Introduction to Global Variational Geometry, Volume 18 (North-Holland Mathematical Library)

North-Holland Mathematical. LibrarySmrd of Aduisory Edi&rs: M. Artin, H. Bass,J. Eells, W. Feit, P. J. Freyd, F. W. Geh...

Author: Demeter Krupka

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(22)

Rm

where c depends only on n and p (but not on N). f is of type (3). Step 3. For the operator g N ,the limit process N + 00 is considered. Since [Lp(a2)]' = L P ~ ( lin 2 )the sense of the formulas (2.2.2/24)and (2.2.2/25),formula (22) is eqmvalent to

and equivalent to

1

C 1( P K k , j ) ( f ) k,j= -w R 00

'

F f j ( E )F - l h k ( 8

Whence it follows that This proves the part (a) of the theorem.

5 cBllfI14(Za)IlhllLp/(u

2.2.4. Multiplier Theorem

165

Step 4. To prove the part (b), we consider again the operator g Ndefined in (13) and (14). Now, it holds U t j ( s ) = 0 if k j . Since

llQN(s

- y ) - QN(s)~~L(~r,= Z,)

k

la&(z

- 3) - Qsc”lk(,)l

(21) remains true after replacing L(l,, I,) by L(lr, Zr). In the same manner, it follows l l ~ ( s ) l l ~ ( t , ,E t ,L1(R,J. ) Considering (22) with p = r for f ( j ) ( s ) = (. . ., 0 , f j ( s )0, , .. .), and summing the so obtained estimates, we find that m

But now, the assumptions of Theorem 2.2.2 with A, = A, = I,, q = 1, and p = r are satisfied. Hence, it holds (22) with 1, instead of I , . Using the considerations of the third step with 1, instead of I,, one obtains now the part (b) of the theorem. Remark 1. Both the theorem and the above proof go back to H. TFLIEBEL [19], Theorem 3.5. But both the theorem and the proof are generalizations of a result [2], Theorem 2.5, for the scalar-valued case to the vectorfound in L. HORMANDER valued case. The proof shows that one can weaken the assumption (4) for the proof of part (b). Further it is easy t o see that the theorem and the proof remain valid, if D”(FKk,j)(t)are regular distributions satisfying (4). Remark 2. Giving up the representation (2) as an integral operator, one can generalize the theorem. The remarks in Subsection 2.2.1 show that (2) may be represented as m

Wf) (4 = {(en)+F-l ( j = c - FKk,jFij)} -m
Now, T = and if for all f

=

is said to be a multiplier-matrix of type ( p , p),if Tk,j ES’(R,,), with f j E X(R,) and f j = 0 for ljl > N we have

where Cis a number independent of f . By the proof of the theorem, it follows immediately that T is a multiplier-matrix of type ( p , p ) , where 1 < p < CQ, if (4) is satisfied with Tic,jinstead of FKk,,i.A similar assertioia holds for the generalization of the part (b) of the theorem. This version explains also the notation “multiplier”. R e m a r k 3. * The assumption (4) is satisfied, if there exists a positive number B such that for all multi-indices 01 with 0

5

For the scalar case, this coincides essentially with the condition originally formulated by S. G. MICHLTN[l, 2,3]. The condition (4) or (24), respectively, may be replaced for the scalar case by the following assumption: For x = ( x , , . . . , x,,), where xj + 0 ,

166

2.2. Integral Operatom and Fourier Multipliers

j = 1, . . . , n, K ( x ) has continuous partial derivatives up to the order n inclusively. Further there exists a positive number B such that for all multi-indices a = (a1,. . ., u,,) where mj is either 0 or 1, and for all x = ( x l , . . .,xn) where xj 0, j = 1, . . ., n,

+

141. . . *WPK(x)I 5 B . Then X E L(L, , L p )and IlXll I; BB) where B depends only on n and p . This theorem [2] and it is a generalization of the original formulation by is due to P. I. LIZORKIN S. G. MICHLIN.A proof of this theorem may be found also in S. M. NIKOL'SKIJ [7], p. 69. These statements remain valid also for the generalization described in Remark2 (with respect to the scalar case). R e m a r k 4. * The first multiplier theorem for trigonometrical series was proved by J. MARCINKIEWICZ [2] in 1939. Using these results, S. G. MICFXLIN [l, 2 , 3 ] obtained similar results for Fourier integrals (scalar case). In the framework of the theory of translation-invariant operators, L. HORMANDER [2] described further generalizations and simplifications. Remarks 2 shows that the statement of the theorem for the scalar case is equivalent to IIF-YFK * Ff)ll~p 6 B B l l f l l ~ ~f, € S ( R n ) *

(25)

FK is called (Fourier) multiplier of type ( p ,p ) . Generalizing this notion, one denotes by M; the set of all distributions T E S' such that for all f E S where Cisindependent of f . Here 1 < p, q c co. The elements of M! are said to be (Fourier) multipliers of type ( p , q). It holds that and

Mi=L,,

M$cL,

if

1 c p < co

1 1 1 1 1 0 Mq P

= M$

if

such that for all R > 0 and all a with

B IPT(x)l 5 12116( for all x

1011

5

E R,,

1

+

with x

El

- we have

+ 0.

This is the usual formulation of the multiplier theorems of M I C ~ I K and HORYANDER. Generalizations are given by P. I. LIZORKIN [ 2 , 5 , 7 , 8 ] , where the last two papers [l,21, are surveys. Further results can be found .in L. CATTABRIGA[l], P. KRI~E S. IGARI, S. KURATSUBO [l])and G. 0. OKIKIOLU[l].In the paper by J. T. SOHWARTZ [l], these considerations axe carried over to the vector-valued case. Generalizations of these investigations are given by W. LITTMAN, C. MCCARTHY,N. M. R ~ R [l] E

2.2.4. Multiplier Theorem

167

and N. M. RIVI~RE[l], where these papers contain rather general results. Further, we refer also to papers written by P. I. LIZORKIN [lo, 111. Multipliers in L,-spaces [l].Tceatments of with weights are considered by P. KREE[2] and A. I. KAMZOLOV aspects of the theory of multipliers can be found in E. M. STEIN[a, 51 and R. LARSEN [l]. The book written by R. LARSEN contains a comprehensive bibliography. Remark 5. * The connection between interpolation theory and the theory of multipliers was considered by J. PEETRE[14], W. LITTMAN [l], and J. LOFSTROM [l]. The aim is to weaken the assumption (4) of the theorem in dependence on p . For this purpose, one “interpolates” between ‘ ‘ M ; = L,” and “ ( 4 ) ” and contains conditions depending on p . For details, we refer to the cited papers. See also C. FEFFERMAN, E. M. STEIN[l] and R. JOHNSON [2]. Remark 6. Clearly, one can extend the definition of multipliers in the sense of (26) to p = q = 1 and p = q = 00. The set of all these multipliers is denoted by M i and M g ,respectively. Since these cases are not important for the later considerations, we do not go into detail here. But we formulate an interesting result: M i = M z is the set of all finite Borel-measures i n R,. A proof can be found in E. M. STEIN[5], p. 94/95. Now we consider a special multiplier which is not contained in the last theorem. Lemma. If Q = {x I x = ( x l , . . ., x,), Ixjl < a,}, aj > 0 , then the characteristic function xQ(x)of Q belongs to M! for all p with 1 < p < 00. Proof. Step 1. First, we consider the one-dimensional case. It holds that

