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P ’*@) (Rn) = F;,p(Rn) = Bi,p(Rn) and F;,q,(p)(Rn)= p;,q(Rn)* To prove the independence of the spaces of the choice of the system {pk}goE @ in the same manner as in Theorem 2.3.2, one needs Theorem 2.2.4(b) with LP(lq,J [resp. LP,JZq)]instead of LP(lq).But this follows by interpolation from Theorem 2.2.4(b), Theorem 1.18.4 with po = p1 = p , and Theorem 1.18.3/2 [resp. Theorem 2.2.4(b) and Theorem l.lS.S/Z]. Afterwards, one can show as in the seventh step of the proof of Theorem 2.3.2 that F $ ’ ) ( R , , and ) Fi,q,(r)(Rn) are Banach spaces. (Since both types of spaces are interpolation spaces, one obtains the last assertion also as a consequence of the following theorem.) Theorem 1. Let -a<so,sl < co, 1 < p o , p l , q o , q l c 00, 0 c 8 < 1, and s = (1
- ep,,
+ esl,
1
1-8
P
Po
+-P81
-= -
and
1
1-8
q
!lo
-=-
+ -.e
Q1
(1)
2.4.2. Interpolation of the Spaces F;,JR,,)
CF&.(Rn), Fz,ql(Rn)1e= Fi,q(Rn)
185
(7)
-
Proof. The proof is analogous t o the proof of Theorem 2.4.1. The formula correspondinn t o (2.4.1/10) is llfll ~ ( { ( ~ ~ , ~ ~ f f n ) , ~ ~ , ~ , Hf ( ~ n* )~) }~) } l l ~ ( { ~ ~ ~ ( ~ (a) is a consequence of Theorem 1.18.4 and Theorem 1.18.2. One obtains (b) from Theorem 1.18.4 and Theorem 1.18.3/2, (c) follows from Theorem l.l8.6/2. Finally, (d) is a consequence of Theorem 1.18.4 and Theorem 1.18.1 with Aj = 2 j S 4 and Bj = 2jW, where C denotes the complex plane. R e m a r k 2. The theorem is fairly general and contains a number of interesting special cases. We remind of Bi,p(Rn)= Fi,p(Rn)and H;(Rn) = Fi,2(Rn). ( a ) I f - m < s O , s l < c o , 1 < p 0 , p 1 < ~ , a n d O < 8 < 1 , t h e n i t f o l l o w s f r o m(2) and (4) that
(B2a,po(Rn), B2x,p,(Rrz))e,p = Bi,p(XJ where s and p are determined by (1). This is also a special case of (2.4.1/7). 9
(b) If -co < so,sl < a,so from (2) that
+- sl, 1 < p o , p l
where s and p are again determined by (1). This formula is also animmediate conseL,,l(Rn))e,p = Lp(R,,)and Theorem 2.3.4. quence of (LpO(Rn),
(d) If -a < so, s1 < co, 1 < p,,, pl < that
03,
[H;,(Rn), H21(fL)le= Hi(&),
where s and p are determined by (1).
and 0 < 8 < 1, then it follows f~.om(7) (11)
~),~~
186
2.4. Interpolation Theory for the Spaces Bi,,(R,) and B';,&,)
(e) If -a < so,s1 < co, 1 < p o , p1 < co, and 0 < 8 < 1, then it follows from (7) that (12) [Hz(Rn),B2,p1(Rn)10= Fi,q(Rn) 9
1 1-8 where s and p are determined by (1) and - = ____ q
e +--. P1
R e m a r k 3. * Essentially, the theorem goes back to H. TRIEBEL[19]. The corresponding results are proved there directly, without use of Theorem 1.2.4. But numerous special cases are known before. As mentioned in Remark 2.4.115, formula (8) (as a special case of (2.4.1/7))is due t o P. GRISVARD [4]. The second part of (9) was formulated by J. PEETRE [15] without proof. (11) can be found in A. P. CALDER~N [3], and also in M. SCHECHTER [5, 61. T h e o r e m 2. Let -co < so,sl < co, so sl, 1 < p < 00, 1 5 q o , q l , q 6 co, and 0 < 8 < 1. If s = (1 - 0 ) so + 8sl, then
+
(BzqI(Rn) B2,qt(Rn))o ,q = (B2qo(Rn)F2,ql(Rn))e ,q
(13)
9
= (F2qa(Rn), F;q,(Rn))o,q = Bi,q(Rn)*
Proof. The first part of (13) coincides with (2.4.1/3). Then the other parts follow from the inclusion properties (2.3.2/4). R e m a r k 4. Interesting special cases of (13) are (Bzqo(&),H;(Rn))e,q = (H:(Rn), f$(Rn))e,q = Bi,q(Rn) 9
(14)
where the parameters have the same meaning as in the theorem. If additionally 0 5 s1 < co,then (Bzqo(Rn),W;(Rn))o,q = (H:(Rn), W2(Rn))o,q= Bi,q(Rn)*
For 0
s so, s1
-, 2
Bi,q(Rn) = (f I f
11f
E S'(Rn),
II~IIE~,~ = II~"-$ ~-'[(+ l
1512)" e - t l c l " f ~ ~ ~ L : ( L p ,
a). (8)
l ELa is an equivalent norm in the space Bi,q(R,).
Proof. Step 1. Part (a) is a consequence of Theorem 1.14.5, the last lemma, formula ( 5 ) ,and
(9) (Lp(Bn),D(A"'))O,q= (Lp(Rn),W;m(Rn)),,q= Bi,q(Rn), 2me = 8. Step 2. Considering the semi-group e-tW(t), then (1.13.1/1) with /? = -1 holds. The corresponding infinitesimal operator is A - E . We use the known formulas
192
2.5. Equivalent Norms in the Spaces BiJR,)
(see for instance H. TRIEBEL[17], p. 100/101). Then one obtains by (2.2.1/4) that
F( W ( t )f ) = (2t)-%F (e-*) Ff = ce-tlel'. F f . Now it follows for s > 0 from Theorem 1.14.5 that
=
clltm-+F-l(l
+
1~12)m e-tlela
FfIIL:(Lp).
e is a poaitive number, and if m is a natural number e > - , then it follows from Theorem 2.3.4 that
If s is an arbitrary real number, if with m
-
2
-
+ IE12)F-FFfIb;,g
~ ~ f ~ (JF-l(l ~ B ~ a
a
P
P
IItrn-VF-l(l
I & ae-tl(la FfIIG(Lp). + Ill2),+ 2
Now, one obtains (8). Remark 1. We shall describe the motive for the introduction of the norm Uf11(*), . BPl In the half-space R:+' = {(x,t ) I x E R,, 0 < t < m} classical solutions ~ ( zt ), of
au
the heat conduction equation - = Au are considered. The "boundary values" at
u(x,0 ) are of interest and also the behaviour of u(x,t ) near the boundary. On the
basis of Theorem 1.14.5, it is easy to see that one can replace in (7) the integration over (0, m) with respect to t by an integration over (0, 8 ) . Here, 6 > 0 is an arbitrary number. Hence, for the finiteness of l l f l l ( 4 ~ , only the behrtviour of f near t = 0 is BP.,
important. But then it follows that f belongs to BiJR,) if and only if the corresponding solution W(t) f of the heat conduction equation has a behaviour near the boundary t = 0 such that l[fll(4! < m holds. Analogously, in the following subsection, BP.,
we shall consider the behaviour of functions near t = 0 which are harmonic in the half-space R,++l. Remark 2. * A systematic treatment of the spaces B;,*(R,,)in the sense of this subsection can be found in M.H.TAIBLESON[l] and T.M. FLETT[l]. In these papers, formula (7)(resp. the corresponding formula (2.5.3/6) by the consideration of boundary values for harmonic functions) iu used as definition of these spaces. On this way one can obtain a large number of interesting theorems of the theory of SobolevSlobodeckij-Besov spaces. M. H. TAIBLESONbased on papers by S. BOOHNER, K. CRANDRASEKHARAN [l]; I. I. HIRSCHMAN, D. V. WIDDER[l]; and E. M. STEIN [2]. Further references will be given in Remark 2.5.313.
2.5.3.
Equivalent Norms and Cauehy-Poisson Semi-Groups
In Subsection 2.5.2, we determined the spaces Bi,q(R,)by the behaviour near the boundary of solutionsof the heat conduction equation. Then it is natural to investigate the boundary values u(x,0) of harmonic functions u(x,t) in the half-space R,++l = {(x,t ) I x E R,; t > 0). We shall follow the line of the last subsection.
2.5.3. Equivalent Norms and Cauchy-Poisson Semi-Groups
193
Setting P(0) = E , then {P(t)}ost), too. Since (I€) + l ) - l ( l + IEl2)+
-
is also a multiplier, F-l(It1 + l ) - l Ff belongs t o Wb(R,) for f E Lp(R,).Whence it follows D ( A ) = Wk(B,,).With the aid of ( 2 ) and It[( - l)k A%+ El f 11 w: Ilf 11 ++I P one obtains that D ( A j ) = W{,(R,,), j = 1 , 2 , . (11)
..
13*
196
2.5. Equivalent Norms in the Spaces B;,JRm)
For f E S(R,J and for the natural numbers k and rn with k < rn, it follows that
F ( - A + E)nr-kF-I-P(-A + E + tE)-'" f = (It1 + l)m-k(lt] + 1 + t)-" F f , 0 < t < 00. If a is a complex number with - k < Re a < nL - k , then Definition 1.15.1 and Lemma 1.15.1 yield
(-A
+ E)" f
1 ta+k-'F-' [(IEl + 1)rrL-k(14+ 1 + t)-" m
= bk,n,
F M ) ]dt.
0
F-l may be written in front of the integral. Now, using by fixed [Example 1.15.1(a) with e = 151 + 1, it follows that
(-A
+ E)" f
=
F-'(Itl
+ 1)"F f ,
f E S(R,,).
(12)
Particularly, by Remark 2.2.414,
Il(-A
+ E ) " f f l l ~S ~cllfllLp,
--OO
0 there exists a number C(E)> 0 such that for all z = (zl,. . . z,,)
...
)
)
R e m a r k . We recall the well-known theorem of PALEY-WIENER-SCHWARTZ ; see L. SCEIWARTZ [ l ,111, p. 128; K. YOSIDA[l,VI.41, or L. HORMANDER [3, Theorem 1.7.71. This theorem shows that a function f E Lp(R,,)belongs t o Mp,t if [7], 3.2.6.) To consider and onlyif suppB'f c (tAI 151 2 t}. (See also S. M. NIKOL'SKIJ the rate of approximation of functions f E Lp(Rn)by functions belonging t o M p , ,, we introduce Ep(t, f ) = inf Ilf - gllr,cw. (2) b'EMp,t
The behaviour of Ep(t,f ) if t + co is of special interest. The following theorem shows that one can characterize the elements of the Besov spaces with the aid of their behaviour by approximation by functions belonging to M p , t . T h e o r e m . Let 0 < s
q 2 p > 1, 1
6 r, p 6
00
n
n
P
! I
a n d s - -> t --,then
B;,r(Bn) c Bi,p(Rn).
If one restricts the values of r and p t,o 1 < r , p
0 , then
BP9r + + ~ + ~ ( R c~~)( R , ) .
(13)
If one restricts the value of r to 1 < r < co, then
F$p , r +1+8(R,) c C'(Rn).
(14)
R e m a r k 2. We note some interesting special cases of the last theorem and the last remark. Using Fb,,(R,,) = Hi(R,,)it follows that and
H;(Rn)c Hb(R,,),
00
> q 2 p > 1,
.S
tL
- -2 t
P -
IL
- -, 4
H f + t ( R i Jc C'(R,,), co > p > 1, 0 < t =I= integer.
(16)
After replacing t by t + E , where E > 0 , on the left-hand side of (16), this formula. holds for all t 2 0 . R e m a r k 3. We describe a simple conclusion, important for the later considerations. It holds that
H;(Rn) u Bi,p(Rn)c Hi(R,,)A Bi,p(Rt,),
12
12
> 1, s - - 2 t - -. P Y
>Q >
(17)
(2) and (15) yield that (17) is a consequence of tlie sharper embedding theorems
H;(Rn) c Bi,p(Rn) and Bi,q(Rn)c H i ( R , ! ) . (18) Using (2.4.2/10) and Theorem 2.4.2/2, the first formula follows by interpolation (-, .)e,p of the embedding theorems H:,(Rn) c H$(Rn),
PO
> p > PI,
I/
J
- -= PJ
tJ
n
- -. q
2.8.2. Other Proofs of Theorem 2.8.l(a)
207
Similarly, the second formula follows by interpolation (., *)8,4 of the embedding theorems n n H;j(R,) c Hij(&), qo > q > q l , ~j - - = t - -.
P
qj
An important special case of (17) (which is also interesting from the historical point of view) is n ?L - 2 t - -. W”,R,) c W;(R,,), 03 > q 2 p > 1, s (19) p R e m a r k 4. With the aid of the method of the last remark, it is easy to see that (2) is a consequence of (3).
-
R e m a r k 5. The question arises whether the statements of the theorem may be improved. S. M. NIKOL’SKIJ [7] has shown in Chapter 7 of his book that (2) and (5) (and hence by Theorem 2.3.2 also (3), (4), and (6)) are untrue if one replaces t by t + E on the right-hand sides where E is an arbitrary positive number. I n this sense, the theorem is not improvable. This is the reason why, by given numbers n n co > q 2 p > 1 and -co < s < co,the number t = s - - + - is called the
P
q
limit exponent. S. M. NIKOL’SKIJbased on papers writ,ten by S. M. NIKOL’SKIJ[l,21, T. I. AMANOV [l], and P. PILIKA [l]. A further improvement of these negative results
is due to M. H. TAIBLESON[l,I], Theorem 19. He showed that (2) is untrue if one replaces r by Q with e < r on the right-hand side. That this is an improvement of the even mentioned results of NIKOL’SKIJis a consequence of Theorem 2.3.2.
R e m a r k 6. * From the very beginning the theory of function spaces (and spaces of distributions) was closely related to embedding theorems of the above type and of the type treated in Section 2.9. If s 2 0 and t 2 0 are integers, then (19) coincides [3] in 1938, where essentially with an embedding theorem proved by S. L. SOBOLEV the above formulation contains an improvement with respect to the limit exponents [l]. Before this time, G. H. HARDY, J. L. LITTLEWOOD [l] due to V. I. KONDRAEOV had proved similar embedding theorems for n = 1. Embedding theorems of type (14), specialized on Sobolev spaces and on Holder spaces, where t is an integer, are [3]. A comprehensive treatment of these results can be also due to S. L. SOBOLEV [4]. The embedding of Sobolev spaces into general Holder found in S. L. SOBOLEV [l]. The embedding spaces was considered by C. B. MORREY[l] and L. NIRENBERG theorems for the spaces B;,,(R,,) are proved by S. M. NIKOL’SKIJ[2] and the embedare proved by 0. V. BESOV[2]. Further refding theorems for the spaces Bi,,(R,,) erences can be found in s. M. NIKOL’SKIJ[7]. The proof described above goes back to H. TRIEBEL [ZO]. J. PEETRE [12] has given a similar proof for the part (a). Further references can be found a t the beginning of the following subsection and in Remark 2.9.412.
2.8.2.
Other Proofs of Theorem 2.8.l(a)
I n contrast t o the investigations in Section 2.9, where the embedding on the boundary is treated, we used interpolation theory for the proof of Theorem 2.8.1 very little. Particularly, we proved (2.8.1/2) directfly. The use of interpolation theory
208
2.8. Embed-
Theorems for Different Metrics
for the proof of embedding theorems of the type of the Theorem 2.8.1 goes back t o P. GRISVARD[4]and J. PEETRE [:I]. Afterwards, A. YOSHIKAWA [l, 3,4,5],T. MuRAMATU [2],and H. KOMATSU [6,7] proved embedding theorems in the framework of abstract interpolation theory containing a large number of well-known concrete embedding theorems for (isotropic and anisotropic) function spaces as special cases. We have made some remarks in this direction in Subsection 1.19.6.In this subsection, we give another two proofs of Theorem 2.8.l(a)closely related t o the ideas developed T. MURAMATU,and H. KOMATSU. by A. YOSRIHAWA, L e m m a . Let m > q > p > 1. I f W ( t )has the s a w meanin.g as in Lemma 2.5.2, then there exists a number c > 0 such that for all t with 0 < t < CO
Proof. Theorem 1.18.9/1 yields
1
where - = 1
e
1 1 -+ -. 1 3 4
Now one obtains (1).
S e c o n d proof of T h e o r e m 2.8.l(a).Theorem 2.3.4 shows that we may assume t > 0 without loss of generality. The spaces B;,JR,,) and B;,JR,,) are normed in the sense of Theorem 2.5.2 with the aid of the analytic semi-group e-*W(t).If we use e-.W(t) in (2.5.2/7) instead of W ( t ) ,then Theorem 1.14.5 shows that we may omit the term IlfllL,. By the same theorem applied t o the case considered here, it follows that
Using the last lemma on0 obtains for f E B;,JR,,)
R e m a r k 1. The idea of the proof, carried over by A. YOSHIKAWA, T. MURAMATU, and H. KOMATSU t o the abstract case, may be described in the following way. I n two Banach spaces A , (= LJR,)) and A , (= LJR,)), two analytic semi-groups coinciding on A , n A , are considered. The corresponding interpolation spaces ( A j ,D(AT))o,r, j = 1,2,may be obtained by Theorem 1.14.5.If one is able t o prove a relation of the type (1)connecting the two semi-groups, then one obtains immediately embedding theorems on the basis of the described procedure. We want t o show that this method works also if the semi-group is not an analytic one. For this'purpose we describe a variant of the last proof which essentially coincides with the original [l]. method of A. YOSHIKAWA
2.8.2. Other Proofs of Theorem 2.8.l((t)
209
T h i r d proof of T h e o r e m 2.8.l(a).If W ( t ) has the same meaning as in the Lemma and if A denotes the corresponding infinitesimal operator, then it follows, by Theorem 1.13.1/1and the last Lemma,
0
E LJR,,),il
> 0,
3
> q > p > 1, and 0 < - - - - < 1. The estimate 2 P remains valid after replacing W ( t )by e-.W(t) and A by A - E . Now, we may apply Theorem 1.14.3 with d = - A + E instead of A . Theorem 2.3.4 shows that, for the proof of (2.8.1/2),we may assume t > 2, without loss of generality. But Here f
oc)
then one obtains for f E B8p,,(R,) by (2.5.2/9)and by Theorem 1.14.3 with k = 0 and with sufficiently large 1
It follows by iteration that also for this method the additional assumption
(L - $) < 1 is not necessary.
2 2 ,
R e m a r k 2. I n these considerations, one may replace the semi-group { W(r)}oSr -1. The space WK3.nn(Ri) may be normed by (1) and (2) after replacing there R,, by R i and R, by [0, co).
2.9. Direct and Inverse Embedding Theorems (Embedding on the Boundary)
214
(b) I n the ,sense of Definition 1.8.1/1, WgxZ(R:) is a closed subspace of
Proof. One may carry over immediately the proof of Lemma 2.9.1/1. This proves (a). Further, one obtains (b) in the same manner as in the first step of the proof of Lemma 2.9.112. R e m a r k 2. I n contrast to Lemma 2.9.112, we did not prove that t.he spaces
are equal. But the method of the second step of the proof of Lemma 2.9.112 shows that at least for 0 6 a < co
If s is a real number then we use the previous notation
+ {s}, [s] integer, O 5 {s} < 1 , s = [sl- + {s>+, Is]- int,eger, O < {s}+ 5 1 . T h e o r e m 1. Let m = 1 , 2 , . . ., 1 < p < co and = (xl,. . ., xnbl). s = [s]
and
(a) If -1
-=
XI
01
83f
< m p - 1 , then 83,
[
= {(XI,O),
af -(x',
a[ Iil-bfl]-f P
0). . . .,
ax,,
i
[m-a+l]-(XI,0)
is a retraction from W;,zv,,a(R,i) (resp. WEZ;(RL))onto
p
:
(5)
(b) If -1 < a < m p - 1, and if W;,f(R;) denotes the completion of C,jO(Ri)in Wr,2a(R:),then
+;zp: =) {f I f wg,p;), E
(c) I f a
2 mp
%f
=
01.
(6)
- 1, then CF(RA)is dense in Wg,;(RA).
P r o o f . Xtep 1. The Lcmma shows that one may restrict oneself in the proof of (a) to the space W&$R;). The Lemma and Theorem 1.8.5(a) yield that 83 is a linear continuous operator from Wg,a(R:) into n
j=O
2.9.2. The Spaces W,".l,,lD(B,) and Wgz;(Ei)
215
Step 2. Let -1 < oc < mp - 1. Clearly, it holds %f = 0 for f E @Ez;(R;). To prove the reversion, we first assume that f is the restriction of a function belonging to C e ( R , , )and that %f = 0. Then we have
and
'
(x)= O(1) for j az;
> x . Analogously to (2.9.1/9),it follows that
Ilf(z) - Xa(zn)f(z)l&#f(B;) c
21
j-
2;-"'"+xP+P
dzI1 -
c'jla-r"p+xp+p+l.
(7)
0
Ifm--
+ integer, then the right-hand side tends to zero if
+
I , .
it follows f
E
1.10. Whence
WgZna(R;t). If m - -is an integer, thcn the right-hand side of +-
( 7 ) is uniformly bounded. Particularly,
{
do a"l-jf
ajy;,+J *
< 1< 1
is a bounded
set in the reflexive Banach space Lp,z$R;t),and hence it is a weakly compact set, Now we may assume without loss of generality that f ( x ) is a real function. If we interpret Lp,Z;(Ri)as a real Banach space then the theorem of S. MAZUR (see K. YOSIDA [I], Chapter 5, Section 1, Theorem 2) is applicable. Whence it follows that for fixed j = 1, . . ., m suitable convex linear combinations of
i = 1, . . ., m.
tend to zero in Lp,x;(Ri).Here, & L O if k + co. Then suitable convex linear combinations of f(x) - x d k ( x ) f ( x ) tend t o zero in WFz;(RA). Whence it follows
f
E
w;ZpA).
Step 3. Let -1 < OL -c mp - 1, f E WEz;(RL) and %f = 0, where it is not assumed that f is the restriction of a function belonging t o C$(R,). Clearly, the restrictions of functions of CF(R,,) t o R: are a dense subset in WgX;(RA).If f l ( z ) are functions of such a type, and if f l + f in W&;(R;t), then %fL-+ 0 in
fi B ~ ~ - T - .' ( R , , - . l if) a+l
J =o
1 -+
03.
Let x ( t ) 2 0 be an infinitely differentiable function defined in [0, 00) such that ~ ( t= ) 1
for 0
t
5
1 and ~ ( t =) 0 for t > 2 .
216
2.9. Direct and Inverse Embedding Theorems (Embedding on the Boundary) -
ajf 1
is infinitely differentiable in R:. Further we have hj,L(z’,0) = -(x‘,O).
k
=
0, 1 , . . ., j , Theorem 2.5.3, Theorem 1.14.5, and (2.5.3/11) yield
i3Xi
For
.-.
Considering
c ayLoJ(z’, X
io,,(z)=
a3
r-0
> Lo > A, >
. . . > A,
> 0,
(9)
one may determine the coefficients aLo)in such a way that
For &,l(z) a formula analogous t o (8) with j = 0 holds. Similarly, one sets for j = 1,. .. , x ,
By a suitable choice of ay), one obtains that
Then K(z) = fi(z)
-
9 ij,,(x) is also an approximation of f
j-0
and it holds !JIh(z)
= 0. By a suitable choice of xl(z‘)E C ~ ( R , _ , )the , same is true for xl(z’)i ( x ) . To functions of such a type, the considerations of the second step are applicable.
Step 4. Let -1 < a < mp - 1. We prove that % is a retraction. By the first step, it is sufficient t o construct a corresponding coretraction. Let a+l
hi(&) E B;;Y-’
.
(Rn-,), j = 0, . . ., x ,
with
x =
2.9.2. The Spaces WEIZn,.(R,,)and W;,:(Ri) .
a+l
If one takes into consideration that C r ( R n - l )is dense in B p , p P method of the last step yields that G , x;: G{ho * * * hx} = ~ ( x n C ) 7Z. a!J)p(&xn) hj(s’) ni---j
9
7
j=o
1.
217
(R?,-J,then the (12)
9
r=j
(after completion) is such a coretraction. Here, the coefficients a:? have the same meaning as in the last step. Xtep 5 . Let IX 2 m p - 1. Since the restrictions t o R i of functions belonging to C r ( R n )are dense in W‘&.nO(R:), one may restrict oneself to functions of such a type. Now, for the proof of (c) one may conclude in the same manner as in the second step. R e m a r k 3. Formula (12) is of special importance for the later considerations. It yields the explicit description of a coretraction t 3 corresponding to 8 having many good properties. In contrast t o the proof of Theorem 2.9.1, we used Theorem 1.8.5(a) by the above proof not in its full extent. But with the aid of Remark 2, one obtains in Theorem 1.8.5(a) for 0 5 LY < 03 a direct proof of the part (a) of the theorem. To make easier the later investigations, we modify the definitions of WEIzn,=(Rn) and Wgxna(R;). D e f i n i t i o n 2. Under the same hypotheses as in Definition 1 , WElxmla(Rn) denotes the completion of S(R,) i n the norm a
I
WgZnO(RA) is the restriction of @gIxn,.(Rn) to R:. It will be shown that the difference t o the spaces considered before is immaterial. T h e o r e m 2. I f one replaces W;lrm,a(Rn)by Wglxnla(Rn) and Wgz:(Ri) by Wgzma(R:),then all the statements of Theorem 1 are true. Proof. Clearly, 8 defined in (5) is a continuous mapping in the described sense. The considerations of the steps 2-5 of the proof of Theorem 1 are true for the spaces W ~ l z n l ~ (and R n )@Ez.(Ri), too. I n particular, (8) remains valid (and this is the basis for the further considerations) if one replaces W by 9. I
-
R e m a r k 4. Again, the proof shows that the construction G in (12) gives rise to far-reaching conclusions. We shall return t o t,his point later on. R e m a r k 5.* As mentioned above, we need the Sobolev spaces with weights introduced here for the investigation of the Lebesgue-Besov spaces without weights. Such relations were considered in H. TRIEBEL[21]. The spaces defined there, however, are different from the spaces W:,.(R,+), the results are similar. The formula analogous t o (6) was obt,ained there also in the “crit(ica1” cases m -
- integer P by a n explicit construction of approximating functions. An idea similar to the [2]. Embedding theorems second step of the above proof can be found in P. GRISVARD of the above type are known in the literature and go back essentially to L. D. KuDRJAVCEV [l], S. V. USPENSKIJ [2], and P. GRISVARD [2]. At the moment we do not give further references and refer t o Subsection 3.10.1. ~
+
218
2.9. Direct and Inverse Embedding Theorems (Embedding on the Boundary)
2.9.3.
Direct and Inverse Embedding Theorems for the Lebesgue-Besov Spaces ( I = fl 1)
-
We recall that the Lebesgue spaces Hi(R,,) and the Besov spaces Bi,q(Rn)contain as special cases the Sobolev-Slobodeckij spaces TV”,R,,),- co < s < co, 1 < p < 03, 15q co. The Lebesgue spaces Hi@,,) are special cases of the spaces Fi,q(R,).
s
s
D e f i n i t i o n . For -co < s < co, 1 < p < co,and 1 5 q 03, BiJR:), resp. H$(R:),denotes the restriction of B&(R,,),resp. Hi(R,,),to R: in the sense of Definition 2.9.1. R e m a r k 1. As factor spaces, B;,,(R,t)and H;(Ri) are Banach spaces. I n the following Section 2.10, we shall be concerned with these spaces in more detail. L e m m a . For -co < s < co,1 < p < 03, and 1 5 q co,the restriction operator from Bi,q(Rn) onto Bp,,(R:) and from H;(R,,)onto H;(Ri) is a retraction. If N is a natural number, then there exists a corresponding coretraction independent of p , q, and s, where 1 < p < 00, 1 5 q 2 co and Is1 < N. P r o o f . We modify the method of the proof t o Lemma 3.9.1/1. Let 0 < 1, < 1, < . . . < ,I2,+, < 03 and p E S(R,,).Further, let
s
Then there exist (uniquely determined) numbers bj , j = 1, . . . , 2 N + 2, such that
aGl&’, Setting
ax;
0) ajG,p(x‘, 0) ajp(z’, 0) , j = O , l , ..., N. ax; ax;
(GI/) (94= f(-Gp),
91 E s(GI)>
(2)
(3)
for f E S‘(R,J,then it follows that this definition for f E S(Rn)coincides with (1). The function cp - G2p, E S(Rn),and all its derivatives, up t o the order N inclusively, vanish for xn = 0. By Theorem 2.9.l(b), one may approximate the restriction of p - E2q t o R: in the space Wt(R;)by functions y,(x) belonging t o CF(Ri).But then one may approximate these restrictions also in the spaces B i J R i ) , Is1 < N, and H”,(R;), Is1 5 N, by the same functions lyi. For g E B;,,(RA),where Is1 < N , and g E Hi(Ri),where Is1 5 N, we set
The convergence and the independence of the choice of the sequence {yj}F1can be obtained as follows. If f E B i J R J , where Is1 < N, or if f E Hi(Rn),where 191 5 N, such that f coincides with g on Ri , then Theorem 2.6.1 yields Ig(Wj)l =
lf(qi)l 5 IlfIIBi,g l l ~ j l l ~ > S~ g~ ,l l f l l ~ ~l , l, ~ i l l ~ ~
< co) and similarly for H;(R,J. The beginning and the end of the last formula are also true for q = co. We show that (5 is a coretraction with the desired properties.
(q
2.9.3. Embedding Theorems for the Lebesgue-Besov Spaces (1 = n - 1)
Let g
Y
E H;”(R:).
E S(Rn)
If f
has the sanie meaning as above, it follows for
E H;’’(R,J
KG9) (941 = lim J+m
=
219
If(Y,)l 5
J+m
IlfII1y lY,llr1;,
IlfIIII;fl 1191 - WIu;+?;)
5 cllfll”;N
llVIlfI~(l?*).
Hence, Gg E HiN(Rn). Construction of the infimum with respect to f shows that lIw1I;yR;)
5
(5)
cll9llfI;fl(fz;).
For g E HF(RA)= WF(RA)one obtains that
if X E R ; , (Gig) (x) if x E R, - R: .
(6’9) (x)= ( g ( l )
By the considerations of Lemma 2.9.1/1, it follows Eg E Hf(R,,)and
cllgllH:(Ri)* (6) (4), ( 5 ) , and (6) yield that E for Hi*’(RA)is a coretraction corresponding to the restriction operator. Now one obtains the lemma by interpolation from Theorem 1.2.4, (2.4.2/11), and (2.4.2/14). R e m a r k 2. As an immediate consequence of the last considerations, it follows ~ ~ @ ~ ~ / 1 ~ ( & )
and
(Hf(RA), H,N(R:))e,,= B:?o)-xe(R:) [Hf(RA), H,”(R:)le
=
H,N(’-e)-*’e(RA).
We shall return t o this point in more detail later on, see Theorem 2.10.1. We use the abbreviations given in (2.9.2/4).Further let x‘ = (xl, . . x’,-~). .)
1 T h e o r e m . (a) Let 1 < p < co and- < s < P
T h e n 8,
00.
i s a retraction from Hi(R,,), resp. Hi(RA), onto
[.- $1 -
n
j=O
l$(RA) the completion of C r ( R i )in H;(R:), timi =
(b) Let 1
{f I f
E Hi(RA),%f =
< p < co, 1 5 q 5
00,
and-
1 2)
1
.
BkpT-J(R,,-l).Denoting by (10)
0).
< R < co. T h e n 8 from (9) i s a retraction
[.- $1 -
1
.
-l). by &,JRi) the from B;JRn), resp. B;,,(RL), onto JJ B ~ q ~ - J ( R l lDenoting i=O
220
2.9. Direct and Inverse Embedding Theorems (Embedding on the Boundary)
(c) Let 1 < p < co and
1 5 s < 00.
P
Then
a [" 7'1 f axn [s-
$1
(x',0) is a continuous mapping
1 from B;,l(Rfl), resp. B;,l(BA),onto Bi:L$}(R,z-l)if s - - += integer, and a continuous p 1 mapping from B;,l(Rn),resp. B;,l(RA),into L,,(R,,-l)if s - - = integer. I t holds
P
&,m = {f I f E B;,l(R:), %f = 01 1 .
(12)
1 1 ifand only i f s - -2s not a n integer. (Inthe case s = -that means Bzl(R:)
P
P
0 1
+ B;,l @A).)
1 1 < q < co, and -co < s =< -, C r ( R i )i s dense in B&(RA) 1 ?, and dense in H:(R:). For - co < s < -, CP(RL)i s dense in B;,l(Ri). ( d ) For 1 < p
j + -. Then i t ajf(x', 0) p follows by interpolation from (14) and Theorem 4.9.1 that is a linear and ax;% continuous mapping from =
1 ni-j- b ; ; P ( R f Awpm(R,,))o,qinto (Lp(Rrl-l)? BP$
(L)L, -is a linear [.- $1 P 1 . and continuous operator from B&(Rfi)into B S - p - J(R,,-l).By the Lemma, i t
0 < 8 < 1, 1
q
5
00.
n
j-0
PdI
follows the corresponding assertion for the spaces B&(R:). Step 3. Now we prove the first parts of (a) and of (b) (and hence also the first part 1 of (c) for s - - =t= integer). If P ( t ) has the same meaning as in Lemma 2.5.3, where
P
22 1
2.9.3. Embedding Theorems for the Lebesgue-Besov Spaces (1 = n - 1)
one replaces n by n - 1, then the considerations of the fourth step of the proof of 1
Theorem 2.9.2 show that Bog = ~(z,) P(z,) g(z') is a coretraction from B ~ ~ ~ ( R , - l ) into WP(R:), m = 1 , 2 , . . . The corresponding retraction is fo%,, = f(x', 0). But 1 m- -
then the Lemma yields that R0is also a retraction from WF(R,) onto B p , pp (R,l-l). Since Go is independent of m and p , it follows by Theorem 1.2.4, Theorem 2.4.2/2, 1
(2.4.2/11), and (2.4.1/8) that !)lois a retraction from B;,,(R,) onto B i i F ( R , l - l )and 1
from Hi(R,,)onto B6iF(R,l-1).Here s > 1, 1 < p c 03, 1 5 q 5 03. By the Lemma one may replace R, by RA. Now, we must prove the corresponding assertions for 1 - < s 5 1. We denote the (n 1)-dimensional Fourier transform with respect to
P
x'
-
=
(xl, . . ., z , , - ~by ) F f l - l .Similarly Fit1.Now (2.5.3/9) yields
X(G)P(z,l)g(z') =
C F ; ! ~+ ( ~1
~ ~ . F1 f l -~ ~ () x nF~ )i l l e-~L'~~~F,L-l.F;!l(l + 1x'12)-mFn-1g(x').(15)
Here, m is a natural number. The last part is a mapping from B;i$(R,l-l) onto s+an&-L B , , p (Rn-l). By the above considerations, the middle is a mapping from s+am- L B , , P (R,,-l) into B;Tim(RA).Finally, one obtains by the first part a mapping into B;,,(R;). Similarly, one concludes for the Lebesgue spaces. Together with the second 1 step, it follows that, for - c s c 03, J!$, is a retraction from B;,,(R,), resp. B;,,(R;), ?, 1 onto Bii$(R,l-l). If 2 4 p < co and - c s c 03, then one obtains by Hdp(Rll) 23 1 c B;,p(R,) that !Jlois a retraction from H;(R,f),resp. H;((R;), onto BirpT(Rn-l). 1 For 1 c p c 2 and - c s c 1, the desired assertions for the spaces H;(R,), resp.
P
H;(R;), are consequences of the interpolation formulas
1
< (I < 1, 1 c po < 2, 0 c 0 < 1, and-
1
1-8
o ; see +2
(2.4.2/11), 2 ?, PO (2.4.1/8), and Theorem 1.2.4. (The case H;(R,,) = Wi(R,) is treated in Theorem 2.9.1.) Now, the method of the third and the fourth step of the proof of Theorem 2.9.2 with oc = 0 shows that 8 is a retraction; see (2.9.2/10), (2.9.2/11) and (2.9.2/12). (One may apply (15) although there is the multiplication with xi in (2.9.2/12) and (2.9.2/10).)
where-
=
~
Step 4. We prove (10) and (11).Let f E B;,,(RA), 03 > q > 1, resp. f E Hd,(RA), with 8f = 0. The considerations ia the third step of the proof of Theorem 2.9.2
222
2.9 Direct and Inverse Embedding Theorems (Embedding on the Boundary)
and the above results show that we may assume without loss of generality that f is the restriction of a function belonging to CF(Rn).Then
Let xA(t) be the functions defined in (2.9.1/8).Then for two integers 0 it holds that
5 m, < s < m2
By interpolation, it follows that
Similarly for Hi(R:). That the spaces over R: have the same interpolation properties as the spaces over R, is a consequence of the Lemma and of Theorem 1.2.4; see also 1 Theorem 2.10.1. If s - - + integer, whence it follows the desired result inclusively 1 P (12). Let s - - be a n integer. Then { f ( x )- %A@,) f ( x ) } O < l < iis a bounded set in
P
Bi,,(R+,),resp. Hi(R:). Since B;,,(R:) and Hi(R,+)are complemented subspaces of B;,,(R,) and H;(R,) respectively, it follows by Theorem 2.6.1 and Remark 2.6.1/2
that they are reflexive Banach spaces. Hence the above set is weakly compact. Now one obtains the desired assertion in the same manner as in the second step of the proof of Theorem 2.9.2. 1 Step 5. We prove the last part of (c). If q = 1 and if 0 < s - - is not an integer, p 1 then the desired assertion follows analogously t o (16). If 0 5 s - - is an integer, P then the first part of (c) yields that (12) cannot be true. Step 6 . For the proof of (d), one may restrict oneself again t o restrictions of func1
tions belonging t o CF(R,,). Then it is sufficient to consider the space BK,(R:) where 1 < q < co. Since this is a reflexive space, one obttains the desired assertion in the same manner as in the third step, see (16). R e m a r k 3. The proof shows that we essentially used the considerations on Sobolev spaces with weights only in the third step. 1 R e m a r k 4. Part (c) shows that B;,,(R:) for s - - = 0 , 1 , 2 , . . . is not a re-
P
flexive space. Namely, if we suppose that BA,(R:) is a reflexive space, then we could apply the method of the fourth step, and we would obtain a contradiction to (c). Similar statements for the spaces Bi,l(R,l)are proved in Remark 2.6.112. As complemented subspaces of reflexive spaces, H",RA) and Bi,,(R:) are also reflexive spaces, -co < s < co, 1 < ?, < co, 1 < q < co. R e m a r k 5 . If one extends a function of Wg,,$R:) to R, - R,t by zero then one obtains a function belonging t o W ~ l s , , O ( R The , , ) . corresponding question for the spaces B;,,(R:) and B',(R;) is much more complicated. Later on we shall see in Theorem 2.10.3 that a corresponding assertion holds for 6he spaces i ; , J R ; ) and
2.9.4. Embedding Theorems for Lebesgue-Besov Spaces (General Case)
223
1
H“,R:) if - 1 < s - - is not an integer and 1 5 q < 03. The functions of Bi,q(RA), P 1 however, where 1 < p , q < co and s - - = 0, 1 , 2 , . . . cannot be extended in
P
this manner to functions belonging t o B;,q(R,). See also Remark 4.3.213. R e m a r k 6. References for the problems considered here will be given in Remark 2.9.412.
Direct and Inverse Embedding Theorems for Lebesgue-Besov Spaces (General Case)
2.9.4.
. ., n - 1, we set R1 = {x I x = (xl,.. . , x L , 0 , . . .,O)},x‘= ( x l , ...,xl), alalf(x) and x” = ( x [ + .~., ., xn). Further we write Dz,.f(x) = aq:;l. . . ax? where a = ( O L ~ +.~.,.,a,,) is a multi-index. n-1 T h e o r e m . (a) Let 1 < p < co and ___ < s < co. Then 8, For 1 = 1 , .
P
is a retraction from H;(R,,) onto
Any function of Hi(R,,) with % = 0 can be approximated by functions belonging to Cp(R,Jand vanishing in a neighbourhood of Rl. n-1 ( b ) L e t l < p < c o , l s q s c o , and< s < 00. Then 8 from (1) i s a retraction from B;,JRr,)onto P
For 1 < q < co, any function of B&(R,,) with %f = 0 can be approximated by funcn-1 tions belonging to C”,R,) and vanishing in a neighbourhood of Rl . If s - -is not I, a n integer, then the same is true for q = 1 , too. n-1 (c) Let 1 < p < 00 and s = 0, 1, 2 , . . . Then DZ,,f(x’, 0 ) with la1 = n-1 r, s-i s a linearly continuous mapping from B;,l(R,,)into Lp(R1). ~
P
Proof. Step 1. Since the product of retractions is again a retraction, one obtains the first parts of (a) and (b) by iterative application of Theorem 2.9.3.
Step 2. Let f E Hi(R,), resp. f E BiJR,,) with 1 < q < 00, and %f = 0. Iterative application of the third step of the proof of Theorem 2.9.2 and the beginning of the
224
2.9. Direct end Inverse Embedding Theorems (Embedding on the Boundary)
fourth step of the proof of Theorem 2.9.3 show that we may assume f where N is an arbitrary natural number. Then
If m, and m2 are two integers with 0
-s+
f(4IIB;,*(n,)
Cf(R,)
s m, < s < m 2 , then it follows that
By interpolation one obtains that
Ilf(4 - XA(””)
E
IC I .
n-c + [.- $1 p
-+I
All the further considerations are the same as in the fourth step of the proof of Theorem 2.9.3. Here, one has t o mollify the function x l ( x ” ) f(x) with the aid of Sobolev’s mollification method described in the first step of the proof of Lemma2.5.1. This proves (a) and (b). Step 3. Using Theorem 2.8.l(c),
one proves part (c)in the same manner as in the first step of the proof of Theorem 2.9.3. R e m a r k 1. As mentioned above, the statement that 02 i a retraction contains the direct embedding theorems as well as the inverse embedding theorems. The continuous embedding described by 02 is called the direct embedding theorem, the existence of a corresponding coretraction is called the inverse embedding theorem. R e m a r k 2. * Some references with respect t o the theory of embedding theorems are given in Remark 2.8.116. The embedding theorems formulated in 2.9.1, 2.9.3, and in this subsection are of fundamental interest for the theory of Lebesgue-Besov spaces. For the case of Hilbert spaces, that means H;(R,) = Bi,2(R,l), the embedding on the boundary was investigated by N. ARONSZAJN [l], L. N. SLOBODECKIJ[l], and G. FRODI [l, 21. A complete treatment of this case can also be found in J. L. LIONS,E. MAGENES[2, I], where the considerations are based on J. L. LIONS [l]. The extension of these results t o the spaces Hi(R,) goes back t o E. M. STEIN[3,II], N. ARONSZAJN,F. MULLA,P. SZEPTYCKI [l], and P. I. LIZORKIN[3]. The corresponding considerations for the spaces W;(R,,) are due to s. V. USPENSKIJ[l, 21, and P. GRISVARD [3], for the spaces B i , m ( R , lare ) due t o S. M. NIKOL’SKIJ[2, 31, and for the spaces Bi,q(R,,)are due t o 0. V. BESOV[2]. I n this context we also refer to J. L. LIONS,E. MAGENES[l, 1111 and E. MAGENES [l], and t o the comprehensive treatments by J. DENY,J. L. LIONS[l], S. M. NIKOL’SKIJ[5,7], and J. NEFAS[2]. There are given further references. I n particular, in S. M. NIKOL’SKIJ[7] there can be found many references t o papers by Soviet mathematicians, e. g. T. I. AMANOV, V. P. IL’IN, P. L. LIZORKIN,S. M. NIKOL’SKIJ,V. A. SOLONNIKOV, I q 2 p > 1 , 1 5 r P P
5 00,
(2)
and from H",R,,)
n l into B;,p(Rl), 0 < u = s - - + -, 2 , q
>q
00
>= p
> 1.
