An elliptic boundary problem for a system involving a discontinuous weight

manuscripta mathematica manuscript No.

(will be inserted by the editor)

R. DenkM. FaiermanM. M¨oller

An elliptic boundary problem for a system involving a discontinuous weight

Received: date / Revised version: date

Abstract. In a recent paper, Agranovich, Denk and Faierman dealt with a priori esti- mates, completeness, Abel-Lidskii summability, and eigenvalue asymptotics for scalar el- liptic boundary eigenvalue problems involving discontinuous weights. Here we extend these results to the matrix valued case with a diagonal discontinuous weight matrix. The given region is subdivided into subregions on which the weights are continuous. Whereas in the scalar case the usual ellipticity conditions suffice to obtain a priori estimates, a counterex- ample shows that here transmission conditions at the boundaries of the subregions are also needed.

1. Introduction

The object of this paper is to extend the results of [6] for a scalar boundary problem involving a discontinuous weight function to that for a system. Accordingly, we shall be concerned here with the boundary problem

A(x;D)u(x) !(x)u(x)=f(x)in^; (1.1)

B

j

(x;D)u(x)=g

j

(x)on for^{j =1;:::;mN;} (1.2) whereis a bounded region in^Rn,^{n 2}, with boundary ,^{m; N 2 N} with

N 2,^{u(x) = u1(x);:::;uN(x)}

T and^{f(x) = f1(x);:::;fN(x)}

T are

N 1matrix functions defined in,^T denotes transpose, the^gj(x) are scalar functions defined on ,^A(x;D) is an ^{N N} matrix operator whose entries

A

jk

(x;D)are linear differential operators defined onof order not exceeding

2m,^Bj(x;D),^1jmN, is a^1Nmatrix operator whose entries^Bjk(x;D) are linear differential operators defined on of order not exceeding^mj

<2m, and^!(x)=diag ^!1

(x);:::;!

N (x)

with the^!j

(x)being real–valued functions in^L1

(). Our assumptions concerning the problem (1.1), (1.2) will be made pre- cise in Section 2.

Let us mention at this point some of our assumptions concerning the prob- lem (1.1), (1.2). We suppose that there exists the sequence^frgN01 ,^{N0 2N}, of R. Denk: NWF I–Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany M. Faierman: School of Mathematics, University of the Witwatersrand, WITS, 2050, South Africa, e-mail: 036mef@cosmos.wits.ac.za

M. M¨oller: School of Mathematics, University of the Witwatersrand, WITS, 2050, South Africa, e-mail: 036man@cosmos.wits.ac.za

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/5065/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-50653

non-empty subregions ofwhose closures are also contained inand pairwise disjoint. Furthermore, if we let0

=n S

N

0

r=1

r, then we suppose that for each

r,^0rN0, the^!j

(x)are continuous inrand extend by continuity to contin- uous non-vanishing functions inr. Thus we allow^!(x)to have discontinuities at the boundaries rof ther,^r=1;:::;N0.

Under our given assumptions we shall proceed as in [6]. Firstly we shall es- tablish our basic theorem, Theorem 2.4, concerning the existence and uniqueness of and a priori estimates for solutions of the boundary problem (1.1), (1.2) in an

L

p Sobolev space setting ^{(1 < p < 1)}. Then turning to the spectral problem associated with (1.1), (1.2), we let^Apdenote the operator induced in^Lp()N by (1.1) and the boundary conditions (1.2) with the^gj=0, let^Vpdenote the operator of multiplication induced in^Lp()Nby^!(x), and interpret the spectral problem (1.1), (1.2) (with the^{gj =0}) as the spectral problem for the pencil^{A2 V2}act- ing in^L2()N, or equivalently, as the spectral problem for the operator^V2 ¹

A

2

acting in^L2()N. By appealing to our basic theorem, we shall then derive results concerning completeness and Abel–Lidskii summability of the principal vectors of^V2 ¹

A

2 in various function spaces as well as concerning the angular distribu- tion of its eigenvalues. Finally, by appealing again to our basic theorem, we shall derive rough and principal asymptotic estimates for the eigenvalues of^V2 ¹

A

2. We have just seen that all the results derived here concerning the spectral prop- erties of the problem (1.1), (1.2) depend upon our basic theorem, Theorem 2.4.

