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

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4.8. Construction of m-Dissipative Extensions in LP(S2) and C(S2)

237

m-dissipative in LP(l ). The case E = C(l) (Cr(S2) if Dirichlet boundary

conditions are used) is handled in the same way. It remains to prove the

duality relations (4.8.6). From the Lagrange identity (4.5.40) we deduce that

Ao(/3) c Ao(/3')*, thus Ao(/3) c A' (ft')* =Ao (/i')*. However, we know

that AO (/3) is m-dissipative, and so is Ao (,6) * after Theorem 2.3.10, since

LP(Q) is reflexive for 1 < p < oo. Hence the resolvent of both operators

exists for A > 0 and they must then coincide (see the end of the proof of

Lemma 2.3.9). This completes the proof.

The precedent LP theory is not entirely satisfactory. In fact, the

characterization of the domain of

AP(/3)= Ao(/3) (=Ao(f')*ifE=L"(SZ), 1<p<oo) (4.8.7)

is somewhat imprecise (compare with the L2 case in Corollary 4.7.8). This

objection will be surmounted with the help of the following result.

4.8.4

Theorem. Let 0 be a bounded domain of class C(2), and let

Aou= EEajk(x)DJDku+Y_ bj (x) Diu + c(x) u, (4.8.8)

where the coefficients ajk, bj, c, are continuous in S2; assume AO is uniformly

elliptic, that is, there exists K>tt t0 such that

LLajk(x)SjEk>KI512 (xE0,EERm)

(4.8.9)

and let 8 be the Dirichlet boundary condition. Finally, let I < p < cc. Then

there exists a constant C (depending only on p, K, 0, the modulus of continuity

of ajk on 0 and the maximum of I ajk I

I bj 1,

1 c I

on S2) such that

Ilullw2,P(,Q)<C(IIAouIILP(a)+IIuIIW1'P(si))

(4.8.10)

for all u in C(2)( 9),8, the set of all u E C(2)(S2) that satisfy l3 everywhere on F.

The same conclusion holds for boundary conditions /3 of type (I) with the added

assumption that the ajk are continuously differentiable in S2 and that y is

continuously differentiable on F. In this case the constant C depends on p, K, 0,

the modulus of continuity of ajk and of D`a jk, the maximum of I y I and I D"y I

on r (Dµ any first-order derivative on F) and the maximum of I ajk 1,

I D'ajk I

lbjl, Icl on 2.

The basic ingredient in the proof will be the following result on

Fourier multipliers. We denote by f u the Fourier transform of u (see

Section 8 for definition and properties).

4.8.5 Theorem.

Let m = m(a) be a function k times continuously

differentiable for a 0 in R a , with k > in /2. Assume that for every differen-

tial monomial D", a = (a,_., a,,,) with I a I = a, + + am < k, we have

IDam(a)l < Blal - ICI. (4.8.11)

Finally, let 1 < p < oo. Then there exists a constant C (depending only on B

238 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

and p) such that

II:-'(mVu)IILP(R.)<CIIuIILP(R-)

(4.8.12)

for every u E LP(Rm)n L2(Rm).

The requirement that u E L2(U8 m) is made to assure existence of q u:

since (4.8.11) for a = 0 implies that m is bounded, m S u E L2(R Q) and

'(m fu) exists as well. For a proof of Theorem 4.8.5 see Stein [1970: 1, p.

96] or Hormander [1960: 1].

With the help of this result we establish several estimates for first

and second derivatives of the inhomogeneous Laplace equation."

4.8.6 Theorem.

Let u be twice continuously differentiable with sup-

port contained in I x I < M and let 1 < p < oo. Then there exists a constant C.

(depending only on p, M) such that

IIuII W'(Rm) < CoII iuII LP(R.)

(4.8.13)

where 0 = E(DJ)2 is the Laplacian.

Proof. Let f = Au. We have

IJ(D'Dku)((,)=

f(a)

I II

for 1 < j, k < m, where mik (v) = aiak I a I -2 is easily seen to satisfy (4.8.11),

hence the estimates for the second derivatives follow from Theorem 4.8.5

(note, incidentally, that these estimates do not depend on M). To bound the

first derivatives and u itself, it is obviously sufficient to prove that an

estimate of the form

II9IILP(Rm) < C

IID'roIILP(R-)

J=1

(4.8.14)

holds for qq continuously differentiable with support in

I x I < M, with C

depending only on p and M. To establish (4.8.14) we use the formula

x. - y.