(3)
See &o the first formula in (2.8.1118). By (3) and (2.3.3/8),it follows for 1 < p 5 2 - q < 00 and for 1 < p 5 q 5 2 that %, is also a continuous operator from I H;(R,,) 15
Triebel, Interpolation
into HZ(Ri), 0 < 0
=s
l -. ?,!I
n
--+
(4)
226
2.10. The Spaces H;(K)and B;,,(R:)
It is possible t o show that (4) essentially remains valid if one only assumes 1 < p < 00 : !I? is , a, continuous operator from n
1
H;(Rn) into HG(Rl), 0 5 0 = s - - + -, P P
co > q > p > 1 .
(5)
Clearly, 0 may be replaced by 0 5 u. Formula ( 5 ) is a consequence of (2.8.1/17) and of (3) where p = q. The statement contains as a special case (s and 17 integers) a n embedding theorem of S. L. SOBOLEV [3], inclusively improvements by V.I. KOND R A ~ O V[l] and V. P. IL'IN[l]. The above formulation goes back to E. &I. STEIN[3] and P. I. LIZORKIN [2]. (In the case 1 = n the sharper formula (2.8.1/17) is true.) R e m a r k 4. We did not consider embedding theorems on the boundary for the spaces F;,q(R,J.The reason is the following. I n the proofs of the theorems we used essentially that Hi(R,) and Bi,q(Rn)are interpolation spaces of Sobolev spaces. This is not the case for all spaces F;,,(R,). But by the last theorem and by Theorem 2.3.2, it follows immediately that % from (1) is a retraction from Fi,q(R,)onto
if q is a number between 2 and p .
2.10.
The Spaces H;(R:) and BiJR:)
The spaces B",,(RA) and H;(RA) are introduced in Definition 2.9.3. Special cases are the Sobolev-Slobodeckij spaces W;(R:) = H;(R+,)for s = 0, 1,2,,. . ., where W!(Ri) = L,(R+,),and W;(R:) = Bi,,(R:) for 0 < s + integer. I n the theorems in Sections 2.9.1 and 2.9.3, embedding properties of these spaces on the boundary {x I x, = 0) are considered. The same theorems contain also characterizations of the spaces &;(R+,)and &,q(R:).In this section, we shall describe interpolation properties of these spaces. Further we shall determine the dual spaces. Equivalent norms will be investigated in Subsection 4.4.1.
2.10.1.
Interpolation of the Spaces H,"(R:) and Bi,q(R:)
T h e o r e m . If one replaces R, by R: then the formulas (2.4.1/3), (2.4.1/7), (2.4.1/8), (2.4.2/9), (2.4.2/10), (2.4.2/11), and (2.4.2/14) (inclusively the special w e s (2.4.2/15) and (2.4.2/16))are valid under the corresponding hypotheses for the parameters. Further
(B&.(R3 B2q1(R9)o,q= Bi,q(Ri) under the hypotheses of Theorem 2.4212. Proof. The proof is a consequence of the given formulas (inclusively Theorem 2.4.2/2), Lemma 2.9.3, and Theorem 1.2.4.
2.102. Duality Theory [Part I]
227
R e m a r k . Clearly, one may carry over further formulas of Section 2.4, for instance (2.4.1/9) or (2.4.1/12) (after definition of the corresponding spaces). P r o b l e m . Analogously t o the spaces Bi,q(R:),one may define the spaces Fi,,(R:) by restriction of the spaces Fi,q(R,)t o R,+, -CO < s < CO, 1 < p < CO, and 1 < q < 03. The question arises whether one may carry over the interpolation theorems of 2.4.2 to the spaces F;,&R;).For this purpose, one would need an extension of Lemma 2.9.3 to spaces of such a type. Some difficulties, however, arise. A clarification of this question is of interest. One obtains a partial affirmative answer by interpolation from (2.4.2/12) and Theorem 1.2.4. I n this way, one obtains only spaces Fi,q(Rn)
Duality Theory [Part I]
2.10.2.
The spaces H;(R+,) and B;,q(R:) are defined as restrictions of the spaces Hi(R,) and Bi,q(Rn). For s < 0, it is for applications of interpolation theory to elliptic differential operators sometimes useful to start with another definition. J. L. LIONS,E. MAGENES [l, particularly 111; 21 put 1 1 W,:(RA) = (h’i(RA))’, 0 < s < CO, 1 < p < CO, - - = 1 .
P
+
P‘
Here the dual spaces must be understood in the sense of the couple {C,”(Ri)= D(R,+), D‘(Ri))of dual spaces, see Section 2.6. Similarly, one may define the spaces H,:(R,+) and B;;,q,(R:)for s < 0. I n this subsection we shall show that these two definitions coincide with exeption of “singular” cases. Later on, we shall make no frequent use of the results of this subsection. This is the reason why we shall quote a plausible [l] without proof, see also R. S. STRICHARTZ [l]. but non-trivial result of E. SHAMIR 1 L e m m a (E. SHAMIR [l], R. S. STRICHARTZ [l]). Let 1 < p < co and 0 5 s < -.
P
If x,(x) denotes the characteristic function of R:, then x,(z) f ( z )is a linear and continuous operator from Hi(R,,)into itself. R e m a r k 1. With the aid of the methods of the proof of Theorem 2.9.4, it follows that o m may approximate any function belonging t o Hi(R,,),where 1 < p < 00, 0
1
5 s 5 -, by functions of CZ(R,J vanishing in a neighbourhood of (P
Then the question arises whether for f E H ~ ( R ,also ,)
x+f belongs 1
{z I xn = O } .
t o H”,R,) and
whether the so obtained operator is continuous. For s < - , the Lemma gives an
P
affirmative answer. Theorem 2.9.4 s h o w that the same question is meaningless for 1 s > -, since in this case there exist boundary values which are generally not identi-
P
cal zero. By the investigations in R. SEELEY[2,3] it follows that also in the singular case 15*
8
=
1 the operat>orxtf is not continuous in H;(R,).
P
(See also the proof of the
228
2.10. The Spaces Hi(R:) and Bi,,(R:)
following theorem.) A proof of the Lemma for the important special case p = 2 can be found in J. L. LIONS,E. MAGENES[2, I]. Using the inequality (3.2.6/6), one may 1 prove the Lemma for the spaces Wi(R,,),1 < p < 00, 0 5 s < -, instead of the P spaces Hi(R,) in a simple way. Namely, if f ( x ) E CF(R,), then
1 If(x)
Ix --y fp( Y )+I P v dx dy
llx+(x)f(z)\l!V~(R,,) 5 R:
x
+ cllfllgp(R,)+ c
1
If(x)l” x i s P d x .
It:
R:
Adding to the right-hand side of (3.2.6/6) the term I I ~ l l & o , ~ , ) ,then it follows by 1 completion that (3.2.6/6)with 1 = s < -holds for f(x’,t ) , too*). Now one obtains P the desired estimate in the same manner as after (3.2.6/6).Using the above lemma for W;(Rn)instead of Hi(R,), then one may prove the following theorem for the spaces B;,,(R,J, but not for the spaces Hi(R,). 1 1 1 T h e o r e m 1. Let 1 5 q < co, 1 < p < co, - + - = 1, and -- < s < co 1 P PI PI with s - - O , . l , 2 , . . . Then E, P
=+
is a coretraction from Bi,,(R:) into Bi,,(R,,) and from H;(R+,)into H;(R,). For --1 < s 5 0 , one has to interpret (1) i n the sense of the theory of distributions of mf r
E).
D’(RA)and of D’(R,,If N is a given natural number then there existsacorresponding retraction from B;,,(R,) onto B;,,(R;) and from H;(R,) onto &(R:) independent 1 1 of 1 S q < 00, 1 < p < CO, and -- < s < N With s - - 0 , 1 , 2 , . . . P’ ZJ 1 1 1 Proof. Step 1 . Let 1 < p < co, 0 5 s < -, and - - = 1. Then for 1, P P’ rp E C;(R,), it holds x + ( x )rp(z)E Lpc(R,)c H;!(R,). By
+
+
(x+V>Y)La(R,) = (Y,
X+W)La(R,)
or y E CF(R,), the above lemma, and Theorem 2.6.1, it follows that x+ may be extended t o a linear continuous operator from H;f(R,) into itself. This operator is described by (1) if one interpretes (1) in the sense of distributions of D’(R:) and D’(R,, One obtains by interpolation that is also a linear continuous operator 1 1 < s < - , 1 5 q 5 co. from B;,,(R,,)into itself, -
- q).
x+
1,
?,
*) See the method of the proof of Theorem 2.9.3, Step 4.
229
2.10.2. Duality Theory [Part I]
Step 2. Let 1
(Ri)into H>(R,,), 1 -1 < s - - + 0 , 1 , 2 , ...
P
1 Step3.Letm = 0 , 1 , 2 , . . ., - -1 < uo,u1 j +--, onesets
s s
2,
Theorem 2.9.3 and the methods developed there show that B;,q,y,(RA), resp. Hi,yj(RA), is a complemented subspace of Bi,q(RA),resp. Hi(R:). If N is a given positive number, then the corresponding projections are independent of p , q, and s < N . Since projections are special retractions, it follows by Theorem 1.2.4 (or Theorem 1.17.1/1) that all interpolation theorems for the spaces Bi,q(Rn)and Hi(R,,)may be carried 1 over t o the spaces Bia,y,(RA)and Hi,7,i(RA)if co > s > j -. Here, exceptional P 1 cases do not exist. But for s < j - , corresponding assertions without exceptions do not hold, see Subsection 4.3.3. p For the later applications i t will be useful t o generalize Theorem 2.10.2/1.
+
+
T h e o r e m 2. Let 1 qs co, 1 < p < co, and -co < s < co. Then G from (2.10.2/1) is a coretraction from i i ; , q ( ~into ; ) B ~ J R , )and from H"",(R:) into H;(R,J. If N is a natural number, then there exists a corresponding retraction from B;,q(Rn)onto and from H;(R,,) onto k p ( ~ independent ; ) , of 1 q 5 co, 1 < p < 00, and Is1 < N .
e,,(~;)
P r o o f . Let LTI be a retraction in the sense of the last part of Theorem 2.10.2/1 (here one has to replace the number N mentioned there by a sufficiently large natural number). Now let N be a natural number. If J , has the meaning of (2.10.3/2),then it follows by Theorem 2.10.2/1 and Theorem 2.10.3 that '%N
= JN?XJ-,y
is a retraction from HF(Rn)onto d f ( R ; ) and from H;'"(Rn)onto ZiN(R,+) belonging to the coretraction G = J,vGJ-s. (Here we used that J , has the same properties as I,s from Theorem 2.3.4, see the first step in the proof of Theorem 2.10.3.) Now one obtains the theorem by Theorem 1, the corresponding interpolation formulas for the spaces over R,, and Theorem 1.2.4.
2.10.6.
Duality Theory [Part 111
One may develop a duality theory for the spaces %,,(R,+) and *p(R,f) from Definition 2.10.3 in a simple way. Here, one has to interpret the dual spaces in the same way as in Subsection 2.10.2. Remark 2.10.3/1 shows that this is meaningful (q < a in
B;,q(R;)).
2.10.5. Duality Theory [Part 111
Theorem 1. Let -a < s < 1 1 1 -+-=-+-=1
P
P’
q
00,
1 < p < a,1
q
235
< 00,and
1 qr
Proof. By definition &p,q(R,,),resp. *p(Rn), is a closed subspace of B;,q(Rn), resp. Hi(R,,).Then it follows by Theorem 2.6.1 that (&,q(
R:))‘ = B;Iq.(R,)/B;?,,.(R,) = B;Iq,( R i ).
Similarly for &(R;). R e m a r k 1. One may give another proof of the theorem if one uses Theorem 2.10.412 and the methods developed in the proof of Theorem 2.10.212. 1 It is assumed Theorem 2. Let 1 < p < 00, 1 < q < 00, and -a < s P that (1) i s satisfied. Then
-.
( B ; , ~ ( R ; )= ) !5;,,.(~;)and (H;(R;))’ = #pi).
(3)
Proof. Theorem 2.9.3(d) yields that CF(RA) is dense in Bi,q(RL)and in H”,(R;). Hence, one may construct the dual spaces in the above sense. By Theorem 2.11.3, it follows that g;?,,,(R;t)and 8;?(R:) are reflexive Banach spaces. Then (3) is a consequence of (2). R e m a r k 2. Using Theorem 2.10.3, it follows that Theorem 1 and Theorem 2 are 1 1 extensions of Theorem 2.10.212. For s < - 1 + - = - - the spaces 8;?,qt(Ri)
P
P‘ ’
and Bpf,,.(R;) are different; the same holds for the spaces & , ; ( R i )and H;?(R:). This is a consequence of Theorem 2.9.3 and the considerations in Remark 2.10.3/1. But then it follows by (2) and (3) that
%,q(R9 -00<s
1. The spaces defined in this way coincide with the spaces B;,q(R,) 1
and FiBq(R,,). In particular, one may choose for s > 0 the number c = 27. Then one obtains a formulation which allows generalizations t o the anisotropic case. Let (s)=(sl ,...,sn), O<s, 0 be given. If one chooses 0 < r,~= V ( E ) and afterwards 0 < 2t < r , ~sufficiently small, and if Qq has the same meaning as in (3.2.1/1), then it follows from the properties of the function a(x) that
Using Sobolev’s mollification method described in the first step of the proof of Lemma2.5.1, then it follows that f(s+ 2ty) E W r ( o t ) can be approximated in W y ( Q )by functions belonging t o Cw(Q).At the same time this is also an approximation in Y ( Q a).;
3.2.2. Properties of the Spaces Vr(l2; u)
249
Step 3. To prove (c) we use the balls Kj,j = 1, . . ., N , from Definition 3.2.1/2 and determine a domain w such tha$ 6 c Q and
ww
3
Q. Choosing the
balls sufficiently small, then the following constructibi are possible without contradictions. Let yo(x) E C$(w) and y,(z) E C$(Kj)be a resolution of unity with respect to 9, that means
O j y j ( x ) s l for j = O , l , ...,IV
and
N
c y j ( z ) = l for
XEQ
j=O
(1) (outside of w or K,,respectively, the functions yj(x) are extended by zero). The existence of such a system of functions may be proved in an easy way analogous to [17], p. 47. If f(z)E WF(Q;a), then the construction (2.3.1/12); see also H. TRIEBEL yj(x)f(x)belongs also t o W r ( Q ;a). Now, one approximates yo(”)f(x)in the same manner as in the second step with the aid of Sobolev’s mollification method. To approximate yj(x)f(x),j = 1, . . .,N , we suppose without loss of generality that a(z) = e(d(z)) in the sense of formula (3.2.1/5). Choosing a suitable cone K j , then (Q n K j ) Kj becomes a n unbounded domain of cone-type. One may extend a(x) outside of supp yj n Q such that one obtains a weight function of type 1. Extending yjf outside of Q n K j , but in (Q n K,)+ Kj,by zero then, using the methods of part (b), the obtained function may be approximated by functions belonging to Cm((S2n Kj) Kj). Multiplying these functions with ~ J x E) C$(Kj), v,(x) = 1 for x E supp y,, and using the above resolution of the identity, then one obtains the desired approximation by composition.
+
+
R e m a r k 1. The method described in the last step of the proof is of fundamental importance. It is called the method of local coordinates or local charts, since the considerations in the domain 9 are carried over with the aid of Definition 3.2.112 t o local investigations. This makes also clear the meaning of Definition 3.2.112. Later on, we shall use this method several times. R e m a r k 2. The proof of the theorem uses essentially that the weight function a(x) is monotonically decreasing if x tends t o the boundary. The question arises whether similar results are true for weight functions of type 2 and type 4. We quote an interesting result obtained by 0. V. BESOV,A. KUFNER[l] (see also 0. V. BESOV, J. KADLEC,A. KUFNER[l]): Let Q be a bounded t?’-domain, and let a(x)be a weight function of type 4. If e ( t )belongs t o L,((O,A)), thenCO”(Q)is dense in WF(Q;a). If e ( t )is not an element of L,((O,A)), and if there exist numbers a > 1 and b > 0 such that
e(t) 6 be(ut) if 0 < t < Aa-1,
(2)
then C$(Q) is dense in W;(Q;a). ( 2 ) is a growth condition near the boundary, analogous t o (3.2.1/2). See also Remark 3.2.1/4. The case e ( t ) = tXis of special interest. is dense in W;(Q; dx(z))for The theorem and the above remarks show that x > - 1, while C$(Q) is dense in W;(Q; dx(x))for x 5 - 1, see also Subsection 2.9.2. Further, by Theorem 2.9.2/2 and the method of local coordinates, it follows that C$(Q) is dense in W;(Q; dx(x))for x 2 mp - 1. I n this connection we refer also to [l]. the paper by G . N. JAKOVLEV
c”(Q)
250
3.2. Definitions and Fundamental Properties
3.2.3.
The Spaces B;,JQ; e;’ 8”) and H;(Q;
ec; 8”)
If SZ c R,,is an arbitrary domain then Lt‘(SZ) denotes the set of all complex-valued functions defined in SZ, such that If(x)lP is locally integrable; 1 p < oc). The functions of L’,””(sZ) are extended by zero outside of SZ. Further, Cm(Q) denotes the set of all complex-valued infinitely differentiable functions defined in
a.
D e f i n i t i o n 1. Let SZ be a n arbitrary domain pin R,. Further let e ( x ) E C”(S2) be a positive function satisfying
IVe(x)I 5 ce2(x) for a suitable number c , such that for any positive number K there exist numbers and rk > 0 with e(x)> K
for d ( x )
E~
or
1x1 2
rJi
( d ( x )is the distance to the boundary). One sets 52Cj)
= {X
1 x EL?,e(z) < 2 J } ,
j = N,N
(1) &k
>0 (2)
(xEQ).
+ 1, .
where N is sufficiently large such that Q@“+= 0,and
Then y/ ( = Y(Q;e))denotes the set of systems of functions { ~ ~ ( x ) } ,satisfying ??~ 0
5 y,(x) 5
m
1, y j ( z )E C,“(sZj),
C yj(x)= 1 j=N
x
for
E
52
(4)
(here y j ( x )is extended by zero outside of A j ) for which positive numbers c ( y ) exist such that for all multi-indices y
lD’yj(~)I5 C(Y) 2 J l Y 1 , j
=
N,N
+ 1, . . ., 0
0 .
R e m a r k 2. To show that the definition is meaningful, one must ensure that Y is not empty. For this pirpose we prove that the domains have the same properties as the domains W ( J ) from (6). For a given number j there exist two points x1 and x2 for which @(XI) = 21, p(x,) = 21+l, and a point z = Ox, + (1 - 0) x2 EL!(/+,)- Q ( J ) , 0 < 8 < 1 , such that
where c1 and cp are suitable positive numbers. Now, it is easy to see that there exist m , ~ the required properties. For instance, if one systems of functions ( y , ( ~ ) } Jwith starts with the characteristic functions for Q ( / + l )- Q ( l ) , j = N , N + 1 , . . ., and for Qi.’), respectively, and if one uses the method of Remark 1, then one obtains a ~N If one denotes the distance of the set supp yj to system of functions { y j ( ~ ) E} Y. the boundary of QJby dJ , then one may additionally obtain by this construction that d, 2 c’2-j,
i
=
N,N
+ 1 , . . .,
(8)
where ct is a suitable positive number, see also H. TRIEBEL [lo], p. 117. D e f i n i t i o n 2. Let 9 be a n arbitrary domain, and let e ( x )be a function in the sense Definition 1. Further let E Y. (a) Let 1 < p < co and s 2 0. Further let ,u and v be two real numbers such that v 2 ,u + sp. Then one sets of
(For s = 0 it is assumed that ,u = Y, further H:(Q; e”; e”)= LJQ; @).) co,.s > 0. Further let ,u and v be two real numbers such (b) Let 1 < p < co, 1 q that v 2 ,u + s p . Then one sets (10)
252
3.2. Definitions and Fundamental Properties
(c) One sets
w g 9 ;e”; e”)=
H>(Q;@; e”) B;,q($2;p ; e’)
0 , 1 , 2 , . . .,
for s
=
for
<s
0
+ integer.
R e m a r k 3. The spaces Hi(R,,)and Bi,q(R,,)have the same meaning as in the second chapter, (11)corresponds t o (2.3.1/7). To justify the definition, one has t o show that e”; e”)and Bi,q(O;@‘; e‘) are independent of the choice of {y,}ZN the spaces E Y (in the sense of equivalent norms). Here, we shall need the assumption v 2 ,u + sp. The system !$‘plays a similar role for the spaces el; e’) and BZ,q(Q;e”;e’) as the system @ from Definition 2.3.112 for the Lebesgue-Besov spaces in R, without weights. On the basis of the (comparatively complicated) description of the spaces H;(R,) and B;,*(R,Jin Subsections2.3.1 and 2.3.2 we were able to reduce the theory of these spaces t o the theory of the L,-spsces, 1,-spaccs, and multiplier theorems. Afterwards, we obtained the usual equivalent norms for these spaces. As to the spaces H;(Q;e”; Q’) and B;,q(Q;@; Q’), we shall follow a similar way. The above definition allows a reduction of these spaces to the simpler spaces H8p(Rn)and Bi,q(R,L). Further, in Theorem 3.2.412 and Theorem 3.2.413, we shall obtain aimple equlvalent norms for the spaces W;(O; p p ; e’). 3.2.4.
Properties of the Spaces Bi,,(Jd; e”; B y ) and Hi(Q2; @”; e”)
L e m m a 1. Let s > 0, 1 < p < oc) and 1 5 q 5 a.Further let ,u and v be two real numbers such that v >= p + sp. If {yj(x)}SNE Y (Definition 3.2.3/1), then there ezists a positive number c such that
+ 2jvIIvj!I12p(nn)5 c (2jpIIfII;;*(Bfl) + 2j’llfll:p(R,)) for all f E B;,q(R,Jand all j = N , N + 1, . . ., resp. 2j”IIyjfII:;&(Rfl)
2j”l\vjfllg;(~~) + 2.i”Ilvjf\I!p(R,)
+
5
c (2jpl\f\l;p(R,,) + 2”11fl12p(R.))
(1a)
(1b)
for all f E H;(R,) and all j = N , N 1, . . . Proof. Step 1. Let f E Bi,q(R,,).Using t,he transformation of coordinates x = 2-jy and setting f(x)= f(2-jy) = f(y), resp. yj(x) = yj(2-Jy) = ijP( J Y) then it follows by (2.5.1/12) with the notations introduced there that
3.2.4. Properties of the Spaces BiJQ; ej';
e') and H;(Q;
ej';
e')
253
It follows by interpolation that (4)remains true after replacing WF]+'(R,,)by B;,q(R,) or H;(R,), respectively. Now, using v 2 ,u + sp, one obtains that ( 2 ) is also valid if one replaces y j i on the right-hand side by f. Returning to the original coordinates x and using again v 2 ,u + sp, it follows (1a). Step 2. To prove (1b), we note that
are multipliers, see Remark 22.414. Hence, H i ( R , ) may be normed by
11F-'151"
FfIILp(ll,)
+
IlfIILp(Rn)*
Now, the formula analogous to ( 2 ) has the form 2 J ' I I v j f llT,;(Rn)
+ 2J'llvjf
ll%p(Rn)
5 c2i'-inllyjfllip(R,) + c2jF+ jsp-jn IIF-~I~IS
Fijj?IIpLp(R,)
*
Using (4)with H;(R,) instead of WFl+l(R,) then one obtains ( l b ) analogously to the considerations in the first step.
T h e o r e m 1 . The spaces H",Q; @';e') and B;JQ; @; e") from Definition 3.2.312 are Banuch spaces. These spaces are independent of the choice of {yj(x)}joo,NE Y (equivalent norms). C c ( 9 ) is a dense subset i n B ; , J 9 ; e"; @), s > 0, 1 < p < 00, 1 5 q < co,and i n H i @ ; e"; e"), s 2 0, 1 < p < co.
Step 1. Let { y j ( x ) } ? E Y and {p!i(x)}iqo,N =Proof. 0, it follows for f E B ; @ , .'. .p 9 ,.e') by Lemma 1 E Y . Setting Y ~ , , - ~ ( Z )= yK.-l(s)
Whence it follows the independence of B;,,(Q; @';e') of the choice of {yj(x)};N E Y. One concludes analogously for the spaces H;(O;e"; e'). Step 2. If {f/ 0 is a given number. Applying Lemma 2 t o the second term on the rightj i 2 hand side of (10) where one may replace f by f yk) , choosing E in ( 11 ) sufficiently
Here
E
c ( small, transforming the corresponding terms t o the left-hand side of ( l o ) ,and using k=j-2
v 2p
+ sp, then one obtains the reversion t o (8). Analogously, one concludes for
s = 0,1,2,..
.
Theorem 3. Let the hypotheses of Theorem 2 be valid. Further let 0
t
xt=p;+v-=
s-t
rU+(v-p)-.
5t
s s and
s - t S
(a)If t is a n integey, then there exists a positive number c such that for all f belonging to J q Q ;e"; e')
c 1e " W l D " f ( 4 I P dx 5 cllfll&;(*;ew;e")*
14 = P I
If t is not an integer then
( b )It holds
(13a)
3.2.4. Properties of the Spaces Bi,,(SZ;d ;e’) and H i @ ; @;
e’)
257
Here ~ ~ f / ~ $ p ( n ; e , , ~ e v ) is a n equivalent norm i n W;(SZ; e”; e’).
Proof. Step 1. Let t += integer and let f E W;(SZ; e”; 8’). Similarly to the first step of the proof of Theorem 2, one obtains that
+ s p it follows that xt, + pt, 2 xI. + p.t% for 0 Particularly we have xt + { t } p 5 x p l . If
From v 2 p
t, 5 t , 5 s. (16) one replaces j p + p ( s - t ) j in (11) by x,j, then one obtains (13b). If t is a n integer, then one concludes analogously. Step 2. By the first step, it is sufficient for the proof of (b) of show that a function fE where Ilf II < co,is an element of W”,SZ; e”; e’). For 0 < s 1, this is a consequence of the second step of the proof of Theorem 2, since for these cases the last considerations of the second step of the proof of Theorem 2 are not necessary. (In these considerations, we used essentially that I(f I(w;(n;Pw;PY) < 00 is known.) Using (16), one obtains the desired assertion for s = 1 , 2 , . . . in an easy way. Finally, let 1 < s integer. Then we use (11) with p + {s}p (5qs1) instead of p, [s] instead of s, end t = 171 + is}, as well as (10). R e m a r k 1. The theorems of this subsection justify Definition 3.2.312 and Remark 3.2.313. I n the further considerations, we shall be based essentially on the formulas in Definition 3.2.312. Further, we emphasize the difference in the formuletions in Theorem 2 and in Theorem 3(b). By the second step of the proof of Theorem 2, it is not clear whether (in the sense of Theorem 3(b)) any function f E Lp(Q)with ~ ~ ~ ~ ~ % ; m2 > m, 2 0,1 < p < co, 1 q 5 00, 0 < 8 < 1 and s = (1 - 8)m, + Om,. If k and 1 are integers such that Osk<sandl>s-k,andifO # sz}
1
and D"u instead of - . (For
a$
3.3. Interpolation l'ieory for the Spaces W'#2;
270
0)
for h E K , then one obtains that,
Using the transformation of coordinates x - lh = y in the last term and putting the result in formula (9), then one obtains the middle part of ( 7 ) . By (3.2.1/2) and (3.2.1/3'), it follows for the other terms of (10) that
Here c1 and c2 are two suitable positive numbers. Putting these formulas in (9),then it follows that
l l 4 (*(n;u),GyQ
;u))o,g
(with the corresponding modifications for q = co). Now it is easy t o see that there exists a positive number c such that, @I'll
c
Q(h)
c QI"1.
Here c is independent of h E M,. Whence it follows (7). Similarly one proves (8). R e m a r k . We consider t,he special case (~(x) = d-"(x), x 2 0. If p = q , then i t W follows by (5) that
n
f
nr
s
m
= )u(x)~P,d-"~(x) ft-l-sp dt dx = c d - X ~ - s P ( lu(x)ll'dz. ~) n n d(.l.) (12)
This is an essential simplification of the last terms in (7) and (8).For the case x = 0, we shall return t o this point later on.
The Spaces W;(sZ; a) with Weight Functions of Type 3
3.3.3.
Let D c R,,be a bounded C"-domain. Then Q t ,0 < t < co, has the same meaning h E I?,, and as in (3.2.1/1). For z E aQ, the inner normal is denoted by Y;. For 0
+
7c
0 < E < -, one sets 2 Q,,,e,t
=
(Q - 91)u ({z I z E aQ; 0 5 ( J L ,
Y;)
< E> x ( 0 , t l ) .
(1)
3.3.3. The Spaces W,”(B; a) with Weight Functions of Type 3
27 1
Here ( h , v,) denotes the angle between the vectors h and v, (see Fig. 2), and (0, t ] must be taken in the direction of the inner normal. Hence Q,L,e,t is the union of the 1 and that part of SZt where the directions of h and v, are “inner” domain SZ - 9 near each other.
JI
6,t
u
ch,vz7
Fig. 2
Theorem. Let SZ c R, be a bounded C”-domin. Let ~ ( x be ) a weight function type 3. Further let m, and m2 be integers, 00 > m2 > m, 2 0, 1 < p < 00, 1 5 q 5 00,0 < 0 < 1,ands = ( 1 - O)m, + 0m2.Ifkand1areintegerssuchthatO 5 k < s and 1 > s - k, and if 6, E and t are sufficiently small positive numbers, then of
B;,*(Q;0 ) = (W?(Q;
For q
=
co one must replace
4 q ? ( Q ;U))o,q
[ 1I-lqG]
by sup
lhlS8
IhlSd
1.1.)
All these norms are
equivalent norms in the space B8p,q(SZ;a). Proof. The balls Kj,the domain w and the functions yj(x)have the same meaning as in the third step of the proof of Theorem 3.2.2. For u E ern@)
-
N
I I ~ I I ~ ( Q ;j =~CO) lly.jUIItv~(Q;o). By interpolation it follows that N
272
3.3. Interpolation Theory for the Spaces H’lip”(Q; a)
If the domains wj = ($2 n K,;) + Kj, j = 1, . . ., N , have the same meaning as in the third step of the proof of Theorem 3.2.2, and if one chooses a suitable unbounded domain of cone-type wo 3 w, then it follows from the explicit form of the K-functional (see 1.3.1) K(t, y j ~W, p ( Q ;u ) , Vp(Q; u ) ) K ( t , ~ j uW?(wj, , u ) , W?(wj, 0 ) ) . Here the weight function u(x)and u ( x )are extended similarly t o the third step of the proof of Theorem 3.2.2. Hence, one obtains by Theorem 3.3.1 that
-
IIy j IIBL~W; ~ u)
N
IIy j u I l B L ( w j ;u ) (6)
where M6 = MLj) is independent of j . (For q = ca one has t o modify.) Now we use the formula
d i @ w )(4 =
c (A24 c C r , d ( d i - W + dy). 1
r=O
1
(5
(2) d=O
One proves this formula for 1 = 1 directly, and for I > 1 by induction. Whence it follows for j = 1 , . ., N that
.
For j = 0 one has t o replace K j n $2 by w. Now, if 0 < s by (5)-(8). Let s 2 1. It follows from (5)-(8) that
< 1, then one obtains (3)
I n the sense of induction, we suppose that (3) is proved for 0 < s < M where M is a natural number. Let M 5; s < M + 1. Then (8) and (9) yield that the second summand in (9) can be estimated by IIuIIBs,( R ; u ) , where s’ is a n arbitrary number PA
such that s - 1 < s’ < s. From (1.3.315) it follows that for any positive number q there exists a number c,, such that (10) I I U l l B ~ , ( O ; u ) 5 V \ l ‘ l \ B ~ , q ( Q g u ) + CqIIUIIL,(R;o)* (9) and (10) prove (3). Similarly one proves (4).
R e m a r k . For the important special case u ( z ) = 1 we shall obtain further equivalent norma, later on. I n particular, it will be possible to replace the rather compliby simpler domains. See Theorem 4.4.211 and Theorem 4.4.212. cated domains $2J,,c3t
3.4.1. Preparatory Lemma
3.3.4.
273
The Spaces w:(sd; a) with Weight Functions of Type 4
Let 9 c R, be a bounded C”-domain. Then Qt, 0 < t c a,has the same meaning as in (3.2.1/1), ll~ll(~,,,,~,~, was defined in (3.3.2/5). Further we shall use Qh,8,t from (3.3.3/1).If a(z) is a weight function of type 4, then we set analogously t o (3.3.214)
with the usual modification for q = 00. Here 1 < p < 00, 1 5 q 6 03, s > 0, and s < 1 = integer. Theorem. Let 51 c R, be a bounded C”-domain, and let u(x)be a weight function of type 4 . Further let m, and m, be integers, 00 > m, > m, >= 0, 1 < p < my 1 s q s 0 3 , 0 < O < l , a n d s = ( 1 -O)rn,+Bnt,.Ifkandlareintegerssw;hthat 0 k < s and 1 > s - k, and if 8,s, and t are sufficiently small positive numbers, then (JQP(Q; a), ~ T ( Q u))8,q ; = IuE a), b ~ ~ $ , ? d , e , t ) < a), (2) r = 1 , 2 , where
~E(Q;
llull
..
*(2)
(k,I,d, (It,)
=
IIuIIL,(O; U)
+ I acl J k[IlPullb;ik(O
;(I), d, c, t
+ IID“u 11 b k , p . q, dl
*
(4)
All these norm are equivalent n o m in ( ~(Q; 7a), Wp(Q;a))o,q. Proof. The proof is the same as the proof of Theorem 3.3.3. By the formula analogous to (3.3.3/6),one must take now (3.3.217)or (3.3.2/8), respectively.
3.4.
Interpolation Theory for the Spaces B;,*(sd; 8’; 8”)and H;(sd; BL;e”)
This section is concerned with the interpolation theory for the spaces Bi,q(Q;Q”; Q’) and H;(Q; Q”; Q”) defined in Subsection 3.2.3. By Definition 3.2.312, the spaces W;(Q; Q”; e’) are special cases for which we obtained new representations in Theorem 3.2.412 and Theorem 3.2.413. We shall apply the results in the theory of strongly degenerate elliptic differential operators.
3.4.1.
Preparatory Lemma
For sake of convenience, we denote temporarily by Q; one of the spaces B;:(R,) or H;(Rn),s > 0, 1 < p < co, 1 5 q 5 co. Since the number q does not play any role in the following lemma, we do not notice this index. Further we write Q; = LJR,). 18
Triebel, Interpolation
3.4. Interpolation Theory for the Spaces Bi,(L?;e!‘;0”)and H i ( s 2 ; @;
274
el’)
If s 2 0 , 1 < p < co, and v 2 p are real numbers, then for j = 1 , 2 , . . . in Gi the equivalent norms
I(fI/$p’”)
.P
2JpIlfllGi
=
+
”’ .v
(1)
IlfllLp
are introduced. To avoid confusion, we shall write a;(j,p, v ) . Further, for s = 0, it is assumed p = v. Lemma. Let s1 2 0, s2 2 0, 1 < p1 < co, 1 < p 2 < co. Further, let v1 2 p1 and v2 2 pa be real numbers such that
- ~ 1 ~)
(PI
2
=~ (PZ 2
- 82) ~ $ 1 -
(2)
Let F({., be either the complex interpolation functor [., *]e or the real interpolation functor (., .)e,, where 0 < 8 < 1 and 1 5 q 5 co. One sets a})
s1 0 and (I n the =case0 one sets p1
s1 = s2
given spces
> 0 one sets p1 = v1 and
E, in the m
e sap2 = v1,,u2 = v2 , a d p = v . ) Further it is assumed that for the and G:, there exists a suitable space G; such that
=
Then
s2
=
81131
-
.
j = 1 , 2 , . . Here “ ” means that the left-hand side of ( 6 )can be estimated by the righthand side with the aid of a constant independent of j, and vice versa. Proof. Step 1. We start with a general remark. Let {A,, A,} be an interpolation couple. If c, > 0, then c,Al denotes the Banach space A , equipped with the norm clII.IIA,. Similarly, one must understand c2A2for c2 > 0. Then F({ClA,, czA2)) = c:4c:F({4, A,}). (7) For the real method, this is an immediate consequence of Definition 1.3.2. If F ( A , , A,, 0) has the same meaning as in Definition 1.9.1, then it is easy t o see that g(z) = c:-zca’f(z) is an isometric mapping from F(c,Al, complex method is a consequence of
cA2,0 ) onto F ( A , , A,,
0). Now, (7) for the
Ilg(e)II[A,,A ~ l e= I l f ( e ) I l [ c ~ A csAsle. ~,
Step 2. If s1 = s2
=
0, then ( 6 ) follows immediately from (7).Let s2
> 0. Setting
then one obtains by Theorem 2.5.1 and Theorem 2.3.3(a) (see also the equivalent norm
3.4.2. Interpolation Theorem
275
in Hi(R,,) described in the second step of the proof of Lemma 3.2.4/1) that
and a similar formula for Ct:,. Using (7)where
then it follows that After returning t o the original coordinates, one obtains (6). R e m a r k . The lemma is a generalization of Lemma 4.2 in H. TRIEBEL [22,II]. It is the basis for the later considerations. Then we shall use several examples for formula (5).
3.4.2.
Interpolation Theorem
With the aid of Lemma 3.4.1 and the results of Chapter 2, one may prove a rather general interpolation theorem. I n this subsection D c R, is an arbitrary domain; p ( x ) denotes a weight function in the sense of Definition 3.2.311. The spaces H;(Q; $‘; e’) and B;,,(SZ; ep;e’) are introduced in Definition 3.2.312. Theorem. Let s1 0, s2 2 0, 1 < p1 < a,1 < pa < co. Further let v1 2 p1 + sIpl and v2 2 p2 + s2p2are real numbers such that (3.4.112) holds. For 0 < 8 < 1, the sazcnabers s, p , v , and p have the same meaning as in (3.4.113)and (3.4.114)(inclusively the special cases marked there). (a) Let additionally s1 + s 2 , p1 = pa = p , and 1 q l , q2 5 00. Then
(B2q1(Q;
@”I),
B&,(Q; P ;e y ’ ) ) e , p
= (B$,,JQ;e”1; e’l),H;(Q;
eP2; @y~))B,I>
H;(SZ; e”*;e v ~ ) ) = ~ ,Bi,p(Q; p p”; e”). (1) I n the cme of the B-spaces, it is assumed that s1 or s2, respectively, are positive numbers. ( b )Let additionally s1 > 0, s2 > 0,1 5 q l , qa < co and = (H$(Q;
@”I),
1-8 e -+-=--.
Then
P1
1-8
then 18*
q2
(B;l,81(Q; @PI;
( c ) Let additionally q1
1 P
s1
@“1),
(2)
BZ,,,(Q; P ; Q”9)o.p
> 0 , s2 > 0, 1
+ -e= Pe
[B21,,,(Q;~’‘1;
=q , p m
s q1 < co, 1 5
92
ep; e7.
5 co. I f
1 P
,0”1),
(3)
’ B;;s,48(Q; e w z ; ~ ” ~ )= l eB;,,(Q; e”; e’).
(4) (5)
3.4. Interpolation Theory for the Spaces Bi,,(Q;e';
276
e')
and H;(Q; $'; e')
[H"d,(Q; @ I ; @"I), H;,(Q; @ " a ; e"*)le= H",Q; f; e"). (8) Proof. Step 1 . We prove (8). Let {yj(x)}FNE !P be a system of functions in the sense of Definition 3.2.3/1, having the property (3.2.3/8). Then there exists a system such that v,(z)E C,"(Oj), of functions {~)i(x)}im,~ JyI < co , IDuy,(z)l c ( y ) 2jl''l for j = N , N + 1, . . ., 0 (9)
s
s
Ipj(2) = 1 if 5 E supp yj, j = N , N + 1, . . . In the sense of Subsection 3.4.1 and (2.4.2/11) we write G; = H;(R,), further, G;(j, ,u, v ) has the meaning introduced there. Then by Definition 3.2.3/2, S ,
Si = { y j ( z )/ ( ~ ) } F N ,
(10)
is a linear bounded operator from H Z ( Q ; p,e"r) into Zp,(QX(j, p r , v r ) ) , r = 1 , 2 . Further, it follows from the proof of Lemma 3.2.4/1 that R ,
is a linear bounded operator from l,,,(GZ(j, p r 7v r ) ) into H Z ( Q ; p;e'r), r = 1 , 2 . It holds RS = E . Hence R is a retraction. Now (8) is a consequence of Theorem 1.2.4, (1.18.1/4), (2.4.2/11), and Lemma 3.4.1. Step 2. The proof of all the other assertions of the theorem is completely analogous. Instead of (2.4.2/11),one has to use (2.4.2/13)for (a), (2.4.1/7)for (b), (2.4.1/8) for (c), (2.4.2/9)for (d), and (2.4.2/10) for (e). R e m a r k 1. We note a special case interesting for the later considerations. Let s1 2 0, s2 2 0, s1 =I= s2, 1 < p1 < m, 1 < p2 < co. Further, let v1 2 ,ul+ slpl and vp 2 p2 szp2 such that (3.4.1/2) holds. For 0 < 8 < 1, the numbers s, p , v , and ,u are determined by (3.4.1/3)and (3.4.1/4)(inclusively the special cases marked there). Then we have
+
(W2JQ;
@"I;
y?,(Q; ;
eV1),
@"P
@y2))&p
=
B;,,(Q; @";
e') .
(12)
This follows by (3) and (6). If s is not an integer, then one may replace B& by W; on the right-hand side of (12). R e m a r k 2. For 0 < 8 < 1, one obtains by (7) and (8) that where
-
(Lpl(Q;Q P 1 ) 7 1 - 1-8
P
P1
Lp2(Q; P ) ) o , p
+-e
P2
=
[Lpl(Q;@'I),
L p B ( Q ; @"2)le
and -P= ( 1 - 0 ) - + 8Pl- .
P
This is a special case of Theorem 1.18.5.
P1
P2
P2
= LJQ;
e")
(13)
3.4.3. Interpolation of the Spaces W;(sZ; 0"; e') with v < p
+ 81,
277
R e m a r k 3. On the basis of the interpolation theorems from the Subsections 2.4.1 and 2.4.2, one may immediately obtain further interpolation theorems similar to that ones in the last theorem. I n particular, one can develop an interpolation theory for spaces F;,q with weights. The above theorem, is a generalization of Theorem 4.3 by H. TRIEBEL [22,II].