If f

=

E S ( R l ) ,it

follows that

-a

1

m

sina(x - y ) f(Y) dY 2 - Y

--m

sin ax 1 cos ax -cos a y f ( y ) dy - 1irnIT 7L 2 - Y L I-?’-ylZe

e l 0

-

-1 I

Theorem 2.2.312 yields that xr-.,.l belongs to M$. Step 2. We consider the case of several dimensions. Temporarily we set (?if)

(21 3

..

* 3

Xj-1,

6 ,xj+1,. . ., x,)

1 e-izjEjf(x)d x j , a,

= (2n)-+

f

E S(R,),

--03

and extend the definition of Fj to S‘(R,) in the usual way. Similarly, we define F;l. If x,(x) denotes the characteristic function of the set {x I -aj < xj < uj), then n

XQ(X) =

Il Xj(x)J=1

Let f

E S(Rn).Uaing

=

the first step, one obtains that

II IIFy'XjFjf(x1,

* * - 9

s cllf

IILp(Rn) *

xj-1,xjs z j + l ,

-

* *>

zn)IILp(Ri)II~(R,,-i)

..

Since F-lXQFf = F-1x1FF-1x2FF-1. Ff, the desired result follows by iteration. R e m a r k 7. The question arises for which domains 52 c R, the characteristic function x&) belongs to M ; . C. FEFFERMAN [2] has shown that the chracteristic function of a ball is an element of M; ,if and only if p = 2. This result was extended [4] t o a large class of closed sets in R,. One can generalize the by B. S. &JAGIN question and ask for which values of p the function

belongs to M$ . A summary of results in this direction can be found in C. FEFFERW [3]. Sufficient conditions for multipliers in dependence on p are also given by W. TREBELS

[4].

I n this section, the spaces Bi,q(Rn)and F;,JRn) containing the spaces HJR,) and W;(Rn)as special cases are introduced. At the same time, we shall prove a number of important properties. As it will be shown later, Bi,q(Rn)are the Besov spaces, H;(R,) are the Lebesgue (Liouville, Bessel-potential) spaces, and W;(Rn)are the SobolevSlobodeckij spaces.

2.3.1.

Definitions

s

The spaces ,?;(A),where 1 5 p co, - 00 < s < 00, have the same meaning as in (1.18.2/1).If A is the complex plane, we will write Zi instead of Zi(A).As usual, the support of the distribution f is denoted by supp f . The convergence of a series in S' is marked by . F and F-l are the Fourier transform and the inverse Fourier transform in S', respectively. Further we set Mj

= {E

15 E R,,, 2j-l 5 151 I - Zit'},

Mo = {5 I 5 E Rn 151 S 2 ) * 9

j = 1,2,.

..,

(1)

2.3.1. Definitions

169

andfor -co < s < c o , l < p < co,andq= coonesets

(c) For -co < s < co and 1 < p <

H;(Rn) = Fi,dRn)(d) For 1 < p < 00 one sets

00

one sets

if s = O , 1 , 2 ,..., Wi(R,) = (Hi(Rn) (7) B;,p(R,,) if 0 < s =l=integer. R e m a r k 1. The considerations in Subsection 2.2.1 show that any distribution aj E S ( R n )with supp Faj c Mj is an analytic function. Sometimes we shall make use of a, = aj(s)E Cm(R,,). R e m a r k 2. * The spaces Wi((Rn)with s = 1 , 2 , 3 , . . . coincide with the well[l,2 , 3 , 4 ] in 1935-1938. The known Sobolev spaces introduced by s. L. SOBOLEV extension of the definition of the spaces W;(R,) to values s with 0 < s =I= integer (Slobodeckij spaces) is closely related to the investigation of boundary values of [l], L. N. SLOBOfunctions belonging to Sobolev spaces. We refer to N. ARONSZAJN DECKIJ [l], and E. GAGLIARDO [l]. As it will be shown later, the spaces with s > 0 coincide with the well-known Lebesgue (or Liouville, or Bessel-potential) K. T. SMITH [l] and A. P. CALDER6N [2]. spaces introduced by N. ARONSZAJN, The spaces B;,q(RIL) with s > 0 are the Besov spaces, see 0. V. BESOV[l, 21. The definition of the spaces Bi,q(R,)described above was given by H. TRIEBEL[19] and is closely related to corresponding definitions formulated by S. M. NIXOL'SKIJ [7]