However, with regards to Theorem 2.4, certain differences arise between the scalar case considered in [6] and the matrix case considered here. To indicate these differ- ences let us mention some of the conditions required for the validity of Theorem 2.4. Firstly we impose the condition that there exists the closed sector^L in the

-plane with vertex at the origin such that for^r=0;:::;N0,

det Æ

A(x;) !(x)

6=0for^(x;)2r R

n and^2Lif^{jj + jj6=0;} (1.3) where

Æ

A(x;)denotes the principal symbol of^A(x;D). Secondly we impose the condition that the system of boundary operators^fBj

(x;D)g mN

1 covers the oper- ator^{A(x;D) !(x)}on in the sense that Condition 2 of Definition 2.3 is sat- isfied. Then apart from certain smoothness assumptions concerning the operators and boundaries involved, these two conditions suffice for the validity of Theorem 2.4 in the scalar case, but not for the matrix case considered here. To clarify this statement, let us mention again that in this work we treat the problem (1.1), (1.2) in a Sobolev space setting, namely in the space^u2Wp^2m

()

N. Hence, because of the discontinuities of^!(x)at the boundaries ^r, the boundary problem (1.1), (1.2) is in fact to be interpreted as the boundary problem: (1.1a) (which is obtained from (1.1) by replacingthere by

Æ

= S

N

0

r=0

r), (1.2), and

B (r)

j;`

(x;D)u(x)=0on rfor^j=1;:::;N and^{`=0;:::;(2m 1);} (1.4)

r=1;:::;N

0, where^B(r)

j;`

(x;D)=@

`

u

(r)

j

(x) @

`

u

(0)

j

(x),^{u(r)(x)=u(x)j}r

for^{r = 0;:::;N}0, and^@ denotes differentiation along the interior normal to

r. Then the third condition we require for the validity of Theorem 2.4 is that the

system of boundary operators (1.4) covers the operator^{A(x;D) !(x)}on rfor

r=1;:::;N

0in the sense that Condition 3 of Definition 2.3 is satisfied. However for the scalar case it is shown in [6, Section 7] that our third condition is redundant, i.e., the boundary operators (1.4) are absolutely elliptic in the sense of [2], while on the other hand this is certainly not the case for the matrix problem considered here, as we shall show by means of an example (see Example A.4 of Appendix A).

Thus in the matrix case our third condition imposes a restriction upon the set of

A(x;D);!(x)

for which the first two conditions cited above are satisfied. One important case where the first two conditions suffice for the validity of Theorem 2.4 occurs when

Æ

A(x;), ^{2 Rnnf0g}, is positive definite at all points of ^r,

r =1;:::;N

0(see Proposition A.1). Another such case occurs when the matrix operator

Æ

A(x;D)is triangular at all points of ^r,^r=1;:::;N0(see Proposition A.3).

To indicate a further difference between the scalar problem considered in [6]

and the matrix problem considered here, let us point out that in this work only lim- ited smoothness assumptions are made at first concerning the operators and bound- aries involved (see Condition (a) of Definition 2.2). These smoothness conditions, together with the conditions cited above, ensure the validity of our basic theorem.

However the basic theorem alone does not suffice in allowing us to arrive at the eigenvalue asymptotics for the problem (1.1), (1.2). Indeed, in order to achieve this end, we will employ the method of [6] for obtaining eigenvalue asymptotics under limited smoothness assumptions. However this method requires not only the validity of Theorem 2.4, but requires as well that the boundary problem formally adjoint to (1.1), (1.2),

A 0

(x;D)u(x) !(x)u(x)=f(x)in^; (1.5)

B 0

j

(x;D)u(x)=g

j

(x)on for^j=1;:::;mN; (1.6) is well defined and that the analogue of Theorem 2.4 for the problem (1.5), (1.6) holds also. Here^A0(x;D)is the formal adjoint of ^A(x;D) and^B0j

(x;D), ¹

jmN, is a^1Nmatrix operator whose entries are linear differential operators defined on of order not exceeding^m0j

< 2m(see Proposition A.6). Thus in this paper we shall impose further conditions (see Condition (b) of Definition 2.2) which will ensure that the problem (1.5), (1.6) is well defined and that^A0(x;D) and the^Bj⁰

(x;D)satisfy the same smoothness assumptions as do^A(x;D)and the

B

j

(x;D)(see Proposition A.6). Then turning to the validity of the basic theorem for (1.5), (1.6), it is clear that the first two conditions cited above for the problem (1.1), (1.2) imply their analogues for the problem (1.5), (1.6). The third condition that we require is that the system of boundary operators (1.4) covers the operator