3.4.3. The spaces
@;(a; e"; e") with v < p + sp < p + sp are introduced in Definition 3.2.6.
Interpolation of the Spaces
*@;e')
ep; with v Let SZ c R, be a bounded C"-domain. Let e ( x ) be a weight function i n the
Theorem. sense of Definition 3.2.3/1 such that e-'(x) d ( z )near the boundary (see Remark 3.2.3/1). Further, let s1 2 0 , s2 2 0, s1 s2, 1 < p 1 < co, 1 < p 2 < 00, v1 < p1 SIPl, v2 < ,u2+ s2p2such that 1 1 + kjpj where lc; = 0, . . ., [ s i ] - 1 {s,} == ! - and p,; + sypj
+
+
N
+
Pj
for j = 1 , 2 . For 0 < 8 < 1 , one sets
(+,(sz;
@"I;
@"I),
+;p;p ;@ " ' ) ) B , p = B;,p(SZ;e";
(b) If s1 and s2 are not integers, then
[+;l(s2;
@"I; @"I),
@;*(SZ;
=
e"2:
If s1 and s2 are integers, then
[+;l(SZ;
p;
@''I),
its;,(Q;
Proof. Theorem 3.2.6 yields
e'"2;
@",)Is
@"+SP).
(3)
q p ( S Z ; e"; @"+SP).
(4)
e";
(5)
= H;(SZ;
@"+SP).
P Jp(j Q ; e P j ' l ;e'j) = w'j(SZ; Pj epj; e"j+sjpj), j = 1,2. (6) Now, (3) is a consequence of (3.4.2/12).If s1 and s2 are not integers, then one obtains (4) from (3.4.2/5).If s1 and s2 are integers, then (3.4.2/8) yields (5). R e m a r k 1. It follows from (2.4.2/12) that the distinction between (4) and (5) is necessary if one wants to obtain a description of the interpolation spaces by H spaces and B-spaces.
R e m a r k 2. As mentioned in Remark 3.2.6/6 and Theorem 3.2.6, the index v 1 in the notation of the spaces e"; e") is immaterial for the cases { s ) =+ -
+
Te(Q;
P
and - 00 < v < s p p. We consider two important special cases. (a) If s is not an 1 integer, { s ) - , and
+ p
P
+ sp =+ 1 + k p
where k = 0 , . . ., [s] - 1 ,
(7 )
3.6. Embedding Theorems for Different Metrics
278
then it follows by (3) and (3.2.6/20) under the hypotheses of the theorem that
(kip;
+;p;e”.;
= @‘”,(Q; @”;
e’).
(8) Here v is an arbitrary number such that -co < v < p + sp. (b) If s is an integer, and if (7) holds, then it follows (under the hypotheses of the theorem) for the integers s1 and s2 by ( 5 ) that @”I;
@I),
[k;l(~; pi; p),FV;,(Q;
p a ;
Here v is an arbitrary number such that
kP( e”;~ e”).; - 03 < v < p + sp.
e’z)le =
(9)
R e m a r k 3. The case pj = 0 is of special interest; j = 1 , 2 . Then one can also choose vj = 0, j = 1 , 2 . One obtains an interpolation theory for the spaces ki(Q). 1 (1) reduces itself to { s j } + -. We shall return to this question later on. Pj
+
R e m a r k 4. A n interpolation theory for the spaces W’“,(Q;$; e”), v < p sp, seems t o be more complicated. Some special results are obtained in C . GOUDJO[l] and A. FAVINI [7].
Einhedding Theorems for Different Metrics
3.6.
Similarly to Section 2.8, we describe embedding theorems for different metrics for the spaces considered in this chapter. The embedding on the boundary will be treated in the next section.
Thc Spaces Bi,,(D; 8’;
3.6.1.
Q’)
and Hi(&?;Q”;
e‘)
The spaces W;(Q; e”; e’) are special cases of the spaces Bi,,(Q;@; e’) and p”; e’), see formula (3.2.3/11). The theorems of Subsection 3.2.4 show that these spaces are of special importance. Theorem. Let Q c R, be an arbitrary domain. Let e ( x )be a function in the sense of Definition 3.2.311. (a) Lett 2 0, n )E s -- = t - -, co > q 2 p > 1, (1)
and v
2 p + sp. Then
and
where (For t = 0 ,
P
4
3.5.2. The Spaces Wi(J2;eL(;e’) with
(b) Let 1
5 r 5 co and t > 0. Further let
(1) be valid. Let v
i s true. Then
Y
0, then it follows by the interpolation properties of the spaces W; that Further we have
(2) is a consequence of (6) and (7). Step 2. The proofs of (3) and (5) are similar. One has to use (2.8.1/15) and (2.8.1/2). Here, W;-’ in (6) and (7)may be replaced by B;;,;”1. Remark. For the important special case
v 3.6.2.
=
,u
+ sp
it holds
t=
+ tq.
x
The Spaces +;(a;8”;B y )with v
(8)
< p + sp
The spaces @,(SZ; e”; e’) are introduced in Definition 3.2.6. Theorem. Let 52 c R,, be a bounded Cw-do7nain.Let e(x) be a weight function in the sense of Definition 3.2.3/1 and Remark 3.2.311 such that e-l(x) d ( x )neur the boundary. (a) Let
-
c o > q ~ p > l ,s
I f -co < v < p where
+ spand
~
t
~ -0c o ,< p < < ,
-a < t < x
+p(sz; e”; @’) c Fk;(Q; e x ; @)
+ tq, then
-x =, - u
q
n
P’
n q (b) Suppose that (1) i s valid. Further let 1 {s) =I= -, ,u s p $. 1 k p where k = 0 , . . ., [s] - 1 , s-->t--. ?,
P
+
+
1 {t}+-,x+tq+1+1q P
where l = O
,..., [ t l - l .
(1)
280
3.6. Direct and Inverse Embedding Theorems (Embedding on the Boundary)
then (2) is true. Proof. The second part of the theorem is an easy consequence of Theorem 3.2.6, Theorem 3.5.1, and Remark 3.5.1. The spaces W$2; @; e r + s q ) are monotonical with respect t o s, similarly Wl,(Q; ex;exftq).This is also valid for the spaces ep; e”), resp. *q(12; ex;Q‘), if s, resp. t , belongs to the interval [k,k l ) , k = 0, 1, . . . For the spaces W ; ( 9 ; efl; e”+sp) resp. W:(Q; f‘; @‘+tq) a corresponding assertion is true for variation of p, resp. x . Similarly for W ; ( S ; e”; e”), resp. $$2; ex;Q‘), if s and t are integers. Taking into consideration that the parameters in (3) may be changed a little, then the part (a) is a consequence of part (b).
+
3.6.
@,(a;
Direct and Inverse Embedding Theorems (Embedding on the Boundary)
This section is the counterpart t o Section 2.9. Here we restrict ourselves t o the
description of boundary values on (n - 1)-dimensional boundaries. One can extend the investigations on boundary values t o I-dimensional manifolds in the sense of l < n - 1. Since the previously used method from the first Subsection 2.9.4; 1 step of the proof of Theorem 2.9.4 is generally applicable, we omit here explicit formulations.
3.6.1.
Direct and Inverse Embedding Theorems for the Spaces WF(sd; d”(r))
If SZ c R, is a bounded C”-domain, and if one denotes, as before, the distance of a point x E 9 t o the boundary aQ by d ( x ) , then the boundary values for the spaces WT(Q; d”(x)), m = 1 , 2 , . . ., 1 < p < co, -1 < dc < 00, are considered in this subsection. These spaces are described in Definition 3.2.114. For 0 dc < 00, d”(x) is a weight function of type 3 in the sense of Definition 3.2.113. For - 1 < 01 5 0, d”(x) is a weight function of type 4.0.V. BESOV, A. KUFNER[l] have shown that C$’ (9) is dense in the corresponding spaces W F ( 9 ;d b ( x ) )with dc 5 -1. This explains the above restriction t o -1 < dc < 03 in the consideration of boundary values. For the formulation of the embedding theorem, some preliminaries are needed. L,(aQ) has the usual meaning. The measure on a 9 is the induced one by the Lebesgue
measure in R,.
Definition. Let 9 t R, be a bounded C”-domain. The balls K j , j = 1, . . ., N ,
and the functions f ” ( x ) have the same meaning as in Definition 3.2.112. Further let jW1(y) be the corresponding inverse functions. The functions yj(x),j = 1, . . ., N , are
Theorems for the Spaces WyPm(Q;d”(z))
3.6.1. Embed-
defined i n (3.2.2/1). Let 0 < s < co, 1 < p < co, 1 6 q 6 B;,,(aQ) = {f
I I f IZ;,(an)
00.
28 1
Then one sets
I f €Lp(aQ),( v j f()f ‘ ” - W )EB;,q(R,c-i),i = 1, . . .)N ) ,
(1)
A’
=
C II ( v j f ()f (’)-‘(Y)) IB;,~(R,,-,)*
(2)
j=1
(Here the functions (y j f)( f ( j ) - l ( y ) ) a r extended e by zero outside of the image of KjnaQ.) Lemma. B;,,(aQ) i s a Banach space. I n the sense of equivalent norms, B;,?(aQ)is independent of the choice of the balls K , as well as of the choice of the functions f (J)(x) and lyi ( 5 ) .
Proof. Let two coverings of the boundary aQ with balls be given. Then one finds a covering finer than the two given ones. Then one proves easily the independence of the balls K j and the functions f ( j ) ( x )and yj(x).Afterwards it follows that B;,,(aQ) is a Banach space. R e m a r k 1. Clearly, one can define similarly spaces H p Q ) , 00 > s > 0, 1 < p < m. As a special case one obtains W;(aQ) = H p Q ) for s = 1 , 2 , . . , and W”,aQ) = B;,,(aQ) for 0 < s integer. For the description of the boundary values we shall need only the spaces B;,,(aQ).
+
Theorem. Let Q c R, be a bounded C”-domain. v denotes the (outer) normal on aQ, and f Ian denotes the boundary value of the function f . Further let m = 1 , 2 , . . . andl p from the above listed cases. It is not hard to see that the above method includes all the cases of the theorem. Step 4. Let p = q. We prove (f). Theorem 3.4.2 yields that
Wlp)(Q;
@l-qP+&.
, e (1-
&l+ZV+rnp(l-Z))
belongs t o
~ ( 6 w;(Q; , e P ; e Q + n t p ) , L J Q ; p v ) ) n ~ ( 6 w;(o; , 9 ;p + m p ) , LJQ; e v ) ) , 0 < 8 < 1, m = 2 , 3 , . . . Now if one determines 6 such that m ( l - 6) is a natural number, then (f) is a consequence of Theorem 3.8.2 and Theorem 1.16.3/1(b).
R e m a r k 1. The theorem coincides essentially with Theorem 4(a) in H. TRIEBEL ~71. R e m a r k 2. The proof shows that one may replace W i in the formulation of the theorem by H ; . R e m a r k 3. If one uses Theorem 3.8.2(f) and Theorem 1.16.3/2, then one may obtain a similar theorem (without (f)) for the widths d j . Since the method is clear, we omit an explicit formulation. R e m a r k 4. Compact embeddings for spaces without weights are treated in more detail later on. See Section 4.10. There are also given further references. R e m a r k 5. The above theorem and Theorem 3.8.2 for p = q = 2 are of special interest in the framework of the theory of degenerate elliptic differential equations. With their aid, one may easily obtain assertions on the asymptotic distribution of eigenvalues. See Subsection 7.8.3.
3.9.
The Spaces w;,,(R,)
The previous sections are concerned with two rather general classes of SobolevSlobodeckij-Lebesgue-Besov spaces with weights, Definition 3.2.114, Definition 3.2.312, and Theorem 3.2.412. The methods used for the investigation of the spaces Bi,,(SZ;ep;e') and H i ( Q ; ep;e') are also applicable to other classes of spaces with weights. I n this section, we consider the spaces wi,,(R,) coinciding essentially with [2,3]. These spaces are similar to the the spaces introduced by L. D. KUDRJAVCEV spaces W$2; ep;e') of Sobolev-Slobodeckij type. The equivalent norms for the spaces w;,JR,) of Subsection 3.9.1 permit in an easy way t o introduce corresponding spaces of Lebesgue-Besov type. We shall not consider here these generalizations,
3.9.1. Definition and Equivalent Norms
299
although they would give a more complete insight into the theory. Further, some interesting problems, e.g. embedding theorems for different metrics, are not treated here, although the previously developed methods seems to be strong enough for such considerations. We shall not try to obtain the most general results. The spaces w;,+(R,),as well as generalizations of these spaces, are considered by several authors. [2, 3, 41. Further references I n particular, we refer t o the papers by B. HANOUZET will be given in Subsection 3.10.3. A more elaborated and generalized version of [29]. this rather short written section can be found in H. TRIEBEL
3.9.1.
Definition and Equivalent Norms
D e f i n i t i o n 1. For x = ( x l , . . ., x f l )EH, om sets 1 < p < CO, -co < ,u < co,and s = 0 , 1 , 2 , . . . l e t
e ( x ) = (1
+ 1x12)+.
For
R e m a r k 1. As mentioned in the introduction these spaces coincide essentially [2, 31. It follows that, in dependence with spaces considered in L. D. KUDRJAVCEV on ,u, polynomials of different degrees belong t o the spaces w ~ J R , ) .This is one of the main motives for treating these spaces. If 52 c R,, is a (bounded or unbounded) domain (of special interest is 9 = R:), then one may define w;,+(Q)as the restriction t o Q. See for instance B. HANOUZET [2, 3,4]. of w;,+(Rm) D e f i n i t i o n 2 . Let
K ,, . - {5 121-1 c 1x1 < 2j+2}, j = 1 , 2 , . . ., I i , = {X I 1x1 c 2}. Then 2 denotes the set of all systems of functions where
{cj(x)}so,
m
(here c j ( x ) is extended by zero outside of K j ) ,for which there exist positive numbers c(y) such that for all multi-indices y ID’[j(x)I c ( y ) 2-jlV1, i = 0, 1, 2, . . . (4) R e m a r k 2. See Definition 3.2.3/1. Similarly to the considerations in Remark 3.2.312, it follows that there exist systems of functions with the required properties. T h e o r e m . Let 1 < p c a,-co < ,u c co,and {cj(x))F0€ 2 . (a) If 0 6 s < 00, then w;,+(R,) is a Banach space, andCF(R,) is dense in w;,+(R,).
300
3.9. The Spaces w;JRn)
(b) If = 0,1 , 2 , . . ., then
(c) If 0 < s
= [s]
+ {a},
[s] i s an integer, 0 < {s}
< 1, then
P r o o f . Step 1 . One proves by standard arguments that U).p,p(R,) is a Banach space. Step 2. Let 1 < p < co and 0 < {s} < 1. Then there exists a number c independent of j and y E K j such that
(7) is analogous to (3.2.4/6).The proof may be carried over from that place. If loci = 0, then one may replace DaCj by 1. Step 3. Let f E W ; , J R ~Then ) . f E W>”‘(Rn).We set K j = K,-l v K i+ l, j = 1,2,..., and KO = KOv K , . For 0 < {s} < 1, one obtains that m
,.
The last term can be estimated by the first term on the right-hand side. Using (7) for lul = 0 , with Kj instead of K j and with 1 instead of DaCj,then one obtains that
By a homogeneity argument (one replaces x by
g ( z ) E W;(Rn)that
E -(s -la l) p
f l o ” g ( x ) p ax Rn
EX),
it follows for
loci 6
[s] and for
30 1
3.9.1. Definition and Equivalent Norms
Here E > 0 is an arbitrary number, c > 0 is independent of and E = 2J, it follows by (8) and (9) that IlfllW Pi , p ( R , )
E.
For g(z) = (&)
5 cllfllX,p(~m) *
(2)
(10)
= 0, then one concludes similarly. 8tep 4. Let again f E uPp,?(Rn). If s is an integer then i t follows immediately from the properties of the functions c, that
If {s}
This proves (5). Now, let {s} > 0. Using (7) and (11) with [a] instead of s, then one obtains, similarly t o the third step, that
If. -[.I+
(4P f ( 4 - e
c
I z - yl"f("1P
+ c Ia I 5 [.q]
la1
(Y)W(Y)I
P
as*.
Rn X Rn (12) We apply the method of the third step t o the terms with la1 < [s]in the last summand and obtain a formula analogous to (8).Now we use
(13) R,
instead of (9). Setting E = 21, then the terms with la1 < [s] in (12) can be estimated in the desired way. Whence it follows (11). Step 5. Using the norms l]fl\2;#(Rfl) and the above technique of estimates, then it follows that C: is dense in w;,+(R,,). R e m a r k 3. The spaces w;,+(Rn)are similar to the spaces Wdp(Q; 9 ; e') of Theorem 3.2.412. The theorem shows how to define Lebesgue-Besov spaces h;,JR,,) and b;,JRn), see Definition 32.312. One has to replace the first summand on the righthand sides of (5) and (6) by the homogeneous parts of top-order differentiation of the norms of B;JRn), resp. H i ( R , J . I n the case of the spaces BiJR,), one can take for this purpose the second summand on the right-hand sides of (2.5.1/10),resp. (2.5.1/11), where 6 = co. In the case of the spaces H;(R,J, one can choose IIF-lI~I~ FfllLp. R e m a r k 4.*) Using a modified Hardy inequality of type (32.614) then one can estimate the second term on the right-hand side of ( 5 ) by the first one. Hence for l < p < c o , - c o < p < c o , a n d s = 0 , 1 , 2 , . . . itholds
*) A more extended version of this remark ran be found in H. TRIEBEL [29]. In particular, 1 (15) is also valid for the singular cases {s) = -.
P
302
3.9. The Spaces wi,,(R,)
Using (3.2.6/6)and the considerations after (3.2.6/6),then one obtains for 1 < p < co, 1 - 03 < p < 00, 0 < s = [s] {s}, [s] is an integer, 0 < {s} < 1, and {s} - that
+
+
P
(14)and (15) are simpler than (5) and (6). But for the later considerations, 1 the formulas (5) and (6) are more convenient. Moreover, the singular cases {s} = are included there. 1, f
E w;JR,J.
3.9.2.
Interpolation Theory
With the aid of Theorem 3.9.1, one may carry over the previous considerations for the spaces Wi(S2;@; e') (resp. H;(SZ; e'; e') and B;,,(SZ;@; e")) t o the spaces w;,,(R,) (resp. hi,p(Rn)and b&,(Rrt)).We shall not describe the most general cases here. T h e o r e m . Let 1 < p < co, -co < pl < co, -co < p2 < co, Ojs,<s,,,(R,,)and b;,q,p(Rr,), as sketched in Remark 3.9.1/3, then one may carry over Lemma 3.4.1 and Theorem 3.4.2 t o these spaces. I n this way, one obtains essential generalizations of the above theorem. I n [7] and H. TRIEBEL[29]. connection with the theorem we refer also t o A. FAVINI
3.9.3.
Direct and Inverse Embedding Theorems (Embedding on the Boundary)
It will be shown that it is not very hard t o describe the behaviour of functions of W;,~(R,) near boundaries on the pattern of Subsect,ion 3.6.4. For t,his purpose we introduce the spaces w;+(R;). D e f i n i t i o n . Let 1 < p < co, 0 -I s < co and -co < p < 00. Then wi,y(R;) denotes the restriction of w;,+(R,) to Ri . The completion of CT ( R i ) in wi,,(RA) wzll be denoted by +,JR;).
3.9.3. Direct and Inverse Embedding Theorems (Embedding on the Boundary)
303
R e m a r k 1. The norm in w;,,(RL) is defined similarly t o (3.6.413).Clearly, w“,,p(RL) (and hence also &p,p(RA)) is a Banach space. Using the extension method of Lemma 2.9.1/1, then it follows that for s = 0, 1 , 2 , . . . the restriction operator from w;,+(Rn) onto w;,JRA) is a retraction. Then Theorem 1.2.4 and Theorem 3.9.2 yield that the J w:,+(R;) is a retraction for all 0 < s < 00. restriction operator from W ; , ~ ( R ,onto At the same time, one obtains from Theorem 1.2.4 and Theorem 3.9.2 that (3.9.2/1) is also valid after replacing R,, by RA .
As before we write
XI
= (xl, .
. ., x,,-~).
T h e o r e m . Let 1 < I , < co, -co < ,u < co,and
3f = 1
[
af f(x’,O),-(x’,
ax,*
0)) . . .,
a [.-$1-
[.,-$I-
ax,,
i
(x‘, 0)
.
< *1 (a)If - < s < co and { s } + - , then ‘93 is a retraction from w;,JRn), resp. w;,JRL),
fo‘5 0, which needs only weak smoothness assumptions for the boundary of 52. Lemma 1. If a > 0 and h > 0 , then Kh = {x I x = (x',x,,)E a,,; 0 < x,, < h ; 12'1 < ax,,}denotes a cone of the height h. Then for each natural number 1 there exist functions O,(z), el(%), . . .,O,(z) such that any function f(x)E C$(R,) can be repre-senkd in the fm f(x) = j ~ y ) f (+x Y ) d Y
+2
aLf@ a$+ Y ) d y ,
Se,(Y)
j= 1
x E R,.
(1)
Kh
Kh
Here O,(z) E C$(Kh). Further, Oj(z),j = 1, . . .,n, are infinitely differentiable functions in Kh such that for suitable numbers E > 0 and 0 < h, < h, < h
and
Oj(x) = 0 for {zI x E K h ;( a - E ) z,,< 151 ' < ax,} h, < Z , < h} U {Z I X E Kh; Oj@)
=
(i) I Kh;
1 ~ 1 ~ - ~ y ~ for
0 < xn < h,}
{Z z E
(3)
where yj are infinitely differentiable functions on the surface of the unit ball. Proof. The surface of the unit ball is denoted by w,. Let 0 6 y ( v ) 1 be an infinitely differentiable not-identically vanishing function defined on w, and having a compact support in onn K, . On a ray beginning at the origin, y ( v ) is extended constantly, we write y
. (3v(t)
Further, let v ( t )be an infinitely differentiable function
defined in [0, 00) such that = 1 for 0 < t < h, and y ( t ) = 0 for h, < t < 0 < h, < h, < h. For v €0, and k = In it follows by partial integration
00,
m
0
011
where c is a suitable number. Let tv = y. Then dy = lyln-l dv dt holds. Calculating the derivatives in (4), one obtains terms P f ( x y ) , 5 k. By partial integration, the terms with la1 < k yield the first summand on the right-hand side of (1).
+
4.2.3. Second Extension Method
313
If 1011 = k, then a t least one of the numbers aj in a = (a1,. . .,a,) is larger than or equal t o 1. Now one obtains (1)by a suitable (k - &fold partial integration. R e m a r k 1. Let Q be a domain in R, . Let f ( z )be an infinitely differentiable function in SZ + K,, . Then (1) remains valid for x E Q. R e m a r k 2. By approximation, for instance with the aid of Sobolev’s mollification method described in the first step of the proof of Lemma 2.5.1, and by alimit process, it follows that (1) remains true for 1-fold continuously differentiable functions f(x) defined in R, (or in a neighbourhood of SZ + K , in the sense of Remark 1). R e m a r k 3. * Representations of the type (1) play an important role in the theory of the Sobolev-Slobodeckij-Besovspaces. With their aid one can prove, on the basis of the theory of fractional and singular integrals, embedding theorems for different metrics, extension theorems, and estimates of “ mixed” derivatives by “pure” ones (see the following theorem and Theorem 4.2.4). The treatment given in this book is based on other principles. I n this subsection, and in the following one, however, we describe some of the main aspects of this method, and derive some conclusions. The idea t o use representations of the type (1) is due t o s. L. SOBOLEV [4]. Afterwards, there are derived more general representation formulas. Beside of derivations there are used differences and derivation-differences. Further, there are obtained representations which are the basis for investigations on anisotropic spaces (with and without weights). I n this connection, we refer t o the papers by V. P. IL‘IN [3, 4, 5, 61, V. P. IL’IN, V. A. SOLONNIEOV [l], 0. V. BESOV[3, 61, 0. V. BESOV, V. P. IL‘IN[l],R. S. STRICHARTZ [2], and T. MURAMATU[l, 3,4,5]. L e m m a 2. Let ~ ( vbe) a n infinitely differentiable function defined on the surface of the unit ball 0,. Then
($)f ( x - Y ) dy,
f
g(x) = / I Y I - ~ + ~
(5)
E Cz(Rn)r
R,
is a continuously differentiable function i n R, and
(6) 01
j = 1, . . ., n . Here the first summand on the right-hand side is a singular integral in the sense of Theorem 2.2.312. Proof. Step 1. Clearly,
has the properties (2.2.316)and (2.2.3/8).To check (2.2.3/7),we remark that
J’
Wnn(4X1’
k ( y ) dv = 0)
J
(XIXI=
-
k ( y ) d y 1 - * dyl-1 dyl+, * 1)
*
0
dyne
A corresponding formula is true for u, n { x I x1 < O ] and { x I x1 = -1}. But then (2.2.3/7)is a consequence of the special form of k ( y ) .This proves that the first summand on the right-hand side of (6) is a singular integral in the sense of Theorem 2.2.312.
314
4.2. Definitions, Extension Theorems
Step 2. One obtains (6) by partial integration of
R e m a r k 4.For the proof of the lemma it is sufficient, that f(x) is a continuously differentiable function with compact support. Further, one can weaken also these assumptions. So, it is sufficient if f(x) belongs t o a Holder class and if f ( x )tends to zero for 1x1 -, co rapidly enough. See S. G. M~CHLIN[3], 8. D e f i n i t i o n . A bounded domain SZ c R,, is said to be a domain of cone-type if there exist domuins U,, . . ., U*, and cones C , , . . ., CAWwhich may be carried over by rotations into the cone Kh from Lemma 1 , such that M
U Uic a f 2 ,
j- 1
(UjnSZ)
+ Cj c SZ,
j = 1 , . . ., M .
(7)
R e m a r k 6. The definition is essentially due t o S. AGMON[3], p. 11. (S. ACIMON, however, considers several types of cone conditions.) Clearly, a large claw of bounded m = 1, 2,. . , domains satisfies the cone-condition, e.g. cubes or bounded CfrL-domains, described in Definition 3.2.112. One may extend the definition (and hence also the results based on it) t o unbounded domains. But we do not go into detail here and [2]. Further, we quote the book by E. M. STEIN[5], p. 189, refer t o R. S. STRICHARTZ where a modified definition is given. R e m a r k 6. Cone-conditions are very useful in the theory of embedding theorems and extension theorems. We shall return t o this point, iater on, in Remark 4.6.2.
.
T h e o r e m . Let f2 c R,, be a bounded domain of cone-type. Then for 0 < s < co, 1 < p < co, and 1 q 5 03, the restriction operator from B;,q(Rn)onto Bi,q(Q)and from H;(R,,) onto H;(SZ), respectively, is a retraction. If N is a natural number, then there exists a corresponding coretraction independent of 1 < p < 00, 1 q 5 co, and 0 < s < N , which is also a coretraction from LJSZ) into LJR,). Proof. Step 1. At first, we construct an extension operator from Wr(Q)into W f ( R n ) Similarly . t o the third step of the proof of Theorem 3.2.2, we determine functions y j ( x ) ,j = 0 , 1, . . ., M (resolution of unity). Here one has t o replace Kj from Theorem 3.2.2 by U j . Choosing the cones Ci from (7)sufficiently small, then the following considerations can be made without contradiction. Let f ( x ) be a restriction of a function, belonging t o CF(Rn),t o Then one sets
s
s
a.
k = 1, 2, . . ., n. Further, let in the sense of Lemma 1
4.2.4. Equivalent Norms in WT(l2)
315
The functions Bi!)(y) have the properties listed in Lemma 1. By Remark 1,
( K j f )(x)= y j ( x )f(x) if x E Q, j = 0, 1, . . . , M . (10) The kernels BiJ)(y)and their derivatives up to the order N - 1 are integrable functions in R,. Theorem 1.18.911 yields Now, we have t o estimate the derivatives P K i f of the order
lyl-n+lv(
6)
in ( 5 ) by
- y) where
D ” ~ j ~ (
IyI = N
- 1 and k
=
=
N . Replacing
1, . . ., n, then (6)
(after a corresponding modification) and the statement of Lemma 2 are also valid. Approximating gi’)(x) in Lp(R,) by functions belonging t o CF(R,), applying the modified Lemma 2 , and using a limit process, then it follows from Theorem 2.2.311 and Theorem 2.2.312 The definition of the spaces W:(Q) yields that the restrictions of functions, belonging to Ce(R,),t o Q are dense in WF(Q).Then it follows by (10) and (12), that $5, M
Gf = j =CO K j f ,
(13)
(after extension) is a coretraction from W:(Q) into Wf(R,) corresponding t o the restriction operator. Step 2. We want t o show that k5 from (13) is also an extension operator from LJQ) into LP(R,J.Similarly t o the proof of Lemma 2 and the above described modification, one can remove in (1) the partial derivations, in analogy t o (6). By Lemma 2, one obtains il sum of singular integrals and a term c f(x).Now applying the same method as in the first step, then it follows that k5 (after this transformation) is a n extension operator from Lp(Q)into Lp(R,). Step 3. Now, the theorem is a consequence of the last two steps, Theorem 1.2.4, (2.4.2/11), and (2.4.2114). R e m a r k 7. * The idea t o use singular integrals for the proof of extension theorems for Sobolev spaces is due t o A. P. CALDER~N[l,21. An important modification of the extension method is treated in E. M. STEIN[5] p. 181. The method of E. M. STEIN has two advantages in comparison with the method described here: (a) It is also applicable t o the limit cases p = 1 and p = co,not considered here. (b) There exists a coretraction independent of N . Further references are given in Remark 3. R e m a r k 8. One can extend the theorem t o special unbounded domains. Examples are the unbounded domains of cone-type from Definition 3.2.111.
4.2.4.
Equivalent Norms in W :(S)
I n Theorem 3.2.5 and Lemma 3.8.1/1, equivalent norms in W r ( Q ) are described. With the aid of the considerations of the last subsection, however, one can obtain sharper results.
316
4.3. Interpolation Theory
Theorem. Let ~2c R, be a bounded domain of cone-type. Further, let m = 1, 2, . . . and 1 < p < co. Then
are equivalent norms in WF(L2). n , an equivalent norm in Wr(S2). Hence, Proof. (4.2.3/12) yields that ~ ~ f ~ ~ $ m ( is P for the proof of the theorem, it is sufficient t o show that
Ilf II $,;
5 cllf lI$;[*).
(4)
If the domains Uj and the cones Cj have the same meaning as in Definition 4.2.3, then it follows by Lemma 4.2.311, Lemma 4.2.312, and the method of the proof of Theorem 4.2.3
R e m a r k 1. * (3) is similar t o (2.3.317). Inequalities of type (4) are veryimportant for the theory of elliptic differential operators. (4) is due t o K. T. SMITH[l]. One may generalize the problem and ask under which conditions it holds that
Here Pi are polynomials with constant or variable coefficients. Further, one may [l], consider similar problems in Slobodeckij-Besov spaces. We refer to K. T. SMITH 0. V. BESOV[4,5], V. P. IL’IN [4, 6,7], 0. V. BESOV,V. P. IL’IN [l],K. K. GOLOVKIN [a], R. S. STRICHARTZ [Z], G. G. KAZARJAN [l,21, I. V. GEL’MAN, V. G. MAZ’JA [l, 21, and J. BOMAN[l]. See also Remark 1.13.4/2. The results in D. ORNSTEIN[l] show that (1) and (3) for p = 1 (generally) are not equivalent. R e m a r k 2. By Remark 4.2.3/8, it follows that the theorem is also valid for unbounded domains of cone-type in the sense of Definition 3.2.1/1, too. I n particular, it holds for SZ = R,+.
4.3.
Interpolation Theory
As mentioned in the introduction of this chapter, important parts of the theory of Sobolev-Besov spaces over domains may be obtained in an easy way from the former considerations. The extension theorems of the last section are essential in this connection.
4.3.2. The Spaces
4.3.1.
iisq(SZ), %,,(Sa), ii(Q), and $p(sZ)
317
The Spaces B;,JQ) and H,S(Ja)
T h e o r e m 1. Let Q c R,, be a bounded C”-domain. Then, after replacing R,, by 0, the formulas (2.4.1/3),(2.4.1/7),(2.4.1/8),(2.4.2/9),(2.4.2/10),(2.4.2/11),and (2.4.2/14) (inclusively the special cases (2.4.2115) and (2.4.2116)) are also valid under the corresponding hypotheses for the parameters. Further, under the hypotheses of Theorem 2.4.212, it holds that (B;qa(Q)>q , q , ( Q ) ) o , q = Bi,q(Q) Pro of. The proof is an immediate consequence of the marked formulas (inclusively Theorem 2.4.2/2), Theorem 4.2.2, and Theorem 1.2.4. R e m a r k 1. The theorem is similar to Theorem 2.10.1. T h e o r e m 2. Let Q c Rn be a bounded domain of cone-type in the seme of Definition 4.2.3. Then, by restriction to s > 0 for the spaces B;,,(Q) and to s 2 0 for the spaces H i @ ) , all the statements of Theorem 1 are true. Proof. Similarly to Theorem 1, the proof is a consequence of Theorem 4.2.3 and Theorem 1.2.4. R e m a r k 2. Remark 4.2.318 yields that one may extend Theorem 2 to unbounded domains of cone-type in the sense of Definition 3.2.1/1.
R e m a r k 3. * The interpolation theory for Sobolev-Slobodeckijspaces over domains has been developed in J. L. LIONS,E. MAGENES[l ; 111-V] and E. MAGENES[l]. The results obtained there are special cases of the above theorems.
4.3.2.
The Spaces hi,,(Q),i;JQ), &;(Q), and $(B)
To obtain a theorem similar to Theorem 4.3.1/1 for the spaces and i;(Q), we use the same method as in Section 2.10. At first we define the spaces Pp,,(l2) and
H”;(a). 1
Definition. Let Q c R,, be a bounded Cm-domain. Further, let q 5 co,and 1 < p < CO. Then one sets
--OO
< s < -OO,
{ f I f E Bi,q(Rn),SUPP f c 01 E;(Q)= { f I f E H,”(R,,),supp f = a>-
-&,q(Q)
=
7
(1 a )
(1b) R e m a r k 1. The definition is similar to Definition 2.10.3. We shall consider the spaces 8i,q(Q) and r?i(Q) not only as spaces defined over Q, but also as closed subspaces of B;,q(R,,)and Hi(R,,),respectively. Similarly, one may define Pp,q(R,, and H;(Rn It is easy to see that
-a)
a).
H;(Q) = H;(Rn)Ifij(Rn- a). Bi,q(Q) = B;,q(W/8i,q(RnThe spaces &i,q(Q) and &;(Q) have been described in Definition 4.2.1/2. T h e o r e m 1. Let Q c R,, be a bounded C”-domain. 1 (a) If 1 < p c 0 0 , l < q < 00, and -00 < s -, then 1,
318
4.3. Interpolation Theory
4,q(9)
B;,,(SZ) = and H,"(SZ)= A;@). 1 If 1 c p c 03 and -co < s c -, then
P
B;,l(Q) = &,,(a). (b) I f 1 < p < 00, 1 q c 00, and -co c s c and dense i n fii(~). ~t holds that
s
&,JSZ) (c) If 1 < p
0 such that for all f E CF(52)
holds (with the usual modification for q = a). (4)yields, that (8)?w& fop. 1 < < co, 1 1 q < co, s > 0, and s - - =+ integer. Comparison with (7) gives the interesting special case P
s
J d-.*"(x)lf(4l" dx s cllfll;~
R
for
s
> 0,
s
Pd
tQ)S
f E C?(Q),
(9)
1
- - =i=0, 1 2. . . ., 1 < p < co. This essentially coincides with P
Lemma 3.2.6/1(b).See P. GRISVARD[2], Lemma 4.1. With the aid of (8),one may give also a partical answer t o the first problem. It holds 1 I?;,,(Q) for s = k + -, k = 0 , 1 , 2 , . . .,
4 , g +( ~ ) 1 0 , 1 < p < co, and 1 q 5 co,one sets
4.4.1.Sobolev-BesovSpaces in Domains of Cone-Type
H ; , ( B , ) ( ~=) {f I f
E H;(Q),
Bjfldn = 0
for
mj
7
mk, 1 c c CO, 1 5 p 5 CO, and 0 c 0 < 1. 1 (a) If there does not exist a number mi, j = 1, . . . , k, such that me - - = mi, then
P
(b) Let ml
= me
1 - -.
P
Extending the coefficients b,,,(x) and their first derivatives
continuously to 9,then ( h p ( ~ )H , F{B,l(Q))e,p [LP(9),
HF{B,)(S2)]o
=
=
(f I f
(f I f
1
E B;~,(B,J(Q), ~
B~(Q)]
l Ef
9
‘F(9)]
(8)
1
HiTB,)(9)7
B,f
*
(9)
R e m a r k 3. (6) and (8) have been proved by P. GRISVARD[5,6,7], while (7)and (9) are due to R. SEELEY [2,3].We refer also t o E. HUGHES [l]. R e m a r k 4. (8) may be written in the form (hp(Q),
H E ( B , ) ( Q ) ) e , p = {f
I f E B : ~ , ~ ~ , J, (dQ -W ) , I B J ( X ) I P ~ ~ < a>(10) n
The equivalence of (8) and (10) is a consequence of (4.3.2/7). This formulation coincides with that one by P. GRISVARD.
4.4.
Equivalent Norms in Sobolev-Besov Spaces
On the basis of the interpolation theory developed in the last section and of the corresponding considerations in the third chapter, one may obtain numerous equivalent norms in the spaces W i ( 9 )and B;,,(Q),s > 0.
4.4.1.
Sobolev-Besov Spaces in Domains of Cone-Type
T h e o r e m . Let 9 c R, be an unbounded domain of cone-type in the sense of Definition 3.2.111. 2 1 Triebel, Interpolatiou
322
4.4. Equivalent Norms in Sobolev-Besov Spaces
(a) If 1 < p < 03 and m = 1, 2 , . . ., then (4.2.4/1), (4.2.4/2), and (4.2.4/3) are equivalent norms in W r ( Q ) . (b) Let 0 < s < 03, 1 < p < co,and 1 5 q co. Suppose that M a ,where 0 < 6 5 03, has the same meaning as in (3.3.1/3).Further, if k and 1 are integers such that
O s k < s and then one sets for h E R,
n 1
Q ~=~ , (X ~ Ix j=O
Then
(11-
(1)
l>s-k,
+ jh E Q } .
are equivalent norms in B;,#2) for all admissible numbers 6 , k , and 1. (For q = 00 one has to replace l q m dh ) 1 by sup I I.) Further, one may replace in (4) and ( 6 )
c
-
5k
101
Proof. Step 1.Part (a) is a consequence of Theorem 4.2.4 and Remark 4.2.4/2. Step 2. Part (a), Definition 3.2.1/4, Theorem 3.2.2(b), the extension method of Theorem 4.2.3, and Remark 4.2.3/8 show that W F ( 9 ) = W r ( Q ; a ) with a(z) e 1. But then Theorem 3.3.1, Theorem 4.3.1/1, and Remark 4.3.1/2 yield that l l f l l ~ ~ , g ~ n ) for r = 1 , 2 are equivalent norms in B;,q(9).It follows from Remark 3.3.1/2 that 6 = co is an admissible value. Since (4) for C can be estimated from above and la1 = k
from below by these two norms, it is also an equivalent norm. Definition 4.2.1/1
Hence,
Ifl:
tn,, where r = 3 , 4 , are also equivalent norms.
PI
R e m a r k 1. * Since one needs for the construction of the norms only points of 9, the theorem gives an “inner” description of the spaces W T ( 9 ) and B;,,(9). Of special importance are the norms (5) and ( 6 ) ,since they are more natural for spaces without weights than the norms (3) and (4). Historical remarks and references are given in the second chapter; in particular, see Remark 2.3.112. Further, we refer t o Remark 4.2.3/3 and Remark 4.4.2/1.
4.4.2. Sobolev-Besov Spaces in Bounded Domains
323
R e m a r k 2. The theorem is similar to Theorem 2.5.1.As in Remark 2.5.1/3,one shall try t o choose the number 1 in (1)as small as possible. The best result which may be obtained is k = [s]- and 1 = 1 + [{s}+]. I n particular, it follows for W,S(SZ) = B;,JQ) where s is not an integer,
c
Here one can replace
lal-Csl
by
in (8) similarly t o (5).
c
la15 Csl
. Further, one can modify the second summand
R e m a r k 3. * A decomposition of a domain SZ in two sub-domains Q, and 0, yields, for the Sobolev spaces W;(SZ) where s is an integer,
llfll p
p )
s c(llfll
Wp,)
+ llfll ,“p,)).
The question arises, whether the Slobodeckij spaces Wj(f2) where s is not an integer have the same property. A n affirmative answer can be found in V. I. BURENKOV [3,4].See also M.6. BIRMAN, M. Z. SOLOMJAK [6].The corresponding considerations 1 for the spaces Hi(R,,)where s - -is not an integer, with respect to the sub-domains P R,+and R ; , are given in E. SHAMIR [l].
Sobolev-Besov Spaces in Bounded Domains
4.4.2.
T h e o r e m 1. Let SZ c R,,be a bounded C”-domain. (a) The statements of Theorem 4.4.l(a)are true. (b)Let 0 < s < 00, 1 < p < 00, and 1 q 2 00. Further, suppose that Qh,s,t has and that k and 1 are integers such that (4.4.111)holds. the same meaning as in (3.3.3/1), Then
are equivalent norms in BE,@) for all numbers k and 1 and all sufficiently small positive numbers 6, 8, and t . (For q = Further,
~
~
OM W Y re$)&
f
~ and ~ ~Ilfll$ ~
c
l4Sk
in
00,
*
by $; ; I
1%)
lhl56
, ( R~)
P.9
from, D )
Theorem 4.4.1 are equivalent norms in B;&2).
Ilfllk‘lP.l ( n ) a d Ilfl;;;,Qtu)bY
c
lal=k
(c) R e e i n g IlfllB. (R) in (4.3.2/6), resp. (4.3.2/7), by PeQ then one obtains equivalent norms in
ql,q(0).