and J. PEETRE [15]. The definition of the spaces Fi,q(R,)goes back t o H. TRIEBEL [19]. I n this connection, we refer t o the papers by P. I. LIZORKIN [12,14], and the spaces L;,q(R,) introduced there which are similar t o the spaces Fi,q(R,), and by J. PEETRE [39]. Finally, we mention the spaces introduced by S. M. NIKOL’SKIJ [2, 3,4] (and often called Nikol‘skij spaces) which coincide with the spaces Bi,m(Rn). In this connection, we remark that (in contrast to the notat,ions used here) the Nikol’skij spaces are often denoted by Hi(R,) and the Lebesgue spaces by Li(R,). Many references, particularly t o papers of Soviet mathematicians (0.V. BESOV, V. I. BURENKOV, K. K. GOLOVKIN, V. P. IL’IN, L. D. KUDRJAVCEV,P. I. LIZORKIN, S. M. NIKOL’SKIJ, S . V. USPENSKIJ, . . .), can be found in S. M. NIKOL’SKIJ[7]. Further we refer t o the survey papers by N. ARONSZAJN, P. MULLA,P. SZEPTYCKI [l], R. A. ADAMS,N. ARONSZAJN, K. T. S ~ [l], H and E. MAGENES[l]. Purther references, particularly t o the spaces B;,,,(R,)with s 5 0, are given in Remark 2.3.412. See also Remark 2.3.2/1. R e m a r k 3. By the definition of the spaces Bi,q(R,,),ff;(R,J,and W;,(R,J,we restrict ourselves to the case 1 0 these spaces coincide with the Besov spaces, Lebesgue spaces, and Sobolev-Slobodeckij spaces, respectively. But for these types of spaces, the limit cases p = 1 and p = co are also of interest. One could try to carry over Definition 1 (or one of the later norms equivalent t o the above oneif 1 < p < co) t o these cases. We do not go into detail here. but we add some remarks : 1. The later considerations are based on Theorem 2.2.4 which is valid only for 1 < p < 03. By an extension of the considerations to p = 1 and p = co,one has t o use Remark 2.2.416 (and a generalization t o the vector-valued case). 2. By the treatment of the spaces with p = 1 and p = 00, there arise new difficulties. The results obtained for 1 < p < 03 are not always true for these limit cases. Particularly, there are several non-equivalent possibilities of definition. (See Remark 2.3.3/5 and Remark 1.13.4/2.) 3. One of the main aims of this book is the application of the theory of function spaces t o elliptic differential operators. But here again a restriction to 1 < p < 00 is meaningful. 4. The definition of the spaces BL,m(R,,)in analogy t o the norms in Theorem 2.5.1 is meaningful and gives interesting spaces. See for instance E. M. STEW[5], p. 141. The spacea BL,m(R,) are closely related to the Holder spaces Ct(R,,) and V t ( R )defined in Section 2.7. The latter spaces are also necessary for the formulation of the embedding theorems in Section 2.8. R e m a r k 4. * The main aim of this chapter is the systematic consideration of the spaces Bi,q(Rn),F&(Rn), Hi(R,,), and W;(R,) (embedding theorems, traces on the boundary, interpolation theory, equivalent norms, duality theory, structure theory, . . .). At the same time, we shall be concerned with the restriction of these spaces t o R,+= {x I x E R n ;x , ~> O } . Further, we consider some properties of Holder spaces. I n the later chapters, we deal with the restriction of the mentioned spaces t o arbitrary domains L?c R,. Furthermore, Sobolev-Besov spaces with weights are considered. The extensive theory of anisotropic spaces is not treated here, but there are some remarks in the last section of this chapter. To this part of the theory, we do not give references here, although numerous new publications have appeared since the publication of S. M. NIKOL‘SKIJ[7], where a bibliography of this subject up t o 1967-69 can be found. But we want t o mention here shortly some classes of isotropic spaces which are important but not conaidered later on. Lebesgue-Besov spaces

2.3.1. Definitions

171

defined on general structures are introduced by M. H. TAIBLESON [2] and vectorvalued Sobolev-Slobodeckij-Besov spaces (with values in Hilbert spaces and Banach [l], P. GRISVARD [4,6], N. N. FROLOV [l] and other spaces) are treated by J. WLOKA mathemat,icians. Interpolation theorems and embedding theorems can be found in P. GRISVARD [4,6]. Sobolev-Besov spaces WL and B;,q with 0 < p < 1and 0 < p < 00 are considered by J. PEETRE [32,35,36] (this includes interpolation theorems, duality theorems, and embedding theorems) and T. M. FLEW.[2] (duality theory). [36] is an extension of Definition The definition of the spaces Bi,q by J. PEETRE l(a) and Theorem 2.3.2(a) to the values 0 < p < oc) and 0 < q < a.An inter[3]) can be found in polation theory for the Hormander spaces (see L. HORMANDER M. SCHECHTER [5,6]. We mention also the Morrey spaces, John-Nirenberg spaces, and the Campanato spaces. A short summary and a description of interpolation [l] and J. PEETRE [20]. The interpolation properties are given by G. STAMPACCHIA theory of these spaces was developed by S. CAMPANATO, M. K. V. MIJRTHY,S. SPANNE, References can be found in G. STAMPACCHIA [11 and J. PEETRE and G. STAMPACCHIA. [20]. As it will be shown later, the Lebesgue spaces are closely related to (elliptic) Bessel-potentials.Similar considerationsfor Lebesgue spaces based on parabolic potentials are given by R. J. BAQBY [l]. In this paper, there is also given an interpolation theory for spaces of such a type (complex method). C. S. HERZ[l] and R. JOHNSON [l I introduced spaces defined with the aid of Riesz potentials. Finally, we mention E. M. STEIN[l]. There are the theory of the Hp-spaces treated by C. FEFFERMAN, also interpolation results for the complex method. Similar results for the real interpolation method are obtained by N. M. RIVIERE, Y. SAGHER [l] and C. FEFFERMAN, N. M. RIVIERE, Y. SAQHER [l]. See also M. ZAFRAN [l]. For the later considerations special systems of functions are of interest. Definition 2. If N i s a natural number, then @N denotes the set of all SY8bn-S of functions ( q ~ , ~ ( x with ) } ~ -the ~ following properties :

There exists a positive number c1 such that

c 00

c1

5 k=O

(FQ)k)

(8.

(9)

For aity multi-index a,there exists a poaitive number c 2 ( a ) such that

F u r t h r , one sets @ =

u GN.

N=1

Example. It is not difficult to describe examples of systems of functions of the above type. Let p(z) E X(R,) be a function with (Fv) (5) 2 0 and

SUPPFF c (5 I 5 E Rn, 2-N 5 151 5 2N),

( ~ 9(5) ) > 0 if

1

-5

12

-

-

IEI 5 1 2 .

(11)

and choosing eo in a suitable way, then ( @ k } g a E @N and it holds that

z 00

k=O

( F e d (5) = c.

(13)

We shall make use of these examples later on.

2.3.2.

The Spaces Bi,q(Rn)and Fi,q(Rn)

As usual, CF (R,) denotes the set of all complex-valued infinitely differentiable functions with compact support defined in R, . “c” means alwaysa continuousembedding. Theorem. (a) Let -00 < s < co, 1 < p < 00, and 1 q a.Then BRT;6q(Rn) is a Banach space. Further, for a n y system of functions {Q)k}& E @ we have

Bi,q(Rn)= {f I f

E s’(Rn)

; Ilf

=

Il{f * ~ k } I I $ ( L P )< a)-

(1)

Here, llfll$;,g is an equivalent mmn in the space Bi,q(Rn).If q < co, then C$’(R,) and S(R,) are dense in B&(R,). On the other hand, C$(R,) and X(R,) are not dense in

q, m(Rn).

(b) Let -CQ < s < co, 1 c p < co, and 1 < q < C Q . Then F;,q(Rn) is a Banach space. Further, for a n y system of functions {Q)k)&, E @, we have Fi,q(Rn) = { f I f Here, 11f Fi,q(Rn)*

I I f II*F.,,*= II{/ * Y ~IIL~(I:I ) < 00) -

E R(Rn),

(2)

is a n equivalent norm in the space Fiiq(Rn).C;(R,) and X(R,) are dense in

2.3.2. The Spaces Bi,q(Rn)and Fi,((R,,)

Proof. Step 1. We prove (2). Let f

E Fi,q(Rn)and

173

let

m

be a representation in the sense of Definition 2.3.1/1(b).Let { Q ) k } & ) E @ N . Then aj * p)k = F-lF(aj * Q)k) = F-'(2n)"12 FQ)$Uj = 0 , if either j < k - N - 1 or j > k + N + 1. Using the remarks in 2.2.1, whence it follows that

f * p)k F I c -0

a,

aj

* qk

c

k+N+1

=

j=k-N-1

(aj * vk) (2).

