Fattorini H.O., Kerber A. The Cauchy Problem

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4.9. Analyticity of Solution Operators

We have

- (p -2) ffI

uip-4(E EajkRe(uDju)Re(uDku)) dx

+ S(p -2)feI

p-4(E EajkRe(uDju) Im(uDku)) dx

L,ajkDJuDku)dx

Iuip-4(y

LajkJjk) dx,

257

(4.9.5)

where

Jjk =

(uD'u)(uDku)

+ (p - 2)(Re(uDju) Re(iiDk u) ± S Re(uDju) Im(uD ku)}.

(4.9.6)

Let z1,. .. , z,,, be arbitrary complex numbers. Consider the matrix

Z = (z jk) with elements

Zjk = ZjZk + a((Rezj)(Rezk)±8(Rezj)(Im zk)),

where a, S are real constants with a> -1. If are real, we have

L.r

which is nonnegative if

1+81 4,

(a+2)2

(4.9.7)

in view of (4.3.6) and following comments; it follows then that Z is positive

definite. As in the one-dimensional case, it is easy to see that if the

inequality (4.9.7) is strict, then Z will satisfy EEZ jk

T I z

121

12 for all

Z1,...,z,,,, where T > 0. We apply this to the matrix F = (f k) in

(4.9.6): the uniform ellipticity assumption and a slight modification of

Lemma 4.4.2 imply 19 that if S satisfies (4.9.7) strictly with a = p - 2,

E EajkJjk i TI

uI2E,IDju12.

(4.9.8)

We estimate next the other terms in (4.9.4). Consider a real vector field

p = (p ... , pm) continuously differentiable in S2 and such that (p, v) =

19Lemma 4.4.1 is applied here to the "symmetrized" matrix fjk = (Z(fjk + fkj)) rather

than to (fjk) itself.

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Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

y + p -'b on t, where v is the outer normal vector on F. Then div(I u I Pp)_

Iulpdivp+pI ulp-2Epj

By virtue of the divergence theorem,

1r(-y+ I b)Iulpdal

f (IuIpp,v)do

fJdivpI lulpdx+p f;l ulp-2(EIp;I 19D'ul) dx

p_ .I

uIp-'(E Ip,I IDiu12) dx+ f

(IdivpI+

2 IPjI)I ulpdx,

(4.9.9)

where we have used the fact that

21u1IDju15e2IDju12+e-21u12. (4.9.10)

This inequality is also used in an obvious way to estimate the fourth volume

integral in (4.9.4). Taking e > 0 sufficiently small in the resulting inequality

and in (4.9.9) and dividing by llullp-2, we obtain an inequality of the form

Re(O(u), Ao(/3)u) < ± 8ImKO(u), Ao(/3)u)+ w11u112,

(4.9.11)

which is then extended to D(Ap($)) in the usual way. The fact that

(XI - Ap(/3))D(Ap(/3)) = LP(S2) is established in the same way as in

Theorem 4.8.3 (note that the dissipativity conditions there play no role in

this). Inequality (4.9.3) is proved like in the case m = 1.

It can be shown as in Section 4.3 that an estimate of the type of

(4.9.3) will not hold if

I I > (pp

4.9.2

Remark. The characterization of the domains of the operators

Ap(/3) in Corollary 4.8.10 extends to the present case. The proof is the same.

We extend next the analyticity results in Section 4.3 for the cases

p =1 and p = oo to the present multidimensional situation, beginning with

the following auxiliary result.

4.9.3 Theorem. Under the assumptions in Theorem 4.9.1 for p > 2

(a) the operator A 1(l3) = Ao (/3) belongs to C+ in L'(2); (b) the operator

A.(/3) = A0(/3) belongs to (2+ in C(2) (Cr(SZ) if the boundary condition is

of type (II)).