9)(x),Y,

fRmlxi ylmDJ (y) dy (4.8.15)

(where co. = 277m /2/ 17(m /2) is the hyperarea of the unit sphere in R m ),

which can be easily established as follows (see Stein [1970: 1, p. 125]). Let

ERm, IEI=1. Then

f00(gradqp (x-it), )dt.

"Theorem 4.8.6 can also be proved with the help of the Riesz transforms and the

Calderon-Zygmund theorem (Stein[ 1970: 1, p. 59]).

4.8. Construction of m-Dissipative Extensions in LP(S2) and C(S2) 239

Integrating this equality on ICI =1, (4.8.15) results. We can rewrite this

formula as follows:

q,(x)_E(Pi*0 W,

where PP (x) = w,-,, 'X(x)xilxI -m, X the characteristic function of I xl < 2M.

Since each Pi belongs to L' (R m ), the estimate follows from Young's Theo-

rem 8.1. This ends the proof.

4.8.7

Corollary. Let R

(x e Qt m; x> 0) and let u be a twice

continuously differentiable function in R + with support contained in

I x I < M

and such that u(x) = 0 for xm = 0. Let 1 < p < oo. Then there exists a

constant Co (depending only on p, M) such that

IIUIIwz-P(R

Coii1UIILP(R+).

(4.8.16)

Proof.

Extend the function u to IRm by setting u(z, x,,,) =

- u (z, - xm) for xm < 0, where z = (x...... xm _ 1) E IR m -' . It is clear that u

is once (but in general not twice) continuously differentiable; in fact,

although DiDku exists and is continuous if j m or k * m, (Dm)2u may be

discontinuous as x passes across xm = 0. However, if qq E 6D,

fu(.z, xm)(Dm)2gp(.z, xm) dzdxm

u(x,xm)((Dm)2q(x,xm)-(Dm)2p(.X,-xm))djdxm

m , 0

= f (Dm)ZU(x,xm)( (x,xm) 9P(x,-xm))dxdxmm > o

_ fRm(D"')2U(x,

xm).p (x, xm) dxdxm,

(4.8.17)

where (D m) 2 u(x) has been extended to all of R m in the same fashion as u;

if f = Du is extended in a similar manner, we have

Du = f (4.8.18)

in R', in the sense of distributions. Since the extended u belongs to W2',

we can approximate it in the

W2.n-norm by

a sequence in 6D such that

q) = 0 if I x I > M + 1. Making use of Theorem 4.8.6 (note that Oq) -> f = Du

in LP(U3m)) and taking limits, the result follows.

We treat next the case of boundary conditions of type (I).

4.8.8 Theorem. Let u (resp. t') be twice (resp. once) continuously

differentiable in O2+ with supported contained in

I xl < M. Assume Dmu(z,0)

= t' (z, 0), and let I < p < oo. Then there exists Co (depending only on p, M)

such that

IIUII w2-P(R+) <

Co(IIDUIILP(R+) +11 01

WI.P(R+))

(4.8.19)

240 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

The proof is sketched at the end of this section (Example 4.8.21).

Proof of Theorem 4.8.4. We begin with the case where 8 is the

Dirichlet boundary condition. Availing ourselves of the observations at the

beginning of Section 4.5 we select a cover of 0 consisting of a finite number

of interior patches V11, V2,... and boundary patches V b, v b,

... and denote

b, b.

by X

X 21

, X i X 2

the partition of unity subordinated to the cover

(V', Vk ). Clearly we can choose both the interior and boundary patches so

small that if x and x' belong to one of them,

Iajk(x)-ajk(x')I <E,

(4.8.20)

where e > 0 is a constant to be specified later. We examine the situation in

each patch.

(a) Interior patches. At one of these, say V'=Sm(x, p), the func-

tion u' = X' u satisfies

Aou'=g',

(4.8.21)

where

g'=X'AOU+( ajkD'DkX'+EbD'X')u+2FY_ajkDFX`Dku.