The functions ak(z)belong to Lp(Rn).Now, we approximate these functions,

CF(R,,) 3 akBE 2 a,+ if E 1 0 . (6) If we set K j , j ( z )= ~ ) j ( for z ) j = 0, 1,2, . . . and Kj,j(z) = 0 for j < 0, then, for fixed E, r, and M, Theorem 2.2.4(b) is applicable ((2.2.4124) is an immediate consequence of (2.3.1/8) and (2.3.1/10)).One obtains that instead of akfr,€.) (Here, the numbers 2sj do not give any trouble, one considers 28jak+r,E The number c is independent of E , r , and M. A limit process shows that the last ineWith quality is true for E = 0, too, and hence it holds also for akcr instead of ak+r,E. M = 00 on the right-hand side, it follows that

Il{f * Q)k}llLp(2i) 5 cll{ak}IILp(li)* Construction of the infimum yields

Il{f * v k } k ) l h ( l i ) s

(7)

CllfIIF;,,.

Step 2. We prove the conversion to (7). We suppose that the left-hand side of (7) is finite. Let e(z) E CZ(R,) be a real function such that e(z) = 1 if 2-N 5 1x1 5 2N, e(z) E CF((5 1 2 - N - 1 < 151 < 2 N + ' } ) . Let &(Z) = @(2-'Z) if k = 1,2,. . . We choose a real function po(x)such that

eo(4 Let

{Q)k}&,E @N

s

= 1 if 1 4 2N, and let K j , j ( z )= 0 for

( F K j , j () z ) = (2n)-n(

03

C

1=0

e o ( 4 E CI?({E I 151 < 2N+'}). < 0, and

Fqz)-'ej(s), j = 0 , 1 , 2 , .

Then the assumptions of Theorem 2.2.4(b) are satisfied. Let vk = Q)k

* Kk,k

n

(Fvk = (2n)TpvkFKk,k).

..

174

2.3. The Spaces Bp,q(Rn),li'i,q(Rn),H;(R,,L and W;(Rn)

Then by Theorem 2.2.4(b) (after a continuity argument analogously to the first step with respect to the functions Q)k * f E Lp)

we find that

* qk,the desired inequality is a consequence of 5 Il{ak)\ILp(l;) c\l{f * Q ) k ) l l $ ( 1 : ) *

Let N = 1. Choosing ak = f IlfIIF:,,

Now let N > 1. We consider a system n

{ ~ ~E } gfor~which (2.3.1/13) with

(9) xk

instead of ,ok and c = (2n)-T holds. If Kk,k= x k for k = 0, 1 , 2 , . . . and Kk,k= 0 for k < 0, then the assumptions of Theorem 2.2.4(b)are satisfied. Hence, it holds that and

l({f*Wk*Xk+r}IIL,(Zi) s C ( I { ' / ' k * f } l l ~ ~ ( l : ) ,

c

k+lV+l

f*yk=

l=k-N-1

r = - N - l , . - - ? N +1,

f * Y k *XI.

(Here x1 = 0 for 1 < 0.) Setting

and

SUPpUk m

t

M/<, k

c %(z) =

k=O

m

k=O

(

=

0, 1, 2 , . . .

lV+ 1

r=-N-1

(see (2.3.1/1))

f * yk * X k t r )

c (f * W

=

k=O

Yk)

7f

(8) and (10) prove (9). Now, together with the first step, one obtains (2).

-

Step 3. The proof of (1) follows the same line. One has to replace I( llLp(p)by (1 * llgLp). Further, it is sufficient to use the scalar case of Theorem 2.2.4(b)where the number c in IlYk

* ak+rllLp

Cllak+rllLp

is independent of k. The limiting cases q = 1 and q = 00 do not give any trouble in these considerations.

2.3.2.The Spaces Bi,(R,) and FiJR,)

176

Step 4. We prove (3). The third, fourth, and fifth inclusion follow immediately by the monotony of the 1,-spaces. The second (and hence also the last but one) inclusion is a consequence of II{2(s+e)jII f * vjIIj}IIlm.

I I ~ I I BI ~ ,C~I I { ~ ~ ~*I IvjIILp}IIh / 5

Here f E Bi:&(R,,).If we choose a suitable system {qj]T4 E @ (for instance such one as in Example 2.3.1), then the first inclusion for f E ,S(Bn)(and sufficiently large a) follows from 2J(s+qF--1(Ffpj * Ff)llLp

5 - C12j'8+8)(( (1

x

C

[lul82n

+ 1212)" P - l ( F f

sup (1

XER,

+

*

F y j ) I(L ,

IZ12)-o+s+e

(DdPcpj(S)l].

Let {qj}so E @ be a system described in Example 2.3.1 where (2.3.1/13) holds, with n

of e k and with c = (zn)-T. Let { y j } z 0E @ with (Fyj) (5) = 1 for 5 E supp Fqj. Then the last inclusion is a consequence of

'pk instead

and of the just proved first inclusion. Step 5. We prove (d). By the monotony of the 1,-spaces and the trivial assertion B&(Rn) = F;,p(Rl,),one obtains the right-hand aide of (4a) and the left-hand side of (4b). If {yj}j?o E @, then the left-hand side of (4a) follows from

176

2.3. The Spaces BiJR,,), Pi,p(R,,),Hi(Rm),and Wi(Rm)

6 c'lIfllF;,q. Step 6 . The last two steps show that (3) is true, too, after replacing B by F . Step 7. Using (1)and (2), we find out that Bi,q(R,,)and Fi,q(R,,)are normed spaces. To prove the completeness of BP,q(R,,),we consider two systems of functions {pj);~ E @ and { y i s o E CP. Here, {p,>sosatisfies (2.3.1/13) with pj inatead of and with c = (2n)-T, and it holds Fyj(6) = 1 for and

S(f) = {f

* pj>jmpo

9

S

It follows that

E L(Bi,q(Rn), l@p))

m

R ( { f j l )7 S

E supp Fp,.

C vj(z)* IjCz), j=o

R

(12)

E L(li(Lp), Bi,q(Rn)).