Proof As in the case m = 1, (a) is proved by renorming the space

LP(Q) by means of a continuous, positive weight function p defined in SZ:

I/p

llullp =

(fIuxIPPxPdx)

. (4.9.12)

4.9. Analyticity of Solution Operators 259

The space LP(SZ) equipped with this norm will be denoted LP(S2)P. The

identification of the dual space is the same as that in the proof of Theorem

4.3.4; we use the same notation for duality maps and for application of

functionals to elements of LP(Q)P. Assuming that p is twice continuously

differentiable, we obtain

II upIl

p-2Re(9 (u), Ao(l3)u)P

= Ilupll

p-2Re(pe(pu), Ao(Q)u)

Ref

-fr(y+

pb)lulpppda

-(p-2)fstl

ajkRe(iW'u)Re(uDku))ppdx

-fal

ulp-2(Y EajkDJUDku)ppdx

P4 I ulp-2(E Y_ ajkRe(uD'u)Dkp)pp-'dx

,,(pc-E D'bj)lulpppdx

f(Y_bjD'p)lulPpP-'dx. (4.9.13)

+p f

We transform now the third volume integral keeping in mind that

P1 U Ip - 2 Re(uD'u) = D' I u I P and using the divergence theorem for the vector

U = (U) of components

Uj=IuIppp-'YajkDkp

(1< j<m).

Once this is done, the right-hand side of (4.9.13) can be written in the form

fr(y-

PD°p+

pb)luIPppdo

-(p-2) flulp-4{EEaikRe(uD'u)Re(uDku))ppdx

f I uip-2(Y

,ajkDJuDku)ppdx

+ f

{E E

Dj(alkpp-'Dkp)

+c-

p ED1b, )pp-(Y_bjD'p)pp-'lluIpdx.

(4.9.14)

We can now choose D"p at the boundary in such a way that the quantity

between curly brackets in the surface integral in (4.9.14) is nonpositive.

Noting that the first two volume integrals combine to yield a nonpositive

260

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

amount we can bound the right-hand side of (4.9.14) by an expression of the

form to IIu1IP, where w does not depend on p. Using a limiting argument as in

the one-dimensional case, we obtain

Re<u*, Ao($)u)p < wllul12

(4.9.15)

where the expression between brackets indicates application of the func-

tional u* E O(u) C L' (2) to A0(/3)u E L' (Q),) and II - IIP

is the norm of

L'(S2)P. The customary approximation argument shows that (4.9.15) -holds

for A,(/3) =Ao(/3). The fact that (XI - A,(/3))D(A,(/3)) = L'(9) for A > 0

large enough is shown as in Theorem 4.8.3. Renorming of the space L'(2) is

of course unnecessary when /3 is the Dirichlet boundary condition.

We prove next (b) renorming the spaces C(R) or Cr() by means of

Ilullp = maxlu(x)I P(x),

(4.9.16)

where p is a continuous positive function in S2; the identification of the

duals is achieved along the lines of Section 4.3. If /3 is of type (I) and

condition (4.8.4) (for p = oo) is not satisfied, we select p twice continuously

differentiable and such that

D°p(x)+y(x)p(x) <0 (x E r).

(4.9.17)

Let U E C(2)( Q),, u $ 0 and mp(u) = (x;

l u(x)I p(x) = Ilullp) so that any

p. E Op(u) is supported by mp(u) and is such that upµ (or uµ) is a positive

measure in m(u). Since D"u(x) = y(x)u(x) at the boundary, up satisfies

the boundary condition

D"(u(x)p(x)) =yp(x)(u(x)p(x))

(x E F), (4.9.18)

where

yP(x) = y(x)+ P(x) D"p(x) < 0 (x E F). (4.9.19)

The argument employed in Section 4.5 shows that mp(u) does not meet the

boundary r if yp (x) < 0 everywhere on t so that

D i I upl2(x) = 0,

gC(x;

IuPI2) %0 (x E m(u)). (4.9.20)

On the other hand, if yy(x) = 0 for some x E r, mp(u) may contain points

of F, but we can prove in the same way as in Section 4.5 that (4.9.20) holds.

Writing rl = p2 and u = u, + iu2 with u,, u2 real, we obtain

D'I upI2 = 2(u,D-'u, + u2DJu2)rl +(uI + u22)Djrl,

D'Dkl upl2 = 2(u,DJDku, + u2DJDku2)71+2(D-'u,Dku, + Diu2Dku2) q

+2(u,DJu, + u2DJu2)Dkq +2(u,Dku, + u2Dku2)D'11

+(U2 +u22)DiDkrl.