(4.8.22)

Denote by A0

the principal part of the operator A0 frozen at x,

m m

AO,zu= E E ajk(x)D-'Dku. (4.8.23)

j=1 k=1

We perform an affine change of variables x - q = L(x - x) in V' in such a

way that the quadratic form becomes

in the new vari-

ables; clearly this change can be made in such a way that

I D''7k I I ID'nk

'I < N,

(4.8.24)

where N depends only on the ellipticity constant Ic and the maximum of the

I ajkI We can then write (4.8.2 1) in the new coordinates as follows:

A0u''=Oil'+EL(ajk(n)-ajk(0))DjDkii +EbjDju'+cu'=g',

(4.8.25)

where A0 is the operator A0 in the variables g1,...,. qm (see (4.5.21)) and u', g'

are u', g' as functions of the inj. We choose now the constant e in (4.8.20) in

such a way that

I ajk('n)- ajk(0) I < 1/2m2C0 when 71 E P'= L(V' - x), Co

the constant in Theorem 4.8.6 (this is clearly possible since (4.8.24) holds

independently of x and the ajk are uniformly continuous in il). We make

then use of (4.8.25) and of Theorem 4.8.6 to conclude that

IIu'IIw2.P(V') <211u'IIW2.P(V')+C"(IIS'IILP( i)+Ilu`IIWI.P(Vi)),

(4.8.26)

4.8. Construction of m-Dissipative Extensions in LP(f) and C(&2)

241

where C' is a positive constant. We obtain from this inequality that

110'

I I

WZ.v(V') <

C"(IIg'IILP(V') +ll u'II W1'P(V'))

Hence, after a change of variables, an examination of (4.8.22) and an

application of Lemma 4.7.5,

Il x'ull w2'P(st) = IIx'ullW2.P(V') < C(II AouII LP(V') +II uII WL'P(Vi) )

C(IIAouIILP(sz)+IIuIIWl.P(a)),

(4.8.27)

where it is easy to see that C depends only on the constant N in (4.8.24), on

the constant Co, and on the maxima of the I ajk l , the I bj I and I c I on SZ (see

equality (4.8.25) and its consequence (4.8.26)).

(b) Boundary patches.

At one of these (strictly speaking, in Vb n S7)

the function ub = xbu satisfies

Aoub = gb, (4.8.28)

where gb can be expressed from xb and u in the same way as g' from x' and

u in (4.8.22), and

ub(x)=0 (xErnVb). (4.8.29)

Let now q be the map associated to the patch Vb and let Ao be the operator

AO in the variables qt,...,. q,,,. Then Ao is uniformly elliptic in S' = S'(0,1)

(with ellipticity constant K depending on K and q as seen in (4.7.25) and

following remarks). As in the case of AO, we define 40 ,0 to be the principal

part of the operator Ao frozen at 0

Ao,o = Y-

Y- ajk(0)D'Dk.

(4.8.30)

j=1 k=1

We perform now two additional changes of variables. The first is a linear

change 71 - ' = L reducing the quadratic form

to canonical

form

clearly (4.8.24) and following comments hold for L, the operator

Ao,o transforms into A and the hyperplane elm = 0 into some hyperplane H.

Finally, we select an orthogonal transformation n'- = Uj', which trans-

forms H back into 1m = 0 (since the transformation is orthogonal, the

principal part of the transformed operator will still be A). We consider then

the total change of variables = ULq under which the equation (4.8.30)

transforms (in obvious notation) into

Ao(lb = gb

0), (4.8.31)

Zlb=0 (4.8.32)

with the support of u contained in

I 1 I < M (M only depending on xb, L).

We can argue then essentially as after (4.8.25) (but using now Corollary

4.8.7) to obtain

Ilxbullw2.P(U) <C(IIAouIILP(O)+IIUIIWl.P(u)).

(4.8.33)

242 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

We note that at the heart of this argument lies the fact that there exists an

absolute constant N such that F can be covered with boundary patches of

arbitrarily small diameter (chosen independently of N) and such that I D-'rik

I < N for the associated maps rl. This follows easily from compact-

ness of IF and will also be used in the treatment of other boundary

conditions.

We deal next with boundary conditions of type (I). The treatment of

interior patches is obviously the same, thus we only have to figure out what

happens at the boundary patches. We take them so small that if x and x'

belong to one of them we have

I ajk (x) - afk (x')I , I D'ajk (x) - D'ajk (x')I < e.