(13)

The convergence of the series (13) in S'(R,,) can be obtained analogously to the estimate (ll),where one has to take into consideration 1; c li-". Further, one has to useTheorem 2.2.4(b) in the same manner as in the first steps. Now, it is not hard to see that S is a coretraction and R is a retraction in the sense of Definition 1.2.4. By Theorem 1.2.4, it follows that S is an isomorphic mapping from B;,q(R,) onto a complemented (and hence closed) subspace of li(Lp).Whence one obtains the completeness of Bi,q(R,)and (in a similar way) of F;,q(Rn). Step 8. If 1 < q < 00 (resp. 1 6 q < a), and if the functions uj(z) have the same that meaning as in Definition 2.3.1/1, then it follows by the theorem of LEBESQUE the functions

M

C

j=O

u,(x) are dense in F;,*(Rn)(resp. in Bi,q(Rn)).Approximation of

q(z)in LJR,,) by functions belonging to C$(R,) or to S(R,) shows that both C$(R,,) and S(R,) are dense subsets in F;,,(R,,) (resp. in Bi,q(Rn)).Finally, it remains to prove that S(Rn)(and hence also C$(R,,))is not dense in B;,,,(R,,). The mapping properties of F and the first inclusion in (3) show that it is sufficient to prove that the functions p E S(Rn)with Fp, E Cg(Rn)are not dense in B;,,,(R,,). Now, let {ui},To be a sequence of functions with

supp uj c N j (see (2.3.1/1)) and 2sj~~u,~~Lp = 1. Then the consideratiom after formula (13) show that

a,

C

j=O

u,(z) belongs to

B;,m(Rn).On the other hand, it is not hard to see that it is impossible to approximate f by functions of the above type. Remark 1. Norms of the type (1)for the spaces BiJR,,) are first used by J. PEETRE [El, while (2.3.1/2a) essentially goes back to S. M. NIEOL'SKIJ [7]; see also Remark 2.3.1/2. In J. PEETRE [12], there are shortly mentioned norms of type (2). The

2.3.3. The Spaces H;(RJ

177

described proof is a modification of corresponding considerations by H. TRIEBEL r191.*) Definition. T h completion of CZ(R,) in Bi,m(Rn),-03 < s < 00, 1 < p < 03, is denoted by S;,a(Rn). R e m a r k 2. The part (a) of the theorem shows that h;,w(Rn)does not coincide with Further, it follows by (3) that S(R,,)is dense in g i , w ( R n )too. , Bi,m(Rn).

The Spaces Hi(&)

2.3.3.

The spaces Hi(R,,)have been defined in (2.3.1/6). The following theorem shows that these spaces are of special interest. Theorem. (a) Let

-00

< s < 00 and 1 < p < 00. Then 6

I f E S'(Rn),IIjIIH; = IIP-Vl + IzIa)-ji-WtILp< a}Here, IlfllHI is an equivalent norm to llfll .

Hi(%)

P

(b) I f 8 = 1,2,3,

Here,

= {f

. . .and 1 < p

<

5.2

03,

then

~ l f ~ ~ wis, an equivalent norm to IlfllH;. ?

Proof. Step 1. Let f E S'(Rn)be an element such that 9 = F-l( 1

Let

(1)

{cpj};o

E @.

+ lzI2)FFf 8

I

HP

< 03. We set

E Lp(Rn).

We shall use Theorem 2.2.4(a) with

~ j , ~ (=z~-1(28j(1 ) + Kj,k

llfll

=0

~zla)-$pqj) if j

= 0,1,2,

. . .,

otherwise,

and = {gj],Cm with go = g and gj = 0 otherwise. It follows (after a continuity argument as in the first step of the proof of Theorem 2.3.2)

IIf IIF;,, S cII{F-YFf Fqj

IIL,,(z,) = cII{P-YPKj,o * Pg)}I I ~ ( u * g}I1~p(tr)S ~ " l l ~=l~l "~I l f l l ~ ~ * . *

= c'II{Kj,o

*

2''))

(3)

Step 2. We prove the conversion to (3). Let f €F;,g(Rn)= Hi(R,,).We choose a system { ~ j } , ? E ~ @ such that

*) Recently, J. PEETRE [38] considered applications of the spaces Pin(B,,).

12 Triebel, Interpolation

(see Example 2.3.1). Further, we choose a second system (~,}i”,~ E @ such that

( P y j ) (6)= 1 if 5 E SUPP Pvj

Then for and

fi

=

2j.f

-

if j = 0, 1 , 2 , . . .,

* vj

= 0 otherwise,

fj

K , , j ( x ) = y j ( z ) if j = 0, 1 , 2 , . . ., Kk,,(z) = 0 otherwise,

the assumptions of Theorem 2.2.4(a) are satisfied. It follows (after a continuity

and

c 2”jf * 03

vj

j-0

* yj

n

=

(2n)P

c 28jF-l(Pf m

j=O

*

Ffpj)

Whence it follows the conversion t o ( 3 ) and hence the proof of (a). Step 3. If s is a natural number, then it is not hard t o see that

< p < m, in the sense of Remark 2.2.414. For

is a multiplier belonging t o M ! , 1

f

E S(Rn),it

follows that

5 c’IlF-l(1 + I z l ” Q y I L p . 8

IlO“fllr,

=

cIlF-l*..

.z.FfllLp

Now, one obtains in the usual way by completion llfll ,I cllfllHi for f WP Step 4 . To prove the converse inequality, we consider

E H;(Rn).

Further, let e ( t ) be an infinitely differentiable function on R , such that e(t) = 0 for It[ 5 3, e ( t ) = 1 for t 2 1 , e(t) 2 0 for t 2 0, and e(t) = -e(-t). Then 8

179

2.3.3. The Spaces Hi(R,)

is a multiplier belonging to M; in the sense of Remark 2.2.414. Hence, one obtains

s c’llf

+ c‘ j =c1 11 llFil@”(ti) ~ ~ F ~ f [ I L p ( R 1 ) I I L p (- R n _ , ) n

llLp(Rm)

Here, F , (resp. Pi’) denotes the one-dimensional (inverse) Fourier transform. Since $ ( t ) is a one-dimensional multiplier and aince it holds Fil@i‘lf obtains

=

c- auf

a$ ’

one (5)

Together with the third step, this yields the part (b) of the theorem. R e m a r k 1. * For s = 0 one obtains F:,2(R,L)= H:(Rn) = L,,(R,) and