4.9. Analyticity of Solution Operators

261

Accordingly, it follows from (4.9.20) that if x E m"(u),

Re(u 'DJu)

= IIuII

-2(u1Dju1

+ u2Dju2)

iIIuII

21uI2,n-'Dj'n

'Djr1. (4.9.21)

On the other hand, again for x e mp(u ), we have

Re(u-'DjDku) = 11 u11-2(u1DJDku, + u2DjDku2)

=iIIuII-2,n-1D,DkIupI2 -Ilull-2(DjuiDku,

+ Dju2Dku2)

IIuII-2( u1Dju1

+ u2Dju2 )')-'Dk71

-IIuII-2(

u2Dku2)71-1Dj,n

-iIIuII-2(u2 +ufl

'DjDkq

II U11

-2,n

'DjDkI uPI2 -IIuII -2(Diu,Dku, + Dju2Dku2)

+,n-2DjgDkr1-

(4.9.22)

Accordingly, if µ E 0,(U),

Re(µ,Ao(/3)u)=f Re(u-'Ao(/3)u)puds

wIIU1122

(4.9.23)

mP(u)

showing that A0 - wI is dissipative. The same conclusion then holds for

A.(#) = A0 (P), and we prove in the same fashion as in Theorem 4.8.3 that

(XI - A.($))D(A,,(/3)) = C(S2) (or Cr() for the Dirichlet boundary

condition).

4.9.4

Remark. The characterization of the domain of A.(#) in Theorem

4.8.11 extends to the present case. So does the identification of D(A1(/3)) in

Theorem 4.8.17.

The following result generalizes the pointwise analyticity conclusions

of Theorem 4.3.5 to the present case.

4.9.5

Theorem. Let 2 be a bounded domain of class C(0°), A0 an

operator of class C(OO) in 2 satisfying the uniform ellipticity assumption (4.8.9),

/3 a boundary condition of type (II) or of type (I) with y E C°°(r). Finally, let u

be an element of LP(1), 1 < p < oo (1 < p < oo if /3 is of type (I)) or of C(S2)

(Cr,(3) if /3 is of type (II)). Then u(t, x) = (SP(t)u)(x) = (exp(tAP(/3))}u(x),

after eventual modification in an x-null set for each t, can be extended

to a function uQ, x) analytic in 2+(7r/2-) Moreover uQ, ) E C(°°)(Sl)

(C(°°)(S2)rl Cr() when /3 is the Dirichlet boundary condition) and for every

E > 0, every 9) with 0 < T < it/2 and every integer k = 0,1, ... ,

there exist

constants C, w depending on e, qp, k, such that

E E(q

E),

(4.9.24)

262

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

where the norm on the right-hand side is the LP norm if 1 < p < 00 or the C

norm if p = no.

It is obviously sufficient to prove the theorem in the most unfavor-

able situation, that where u E L'(9); as in the one-dimensional case the

argument is based on repeated application of Sobolev's imbedding theorem,

although the details are somewhat more complicated. We begin by selecting

two integers j and I such that

j > m/4, 21 > k + m/2. (4.9.25)

Let A > 0 be sufficiently large (so that R(A; A,($)) exists). We have

Si(t)u = R(A; A,(/3))"(AI - A,

(/3))'+'S1(t)R(A;

A,(l3))'u.

(4.9.26)

It was shown in Theorem 4.8.18 that if p < m /(m - 1), then D(A, (/3 )) c

W', P (9) and inequality (4.8.69) holds. By Theorem 4.8.16, W', P (S2) c

LmPAm-P)(0). Arguing exactly as in the proof of Theorem 4.3.5, we show

that if I < r < s < no, we have Ar(/3)

AS($) (both operators thought of as

acting in L'(0)), hence_R(A; AS($)) is the restriction of R(X; Ar($)) to

LS(S2) (to C(s) or Cr(SZ) if s = oo). We can then write R(A; A,($))2u =

R(A; Aq(13))R(A; A,(/3))u with q = mp/(m - p), and it follows then from

Corollary 4.8.10 (see inequality (4.8.40)) that R (X; A1(/3))2u E W2, q( 0) and

IIR(X; A,(/3))2U11wz.vlg) <

(4.9.27)

Applying again Theorem 4.8.16 we see that R (X; A, (/3 )) 2 u E L'(52) with

r = mq/(m - 2q) = mp/(m - 3p) and

IIR(A;

(4.9.28)

Arguing repeatedly in the same way, we deduce that R(A; A,(/3))-'u E LS(S2)

with s = mp/(m -(2 j -1)p) and

IIR(A;A,(l3))'uIIL'(9)<CIIuIIL'(n).