(4.8.34)

The constant e > 0 will be specified later.

here the change of variables rj (but drop the hat from q for simplicity of

notation); as before,

over a function denotes the action of the change of

variables. We argue as in (b) but we must examine carefully the (local)

transformation of the boundary condition P. Because of the invariance of

the conormal derivative (see (4.5.28)), the function u satisfies

D"Ub(rl)=P(rl)Y(rl)ub(rl) (fm=0),

(4.8.35)

where p(q) is the normalization factor in (4.5.28) and n is the conormal

vector (with respect to A0) at the corresponding point of the hyperplane

elm = 0. Now, since the principal part of Ao at q = 0 is the Laplacian,

n=(0,...,0,-1)andD"=-Dmatq =0. We have

- Dmab(71) = - (D" + D"') ab(rl)+P(rl)Y(t1) ab(rl )

=h,(q)+h2(q)

(rlm=0),

(4.8.36)

where D"ub(q) _ Edjm(,q)Djicb(rl). We extend h, and h2 to rim > 0 set-

ting y(rl,,...,rlm_,,rlm)=Y(rl,,...,rlm_,,0) and continue p in the same

fashion. Then, taking e sufficiently small in (4.8.34) we can assure that

IIh,IIw"P(R+) = II(D" + Dm)ubIl w"P(R+) < (l/4Co)II4bllw2 P(R+),

(4.8.37)

where Co is the constant in Theorem 4.8.8. On the other hand, there

obviously exists a constant C' such that

11h211 w'-P(R+) < IIPYubII W"P(R+) < C'Ilubli W'P(R+)'

(4.8.38)

We combine the last two inequalities with (4.8.25) and Theorem 4.8.8 and

obtain an inequality of the type of (4.8.26) for ub. After the inverse change

of variables, (4.8.33) results, the constant C satisfying the stipulations in the

statement of Theorem 4.8.4. Putting together inequalities (4.8.27) for the

interior patches and (4.8.33) for the boundary patches, we obtain (4.8.10),

thus completing the proof of Theorem 4.8.4.

4.8. Construction of m-Dissipative Extensions in LP(S2) and C(S2) 243

Inequality (4.8.10) can be cast into a more useful form if the

boundary condition /3 is of type (II) or if /3

is of type (I) and the

dissipativity condition (4.8.4) is satisfied for the p in question. We shall use

the following result:

4.8.9 Lemma.

Let 2 be a bounded domain of class C = C(D,

1 < p < oo, E > 0. Then there exists a constant C depending only on 9, p, E

such that

Ilullw'.P(t) < EIIuIIw=.P(o)+CllullLP(a).

For a proof (under considerably weaker hypotheses) see Adams

[1975: 1, p. 75], where a more general result involving higher order Sobolev

norms can also be found.

We make use of Lemma 4.8.9 with E < 1/C, C the constant in

(4.8.10), and obtain (with a different constant)

IIullwz o(u) s C(llAoullLP(SZ)+llullLP(sl)).

(4.8.39)

We note next that AO - wpI is dissipative if top is given by (4.8.5); therefore,

if A>

IIulILP(Q)OX -wp)-'ll(XI-AO)ullLP(St)

(see (3.1.9)). Combining these two last inequalities, we obtain, again with a

different constant,

Hull

w2'P(a)<CII(AI-A0)ullLP(si)

(4.8.40)

(A large enough) for u E C(2)(SZ)s. In view of Lemma 4.8.1, inequality

(4.8.40) (as well as (4.8.10), of course) can be extended to u E W2,p($2)R

using an obvious approximation argument.

The main application of Theorem 4.8.4 will be a precise characteriza-

tion of the domain of Ap(p), the closure of Ao(/3) in L", 1 < p < oo. As a

bonus, we shall also obtain a similar characterization in the C(S2) case.

4.8.10 Corollary. Let 9, AO satisfy the assumptions in Theorem

4.8.3. Then

aD(AP(Q))

D( A0(/3) )

= w2'([),

where W 2, p (SZ )s is the space of all u E W 2, p(S2) that satisfy the boundary

condition /3 at I' in the sense of Lemma 4.8.1.