IIfII L, II{f * vj}IILp,12) {pj}jao_oE @ * (6) Theorems of such a type are very important in functional analysis. They are called theorems of Paley-Littlewood type. Treatments on several aspects of theorems of Paley-Littlewood type can be found in E.M.STEIN [a] and W.LITTUN, C. MCCARTHY, N. M. RIVIERE [l]. R e m a r k 2. I n the last two steps, we proved a little more than stated in the theorem, namely N

where l l f l l , , , and l l f l l ~ , are equivalent norms. Here s = 1, 2, . . . and 1 < p < 00. P P Assertions of such a type on equivalent norms are very important. Later on, we shall prove numerous theorems of such a type. R e m a r k 3. * The last theorem describes the usual definition of the Sobolev spaces W”,R,,)and the Lebesgue spaces H i (R,,).References may be found in Remark 2.3.1/2. Later on, we shall obtain for the spaces B;JR,J equivalent norms, too. It is also possible t o find equivalent norms for the spaces Hi(Rn) if s integer with the aid of singular integrals, hypersingular integrals and differences. We refer t o [l,21, P. I. LIZORKIN[9], C. FEFFERMAN [l], E. M. STEIN [ 3 ] , R. S. STRICHARTZ and R. L. WHEEDEN[I, 2,3]. In the framework of approximation theory, one obtains also numerous equivalent norms in the spaces HE(Rn).We referto P. L. BmZER,E. GORLICH[l], E. GORLICH[l], and W. TREBELS[1, 2,3]. (See also P. L. B m ZER,W. TREBELS [l] and P. L. BUTZER, H. BERENS[l].)Similar considerations for anisotropic spaces can be found in R. J. BAGBY[2]. R e m a r k 4. * For -a < s < 00, one obtains by Theorem 2.3.2(d)

+

Bi,2(Rn)c fl;(Rr,) c Bi,p(Rn), 2 S P < a ) q.p(R,,= ) ffp,,) = q),2(&), 1 < P 5 2, 12*

(8)

and Hi(R,) = B;,p(Rn).One can replace Hi by W: in (8). Further, together with (2.3.2/3), one obtains that

HFe(R,),W F ( R n )c B;,q(Rn)c HO,-"(R,),W",-"(R,), (9) -co < s < 00, 1 0 , l 5 q 5 00. Inclusion properties of type (8) and (9) for Sobolev-Slobodeckij-Lebesgue-Besov spaces are well-known. We refer [ l ] (one-dimensional case for periodical functions); 0. V. BESOV to I. I. HIRSCHMAN [2]; A. P. CALDER~N [2]; S. V. USPENSKIJ[l]; J. L. LIONS,E. MAGENES [l,1111; M. H. TAIBLESON[ l , I], Theorem 15; P. I. LIZORKIN[ 4 ] ; and N. ARONSZAJN, F. MULLA, P. SZEPTYCKI [ l ] . An improvement of (8) is impossible in the following 00. sense. The embedding H;(R,) c Bi,q(Rn)is true if and only if max [2, p ] q Similarly, Hi(R,) 3 Bi,q(Rn)holds if and only if 1 5 q 5 min [2, p ] . We refer t o M. H. TAIBLESON [ l , I], Theorem 20; and K. K. GOLOVEIN[3].

s

R e m a r k 5. I n Remark 2.3.113, we mentioned the possibility to extend the considerations t o the limit cases p = 1 and p = co. If Hi(R,) and H",R,) are defined by (1)) and if Wi(R,) and W$(R,) are defined by (2), where s = 1 , 2 , 3 , . ., then Hi(Rn)9 Wi(R,) and HS,(R,) WL(R,) for s = 1 , 2 , 3 , . . . and n 2 2. (For 1 < p < co these spaces coincide, respectively.) For n = 1 and s = 2 , 4 , 6, . . ., we have Hi(R1) = W$(Rl)and Hd,(R1)= W$(Rl),while for n = 1 and s = 1 , 3 , 5 , . . . there is Hi(Rl) $: Wi(R1) and H",(Rl) W$(Rl). We refer t o E. M. STEIN [5], pp. 135/160. Further, it follows by Remark 1.13.412 that formula (7) for p = 1 and p = 00 is untrue, too.

.

+

+

2.3.4.

Lift Property

It is easy t o see that

+ lx12)"B'f, 8

-a < s < 00, (1) is a continuous one-to-one mapping from S(R,) onto S(R,) and from S'(R,) onto S'(R,).*) It holds Iil = I-,. T h e o r e m . Let -co < s, 0 < 00 and 1 < p < co. Then I , is a continuousone-to-one mapping from F,",,(R,,)onto F;Z(R,,), 1 < q < 00, and fTom B;,q(Rn)onto B;;:(R,), 15qsco. P r o o f . Let {qk)& E @. If

I,f = F-I(l

yk

=

* F-l

~ph

2ks

(2)

+

((1 I x , 2 d then {yk)&, belongs also to @. It follows t h a t

I,?/* yk

n

=

F-'((2n)TPyA FIJ)

= F-'((2n)" 2"Fqk

n

. Ff) = (2n)T2ksf * ~ ) k .

*) As usual, the notation "onto" means that the range of the operator coincides with the whole space.

2.4.1. Interpolation of the Spaces Bi,,(R,)

181

Now, one obtains by (2.3.2/1) and (2.3.2/2) that

-

l l M J sP d- * IlfIl$,p =

and

- lIfllBU .

IILfllga-.

PIP

PIP

R e m a r k 1. It follows immediately by (2.3.3/1) and (1) that the spaces Hi(R,) Fi,2(R,)have the lift property described in the theorem.

R e m a r k 2. * We just mentioned that the spaces B;,q(Rr,)for s > 0 coincide with the well-known Besov spaces. But on the basis of the usual definition of these spaces it is more difficult to prove the l i f t property. Conversely, the l i f t property for the spaces B;,q(R,) with s > 0 was used t o define the spaces Bi,q(R,)for s j0. Several authors discussed tJhequestion what the spaces B;,q(R,) are. In this context, we refer to [2]; N. ARONSZAJN, F. MULLA, P. SZEPTYCKI El]; M. H. TAIBLESON A. P. CALDER~N [l, I]; S. M. NIKOL'SKIJ,J. L. LIONS, P. I. LIZORKIN[l]; and S. M. NIKOL'SKIJ ["I, Theorem 8.9.1.

2.4.

Interpolation Theory for the Spaces Bi,q(Rn)and J",q(Rn)

One of the main aims of this chapter is t o give a treatment of the spaces B;,q(R,,)and F;,q(R,) (containing the spaces Hi(R,,) and Wi(R,,)as special cases) on the basis of

interpolation theory. I n this section, we shall prove several interpolation theorems which are of fundamental importance for the later considerations.

2.4.1.