(4.9.29)

Since p < m/(m - 1) can be taken arbitrarily close to m/(m - 1), we may

assume mp/(m -(2j -1) p) is as close as we wish to m Am - 2j) so that,

in view of the first inequality (4.9.25), s > 2; hence R(A; A,(/3))Ju E L2(S2),

and we may take s = 2 in (4.9.29). This means that S,(t) on the right-hand

side of (4.9.26) can be replaced by S2(t) and we can then use Theorem 4.9.1

in the case p = 2. Writing (Al - A,($))i+'S,(t) = (Al - A2($W+'S2(t)

((AI - A2(f3))S2(t/(j + l)))1+' and making use of (4.1.18), we obtain an

inequality of the type of (4.9.24) for (Al - AI(/3))J+'Si(t)u, but in the

L2(S2) norm. Inequality (4.9.24) is now obtained as follows. It results from

Theorem (4.7.12) that

if v E L2(S2), then R(A; A2(/3))'v E W21'2(2) _

H21(Q) and

IIR(A; A2($))'vIIH2I(t-)<CIIvIIL2(g).

4.10. Positivity and Compactness of Solution Operators

263

We apply then Corollary 4.7.16. The treatment of boundary conditions of

type (I) is entirely similar.

4.10.

POSITIVITY AND COMPACTNESS

OF SOLUTION OPERATORS

As pointed out in Section 3.7 the real spaces L"(2) (1 < p < oo) and

C(I), Cr() are Banach lattices with respect to their natural order relations

(u < v if u(x) < v(x) everywhere in f2 in the spaces C(S2) and Cr(SA) with

"almost everywhere" instead of "everywhere" in the spaces LP(2)).

We study in the sequel positivity of the solution operators Si,, in LP(Q)

(1 < p < oo) and of S in C(S2) or Cr() using the theory of dispersive

operators in Section 3.7. As in the case m = 1, we center the treatment in the

case E = C(2) where verification of dispersivity is immediate. We have seen

(Lemma 3.7.1) that all duality maps in C(f2) (hence in Cr(1)) are proper.

Actually, we shall use this result not in C(f) with its original supremum

norm but in relation to the weighted norm (4.9.16); the proof is the same.

4.10.1 Theorem. Let k > m/2, f2 a bounded domain of class

C(k+2)

A0 an operator of class C(k+') in 11 satisfying the uniform ellipticity assumption

(4.8.9), /3 a boundary condition of type (II) or of type (I) with y E 0k+1)(I').

Then if Ap(/3)=Ao(a) (Ao(/3) thought of as an operator in the real spaces

LP(S2) and C(S2)) the operator SS(t) = exp(tAp(/3)) is positive for all t > 0,

i.e.,for every u E LP(2) (C(3), Cr(S3)) such that u(x) > 0 a.e. in 2 (u(x) > 0

in 37), we have

SS(t)u(x) >,0 a.e. in S2

(S(t)u(x) > 0 in S2).

The same positivity properly holds for the resolvents R(X; AP (ft)) (1 < p < cc).

Proof We begin by showing that (a translation of) the operator

Ao(/3) is dispersive in C(3) (see Section 3.7, especially (3.7.27) and preced-

ing comments), where we assume C(S3) nonmed with (4.9.16), the weight

function p satisfying condition (4.9.17). The subsequent computation is

essentially the same as that leading to (4.9.23), but considerably simplified

due to the use of real functions throughout, thus we include the details

below. Let u be a function in C(2)(S2)0 such that u+ * 0 (i.e., such that

u(x) > 0 at some point of S2)) and let µ E E, (u) so that tt E E(0) is a

positive measure with support in m,(u+) = (x E S2; u+(x)p(x) = Ilu+ IIP). If

x E m,(u+), then D'(up) = (D'u)p + uDip = 0 and the Hessian matrix

%(x; up) is negative definite. Since DJDk(up) = (DJDku)p + DjuDkp +

DkuDip + uD'Dkp, we easily show that

(µ, Aou) = f (E Y_ajkD-'Dku + Y_ b. Diu + cu) p dtt

-, (u+)