21*
( j I P$)+
one has to replace
*
llfll~~,Q(Q), where
j = 1, 2, 3, 4,
324
4.5. The Holder Spaces c'(f2)
P r o o f . Using Theorem 3.3.3 and Theorem 3.2.2(c), then one obtains (a) and (b) in the same manner as in the proof of Theorem 4.4.1. Now, part (c) is a consequence of (4.3.2/6). R e m a r k 1. * Ilfl ;: tn) and l l f l \ ~ ~ , c ( n ) are of special importance. The norm \lfll(4! PIC B P , P coincides essentially with the definition given in T. MURAMATU[3, 41. Here we defined the spaces Bi,,(Q) and H;(Q) as the restriction of the corresponding spaces over R,,to 52. To carry over the results for the spaces over R,,to the spaces over SZ, one needs some smoothness assumptions for the domains 52 (bounded or unbounded domains of cone-type). If these smoothness assumptions are not satisfied, then it is meaningful to define the spaces Bi,,(52) in a direct way, e.g. with the aid of IlfllB,(4) Pd
similarly for WT(52). We do not go into detail here and refer to V. P. IL'IN[3], T. MURAMATU[3,4], and the papers quoted in Remark 4.2.3/3, which a t least partly contain considerations in this direction, too. Further, we refer to Remark 4.6.2. T h e o r e m 2. Let 52 c R, be a bounded domain of cone-type in the seme of Definition 4.2.3. (a) All the statements of Theorem 4.4.l(a) are true. (b)Let 0 < s < 00, 1 < p < co, and 1 S q co. Then ll/ll;~,c(n) and lfl:! P.C tQ, from Theorem 4.4.1 are equivalent norms in B;,,(Q).Here k and 1 are integers such that (4.4.1/1)holds. Proof. Part (a) is a consequence of Theorem 4.2.4. It follows from Theorem 4.3.1/2 and the proof of Theorem 3.3.3 that there exist equivalent norms ~ ~ f ~ ~ ~ ~ , c ( Q ~ , = Bi,,(SZ;a), where a(z) = 1, such that (4.4.1/7) holds. (The (r) norms ~ ~ f ~ ~ B i , c (however, n), where r = 1 , 2 , cannot be described in the previous manner.) Whence it follows (b).
r = 1 , 2 , of B;,$2)
R e m a r k 2. The considerations of Remark 4.4.1/2 are also valid. Formula (4.4.1/8)is of special importance. R e m a r k 3. Clearly,
where 0 < u < s and 1
4.5.
St5
00
are equivalent norms in B;JSZ).
The Holder Spaces C(Q)
We develop the theory of the Holder spaces defined over domains in analogy to Section 2.7. I n the introduction to Section 2.7, we described the motives for considering the Holder spaces in this book.
4.5.2. 1nterpolat)ionand Equivalent Norms
4.6.1.
325
Definition and Extension Theorem
The notations used here have t,he same meaning as in Subsection 2.7.1. D e f i n i t i o n . Let Q c R, be a bounded C”-domain. If t 2 0 , then C l ( Q ) denotes the restriction of CL(R,)to Q. If t > 0, then W(Q) denotes the restriction of W(R,,)to Q. R e m a r k 1. ct(Q)and W ( Q ) are Banach spaces, since they are factor spaces normed in the usual way by
If t is an integer, then one denotes To avoid confusions, we chose the notation et(Q). the set of all t-fold continuously differentiable functions defined in SZ often by Ct(Q). It holds cf(Q) C1(Q). Theorem. Let Q c R, be a bounded C”-domain. Then the restriction operator from C/(R,) onto C/(Q) for t 2 0 and from W(R,) onto W ( Q )for t > 0 is a retraction. If N i s a given natural number, then there exists a correpnding coretraction independent of OSt 1, then it follows by (4) and (5)
See Remark 2.8.1/3. Formula (8) is also of interest from a historical point of view. See Remark 2.8.1/6. R e m a r k 2. The theorem is of special interest for domains having the extension property (that means that the restriction operator from R,, onto 52 is a retraction). Theorem 4.2.3 and Remark 4.2.3/8 yield that this is the case for unbounded and bounded domains of cone-type. We shall return t o this problem in Remark 4.6.2.
46.2.
Embedding Theorems for Bounded Domains
If Q is a bounded domain, then one may generalize Theorem 4.6.1. Lemma. Let y(z ) E C ~ ( R , )Further, . let - co < s < coy 1 5 r 4 00, and 1 < q p < co. Then A ,
(4) (4= Y ( 4 fW' is a continuous m a w n gfrom B;,JR,,)into B;,JR,,)and from H;(R,,)into H;(R,,). Proof. If N is a natural number, then the lemma for the spaces Hf(Rn)and Ht(R,,) is a consequence of (2.3.3/2)and Holder's inequality. Using (2.6.1/1), then it follows that a corresponding assertion is true for the spaces HiN(R,,)and HiN(R,).Now one obtains the lemma by interpolation on the basis of (2.4.2/11) and (2.4.2/14). T h e o r e m . Let Q c R,,be a bounded domain. (a) Let 00 > q , p > 1, 1 5 r 5 03, and -00 < t 5 s < 00. Suppose that (4.6.1/3) is valid. Then (4.6.1/4) holds. (b) Let co > q , p > 1 and - co < t < s < co, such that (4.6.1/3) is valid. Then (4.6.1/ 5 ) holds.
4.7.1. Direct and Inverse Embedding Theorems ( I = n
- 1)
329
co,and t < s, then (4.6.1/4), resp. (4.6.1/5), Proof. Fixing 1 < p < 00, 1 5 r are valid for all q such that (4.6.113) and co > q > p > 1 hold. Now,choosing a function y ( x )E C r ( R n ) ,identical 1 in a neighbourhood of 9,then it follows by the and (4.6.1/5) are also true for p 2 q > 1. I n the case of above lemma that (4.6.1/4) formula (4.6.114)one can also start with t = s and p = q. R e m a r k . * We return t o Remark 4.6.1/2.The Lebesgue-Besov spaces H i ( 9 ) and Bi,q(Q)are determined as restrictions of the spaces Hi(R,,) and B&(R,,), respectively. For bounded domains of cone-type (and similarly for unbounded domains of conetype), we obtained, by Theorem 4.2.4 (Remark 4.2.412)and Theorem 4.4.212(Theorem 4.4.l),descriptions of the spaces W ; ( S ) and Bi&?) for s > 0 containing only values of the function f ( x ) with x E 9.This was based on the fact that these domains have the extension property. If a domain has not this property, then Definition 4.2.1/1 is not useful. I n this case, it seems to be meaningful t o use (4.2.4/1) and (4.4.1/6) (or modifications) as definitions. The spaces W ; ( 9 ) and B i , q ( 9 )defined in such a way are not necessarily the same as the corresponding spaces in Definition 4.2.1/1. But then the above theorem and Theorem 4.6.1 need not be true for these modified spaces. Denoting by Cm(9) the set of all infinitely differentiable functions defined in 0,then N. MEYERS,J. SERRIN[l]and T.MURAMATU [3]have shown that also by this modified definition Cm(9)n W ; ( 9 ) is dense in W$2) and Cm(9)n Bi,,(Q) is . also V. I. BURENKOV [5]. From such a point of view cone-condidense in B i , q ( S ) See tions for domains 9 are natural assumptions if one wants t o formulate embedding theorems with respect t o 9 in the same manner as in R, . V. I . BURENKOV [2]proved that for domains, where the cone-condition is not satisfied, the embedding theoremp (for the modified spaces) are not valid in their full extent. Embedding theorems in arbitrary domains are considered in V. P. IL'IN[3], T. MURAMATU [3,4,5],and E. M~LLER-PFEIFFER, A. WEBER [l] without use of the extension property. Farreaching investigations on the validity of embedding theorems and on related topics in arbitrary domains can be found in V. G.MAZ'JA [l, 2,3,4].The criteria on the validity of embedding theorems proved by him have necessary and sufficient character. In this connection we refer also to R. A. ADAMS[5],E.A. STORO~ENKO [l], R. ANDERSSON [l],and V.I. BURENKOV [6].
4.7.
Direct and Inverse Embedding Theorems (Embedding on the Boundary)
I n this section, we carry over the results of Subsection 2.9.3and Subsection 2.9.4. The obtained theorems are of great importance for the theory of boundary value problems for differential operators. 4.7.1.
Direct and Inverse Embedding Theorems ( I = n
- 1)
For bounded Cm-domains9 the spaces B;,,(aQ),0 < s < 0 3 , l < p < 03,l 5 q5 co, are determined in Definition 3.6.1.As in Theorem 3.6.1,the outer normal with respect to a 9 is denoted by v. Further, flan is the boundary value of the function 1.
4.7. Direct and Inverse Embedding Theorems (Embedding on the Boundary)
330
Theorem. Let
SZ c R,,be a bounded Cm-domain.
(a) Let 1 < p < co and-
1
P
<s
0 and y 2 0 such that IlA~Il5 c e y l t l . (6) R e m a r k 2 . * The theorem is the basis for the structure theory in the next subsections. The statement that A , + QEis a positive operator in L,(S) is due to S. AGMON [2]. The proof of (6) is deep and goes back to R. SEELEY[l], see also R. SEELEY[2]. Before, D. FUJIWARA [l, 2,3] and N. SHIMAKURA [2] had proved similar results for elliptic differential operators of second order. See also Remark 2.5.312. Part (a) [2] and the well-known a-priori-estimates for follows also from the paper by S. AGMON We shall return elliptic differential operators and complemented systems {B,}yx1. to this point later on, see 5.4. There are also given references. R e m a r k 3. We describe an example which can also be found in S. AGMON[ Z ] . If k = 0 , 1 , . . ., m, then Bjf
ak+j-lf
=avh’+j-l ? c
i
= 1,.
. ., m ,
4.9.2. Scales
335
is a complemented system with respect t o Af = ( - A ) m f . Further, A and {B;}Fl satisfy the assumptions of the theorem. See also Remark 5.2.114.
4.9.2.
Scales
D e f i n i t i o n . (a) A set of Banach spaces {Bl}-, t, 2 t, > - 03, is said to be a two-side scale, if for any number N > 0 there exists a set of linear operators {AiN)}Osrs2,V such that AiN) i s a n isomorphic mapping from B, onto Bt-,, - N 5 t 5 N , -Ar 5 t - t 5 N . Further it is assumed that
AhK) = E and A ( N ) A 7, ( N =) A::;)=, (t,+ t, 5 2 N ) . (1) (Here A$:) maps BL onto Bl-T2 and afterwards A i r ) maps Bl-12onto Bl-rl-T2, It1 5 iV, t - tl - t 2 2 - N . ) (b) A set of Banach spaces {B,},,,, ,, resp. {B,}o,t, m , where B,, c B,, for co > t, 2 t ( 2 , O is said to be a n one-side scale, if for any number N > 0 there exists a set of linear operators {A$-")}OsI~~)AT such that ALN)is a n isomorphic mapping from Bi onto B1-?,0 ( 5 )t 6 N , 0 ( 5 )t - t 5 N . Further, it is assumed that (1) holds for the corresponding values of the parameters. R e m a r k 1. If {B,}-,,,,q(Q) for -a < s c O}
are two-side scales. P r o o f . Step 1. Let A and {Bi}cl be the differential operator and the boundary value operators described in Remark 4.9.113. Applying Theorem 4.9.l(b)to A + eE, where e is sufficiently large, then it follows that the assumptions of Theorem 1.15.3 are satisfied. Hence, it holds for 0 < 8 < 1 that
336
4.9. Structure Theory
Choosing k
=m
+ eE)') = CLJQ), ~ + % b , l ( ~ ) I e . in Remark 4.9.1/3, then one obtains by Theorem 4.3.3 that
~ ( ( A P
D((A,
+ @)')
=
H;"'(Q)
where 2m8 < m
1 +.
(2)
P
Now, Theorem 1.15.2 yields that { H ~ ( Q ) }-O s , s O,then
P
n
a(&; I ; Wi(SZ),Lp(52))5 c&-T.
(b) If 0 < a < 1, then there exists a positive number c, such that R e m a r k 1. Part (a) is due to M. 8. BIRMAN, M.Z. SOLOMJAK [2]. It ia essential n n that (1) holds also for r, > q, provided that t - - > --. M. 8. BIRMAN, M. Z. SoU
V
[2] proved (1) with the aid of the method of piecewise approximation by polynomials. (See the proof of Theorem 3.8.1, and the two previous subsections.) Now the above used linear approximation is not sufficient, there are needed new considerations. Formula (2) for a cube-shaped domain Q instead of 52 was proved by G. F. CLEMENTS[2]. Here, Fb(Q) has the same meaning as in Definition 4.5.2. With the aid of the previous extension method, it is easy to see that (2) is also valid for bounded Cw-domains. T h e o r e m . Let 52 c Rn be a bounded C'"-domain. Further let 1 < p , q < co, n n -co < s < t < co,t - - > s - -, and 1 6 b, e 6 00. Then LOMJAK
9
P
H(E;1,q p ( Q ) ,B;,&Q))
-
-- n
t-s * (3) Here, one can replace Bi,$2) by H;(SZ) and/or BiJSZ) by H#2). P r o o f . Step 1. For fixed p and q, we apply Theorem 1.16.2/1(a) to the interpolation couple { W;(SZ), Wk+'(SZ)) where 6 is a sufficiently small positive number. Then (1)yields
n
H ( c ; I , B;,,(SZ), 5 CE-T, (4) n n provided that t - - > --, t > 0, and 1 5 u 5 00. It follows from Theorem 4.9.2 q P and Remark 4.9.2/3 that (4) remains valid if one replaces BiJSZ) by and LJQ) by H i @ ) , x 2 0. Applying Theorem 1.16.2/2(a), one obtains that one can 23*
0
356
4.10. Qualitative Properties of Embedding Operators
s s
replace H i @ ) by B;,,(Q), 1 c, 00. Theorem 1.16.1/2 yields that a similar estimate holds for BzJQ) instead of Bi:z(Q) and BF,,(O) instead of B;,,(SZ). (See also the second step of the proof of Theorem 4.10.2.) Applying again Theorem 4.9.2, one obtains that n
H(E; I , Bi,,(Q), BZ,,(Q)) 5
CE-t-S.
Step 2. Formula (2) yields n
H ( & ;I, w;(Q), L,(Q))2 c s - a where O < , x < l ,
n
l < p , q < c o , and a - - >
4
n
--. P
Let l < p s q < c o .
Applying Theorem 1.16.2/1(b) t o the interpolation couple {Lq(Q),W t ( Q ) } , where x > a,and using (4) and ( 5 ) ,then one obtains that
s
- _-
H(E; I,B ; , m , Lp(Q))
n
E
for 1 5 0 co. Using again Theorem 4.9.2 and Theorem 1.16.2/2(b),then it follows in the same manner as above that
- __
H(E; I , q , u ( Q )B;,,(Q)) , E t-S (6) where 00 > t > s > 0, 1 5 0,c, 5 co. Theorem 4.3.212 and the above method show that a corresponding formula holds, with B;,,(Q) instead of B;,,(Q) and pp,,(Q) n instead of BSJQ). Afterwards, one can also admit p > q, provided that t ; 7
n
'1
> s - - . Theorem 4.9.2 and Remark 4.9.213 yield that (6) is also true for 00 > t P n n > s > - c o , t - - > s - - , 1 5 b,@ s 00. P Step 3. Formula (3) for and/or H i @ ) is a consequence of (4.6.112). R e m a r k 2. Theorem 1.16.1/2 shows that it is sufficient t o know (1) for 0 < t < 6
where 6 is a sufficiently small positive number. A stronger use of interpolation theorems (formula (2.4.2/11)with Q instead of Rn)shows that for many (but not all) pairs of parameters p , q it is sufficient to know (1) for t = 1. R e m a r k 3. * As a special case of (3), one obtains that
H ( & ;I, w;(Q),W$2)) 0
5 s < t < co,1 < p,q
4
s
--.n P
For1 < p
q < co this formula
is due to A. MOSTEFAI[l]. The case s = 0 and p = q can also be found in G. G . LORENTZ [3]. We just mentioned that (1) goes back t o M. g. BIRMAN, M. Z. SOLOMJAK
[2]. Further, M. 6. BIRMAN, M. Z. SOLOMJAK [l, 21 proved t h a t
n
for 1 < q < co and t - - > 0. Now, using the above method, then it follows that
4
357
4.11.1. “Periodic” Spaces
H ( & ;I , B;,@), for 1 < q < co, 1
0y.n))
--n N
(9)
&
n
s s co, and t - > 0. (8) and (9) are modifications of the 4 (T
well-known result due t,o A. N. KOLMOGOROV, V. M. TICHOMIROV [l],
El(&; I , C‘(.n), EO(Q))
- _- , n
&
0 < t < co.
(10)
See also G. G. LORENTZ [2]. Formula (3) is a generalization of corresponding results in H. TRIEBEL[15,27]. Estimates of the &-entropyof embedding operators in aniso[2,3]. BORZOV uses tropic Sobolev-Slobodeckij spaces can be found in V. V. BORZOV also the method of piecewise approximation by polynomials of M. 8. BIRMAN, M. Z. SOLOMJAE. On the basis of Theorem 2.13.2 the results by V. V. BORZOV are [27]. Investigations on the &-entropyin functions spaces generalized in H. TRIEBEL and on relations between &-entropyand width numbers can be found in G. G. LORENTZ [2,3], J. W. JEROME, L. L. SCHUMAKER [l], and G. F. CLEMENTS [l]. The [2,3] contain many references. papers written by G. G. LORENTZ
Complements
4.11.
I n this section, we describe very briefly some aspects of the theory of function spaces which did not play any role in the previous considerations. Essentially, we restrict ourselves t o references. On this connection, we refer t o the survey paper by 0. V. BESOV, V. P. IL’IN,L. D. KUDRJAVCEV, P. I. LIZORKIN, S. M. NIKOL’SRIJ[l]. There can be found a short description of some aspects of the theory of functions spaces as well as many references (the bibliography contains items up t o 1968). See also Section 3.10.
4.11.1.
“
Periodic” Spaces
Let
T
=
(ZI z = (z~, . . ., x,))E R,,, 0 5 ~j 5 2n, j = 1 , . . .,
.>
be the n-dimensional torus (opposite points on the “boundaries” are identified). C m ( T )= D ( T ) is the set of all complex-valued (periodically) infinitely differentiable functions defined on T . Further, D’(T) denotes the set of all linear and continuous functionals over C m ( T ) (periodic distributions). For abbreviation we write
f Any f
=
E D‘(T)
(kl,. . . , k,J, kj are integers,
n
It1
=
C lkjl, j=1
can be represented in D’(T) as a Fourier series
and f .x =
n
C j=1
kjxj.
358
4.11. Complements
Now, similarly t o (2.3.3/1), one may introduce “periodic” Lebesgue spaces Hi,,, ,
- co < s < 00,l < p
0 such that for all x E Q
AU
=-
In a similar way, one may define uniformly elliptic i n 0.For our purpose, the difference between elliptic and uniformly elliptic is not important : If L?is a bounded C”-
5.2.1. Definitions
domain, then (2) coincides with (6) where 1 = 2m and x €0. If 9 = R: shall be concerned only with operators having constant coefficients.
363 then we
D e f i n i t i o n 2. Let D be either R: or a bounded C"-domain in R,,. Let B i ,
(Bju)( x ) =
X
la1d lnj
bj,a(x)D * ~ ( x ) , bi,a(x) E crn(aQ)7 *)
(7 1
j = 1, . . ., E , be differential operators defined on aQ, where bj,&are constants in the case Q = RA. Then {Bj}:=l i s said to be a normal system if
0 5 m, < m, < . . . < mk and if for each normal vector vz on aD, where x
C
la1 = I I l j
bj,,(x)
V:
f 0, j =
(8) E
aD, it holds
1 , . . ., k.
(9)
Reinark 2. Essentially, this definition coincides with Definition 4.3.311. D e f i n i t i o n 3. Let D be either R: or a bounded C"-domain in R n , and let A be a proprly elliptic differential operators in the sense of Definition 1. Furthermore, let {Bj}J.'ilbe differential operators on the boundary aD in the sewe of (7), where bj,a are consta& in the m e 52 = R:. Then {Bi}pli s said to be a complemented system (with respect to -4) if for each x E aS2, the corresponding nornml vector v, 0 , and each tangential vector t, f 0 in the p i n t x with respect to aS2, the polynomiak in the variable t
+
bj(z; t x
+ tvz) =
C
la1 =in,
+
bj,a(x)(tr tvz)",
ex,
( 10)
are linearly independent modulo a+(x; v . ~t), , where the last polynomial has the meaning of Definition 1. R e m a r k 3. This definition coincides essentially with the second part of Definition 4.9.l(b).The polynomials b j ( x ;tx+ tv,), j = 1, . . ., m, are said to be linearly inde~ fixed, if the identity pendent iiiodulo a+(x;Ex, v,, z), where x E a 0 , t,,and v , are int Ill C cjbj(x;t.c Z V . ~ )= f ' ( r ) a + ( x ;& 7 v.z, t), (11) .j = 1
+
where c l , . . . , c,, are complex numbers, and P ( t ) is a polynomial in t,holds only in the t(ririvia1case c1 = c2 = . . . = c,,, = 0. The question arises what happens when a+(x;&, v,, t)in the complementing condition is replaced by a-(x; Ex, v , ~t). , Since a(x, t + tq) = a(x, -6 - tq),it follows that
a-(2; -t,q, -t) = ( -l)ma+(x;E , q ,t). (12) From t8hisrelation, one obtains easily that a+(x;tx, v , ~t) , in the above definibion may be replaced by a-(x; tz,vx , t).(If &. is a tangential vector, then also - 5, .) D e f i n i t i o n 4. Let D be either Ri or a bounded C"-domain in RI,. The differential operator (1) and the boundary operators ( 7 ) are said to be regular elliptic, provided that ( a ) A is properly elliptic (Definition l), (b) it h.olds k = m in Definition 2 and {Bi}Jilis a normal system where mj 5 2m - 1, j = 1, . . ., In, (c) the complementing condition from Definition 3 is satisfied. *) Cm(dR)is the set of all infinitely differentiable complex functions on aR.
(13)
364
5.2. Regular Elliptic Differential Operators
R e m a r k 4. We describe an interesting and important example. Let px E Rn,
x E 8 9 , be vectors on 8 0 , where the components of pL5are C”-functions on dSZ. It is assumed that px is not a tangential vector in x E aQ. Let A be a properly elliptic
differential operator, Definition 1, and let k be a fixed number, k = 0, 1, . . ., m.
is a normal system, satisfying the complementing condition, Definition 3. (As before, b,,a is constant in the case Q = R:.) Hence, A , B , , . . ., B,, is regular elliptic. For k = 0, this is a n immediate consequence of (ll),because the polynomial on the left-hand side of (11) has only the degree m - 1. Let k 2 1. For fixed x E dQ, rach C Z ~ ) Here ~. c, and d , summand on the left-hand side of (11) contains the factor (CJ are real numbers, cI, 0. But a+ has no real roots, and so the factor ( c , t + dJL must be a factor of P ( t ) . Dividing both sides with this factor, one has the same situation as in the case k = 0. Boundary value systems {B,}Zl satisfying the complementing condition for all properly elliptic differential operators are considered in [l]. See also S. AGMON[2]. Of special interest are the operators L. HORMANDER
+
+
ak+J-lu
B,u = ayk+j-l ’ where vz is the normal vector. For k = 0, one obtains Dirichlet’s X
boundary value problem. See also Remark 4.9.1/3. R e m a r k 5.* Extensive references can be found in J. L. LIONS,E. MAGENES[ Z , I] and Ju. M. BEREZANSKIJ [l]. So we restrict ourselves to few quotations. The main notations (root-condition, complementing condition) are due t o JA.B. LOPATINSKIJ [l], Z. JA.SAPIRO[l I, M. SCHECHTER [ Z ] , and S. AGMON,A. DOUGLIS,L. NIRENBERG [1, I].The definition of normal systems was given in i”.A ~ o r j s z a ~ h A. - , N. IMILGRAM [I].
5.2.2.
Elliptic Operators
Throughout this subsection, A ,
AU = C aaD”u, lnl=2nr
denotes a properly elliptic differential operator having only constant top-order coefficients. a(6) = a(x, t),a + ( t ,q, t) = a+(x;t , q , t),a - ( t q t)= a-(x- 5 q t) and the roohs ti and t; have the same meaning as in Definition 5.2.1/1. We shall prove here some statements of preliminary character. Now, let .$ = (El,. . . , .$+,, 0) = (E’, 0), 6’ E R,L--I and 9 = (0, . . ., 0, 1). The corresponding roots of a([ tq) = 0 are denoted by ti(t’)and 9
,
ti(r).
L e m m a 1. (a) There exist two positive numbers c1 and c2 such that
9
9
+
7
,
365
5.2.2. Elliptic Operators
where a,'(E') are analytic functions for 5' E Rn-l, 5' (c) It hoZds
ak+(LE')= Lka;(F),
a-(-E',
-t) =
+ 0. Further
L > 0.
(3)
(-l)ma+([',t).
(4)
+
P r o o f . If 16'1 = 1, then (1) holds. Furthermore, a(5 zq) = 0 is satisfied if and O.rea1. From this fact, it follows (1) (general case), (3), only if a(LE + Ltq) = 0, L and (4) (here one must use L = - 1). Finally, the analytic dependence of a: (6')on 0 is a classical fact proved in complex function theory. [We give here a short proof of the last assertion. First we remark the well-known fact that the roots t,'(f') depend continuously on t',where F 0. So zi(6') (resp. t k ( 5 ' ) ) will be in theupper (resp. lower) half-plane if 6' has complex components with small imaginary parts. If tl,. . ., are fixed, then a ( l ' ,t)= 0 (as a polynomial of [n-l and t)determines an algebraic function. By complex function theory, the elementar-symmetric functions of the roots ti(c')(resp. t i ( ? ) )depend analytically ( = holomorphically) on &-l. But these functions coincide with %+(6')(resp. a i ( F ) ) .Hence, a;(F) (and so also ai(6'))depends analytically on each of the variables El, . . . , separately. But such a function is also an analytic function of El, . . ., & - l , varying simultaneously. The last fact follows from complex function theory of one variable if one applies Cauchy's integral representation formula separately to El,. . ., [ , 1 - 1 .] L e m m a 2. Let 1 < p < 03, and let s be a real number. Then
+
+
+
IlF-la+(t', 6,) Ff I I H p , ) is an equivalent norm in
is an equivalent norm in
H%'"(Rn)n {f I f E fJ'(Rn)7( F f )(t',t n ) = 0 for Proof. Using Theorem 2.3.3 we have
IE'I < I}.
IIF-l(l + 1E12)"'2 a+(6',6,) p flILp(Rn)* Let v ( t )be an infinitely differentiable function in [0, a), where
IIF-'a+(E', En) F f IIH;(R,)
v ( t ) = 0 for 0 5 t
2 9,
and y ( t ) = 1 for
15 t
0 , 5' 0 . Now the lest relation and the analytic dependence of cj,l(l') show that cj,l(t')(1
+ l 5 ' l ~ ) ~ ~ n j - ~ ) ~=(cj,l(+) l5'l)
l t ' l a ) + ( l ~ ' ~ -p(lt'1) ~)
lt'l-mj+l(1+
satisfies the condition for a multiplier, Remark 2.2.414. Temporarily, we denote by F' the (n - 1)-dimensional Fourier transform with respect t o the n - 1 coordinates x l , . ., zn-, of u ( x ) = u ( x l , . ., x~,-.~, xn). Hence, F'(u(x)) = (F'u) ( 5 ' 9 X n ) , 5' E Rn-1.
.
.
.
Lemma. 2. Let u ( x )be the ratriction t o q o f a function belonging to S(Rn).Let
Then
(Au)( 2 ) =
C
a,Dau(s) = 0 for xlz 2 0 .
la1= 2m
368
5.3. A-Priori-Estimates
Here a‘ = (a1,. . ., an-1). For fixed 6‘E Rn-l, this is a n ordinary differential equation in [0, 00) with constant coefficients, solved in the usual way by a linear combination of exponential functions eirx- (maybe multiplied with powers of x,, in the case of degenerate roots of the characteristic equation). Putting e i 7 ~ nin (7) it follows that z must be one of the roots z,f(p) described in Lemma 5.2.2/1. But (Flu) (t‘,x,,) E X(Rn) and so (B’u) ( p ,x,,) is bounded on [0, 03) for fixed 6‘.This shows that (Flu)(p,x,,) is a linear combination of ei7ixn(maybe multiplied with powers of x,,) only. Using that (7) is equivalent t o
i
a+ ( F , T 2.
then it follows that
(Here, a* (l‘,
-)axna a- (r,7i -)3x1,a
(Flu)(p,x,,)
=
0,
&)
are the polynomials of Lemma 5.2.2/1, where z is replaced by
I
h
wit But the right-hand side coincides essentially (beside the factor imr(2n)(n-’)/2 (F’Bp)(l’,x,,). If x,, = 0, then one obtains the lemma. R e m a r k . The proof shows that one can weaken the smoothness assumptions for u(x). It is sufficient to suppose that u(x) is the restriction t o z of a function w(x), provided that M is a sufficiently large natural number.
5.3.
A-Priori-Estimates
The aim of this section is the proof of the a-priori-estimate for general regular elliptic boundary value problems in the framework of the &theory. Such estimates can be obtained on the basis of singular integrals or on the basis of multiplier theorems in L,-spaces. The first version can be found in the fundamental paper by S. AGMON, A.DOUGLIS, L.NIRENBERG[1,I]. We shall be concerned here with the second [l]. method. In this section, we follow the paper by L. AREERYD 5.3.1.
The Spaces H Y ( R T )
We start with auxiliary considerations on special anisotropic spaces of Lebesgue-type. D e f i n i t i o n . Let 1 < p < a,-a < s < 00, and - m < r < 03. Then H;‘(Bn) = {f I f
E X’(Rn)7 IIP-Vl
+ It12P2(1 + 1E‘12)r’2.Zi’fllL,
0 and v ( x ) = 0 for x,, < 0, then (F-lv(It'1)Fv) ( 2 ) = 0 for xn < 0. It follows (again by using a multiplier theorem) that
Theorem 2.9.1. The same theorem shows that there exists a function v E HF+*(RA) such that ajv a h -(X',O) = -(x',O), j = 0,. . , m - 1, (21) axd ax;l
.
5.3.3. A-Priori-Estimates[Part 111
373
This proves (11).
5.3.3.
,constant coefficients,
A-Priori-Estimates [Part 11. R: general boundary problem]
.
Theorem. Let A , B,, . ,, B , be a regular elliptic problem in RA, Definition 5.2.114, where Au = C aaDau, Bju = C bj,,Pu, j = 1 , . . . , m , (1) I 4= m j
la1 = 2 m
are differential operators having only constant top-order coefficients. Let 1 < p < cx) and s = 0, 1, 2, . . . Then there exists a positive number c such that for all U E Wirn+'(RA)
llull
WEm+'(R:) n1
S c (llAull~i(~:)+ C ll(Bju) j=l
(2'3
O ) I I BPIP ~ ~ + ' - ~ J - (Rah1) -
+ IlulIWF+'-'(R:)) -
(2) P r o o f . Step 1 . Let w be the restriction to R i of a function satisfying (52.319) with sufficient large M . Let ( A w )(2) = 0 for xn 2 0 and
(F'w) (6')~~) = 0 for
15'1 < 1,
and xn
2 0,
(3)
where F' is the (n - 1)-dimensional Fourier transform, described in front of Lemma 52.312. This lemma and the notations introduced in Lemma 5.2.311 yield in
(F'Bjw)(t',0) C ~ , ~ ( E ' ) ,1 = 0, . . .,m - 1. . a 4 (F'w) (6')0) = c j C = l a1
Hence.
c
i n - 1 ni
-c I
C l \ p - ' C j , z ( E ' ) P'B,u'(5'7 o)lI~:~++.-"-f(~,,-~).
Z=O j = 1
We want to apply Lemma 5.2.311. Since (5.2.315) is a multiplier in .L,,(RnPl), so it is also a multiplier in HXp(R,t-l),x real, and so, by interpolation, also in B;,p(Bn-l). Further, by (3), it holds (F'Bjw) (t',x , ~ )= 0 for 15'1 < 1 and z,,>= 0. Now we use
374
5.3. A-Priori-Estimates 1
Lemma 5.2.311 with Biy;s-l-- P (R,,-,) instead of Lp(Rr1-J,
c
C c IlF'-'(l + 1=0 j = 1
m-1
- c' I
m
l~'")+"'''j)
F'Bjw(E', O)lle"+'-'-p,p
1 P(RI-11
c II(B,w) (x',0)II~2m+s-mj-' nr
9.P
j=l
P (%-I).
(4)
I n the last estimate, we used the lifting property described in Theorem 2.3.4. Step 2. We prove ( 2 ) . It will be sufficient t o consider the restriction to R,+of functions belonging t o S(R,,). (This follows from the fact that the restriction of S(R,) to R: is dense in W F + s ( R i )and , that the right-hand side of ( 2 )can be estimated from above by IlullW;m+'(q.) Let u E S(R,). We use the decomposition (5.3.2/16). Clearly, F and F-' may be replaced by F' and F'-l) respectively. We use the extension operator S, described in the proof of Lemma2.9.1/1, where the number N , appearing there, is chosen sufficiently large. Let V(Z)
= F-'u-l(E) F[SAu,].
(5)
Here a ( [ )is the polynomial belonging to A , Subsection 5.2.2. Using the explicit form of S, formula (2.9.1/4),it follows that W(X) =
F-la-l([) F[SF'-'(l
=
F-'u-l(E) FF'-l(l
=
F-'u-'(~)( 1
= F'-l(l
- ~ ( l e ' l ) )F'Au]
- y(IE'1))F'SAU
- ~(15'1))FSAU
- T(~E'~))F'F-'U-'(E) ~ ( 5 'FSAU. )
(6)
Here y ( t ' ) is an infinitely differentiable function, vanishing near the origin and equals 1 on the support of 1 - ~(16'1). SAu satisfies (5.2.3/9),where M depends on N , (2.9.1/4).Now the same is true for FSAu, perhaps with a n other value of M . This follows by elementary computation, see for instance H . TRIEBEL[17], p. 99, formula (10.6).Consequently,
where M depends on N ; M ( N ) + co for N
+
co. Furthermore, ( 5 )and ( 6 )yield
(Aw) (x)= (Au,) (x) for x,, 2 0 ,
(P'v)(E', x,,) = 0 for
IE'I < 1 .
If w = u1 - v , then (7) and ( 9 )hold, where v is replaced by w and ( A w )(z) = 0 for x,,2 0.
(8) (9)
5.3.4. A - P r i o r i - ~ s t ~ m [Part a ~ 1111
375
Consequently, (4) holds for such a function 20. This @eIds
The above ~ ~ i d e r a tshow i o ~that one may replace u1 by w on the ~ ~ h t - h a nside d of the lsst estimate: liffuiilW:(R:)
$ llF'-'(L
- y(IE'I))F'flA%llW;(R,)
and a c ~ ~ ~ e ps t o~ for an ~~~ ~ u ~ l ~ ~ thia follows from the multiplier property of (1 r n e n ~ aot $he ~ ~end of the first step. Hence,
C I ~ ~ U I It ~ ( R ~ ~
By the last expressions in (12)
~ ~ ~ ~ - i ( f i ~ ) .
- cp(lt'1))in the spaces q,p(Rn-l),
For u,, we may uae (5.3.2/18),where H T m ( R i )is replaced by Wfi+am(Ri). Since u = pbo u,, we get (2).
+
8.3.4.
A-Priori-E8timates [Part 111. Bounded domain, variable ooetrieients, general bound^ ~ r o b l e m ~
heo or em. Let D be a ~ # ~ ~ A , B,,n . . .,.B , be regflla~e ~ D e f i n i t h 5.2.114. k t 1 < p < 00 and 8 = 0, 1, 2, , . . The% there e&t two po&tive ~~
~
376
5.3. A-Priori-Estimates
numbers c1 and c2 such that for all u E W;m+s(Q) C~IIUII lvzm+*(n)
s IIAUIIwi(Q)+ s CzII4Iw~+'(Q)-
rn
+ c IIBjullsam+a-mj-'
IIUIIL,(~)
PBP
j=l
P
can)
(1) Proof. Step 1 . The last part of (1) is a consequence of Theorem 4.7.1. Xtep 2. To prove the first part of (l),we shall construct a resolution of unity.
Let Kj be open balls, j = 1 , ...,N , with sufficiently small diameters, where Let yj(z) E c,"(Kj), 0
N
ac IJ Kj . j=1
N
5 yj(x) 5 1 for j = 1 , . . ., N , and C yj(x) = 1 for x E Q . (2) j=l
(See the third step of the proof of Theorem 3.2.2). If aS n K j 9 0, then we suppose that the balls Ki are of the type described in Definition 3.2.112. But first we are Then yju E Wpm+s(Rn). Let concerned with the case aQ n Kj = 0. Let u E W~'"+"(S). xj be the centre of K j . Using the ellipticity condition and the multiplier properties of the corresponding polynomial a(sj, l ) ,Definition 5.2.111, it follows
n yju IIw?(n)
= II Yju II W:~+'(R,,)
5 clIS-Wxj, 5 ) SvjuII w;(R,,)
+ CII~Y~UIIW~(R,,)
5 CIIAyjUIIwi(n)+ cIal=2m C II(aa(z) - aa(xj))D"yjuIlwi(n) + c la1C< 2niIlaa(z) D"yjuIlw;(n) + CIIYj~llw~(n)*
If the diameter of Kj is sufficiently small, then the second term on the right-hand side can be estimated by ~ J I y j u l l w ~ ywhere ~ ) , E is a small positive number. The third and the fourth term can be estimated by c' 11 yju 11 ~ ; m - l + r (n). This proves II~jUlIwi~+*(n) i c (IIAyjullw;(n) + I l ~ l l ~ ~ m - 1 + ~ ) ) Since Ayju = yjAu + lower terms, it follows by the same argumentation that
+
IIyjuIIwim+'(n) 5 cllAullw;(n) cllull w;~+'-'(Q)* (3) Step 3. We consider the second case, that means aQ n Kj 0. Then we have, by Definition 3.2.112, a Cm-one-to-one-mapy(x) from SZ n Kj onto a domain o in R, such that the image of aS n Kj is a part of the hyperplane {y I yn = 01. Let again u ( z )E WPm+"(Q) and yi(x) be the function described above. The transformed functions are indicated by ', so u'(y) and y,!(y) are the transformed functions of u(z) and yj(z),respectively. Let (Ayju)' (Y) = A'Y;(Y) u'(Y) = C ah(?/)D'YJY) u'(Y) Y E 6 (4) b l 5 2m
+
9
9
(5) For sake of simplicity we assume 0 E 3 0 and that 0 is the centre of Kj (here j is fixed for the further considerations). Suppose that aL2 near the origin may be represented
5.3.4. A-Priori-Estimates[Part 1111
af f ( 0 ) = 0, - ( O ) x,, = f(xl, . . ., x,&-~), axk
=0
for k
= 1,.
377
. ., n - 1.
Then the transformation y(x) may be described by yk = 8 , k = 1, . . ., n - 1, and y,, = x,, - j(xl,. . . , x,,-~).This shows that the Jacobian for the origin coincides with the unit-matrix. It holds ui(0)= uJ0) for ldcl = 2m and &(O) = bl,c(0)for IpI = m l . Hence,
Here, holds
E
and E' are arbitrary positive numbers (depending on the diameter of Kj). It
IIY.~~II~P,",+'-+(~~) s ClllyiullWim+*(n)-
Choosing E and E' sufficiently small and using respectively
Ayp
=
yjAu
+ lower terms,
Blyju = yjBlu + lower terms,
it follows that
+
rn
1
Il~juIlw~m+'(n) S cll Aull w ~ ( Q ) c1=1 C IIB~Ull~2m+'-mj-P can) PIP
378
5.4. L,-Theory in Sobolev Spaces
one obtains the left-hand side of (1). R e m a r k . * The theorem is one of the main results for regular elliptic problems. As mentioned in the introduction, the proofs of the results of this section are based on L. ARRERYD[l].The last theorem, respectively special cases of it, are also proved by S. AQMON,A. DOUQLIS, L. NIRENBERQ [l, I], F. E. BROWDER [2, 41, A. I. KOBELEV [l], S. AQMON[l], and L. NIRENBERQ [l]. The main tools for the proofs are the theory of singular integrals (see 2.2.3), and the (scalar) multiplier theorems from Subsection 2.2.4. Further we refer to L. H~RMANDER [l], M. SCHECHTER [l], J. PEETRE [l,21, L. N. SLOBODECKIJ [2, 3,4], and E. B. FABES,N. M. RMERE [Z]. For p = 2, the proofs become easier. Systematic treatments of the L,-theory can [l]. Further be found in J. L. LIONS,E. MAQENES[2, I] and Ju. M. BEREZANSKIJ we refer to C. G. SIMADER [l],who developes the L,-theory for the Dirichlet problem.
6.4.
L,-Theory in Sobolev Spaces
The classical Lp-theory for general regular elliptic problems was developed at the end of the fifties and a t the beginning of the sixties. One of the main results, the a-priori-estimate, was proved in Subsection 5.3.4. I n this section we derive the most important features of the Lp-theory in Sobolev spaces needed for the later considerations.
6.4.1.
Smoothness Properties
Theorem. Let Q be a bounded C"-domain. Let A , B,, Definition 5.2.1/4. Let 1 < p < oc) and s = 1, 2, . . If
.
. . ., B ,
be regular elliptic,
1
u E ppm(Q), Au E W;(Q), and BjuE B::i+"-'nj-P (am, then u i s an element of
(1)
W;"'+"(Q).