Interpolation of the Spaces B&(Rn)

For a comprehensive description of the interpolation properties of the spaces B;,q(Rr,), we introduce the spaces Bi,q,(r)(R,J and B;$)(R,) generalizing the spaces given in Definition 2.3.1/l(a) and Theorem 2.3.2(a). These new spaces are not important for the later considerations. Later on, we shall be interested only in interpolation spaces of B;JRrZ)and Fi,q(Rn)which are also spaces of the type B;,q(Rn)or F;,q(Rn). The scalar (complex) Lorentz-sequence spaces introduced in Subsection 1.18.3 are denoted by lq,r,and the scalar (complex) Lorentz spaces defined in R, with respect to the Lebesgue-measure, introduced in Subsecttion 1.18.6, are denoted by Lq,r.rP has the same meaning as in Definition 2.3.1/2. D e f i n i t i o n . Let (Q?,~)?=~ E @. Further let l j r j c o . (a) For 1 5 q 5 co one sets

B;,q,(r)(Rtr)= {f I f E

fif(&t),

IIfIIgig(,) 8

.

03

< s < co, 1 < p < co, and

= II(2"'IIf

* p j IIL~,,}

111,

< a}.

(1)

(b) For 1 < q < co one sets

Bg;)(Rn) = {f I f

E S'(Rn), llfIlB.*(*) = P.P

ll{28j.llf

* ~;llLp}llzg,~ < .I.

(2)

182

2.4. Interpolation Theory for the Spaces B:,q(Bn)and F:,,(R,,)

R e m a r k 1. Clearly, it holds

B;.q.(P)(R,l) = q,q(R,J and q,Ip’(R,,) = q , q ( R , l ) . Using the previous methods (proof of Theorem 2.3.2, particularly Step,7),we realize that Bi,q,(r)(R,,) and B:$’(R,,) are Banach spaces, independent of the choice of (qk}gO E@. To prove the last assertion, we remark that Theorem 2.2.4 is also true for the Lorentz spaces Lp,ras follows immediately by interpolation on the basis of Theorem 2.2.4 and Theorem l.l8.6/2. Now, the method of the proof of Theorem 2.3.2 and the scalar case of Theorem 2.2.4 with Lp,rinstead of Lp give the desired statement’. We remind of the Definition 2.3.2 of the spaces &,,a(Rll). T h e o r e m . (a) Let -co < so, s1 < co, so and 0 < 8 < 1. Then

$1 sl,

1

(Bzq,(Rn),B2ql(RI1))@,q = Bi,q(Rn) where

(b) Let -co < s < 0 < 8 < 1. Then

s

=

(1 -

00,

1 < p < co, 1 5 qo,ql

e)so + esl ,

1

= (1

1

5 qo,ql,q 5

- 8 ) $0

+ 881.

00,

(3)

S co, qo $1 ql, 1 5 1’5 00, and

e

1-0

5

00,

-4 Po +-,q1

and

1

-P= -

1-0 PO

+ -.8

P1

(6)

If additionally p = q, then (7)

(Bz,q,(Rn),B23,q,(%))e,q= Bi,p(&).

(d) Let -co < so, s1 < co, 1 0 < 8 < 1. Then

5 qo < co, 1 S q1 5

00,

1 < p o , p l < co, and

[B;,qe(Rii))B2,q1(Rn)le= Bi,q(Rn) with (6)* (e) Let

-00

< so,sl <

co,so $: sl, 1

(8)

< p < co,and 0 < 8 < 1. Then

[ B ~ m ( R n ) , B ~ m ( ~ n=) &,,(R,J ~e

where

=

(1 - @ s o

+ 88,.

(9)

Proof. s t e p 1. We use the two systems {P)k)& E @ and {v k },& E @ introduced in Step 7 of the proof of Theorem 2.3.2, and we use the operators S and R defined in (2.3.2/12) and (2.3.2/13). We remarked there that S is a coretraction from B;,q(Rn) into l;(Lp)and R is a corresponding retraction from li(Lp)onto B;,q(Rn).The last B2,,ql(R,L)) is an interpolation couple (in the part of (2.3.2/3) shows that {B;,qo(R,l), If P is an sense of the notations of Subsect’ion 1.2.1 one can choose a2 = S’(RII)). interpolation functor, then one obtains by Theorem 1.2.4 that

2.4.1.Interpolation of t h e Spaces B;,JR,,)

183

Step 2. All the statements of the theorem are consequences of (10)by specialization. (a) Part (a) of the theorem follows from Theorem 1.18.2. (b) One obtains part (b) of the theorem from Theorem 1.18.3/2 if one takes into consideration that one can in (1.18.3/8) by (and similarly for the other spaces). (c) To prove (c), replace we use (1.18.1/3) where A,i = 2jSoLp,(Rn) and Bj = 2J"lLp,(R,,). One obtains as an easy consequence of Definition 1.3.2 and Theorem 1.18.6/2 that

-

Here, means that the numbers of estimates are independent of i. Now, it follows (c) if one takes into consideration Bi,p,(p,(Rn) = B;,p(R,,).(d) Assuming q1 < 03, then (d) follows from (1.18.1/4) similarly t o (c). Here, one has to use a formula corresponding to (ll),see (1.18.1/15) and (1.18.6/15). For q1 = co, one obtains the desired result from (1.18.1/12). (e) To prove (e), we use (1.18.1/16). We must show that CF(Rn)is dense in the interpolation space. Let { y k } E oE @ be a system such that f

5 f * y j , see Example m

=

S' i=O

2.3.1. Then

2 f * y j approximates f in the M

=

i-0

norm of the interpolation space. But such functions can be approximated by functions belonging to Cg(Rn)in the desired way. R e m a r k 2. We just mentioned that we are above all interested in interpolation results which lead to spaces of type BiJR"): (3), (7), (8), and (9). Formula (9) is of special interest, since it shows that &&(Rn) may be obtained by interpolation of R,m(Rn)R e m a r k 3. By the extension of (9) t o different values of p , there arise new difficulties. But a t least, one obtains for - co < so, s1 < 00, 1 < p o , p1 < co,and 0 < 8 < 1 with the aid of the above method by (1.18.1/13) and (1.18.1/14) that

where s

=

(1

- 0 ) so + 88, and-1 P

=

1-8 e f-. PO Pl

R e m a r k 4. I n contrast t o (3), the parameter q in (5) is determined by q o , q l , and 8. The question arises whether one is able to determine the interpolation spaces on the left-hand side of (5) if the second index of the real interpolation method does not coincide with q. We quote a result by J. PEETRE [15], Theorem 1 : If the parameters have the same meaning as in part (c) of the theorem, inclusively formula (6), and if 1 5 r 5 co,then

Bi,min(q,r),(v(k) c (Bz,q,(Rn),B;,q,(Rn))e,rc Bi,maxcq,n,(r,(Rn) * Here, the parameters min (q, r ) and max (q, r ) cannot be improved, a t least in the 12

n

PO

P1

case po < p1 and so - - 5s1--.

