984), 159- I84

I ~ O N Y , J . - M.: Calcul Syrnbolique et Propagation des SingularitiC pour les Equations aux 1)6riv6es Partielles Non Linbaires, Ann. Sci. Ecole Norm. Sup. 14 (1981), 209-246

I jONY, J . - M . ' Second Microlocalization and Propagation of Singularities for Semi -Linear I lyperbolic Equations. In: MIZOHATA, S. (Ed.), Hyperbolic Equations and Related Topics, 'I'aniguchi Syrnp. HERT, Katata, 11 - 49, 1984

I % O U R D A U D , G.: Une Alghbre Maxirnale d' Operateurs Pseudo - Diffhrentiels, Comm. Partial Ilifferential Equations 13 (1988), 1059- 1083 CHAZARIN, I., and PIRIOU, A,: Introduction to the Theory of Linear Partial Differential Q u a - lions, Stud. Math. Appl., Vol. 14, North - Holland, Amsterdam, 1982

CHEN, S.: Pseudodifferential Operators with Finitely Smooth Symbols and Their Applications 1.0 Quasilinear Equations, Nonlinear Anal. 6 (1982), 1193- 1206

DEFANT, A . , and FLORET, K.: Tensor Norms and Operator Ideals, Math. Studies, Vol. 176, North - Holland, Amsterdam, 1993

I i I R s c H d m N , T.: Pseudo- Differential Operators and Asymptotics on Manifolds with Corners, Va. Supplement, Pseudo- Differential Operator Calculus with Exit Conditions, R- Math -04/91, Report, Karl - Weierstrafl- Institut fur Mathematik, Berlin, 1991

HORMANDER, L.: Pseudo- Differential Operators, Comrn. Pure Appl. Math. 18 (1965), 501 -517

HORMANDER, L.: The Analysis of Linear Partial Differential Operatom 111, Grundlehren Math. Wiss., Vol. 274, Springer -Verlag, Berlin, 1985

1121 IjORMANDER, L.: Pseudo-Differential Operators of Type 1,l. Comm. Partid Differential Q u a - t,ions 13 (1988), 1085-1111

I131 JARCHOW, H : Locally Convex Spaces, Math. Leitfiden, Teubner, Stuttgart, 1981

11.11 I C O H N , J . J., and NIRENBERG, L.. An algebra of Pseudo-Differential Operators, Comm. Pure Appl. Math. 18 (1965), 269-305

1151 K U M A N O - GO, H., and NAGASE, M.: Pseudo- Differential Operators with Nonregular Symbols and Applications, Funkcial. Ekvac. 21 (1978), 151 - 192

I I fi] MARSCHALL, J .: Pseudo- Differential Operators with Coefficients in Sobolev Spaces, 3 a n s . Amer. Math. SOC. 307 (1988), 335-361

[ 17) MEYER, Y .: Remarque sur un Thhorhrne de J. - M. Bony, Rend. Circ. Mat. Palermo (2) Suppl. 1

[ 181 NACASE, M. : The LP - Boundedness of Pseudo- Differential Operators with Non- Regular Sym-

1191 REMPEL, S., and SCHULZE, B.-W.: Index Theory of Elliptic Boundary Problems, Math.

(1981), 1-20

bols, Comrn. Partial Differential Equations 2 (1977), 1045- 1061

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1201 SCHULZE, B. - W.: Pseudo - Differential Boundary Value Problems, Conical Singulariticx ntiil

1211 TAYLOR, M.: Pseudodifferential Operators, Princeton Math. Ser., Vol. 34, Princeton Univ. I ’ Iw ,

(22) TAYLOR, M.: Pseudodifferential Operators and Nonlinear PDE, Progr. Math., Vol. 100, l l l t k

1231 WITT, I.: Non -Linear Hyperbolic Equations in Domains with Conical Points, Math. Restwc 1 1 ,

1241 WITT, I.: On a Topology on Symbol Classes of Type 1,0 (submitted to J. h n c t . Anal.) [25] WITT, I.: A Calculus for Non-Classical Pseudo-Differential Operators with Non-Snicwlli

Aaymptotics, Math. Topics, Vol. 4, Akademie Verlag, Berlin, 1994

Princeton, New Jusey, 1981

hiuser Boaton, Boston, 1991

Vol. 84, Akademie Verlag, Berlin, 1995

Symbols (in preparation)

Arbeitsgruppe “Portielle Differentiolgleichungen und Kompleze Analysis” Llniuersitht Potsdom Institut f ir Mothemotik Postfoch 60 15 53 D - 14415 Potsdom Germany e -moil: ingoompg - ono.uni -potsdom.de

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