Proof. Step 1. The theorem is proved by mathematical induction. Assume that the assertion is true for s - 1, where s is a fixed natural number. Then the hypotheses (1) yield u E W~"'+'-'(Q).We must show u E W;"'+*(Q). Let {yl}L1be the resolution of unity described in the proof of Theorem 5.3.4. Using u E W;m+s-l(Q), it follows y l E ~~p"'+s-'(Q),
Aylu
1
E
Wi(Q), BjyEuE B2"1+8-mjp (3Q). P9P
(2)
Hence, it will be sufficient to prove the assertion for ylu. Assume, without loss of generality, ylu = u. Let K 1n a 0 = 0, where K 1 is the ball belonging t o y1 (Step 2
5.4.1, Smoothness Properties
379
of the proof of Theorem 5.3.4). Let
Assume that Ihl is sufficiently small. We use the a-priori-estimate (5.3.413)for A h p instead of u (resp. yju), and s - 1 instead of s. It follows Ildh,kUII Ivam+i-l (Q)
= < c/IAdh,hUIIw;-l(n)
+
clldh,k~IJ lVam-z+a ( Q ) '
( 4)
Now we claim that the right-hand side of (4) is uniformly bounded with respect to h. By (3), it is sufficient to prove that IIAlr,kwI(w,-l(n), w = Au E Wd,(sZ), and
Ildh,k~II ,+,am-a+. u E W~rn-l+s(sZ), are uniformly b:unded. Clearly, it is sufficient (Q)' to deal with the first case. If Fk is the one-dimensional Fourier transform for the xk-direction, then IlAh,kvII w;-l(n)
Here,
~ ~ ~ ~ ~ L must p ~ Rbel )understood
where the other n
-
as the L,:space with respect to the xk-direction, elhh - 1 1 coordinates are fixed. is a one-dimensional multiplier, hEk
where the number B, appearing in Remark 2 . 2 4 4 , is independent of h. This yields
As remarked before, this proves that the right-hand side of (4) is uniformly bounded. W;m+s-l(Rn) is a reflexive Banach space (actually, it is isomorphic t o L,(R,,)),and so it follows from the uniform boundedness of d/,,pthat a suitable sequence dh,,ku au Hence converges weakly t o w E Wirn+s-1(12).On the other hand, dh,ku+ -. 8' axk au - - - w E W;m+s-l(Q). This proves u E W;m+"(sZ). axk Step 2. Let al2 n K L 0. We use the transformation of co-ordinates y = y(x) described in the third step of the proof of Theorem 5.3.4. Then we obtain the a-prioriestimate (5.3.4/6),wherevju' ands are replaced byu' and s - 1, respectively. Further, the estimate technique used there shows that one may replace A; by A', and Bi,r by B:,
+
IIU'IIW;m+'-l(p)
5c
IIA'u'
I(W;-~(R;)
m
+j 1 llB$'(x', =l
O ) l l ~ ~ " ' + ~ - 9~ (&I-1) t-l-~ P.P
5.4. L,-Theory in Sobolev Spaces
380
It holds
u' E WZ'"+'-'(RII+), A'u' E Wi(R:),
Bj'u' E B:is-mj--
1 p (Rn-1)
-
(8) If k = 1, . .,n - 1, we may apply the method of the first step. One obtains the . want t o show that the right-hand inequality (7) where u' is replaced by A h , k ~ 'We side of the so-obtained inequality is uniformly bounded with respect to h. Clearly,
.
onemayreplace F-l(l
-
s-1
+ 1E12)i-
F i n (5)
,formula (5.3.1/7).Using thelifting a
s - l , it follows that (6) remains true after replacing D by RA. properties of J2 Further, (6) holds also for D = Rn-l. Interpolation yields that (6) is valid for Bi,p(Rn-l), too. Now it follows, in the same way as in the first step, ad E WE"'+s-l(R+) n , k = 1, . . ., 18 - 1. Here we use the fact that W:"'+*-l(R,') is axk a reflexive Banach space. This follows from Theorem 2.11.3. Finally, we must show aui
W;m+"l(R,'). The operator Ab from the third step of the proof of Theorem axn 5.3.4 is properly elliptic.Hence, U'~~,...,~~~~)(O) =+ 0. By continuity, this holds also in a neighbourhood of the origin. Hence, -E
Consequently, u' E WEmta(R,+). Retransformation yields the desired result. R e m a r k . For later applications it will be useful t o describe a simple conclusion of j = 0, 1 , 2 , . . ., denotes the set of all functions belonging to the theorem. W$loC(D), W i ( w ) for all strict subdomains o of D,that means G c D.We shall prove the following assertion: If A is a properly elliptic differential operator, Definition 5.2.1/1, if u E W~m~loc(D) and Au E W$loc(D)where j is a natural number, then u E W~m+j*'Oc(D). Let y E Corn (D). Then
W;m(D) and A(yu) E W;(D). Applying the theorem, we have yu E W~"'+'(D).Hence, u E W~"'+l,'oc(D). Iteration Essentially we did not use the full theorem, but only the yields u E W~m+J~'Oc(D). yu
E
comparatively elementary considerations of the first step of its proof. I n particular, the deep a-priori-estimates near the boundary are not needed here. The above result is known as the Principle of Local Smoothness.
5.4.2.
Adjoint Operators &-Theory)
The L2-theory for regular elliptic problems is simpler than the Lp-theory described here. The main complications arise in the proof of the a-priori-estimates, Theorem 5.3.4. It is possible to show that all results of the L,-theory (at least as far as they are treated here) can be obtained from the corresponding results of the L,theory and Theorem 5.3.4. An example is Theorem 5.4.1, but we gave a direct proof here. A systematic treatment of the La-theory, including all aspects described in this chapter, can be found in J. L. LIONS,E. MAGENES[2, I]. (We refer also to Ju. M. BE-
5.4.2. Adjoint Operators (&-Theory)
REZANSKIJ
38 1
[l], and Chapter 10 in L. HORMANDER [3].) For the later considerations it
will be helpful to use one (but not more) result of the L,-theory.
T h e o r e m . Let D be a bounded C”-domain. Let A , B,, . . ., B , be regular elliptic, Definition 5.2.114. If A , with the domain of definition D ( A 2 )= {uI u E Wgm(Q),Bjula~= 0 for j = 1, . ., m } , A,u = A u , (1) is considered as a n unbounded operator in the Hilbert space L,(Q), then the adjoint operator A$ (in the sense of Hilbert space theory) is given by D ( A l f ) = ( u ~ u ~ W ~ ~ ( D ) , C ,for u ~j~=~l = , ..., 0 m}, A!u=A*u, (2) where A*, G, , . ., C, is also regular elliptic. Here A* is a properly elliptic differential operator of order 2m, and C,, . . ., C,,, are suitable boundary operators. S k e t c h of t h e proof. For a full proof we refer to J. L. LIONS,E. MAOENES[2, I], 2.8.4. To give an insight into the theorem and into the operators appearing there, we describe the main steps of the proof. We denote by A* the operator formally adjoint to A , A*u = C ( -l)lal D * ( ~ ( xU)) .
.
.
la15 2ni
In particular, it holds
J (Au)B dx = J’ u(A*v) dx, u E C$(D), v E C$(D). n n Clearly, A* is also a properly elliptic differential operator of order 2m. Next we notice Green’s formula. There exist differential operators Sj , T i , and Cj , j = 1, . . ., m, Cju =
C
c~,,(x)D%,
(4)
I d 5 rj
where sj,,(x), tj,,(x), and c,,,(x) are coefficients belonging to Cm(aD),and all the numbers 1, , kj , and r i are less than or equal to 2m - 1, such that
J’ (Au)B dx
1 u(A*v) dx n
-
C J’ ( S ~ c,V U - Bju G)ds, Iff
=
(5) j=lan u E Ern@),v E cm(D).Clearly, this is an extension of the usual Green’s formula. Although the proof is not very complicated (in particular the above a-priori-estimates are not needed), it is rather long. We refer t o J. L. LIONS,E. MAOENES[2, I], Chapter2, ., C, are the differential operators of the above theorem. It Theorem 2.1. A*, C,, is possible t o show that C,, . . ., C,,, is a normal system, Definition 5.2.112, and that the complementing condition (with respect to A*) is satisfied, Definition 5.2.113. Hence, A*, C,, . . . , C, is regular elliptic, Definition 5.2.114. Now we define the adjoint (non-homogeneous) problem n
..
A*u = 9, Cjulan = y j , j = 1 , . . ,, m . (6) Let A,*be the corresponding operator given by (2). Extending (5) to functions u and v belonging toW~m(D), it follows immediately that the adjoint operator t o A, (in the sense of the Hilbert space theory) must be an extension of the operator A ; . Denoting the first operator by (A2)*,then A; c (A,)*. The main problem is to show that these two operators coincide. Assume v E D((A,)*) and (A2)*v = f E L,(D).
382
6.4. L,-Theory in Sobolev Spaces
Then it follows from the definition of the adjoint operator that
J (A,u) ij dx = J uj dx, n
R
u E D(A,).
(7)
In particular, this relation holds for all u E: C$(D).But this is the usual definition of a weak solution for A*v = f . If one extends the considerations on the local smoothness given in Remark 5.4.1 (we shall not do it here) then one can prove that such a funcClearly, this is less than the desired result tion v must belong to W~m*'oc(D). v E W;"'(D), and C,vla~= 0 for j = 1, . . ., m. The proof of the smoothness properties near the boundary aD is not so easy. We shall not be concerned with this problem here and refer again t o J. L. LIONS,E. MAGENES [2, I]. R e m a r k . * Further informations on the problems sketched above we shall give in Subsection 5.4.6. To clarify the situation we add here some comments. What we need in the further considerations is not the full theorem, but an important conclusion from it. As will be shown in the following subsection i t is not hard to prove (on the basis of the above theorem) that A , has a closed range of finite codimension. It is the last fact that is needed. It is possible to give direct proofs of this [2], L. HORassertion, also in the framework of L,-theory. We refer to J. PEETRE MANDER [3], M. S. AQRANOVI~', M. I. V I ~ K [l], Ju. M. BEREZANSRIJ [l], M. SCHECHTER [2,3], J. L. LIONS, E. MAOENES [I, in particular VI], and E. MAQENES[I]. 6.4.3.
The Basic Theorem of L,-Theory in Sobolev Spaces
After the description of an important aspect of L,-theory in the last subsection we return t o the L,-theory. First we remind of the notation of @-operators (Noether operators, Fredholm operators, operators with index). A bounded operator A mapping from one Banach space into another one is said t o be a @-operator if its kernel (null space) N ( A ) is finite-dimensional and if its range (image) R ( A ) is closed and of finite codimension. Further, we remind of the spaces W,2,TiB,1(D)explained in Definition 4.3.312. Theorem. Let D be a bounded C"-domain. Let A , B,, . . . , B , be regular elliptic, Definition 52.114. Let 1 < p < 00. Then the operator A ] , ,
Apu = Au, D(A,) = W ; ~ B , ~ ( Q ) , (1) considered as a mapping from D(A,) into Lp(D),is a @-operator. Proof. Step 1. First we prove that the kernel N ( A , ) of A , is finite-dimensional. (5.3.411)yields
-
llull wF(n) II~IIL&?)> E W A p )* (21 N(A,) may be considered as a closed subspace of Lp(s2).By (2) each bounded set in N(A,) (as a subspace of L,(D)) is also bounded in Wy(D). The embedding from ~ p m ( Dinto ) LJD) is compact, Theorem 3.2.5. Consequently, each bounded set in N(A,) is pre-compact. Hence, dim N(A,) < 03. Step 2. We prove that R(A,) is closed. Since N(A,,)(now considered as a subspace of D(A,)) is finite-dimensional, there exists a projection P parallel to N(A,). We claim
s
C,IluIIw;~cn) IlAullLJo, 5 czllull I V p ? )
9
16
EPw4,)
9
(3)
5.4.4. The Operators A,
383
where c1 and c2 are two positive numbers independent of u E PD(A,). Obviously, we must prove only the left-hand side. Assume that there does not exist such a number c1 > 0. Then we may construct a sequence {uj}Zl,u, E PD(A,),
1
llAuj II L ~ ( Q < ) 7 lluj II I V ~ Y O ) 1
*
Let IIu,lltv;mp) = 1 (without loss of generality). Then { u ~ } $ is~pre-compact in
L,(Q), Theorem 3.2.5. Assume (without loss of generality) uj + u in L,(O). Formula (6.3.4/1) yields
Hence, {uj}slis a fundamental sequence in Wi"'(Q), and uj + u in WEm(sZ). Consequently, u E P D ( A p ) , IJU~(~,:"(Q) = 1, AU = 0 . (4) But this is impossible. This proves (3). Now it is an easy consequence of ( 3 ) that R(A,) is closed in Lp(Q). Step 3. To prove that the codimension of R(A,) is finite we start with the case p = 2. Using the above result und Theorem 5.4.2, it follows that (5) L,(Q) = R ( 4 @ WG). Applying the first step t o A*, one obtains the desired assertion. Using Theorem 5.4.1, it follows that
N(AS) c
m
n Wim+a(Q) = Cm(Q).
(6)
s=o
(5) and (6) yield that each function f E C"(Q) may be represented by N
f(z)=
C c j f j ( z )+ g(z) j=1
(7 1
9
where fj(z)E C"(sZ) span N(A,+)and g(z) E Cm(Q)A R(A,). If A& = g , then it follows again by Theorem 5.4.1 that h belongs to C"(S2). In particular, h E D(A,) for all p , 1 < p < 00. This shows that the closure of C"(sZ) A R(A,) in L,,(sZ)is contained in R ( A p ) Together . with (7),this proves that R(A,) has a finite codimension. R e m a r k . Clearly, the last considerations yield a bit more, namely (8) L,(Q = W , )a3 {fM* . f N ( 4 where fl(z),. . . , f m (z) have the above meaning and 6 denotes the sum of conipleY
. 9
9
mented subspaces. 5.4.4.
The Operators A,
I n this subsection and the following one, we prove some comparatively simple conclusions of the last theorems. We recall the notation of an associated eigenvector. Let A be a closed operator in a Banach space. Then 0 u E D ( A ) is said to be an associated eigenvector, if there exist a complex number 1 and a natural number k such that (A - A E ) k= ~ 0, u E D ( A k ) . (1)
+
384
5.4. L,-Theory in Sobolev Spaces
If k is the smallest number with this property, then ( A - AE)k-l u + 0 is a n eigenvector for the eigenvalue A. For fixed A, the dimension of the space of all corresponding associated eigenvectors (and 0) is called the algebraic multiplicity of A. T h e o r e m 1. The operator A , defined in (5.4.3/1) is considered as a n unbounded operator in L,(Q).Here 1 < p < 03. m (a) The locally convex space D ( A ; ) = D(Ak,) (equipped with the semi-norms
n
k=O
IIA;~IIL,,k = 0, 1, 2, . . .) coincides (set-theoretically and topologically) with the closed subspace c$,~,j(Q)= { f I f E ~ ” ( Q ) ,BjAkflan= 0, j = 1, . . ., m, k = 0, 1, ...} (2) of
the locally convex space C”(Q) (equipped with the semi-norms sup lD@f(x)I, 0 aa
s la~l
0 is an arbitrary number. Using (4) and Theorem 5.3.4, one obtains by mathematical induction that
!lAk,uIILp(n)+
-
II~IILp(n)
Ilull w;”Yt(n) , u E
w;).
(5)
Hence, D(Ak,)is a closed subspace of W;mk(Q).Now, it follows from Theorem 4.6.l(e) that D ( A ; ) is a closed subspace of One obtains the boundary values in (2) again by induction with respect t o k. This proves (a). The above considerations yield also that N(A,) and the spaces of associated eigenvectors are contained in C : , ( B , ) ( Q ) , they are independent of p. Step 2. If the resolvent set of A , is not empty, we may assume without loss of generality that 0 is an element of the resolvent set. (5.4.3/3),where PD(A,) = D(A,), and Theorem 3.2.5 yield that A;’ is a compact operator in L,(Q). A complex number 1 0 is an eigenvalue of A , if and only if 1-1is an eigenvalue of A;’. Furthermore, the associated eigenvectors are the same for A , and A;’. Then (c) is a consequence of the Riesz-Schauder theory for compact operators in Banach spaces (see N. DUNFORD, J. T. SCHWARTZ [l,I], Chapter 7).
em(Q).
+
R e m a r k 1. I n connection with the theorem, there arise a number of problems. (a) Of interest are additional assumptions ensuring that the resolvent set of A , is not empty. Conditions of such a type are given in S. AGMON[2], see Theorem 4.9.l(a).
6.4.4. The Operators A,
385
(b) If the resolvent set is not empty, then the distribution of eigenvalues is of interest. Furthermore, on0 may ask whether there exist so many associated eigenvectors that their finite linear combinations are dense in LP(Q).We shall return to these questions later on, see 5.6.2 and 5.6.3. (c) If 1. $ S.ip,then one may ask whether (AP- AE1-l may be represented as (fractional) integral operator. Of interest are assertions on the kernel of these integral operators (Green’s functions). We shall treat this question later on, see 5.6.4. R e m a r k 2. If A , is a self-adjoint operator in L,(Q), then one may apply the methods developed in Chapter 8. One obtains that c 2 , { B j ) ( Q ) (and also cm(Q)) are nuclear spaces which are isomorphic to s, the space of rapidly decreasing sequences. R e m a r k 3. If f E LJQ) is given, and if there exists a function u E D(A,) such that APu = f, then u is said t o be a solution of the homogeneous boundary value problem Au = f , Bjula* = 0 , j = 1 , . . ., m . (6) T h e o r e m 2. Let Q be a bounded C”-domain. Let A , B,, . . ., B,,, be regular elliptic, Definition 5.2.114. Let 1 < p < 00, and s = 0 , 1 , 2 , . . . Then A!),
A(’)u P = Au,
D ( A F ) )=
2m+s
(7)
Wp,(Bj)(Q),
considered as a mapping from Wi:ij;(Q)
into W”,Q) is a @-operator. It holds
N ( A S ) )= N ( A P )c cz,(Bj)(Q),R(AF))= R(A,) n W;(Q). (8) There exist a finite number of linearly independent functions f j ( x )E Cm(Q),j = 1,. ..,N, independent of p and s, such that
w;(Q)= R ( A p )CD {flW, . *
* 3
(9)
fN(4)
( { f , , . . . , fN} is the space spanned by f l , . . . , fN). P r o o f . (8) is an immediate consequence of Theorem 5.4.1 and Theorem 1. In particular, R ( A t ) )is a closed subspace of W;(Q). If fl(x),. . . , flv(x) have the same meaning as in (5.4.3/7) and in (5.4.3/8),then (9) is a consequence of the fact that C”(Q) is dense in W;(Q). This proves that A t ) is a @-operator. Next we prove a lemma, useful for the later considerations. First we remind of the spaces H;,,B,)(Q) and Bi, q,(B,)(Q) described in Definition 4.3.312. Lemma. Let Q be a bounded C”-dmain. Let {Bi};-lbe a lzorrnal system, Definition 1 5.2.112. Let 1 < p < co, 1 q 5 co,and s > mk -. p I; 1 (a) Then {B,u, . . ., Bku}gives a retraction from H;(Q) onto r]:Bi,pl’-mj(i3Q)and from
s
k
1
+
j=1
B i J Q ) onto JJ BiiT-mj(i3Q).There exists a corresponding coretraction C3, independent of 1
j=1
mj(+ -.P1
(b) H ; , J ~ J ( Qis) a compkmented subspace of H i @ ) , and B;,q,{ ~ , ) (is0a)complemented subspace of B;,q(Q). Proof. Step 1. In a neighbourhood of 22, we introduce a system of curvilinear co) the coordinate lines ordinates such that the tangential vectors p l ( x ) , . . ., p , , ( ~ on 25
Triebel, Interpolation
386
5.4. L,-Theory in Sobolev Spaces
are infinitely differcntiable vector-functions. Assume that pU,,(x)= vL is the normal vector, while pl(z), . . ., y,-,(x) are tangential vectors on aQ, here x E aQ. The differential operators Biu may be expressed in these curvilinear coordinates p l , . .. ,p),,
where aj,,(z) E Cm(aQ)and a j ( x )E C"(aQ2). It holds a;@) + 0 for
{B,u, . . . , B,u} is a continuous mapping from H$?)
into
2 E
aQ. Since
nB;;T-"'j(aQ), h.
1
j=1
we
must prove the existence of a corresponding coretraction. Assume that the left-hand 1
( a Q )we . ask for a funcsides of (10) are given functions belonging to B ~ ~ F - f U , L ' Then tion u E H;(SZ) satisfying these relations and for which additionally
I
Now, using (10) and ( l l ) ,one can determine - step by step, r = 0 , . . ., aru ar'u avr an We remark that if -E B:,,(aQ) is known, then also the functions
av
are known. Using Theorem 4.7.1, one may determine a coretraction G. Similarly, one concludes for the spaces B;,,(Q). But for our purpose, it is important to know that G can be constructed in such a way that it is independent of 1 < p < a, 1 1 q m and s > mk -. This is a consequence of the method of local coordi-
+P
nates, Theorem 4.7.1, and the considerations in the third step of the proof of Theorem 2.9.3.
Step 2. E - G{Bl,. . ., B, 0, then At,)q, 15q 2nr+s
AF,)qU = Au, D(At,)q)= Bp,q,(Bj)(Q),
00,
(2)
s+2m
consideled as a mapping from BP,q,(B5)(Q) into B;,q(Q),is a @-operator. It holds
and
N ( A g ) )= N(A!,)q)= N ( A ) c cZ,{B,)(Q), R ( A t ) )= R(A,) n H @ ) , B(A;jq) = R(A,) n B;,,(Q),
(4)
codim R ( A 9 ) )= codim R(Ag\) = codim R(A,) < co.
(5)
(3)
s+ 2111 (b) If A 4 S A , then A - AE is an isomorphic ma?vping from H,,,B,,(Q) onto H$2), s+2m s 2 0, and 1 < p < 00, and an isomorphic mapping from Bp,q,(B51(Q) onto BE,,@), s>O,1 < p < c o , a n d l S q s c o .
P r o o f . If k is a natural number, then Theorem 5.4.4/2 shows that the projection from Wi(L?) onto R(AE)),given by (5.4.4/9),is the restriction of the corresponding projection from Lp(Q)onto R(A,). It follows from Theorem 1.17.1/1, (5.4.4/8),and Theorem 4.3.1/1 that
n R(Ap) (6) [R(AE)), R(Ap)I, = [H;(Q), Lp(Q)Ien R(Ap) = where 0 < 8 < 1 and s = (1 - 8) k. Remark 1.17.2/2, Lemma 5.4.4 (in particular 2
the second step of the proof of this lemma), and again Theorem 1.17.1/1 yield
rHi,T:;(Q)/N(A),
H:JiBj)
)]e = [H:,fE$(Q)9 H i 7 B , ) ( Q ) ] e / N ( A =
HiI$;(L?)/N(A),
(7)
where 8 and s have the same meaning as above. By Theorem 5.4.4/2, one obtains that A is an isomorphic mapping from H ; I g i ( Q ) / N ( A ) onto H ; ( Q ) n R ( A , ) . One concludes similarly for the B-spaces. All the other considerations are the same as in the proof of Theorem 5.4.412. R e m a r k . (4)shows that a.nd B;,,(Q) may be represented by direct sums similarly t,o (5.4.4/9),where one may use the same functions fl, . . . , fi,, as there. 6.6.2.
Non-Homogeneous Boundary Value Problems
All the symbols have the same meaning as in the preceding subsection. T h e o r e m . Let Q be a bounded C”-domain. Let A , B,, . . ., B,n be regular elliptic, Definition 5.2.1/4. (a) If 1 < p < 00 and s 2 0, t h e n a p ,
%!)u
= { A u ; B1u, . . . , B,u},
5.6.2. Non-Homogeneous Boundary Value Problem
considered as a mapping from H F ~ ~ ( into Q ) H;(Q) x
s
n B ~ - " ' ~ - - +@Q), ' i s a @ni
1
p
3=1
oprator. If 1 < p < a1,1 q 5 03, and s > 0, then%:,),, = { A u ;B,u, . . ., B,,u),
%g!q~
39 1
1
111
considered as a mapping from BZ;"(Q) into B"p'q (9)x JlJ= 1 B2f1'-fr'j--+s P ( d 9 ) is a PQ @-operator. It holds
N(%g') = N(%gh) = N ( A ) c C ~ , ~ B , ~ ( Q ) ,
(
R(%g))= H",(SZ) x
n I f1
J=l
R(rU!,)q)= (B;,,(Q) x and
1 B K - l " * -P- + s( d Q ) )
A
(1)
R(%IP,)
(2a)
3
fi B ~ Z - ' " ~ - - (+8'9 ) ) n ~(9ir)) , 1
J=1
codim R(9ig))= codim R(%$!,)= codim R(A,)
p > n. Application of Theorem 5.6.113 gives the desired result. Remark. * The density of finite linear combinations of associated eigenvectors of [l,31 for the Dirichlet elliptic differential operators was proved by F. E. BROWDER problem and by S. AUMON [2] for general boundary value problems. See also G. GEYMONAT, P. GRISVARD [2]. These results are more general then the above result. In particular, the restriction to p = 2 and to operators of type (1) is not necessary. For more general operators, however, hypotheses of other types are needed.
m)-l
5.6.4.
Green Functions of Elliptic Differential Operators
One of the most frequently used methods of the determination of distributions of eigenvalues for differential operators is the investigation,of qualitative properties of the corresponding Green functions. Our method here is the conversion to this procedure. From (5.6.2/1), we derive differentiability properties for the Green functions having necessary and sufficient character. Lemma. Let Q be a bounded Cm-domain.Then for s 2 0 we have
W ~ xQQ) = (WQ)
G L,(Q))n (L,(Q) G WW).
(1)
398
5.6. Distributions of Eigenvalues, Associated Eigenvectors
Proof. Step 1. Let s = 0, 1 , 2 , . . . Since Q x SZ c R,, is a bounded domain of cone-type, (1) is a consequence of Theorem 4.2.4 and of the fact that C"(Q x Q) is dense in both spaces of (1). Step 2. Let 0 < s < m, where m is a natural number. Let B be a self-adjoint positive-definite operator in L,(SZ) with the domain of definition D ( B ) = WT(Q). Since the embedding from W,n'(Q)into L,(Q)is compact, B must be an operator with pure point spectrum. Its eigenvalues (inclusively their multiplicities) are denoted and its corresponding orthonormed eigenfunctions are denot,ed by by { l j & {fj(x)}F1. Then B @ E E @ B is a n operator with pure point spectrum in L,(Q x Q), its eigenvalues are {Aj &}?k,l, its eigenfunctions are { f j ( x )fk(y)}~k-l. B y the first step,
+
+
W r ( Q x SZ) = D ( B @ E
+ E @ B)
(2)
Let s = Om, where 0 < O < 1. Then Theorem 4.3.112 and Theorem 1.18.10 yield
?V&C2 x 52) = [L,(Q x
a), Wg(Q x Q)]' = D ( ( B @ E + E @ B)')
$ L,(Q)n L,(Q) 6D(B') = W;(SZ)$ L,(Q)n L,(SZ) ^o w;(Q).
= D(H)
This proves the lemma. Let A , be the operator described in Theorem 5.4.2. Let A: be the adjoint operator, Theorem 5.4.2. Then &(Q) = WAX) @ N(Az) = R(A,) 6 (3) Denoting the restriction of A , to R(AZ) by A,, D(A,) = D(A,) n R(AZ), then 8, is an isomorphic mapping from D(A,) onto R(A,) = R($).The inverse operator Ail will be considered as a compact mapping from R(A,) into R(AB) (both spaces equipped with the L,-norm).
T h e o r e m . Let SZ be a bounded P-domain. Let A , B,, . . ., B , be regular elliptic, n has the above meaning, then 2;' can be repreDefinition 5.2.114. Let 2m > -. If 2 sented in the form
(A;'/) (2) = f G(z, y) f ( y )d y such that R
if and only if 0
G(z, y)
E
Wg(Q x 52)
(4)
I
e
n < 2m - - . 2
(5)
P r o o f . Step 1. A,AZ and AZA, are self-adjoint operators in L,(Q) (see for instance F. RIESZ, B. SZ.-NAC+Y [I], $119). They have a pure point spectrum. This is a consequence of Theorem 5.4.2 and Theorem 5.4.1. In particular, D(A,A:) and D(ABA,) are closed subspaces of Wim(Q).The positive eigenvalues of ABA, are denoted by { 1 ~ ]the ~ 1corresponding , orthonormed eigenfunctions are {fj(x)}F1.From N(AZA,) = N(A,) and (3) it follows that R(ABA,) = L,(Q) 0 N(A,*A,) = R(AS)
5.6.4. Green Functions of Elliptic Differential Operatom
399
is spanned by { f j ( z ) ] F l Setting . gj(x) = A;lA,fi E D(Ag) it follows that A2A,*gj
=
A7gj
(gj 9
Y
gk)L, =
(6)
dj,k*
Since conversely f j = Ai1A;g,, one obtains in this way all positive eigenvalues and the corresponding spaces of eigenvectors for A,A,*. In particular, R(A,) is spanned by { g j ( z ) } , " i l . For g E R(A,) = D(&l) we have
Hence, in the sense of (4), we write formally
8tev 2. Let
For 0
5
Q
n 2
< 2m - -, it follows that
Since D(ABA,) is a closed subspace of Wt"(Q), containing W;"(Q), i t follows from (5.6.1/6), (5.6.1/5), the proof of Theorem 5.6.2, and (4.10.2/14) that
22I 2m
S'mce n D(AgA,) c
Q
j4lll/12.
-2m m
(10)
< -1, the right-hand side of (9) converges if N + 00. By
Wp(Q), Theorem 1.18.10, and Theorem 4.3.1/2, it follows that
(s
Y) E W,P(Q) La(Q)* Replacing A t A , by A d ; and vice versa, and considering G N ( x ,y) instead of O N ( x ,y), one obtains that
% W Q.)
G ( x ,Y) E L,(Q)
Then (4) is a consequence of the above lemma; 0 Z"L---ll
Step 3 . Assume G ( x , y) E W z
Q
n
< 2m - -.
2 A (Q) @ L,(Q).Let B be a self-adjoint operator in em-2
L,(Q) with the domain of definition D ( B ) = We i t follows that
5
(Q). For g E D ( B ) and f
-G ( z , Y) f(Y) (Bg) (4dY dx
=JQ =
J (9(% BGY.9 Y))L, f o dY
n
E L,(Q),
400
5.6. Distributions of Eigenvalues, Associated Eigenvectors
Consequently,
Gf = j G(x, Y) f ( y ) dy
E D ( B )9
R
B J a(-,Y) f ( Y )dY = J BG(., Y) f(Y) dY R
R
Using .73G(.,y) h1!-
L2(SZ)into W2
x If.
a),it follows that G is it Hilbert-Schmidt operator from
(SZ).Hence, the approximation numbers
(sj(G;
~
2rrr-R ~
0
w2 1
9
2 (Q)lj=,
belong to I , . (1.16.1/28) and (4.10.2114)yield for j 2 1
It follows from (8) and (10) that
-i
sa(G; L 2 ( 0 )L2(SZ)) ,
--2 m
, i 2 1.
Now one obtains that
This is a contradiction to the above assertion. Remark 1. In the proof of the theorem, we followed the treatment given by 72
H. TRIEBEL[5] (see also H. TRIEBEL[l]). One can show that the restriction 2m > 2 is not necessary. In the general case, one has to interpret G(x, y) as a distribution belonging to D ' ( 9 x SZ). But there arise new difficulties. So we do not go into detail here and refer to H. TRIEBEL [5].
Remark 2. * The above considerations are not in the line of the usual investigations for Green functions of elliptic differential operators. These are concerned mainly with the local behaviour of Green functions, the characterization of singularities of G(x, y) in a neighbourhood of x = Y E SZ, and the description of differentiability properties of G(x,y) for x =k y. Roughly speaking, G(x, y) has locally the same behaviour as the well-known fundamental solution of Am. One can extend these considerations to non-homogeneous boundary value problems. We refer to Ju. M. BEREZANSKIJ [l], Ju. P. KRASOVSKIJ[I, 2,3], M. I. MATIJOUK, S. D. EJDEL'MAN El], I. A. KOVALENKO, JA.A. ROJTBERG [l], and T. V. LOSSIEVSKAJA [l]. Remark 3. * The proof shows the close connection between smoothness properties of kernels of integral operators and the approximation numbers of the corresponding operators. First results in this direction can be found in I. C. GOCHBERG, M. G. KREJN [l], 111, 5 10. These results are generalized by V. I. PARASKA [l], P. E. SOBOLEVSKIJ [l], and H. TRIEBEL[a, 121.
401
5.7.1. Lebesgue-BesovSpaces without Weights
5.7.
Boundary Value Problems [Part 111
The main aim of this chapter is the investigation of regular elliptic differential operators in Lebesgue-Besov spaces without weights, Sections 5.3, 5.4, and 5.5. It is possible t o generalize these considerations in several directions. We ‘describe some of them here.
8.7.1.
Lebesgue-Besov Spaces without Weights
Boundary value problems in Lebesgue-Besov spaces without weights are considered in the Sections 5.4 and 5.5. The basic spaces are H“,(SZ) where (T 2 0 , and B&(Q) where (T > 0. A systematic extension of this theory t o spaces where IS 5 0 was gven in J. L. LIONS,E. MAOENES [ l , particularly VI; 2, I], and E. MAGENES[l]. One needs new tools which are out of our line here. But it will be shown that in the framework of the ideas developed here one can extend the theorems of the Sections 5.4 and 5.5 to some negative values of (T.For sake of simplicity, we restrict ourselves to homogeneous boundary value problems and assume additionally that 0 is an element of the resolvent set. T h e o r e m . Let SZ be a bounded C”-domain. Let A , B,, . . ., B , be regular elliptic, Definition 5.2.114. Let 0 # S A (= SAP).Then A is an isomorphic mapping from
H ~ ; ; ; ( D )onto H;(Q) < s < 00, and 1 =< q
and from B ~ , ~ , ( ~ ,onto ) ( S BZ ~) J Q ) where , 1
0 there exists a natural number j ( E ) such that EpxioI+ I I ~ ( x ) for z E a - , W e ) ) , (7b)
(This means that for a n y number lDYas(x)l
s
E
where 9 ( j ) has the same meaning as in Definition 3.2.3/1.) (b) The subclass '&&(9; e ( x ) )of %&(i2; &)) consists of of all operators of %&(9; e(x)) for which there ex%& a positive number 6 > 0 such that
s
DYas(x) =
(8)
o(@x161+lYl-8)
for 0 IpI < 2 m and for all multi-indices y . Remark 2. Clearly, (8) is sharper than (7). Remark 3. %Ev(f2;e ( x ) ) and % E y ( 9 ;e(z)) are comparatively comprehensive classes of degenerate elliptic differential operators. We describe two simple examples. (a) If 0 is an arbitrary bounded domain, and if e-l(z) d ( x ) in the sense of Remark
-
6.2.2. Powers of Strongly Degenerate Elliptic Differential Operators
3.2.311, then Au = f(z) ( - & I
+ e”(z) a,
u
v >. p
407
+ 2?n,
belongs h@”(Q; e(z)). If Q is a bounded Cm-domain,then one can put e(s)= d - l ( z ) near the boundary. (b) If Q = R, , then any operator of the form
AU = (1
+ Izls)*l
u
+ (1 +
q2 > ql,
1 ~ ” ) ” s ~ ~
rlr (R,; (1 + I Z ~ ~ )provided ~), that 6 > 0 is sufficiently small. a a
belongs to the class
R e m a r k 4. * The above definition goes back to H.TRIEBEL[24]. Differential operators of the above type are closely related to investigations on the structure of nuclear function spaces. We shall return to this problem in Chapter 8. In this connection the special case
A ~= L -Au
+ e”(s)
U,
v > 2,
in bounded domains was considered in H. TRIEBEL [2]. Extensions of these investigations to unbounded domains as well as generalizations and improvements can [l,21, B. LANQEMANN [l, 21, D. KNIEPERT [l], be found in E. MULLER-PFEIFFER and H. TRIEBEL [lo, 241. Similar differential operators are considered in L. A. BAQIROV [l] and L. A. BAGIROV, V. I. FEJQIN[l].
6.2.2.
Powers of Strongly Degenerate Elliptic Differential Operators
We need for the later considerations that powers of strongly degenerate operators in the sense of De€inition 6.2.112 are also operators of such a type. L e m m a . (a) If A E%E,(Q; e(z)) in the sense of Definition 6.2.1/2, then Ak E % ~ E ~ ~e(s)) ( Q for ; k = 1,2, . . .
(b) I f A ~iflE,(Q;e(x)),then Ak E&;&(SZ; e(z))for k = 1 , 2 , . . . Proof. The proof is given by induction. Assume that the lemma is true for k = 1 , . . ., j. Then we have
c c nij
Aju
=
1=0 l a l = 2 1
xp’ = y
@4{)(Z a$’(.) )
2rnj - 1 2rn
+P
1 G Y
D*U
+
c
c2nrj
1 = 0, 1 , .
UP@)
DBU,
(1)
. ., 2mj.
The coefficients have the properties mentioned in Definition 6.2.112. xjj)
+ x , ~= x$gi), 1 = 0 , 1 , . . .,2rnj,
s = 0,1,
. . ., 2m,
(2)
yields that the ‘‘ main part ” of A j + L = Aj(Au)has the desired structure (inclusively (6.2.1I S ) ) . Using and
lDe”(s)I5 cex+lyI(z), x real number, xij)
+ x, + 1y1 = xjcl) + ~ y 0 such that e-"(x) E L,(Q),then we have for all p with 1 5 p 5 oc) ( i n the sense of continuous embedding) Se(.x)(Q)c L p ( Q ) . (1) (c) If (1) i s true for a suitable number p with 1 5 p c 00, then there exists a aumber a > 0 such that e-"(x) E L,(Q). Proof. Step 1 . We prove that C$(Q) is dense in Se(&2). If the domains Q(J) have the same meaning as in Definition 32.311, then there exist functions q ~ j ( xE) C ~ ( Q ~ + l ) ) such that v j ( x ) = 1 for x E QCj) end lDyvj(z)l 5 cY2Jlrl (2) for j = N , N + 1, . . . and for all multi-indices y. See Remark 3.2.312. (Outside of Q(J+l)we set p i ( x )= 0.) I f f E Se(,)(Q), then v j ( x )f ( x ) E Corn (Q) approximates the function f(s)in Sec,)(9).It is easy t o see that X,(,)(Q) is an (P)-space. Step 2. The part (b) of the theorem is clear. To prove (c), we show at first that there exists a number b > 0 such that I Q ( j + l ) - Q ( j ) l 5 bj, j = N , N + 1 , . . . (3)
Assume that there does not exist a number b having the property (3). We fix a natural number k and a sequence O < a l < a 2 < . . . < a l < ..., a l + c o if Then there exist natural numbers jz > N + 1 such that IQth+l)
1-oc).
(4)
- Q ( j q > ail, jz+l- j l 2 k . I
We choose k sufficiently large and set
Using Sobolev's mollification method described in the proof of Lemma2.5.1 and setting ( U ) ~ ~ - ~ for ~ ( Zx ) E SZ(jlt3) - SZ(jl-z), v(x) = otherwise ( x E 9 ), ( 0 where c > 0 is sufficiently small, then it follows that v E Cm(sZ)and
6.2.3. Properties of the spaces A!?~&?)
409
for x E Q ( j i + j ) - D(ji-2).Hence v E Se(,,(Q). On the other hand, one obtains by (5)
j
J)
W
Iv(s)lpdx
2C
1= 1 J)Ul+') -&I,
Iv(s)lp dx
W
2 C
-
~ j i l Q ( j ~ " j Q(jljl = 03.
Z=1
This is a contradiction to (1). This proves (3). Now it follows for a > 0 with 2a > b
R e m a r k 1. If 52 is bounded, then e-"(x) belongs t o L,(Q) for all a 2 0. One of the main aims of this chapter is the development of a n Lp-theory for operators from the Definition 6.2.112. For this purpose, (1) is a natural assumption. The theorem shows that (1) is equivalent t o
3a > 0 where ,p-"(x) E L , ( 9 ) .
(6)
R e m a r k 2. We shall see later that under the hypothesis (6) S,(,)(D) is a nuclear (F)-space isomorphic to the space s of rapidly decreasing sequences. It is easy to see that the Schwartz space S(R,,) is a special case. L e m m a . Let D c R,,be a bounded domain, and let e-l(x) d ( x ) be a smoothed distance function i n the seme of Remark 3.2.311. Then it holds set-theoretically and topologically (7) Se(3j(Q) = CO"(52) = { f I / E CF ( ~ n;)SUPPf c N
.n7
where the topology i n CF(52) i s given by the semi-norms sup lDaf(x)l,0 Z€Q
=< la1 < 03.
P r o o f . It is not very hard t o see that Se(,)(D)andZF(52) coincide set-theoretically (here the functions of ~ S ~ ( ~ ) (are 5 2 extended ) by zero outside of 52). Further, E$(Q) is an (F)-space where the topology of Se(,)(52)is finer than the topology of c$(D). Then one obtains as a consequence of the closed graph theorem (see N. DUNFORD, J. T. SCHWARTZ [l,I], 11.2, Theorem 5) that fYe(&) and @(Q) coincide also in the topological sense.
6.3.
A-Priori-Estimates
The main aim of this chapter is t o give a treatment on mapping properties for the operators of Definition 6.2.112 in the framework of an L,-theory. A priori-estimates and an L,-theory for special self-adjoint operators of such a type (Section 6.4) are essential for these investigations. On this basis and with the aid of the SobolevLebesgue-Besov spaces with weights, one can prove in the following sections theorems on isomorphic mappings.
410
6.3. A-Priori-Estimates
6.3.1.
Equivalent Norms in the Spaces W,(sd; 8’;
e’)
Lemma. Let e(z) be a weight function in the sense of Definition 3.2.311. Further, let k = 0, 1, 2, . . ., 1 < p < 00, Y 2 p kp, and {yj(x)],ZN EY(Q;e) in the sense of Definition 3.2.311. Then there exist: (a) Balk K f ) = {z I 1z - zj,ll .c d 2-j} such t h d
+
-
s,
Qjc U Klj) c Q;-luQ j + l , 1=1
j = N,N
+ 1 , . . .,
(QN-l = Qx), where at most L balls have a mn-empty intersection ( L i s a suitabk natural number), and d > 0 i s independent of j. (b) Systems {~$)(X))E’~, j = N , N + 1, . . ., s w h that
ID“p71U)(~)l ~,,2”“’, j
=
N,N
+ 1 , . . .,
1 = 1 , . . ., N ; , IyI
> 0 . (3)
i s an equivalent norm in W”,Q; e”; e’) (Definition 3.2.312, Theorem 3.2.412). Proof. The existence of balls KiJ)and functions yiJ)(x)is a consequence of the properties of the domains Qj,see Remark 3.2.312. Lemma 3.2.411 is also valid for the functions rpjJ)(x).Whence it follows in the same manner as in the proof of Theorem 3.2.411 that (4) is an equivalent norm in W#2; Q ” ; e”). Remark. The magnitude of d determining the radius of the balls can be chosen arbitrarily small. We shall use this fact in the next subsection.
6.3.2.
A-Priori-Estimates
Theorem. Let A E %EV(Q;~ ( s ) )in the sense of Definition 6.2.112, where v 2 0. Further, let x be a real number and 1 < p < 00. Then there exists a real number c1 such that, for every complex number il with Re 3, c l , there exist two positive numbers c2 and c3 with the property that for all u E Wtrn(Q;e X + P ” ; ex+p’)
P r o o f . Step 1. Theorem 3.2.412 and Theorem 3.2.4/3(a) yield that the left-hand side of (1) is true. (Here one needs the assumption v 2 0.)
6.3.2. A-Priori-Estimatee
41 1
Step 2. To prove the right-hand side of (1)we assume temporarily that the support of u E W;rn(12)is contained in one of the balls K g ) of Lemma 6.3.1. Let xj,,.be the centre of KAj).We set rn
A1u
Then me have
AU
=
c c z=o
l al = 21
D% - h,
@xal(Xj,k)ba(Xj,k)
- ;lu = A,u + AZu+ A,u.
Extending u ( x ) oufside of K,O"by zero, then it follows that
and taking into consideration xzl the Fourier transform that
21 +(v - p ) = v , then it follows with the aid of 2m
IlA14Ep(~;e~)
c
> 0 is independent of j, k, and A. Remark 2.2.414 and Definition 6.2.112 yield that
and
are multipliers for Re A 5 0, where the number B appearing in Remark 2 . 2 4 4 is independent of A, j,and k, while it depends on the constant of ellipticity C in (6.2.1/6). Then one obtains by Remark 2.2.414 that
412
6.3. A-Priori-Estimates
where the constant c > 0 depends only on C , but not on A, i, and k. Putting this result in (3), and returning to the original coordinates x in ( 2 ) ,then it follows that
Here cl, c2, c3 ,and c, are poaitive numbers depending only on the constant of ellipticity C , but not on 2, j, and k ; Re1 5 0. To estimate A2u we choose (in the sense of Remark 6.3.1) d sufficiently small. Then one obtains for x E KP) that @ q s )ba(z)-
6
@X?'(Xj,k) ba(Zj,$)l
C2.i("2'+')
Ix - X j , k J
6 E2jXZ'.
Here d = d ( E ) is a given number, E > 0. Theorem 3.2.412 and Theorem 3.2.413 yield
s Here 6c
E"
> 0 is a given number ; d = d ( d r ) > 0. If
w is a bounded domain such that
52, then we have
ll4lw;m-1(")
6
(5)
E " I I ~ l l ~ ( ~ ; e ~ + ~ ~ ; e ~ + ~ ~ ) .