A 0

(x;D) !(x)on rfor^{r = 1;:::;N}0. For the scalar problem considered in [6] this condition is redundant since the boundary operators (1.4) are absolutely elliptic, while for the problem considered here this condition appears to impose a further restriction upon the^A(x;D)and^!(x)which can be dealt with according to our theory. However we shall show in Appendix A (see Proposition A.5) that this is not the case. Indeed, we shall prove that if the system of boundary operators (1.4) covers the operator^{A(x;D) !(x)} on ^{r (1 r N0)}, then it also

covers the operator^{A0(x;D) !(x)}on r. It is precisely here where the fur- ther difference between the scalar and matrix problems appear; for in proving this result we require stronger smoothness assumptions concerning the rthan those supposed in [6] (see the first paragraph of Subsection 8.1 of [6]).

It is of interest to compare our problem to a similar one considered by Sango [17], [18] wherein the spectral properties of the boundary problem (1.1a), (1.2), (1.4) have been investigated. By appealing to the results of [6], Sango derives information concerning this problem similar to those cited above for our prob- lem. However, there are points in [18], and in particular, one which pertains to the treatment of the adjoint problem (the significance of which has already been pointed out in the previous paragraph), which are questionable. Let us fix our at- tention now upon this last point. Assuming only smoothness assumptions which are sufficient to establish the basic theorem (Theorem 2.4) for the system (1.1), (1.2), but certainly not sufficient to even construct the formal adjoint^A0(x;D)of

A(x;D), Sango deals with the adjoint problem by introducing the Assumption, which consists of two parts. In the first part he requires that the boundary operators

fB

j (x;D)g

mN

1 are normal (see [9], [13, Definition 1.4.3, p.50], [16]). However we shall show in the sequel that Sangos’ earlier assumptions imply normality, and hence this first part is redundant. In the second part Sango requires that the bound- ary problem formally adjoint to (1.1a), (1.2), (1.4): (1.5a) (which is obtained from (1.5) by replacingby

Æ

), (1.6), (1.4) is well defined and satisfies at least those smoothness assumptions as were imposed upon the problem (1.1), (1.2) to ensure the validity of the basic theorem. Then he states that the boundary operators (1.4) are not the adjoint boundary operators for the rappearing in the formal adjoint problem of (1.1a), (1.2), (1.4), but claims that if one approaches the problem from local considerations, then they are (of course this does not make sense unless we take it that he means that the two systems of boundary operators are equivalent in the sense of [15, p.121]). Sango gives no proof of this claim, but states that it is proved in [9]. However, we find all of the second part of Assumptionunac- ceptable for the following reasons. Firstly, Sango gives no indication at all of the smoothness assumptions required for the validity of the basic theorem for the ad- joint problem (1.5), (1.6). Secondly, and most importantly, Sango’s assertion that his claim is proved in [9] is false since the result referred to there deals only with boundary operators whose orders do not exceed^{m 1}, while the boundary op- erators Sango has to deal with have orders up to^{2m 1}. Thus, in light of this unproved claim and from a scrutiny of the proofs of his various theorems, one is left in doubt as to whether Sangos’ results concerning the eigenvalue asymptotics are correct.

Finally, in Section 2 of this paper we introduce those assumptions and defini- tions which we require for our work as well as prove the basic theorem, Theorem 2.4. In Section 3 we prove those results cited above concerning completeness, Abel-Lidskii summability, and the angular distribution of the eigenvalues. Section 4 is devoted to the eigenvalue asymptotics, and in Appendix A we prove the vari- ous assertions that have been made above.

2. Preliminaries

In this section we are going to introduce some assumptions and definitions which we require for our work as well as prove the basic theorem, Theorem 2.4. Hence to begin with, we let^{x =(x1;:::;xn)= (x0;xn)}denote a generic point in^Rn and use the notation^{Dj = i@=@xj},^{D = (D1;:::;Dn) = (D0;Dn)},^{D =}

D

1

1 D

n

n

=D 0

0

D

n

n

, ⁼¹

1

n

n

= 0

0

n

n

for ^{= (}1

;:::;

n )=

( 0

;

n )2R

n, where⁼⁽1

;:::;

n )=(

0

;

n

)is a multi-index whose length

P

n

j=1

j is denoted by^jj. We also let^Rn

=fx2R n

jx

n

?0g(in the sequel we shall at times use the notation^R whose meaning is clear). For^1<p<1,

0 s < 1,^{` 2 N</sup>, and<sup>G</sup>an open set in <sup>R`}, we let^Wp^s

(G)denote the usual Sobolev space of order^srelated to^Lp

(G)and let^kks;p;Gdenote the norm in this space.