<wllu+112

(4.10.1)

264

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

showing that A0(/3)- wI is dispersive with respect to any duality map. Since

we have already shown in Theorem 4.9.3 that wI is m-dissipative

for w large enough (the passage from complex to real spaces effected along

the lines of the remarks at the end of Section 3.1) it results from Theorem

3.7.3 that S (t) =

is a positive operator for all t >_ 0. The case

p < oo is handled through an approximation argument. Let v be a nonnega-

tive element of Lp(S2),

a sequence of nonnegative functions in

6D such

that v -> v in Lp(3), A > 0 large enough, u (resp.

the solution of

(XI - Ap(/3))u = v (resp. (XI - Ap(/3))u =

in Lp(S2). Since

C Ap(/3) (see Section 4.9), it follows that u E and that

(XI - v,,. Now, since R(X;

is a positive operator, u > 0

for all n, and it follows from continuity of R(A; Ap(/3)) that u - u in

Lp(S2). Hence u > 0 a.e. in 0, showing that R(A; Ap(/3)) is a positive

operator. The fact that Si(t) is a positive operator follows then from

Theorem 3.7.2. This ends the proof of Theorem 4.10.1.

Going back to complex spaces, we study the compactness of the

solution operators S,(.). We limit ourselves to the following simple result.

4.10.2 Theorem.

Let the assumptions of Theorem 4.9.5 be satisfied,

and let the complex number

(Re t> 0) be fixed. If (uk) is a sequence in

LP(Q) with ilukll < C, then the sequence

contains a subsequence

convergent in the norm of Q2). The same property holds for

in C(S2)

or Cr(S2).

The proof is an immediate consequence of the Arzela-Ascoli theorem

(see Lemma 1.3.3).

Theorem 4.10.2 implies that SS(t) is a compact operator in LP(Q) for

every t > 0 and that S (t) is compact in C(s) or Cr (S2 ). This yields

significant information on spectral properties of AP (p) and A.(#) through

known properties of compact operators (Example 3.10) and the following

result:

4.10.3 Example. Let A be the infinitesimal generator of a strongly con-

tinuous semigroup S(.) in an arbitrary Banach space E. Then: (a) For each

t > 0 we have (µ; ,t = e`s, A E a(A)) c a(S(t)) (see Hille-Phillips [1957: 1, p.

467]). (b) If µ * 0 is an eigenvalue of S(t), at least one of the (countably

many) roots of the equation

e`A = µ

(4.10.2)

is an eigenvalue of A. (c) The generalized eigenspace of S(t) corresponding

to µ is the subspace generated by the generalized eigenspaces of A corre-

sponding to all solutions of (4.10.2) which are eigenvalues of A.

4.10.4

Example.

Let A,(/3) be the operators in Theorems 4.9.5. Then (a)

a(A) consists of a sequence (A,, A2,...) such that IA

I - oc as n - oc. (b)

For any q), 0 < (p < ir, a(A) is contained in the sector I arg A I >

(except for

4.11. Miscellaneous Comments

265

a finite number of elements). (c) Each X

is an eigenvalue of A with finite

generalized multiplicity (see Section 3).

4.10.5 Remark. Throughout the last three sections, we have been forced

to impose additional smoothness conditions on the domain and on the

operator A0 when constructing extensions in LP(Q) (p =1 or p > 2) or in

C(SZ), Cr(SZ). These requirements can in fact be dispensed with, taking

careful note of the "uniform" character of Theorems 4.8.4 and 4.8.12 and

suitably approximating SI by smoother domains and A0 by smoother opera-

tors. In this way, even the assumptions for the range 1 < p < 2 can be

substantially weakened. We give some details below.

Let 9 be a bounded domain of class C(Z), and let A0 be the operator

Aou=Y_ Y_a'k(x)D-'Dku+bj(x)u+c(x)u

(4.10.3)

Then the conclusion of Theorem 4.9.1 (with identification of the domain of

A,(/3) as in Corollary 4.8.9) holds under the assumptions that A0 E 0)(9)

and that y E C(')(F) if /3 is a boundary condition of type (1). The same

conditions insure the validity of Theorem 4.9.3 on the operator A00($), with

identification of domain as in Theorem 4.8.17. In the case of the space

L'(SZ), Theorem 4.9.3 holds if, in addition, bl E 0)(SI) in (4.10.3). Even

weaker assumptions can be handled through perturbation methods; see

Section 5.5.