E l l 4 w;%J)
+ C(E)
5 E'IJ~IIw~~(~;
Il~lILp(w)
ex+wp;e"+vp)
+
ll~ll~,(~; ex).
Here E' > 0 is a given number. (This follows from a formula similar to (4.10.1/13).) Using this estimate and the assumptions on the coefficients as(.), then it follows from Theorem 3.2.412 and Theorem 3.2.413 that
IIA~uIIE~(Q; e x ) S EIIuII&;~(Q; e x + w p ;
ex+vp)
+ C(E) l I ~ l l & ( ~ ;
IIAu - ~
L
~ l I Q ~ ; e x )
ciIIull&im(Q;
ex+pw;ex+pv)
(6)
ex).
Finally, one obtains by (4), ( 5 ) , and (6), and by a suitable choice of
+( ~ 2 14 ~
E
and E" that
~ 3 IIUll!ip(n;ex). )
(7)
where R e d 5 0. Here, the positive numbers c1 and c2 depend only on C , while c3 depends on C and on the constants of estimates for DYbJx) and D Y a s ( x ) in the sense of (6.2.117). (Clearly, these numbers depend also on the fixed domam 52 and on the fixed function e(s).) Step 3. Let u E W26n(SZ;ex+pJ'; e X + " P ) . Using Lemma 6.3.1, then it follows by (7) 00 .vj for c2(LIp - c3 2 0 and u = C C yj&)u that j=iV
k=l
413
6.3.2. A-Priori-Estimates
We have
A(vjq$'~)- 1vjq$~= vj~)($(Au -1 ~ )
+
C
c@,a(x)DYvjd!')flu*
IS1 2m lJlnl$2m-ISI
Using xlal+lp~
(9)
+ la1 < xlslfor (a12 1, then it follows that
C S , & ( X ) D"(yJj'P$) = O(@xl.+I#l@'"l) =
o(@"lBl-a),
where 6 > 0 is a n appropriate number. Similarly t o the second step, one obtains
Putting this in (€9,and choosing E the right-hand side of (1).
C
= 2and Re
2
1 sufficiently small, then one obtains
R e m a r k 1. The proof yields a sharper result than formulated in the theorem: T h r e exist a real number c1 and a positive number c 2 , depending only on the con-stant of ellipticity C i n (6.2.1/6), the constants of estimate for DYbJx) and DYas(x), and the o-behaviour of DYas(x) i n the seme of (6.2.1/7) (and In, e ( x ) ,p , and x ) such that for all complex numbers 1with R e il 5 c1 and all u E Wim(sZ;@ + p P ; e x + p v )
llAu -
~UIIL~(Q;
ex)
2 czllull +yg;
+ c2Vl
px+v)
llullLp(n;ew) -
(11)
Later on, we shall use this sharper version of (1). R e m a r k 2. The assumption v 2 0 is needed only for the proof of the left-hand side of (1). It is easy t o see that v 2 0 is a natural assumption. Namely, if v < 0 and il < 0, then A - ilE belongs t o '%Eo(Q;e ( x ) ) , but not to ?l$(In; e ( x ) ) .In this case the term -ilu belongs t o the ''main part" of A - AE, and one cannot expect i l ~ estimate l of the form (1). Let v >= 0. Since x in (1) is an arbitrary number, it is easy t o see that one can replace ilu by ile"(x)u with 0 5 v. Afterwards it follows that, in such a formulation of the theorem, the assumption v 2 0 is not necessary. We shall return t o this question later on, Theorem 6.5.1.
6.4.
&-Theory for -
-I-gU(x),v
>2
The results of this section and the a-priori-estimates of Section 6.3 are the basis for the further considerations. But the investigations of this section are also of selfcontained interest. Together with Theorem 6.6.1, they are the basis for the structure theory for the spaces Se,,,(In)in the eighth chapter.
-A
414
6.4. &Theory for
6.4.1.
Self- Adjointness
+ e'(z),
Y
>2
Lemma. Let 52 c R, be an arbitrary domain, and let Au = 0 for u E D'(Q). Then u is a harmonic function in the cbsicu2 sense. P r o of. Let p E C$ (52).Then pu can be interpreted in the usual way as a distribution belonging to E'(R,) c S'(R,). (See for instance H. TRIEBEL[17], p. 49/50, p. 103.) From the properties of distributions of E'(R,) and from (2.8.1/16), it follows for y E Corn(R,) that
l(pu)(w)l 6
Cz~~~~IW~(R,)*
c~lIWIICk(J?,)
Here k is a suitable natural number, and 1 is a natural number such that 1 > k Theorem 2.6.1 yields pu E Wiz(Rn). Further we have (in the sense of D'(R,))
n +. 2
Using Theorem 2.3.4 with s = -2 (or the usual rules for Fourier transforms in S'(R,)), then one obtains pu E Wi'+l(R,). Putting this in (l), then it follows m
.
Wi1+2(R,).Iteration and application of (2.8.1/16) yield guu E n Wi(R,) j=-m c C"(R,). Whence it follows the lemma. Remark 1. * Differentiability properties of the above type are well-known in the literature (theorems of Weyl-type, properties of hypo-ellipticity). One may generalize the lemma essentially. See, for instance G. HEUWIQ [l], IV, 3.4/3.5, and the references [3]. given there, and L. HORMANDER Theorem. Under the hypotheses of Definition 6.2.112 the operator A ,
pu
E
+ e'(s)u,
> 2, B ( A ) = C$(Q), is essentially selj-adjoint i n L,(Q).Its closure A is an operator with pure p i n t spectrum. Au = -Au
Y
Proof. Step 1. Clearly, A is a symmetric positive-definite operator. We want to show that it holds N ( ( A orE)*) = (0) for sufficiently large values of a > 0. Then, by well-known theorems, it follows the self-adjointnessof A (see for instance H. TRIEBEL [17], p. 206-207). Let A*v av = 0. Then in the sense of the theory of distributions we have -du e'@) v av = 0, v E L,(S). (2)
+
+
+
+
If o is a bounded Cm-domainsuch that 6 c 52, then -dw
= -&V
- eY(s)vEL&)
(3)
has a solution w E Wg(o)A W i ( o ) . (Dirichlet's boundary value problem for - A . We note that - A satisfies the hypotheses of Theorem 4.9.1. See also Remark 4.9.1/3. Since -d is positive-definite on W i ( o )A @(o),then it follows by Theorem 5.2.3/1, that 0 is an element of the resolvent set.) Applying now the lemma to v - w, then it Eollows v E Wg(o). Now, (3) and Theorem 5.2.2(c) yield that w belongs to Wi(w). Application of the lemma gives w E W,"(o).Using (2.8.1/16),then it follows by iteration w E C"(52).
6.4.1. Self-Adjointness
416
Step 2. For the proof of the self-adjointnessit is sufficientto show that for sufficiently large a > 0 any function w(x)with
-dw
+ e’(x) w + aw
0, w(x) E P ( QA)L2(Q), (4) vanishes identically. Assume without loss of generality that w(x) is a real function. For { ~ j } j ” E, ~!P (Definition 3.2.3/1) we set vj(x.0 = (e’(x)
=
+ a)-* C yl(x)
Then we have
j
1= N
E
Cz(Q), j = N , N
+ 1, *
n
n
Using (4) and
av 2 C - -avJ . -vjw ’l
l i = l axk
axk
1 2
=-
c n
k=l
a,p? - I - ,
axk
a$ axk
then it follows by (5)that
(z(x))2dz
Using the properties of e ( x ) and y&), partial integration yields
J-
n
fi yr)2axs
w2( l = N
J-wz n
2
k=l
Since v > 2, one can make the factor in the last integral arbitrarily small, independently of j, if one chooses a sufficiently large. Transfering this summand to the left-hand side and considering j * a,one obtains w(x) I 0.
Step 3. Now we have to prove that A is an operator with pure point spectrum. It follows from Theorem 6.3.2 and Theorem 3.2.4/1 that D ( A ) = W,”(Q;1 ; e2’). (8) By the theorem of F. RELLICR (see for instance H. TRIEBEL [17], p. 277), one has to prove that the embedding from D(A)into L2(Q)is compact. If xj(x)is the characteristic function of Q ( j ) from Definition 3.2.311, then it follows from the compactness of the embedding from W i ( S ( i )into ) L,(Q(j))(Theorem 3.2.5) that
Mj = { x j ( x )~ ( xI )IIU(~)IIw~(n;l;e’v)5 1) is a precompact set in L2(Q).Now it follows from
\ 11 - X , j ( X ) 1 2 lu(x)12 ax 5 2 4 ’
ir
[ @2”lU12a x fi
that M,; for j 2 j&) is a pre-compact &-netfor the image of the unit ball of D ( A )in L2(Q).Whence one obtains that the embedding from D(A)into L2(Q)is compact.
416
6.4. &-Theory for
+ e'(r). v > 2
-d
Remark 2. * The above proof is due to H. TRIEBEL[2].The used method, in particular estimates of the type ( 6 ) and (7), goes back to E. WIENHOLTZ [ l ] (see also I. M. GLAZMAN [2],Chapter 1 , Theorem 3.5). 6.4.2.
Eigenfunctions
Theorem. If A is the operator of Theorem6.4.1, then the eigenfunctions of A belongto SQ[=) (52) (Definition 6.2.1/1). Proof. Step 1 . We start with preliminaries. Let v E L2(Q),and let
-dv
+ @'(X)
v
=
g
E L,(52)
in the sense of the theory of distributions. We want to show that v belongs to D ( A ) . If 9 E C$(52), then (v, AV)L, = (9, d L , *
Whence it follows v E D(A*) = D ( A ) . Step 2. Let il be an eigenvalue of A, and let u(x)be an eigenfunction, Au Then, in the sense of the theory of distributions, we have
-du
=
ilu.
+ $(z)u = ilu.
(1)
Further, one obtains from (6.4.1/8)and from the first step of the proof of Theorem 6.4.1 that u E W,"(52;1 ; e2') A C@(52). (2) We want to show that @ ( x ) u ( x ) belongs to D ( A ) = % 1( ; e2') $ forIeach ; number oc 2 0. Let 26 = v - 2 > 0. The proof w i l l be given by induction. Assume that e e ( j - l ) ubelongs to D ( A ) for a natural number j . We shall show e"ju E D ( A ) .It holds
Since eE(j-l)u belongs t o D ( A ) ,it follows from (6.4.1/8)and Theorem 3.2.4/3 that
e&j+2I I < = ce8(j-1)+vIuI
E L2(Q),
(4)
Whence it follows that v = eeju and the right-hand side of (3) belong to L,(Q). Now it follows, from the first step, e8juE D ( & . Repeated application of Thec j . Hence e% E D ( A ) for all a 2 0. orem 3.2.413 yields @" E D ( A ) for y Step 3. If u is the eigenfunction of the second step, then we want to show that @DYu belongs to D ( A )for every number oc 2 0 and every multi-index y. We use again induction, firstly by Iyl = 0, 1 , 2 , . . .,and secondly for fixed y by oc = E j , j = 0,1,2,. .
.
417
6.4.3. Domains of Definition of Fractional Powers
Here, E has the same meaning as in the preceding step. Assuming that the statement is true for 0 IyI k - 1, then it follows for I/?I= k that -ADs, + @(z)Dsu = W U+ 1 c,,D"$"'fl-'h E L,(Q).
s
s
llllZ1
The first step and Theorem 3.2.4/3 yield Dsu E D ( A ) .Then one obtains in the same manner as in the second step that e"D% E D ( A ) ,OL 2 0. Step 4 . If the functions y j ( x )have the same meaning as in Definition 3.2.3/1, then it follows from (2.8.1/16) and from the above results
R e m a r k 1. One can show that, for v = 2, the Theorem 6.4.1 as well as the above [2], p. 165. This shows that theorem are generally untrue. We refer t o H. TRIEBEL v > 2 is a natural assumption. R e m a r k 2. Considering the above theorem and Theorem 6.2.3(a) it seems to be meaningful t o extend the domain of definition of the operator A from C$(Q) t o S e ( & ) ( 9I)f. one wants t o remain in the framework of L,-theory (or L,-theory), then (6.2.3/1)must be valid. But Theorem 6.2.3 shows that this is equivalent t o (6.2.3/6). Hence, for a n L,-theory (or L,-theory), (6.2.3/6) is a natural additional assumption. 6.4.3.
Domains of Definition of Fractional Powers, Isomorphic Mappings
I n the following considerations, we shall assume that there exists a number a 2 0 such that e-"(x) E L,(Q) (see Remark 6.4.2/2). T h e o r e m . Let A be the operator of Theorem 6.4.1. Further, let a 2 0 such that
e-"(x) E Ll(Q). (a) For s 2 0 it holds
*(a;
D(A8) = 1 ; e2'"). (b) For s 2 0, the differential expression -Au from @ + a @ ; I ; e 2 8 v + 2 ) onto W,88(Q;1 ; e2'"). (c) It holds
(1)
+ e'(x)u i s an isomorphic
(2) mapping
OD
D(A") =
( D l ( A j ) = AS'~(~)(Q)
j=O
(3)
(set-theoretically and topologically). The differential expression -Au + eV(x)uyields an isowwrphic mapping from Se,, (Q) onto itself. Proof. Step 1 . Let j = 0, 1, 2 , . . . It follows by Theorem 3.2.4/1, Lemma6.2.2, and Theorem 6.3.2 that IIuIID(zj) llull wijm 1; P"") for u E W,2J(Q;1 ; @jr)c ~(.li). (4)
-
27
Triebel, Interpolation
418
6.5. Lp-Theory
On the other hand, one obtains with the aid of (1) t h a t S@(,)(Q) c W,2j(S;1 ; e 2 J V ) .
(5)
But by Theorem 6.4.2, there exists a subset of A ! ~ ~ ( ~namely ) ( Q ) , the set of all finite linear combinations of eigenfunctions, which is dense in D ( A j ) . Then, ( 2 ) with s = j = 0 , 1 , 2 , . . . is a consequence of (4). Step 2. Now one obtains (2), for arbitrary values s 2 0, from Theorem 1.18.10 and from (3.4.218).(It holds HE = W2 .) Since A is positive-definite, whence it follows also the part (b) of the theorem. Step 3. Formula ( 5 ) yields (set-theoretically and topologically)
D(A")'
'Q(.Z)(')
(6)
Now, let u E D(2"). Then Theorem 3.2.413 yields e'DYu E La@)for arbitrary numbers 01 5 0 and arbitrary multi-indices y. Now one obtains in the same manner its in the fourth step of the proof of Theorem 6.4.2 the conversion t o (6). This proves (3). Clearly, -Au + @ ( x )u gives an isomorphic mapping from D(A") = SQ(r)(Q) onto itself.
6.5.
L,-Theory
In this section, anLp-theoryfor operators A belonging to %Ev(Q;e ( x ) )will bedevelop-
ed on the basis of the previous considerations. Differentiability properties and the behaviour near the boundaxy of solutions of the equation Au = f (or Au - Ae"(x)u = f ) will be described as before with the aid of isomorphic mappings between function spaces generated by A (or by A - Ae'(z)).
6.5.1.
A-Priori-Estimates (Generalization of Theorem 6.3.2)
Using the results of Section 6.4, one can generalize Theorem 6.3.2 essentially. Theorem. Let A E 2iEv(Q; e ( x ) ) in the sense of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E L,(Q). Let (T v, 1 < p < 00,
'
0, 1 , 2 , . . . Let z = - > 2. Then there exists a real number c1 m such that for all complex numbers A with Re 1 5 c1 there exist two positive numbers c2 and c3 such that for all u E W:m+2k(Q;px+pJ'; ex+p(v+ks)1
x real, and
k
=
C31JU11w;'"+k'(n;
@ x + P r ; ex+P(v+kr))
2 IIAu - Ae'(x) u l l ~ r e(x ;~q x;+ P * r ) 2 c211uJIw2'"+k'(Q; Q x + P r ; @X+P(Y+kZ)).
Proof. Step 1. We start with preliminaries. Let
Bu
=
(-A
+ @'(X))kU
- '1u, '1 5 0 .
6.6.1. A-Priori-Estimates(Generalizationof Theorem 6.3.2)
419
Theorem 6.4.3 yields that B is a n isomorphic mapping from Se(z)((sz)onto itself. Then it follows from Lemma 6.2.2, Theorem 6.3.2, and Theorem 3.2.411, that B for rj 5 c is a n isomorphic mapping from W;"(S; ex;e x + P T k ) onto LJQ; ex). Step 2. By the first step, it holds for u E C$((S)
IIAu - Ae"(z)ull W F ( Q ; e x ;
ex+pk+)
-
IIB(Au - le"(z)~ ) l l ~ e~x ) .( n ;
(3)
Considering the operators p s ( - A + p y (z))and A , then one has the same situation as in the proof of Lemma 6.2.2. Whence it follows that V-P
(-A
+ @')A = e--
P
=
Iteration yields
BA E 91;;:k'p;
P
ern ( - A
?L,":,',(Sz;
+ em)A v-P
-L
Ee
rn
e(z)).
n1+1
) I l P + ~ , . + ~ ( ( s ze(z)) ; rn
nr
e(4).
Since CT 5 v, the left-hand side of (1) is a consequence of Theorem 6.3.2. Step 3. We prove the right-hand side of (1). Since x is an arbitrary number, one may assume 0 = c 5 Y without loss of generality. If B has the above meaning, then it follows, in the same manner as in the second step, with the aid of the proof of Lemma 6.2.2, that BAu = ABu + 2 afl(z)D8u. (4) 161 < 2m+2k
Here, the last term is a perturbation of BA (or AB) in the sense of (6.2.1/5). Then, from Theorem 6.3.2, the counterpart t o (6.3.2/6),and (3) it follows that
One obtains by the second step that
[(-A
+ e')"
- @I B E q g ; : m + k ) ( Q ; e(4).
Applying Theorem 6.3.2, putting the result in ( 5 ) , and choosing E in a suitable way, then it follows that llAu - ~ u ~ ~ w Fe(x D ; e x;t p k r )
>= c I I J u J ( ~ ~ ( ~ + ~ ) ( Q ;
extwp; extp(v+kr))
- C,IIUIIL,(O; p x t v q .
(7)
(Here we used Y 2 0 and Y > p.) Transfering the last term t o the left-hand side and estimating it with the aid of Theorem 6.3.2, then one obtains the right-hand side of (1). 27*
420
6.5. Lp-Theory
R e m a r k 1. The special choice t
'
=- makes
the proof easier. But probably m the theorem remains also valid if one assumes only t > 2. R e m a r k 2. The proof is based on Theorem 6.3.2, and it uses the special operator -A + e"(x). This shows that one may carry over the important Remark 6.3.2/1: There exist a real number c1 and a positive number c2 depending only on the constant of ellipticity C from (6.2.1/6), the constants of estimates for DYb,(x) and Dyas(x),and the odehaviour of Dyas(x)in the sense of (6.2.117) (and Q, e(z), p , x , k,and a), such that for all complex numbers 1 with Re 1 5 c1 and all u E ~ ~ m + ' c ) ( Qe x;+ P p ; ex+P(v+kr) )
l!Au - A@"(%) u l l w ~ ( n ; ex;
QX+Pk)
2 C211UIIw2(m+.t)(n; e x t p r ; eu+P("+kr)).
(8)
Here, c1 is also independent of k. This improvement w i l l be very useful, later on.
6.6.2.
Isomorphism Theorems
In this subsection there are proved some of the main results of this chapter. Theorem 1. Let A E UEv(Q; e(x)) in the sense of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E Ll(Q).Let 0 < v, 1 < p < co, 1 5 q 5 03, and let x be a real number. Further, let c1 5 0 be a real number in the sense of Theorem 6.5.1 and Remark 6.5.112. (a) If Re 1 5 cl, then A - 1e"(x)gives an isomorphic mapping from tSp(x)(Q)onto S d X ) (Q). (b) If Re 1 5 cl, and if s
2 0 , then A - ?&x)
H ; m + S ( Q ; @ x + p P ; ex+pv+sp
( c )If Re 1
5
cl,
and if
s
'
P r o o f . Step 1 . Let
+ e"(X))"
BU = ( - A
s)onto
H;(Q;
@x+sp
s).
> 0, then A - le'(x) gives an isomorphic mapping from
BS+zm(Q.ex+PP ; @X+PV+sP P.P
gives an isomorphic mapping from
5) onto
B;,,(Q; @*;@ X + s P
s)
u - 1@(x)u = B0u - I@(x) U ,
,
(1)
where t > 2, rj < t m , and 1 < 0. We want to show that B gives an isomorphic mapping from Secz) (Q) onto itself. By Theorem 6.4.3, B,, where D(Bo)= Gm(SZ; 1 ;e Z r m ) ,is self-adjoint and positive-definite. Further, it holds that
D(B:)
=
W,"mk(Q;1 ; eZrkm),k = 1 , 2 , . . .
By the previously developed technique of estimates it follows that
Bku = B ~ u+ Du, IIDUIlr,,(n) 5
EIIuII~?'~Q;
1; e * r h )
+ c b ) Ilull~,~~)
5 E'IIB~UllL,(Q) + C'(&')llUll&(Q).
6.5.2. Isomorphism Theoreme
42 1
Here E > 0 (resp. E' > 0) is a given number. Now, the criterion of self-adjointnessof T. KATO(see H. TRIEBEL[17], p. 209) yields that Bk, where D(BIC)= D(B$),is a self-adjoint positive-definite operator. By Theorem 3.2.411, CF(Q) is dense in D(Bk). Hence, Bk, D(Bk) = Wirnk(Q; 1 ; elrkm),is the k-th power of the positive-definite Now, (6.4.3/3) yields that B self-adjoint operator B , with D ( B ) = Wim(Q;1 ; eZrrn). gives an isomorphic mapping from Spcz. (Q) onto itself. Step 2. Let A be the operator of the theorem, and let B be the operator of (1) V - P where t = n2
and 7 = c - p. We set for 0
a
5
1
Aau = &(A - A@(x))u + (1 - a)@(z) ( B , - Re A * @(x)) u = aAu + (1 - a) @(z)B,u - (an + (1 - a ) Re A) @(z) u .
(2)
+
Since Re (an (1 - a)Re A ) = Re A, it follows by Theorem 6.5.1 and Remark 6.5.1/1 that there exists a number c1 independent of a and k such that (6.5.1/2) holds with IXA (1 - a)e"(z) B, instead of A . Remark 6.5.1/2 yields that c, is independent of a. Assuming that for a given number 0 5 a, c 1 the operator Amogives an isomorphic mapping from
+
R 1 --
J+'2(m+jC)(Q; @ + P P ; e x + P ( v + k r ) )
then it follows that
P
onto R, = W F ( Q ; e x ;
f+Pk),
A,u = f E R, is equivalent to ~1
It holds that
+ A,f(Aa - Am,)u = A,:/
E R,
(4)
IlA;f(Am - ~ m , ~ l l I ~ 5 l +1 I01 ~~ O1ol c < 1 for
la - a01 < C 1
Here, c is independent of 01 and a,. Now one obtains that (4) has a unique solution. For these values of a, there exists A i l . The first step and Theorem 6.5.1 yield that A,' exists in the above sense. Now it follows, by iterated application of the above procedure, that A - Ap'(x) is an isomorphic mapping from R, onto R,.For x = 0 and p = 2, one obtains from (6.4.3/3) that A - A@(x) gives an isomorphic map(Q) onto itself. ping from Step 3. Part (b) and part (c) of tthe theorem are consequences of the second step and of Theorem 3.4.2. Remark 1. As a special case the theorem contains the assertion that A - &"(z) gives an isomorphic mapping from wgrnts
(Q; @ x + p p ; e x + P v + s P q
onto
W> (Q; p; p
~ + ~ p s ) .
For 0 = 0 this result was proved in H. TRIEBEL[24]. R e m a r k 2. The above theorem is comparatively general. We describe two simple conclusions.
422
6.F. Distributions of Eigenvalues, h c i a t e d Eigenvectors
c((Q)
-
1. Let Q c R , ,be a bounded domain, has the meaning of Lemma 6.2.3, and e - l ( x ) d(z) is the smoothed distance function in the sense of Remark 3.2.3/1. Then it follows from the above theorem and from Lemma 6.2.3 that Au + ( u + e"(x)u, v > 2m,
c$
is an isomorphic m p p i n g from (Q)onto itself. (Here one has t o use that A is positivedefinite in L2(Q).)The same holds for @(x) ( u + @"x) u where v - ,u > 2m. 2. If Q = R,,, then Remark 6.2.113 yields that
AU
=
(1
+ lxl2)V1(-d)l~lu + (1 +
is an isonzorphic mapping from
I~(')qzU,
2
> q17
S(R,,) onto itself.
T h e o r e m 2. Let A E 21Ev(Q; e(x))i n the seme of Definition 6.2.112. Further, it i s assumed that there exists a number a 2 0 such that e-"(x) E Ll(Q).Let x be a real number. Then A , considered as a mapping from W;m(Q;e x + P C ; e x + P v ) into Lp(Q:en), is a @-operator with the index O.*) P r o o f . If CT < v and if c1 has the meaning of Theorem 1, then A , considered as a mapping from W?(Q, e X + P P ; QX+P") into &(Q: e x ) , is a @-operator if and only if A ( A - c1@'(x))-l, considered as a mapping from L P @ ):'Q into itself, is a @-operator. This is a consequence of Theorem 1. Both the operators have the same index. It holds that A ( A - CleU(x))-l= E
+ cleu(x)( A - Cl@(Z))-l.
(5)
In the same manner as in the third step of the proof of Theorem 6.4.1, it follows that ZL + @(x) u is a compact mapping from W?(Qn;
@X + P P ;
ex+"")
into Lp(Q;ex).
Then, ~ " ( x( )A - cle"(x))-l is compact operator in L,(Q; ex).Now, (5) yields that A ( A - cleU(x))-l is a @-operator with the index 0. R e m a r k 3. The theorem is the counterpart to Theorem 5.2.2(b).
6.6.
Distributions of Eigenvalues, Associated Eigenvectors, and Green Functions
This section is the counterpart t o Section 5.4. The investigations on distributions of eigenvalues are of interest, later on, in the framework of the structure theory for the spaces h z )
can,.
*) The index of a @-operatoris defined as the difference between the finite codimensionof the range and the finite dimension of the null apace.
6.6.1. Distributions of Eigenvalues
423
Distributions of Eigenvaliies and Domains of Definition of Fractional Powers
6.6.1.
The used symbols have the same meaning as in 5.4.1. T h e o r e m 1. Let the differential expression A belonging to the class '%Ev(!2;e ( x ) ) (Definition 6.2.1/2) be formally self-adjoint. Let v > 0. Further, it is assumed that there exists a number a 2 0 such that e-"(x) E L,(Q).Then A ,
D ( A ) = W$"(O;$"; $'), (1) is a self-adjoint operator, bounded from below, with pure point spectrum in L,(sZ).There exist two positive numbers c1 and c2 such that inin (p,0). I f s 2 0, then D(A") = Wp"(f2; pa""; eaSv), (3) provided that A , without loss of generality, is positivedefinite. P r o o f . Step 1 . Theorem 3.2.4/1 yields that C,"(!2) is dense in D ( A ) . Hence, A is a symmetric operator. Now it follows from Theorem 6.5.211 that A is a self-adjoint operator. (For appropriate A, there holds R ( A - ilE) = L,(Q)).Further, one obtains from Theorem 6.5.211 that (3) is valid for s = I = 0, 1, 2, . . . For general values s 2 0 , (3) follows by interpolation from Theorem 1.18.10 and (3.4.2/8). (It holds Wg = H g . ) One obtains in the same manner as in the third step of the proof of Theorem 6.4.1 that the embedding from D ( A )into L2(f2)is compact. (Here, one uses the assumption v > 0.) By the theorem of F. RELLICH(see for instance H. TRIEBEL [17], p. 277), A is an operator with pure point spectrum. Step 2 . We prove the left-hand side of (2). Let K , and K , be two open balls such that I7,c K , and If, c 0.Let S be an extension operator from W,2"(K1)into fki'"(K2)c D ( A ) in the sense of Theorem 4.2.2. (The extension operator of Theorem 4.2.2 is multiplied with ~ ( xE)C$(K2)where ~ ( x=) 1 in a neighbourhood of K , . The functions are extended by zero outside of K 2 . ) Let R be the restriction resp. from L,@) onto L,(K,). Denoting embedding operator from D ( A )onto Wim(K1), operators by I , then Here ,C
=
I W ~ m ( I \ l ) - t L t ( K ,= )
RID(A)+L2(i$.
Theorem 1.16.1/1 yields sI ( I w, 2mwI)-tf2wI)) 6 cs,(IwwLm))-
The left-hand side of ( 2 ) is proved by Theorem 3.8.1 and Theorem 5.4.1/1 (and (5.4.1/5) and (5.4.1/6)). Step 3. We prove the right-hand side of (2). Let ilbe a (sufficiently large) positive
+
number, and let j A = [log, A;] 1. If Q ( Jhas ) the same meaning as in Definition 3.2.3/1, then Q ( j l ) will be covered by cubes Ql of the side-length d2-jA which are parallel to the axes. By (3.2.3/7), one can choose d > 0 such that
f2Wc
(J Q1 c Q ( J A + ~ )
(4)
424
6.6. Distributions of Eigenvalues, Associated Eigenvectors
(ais independent of in).It holds that Using (3.2.3/7), then it follows that one needs for the above covering a t most a -+-
a
LA 5 c p 2.iA'L I - c21
n
(6)
cubes. It holds that
In the sense of Remark 5.4.113the self-adjoint operators belonging t o the ECilbert spaces Wim(Q;ew;e2') (with respect t o L,(Q)), Wim(Q - U Q 1 ; e2"; e2") (with respect t o L,(Q - U QJ), and f l m ( Q 1 ear, ; eaU)(with respect t o L2(Qz)) are denoted by A,, , A , , and A z, respectively. It holds t h a t j5J-Q) = Lz(Q -
UQd
LA
CB
C @ L2(Qd
>
Here, E is a sufficiently small positive number. Hence N A , ( A )=0. Setting ,ii =min(p,O), one obtains that 2
I I ~ wzm(Q1; II
q2p; e z v )
2
2
2ii -
2
c ~ ~ ~ ~ " ~ ~ I I ~ I~ I ' 'YL1 ~I I~~ (I I Q ~ V~ ~)~ (2 Q , )
> 0, cr > 0. If B is the operator belonging t o the quadratic form IIUll'$im(p,) respect to L,(QJ),then, using (6) and Theorem 5.4.1/2, it follows that
c
(with
If Q is the unit cube and if D is the operator belonging t o the quadratic form I I u ~ ~ $ ; ~ ( Q ) , then NB(7) 5 cND(7).
(9)
This is a consequence of the transformation of coordinates mapping Q1 onto Q, and a comparison of the corresponding quadratic forms. Theorem 5.4.111 and Theorem 3.8.1 yield N&)
II
5 ~ 7 % .Then, it follows from
(8) and
(I))that
6.6.2. Associated Eigenvectors
425
Remark 1. The estimates of the third step can be improved. On the other hand, formula (2) shows clearly the influence of the different parameters, in particular of a and v. An asymptotic formula cannot be expected under these general assumptions. Theorem 2. Let Q c R, be a bounded C”-domain. Let e ( x )be a function in the sense of Definition 6.2.111, where e-l(x) d ( x ) near the boundary. Here d ( x ) denotes the distance of a p i n t x E Q to the boundary. Then it holds for the operator A from Theorem 1, where p > -2m, that n-1 2m if - 2 m < p < - - , n N
2m
n
1% log I
if p = if
-
/A>--,
7
2
9
c > 0.
(10)
2m n
Proof. The theorem is a consequence of ( l ) , Theorem 3.8.2, Theorem 5.4.1/2, Theorem 5.4.1/1, (5.4.1/5), and (5.4.1/6). Remark 2. The theorems show that N ( 1 )
2m
n
N
2% holds, provided that p > - -.
n See Theorem 5.4.2. The differential operators of Theorem 2 are closely related to a special class of Tricomi differential operators. We shall return to this question in Remark 7.8.312.
6.6.2.
Bssociated Eigenvectors
The used notations have the same meaning as in Subsection 5.4.1. Theorem. Let A E 2?Ly(Q; e(x)) in the sense of Definition 6.2.1/2. Further let v > 0 , 1 < p < C Q , and e-a(x) E L,(Q) for an appropriate number a 2 0. Then A with the domain of definition
D ( A ) = W;m(s2;e p r ; e P ” ) is a closed operator in Lp(Q).Its spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The eigenvalues and the associated eigenvectors are independent of p . The associated eigenvectors are elements of Secx.(Q), their linear hull is dense in ~Y~(~)(s2). Further, the linear hull of the associated eigenvectors is dense in all the spaces Wi(Q; ex;e’) where 0 s < co, 1 < q < CQ, - m < x + sq z c m (hence, it is also dense in Lq(Q)). Proof. Step 1. Let p = 2. We set,
s
1
-4u = 2
-
C C z=o Ial=21
b,(x) PU+ P [ ( e x 2 1 ( x )b,(x) u)] + BU
111
[ex21(s)
AU
+ Bu.
(1)
426
6.6. Distributions of Eigenvalues, Associated Eigenvectors
It follows from the proof of Lemma 6.2.2, that A belongs also to kFy(Q;e(z)) and that B is a perturbation operator, where its coefficients have the property (6.2.1/8). A is formally self-adjoint. Hence, it follows by Theorem 6.6.1/1 that A , where D ( A ) = WB,m(Q; p ;p ) , isself-adjointinL,(Q). Let k A where
D(A)=
= 0 , 1 , 2 , . . . Now,
oneobtains by Theorem 6.6.1/1 that
@ z ( k + l ) r ;@Z(k+l)V)
(2) is a self-adjoint operator with pure point spectrum in the Hilbert space H = Wikm(Q;ezkr;eWv)(after introduction of a suitable norm). To apply Theorem 5.4.113 to the operator A with the domain of definition (2), we use the decomposition (1). It holds that w;(h-+l)m(Q;
IlBuIlfI = I ( B U I I M ~ , ~ ~S~ c; ~ *~ 2~/ i~c ; , Z ~ JW@“18’-6(5) ) rll+2nr-1
1
p u 1 2
dz)T,
R
(3)
= Ej
m
V + -(2km + 2m - j ) . m
(4)
Here 6 > 0 is a suitable number. The last formula is a consequence of the technique of estimates developed in the proof of Lemma 6.2.2, in particular (6.2.Z.lZ) and (6.2214). (Here one has t o take into consideration that A belongs not only t o %Ev(Q; e(z)), but also t o $&(Q; e(z)).) Setting 6 = Zvs, then
Without loss of generality let 0 < orem 3.2.413 that
1 s < -. 2m
IIBuIJUS cllull w 22( B t l - l ) m ( n ; e r ( k + i - a ) r :
If I is the embedding operator from w;(k+l)m(Q;
$(k+l)~
;
e2(’”+1’”)
into
+
(5)
Then it follows by (3), (5), and The(6)
ez(~+i-a)v).
J,j7?4(’”tI-*)m(Q;
@Z(ktl-.)p.
ea(rtl-a)v
),
and if ilis a complex number with I m il 0, then B ( A - AE)-l considered as an operator from H into H can be represented a s B(A - ilE)-l = B I ( A - ilE)-l. (7) Here, B on the right-hand side is a bounded operator in the sense of (6), while (d - U3-l is a bounded operator acting from H into D(A) in the sense of (2). Theorem 6.6.1/1 and the proof of Theorem 5.4.111 yield that I from the right-hand side of (7) belongs t o Gr where r is a suitable number, 1 < T < 01). Then it follows from Theorem 5.4.113 that A , with the domain of definition (a), is a closed operator in H where the spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The linear hull of the associated eigenvectors is dense in H .
6.6.3. Green Functions
427
Step 2. Let again p = 2. It follows from the first step that the linear hull of the associattecl eigenvectors of the operator A , where D(A) = qm(5 ear; 2; e”), is dense in the Hilbert space L,(Q).Theorem 6.5.211 and Theorem 3.2.413 yield
D(A”) =
m
n D ( A ~=)
j -0
n 00
p i ;p
~zjm(52;
j-0
j ) =
S~(~)(L?).
(8)
(See t,he third step of the proof of Theorem 6.4.3 and the fourth step of the proof of Theorem 6.4.2.)Here, the last equality must be understood not only set-theoretically, (52) but also topologically. Hence, the associated eigenvectors are elements of SQ(=) = D ( A m ) .Now one obtains that these associated eigenvectors coincide with the associated eigenvectors in Ekm(Q; e 2 k p ; ew’). Together with (8) whence it follows that the linear hull of the associated eigenvectors of the operator A is dense in (52). S t e p 3 . L e t O S s c 0 0 , lc q c oo,and-oo < x + s q s z < c o . T h e n w e h a v e fJ&)
(52) = W p - 2 ; ex;e’) .
ByTheorem 3.2.4/1, C$(52) (and hence also SQcx)(52)) is dense in W$2; e x ; e‘). Now one obt#ainsfrom the second step that the linear hull of the associated eigenvectors of d is dense in W @ ; ex; e‘). Here, in the sense of the theorem all values 1 < p < 00 are admissible. This follows from (8),after replacing there 2 by p , and the fact that ( A - iZE)-l, where D ( A ) = WEm(Q;e p w ; el’”) for suit,able values of ?, is a compact operator in L,(Q). (See Theorem 6.3.2 and Theorem 6.9.1.) R e m a r k . The theorem was formulated in H. TRIEBEL[25] without proof. A variat,ion of the theorem was proved by H . KRETSCHMER [l]. The proof of the theorem is also based on the criterion of I. C. GOCHBERC, M. G. KREJN by H. KRETSCHMER from Theorem 5.4.1/3. 6.6.3.
Green Functions
The methods developed in Subsection 5.4.4 are valid for general operators in Hilbert spaces. In this subsection they are applied t o operators belonging t o %K,(SZ; e(z)). Let, v > 0 and let E L,(SZ) for a suitable number a 2 0. By Theorem 6.5.2/2, ,@p; e2”), is a @-operator every operator A E %K,(SZ; e(z)), where D(A) = V,””(SZ; in L,(Q). With the aid of Theorem 6.5.211, it follows that A* is the formally adjoint operator to A with the domain of definition D(A*) = D(A).In particular, A* is also a @-operator in L,(Q).Now, one can apply the considerationsin front of Theorem 5.4.4 and introduce the operator A,
Au = Au, D(A) = D ( A )n B(A*). (1) A generates an isomorphic mapping from D( A) onto R(A).I n this sense we construct the operator A-l. T h e o r e m 1. Let A E fA;,(SZ; e(z)) (Definition 6.2.1/2), v > 0, and e-“(z) E L,(Q) for a suitable number a 2 0. Further, one sets in t h seme of Theorem 6.6.1/1
7
u = -V a + n + ( v -
a&j.
428
If
6
6.6. Distributions of Eigenvalues, Associated Eigenvectors
< 2 , then k1 can be represented in the form
(A-Y)(4=
1 G ( x ,Y)f ( Y )d?/,
(3)
n
6
G(x, y ) E WT(D; Q’”; Q’”) $ L 2 ( 0 )A L2(D) K”(0;e’”; e”), (4) where 0 t < 2 - cr. Proof. The first step and the beginning of the second step of the proof of Theorem 5.4.4 may be carried over without any changes. The operator A*A is selfadjoint , D ( A * A ) = Wim(D;e4P; e4’).
In the sense of (5.4.4/6), its positive eigenvalues are denoted by 2.. The counterpart to (5.4.4/9) is
Applying Theorem 6.6.1/1 to A*A, and putting il = il; in (6.6.1/2), then one obtains j 5 c(A; + 1). Whence it follows that (5) converges if N + co. By Theorem 6.6.1/1, it holds that
D ((A*A)$)= WGm(D; Q’”; e ” ) . This proves the theorem. (See the second step of the proof of Theorem 5.4.4.) Theorem 2. Let D c R,,be a bounded C”-domain. Let e(x) be a function in the sewe of Definition 6.2.1/1 where Q-l(x) d ( x ) near the boundary. Here, d ( x ) i s the distance of a p i n t x E I2 to the boundary. If A is the operator from Theorem 1 where p > -2m, then A-l can be represented in the form (3),(a), provided that n - 1 2m n 1 0 t m < 2m - m if - 2 m < p < - and 2 m - - > -p---, 2m. p n 2 2 N
+
2m
n 2
5 zm < 2m - -
n
and 2m - - > 0 . if p z - - n 2 Proof. By the proof of Theorem 1, one has to show that (5) converges if N -P 03. Applying Theorem 6.6.112 to the operator A*A with the eigenvalues A;, then one obtains that
0
i .:’ 1 -
j s c + c
AIKlogAj
if if
-2m 0. The assumptions a.reconstructed in such a way that the second term in (3) is a perturbation of A m , k I.f one replaces ( 5 )by
+
+j(X)
= 0(@-2"+J(x))
.,
(7)
then the second term in (3) is no longer only a perturbation of Am,k.In that case the behaviour of Bnl,kis changed essentially. R e m a r k 2 . * The differential operators A",,,;are introduced by H. TRIEBEL [la, I]. For m = k = 1 and p ( x ) = ( x - a ) (b - x ) , one obtains the cla,ssical Legendre differential expression. Here, A m,k and Bm,kare called (generalized) Legendre differential operators. Differential operators of type (3) and ( 7 ) in R , , (0, a), and (a,b) (as well as generalizations) are considered by N. SRIMAKURA [l,4,5] and P. BOLLEY, J. CAMUS[l,3 , 4 , 51.
7.2.2.
Trieomi Differential Operators
If Q c R,,is a bounded C"-domain, then a neighbourhood of the boundary inside of Q, denoted by S, can be represented in the form S = aQ x (0, h ) , 0 < h sufficiently small.
432
7.2. Definitions
Choosing suitable open balls K j such that
N
U K i =I 8,then one can equip K j n S
j=l
with C"-coordinates ( y l J ) ., . ., y!/c1,y:)) = (y")', y$)), where yIL= y g ) is taken in the direction of the inner normal, and y i J ) ,. . ., ~ $ are 1 local ~ C"-coordinates on ,352 n K j . The above representation of S is to be understood in the sense of these coordinates. One sets
Let d ( x ) be the distance of a point x E SZ to the boundary 352. We assume that y, = d ( x ) holds for x E S . Definition 1. Let 52 c R,,be a bounded C w - d m i n . Further, let m = 1 , 2 , . . ., and k = 0,1, . . ., 2 m - 1. Then, the differential operator Bm,k,
is called the Tricomi operator
first type provided that: (a) The coefficients bu(x) are real for la1 = 2m, and there exists a function c(x) > 0 such that for all x E SZ and all 5 E R, (-1P
c
la1 = 2m
of
b,(4
E" 2 c ( 4 I5l2"
(ellipticity condition). (b) In Kj n S the differential expresswn Bm,ku can be represented as
+ IYId
2m
cy(y(j)')Dc(j),vj
+
C
ldls2m-1
ad@(')) @ y i *
(3)
Here, u(x) = wi(y(j))in K j n S. Further, hj = hj(y(j)')2 c > 0 is a Crn-function in K j n S , independent of y,, and a(t) is an infinitely differentiable positive function in [0,h]. Further y = ( y l , . . ., Y , - ~ ,0 ) , and the s e d term in (3) is a regular etliptic positive-definite differential operator of order 2 m in the local coordinates y(J)' with the real-valued C"-coefficients cy d e f h d o n 852. The coefficients dd(y"') belong to Cm(KjA 8).Setting 6 = (al,. . ., d,-l, 6J = ( 8 ,a,), then it is assumed that
(0must be understood in the sense of y, 10). (c) Let
D(Bm,k)= C"(Q
if
k = m, m
D(&,k) = ( u I u E C w ( S ; ) ) ; -
b
a'u n aY;
+ 1 , . . ., 2m - 1 , =0
if
1
=
if
(54
0 , . ..,m - k - 1
k=0,1,
..., m - 1 .