Of particular interest to us is the case where^G is an open bounded set in

R

n and its boundary,^@G, is of class^{C2m 1;1}. For this case we can consider^@G as a ^{C2m 1;1} submanifold of^Rn of codimension 1. For^s an integer satisfying

1s 2m, for^f(Uj

;

j )g

`

1

a^{C2m 1;1}atlas for^@G, and for^fj g

`

1a^{C2m 1;1} partition of unity subordinate to the open covering^fUj

g

`

1of^@G, we define (see [11, p.20], [15, p.35])

W s 1=p

p

(@G)= n

v2D 0

(@G)j(

j v)Æ

1

j 2W

s 1=p

p (R

n 1

);j=1;:::;` o

:

(2.1) It can be shown that the definition (2.1) is independent of the atlas^f(Uj;j)g`

1

and the partition of unity^fjg`1. If we equip^{Wps 1=p(@G)}with the norm

kvk

s 1=p;p;@G

=

`

X

j=1

(

j v)Æ

1

j

s 1=p;p;R n 1

; (2.2)

then ^{Wps 1=p(@G)} is a Banach space, and we can also show that different at- lases and partitions of unity give rise to equivalent norms. We can also charac- terize ^{Wps 1=p(@G)} as the space of boundary values^{v = uj@G}of the vectors

u2W s

p

(G), and if we equip^{Wps 1=p(@G)}with the norm

kvk y

s 1=p;p;@G

=infkuk

s;p;G

; (2.3)

where the infimum is taken over all^{u 2 W}p^s

(G)satisfying^uj

@G

= v, then the norms (2.2) and (2.3) are equivalent.

It will be convenient for us to include a parameter in the norms defined above, namely for^2C we set

kjujk

s;p;G

=kuk

s;p;G +jj

s=2m

kuk

0;p;Gfor^u2Wp^s (G);

kjvjk

s 1=p;p;@G

=kvk

s 1=p;p;@G +jj

(s 1=p)=2m

kvk

0;p;@Gfor^v2Wp^{s 1=p} (@G);

where^kk0;p;@Gdenotes the norm in^Lp (@G).

At times in the sequel we should have to deal with the space^Wp^s (G)

N; and we shall equip this space with either of the norms^jjjjjjW

s

(G)

N,^kkW s

(G)

N, where

jjjujjj

W s

p (G)

N =

P

N

j=1 jjju

j jjj

p

s;p;G

1=p

,^kukW s

p (G)

N =

P

N

j=1 ku

j k

p

s;p;G

1=p

for^{u=(u1;:::;un)T 2W}p^s (G)

N.

Let us now turn to the boundary problem (1.1), (1.2).

Assumption 2.1.We suppose henceforth that all the assertions made in the second paragraph of Section 1 concerning the^rand^!(x)hold.

Next let us write

A

jk

(x;D)= X

jj2m a

jk

(x)D

for^x2and^{1j; kN;}

B

jk

(x;D)= X

jjm

j b

jk

(x)D

for^x2 and^{k=1;:::;N;j=1;:::;mN:}

Definition 2.2.

(a)Minimal smoothness. The boundary problem (1.1), (1.2) will be called min- imally smooth if: (1) and the ^rare of class^{C2m 1;1}; (2) for each pair

j;k,^ajk 2 L

1

()for^{jj 2m}, while the top order coefficients^ajk (x)

(^jj=2m) are continuous in; (3) for each pair^j;k,^bjk 2C

2m mj 1;1

( )

for^jjmj.

(b)Weak smoothness. The boundary problem (1.1), (1.2) will be called weakly smooth if: (1) and the rare of class^C2m;1; (2) for each pair^j;k,^ajk

2

C jj 1;1

()for^{1 jj 2m},^ajk 2 L

1

()otherwise; (3) for each pair

j;k,^bjk 2C

2m m

j 1;1

( )\C jj;1

( )for^jjmj. For^2Rn let

Æ

A(x;)=

Æ

A

jk (x;)

N

j;k =1

for^x2;

Æ

B(x;)=

Æ

B

jk (x;)

j=1;:::;mN

k =1;:::;N

for^{x2 ;} and denote by

Æ

B

j

(x;)the ^j-th row of

Æ

B(x;), where

Æ

A

jk

(x;)and

Æ

B

jk (x;)

denote the principal symbols of^Ajk

(x;D)and^Bjk

(x;D)respectively.