Proof. The fact that /3 makes sense for any u E W2' ([) has been

already established (see Lemma 4.8.1); it is easy to see that W2,p(S2) is a

closed subspace of We make now use of the fact that C(2) (S2),3 =

D(A0(/3)) is dense in W2'p(SZ)'8 (Lemma 4.8.1) and of the equivalence of

the W2'p(S2)-norm and the graph norm II(AI-AO)ullLP(SZ) in D(Ao(/3)),

which is an easy consequence of (4.8.40). The remaining details are very

similar to those for the case p = 2 (Corollary 4.7.8) and are left to the reader.

244 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

We note that Ap(,(3) acts on elements of its domain in the classical sense,

that is, (4.7.13) holds.

We can also identify in a fairly satisfactory way the domain of the

closure of A0(/3) in C(S1) (Cr(S2) if /3 is the Dirichlet boundary condition).

4.911 Theorem.

Let the assumptions of Theorem 4.8.3 be satisfied

for p = oo. Assume that (4.8.4) holds for p = 1, oo, and let be the closure

of A0(/3) in C(11) (in Cr(S2) if /3 is of type (II)). Then consists of

all u E n p, IW2'p(Q)'6 such that12

A0(a)uEC(SZ) (Cr(SZ)if/Iisoftype(II)). (4.8.41)

Every u E belongs to 01w(S2) and satisfies the boundary condition

/3 in the classical sense everywhere on F. The following alternate characteriza-

tion of

also holds: consists of all u E C(S2) (Cr([ )) such

that

understood in the sense of distributions, belongs to C(S2) (Cr(a)),

i.e., such that there exists v =

in C(S2) (Cr(l)) with

fnuA0'(/3')wdx = favwdx

(w E D(A'

C(2)(S2)s')

The proof is based on the #-adjoint theory of Section 2.3. This

theory will be applied to the space E = L'(2) and the operator A = A, (/3') =

A' (0). The first task is to identify the space E # = D (A, (/3') * )

=D(A' (/3')*) in E* = L°°(52), where Ao is the formal adjoint of A0 and /3'

is the adjoint boundary condition. We shall presently show that

E#=

C(S2) if /3 is of type (I),

(4.8.42)

Cr(S2) if /3 is of type (II).

Consider first the case of Dirichlet boundary conditions. Application of the

Lagrange identity (4.5.40) shows that A0(/3) c A' (/3')* so that D(Ao(/3)) _

C(2)(S2)n Cr(S2) c E# and, taking closures, Cr(12) c E#. The opposite

inclusion is not nearly as trivial. Let u E D(A0' (/3')*) (as before, A'(#')

considered as an operator in L'(SZ)). Then there exists f E Loo (9) such that

fuuA' (/i')wdx

=f fwdx (w E D(Ao(/i'))). (4.8.43)

But this implies that u e D(Ap(/3')*) = D(A0with Ao(/3') thought of

as an operator in Lp(S2), p > I and A' ($')*u = f E LP(S2). Since A'(#')*

= Ao (/3) = A p (/3 ), it follows from the previous Corollary 4.8.10 that u E-=

W2'p(S2)1 for all p > 1. Taking p > m, we deduce that u E 0)(0) from

Theorem 4.7.15; since u satisfies the boundary condition /3 on F in the

generalized sense of Lemma 4.8.1, it is not difficult to show that u must

12The definition of A0 is slightly stretched here.

4.8. Construction of m-Dissipative Extensions in LP(u) and C(S2)

245

satisfy /3 in the classical sense. As each u E D(A' (/')*) belongs then to

Cr(S2) the same must be true of each u E E* (the L°° and the C norm

coincide in C(S2)) and the proof of the second equality in (4.8.42) is

complete. The first can be handled in an entirely similar way and details are

omitted.

The identification of A.'(#')# = Ao (/3') # is essentially the same as

that of A,(f3')* examined above (with the only difference that f E Cr(S3) or

C(S2) instead of L'). In particular, every u E D(A o (/3 )#°) satisfies the

stipulations set forth in Theorem 4.8.11. Conversely, let u be a function

enjoying these privileges. Then it is obvious that u must satisfy (4.8.43) with

f = Ao(/3)u and thus belongs to D(A' ($')#) with A' ($')#u = f.