7.2.2. Tricomi Differential Operators
433
R e m a r k 1. (2) is the counterpart t o (4.9.1/2).(3) yields that c(x) tends to zero if x + aQ, provided k > 0. Since the coefficients bJx) are continuous, one may assume y," = d k ( x ) .(5) is similar to (7.2.1/4).The that c ( x )is also continuous. (3)yields C(Z) assumption (4) ensures that the third term on the right-hand side of (3) may be considered as a perturbation of the first and the second term. (See (7.2.1/5).)The is not essential. Namely, if one dependence of the first term in (3) on y i J ) ,. . . , goes over from y i J ) ,. . . , y$!ll t o a new system of Cm-coordinates # ) , . . . , ijgL1 in h. h". aQ n I[,;, then one obtains 2 = 4 , where hi = hj(#), . . . , is a suitable
-
-
Sj
-
Sj
function and gj is the Jacobian. D e f i n i t i o n 2 . Let Q c R,, be a bounded C"-domain, and let a(x) E ~ ~ ( beQ a) positive function such that ~ ( x= ) d(x) near the boundary. Further, let {aj,k(x)}~;;,:+ be a positive-definite symmetric matrix having real coefficients aj,k(x)E c"(Q)such that
c ai,k(x) n
j,k =1
6j6k
2
c1t12,
(6)
where c > 0 is a suitable number independent of x E Q and 5 E R,, . Then the differential operator A , AU = - aj,k(x)u(z), (7)
c
)
axk axj a ( D ( A ) = C"(Q), (8) is said to be a Tricomi operator of second type. R e m a r k 2. The last definition can be extended in an eaay way t o differential operators of order 2m. While the Tricomi operators of first type degenerate only in direction of the normal, (7) shows that the Tricomi operators of second type degenerate uniformly in all directions of the coordinates. C. GOULAOUIC R e m a r k 3. * Definition 2 is due essentially to M. S. BAOUENDI, [2,31. These operators are generalized by several authors. The generalizations deal with more general weight functions, with other types of domains and with differential C. GOULAOUIC, B. HANOUZET operat,ors of higher order. We refer t o M. S. BAOUENDI, [l], S. BENACHOUR [l], P. BOLLEY,J. CAMUS[2,3, 61, M. DERRIDJ,C. Z ~ L [l, Y 21, B. HAXOUZET [l], N. SHIMAKURA [l, 4, 51, and C. ZUILY [l,21. Special cases of Definition 1 are due to H. TRIEBEL[7, 111. See also P. BOLLEY,J. CAMUS [2], and M. S. BAOUENDI, C. GOULAOUIC [S]. Special Tricomi differential operators are used [2, 31 for the explicit construction of orthonormal by M. GUILLEMOT-TEISSIER bases in L2(Kn)where K,, denotes the unit ball in R,,. Degenerate elliptic differential equations which have a behaviour as the Eulerian differential equation near the [l,2,3]. boundary are considered by A. V. FURSIKOV j , k *= l
7.3.
Inequalities, Equivalent Norms, and Isomorphic Mappings
This section is concerned with some preliminaries. There are proved a number of important integral inequalities for smooth functions defined on the interval (a, b ) , and there are described equivalent norms for the spaces WF((a,b ) , d"(s)). Further, 28
Triebel, Interpolation
434
7.3. Inequalitiee, Equivalent Norms, and Isomorphic Mappings
there are considered isomorphic mappings between these spaces. This section is the basis for the investigations on Legendre differential operators. May be some results are also of self-contained interest.
Integral Inequalities [Part I]
7.3.1.
Lemma 1. Let - oc) < a < b
6 > 0 such that 03,
J ~ I ( ( Y V U dx ) ( "+~ )1~ lu12 ~ dx 6
b-8
a+8
a
- 1p+21~(m))2 +1 b-8
b
)uI2d x ,
dx
(1)
a+8
fl
provided that one of the two following conditions is satisfied: (a) -1 < x 9 2m - 1, u(x)E C m ( ( a ,b ) ) ,
x+3
.
(b) x < -1, u ( z ) E Cm((a,b ) ) , u ( x ) = o ( p n L - T ( x ) ) ("-" means that the right-hand side can be estimated from above and from below with the aid of positive constants by the left-hand side).
Proof.Step1. Westartwithpreliminaries. Let u ( x ) ~ c ~ (b)) ( afor , the casex > -1, and let u(x)E Cg((a,6 ) ) for the case x < -1. Further, let qa(x)E em((a, b ) ) , with 0 5 q a ( x ) 5 1, be a function identical 1 in a right-side neighbourhood of a and . a suitable choice vanishing in a left-side neighbourhood of b. Similarly ~ ) b ( x )After of va and 96,it follows from Remark 3.2.611 that b
J pxp1u12 a x a
s (1 + I 4(1 - (x
6
+ E2)
+ lp'(b)1x+2Iu9bl2 (b - x)") + +1 luI2dz b-8,
(Ip'(a)1x+2 IupaI2 (5 - a).
E1)J a
1 b
+ 1)2 a
+ i(uq6)~i2) px+2dx + c
(i(uqa)'i2
using I(U%)'I2
1
C a
8:
b-8,
wdx.
a+4
s (1 +
Es)
b'I29: + 4 8 3 ) luI2(v:)2,
then one obt'ains 6
b
6-6.
Here E~ > 0 are given numbers (one has to choose qa and qbin an appropriate way). The constants c and 6j > 0 depend on & k .
7.3.1. Integral Inequalities [Part I]
435
Step 2. Let again u E em((a,6)) for x > - 1 and u E C$( (a,b ) ) for x < - 1 . Then we have m ( p )= ~ [I)u(~) + mpfu(m-l)]+ C cjp(i)t@-i) (3) j=f
=
+ D2u.
D,u
One obtains by partial integration that b
b
J- p”lD1u)2a x = j a
+
Cl)“+yu(”)12 (m2
- m(x + 1 ) ) pp’21u(R’-l)l2- mp+1pf’lu(m-1)12]a x .
a
Now we can apply ( 2 ) with u ( m - 1 ) instead of u to the middle term on the right-hand side. Using d
J- ~ u ( q 2 a 5 x E
d
j
d
IU(m)12dx + C ( E ) J- lu12dx
for j = 0, 1 , . . . , m
(4)
C
C
C
- 1,
b
b
b-8
J- px+11p‘f1- IU(n-1)12d x 6 & j p+21u(”’)12ax + J and
lul2 a x ,
a+8
a
a
then it follows with a suitable choice of 6 > 0 that b
b
b-6
j plD1u12a x + JU
lul2
ax
a+8
b-6
J- px+21U(m)12a x + J-
N
lu12
ax.
(5)
a+6
a
Using the estimates h
b
h
h
b
b-8
J- $Pp2uyax 5 E‘ J px+2Iu(m)12a x + C ’ ( E f ) Ja
a
lu12
ax,
a+8
where E > 0 and E’ > 0 are arbitrary numbers, then (1) is a consequence of (4). xt3 Step 3. Let x < - 1 , and let u ( z )E C”((a,b ) ) such that u ( x ) = o ( p ” - T ( s ) ) . Whence it follows that u(z) = o(p“-
[TI+l).
Let x ~ ( xE) @‘((a, b ) ) , where x l ( x ) = 1 for x E (a + 21, b - 2A), xA(x)= 0 for x E (a,a 1)u (b - 1,b ) , and Ix$ (x)l c1-j. Here 1 > 0 is sufficiently small. and considering 110,then one obtains ( 1 ) for u. Applying ( 1 ) to u x ~
+
28*
436
7.3. Inequalities, Equivalent Norms, and Isomorphic Mappings
L e m m a 2. Let -00 < a < b < 00, j = 1 , 2 , . . ., and 1 = 0, 1 , 2 , . . . Further, let p(x) be the function from Definition 7.2.1. Then
are eqztivalent norms on (?((a, b ) ) . P r o of. Repeated application of Lemma 1 yields that every norm of (6) is equivalent to
(;
[p2yu(J)12+
lu12]dz
>+
.
L e m m a 3. Let -00 < a < b < 00, m = 1 , 2 , . . ., and Further, let p ( x ) be the function of Definition 7.2.1. Then b
j
palzL(m)12 dz
+
U
b-6
1
lUl2 dz
b
J’
(palu(m)12
a
ai-6
-00
-1, oc > -1, and u ( x ) E C”((a,b ) ) !
+
a+l
-1, -3, -5, . . . , u ( z )E Cm((a,b ) ) , u(z)= o ( p m - T (z)). (b) P r o o f . Step 1.Let u(z)E em(@, b ) ) for p > - 1 and u(z)E C$((a,b ) ) for /? < -1. Remark 3.2.611 yields in the previous manner that b
J
#1U(2dX
b
\
5c
a
b-6
flyU‘12dx
+c J
lu12dz.
a+d
a
Using (4), then, in the case (a), iteration of (8) gives (7). Step 2. We consider the case (b). Again, it follows by iteration that (7) holds for oc+1 u E CF((a,b ) ) . One has m 2 - -. Now one obtains (7) for arbitrary 2 2 functions in the sense of (b) with the aid of the same limit process as in the third step of the proof of Lemma 1. R e m a r k . It is easy t o see that (7) is true, too, for 9, = -1, -3, -5, . . .,
’
+
~
-
00
< oc < 2m
+ p, and u(x)E C”((a, b ) ) , u(z)= o ( p m - T ( z ) ) . a+l
L e m m a 4. Let -00 < a < b < 00, m = 1 , 2 , . . ., and -a < oc < 00. Further, let p ( x ) be the function of Definition 7.2.1. Then there exists a positive number c such that for a2l u E C$((a,b ) ) b
c
J’ luladx 5 a
b
1 palu(m’J2dx
if and only if --co
is satisfied.
(9)
0
< d
0 is an arbitrary number, and from the compactness of the embedding from “?((a + 6, b - 6)) into L2((a 6, b - d)), that the embedding from Fbr((a,b ) , pa) into L2((a,b ) ) is also compact. Assuming that there does not exist any number c > 0 with the property (9),then there exists a sequence uj E C$((u,b)), i = 1,2, . . . , such that
+
b
I I U ~ ( ( ~ , ( , ~ , ~ )=) 1 and J’ palujm)12dz
--f
0 if
(12)
j + m.
U
This sequence is bounded in @?((a, b ) , pa) and hence (without loss of generality) convergent in &((a, b ) ) . Now, it follows from (11)and (12) that {uj)Ti is also convergent in Jkr((a,b), pu). If u E @r((a, b ) , pa) denotes thelimit, then (12) yields that u = Pnl-l is a polynomial of degree m - 1, and that ~ ~ P f , i - l = ~ ~1 holds. ~ 2 ~ ~One a,~~) obtains from Theorem 3.6.1 that Since
-1-
0,.
. ., [m -
[ m - , a- 3+ l - = m - [ - - -o; ci -+]l- l > m -
I:[
Pf!-i(u)= P$t4,(b)= o for I
=
it follows Pll,.-l= 0. This is a contradiction.
“3+
Step 2. Let m 2 2 -
P,&)
= [(z
O L + l
2
.
- -2
1. Theorem 3.6.1 yields that m
- a ) ( b - x ) , l l l - [ T 1-l
belongs t o g g ( ( a ,b ) , pa). Assuming that (9) is true, then (9) must be valid also for this funct,ion. This is a contradiction.
7.3.2.
Properties of the Spaces W r ( ( a ,b ) , p a )
The spaces W r ( ( a ,b ) , pa) and &?((a, b ) , pa) have been described in Definition 3.2.1/4, and the spaces 0 ((a,b ) ) in Definition4.5.1. Here - co < a < b < co, and
438
7.3. Inequalities, Equivalent Norma, and Isomorphic Mappings
p ( z ) has the same meaning as in Definition7.2.1. Theorem 3.2.2 yields that c "( ( a , b)) is dense in (a,b), p'), a 2 0.
m(
s
T h e o r e m 1. (a) Let 0 a < 2m - 1, and let j = m holds in the sense of continuous embedding that (b) If 0
s a < 00, then dim [ W g ((a, b ) , jP)8 @?((a,b ) ,p')] = max
(c) If0
5 LY < 2m, then the embedding from WE((a,b), pu)into L2((a,b)) is compact.
Proof. Step 1. If va has the same meaning as in the first step of the proof of Lemma 7.3.111, then b
u(x)= -
1 (uva)'(y) dy,
z
u E Cm((a,b ) ) , x near a .
(3)
From Holder's inequality and from the iterative application of (7.3.1/2), it follows for 0 5 t < 1 that b
b
1
lu(x)12 5 c1 p"(lu'12 n
+ lu12) dy 6 J p"+2m-2(Iu(m)12+ lul2) dy. c2
a
(4)
Here we used (7.3.1/4). Formula (4) holds for all x E [a,b]. Replacing u by and - j, then one obtains (1). Step 2. Theorem 3.6.l(b) yields that (2) is true for a 2 2m - 1. Taking into consideration m by m
and the fact that Cm((a,b)) is dense in W r ( ( a ,b), p'), then ( 2 )for 0 2 a < 2m - 1 is a consequence of Theorem 3.6.l(a). Step 3. One proves the compactness of the embedding from WF((a,b), p') into L2((a,6)) if 0 5 a < 2m in the same manner as in the first step of the proof of Lemma 7.3.114. T h e o r e m 2. If {Pk.-l)denotes the set of all plylzom~als,the degrees of which are at most k - 1, where k = 1 , 2 , . . ., then one sets
Lbk)((a,a))
=
L,((a, b ) ) 0 {%I],
Lbo)((a, 6)) = L2((a,b ) ) . Further, let m = 1, 2, . . . (a) I n the space Wgm((a,b ) , p2,znb) n @)(,((a, b ) ) , where k = 0, 1, . . ., 2m, and in the s p e Gp((u,b ) , p 2 m ) it holds that
7.3.3. Mappinga in c ( ( a ,a), p a )
439
(b) In the q m e w((a, b), p m )A Lh"')((a,b ) ) it holds t h t
s pmlw("')lzax - llull b
2
W:(ta,b),p)
a
-
(6)
Proof. Step 1. By Lemma 7.3.112, for the proof of ( 5 ) one has to verify that b
1
b lul2
ax
c
a
a
(7)
I(pnlu(k))@m-k)12 ax
if u E Em( (a, b ) , p 2 m ) A hik)(@, b ) ) , resp. if u E @:"'((a, b ) ,p 2 n L ) . Assume that there does not exist a number c > 0 with the property (7).Then there exists a sequence uj E
j
=
~ i m ( ( ab ), , ~ P J , ,A) ~ ~ k ) (b()a) ,resp. , uj E @m((a, b ) , p z m ) ,
1 , 2 , ..., suchthat IIujI)L,((a,b))=
b
1 and
11(pm~(ik))(2m-k)12 dx -+ 0
a
if
i
co.
(8)
Applying Theorem l(c), then one can assume without loss of generality that the sequence {uj}gl converges in &((a, b)). Then (8) yields that { u j > s l converges also in wp((a,b), p2m). If u E ~ i m ( ( ab ,) , p 2 m ) n ~ b k ) ( ( ub, ) ) , resp. u E +:m((u, 61,
gm),
denotes the limit function, then 1 = ~ ~ ~ ~ ~ L , ( k and z,b))
PmU(k)
= P2m-k-1.
+
(9)
(Pi is a polynomial, the degree of which is a t most i, P-, = 0). If P 2 m - k - l 0, then either k k l k 1 pFuck) (x - u ) ~ ,where x 5 m - - - - + - - m = - 2 2 2 2'
-
or a corresponding relation a t the point b holds. Now, iteration of (7.3.118)shows k
that pTu(")E L,((a, b ) ) . This is a contradiction. Hence, one obtains P 2 m - k - l = 0 and ZL = P , i - l . Since either the function u is orthogonal to { P k - , } ,or by Theorem 3.6.1 u ( j ) ( a ) = u c j ) ( b ) = 0 if j = 0, . . .,m - 1, it follows u I 0. This is a contradiction to (9). Whence it follows (5). Step 2. One proves (6) also indirectly, similarly to the first step.
7.3.3.
Mappings in WT((a,b ) , p )
Beside W g ( ( a ,b), pa) and ~ ( ( ub), ,pad),we shall need the spaces Whn((a,b), pa,2)") from Definition 3.2.312 (andTheorem 3.2.412). Here - 00 < a < b < co,m = 1 , 2 , ... and LY - 2m. See Remark 3.2.311. The space Wi((a,b), p , p-l) is of special interest. Theorem 3.2.6 and Remark 3.2.615 show that this space holds a special position.
440
7.3. Inequalities, Equivalent Norms, and Isomorphic Mappings
Theorem. Lp)((a,b ) ) has the same meaning as in Theorem 7.3.212. nr
(a) Cf,,u = p T u(m), m = 1, 2, . . ., gives an isomorphic mapping from and from,
W?( (a,b ) , p q n
W)( (a, b ) )
onto
L,( (a,6 ) )
Wg+I((a,b ) , pnt+l) n L(,“)( (a,b ) ) onto
Wi((a,b ) , p , p - 1 ) .
(b) C, gives an isomorphic mapping from
~ i ( ( ab ),, pa) A L B B ) (b()~) , onto ~ i ( ( ab),, p 2 ) . Proof. Step 1. Theorem 7.3.2(b) yields that C , is an isomorphic mapping from Wg((a,b ) , p i ~ tn ) L e ) ( ( ab , ) ) onto a subspace of L,((a,b ) ) . Since R(C,) 3 C$((a, b ) ) , it follows that GI,,is a mapping onto &((a, b ) ) . Step 2. For u E C$( (a, b ) ) , it follows by partial integration that
-J
b
2
l l u l l l l ~ ( ( ~ . ~ ~ , p ,(PlU’I2 ~~~) --. ~ ( 1 ” )
m2 4
instead of u,then one obtains
b
( ~ c f f2, u ( ~ W ~ ( t a , j-b ~[I)m+llU(”‘+lq2 ,p,~~l) + a
It follows in the same manner as above that in
Wg+1((a,b ) , pm+1)n Llm)( (a, b ) )
p,rl-1IU(”’q2]
ax.
(2)
7.4.1. Self-Adjoint and Positive-Definite Operators
44 1
the right-hand side of (2) is equivalent t o IIull&;+] ((a,b),p’”’’). Since C$((a, 6 ) ) is a part of the range of C,,,, and since C$( (a,b ) ) is dense in W;((a,b), p , p - l ) , (a) is a consequence of the two preceding steps. Step3. Let U E Cm((a,b))nLiz)( (a,b)).ByTheorem3.6.1, C,ubelongsto @((a,b),p2). One obtains from (7.3.2/5)that
-
b
I(pCZU)’’12
IIC2UIl~((n,6~,,~Z)
a
b
N
J I(~~U’’)’’I~
N
IIUII%i( m - 1, hence 2k > 3m - 1 , then one obtains Pm-l = 0. Consequently v E D(Am,k). If 2(k - m ) 5 m - 1 , then
From this, one can determine the local behaviour of vO)(x)near x = a and x = b. Now, using A$,kv= 0, it follows for u E D(A,,,) = Ccp((a,b ) ) by partial integration b
b
0 = J” ( -
l ) f l l ( p k u ( q ( n *2))
a
a x = J” Pm-l(x)U ( m ) ax. a
Then it holds PmVl= 0. Hence, v E D(Am,,J. Step 3. We consider the case k c m. Now it follows from (2) by integration, that vU)(z)is regular near a and b for j = 0: . . .,m - k - 1 , while w ( ~ ) ( z )has a behaviour as log po(x)for j = m - k, and a behaviour as ~ ? - ~ - j ( for x ) j = m - k + 1 , . . . ,m. One obtains for u E D(Am,]Jby partial integration that b
0 =J
v
(pk~(m))(m) dx
a m-k-1
=
( - 1)‘
6
( ~ ~ ( n l ) ) ( ~ - r W(r)l: - - 1 )
r-0 m-k-1
=
r-0
+ ( - 1 ) m j” @u(m)v ~ ax) a
(- l ) r
( @ u ( m ) ) ( n * - r - l )&)lj:
+ (-
c (-
k-1
l)m
s-0
(3)
l ) S p g : l U ( m - s - l ) la. b
If u E D(Arn,k), where u(J)(a) = d j , m - k , is afunctionvanishingidenticauynear b, then (3) yields PE:i)(a) = 0. Similarly, one obtains P:Ll(a) = P:Ll(b) = 0 if s = 0,..., k 1 , and afterwards w(r)(u) = w ( ‘ ~ ( b )= 0 if r = 0, . . ., m - k - 1 . (4)
-
If 2k > m - 1, then Pnl-l = 0. Then one obtains by (2) and ( 4 ) v 2k 5 m - 1, then
If
E D(Am,k).
Together with (4),it follows again w E D(Ana,k). Hence, An,,,,is self-adjoint. From the first step, it follows that A m , k is an operator with pure point spectrum, the eigenvalues of which are nonnegative. Step 4. To prove (b),we may assume, without loss of generality, p = p , . If 2k m, then it follows from (5)and the previous considerations w = p2m-2k-l(x). On the other hand, one obtains by (4)w = O(pT-’ 0
-
'(%) (5 a)",
= C,
9
> 0,
If m = 1 , 2 , . . . and 0 5 k,, kb 5 2m A,,k.,kbU
= (-
lim
1 ' t I,
p(z)
=
(b - 2 ) " b
cb > 0 .
- 1, then one sets
( q ( 4 U(m))(m)
9
D(A"&kb) = { u l u € C " ( ( a , b ) ) ,
U ( j ) ( U )= 0
for j = 0,... ) m - k,,- 1 ,
u(jj(b) = 0 for j = 0,. . . ,m- k, - 1).
(If 12, 2 m or if kI,2 m, then the boundary conditions are omitted.) Using the theory of deficiency indices for ordinary differential operators, as it can be found in M.A.NAJMARK [I], then it is easy to see that the operators Am,ka,ks are also essentially selfadjoint : Namely, if one considers the operator A%!k )
Ak!,% = (- 1)"~ (pk(z)~ ( l a ) ) ( r f l ) , D(A$!k)
=(U
I U € c m ( ( U ) b ) ) )U ( j ) ( a ) = 0
(T)
a+b
u(j)
=
o
for j = 0,. . ., m - k - 1 , for j = 0,.. ., 2m - 1
(withthe usual modification for k 2 m)in the space L, ((a,
q))
and a correspond-
ing operator then it holds for the deficiency &dices (see'M. A. NAJMARK [l], S 17.5) ( 0 , O ) = Def Am,/c= Def A!& + Def A ,!: - 2m = 2 Def Ag!k - 2m.
Whence it follows that DefAln,k,,lcb = DefA$tka + DefAg!rib - 2m = 0 . Hence, Al,,,ka,kb is essentially self-adjoint.
7.4.2.
The Minimal Operator Am,k
Definition. Under the hypotheses of Definition 7.2.1 one sets
Remark 1. Ordinary and partial differential operators in domains 52 c R, with the domain of definition C,"(Q) are called minimal differential operators.
444
7.4. Self-Adjoint Legendre Differential Operators
Theorem. (a) All,,h.coincides with Friedrichs's extension of AI+ if and only i f either k = 0, or k = 1, or k = 2m - 1. (b) If Def A,,,, denotes the deficiency index of
t?ben
Def A,,,.,,= (2m, 2m,)
if
Def A,,,, = (4m - 2k, 4m - 2k) if
k = 0,. ..,m,
k = m, . . ., 2m - 1 .
(1)
Proof. Step 1. (7.4.1/1)yields that kr((a,b), 9) is the energetic spaceof the operator A,,,(. From the usual method of the construction of Friedrichs's extension for operators bounded from below (see for instance H. TRIEBEL[17], p. 215), it follows that A,,,,. is Friedrichs's extension of Al,l,kif and only if D(All,,k)= k q a , a),
13k).
(2)
Let k 2 m. Then Theorem 3.6.1 yields that (2) holds if and only if k = 2m 0 4 k < m. By Definition 7.2.1 and Theorem 3.6.1, (2) holds if and only if
[m -
m - k - 1 2
-
k + l
-
1. Let
= m - [ T k] -+l l
Whence it follows k = 0 or k = 1.
Step 2. If u
E
ern(@,b)), then one obtains byrepeated application of Lemma 7.3.1/1
b
j
b
+
dx
(l(~u(~'~))("~ lul2) )12
a
J'
(pwlU(2l,r)12
+ lu12) dz.
(31
a
Whence it follows that -
=
~ ( ~ 1 1 1 , , , )
By Theorem 3.6.1, u
kP((a,b ) , p 2 k )
E Cm((a,b))
*
kim((a, b), pm) if and only if j = 0, . . ., 2m - k - 1 .
belongs to
u(j)(a)= u ( j ) ( b )= O if
Since A,>k is self-adjoint, now one obtains that
b), $)2k) Defim,/,= dimD(Am,k)/giam((a, (2m, 2m)
=(
(2(2m - k), 2(2m -
k))
A
D(Arn,k)
if
O s k < m ,
if
m
5k 2
2m
-
1.
Remark 2. * (1) yields that in the interval (a, b) there exist minimal differential operators of order 2m with the deficiency index (r, r ) where r = 2 , 4 , . . . ,2m. Starting from this result, one can construct in the interval (a, b) minimal differential operators of order 2m with given deficiency index (r,r ) where r = 0, 1, . . . 2m. For this purpose, one uses the methods from Remark 7.4.112. We refer to H. TRIEBEL [14, I ] .The existence of ordinary differentia.1operators of order 2m with the deficiencyindex (r,r ) where r is a given number, r = 0, 1 , . . ., 2m, is well-known. The first [ l ] .A systematic treatment of these problems examples are due to I. M. GLAZMAN El].The case m = 1 goes back to H. WEYL[ l ] . can be found in M. A. NAJMARK
7.4.3. Domains of Definition of the Operators
xi.",, I = 0, 1 , 1,. . .
Domains of Definition of the Operators I$, I = 0, 1, 2,
7.4.3.
445
...
The determination of the domains of definition of the fractional powers A!n9iL, 0 < 8 < 00, of An,,/< is rather complicated. We shall return to this problem in Section 7.7. On the other hand, it is easier t o determine the domains of definition D(A;,/,) 1 3 where 1 = 0, - , 1, - , . . . , provided that 0 5 k m. Concerning this question, there 2
2
s
s
is a deep difference between the cases 0 6 k m and the cases m + 1 5 k 2m - 1. L e m m a . Let A,,,,/,be the closure of the operator Am,/tfrom Definition 7.2.1, where 05k m. If A,,/,u E Cm((a,b ) ) , then u belongs to D(A,,/J c Cm((a,b ) ) .
s
Proof. Step 1. (7.4.2/3) and Theorem 3.6.1 yield
D(A,r2,,c) = {u I u E Wgm((a,b ) , p z k ) , uW(a) = u(J)(b) = 0, j = 0, ..., m - k - 1) (1) if 0 5 k < m and D(ArnJ = Em( (a,b ) , p 2 m ) . Hence, for the proof of the Lemma, it is sufficient t o verify u ( m ) E Cm((a,b ) ) . Step 2. If k < m , then one obtains, from Theorem 7.3.2/1, l ~ ( ~ ) ( 5 z )clif z ~ ( a , b ) . Further, i t holds p%(nt) E Cm((a,6)). Whence it follows d m E ) Cm((a,b ) ) . Let k = m. Then one obtains, from Theorem 7.3.2/1, I U ( ~ - ~ ) ( X ) I 5 c for z E (a, b). Assuming that p ~ n u ( ~En )Cm((a,b ) ) has a root a t a of the order I < m, integration yields that u('n-l)(x) is unbounded in (a,b). This is a contradiction. Whence it follows U P ) E CW((a,b ) ) . T h e o r e m . Let Am,kbe the operator from Definition 7.2.1, where k = 0, 1, . . ., m. (a) It holds m
D(A:,,)
= n~ l=O
=
{U
if
if k = 0 , . . . , m - 1, and
(2k.d
I u E Cm((a,a)), (A;,+)(J)(a) = ( A L , ~ u )(b) ( J )= 0 1 = 0 , 1 , 2,..., and j = O ,..., m - k - 1 )
(Za)
(2b) D(A$,,) = Cm((a,b ) ). (b) If 2 = 0, 1, 2, . . . , then D(A$$.) is the completion of D(A&) in the space WP((% b ) , p k l ) . Proof. Step 1 . It follows from Remark 7.4.1/1 that C"((a,b ) ) 8 N(A,,,) is dense in L,((a,b ) ) 8 N ( A , , J . The operators A;,!, are positive-definite on this subspace, 1 = 1, 2, . . . Hence,
A:,/,% = f E Cm((a,b ) ) 8 N(Am,/ m.
The Spaces
7.4.4.
z$((a, b))
To prepare the later investigations on the structure of nuclear function spaces, we consider in this subsection the spaces c3((a,b)). Definition. Let - 00 < a < b < 03. Further let j 2 1 and 1 be integers such that 0 5 1 5 i. One sets Cj$((a,b ) ) = C m ( ( ab, ) ) for 1 = 0 and
I
" ( ( a , b ) ) = { u ( z ) u E Cm((a,b ) ) , u @ ) ( a= ) u @ ) ( b= ) 0
for
12
1.
for i = n j + n , , wherenl=0,1,2,... andn,=O ,...,1-11
(1)
Remark 1. Using the semi-norms sup Iu@)(x)l, i = 0 , 1 , 2 , . . . , then Cz((a,b ) ) 4a,6)
becomes an (P)-space. Later on, we shall see that Cjs"l((a,b ) ) is a nuclear space, isomorphic to the space s of rapidly decreasing sequences. For j = I , it holds €;((a, b ) ) = @'((G, a)) in the sense of Lemma 6.2.3.
7.4.4. The Spaces
qT((a,b ) )
447
cz(
It is the aim to identify (a,b)) with D ( A m )where , A is an appropriate operator. For this purpose we need a preparation which is also of self-contained interest. Theorem 1. Let -a < a < b < 00, m = 1 , 2 , . . ., and k = 1 , 2 , . . . Purther, let p ( x ) be the function of Definition 7.2.1, where p ( x ) = Ca(x - a ) for a < x < a
+
E
and p ( x ) = Cb(b - x) for b
D ( . L ~ ,= ~ ~{u , ~I)u E Cm((a,b ) ) ,u ( ~ ) ( = u )u(')(b)= 0 for i
=
- E < x < b.
0,. . . , m
-
(2)
+ k - l} ,
(4)
i s an essentially self-adjoint operator in L2((a,b ) ) . The cbsure J n l , k i s a positivedefinite operator with pure p i n t spectrum. It holds
-
.
D(iz,.k) C Wimn((a,b)), n = 1, 2, . . (5) Proof. Step 1.Lemma 7.3.114 (in relation with the methods of approximation from the third step of the proof of Lemma 7.3.1/1) and partial integration yield that dm,I, is a positive-definite operator. Let d2,kv = 0. It holds in the sense of the theory of distributions that (p-kw(m))(m) = 0. Whence it follows v E Cm((a,b)). Now, let u E D(d,,k) be a function, vanishing identically near b such that u(*)'(a) = 8.,2m+k-1. One obtains by partial integration that 0=
I
0
( A m , k U , V)L,((o,b)) = CU(2m+k-1) U 2,0 a
7
c*o-
Hence, v(a) = 0. Similarly v(b) = 0. By the same method one obtains vO)(a)= w(j)(b) = 0 if i = 0, . . ., m - 1. Since = @Pm--I, the function v belongs to D(Am,k). As i m , k is positive-definite, we have v = 0. Whence it follows that A i, k is essentially self-adjoint. Step 2. Using estimates of the type (7.3.1/2) and (7.3.1/4) one verifies easily that for u E Cm((a,b)) L
b
( I(giiv)(j)l2ax
a
b
s c J- ()v(j)l2+
lpkw12)
ax,
a
j = 1 , 2 , . . ., holds. Setting v = p - k ~ ( mwhere ) u E D ( d m , k )then , it follows that b
j , u c m + q 2 ax
c J- (1(p-ku(m))(J)l2 + Iu(m)12)ax.
(6)
a
a
Using again an estimate of type (7.3.1/4), and setting j = m,then one obtains ( 5 ) with n = 1. We shall use induction' and assume that (5) holds for n = 1 , 2 , . . ., 1. Applying ( 6 ) with j = m + 2ml and (7.3.1/4), then it follows that
llull
2
W~m'+zm((a,b))
2
5 czll&,iuIIt,(ta,b))
C1lIim,kuI1 Wimi((a,b))
(71
for u E D(&$). Under the hypothesis
-
21m.k 4 - (Am,k)'+l, -
(8)
7.4. Self-Adjoint Legendre Differential Operators
448
(5) is a consequence of (7). Now it follows in the same manner as above from the theorem of F.RELLICHthat -iim,k is a n operator with pure point spectrum. Step 3. We prove that is essentially self-adjoint. This shows - also that (8)
&$
is valid. Let f E Cm((a,6 ) ) and (-l)m(p-ku(m))(nz) = f where u E D(im,k). Then we have u E Cm((a,6 ) ) and u ( ~=)@g, g E Cm((a,b ) ) . Since ( 5 ) is known for n = 1 whence it follows-(independently of k) that u is an element of D(Am,k).Let
f E Cm((a,b ) ) and ikl,ku = f . Iteration of the last considerations yields u E D(AL,,;). Whence it follows that
is essentially self-adjoint.
i . Setting m = j - 1 T h e o r e m 2. (a) Let j 2 1 and 1 be integers such that 0 1 k = j - 21, and assuming that p ( x ) has the form (2) then 2
and
s s
D(Az,k)= c3( b ) )* (b) Let j then
(9)
3 < 1 < j. 2 1 and I be integers such that 2
-
D(L$,k)
=
Setting m = j - I and k = 21
C3((a,b ) ) .
- j, (10)
Proof. Step 1. I n the case (a) it holds m 2 k. If 1 = 0, then (9) is a consequence of (7.4.3/2b).If 1 > 0, then the special choice of p ( x ) yields that the right-hand side of (7.4.3/2a) coincides with C ~ ( ( U 6 ),) .This proves (9). Step 2. We prove (b). Taking into consideration the special choice of p ( x )it follows
D(iiL,k)= {W [ if
U E COD((&b ) ) ,(>,,ku)(t)
s=O
= {uI u
,..., r - 1
(a) =
( 2 k , p ) ( t(b) ) =0
and t = O , . . . , m + k - 1 )
E COD((a, b ) ) , u(")(a)=
u(")(b)= 0 for
+ k ) n1 + n2 n2 = 0 , . . ., m + k - 11. cr = (2m
where n1 = 0,
It holds 2m
. . ., r
-1
and
+ k = 1. (5) and Theorem 7.3.211 yield
+k =j
and m
D(Z$,J
= Cm((a,b ) ) .
(11)
-
-
If f E D ( i z , k )then , bL&f belongs to Cm((a,b ) ) . Applying the considerations of the third step of the proof of Theorem 1, then it follows f E D ( i k , k ) r, = 1 , 2 , . . . Then one obtains by (11) t h a t 8
-
D(jg,k) =
n D(bh,k)= C3((a, b)). r=O a,
R e m a r k 2. Remark 7.4.3/1, Theorem 7.3.2/1, and (5) yield that the spaces (9) and (10) coincide not only set-theoretically, but also topologically. (The topologies are the same as in Remark 7.4.3/1.)
7.5.1. Associated Eigenvectors of the Operators
7.5.
449
Non-Self -Adjoint Legendre Differential Operators
This section is concerned with the differential operators B,,+ from Definition 7.2.1. The main aim is the investigation of mapping properties and of the problem of the density of associated eigenvectors.
7.6.1.
Associated Eigenvectors of the Operators Bm,k
T h e o r e m . The operators Bnr,kfrom Definition 7.2.1 are closable in L,((a, b)). I t holds
D(B,,,J
=
D(J,,k)
{u I u E Wifra((a, b ) , p 2 k ) , u(j)(a)= u(j)(b)= 0
=
if j = O ,
if k = O , ..., m - 1 , a n d
..., m - k -
1)
D(Brn,k)= D(Arn,k) = Wirn((a, b), pm)
(1 b)
if k = m , . . ., 2m - 1. The spectrum of Brn,kconsists of isolated eigenvalues of finite algebraic multiplicity Without any finite cluster p i n t . The linear hull of the associated eigenvectors i s dense in L,((a, b)). Proof. Step 1. (1) for the operators A m , k is a consequence of (7.4.2/3), Theorem 3.6.1, and Theorem 3.2.2. Step 2. If u E C”((a, b)), then (7.2.1/5) and bj(z) E C”((a,b ) ) yield
1) C
b
2111-1
j=O
If
u E D(AI+),
bj(x)u(j)l/l
5c J
w:((a,b))
c p2max(0,h.-2m+j)lUd)12ax. 2rn
I =O
(2)
then one obtains from (7.3.1/2) by completion
Replacing Wi in (2) by L,, and using again (7.3.1/2), then it follows for u E D(Arn,k)
Here E > 0 is a given number. (4) yields IIBrn,kU1ltz((a,b))
-k
IIUIIiz((a,b))
- 11 Ull%(&,, ~)
(5)
if u E D(Arn,k) = D(Bm,k).One obtains by (5) that Bm,&is closable and that (1) holds.
Step 3. We want to apply Theorem 5.4.1/3. It follows from Theorem 7.3.2/1(c) and the theorem of F. RELLICHthat Am,kis an operator with pure point spectrum. 29
Triebel, Interpolntion
7.5. Non-Self-AdjointLegendre Differential Operators
450
If I is a complex number such that I m I Cu =
2m-1
2
J=o
+ 0,
and if one sets temporarily
b,(x) u ( J )then , CD(xm,k)+L2(Ani,k - IE)-l = I W 2 + l , , C D ( i ~ ~ , ~ ) + I l ' : ( A m-, hAE)-', .
where the operators will be considered as mappings between the indicated spaces. Now the desired assertion follows from (3), (4.10.2/14), and Theorem 5.4.1/3. R e m a r k . We mentioned in Remark 7.2.1/1 that (7.2.1/5) is chosen in such a way that C can be considered as a perturbation of A,+. The above proof is not valid if one replaces (7.2.1/5)by (7.2.1/7). Then C cannot be considered as a perturbation of An1,k.See also Remark 7.2.1/2.
7.6.2.
Isomorphic Mappings
D e f i n i t i o n . Let -00 < a < b < 00, and let Q U , b = (a,b ) x (a,b). It i s assumed that p ( x ) has the meaning of Definition 7.2.1. Further, d ( x , y ) denotes the distance of a p i n t (z,y ) E Q u , b to the boundary dQu,b. For x 2 0 and 0 < s = [s] + ( 8 ) ;[S] integer, O < {s} < 1, one sets
I
E
W'((a, b), pX)
9
(1)
R e m a r k 1. It is easy t o see that d X ( x y, ) min ( p x ( x ) @,(y)). This justifies the notation. If s is not an integer, then one obtains by Theorem 3.3.3 that N
Bi,z((a,4 ,P)= K ( ( a ,b ) , p X ) .
(2)
This relation and the interpolation properties described in Theorem 3.3.3 are the basis for the further considerations. T h e o r e m . Let m = 1 , 2 , 3 , . . . and k = 0 , 1 , 2 , . . ., m. Further let I be a unnplex number, which is not an eigenvalue of the operator Bn1,,< from Theorem 7.5.1. Then the differential expresswn Bm,ku- Iu, where k = 0 , 1, . . ., m - 1, generates for s 2 0 an isomorphic mapping from {u I u E Wgm+s((a, b ) , p2"), u(J)(a)= u ( j ) ( b )= 0 for j = 0, . . ., m - k - 1) (3) onto lVi((a,b ) ) , and the differential expression B,,,u - I u generates for s 2 0 an isomorphic mapping from WPts((a,b), p2m) onto Wg((a,b ) ) .
P r o o f . Step 1. We start with preliminaries. Let p be a complex number, k = 0, . . , m - 1, and A m , p - p u E C"((a, b)). We want t o show that u E D(A,,l,,t) c Cm((a,b)). It follows similarly t o the proof of Lemma 7.4.3 that uE
em((a,b ) ) ,
1J.u("') E Ezm((a,b)).
(4)
7.5.2. Isomorphic Mappings
45 1
Like there, one obtains ~ ( 1 "E) c2m-k((a, b ) ) . Hence u E cZm((a, 6)). Now, one can replace m in (4) by 2m and afterward 2m by 3m. Whence it follows u E csm((u,b ) ) . Iteration yields u E Cm((a,b ) ) and hence u E D(AIIl,h). Step 2. Assume that ,u is not an eigenvalue of A m , h , k = O,1,. ..,m. Let 1 = O,1,2,. .. Then, for u E D(Am,k),one obtains by repeated application of (7.3.1/1) that lIAm,kU
em(@,
- pull 2W.$to,b))
llull
-
2 Wi"+l((a,b),par)
(5)
Since b ) ) is dense in Wk((a,b)), it follows for k = 0 , . . ., m - 1 from the first step that A m , p- p u is an isomorphic mapping from the space ( 3 ) with s = 1 onto @((a, b ) ) . If k = m, then one obtains the corresponding assertion from (7.4.3/2b). Step 3. If A is not an eigenvalue of BIII,/, , then it follows similarly t o ( 5 )that 2
2
~ ~ B I-I Au;u(l ~ ,W/i (~( a ,~b ) ) IlUlI Wimt'((o,6),pzA) (6) if 1 = 0, 1, 2, . . . and u E D(B,,,,k)= D(AnI,h).For abbreviation we set again Cu
2111-1
= J
=o
b,(x) u ( J ) If . ,u is not an eigenvalue of Aln,k, then the dimensions of the
null spaces of the mapping
(CU - AU + ,UU) B m , k U - AU = AI,,,,,U - PU from Wim+'((a, b ) , p2k)n D(A,,,) into Wk((u,b)), and of the mapping
+ (c- AE + pE)
- pE)-'U
(7) from Wk((a,b ) ) into itself, are equal. The same holds for the codimensions of the ranges. Similarly t o the second and t o the third step of the proof of Theorem 7.5.1, i t follows that (C - U8 + pE) ( k t I n , h - pE)-l is a compact operator acting in @';((a,b ) ) . Now, (6) yields that E (C - UC P E ) (-4m,l, - pE)-' is an isomorphic mapping from Wi((a,b ) ) onto itself. Then, Bm,hu- lu is an isomorphic mapping in the sense of the theorem, where s = 1 = 0 , 1 , 2 , . . . Step 4. For k = m and arbitrary values of s 2 0 the theorem is a consequence of ( 2 ) and Theorem 3.3.3. If k < m , then (3) with s = 1 = 0,1,2, . . . has a finite codimension with respect to Wtm+'((a, b ) , pm). Theorem 1.17.1/1is applicable to two such spaces. Now one obtains the theorem for arbitrary values of s 2 0 from (2), Theorem 3.3.3, and Theorem 1.17.1/1. t~
+
(Anz,h
+
R e m a r k 2. Let s = 0 , 1,2, . . . The third step yields that Bm,k,considered as a mapping from ( 3 ) into W i((a ,b ) ) for k < m , and considered as a mapping from g m + s ( ( ab ), , p)into Wi((a,b ) ) for k = m, is a @-operator. The index*) is 0. Using the theorem and the above methods, i t follows that this assertion is true for all s 2 0. R e m a r k 3. The theorem can be generalized. The above proof based on the fact that the considered mappings are restrictions of the operators ii,,,,h and B,,,,,,. In
(4)
this sense, one can for instance replace the basic space L2((a,b ) ) by the spaces D A n1,k from Theorem 7.4.3. Here, the case k = m is of special interest. In this way, one obtains mappings in Wi((a, b ) , p') where 0 equals some (but not all) positive values. *) See p. 422.