Definition 2.3.Let^Lbe a closed sector in the complex plane with vertex at the origin. Then the problem (1.1), (1.2) will be called elliptic with parameter in^Lif it is minimally smooth and the following conditions are satisfied.

1. (1.3) holds for^r=0;:::;N0.

2. Let^x02 . Assume that the boundary problem (1.1), (1.2) is rewritten in terms of the local coordinates at^x0 (here^{x0 !0}and ^!en, where is the interior normal to at^x0 and^(e1;:::;en)denotes the standard basis in^Rn). Then the boundary problem on the half-line

Æ

A(0;

0

;D

n

)v(t) !(0)v(t)=0for^t=xn>0;

Æ

B

j (0;

0

;D

n

)v(t)=0at^t=0for^{j =1;:::;mN;}

jv(t)j!0as^t!1

has only the trivial solution for^{0 2Rn 1} and^2Lif^j0j+jj6=0.

3. Let^{x0 2 r},^{1 r N0}. Assume that the boundary problem (1.1), (1.4) is rewritten in terms of the local coordinates at^x0 (as explained in Condition 2 above, where we now replace by rand points intor). Then the boundary problem on the line

Æ

A(0;

0

;D

n )v

+

(t) ! +

(0)v +

(t)=0for^{t=xn >0;} (2.4)

Æ

A (0;

0

;D

n

)v (t) ! (0)v (t)=0for^t<0; (2.5)

D

`

n v

+

j

(t) D

`

n v

j

(t)=0at^t=0for^j=1;:::;Nand^{`=0;:::;(2m 1)} (2.6)

jv +

(t)j!0as^{t!1; jv (t)j!0}as^{t! 1} (2.7) has only the trivial solution for^{0 2 Rn 1} and ^{2 L}if^{j0j+jj 6= 0}, where

! +

(x)(resp.^{! (x)}) denotes the restriction of^!(x)to some appropriate subset of

R n

+ (resp.^Rn) and the^vj

(t)denote the components of the^v(t). We now come to our basic theorem.

Theorem 2.4.Suppose that the boundary problem(1.1),(1.2)is elliptic with pa- rameter in the sector^L. Then there exists a^{0 =0(p)>0}such that for^2L with^{jj 0}, the boundary problem has a unique solution^{u2 W}p^2m

() N for any^{f 2L}p

()

N and^gj 2W

2m m

j 1=p

( ), and the a priori estimate

N

X

j=1 kju

j jk

2m;p;

C 2

4 N

X

j=1 kf

j k

0;p;

+ mN

X

j=1 kjg

j jk

2m m

j 1=p;p;

3

5

holds, where the constant^Cdoes not depend upon^f, the^gjand.

The proof of this result is essentially based on the explicit knowledge of the resolvent of model problems corresponding to (1.1), (1.2). We therefore start with some remarks on constant coefficient systems without lower order terms in the half-space^Rn+.

In the following, let^A(D)=

P

jj=2m a

D

and^Bj (D)=

P

jj=m

j b

j

D

,

j=1;:::;m, be partial differential operators with constant coefficients

a

=(a jk

)

j;k =1;:::;N

; b j

=(b jk

)

k =1;:::;N

; j=1;:::;mN:

We set^{B(D)=(B1(D);:::;BmN(D))T} and denote by^I the^NN unit ma- trix. Then from an obvious modification of Definition 2.3, the differential operator

A(D) Iis said to be parameter-elliptic in^Lif

det(A() I)6=0 for ^2Rn;2Lwith^jj+jj6=0: (2.8)

The boundary problem^{(A(D) I;B(D))}is said to be parameter-elliptic in^Lif (2.8) holds and if the boundary problem on the half-line

A(

0

;D

n

)v(t) v(t)=0 for^t=xn>0; (2.9)

B

j (

0

;D

n

)v(t)=0 at^t=0for^j=1;:::;mN; (2.10)

jv(t)j!0 as^t!1

has only the trivial solution for^{0 2Rn 1} and^2Lif^j0j+jj6=0.

From a minor modification of the argument of [21, Section 3] we have the following result.