We note finally that since Ao(/3) c A'

(#')# in Cr(U) or C(U), we

must have A,,(/3) =AO(ft) c A'(#')#, hence using once again the argument

at the end of Lemma 2.1.3 we deduce that

A.(#) = A'(/3 )# = A0(/3 )#

(4.8.44)

(Note that, strictly speaking, the expression A'(#')# does not make sense

since A' (/3') itself is not a semigroup generator; however, the meaning is

clear in the present context.)

The computation of the # -adj oint of A.(#) is considerably more

complex and will require additional LP estimates. We shall only consider

the Dirichlet boundary condition.

Let fo, fl,...,fm E LP(S2) for some p >- 1. We shall examine in what

follows solutions of the equation

Aou = fo + E DJf (4.8.45)

satisfying a boundary condition of type (II). The main result on this score is

the following Theorem 4.8.12, where both u and f are smooth and the

solution of (4.8.45) is therefore understood in the classical sense.

4.8.12 Theorem.

Let 2 be abounded domain of class C(2) and let Ao

be given by (4.8.8) with ask, bj continuously differentiable, c continuous in U.

Let l3 be the Dirichlet boundary condition. Finally, let 1 < p < oo. Then there

exists a constant C (depending only on p, K, 0, the modulus of continuity of the

ask, D'a .k and the maximum of I ask I

, I D`aik I , nI bi I ,

I D`bj 1, I c l on S2) such that

if u E and (4.8.45) holds with fo E C(S3), f1,...,fm E 00(U), we have

Ilullw'.o(u)1<

C(Y-

IIf1ILP(u)+IIulILP(u)

(4.8.46)

J=m

The proof runs along the "local reduction to the Laplacian" lines of

Theorem 4.8.4, thus we only sketch the main points leaving some details to

the reader. For the local treatment of the Dirichlet boundary condition, we

need analogues of Theorem 4.8.6 and Corollary 4.8.7. These are:

246 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

4.8.13 Theorem. Let u be twice continuously differentiable, f0 con-

tinuous, f1' .. ,fm continuously differentiable, all functions will support in

I x I < M, and let I < p < oo. Assume that

Du = fo + Y_ Djf.

(4.8.47)

Then there exists a constant Co depending only on p and M such that

IIullW'-P(R"')1<C0 E IlfIILP(R")

j=0

Proof. It follows from (4.8.47) that

(4.8.48)

(Dku)

t 2

J0(.7)+ L

Ffj(a)=IkJO+ E Ijkf .

(4.8.49)

I.71

j=11.71

j=1

Making use of Theorem 4.8.5, we obtain

IlIjkf II LP(R'n) -< Cjkll f II LP(R`)

(4.8.50)

with Cjk only depending on p. To estimate Ik for m >_ 3, we note that if wm is

the hyperarea of the unit sphere in Rm,

Ikfo(x)

(2- m)wm JRm

Ix -

ylm-21fo(y) dy

(this follows essentially from Stein [1957: 1, p. 126]; for the case m = 2 see

Example 4.8.23), and bound this integral using Young's inequality as in the

end of the proof of Theorem 4.8.6, obtaining

lIIkfollLP(tl)1< Ckll.folILP(s1),

(4.8.51)

where Ck only depends on p and M. The desired inequality is now obtained

from this one and (4.8.50), the LP norm of u estimated in terms of the LP

norm of its derivatives again as in Theorem 4.8.6.

4.8.14 Corollary.

Let U E C(2)(U8+),

fo E C(R+), ft,.

'fm E

C(1) (ic) be functions with support in

I x I < M. Assume that u satisfies (4.8.47)

in xm > 0 and that u(x) = 0 for xm = 0. Then there exists CO > 0 depending

only on p, M such that

(lull W1'P(R,) < Co E IIf hlLP(R,).

j=0

(4.8.52)

Proof. We extend u to the entire space setting u(x, xm) =

- u(.z, - xm) (xm <0), where z = (x1,...,xm_ 1) as in the proof of Corollary

4.8.7. The same extension is used for f0,

...

,fm- 1; for fm we use instead the

even extension fm (JC, xm) = f (z, - xm ). Although the extended functions

satisfy (4.8.47) off the hyperplane xm = 0, they do not fit into the hypothe-