29*
452
7.6. Tricomi Differential Operators
7.6.
Tricomi Differential Operators
This section is concerned with properties of Tricomi differential operators from Definition 7.2.211 and Definition 72.212. As a. preparation we consider in 7.6.1 elliptic differential operators on compact Cc”-manifolds.There we restrict ourselves to facts needed later.
7.6.1.
Elliptic Differential Operators on Compact Cm-Manifolds
Compact ( n - 1)-dimensional am-manifoldscan be described in the usual way by “local charts” (systems of local coordinates). (See for instance L. HORMANDER [3], 1.8.1.)But we shall restrict ourselves to boundaries aD of bounded Cm-domains D c R,,(Definition 3.2.1/2).On aD we consider (regular) elliptic differential operators C of order 2m ,which can be represented in local coordinates y$J),. . . , ygll ( j = 1,. . . , N ) bv Here c,,(y(jY), y(j)’ = (yiJ),. . ., y:!ll)*), are complex-valued Cm-functions. It is assumed that the top-order coefficients c,,(y(j)’),IyI = 2m, are real. Further, it is supposed that there exists a number c > 0 such that
for all 6 E R,,-l (ellipticity condition). c is independent of y(j)‘ and j . Further, we need the spaces W$(aD), k = 0, 1, 2. . . . See Definition 3.6.1 and Remark 3.6.1/1. T h e o r e m . Suppose that C from (1))having the domain of definition D(C) = Cm(aD), i s symmetric i n L2(aD).It holds: (a) C is boiinded from below and essentially self-adjoint. (b)
c i s an operator with pure p i n t spectrum and 1
+ N(I)= 1-+
c
A,SA
1 -I=
n-1
+1
Proof. Step 1 . The local charts, covering aQ, are denoted by Uj c R,,-l, j = 1, . . ., N , the corresponding local coordinates are yi@, k = 1, . . ., n - 1. (One E Cm(aQ)be a resolution of unity with remay assume that Uj are balls.) Let spect to U j , hence
x;
N
C x; = 1 j=1
on aQ, x j ~ C g m ( U j ) ,0
sxj 5
1.
*) To avoid confusion in the later considerat,ions,we write y””. See Definition 7.2.211.
(6)
7.6.1. Elliptic Differential Operators on Compact Cm-Manifolds
453
> 0 is a given number, c > 0. Whence it follows that C is semi-bounded. One obtains by the same method and by iterative application of (5.2.2/1) that
E
-
IICkull~-,(~~) + ll~Ilr,,(a~) I l ~ l l w ~ ~ ( a nk) ;= 1 , 2 , . . . 2
2
2
(10)
Step 2 . The method of local coordinates yields that the embedding from W$(aQ) into L,(aQ) is compact. Now one obtains as before by the theorem of F. RELLIUH (see for instance H. TRIEBEL[17], p. 277) that Friedrichs’s extension of C is an operator with pure point spectrum. To verify that Cis essentially self-adjoint, we must show 2 ) . C*v = 0. Now it holds for u~Cg(Uj), that C*v = 0 has theconsequence v ~ C ~ ( a J Let similarly t o ( 8 ) ,
(. f . i ( y ( j ) r (CU) )
(~J[JY)
v(y(jq~ Y ! J Y = 0.
L’j
Since every elliptic operator with Cm-coefficientsis also hypoelliptic (L. HORMANDER [3], 7.4), whence it follows v E Cm(aQ).(See also Lemma 6.4.1, where this assertion is proved for a special case.) Hence, C is essentially self-adjoint. Step 3. Ckwith the domain of definition D(Ck)= C”(aQ) is also an elliptic essentially self-adjoint operator, k = 1 , 2 , . . . Then (4) is a consequence of (10). One obtains (5) from Theorem 4.6.1. Step 4. To prove (3), we use Theorem 5.4.1/1 and estimate s j ( I ; W,2’”(aQ), L2(aS)). There are denoted extension operators by S, restriction operators by R , and embedding operators by I . and (4.10.2/14)yield
454
7.6, Trioomi Differential Operators
On t,he other hand. we have where x1 has the same meaning as in the first step. Using again (4.10.2/14),one obtains the converse estimate to (11). Now, (3) is a consequence of Theorem 5.4.1/1. R e m a r k . * One can ask whether (3) can be reinforced in analogy to the asymptotic [l]. Further, we quote in this formula (5.4.2/3). We refer t o A. N. KO~EVNIKOV connection the papers by S. MINAKSHISUNDARAM, A. PLEIJEL[l], H. P. MCKEAN, I. M. SINGER[l], W. GROMES[2], G. A. SUVORI~ENKOVA [l], and J. FARAUT [l]. The self-adjointness of elliptic differential operators on non-compact Riemannian manifolds is investigated in H. 0. CORDES[l] and A. A. CUMAH[l]. 7.6.2.
Integral Inequalities [Part 111
I n order to prepare of the considerations on Tricomi differential operators of first type, we prove some integral inequalities. The estimates obtained in this subsection are more general than needed in the following investigations. L e m m a 1. Let Q c R, be a bounded C”-domain. S has the Same meaning as in 7.2.2; ( y l , . . . , Y , - ~ , y,J = (y‘, yn) are the local coordinates*) in the seme of 7.2.2. Further, let 1 = 1 , 2 , . . . , 1 < p < m, x real, 1 > 0 real; /? = (PI, . . . ,/?n-l, BIZ) = (p’,B,) multi-index, where IBI =< I , and
1 the m e that both sides of (1) are equal it is assumed additionally 0 + P Then there exists a number c > 0 such that for all u E C$(S)
In
+ 1 , . ..,1.
(0,’. means that there are only derivatives with respect to y1 . . ., y,-J. Proof. Step 1. The balls K, have the same meaning as in 7.2.2. Further, let ~
8
xj(Y’) E cm(aQ), C xj(Y’) j=1
1, ”PP
x,
c (Kj n aQ),
be a resolution of unity on aSZ. It is sufficient t o prove ( 2 ) for xJy’) u ( y ) . A transformation of coordinates yields that (2) can be reduced t o the following problem: There exists a number c > 0 such that for all
v E C$(RA), where supp w c { x I 1x1 6 B ; 0 < zn < a>, the inequality
(3)
*) For sake of simplicity, we do not write the index “j” in the local coordinates (yf),. . .,gicl, y.).
7.6.2. Integral Inequalities [Part 111
455
holds. Here B is a given fixed number. (See (2.3.3/7).) Further it is easy to see that one can restrict oneself to the case
m1 +1 - ae =
1 1, a + - *
1, . . . )1 .
P
Step 2. We prove (4)under the hypothesis (5).Let for k = 1 , 2 ,..., n>.
Q = ( X ~ X E R ,1, < x k < 2 Then one obtains by Theorem 4.2.4 that
Let 1 and p be two real numbers such that 11 = e(l - a).Applying the transformation of coordinates
xn = Py,,
x, = 2'yi
for j = 1 , . . ., n
- 1,
to (6),then one obtains (after changing the notation of the variables) that ~PXP-PP&,-P~S'I
J
ax
x;xp@v(x)l~
QA,O
+ c2-~e(1-0)j'
x ; ; ~ ~ ~ - ~ ) l ax. vl~
(7)
Q40
Here, is a rectangle with the side-lengths 2-P in xn-direction and 2-a in the other directions. The distance of QA,@to the plane { x I x,, = 0) is 2-0. From 11 = @(l - a) and e(x
- /!In) + 1(1 - IP'I) = @(Z
- a)
it follows, that the powers of 2 in ( 7 ) are equal. (7) is also valid after a translation of Qa,e parallel to the plane { x I xn = O}. Adding inequalities, modified in such a way, then
1
x,PXIDBvlPdx
Rn+
Now one obtains ( 4 ) by an iterative application of (3.2.6/4)to the last summand on the right-hand side of (4). Lemma 2. Let SZ c R,, be a bounded C " - d m i n . S has the meaning of 7.2.2, (Yl3 . . Y11-13 Yn) = (Y') Yn) are local coordinates in the sense of 7.2.2.*) Further let . 2
*) See the footnote on p. 454.
466
7.6. Tricomi Differential Operators
I = 1 , 2 , . . ., 1
o > -- red, < I , and P
[
1
> max --, /?,z - (1 - c)] for IP’I 2,
Then there exist numbers c > 0 and and for all E > 0 we have
Q
= (PI,.. ., P n - 1 , B n ) = = 0 and x
2 Bn for
(/?‘~/?ll)
>O.
(8)
> 0 such that for all u E C”(Q), where suppucg,
Proof. (8) yields that (1) is valid with “ 0 ie sufficiently small. Hence, it holds (4) for the set of functions (3), where one can replace x by x - 7 . By approximation (for instance by the method of the second step of the proof of Theorem 2.9.1) one obtains (4) with x
- 11 instead of
x for functions
ah
w E P ‘ ( R ~ where ), -(d,O)
a4
= 0 for
j = O , . . . , 1 - 1 and suppwc{xI1z1$B; O~x,,:,<w}.Let k = O ,..., 1 - 1 . In the sense of induction we assume that (4) with x - 7 instead of x holds for func-
ah
(x’,0) = 0 for j = 0, . . .,k and supp v c {x I 1x1 5 B ; axi ah 0 5 rc,, < co}. Let w E Cm(B,+) with -(XI,0) = 0 for j = 0, . . ., k - 1 (for k = 0 axi this condition is omitted) and supp w c {z 1 1x1 5 B; 0 S x,, < 00). By assumption one can apply (4) (with x - q instead of x ) to
tions v
E Cm(B,+), where -?-
2
W(X) =
We have
w(x) - a \ w(xl, . . . , x,~-~, xnt)dt, i
a =(
j t k
dt)’ 6 1 .
1
The inner integral in the second summand is independent of t (transformation of coordinates tx,l = y,,). Let IB‘I > 0, and let 17 > 0 be sufficiently small. Then we have a
a
J
1
1
at < 1 .
~ - ( x - q ) - ~
457
7.6.3. Self-Adjoint Tricomi Differential Operators of First
Whence it follows
J
fc
Z { ( ~ - V ) I D ~dx IP
c
1 Z{(~-V)~DDW~P dx. R:
As before-mentioned, one can apply (4) with w instead of v and with x - 9 instead of x to the right-hand side. Applying again the same technique of estimates as in (lo), then one obtains (4) with x - 9 instead of x for the function v considered here. It follows by induction that (4) with x - 11 instead of x holds for functions v E P ( R ; ) where supp v c {x I 1x1 5 B, 0 xn < a}.Whence one obtains for functions of such a type that
J x,PX~D’V~P dx R:
EI
J X { ( ~ - ‘ ) ~ @ V ~ P dx + CEi”
R:
I
Io8w1P
dx
{ r l s c R ~ , c ~ : r < 3 ;m1 }
0. Similarly t o (2.4.2/17),one obtains from Theorem 2.10.1 that
p4 > 0. Using Theorem 4.2.4 and Remark 4.2.412,whence it follows by an appropriate choice of E~ = cECf0 that
1CPIDpvlPdx
I 0 is a suitable number and u E D(B,,,,k)( E > 0 can be chosen arbitrarily). Near the boundary, there coincide the main part of Bm,kuand Gu from ( 2 ) . Using the same technique of estimates, then one obtains by (5) and (5.2.211)that
+ Ilullir(Q)
lIull*2,
~~Bfn,ku~~&~)
(7)
E D(Bm,k)*
(6) yields that Bm,kis semi-bounded. Since from the first step is an operator with pure point spectrum, one obtains by (6) that Friedrichs's extension of Bm,kis also a n operator with pure point spectrum. But then B m , k is self-adjoint if and only if
WBZ,,) = D(B"f,k). Step 3. Again we assume that dd(y'j') = O in ('7.2.213).Let BZ,,w = 0. Since every elliptic operator having C"-coefficients is also hypoelliptic (L. HORMANDER [3], 7.4): it follows that w belongs t o C"(l2). To consider the behaviour of w near the boundary, we replace G from (2) by 6, =
&4
[Arn,rn,k
8E
+ E 8 c]u,
D(a",= D(Am,m.k)@ D ( c ) .
Here, A,13,,l,,l~ has the meaning of Remark 7.4.112 (a corresponds t o the part of the boundary d(x) = h and b corresponds t o the part of the boundary d(x) = 0).(This modification is needed only for k < m.) Clearly, & coincides essentially with B,,,,u near the boundary. (Here we use the coordinates (xl,. . ., zn) as well as the local coordinates (yl,. . ., yfl).)If xo(x) and xl(x) have the above meaning, then it follows for w E D ( 6 )and a suitable choice of the function q(x) from Remark 7.4.112 that
-
(h, w)C2(S)=
-
2 ( h x 0 9 W)La(S)
+ (Bm.kWX?
7
'U)La(S) =
(&xi,
w)L,(S)
-
Partial integration yields ( G W , W)L*(S) =
(w, E ) L , ( S ) .
Hence, w belongs to D(G*).24 is essentially self-adjoint and it holds the counterpart t o (5). Whence it follows that v belongs t o the completion of D(Bm,k) in the norm (1). Consequently, w E D(Bnz,k). This shows that BffL,k is self-adjoint. For the case dd(y'") = 0, all the other assertions of the theorem follow from the previous considerations. Step 4. Now we consider the general case. Bf,,liucan be represented in the form
Bm./ n. Now the theorem is a consequence of (2), Lemma 5.4.112, and Theorem 5.4.113. 7.6.6.
The Spaces
z$(Q)
In this subsection the results from 7.4.4 are carried over from one dimension to several dimensions. D e f i n i t i o n . Let sZ c R,, bea bounded C”-domain. F,urther, let j 1and 1 be integers such that 0 6 1 5 j . One sets I7~#2) = C”(S) for 1 = 0 and
1
c~(Q = (u(x) ) u E c”(Q), ;vu where n,
=
laQ
=
o
/or r
0 , 1 , 2 , . . . and n2 = 0,
for 1 2 1. v is the normal with respect to asZ.
=
n,j
+ n2
I
. . ., 1 - 1
462
7.6. Tricomi Differential Operators ~~~~~~
R e m a r k 1. t?Zl(Q)becomes an (P)-spaceif the topology is generated by the seminorms sup IDYu(z)l,IyI 2 0. If j = I, then t?jq;.(Q)= @‘(Q) in the sense of Lemma 6.2.3. zeR T h e o r e m 1. Let SZ c R, be a bounded C”-domain. Further, let m = 1 , 2 , . and k = 1 , 2 , . . . , and
where b,(x) are real functions for
and that
Bm,k
= 2m. I t is assumed that
admits in s the representation (similar to (7.2.2/3)) (1)
where the second term has the same properties as in Definition 7.2.211. If the domain of definition
k,+, with
-
is symmetric in L,(Q), then I?,,,,( i s essentially self-adjoint. i m , k i s a semi-bounded
operator with pure p i n t spectrum. I t holds
-
D(&J c
w;mr(~),
r = 1 ~ 2 ,. ..
(3)
Proof. Using Theorem 7.4.411, then one can carry over the proofs of Theorem 7.6.311 and Theorem 7.6.312 without essential changes. R e m a r k 2. It follows from the method of the first step of the proof of Theorem 7.6.4, that there exist symmetric operators k’m,h with the required properties. T h e o r e m 2. (a) Let j 2 1 and 1 be integers such that 0
i . Setting m 6 152
=
j -1
and k = j - 21, and assuming that Bm,kis a symmetric operator in the seme of Theorem 7.6.312, where a(t) 3 1 in (7.2.2/3),then it holds (set-theoretically and topologically in the sense of Remark 7.6.312)
D(B2,k) = C z ( Q ) .
(4)
i < 1 < j . Setting m 2 1 and 1 be integers such that -
= j - 1 and k = 21 - j , 2 and assuming that Bm,hi s a symmetric operator in the sense of Theorem 1, then it holds (set-theoretically and topologically)
( b )Let j
-
D(gz,k)
=
c$(Qn).
Proof. Step 1. (7.6.318) coincides with (4). Step 2. (5) is a consequence of (7.4.4110) and Theorem 1.
(5)
7.6.6. Tricomi Differential Operators of Second Type
7.6.6.
463
Tricomi Differential Operators of Second Type
Tricomi differential operators of second type have been introduced by M. S. BAOUENDI, C. GOULAOUIC[ 2 , 31. We do not give here a comprehensive treatment and restrict ourselves t o some important facts. T h e o r e m 1. The differential operator A from Definition 7.2.212 is essentially selfadjoint in L,(Q). A is an operator bounded from below with pure p i n t spedrum. For j = 1 , 2 , . . . it holds (1) D ( $ ) = W:J(Qn; a"). R e m a r k 1. A proof of the theorem can be found in M. S. BAOUENDI, C. GOULAOUIC [3]. Using the method of local co-ordinates, Lemma 7.3.1/1, and (7.3.1/2), then it follows easily, that (1) coincides with the formulation given in M. S. BAOUENDI, C. GOULAOUIC [3] for the determination of D(Aj).For n = m = k = 1 and 1 = 2,4,. .. this corresponds with Theorem 7.4.3(b). The considerations in M. S. BAOUENDI, C. GOLJLAOUIC [3] are not restricted t o the self-adjoint case.
Similarly t o (7.5.212) we set for 0 < s
+ integer
Wi(Q; 8 )= B&(Q; d(), x 2 0 ,
(2)
where the spaces Bi,,(Q;#) are defined in Theorem 3.3.3. T h e o r e m 2 . If A is the operator from Theorem 1, and if 1 is not an eigenvalue of A , then A - / E generates for all s 2 0 an isomorphic mapping from
G'"(J-2; a2)
onto W i ( Q ) . R e m a r k 2 . For s = 0 , 1 , 2 , . . . the proof is given in M. S. BAOUENDI, C. GOULAOUIC [3]. If 0 < s is not an integer, then the theorem follows from Theorem 3.3.3 by interpolation. R e m a r k 3. Important generahations of these differential operators are mentioned at the end of introduction 7.1 and in Remark 7.2.213. R e m a r k 4. (1) and Theorem 7.3.2/1 yield D(Aj) c W$I(Q). One obtains by Theorem 4.6.1
n D(Z)= D ( A m )= Cm(sZ) .i-= 1 W
(set-theoretically and topologically).
7.7.
Domains of Definition of Fractional Powers
In this section, domains of definition of fractional powers of self-adjoint Legendre differential operators and self-adjoint Tricomi differential operators are determined. A complete solution of this problem is obtained only for the operators A1,*from
7.7. Domains of Definition of Fractional Powers
464
Definition 7.2.1, containing as special case the classical Legendre differential operator -((z - a ) (b - z)u')'. For general Legendre differential operators, there are some partial results formulated without proof. Further, we sketch how to determine the domains of definition of fractional powers of Tricomi differential operators.
Legendre Differential Operators (rn = k = 1)
7.7.1.
Lemma. Let - co < a < b < co. Further, let p ( x ) be a weight function in the sense of Definition 7.2.1. The s w e s Wi((a,b ) , p", 1) have the meaning of Definition 3.2.6. (a) For s > 0 it holds
C"((a, b ) ) c Wi((a, b ) , p 8 , I), 5 > 0 . (b) Let 0 5 s 1. Then C$((a, b ) ) i s dense in Wl ((a ,b ) , p", 1). Proof. Step 1. To prove (a), one must verify that
s
a
(2 > 0 is sufficiently small). Then we have
a
Using the sequence 1 = 2-k, k = k,, k, + 1, . . ., then it follows easily that fpF belongs t o %((a, b ) ) . Step 2. The definition of the spaces Wi((a,a), p", 1) and the first step yield that cm((a, b ) ) is dense in Wi((a,b ) , p", 1). Then it follows from the first step that @ ( ( a , b ) ) is dense in Wi((a,b ) , p 8 , l), 0 < s < 1. One obtains from Theorem 3.6.1, that CF((a,b ) ) is also dense in W?j((a,b ) , p , 1). 1 R e m a r k 1. The proof yields that pabelongs t o @((a, b ) ) ,provided that (T > s - - , 2 0 < s < 1. On the other hand, one obtains from (3.2.612) by a limit process that pa, 1 1 1 where u s - , is not an element of Wl((a, b ) ) , s =k , then pa,where 2 2 . If s = 2 (T < 0, does not belong t o i@((a,b ) ) = W t ( ( a ,b ) ) , Theorem 4.7.1. Finally, the lest relation yields that 1 = p0 is an element of ~ ( ( ab ) ,) . D e f i n i t i o n . Let -co < a < b < 03. Further, let p ( x ) be the weight function from 1 Definition 7.2.1. I f 0 s -+ - + I , where 1 = 0, 1, 2, . . ., then one sets 2
s
s
K" = K ( ( a ,b ) , ps,1)
(1)
7.7.1. Legendre Differential Operators ( m = k = 1)
465
1 (Definition 3.2.6).If s = - + I , where 1 = 0 , 1,2,. . ., then K 8 is the completion of 2 Cm((a,b ) ) in the norm
'I
6
llullIC# =
+ 1p"'(x)
lU(')(z)ladz
IIUI(W~((a,b),p*,l) [
a
2
-
(2)
R e m a r k 2. By the last lemma the definition is meaningful. To explain the special
1
position of the spaces K', where s = - + I ( I = 0,1,2,. . .), we generalize (2), 2
Here s = [s] + {s}, where [s] is a n integer and 0 6 {s} < 1. The proof of the and with {s) instead of 8) and (3.2.6/2)yield above lemma (with respect t o u([#l)
- llul18*
*
1
for {s} -2 1 Assume that (4)holds also for {s} = -and set again s = I 2 7.3.2/2(b),then it follows for u E I?""( (a,b ) ) n L& (a,b)) that ~~u~~W~(a,b),p*,l)
1
1J b b
p-'lu(')12 a x
5
c
a
a
IP* S
A?/) I axdy
(4u(%) 1% - YI2
+2
+
Iu(Z)12
as,
(5)
a+d
where 6 > 0. Setting p T u @ )= v , then one obtains by (5)that
I p-'lv12 ax 5
I
(4) Theorem
b-d
2
U(')(Y)
1 -.Using
b
(6)
CIlvl1b$((o,6))*
a
This inequality iq valid for dl functions w E C$((a, 6)). By Theorem 4.7.1,C$'((a, 6)) is dense in &((a, b)). But then (6)must be true for all w E @((a, 6)). If v = 1, then 1 one obtains a contradiction. Hence, (4)is not valid for {s} = 2
-.
R e m a r k 3. If s = I = 0,1,2,. . ., then K z = Wi((a,b), #) holds in the sense of Definition 3.2.114.This is a consequence of (7.3.1/2).Then Theorem 7.4.3 yields
= K l for 1=o,i,2,... (7) The determination of the domains of definition D(di,,),where s 2 0, requires the determination of the interpolation spaces (&((a, b)), Kz)0,2,I = 1,2,. . . This task is the counterpart t o the theory of P. GRISVARD and R. SEELEY from Theorem 4.3.3. The special cases from Theorem 4.3.3(b) a,nd (4.3.3/10) correspond t o the special
o(&)
1
cases for the spaces K" , where {s} = L. 2 T h e o r e m . Let Al,i be the operator from Theorem 7.4.1. Then D(&) = K2" if s 2 0 . P r o o f . Step 1. (7)proves (8)if 2s = 0, 1,2,.. . 30
Triebal, Interpolation
466
7.7. Domains of Definition of Fractional Powers
Step 2. Let 1 = 1 , 2 , . . . and let 0 < 8 < 1. Theorem 1.18.10, Theorem 1.17.1/1 (Remark 1.17.1/2), and Theorem 7.3.3 yield
if u E IiztlA @ ( ( a , b ) ) . One obtains by Theorem 3.4.2(d)that (&((a,b)), Wi((a,b), P,~ - * ) ) e , 2 = W!((a,b), pe, p-') Setting s
i + -,e 2
=-
2 7.3.2/2(b)that
then i t follows from ( Z ) , (4),Theorem 3.2.4/2, and Theorem
-
Ilull%(a;,~ IIu1)&,
u E KL+ln @((a, 6)).
(10)
Since Cm((a, b ) ) is dense in D( &,,) and dense in Kas (see the above lemma), it follows by (10) that D(Af,J and K2' coincide set-theoretically. Then they are also equal 1 in the topological sense. Whence it follows (8) for s 2 -. 2 1 Step 3. We must prove the t.heorem for the cases 0 < s < -. It follows from The2 orem 7.7.3 that ctr. = (& + E)-1 p" is an isomorphic mapping from K 2 n hi2)( (a, b ) ) onto K 2 and from K4 n Li2)((a,b ) ) onto a closed subspace r? 'of K4.If &' is the completion of C$'((a, b ) ) in K Q ;Q 2 0, then k4c k4c K4. (11) 1 For 0 < 8 < -, it follows from Theorem 3.2.6, Theorem 3.4.2(d), (7.3.1/2), and (4) 2 that ( 8 2 , i4),,,, = (~;((b a),, p 2 , p-2), w;t((a,b ) , p4, =
w;+2e(((,,
b ) , p2+2e,
p-2-2e)
= i2+2e.
(12)
On the other hand, one obtains from the second step that ( K 2 ,K 4 ) e , 2
=
(13)
K2+20.
The explicit form of the K-functional from 1.3.1, (12), and (13) yield
-
-
ll4lD(A-~~) (14) 1 for u E C F ( ( a ,b ) ) , 0 < 8 < -. Taking into consideration dim K 2 + 2 e / K 2 + z e < co, 2 then it follows from (14) that ( K 2 ,k 4 ) e , 2 = g2+2e is a closed subspace of K2+2' = D(A;?J containing One obtains from the above considerations, Theorem 1.17.1/1 (and Remark 1.17.1/2) that pc" = (Al,l E ) Cu is a n isomorphic mapping l141(K~,L%,2
lIUIlKa+ro
+
7.7.2. Legendre Differential Operators (General Case)
onto a closed subspace of D(A!,,). Using (7.3.2/6), where m pu" E C$'( (a, b ) ) and u E (a,b ) ) A LA2)((a, b ) ) that
c""(
IIP"IID(~!,~)
-
\Iullf,2+2e
-
467
=
2, then it follows for
I I ~ ( ! I Io~ 0 such that
for all mu1t.i-indicesa.Let L? c R,,be an (open) domain. Then u belongs to G,(L?)if u is an element of G J K )for each compact subset I< of The class G,(Q) coincides with the class of all analytic functions. Investigations on regularity properties in Gevrey C. GOULAOUIC[4,6] for Tricomi differential classes are made by M. S. BAOUESDI, C. GOULAOUIC [5,8] for (modified) opera%orsof second type, by M. S. BAOUENDI, Tricomi differential operators of first type (in the sense of Definition 7.2211, where m = k = l ) ,and by P. BOLLEY, J. CAMUS,B. HANOUZET [1,2] for differential operators of type (7.9.1/4).Further, we refer in this connection t o the papers by M. S. BAOUENDI, C. GOULAOUIC “7, 91, M. S. BAOUENDI, C. GOULAOUIC, B. HANOUZET [l], 0. A. OLEJNIK [l], and M. DERRIDJ, C. ZUILY [3].
a.
7.9.3. Further Types of Degenerate Elliptic Differential Equations
7.9.3.
475
Further Types of Degenerate Elliptic Differential Equations
In recent years, the general theory of degenerate elliptic differential equations and the theory of (regular and degenerate) elliptic differential equations in spaces with weights have been developed in a large scale. We cannot give here a survey on the different directions and problems. We mention thc survey papers by J. J. KOHN,L. NIRENBERU [l], and M. K. V. MURTHY,G. STAMPACCHIA [l], which influenced the further developments heavily. Further, me refer t o some papers closely related to the spaces described in Section 3.10: In the papers quoted in Subsection 3.10.3, there are, a t least partly, also considered degenerate elliptic differential operators which are related t o the spaces of Subsection 3.10.3. The spaces from Section 3.9 are very closely related t o the spaces from Subsection 3.10.3. It seems t o be possible t o develop a theory for elliptic differential operators in these spaces, similar to the theory in Chapter 6. The spaces with weights introduced by I. A. KIPRIJANOV (see Subsection 3.10.4) permit a systematic treatment of special degenerate elliptic differential operators. [2, 3, 5, 6, 71 and V. V. KATRACROV [l]. We refer t o I. A. KIPRIJANOV A. KUFNER [2] used the spaces described in Subsection 3.10.5 for the consideration of boundary value problems for elliptic differential operators.
8.
NUCLEAR FUNCTION SPACES
8.1.
Introduction
The aim of this chapter is the investigation of the structure of special nuclear function spaces. We restrict ourselves t o spaces which are closely related t o the considerations of the preceding chapters. A detailed knowledge of the theory of abstract nuclear spaces is not necessary for understanding this chapter. I n Section 8.2, we describe the general background and derive an important structure theorem. On this basis and with the aid of the previous results, there are obtained, in Section 8.3, results on the structure of special nuclear function spaces.
8.2.
The Spaces D ( A m )
In this section, the connection between the general theory of nuclear (F)-spacesand the more special considerations of this chapter is described. Further, we derive a structure theorem for the spaces D ( A ” ) .
8.2.1.
Nuclear (F)-Spaces
It is assumed that the general theory of locally convex spaces and of metric spaces [l], I, 1, K. MAURIN [ 2 ] ,111, or G. KOTHE[l]). is known (see for instance K. YOSIDA A complete locally convex space F is said t o be an (F)-spaceif its topology is generated by a countable set of semi-norms I(allj,j = 1 , 2 , . . . One can assume without loss of generality that
llalll 5 llalls S
*
-
*
5 llallj 5
Ilallj+l 5
..
*
(1)
For our purposes, it will be sufficient at the moment if we suppose that Ilal(j are norms (and not only semi-norms). If
then F becomes a metric space (inclusively its topology). An (F)-spaceis said to be a Monte1 spuce if every bounded set is pre-compact. Fj denotes the Banach space obtained by completion of F in the norm Ila(lj.The identical mapping from F onto itself can be extended to a linear continuous mapping I k , jfrom F,( into Fj, provided that 1 5 j k < 03.
8.2.2. The Structureof the Spaces D ( A W )
A
477
D e f i n i t i o n . (a) Let Bo and B, be Banach spaces. Then a linear continuous operator E L(B,, B,) i s said to be nuclear if it can be represented in the form m
m
(b) An (F)-spaceis said to be a nuclear space if for every natural number j there exists a natural number k = k ( j ) > j such that I k , j i s a nuclear operator from Fkinto Fj . R e m a r k 1. * It is easy t o see, that the definition of the nuclear space is independent of the way how the topology is generated by systems of (semi-)norms of the type (1). Since all the concrete examples treated in the following section are (F)-spaces, we restrict ourselves t o this class of spaces. The notation of nuclear spaces is essentially more comprehensive and includes important spaces which are not (F)-spaces. The (general) nuclear spaces are introduced by A. GROTHENDIECK[l]. Systematic treatments of the theory of nuclear spaces have been given by A. GROTHENDIECK [2] and by A. PIETSCH [3]. L e m m a . The space s 01 rapidly decreasing sequences = { E I E = ( E i ) E l ,61 wmplex, IIEIIj = SUP 1 j I E ~ I < a 1
for j = 0, 1 , 2 , . . .} (4) i s a nuclear (F)-space. Proof. One verifies easily that s is an (F)-space. Let el = ( 0 , . . ., 0, l , O , 0, . . .) where 1 occupies the place with the number 1. If F j and I k P have j the above meaning, and if 1 6 j 5 k < 00, then m
W
IkjE =
Z 1-1
Elel
=
C f t ( 5 )e l . 1-1
(5)
One obtains t,hat IlelllF, = U and ~ ~ f l ~=~ I-". F k ~ (6) Hence, I k , j is nuclear, provided that k 2 j + 2. R e m a r k 2. * The space s is of great importance for the later considerations. The space s holds also a central position in the structure theory for (general) nuclear spaces. If F is a (general) nuclear space, and if L is an arbitrary set of indices, then the is also a (general) nuclear space. Further, each linear subspace Tichonov product (F)L of a (general) nuclear space is also a (general) nuclear space. (See A. PIETSCH [3].) A. GROTHENDIECR[2] conjectured that every (general) nuclear space is isomorphic t o a linear subspace of ( s ) where ~ L is a suitable set of indices and s is the space of Y. KOrapidly decreasing sequences. This conjecture was proved by T. KOMURA, MURA [l]. A proof can also be found in A. PIETSCH [3], 11.1.1, and in S. ROLEWICZ [l]. See also G. KOTHE[2]. 8.2.2.
The Structure of the Spaces D(A")
L e m m a . Let H be a separable (complex) Hilbert space. Let A be a self-adjoint operator acting in H . Then m
D(Arn)= i s an ( F ) - s p .
n D ( A J ) , Ilhll.; = IlhllH + IlAjhIlH,
j-0
j = 1 , 2 , . . .,
(1)
478
8.2. The Spaces D ( A m )
Proof. Aj are closed operators. Whence it follows the proof of the lemma. T h e o r e m . Let H be a separable Hilbert space, and let A be a self-adjoint operator acting in H . (a) D(AO")i s a Montel space if and only if A is a n operator with pure p i n t spectrum. (b) D(A") i s a nuclear (F)-spaceif and only if A is a n operator with pure p i n t spectrum and there exist numbers c > 0 and z > 0 such that N ( 1 ) cl' -i- 1. (2) (Here N ( 1 )has the meaning of (5.4.1/1).) (c) D(A") is isomorphic to the space s of rapidly decreasing sequences if and only if A is a n operator with pure point spectrum and there exist numbers c1 > 0, c2 > 0, tl > 0, and t 2> 0 such that
c13Lt1 + 1 5 N ( 1 ) + 1 5 c21'2
+ 1.
(3) P r o o f . Step 1. Let D(A") be a Montel space, and let { E Q } - m < e 0 and r = [a - l/p]-. Ann. Scuola Norm. Sup. Pisa 27 (1973), 73-96. [22] Interpolation theory for function spaces of Besov type defined in domains, I , 11. Math. Nachr. 67 (1973), 51-85; 58 (1973), 63-86. [23] Uber die Existenz von Schauderbasen in Sobolev-Besov-Riiumen. Isomorphiebeziehungen. Studia Math 46 (1973), 83-100 [24] L,-theory for a class of singular elliptic differential operators. Czechoslovak Math. J. 28 (1973), 525-541. [25] Function spaces and elliptic differential operators. Coll. Intern. C.N.R.S. sur les Bquations aux derivBes partielles linhaires. AstBrisque 2 e t 3. SOC.Math. France 1973, p. 305-324. [26] Structure theory of function spaces. SBm. Goulaouic-Schwartz 1973/74, Exp. 3. [27] I'IHTepnOnRL(B0HHbIe CBOlfCTBa &-3HTPOIlHli H IIOnepeYHAKOB. reOMeTpHYCXKHe X a p a K T e p H CTEKA BJIOXeHHR npOCTpaHCTB t$YHKU& T H n a CO6OneBa-&COBa. M a T e M . C6OPHAK 98 (1979, 27-41. [ZS] Eine Bemerkung zur nicht-kommutativen Interpolation. Math. Nachr. 69 (19x9, 57-60. [29] Spaces of Kudrjavcev type I, 11. J. Math. Anal. Appl. 66 (1976), 253-277, 278-287. TULOVSEIJ, V. X. (T~JIOBCKHZ~, B. H.) [ 11 0 6 aCHMnTOTHYeCKOM PaCIIpeAeJIeHHH CO6CTBeHHblX YHCen BbIPO)KqaH)1WXCR 3JIJIHnTWIBCKHX YpaBHeHHk BTOpOrO nopanKa. M a T e M . C6OpHHK 86 (1971), 76-89. [2] ACEMllTOTHYeCKW P a C n p e n e J I e H A e CO6CTBeHHblX 3HaYeHHfi nHt$@peHI4€i&JIbHbIX YpaBHeHHlf. M a T e M . C6OPHHK 89 (1972), 191-206. TZAFRIRI,L., see LINDENSTRAUSS, J.
UNINSKIJ,A. P. (YHHHCKH~~, A. ll.) [1] Teopemr BJIOXCeHHR LInR KnaCCOB 4YHKLIHG CO CMeIIIarlHOfi HOpMOlf. C A ~MaTeM. . XYPHZIJI 10 (1969), 158-171. UNTERBERGER, A. [l] RBsolution d'bquations mix dBrivBes partielles dans des espaces de distributions d'ordre de rBgularit.4 variable. AM. Inst. Fourier Univ. Grenoble 21 (1971), 85-128. [2] Sobolev spaces of variable order and problems of convexity for partial differential operators with constant coefficients. Coll. Intern. C.N.R.S. sur les Bquations aux dbrivbes partielles IinBaires. AstBrisque 2 e t 3. SOC. Math. France 1973, p. 325-341. USPENSKIJ, s. v. ( Y C n e H C K d , c. B.) [I] r p a H H Y H b I e CBOlfCTBa (PYHKLIHA. AOKJI.aKaA. HayK CCCP 138 (1961), 785-788. [2] 0 TeOpeMaX BJIOXeHHR AJIR BeCOBbIX KJIBCCOB. TpyAbI MBTBM. HH-Ta HM. B. A. ClXKJIOBa aKm. HayK CCCP 61 (1961), 282-303. [3] 0 r e o p e M a x BJIOXCeHHR AJM 0 6 0 6 ~ 1 e H H b l X KnaCCOB w i Co6one~a.C H ~MBTBM. . mypHan 3 (1962), 4 1 8 4 5 . [4] Teopemr BnOXeHHX li n p o n o n x e H u X nnn OAHOrO Knacca @yHKUH8, I, 11. C H ~MaTeM. . XypHaJl 7 (1966), 192-199, 409418. [5] 0 T e o p e M a x B n o x e H l i R t$ymud B rnanrux 06nac.r~~. C ~ O ~ H,,TeopemI HK B n o x e m i n A HX npUnOmeHHn" (Tpynbr CHMII. T e O p e M a M BJIOXeHHR, k K y 1966). HayKa, MOCKBa 1970, CTP. 219-222.
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TABLE OF SYMBOLS
1.
Basic notations in abstract interpolation theory
2.
Abstract interpolation spaces and related spaces
3.
Spaces of L,-type, related norms 1.5.1 1.6.1 1.13.2 1.13.2 1.13.4
4.
s-numbers, ideals 1.16.1 1.16.1 1.16.1 1.16.1 1.19.7
520
Teble of Symbola
Special domains in R,, special sets of functions 2.3.1 2.9.1 2.11.2 3.2.1 3.2.3
@N, @
Y = Y(Q;@)
z
MP,t
2.3.1 3.2.3 3.9.1 2.5.4
Special operations 2.3.4 2.10.3 2.5.2 2.5.3 2.5.1 2.5.1
G { b-
*
a ,
a}
@-operator [a19
(4
[$I-, {4+ N(&
2.9.2 5.2.2 2.5.1 2.5.1 5.4.1
Banach spaces of functions and distributions on R, and R; 2.3.1 2.3.1 2.3.1 2.3.1 2.3.2 2.9.1 2.9.3 2.9.3 2.9.3 2.9.3 2.10.3 2.10.3
2.9.2 2.9.2 2.9.2 2.9.2 2.9.2 3.6.4 3.6.4 3.9.1 3.9.3 3.9.3 2.7.1 2.7.1
Banach spaces of functions and distributions on domains 4.2.1 4.2.1 4.2.1 4.2.1 4.2.1 4.2.1 4.3.2 4.3.2 4.3.3 4.3.3
W,”(Q;a) B;,q(Q; 4 @(Q; a) B;,(aQ) B;,q(Q;e”; e’) H;(Q; e”; e’) w;(Q; e”; @’)
ii..,(Q;e”; e’) w;((a,b ) , P)
K8
3.2.1 3.3.1, 3.3.3 3.2.1 3.6.1 3.2.3 3.2.3 3.2.3, 3.2.6 3.2.6 7.5.2 7.7.1
Table of Symbols
7.4.4 7.6.5 5.2.3 6.2.1 2.2.2 8.2.1 6.2.1
10.
Differential operators 6.2.1 6.2.1 7.2.1
BmA 4n,k B",k
7.2.1, 7.2.2 7.4.4 7.6.6
621
AUTHOR INDEX
ADAMS, R. A. 170,307,329,354,483 AQMON,S. 151, 314, 334, 364, 368, 378, 384, 389, 396, 397, 403, 404, 434, 483 AQRANOVI~,M. S. 382,483 AKILOV,G. P. 141,497 ALIMOV, s. A. 336, 483 T. I. 207,224, 358, 483 AMANOV, A N D E R S O N , R. 329,483 P. J. 353,354,483 ARANDA, L. 368, 378, 484 ARKERYD, &ONSZAJN,N.16, 20, 21, 25, 69, 144, 151, 169, 170, 180, 181, 190, 224, 245, 364, 388, 483, 484 AUDRIN,J.-M. 353, 354, 484 A. 245, 484 AVANTAGGIATI, S. B. 350,484 BABAD~ANOV, R. J. 171, 179,484 BAQBY, L. A. 407, 484 BAGIROV, L. A. 407,484 BAQIROV, BALAKRISHNAN, A. V. 98, 100, 484 S. 236, 237, 484 BANACH, M. S. 118, 433, 460, 463, 471, 472, BAOUENDI, 474, 481,484, 485 BASS,G. I. 396, 485 BEALS,R. 396,485 BEAUZAMY, B. 360, 485 S. 433, 485 BENACHOUR, A. 136, 306, 485 BENEDEK, C. 136, 138, 143, 485 BENNETT, H. 16, 23, 25, 29, 62, 75, 81, 96, 98, BERENS, 100, 127, 132, 143, 144, 150, 179, 190, 485, 488 C. A. 136, 485 BERENSTEIN, Ju.M. 151, 231, 364, 378, 380, BEREZANSKIJ, 382, 388, 394, 400, 458, 486 G. 469,473,486 BERQER, M. S. 354,486 BERQER, BERIEV,A. D. 306,486 O.V. 86, 151, 161, 169, 170, 180, 190, BESOV, 207, 225, 242. 245, 249, 265. 280. 282. 306. 313, 316, 357, 358, 486 ,