4.11. MISCELLANEOUS COMMENTS

The opening remarks of Section 4.1 (in particular, Lemma 4.1.1 and

Corollary 4.1.2) are due to Hille [1950: 11. In particular, Hille proved

Theorem 4.1.8. Theorem 4.1.3 was discovered by Pazy [1968: 2]; earlier

sufficient conditions for a semigroup to be in the class e°° were obtained by

Hille [1950: 1] and Yosida [1958: 1]. The rest of the theory in Section 4.1 is

due to Hille [1950: 1], but the relation between (4.1.10) and the possibility

of extending S to a sector in the complex plane was discovered by Yosida

[1958: 11. Theorem 4.2.1 is due to Hille [1948: 1]. Theorem 4.2.4 was given

by Pazy in his lecture notes [1974: 11. The theory pertaining to the class d

was extended to locally convex spaces by Yosida [1963: 2]; a similar

extension of Theorem 4.1.3 is due to Watanabe [1972: 1]. We include below

some bibliography on semigroups in the classes d and C°°, abstract para-

bolic equations and related matters. See Neuberger [1964:

11, [1970: 1],

[1973: 11, [1973: 3], Prozorovskaja [1967: 11, Certain [1974: 1], Solomjak

[1958: 1], [1959: 11, [1960: 11, Beurling [1970: 1], Pazy [1971: 2], Hasegawa

[1967: 1], Efgrafov [1961: 1], Valikov [1964: 11, and Kato [1970: 2].

The treatment in Section 4.3 uses suggestions in Pazy [1974: 1], in

particular that of showing analyticity by means of Theorem 4.2.4. The

characterization of the "angle of dissipativity" qp, in (4.3.3) is in the author

266

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

[ 1983: 11. The idea of reducing the proof of Theorem 4.3.4 to the dissipative

case by means of weight functions was inspired by the treatment of Mullikin

[1959: 1].

Many (if not all) of the results in the treatment of partial differential

operators in the rest of the chapter are well known to specialists, although it

seems difficult to trace all to their sources. Others are folk theorems. The

theory of assignation of boundary conditions in Section 4.5 is essentially

due to Fichera in [ 1960: 1 ] and previous works; in particular, we owe to him

the division of the boundary IF according to the type of boundary condition

(if any) imposed on each piece r'. An elementary account of the Fichera

theory can be found in Lieberstein [1972: 1]. The only novelty in our

treatment is perhaps the emphasis on obtaining dissipative extensions of A0.

Lemma 4.6.1 is the well-known Lax-Milgram lemma ([1954: 1]). The

rest of Section 4.6.2 follows closely that paper, with the simplifications due

to the fact that we only deal with a second-order elliptic operator. In the

degenerate case, sketched in Remark 4.6.8, the use of "weighted Sobolev

spaces" such as ¶ (Sl) is standard (see Oleinik-Radkevic [1971: 1]). In

Section 4.7 we have followed closely Gilbarg-Trudinger [1977:

11, where

historical references to the material in this and the previous section can be

found. The treatment of the operator A,(#) in LP(), p $ 2 as the closure

of A0(fl) is closely related with Phillips [1959: 1]. Theorem 4.8.4 is a very

particular case of general estimates for higher order elliptic equations due to

Agmon-Douglis-Nirenberg [1959:

11; the proof here incorporates some

obvious simplifications. Theorem 4.8.5 is due to Hormander [ 1960: 11. That

the Agmon-Douglis-Nirenberg estimates imply the characterization of

D(Ap(/3)) in Corollary 4.8.10 for 1 < p < oo is already pointed out in the

work of these authors. The characterization of the operators AP (p) in the

cases p =1, oo has been apparently known for some time, but we have not

attempted to trace each result back to its origin. The case p =1 is treated in

Brezis-Strauss [1973: 11 (see also Massey [1976: 11). For the case E = C, see

Stewart [1974: 1], and references therein; we note that in this last paper

operators of arbitrary order are considered and the emphasis lies on

showing that the operators in question belong to the class d, a task that we

have avoided. The results in Section 4.9 are multidimensional versions of

those in Section 4.4 and can also be found in the author [1983: 1].