Lemma 2.5.Let the boundary problem^{(A(D) I;B(D))}be parameter-elliptic in^Land let^{X =f(0;t;)j02Rn 1; 2L;j0j+jj6=0;t0g}. Then there exists the^{N mN} matrix function^W(0;t;)defined in^X with the following properties:

(1)^W(0;t;) has derivatives of all orders with respect to ⁰ and^t which are jointly continuous in⁰,^t, and;

(2)^{W(0; 1t;2m)=W(0;t;)diag ( m1;:::; mmN)}for^>0; and(3)

(A(

0

;D

t

) I)W( 0

;t;)=0 for^t>0;

B(

0

;D

t )W(

0

;t;)=I

mN at^t=0;

where ^ImN denotes the ^{mN mN} unit matrix. Furthermore, for each

( 0

;t;)2X,^W(0;t;)admits the representation

W( 0

;t;)= Z

+(

0

;) e

it

(A(

0

;) I) 1

M(

0

;;)d;

where ^+(0;) denotes a contour in ^C+ enclosing all the zeros of

det(A(

0

;) I)having positive imaginary part,^M(0;;)denotes an^NmN matrix function having derivatives of all orders with respect to⁰which are jointly continuous in⁰,^t, and, and

M( 0

;; 2m

)= 2m 1

M(

0

;;)diag(

m1

;:::; mmM

)for^>0:

We henceforth let^F0 denote the Fourier transformation in^{Rn 1} (^{x0 ! 0}).

Then from [6, Section 2] we have the following result.

Lemma 2.6.Let^{1 <p <1}and^jj 0

> 0. Then there exists a parameter dependent extension operator

T():

mN

Y

j=1 W

2m mj 1=p

p

(R n 1

)! mN

Y

j=1 W

2m mj

p

(R n

+ )

defined by^{(T()g)(x)=(F0 1F0g)(x)}for^gin the domain of of^T(), where

=(

0

;x

n

;)=expf (j 0

j+jj 1=2m

)x

n

g, such that:

(1)^0ÆT()is the identity operator in

Q

mN

W 2m m

j 1=p

p

(R n 1

), where^0u

denotes the trace of the function^u(defined in^Rn+) on^{Rn 1};

(2)if we denote by^gjand^ujthe components of^gand^T()g, respectively, then

mN

X

j=1 kju

j jk

2m;p;R n

+ C

mN

X

j=1 kjg

j jk

2m m

j

1=p;p;R n 1;

where the constant^Cdoes not depend upon^gand.

The key step for the proof of Theorem 2.4 is the continuity of the pseudodif- ferential operator given by the matrix^W. More precisely, we have the following result.

Lemma 2.7.Let the boundary problem^{(A(D) I;B(D))}be parameter-elliptic in^L. Let^1<p<1and0

>0. For^2Lwith^jj0let

(E()g)(x)=(F 0

1

W(;x

n

;)F 0

g(;x

n ))(x

0

); g=(g

1

;:::;g

mN )

T

:

Then the mapping

E(): mN

Y

j=1 W

2m m

j 1=p

p

(R n 1

) !W 2m

p (R

n

+ )

N

is continuous and if we let ^gj and^uj denote the components of ^g and^E()g, respectively, for^gin the domain of^E(), then the estimate

N

X

j=1 kju

j jk

2m;p;R n

+ C

mN

X

j=1 kjg

j jk

2m m

j

1=p;p;R n 1

holds, where the constant^C>0does not depend upon^uand.

Proof. We observe from the definition that^E()is a pseudodifferential operator in^{Rn 1} depending upon the parameter^xn. Then in order to prove our assertion concerning^E() let us put^{h := T()g} for^{g 2}

Q

mN

j=1 W

2m mj 1=p

p

(R n 1

), where^T()is the extension operator of Lemma 2.6. Following [21] let us fix an with^jj=2mand write

D

E()g(x) = D

(F 0

) 1

W( 0

;x

n

;)(F 0

h)(

0

;0)

= (F 0

) 1

( 0

)

0

D n

n W(

0

;x

n

;)(F 0

h)(

0

;0)

= Z

1

0

@

@ h

(F 0

) 1

( 0

)

0

D n

n W(

0

;x

n

+;)(F 0

h)(

0

;) i

d

= Z

1

0 (F

0

) 1

( 0

)

0

D

n +1

n

W( 0

;x

n

+;)(F 0

h)(

0

;)d

Z

1

0 (F

0

) 1

( 0

)

0

D

n

n W(

0

;x

n

+;)D

n (F

0

h)(

0

;)d

=:

Z

1

v

1 (

0

;x

n

;)d+ Z

1

v

1 (

0

;x

n

;)d=:u

1 +u

2 : (2.11)