R e m a r k 5. * From the very beginning, the interpolation theory for function spaces and for spaces of distributions was closely related t o the development of the abstract interpolation theory. We refer t o E. GAGLIARDO [4], J. L. LIONS [3, I], and J. L. ~ O N S ,E. MAGENES [l, particularly 1111. After preliminary results in the just mentioned papers, formula (3) was proved essentially by J. L. LIONS,J. PEETRE [Z] and E. MAGENES[l]. As a powerful tool, they used strongly continuous semi-

184

2.4. Interpolation Theory for the Spaces B:,(R,,) and F;,q(R,,)

groups of operators in the sense of Section 1.13. Here, in this book, we have the converse situation. On the basis of the just proved formula (3) and the theory developed in Section 1.13, we shall obtain equivalent norms for these spaces later on. Particularly, we shall derive formulas usually used for the definition of the Besov spaces B;,q(Rn).The further development of the interpolation theory of function [3,4] spaces without weights in R,, is characterized by papers by A. P. CALDER~N (complexmethod), J. L. LIONS[lo], J. PEETRE [ l l , 151, P. GRISVARD [a], M. H. TAIBLESON [l,113, and many other papers on interpolation of function spaces (in R, or in domains, with and without weights) which we shall mention later on (see Remark 2.4.2/3).The paper by J. PEETRE [15] and the unpublished lecture notes by J. PEETRE [12] are of special importance for the methods considered here. There is the first description of the methods developed here. Further, there can be found (5)and the generalization described in Remark 4. The interesting special case (7) was proved [4] and M. H. TAIBLESON before by P. GRISVARD[4]. (8) goes back to P. GRISVARD [l,111. Before this time, J. L. LIONS,E. MAOENES[l,V] and J. L. LIONS[lo] had proved special caseg.

2.4.2.

Interpolation of the Spaces FiJZtJ

Analogously to the previous Subsection 2.4.1, we generalize the spaces F;,*(Rn)of Definition 2.3.1/l(b) and introduce the spaces F;,O(R,) and F&(n(Rn).These new spaces are not important for the later considerations. Later on, we are interested or F;JR,,). only in interpolation spaces which coincide with Bdp,q(Rn) Definition. Let ( p k } k ~Eo@. Further let -co < s < co and 1 < p , q < co. (a) One sets

F:,P(Rn) = (fI f ES‘(Rn),IIfII,+(p)P.I =

II

5 r 5 co. One sets F;,q,(r)(Rn)= (f I f E fi‘(Rn),IIfIIpP.%-dr)

=

(b) Let additionally 1

Remark 1. Clearly,

II(2jIf

* pjI}IIlq,pIIz,p< “1.

II II(2”If * pjIHIzqIILp,,< “1.

FP>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 <

03,

and 0 < 8 < 1, then it follows

( B ~ o , p o (H21(Rn))e,p ~n)~ = (H:o(Rn), H21(RJ)e,p= B;,p(Rn) where s and p are again determined by (1).

7

(c) If -co < s < co, 1 Hil(Rrt))e,p= Hi(Rr,)> 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 <

03,

so

(15)

+ sl, one obtains that

(W',.(Rn),W ; ( m ) o , q = q , q ( R n ) f

(16)

R e m a r k 5. Theorem 2.3.3(b) shows that for s = 1 , 2 , 3 , . . . the spaces Wi(Rn) coincide with the usual Sobolev spaces. Further, it holds WE(Rn)= Lp(Rn). But then formula (16) is equivalent with one of the usual definitions of the Besov spaces, namely the definition by interpolation of Sobolev spaces. R e m a r k 6. By (13) and (1.3.3/5),it follows that

where the parameters have the same meaning as in Theorem 2. On the right-hand side of (17), one may replace Bgqoby H;, F2qoor W;, similarly for B2,q,.Estimates of type (17) are often needed rn the theory of elliptic differential operators. Sometimes they are called multiplicative inequalities.

2.5.1. Equivalent Norms and Tranalation Groups

2.5.

187

Equivalent Norms in the Spaces Bi,a(RJ

In Theorem 2.3.3 and in Remark 2.3.312, we have obtained equivalent norms for the Lebesgue spaces Hi(R,J, -co < s < co, and particularly for the Sobolev spaces WT(R,), m = 1 , 2 , 3 , . . ., which coincide with the usual definitions of these spaces. This section is concerned with equivalent norms for the Besov spaces B&(R,,), 0 < s < 00. Formula (2.4.2/16) shows that the spaces BiJR,,) for s > 0 are mterpolation spaces of Sobolev spaces. On this basis, one can describe the main motive of the considerations of this section as follows. I n the space Lp(Rn),we choose a strongly continuous semi-group (resp. n strongly continuous semi-groups commutative t o each other) of operators with the infinitesimal operator A (resp. A,, . .,A,) such that Wr(R,) = D(Ak)(or

.

n D ( A f ) )for suitable natural numbers m and k. 11

=

j-1

Then we can apply the results of Sections 1.13, 1.14, and 1.15, and we obtain a large number of equivalent norms in the interpolation space BE,')"(Rn) = (WF(R,), Lp(Rn))e,q. There are three types of semi-groups under consideration : Translation groups, Gauss-Weierstrass semi-groups, and Cauchy-Poisson semi-groups. Finally, we determine equivalent norms by approximation of functions belonging to Lp(Rn) by smooth functions.

2.6.1.

Equivalent Norms and Translation (froups

I n the space LJR,), there are considered the strongly continuous commutative groups ( G j ( t ) ) j' = 1, . ., n, of isometric operators,

.

-

[aj(t)f l (2)= f ( ~ 1* ,

9

xj-1, xj

+ t,

2 j + 1 9

* * *

9

zn),

f

E Lp(Rn)*

(1)

For our purpose, only the semi-group{aj(t)}O~t<,is of interest. The corresponding infinitesimal operator will be denoted by Aj, its domain of definition by D(Aj). Similarly, AT and D(AY) are defined for m = 1 , 2 , . Further, K m has the same meaning as in Definition 1.13.3. Temporarily we set

..

m

1 , 2 , . . ., j = 1,. . . , m , 1 < p < 00. L e m m a . (a) S(R,) is dense in WgI(R,)and =

where w,,,and (b) It holdsP'hd

llfll *q, are equivalent m w .

188

2.5. Equivalent Norms in the Spaces B;,JR,,)

and n

=

~m

n D ( A ~=) w;(R,). j=1

(5)

Proof. Xtep 1. One obtains formula (3) in the same manner as in the fourth step of the proof of Theorem 2.3.3; see also Remark 2.3.312. If v E CF(RI,)with ~ ( x=) 1 if 1x1 5 1, then i t is not hard t o see that f E WEj(Rn)can be approximated in

(3

Wgj(Rn)by the functions p - f(z) if N -+ 11m11~, = 1. For 0

00.

Let 0

<= w ( z ) E C$'(R,J

with

< h < co and g e L p ( R , , )we , see

This is SOBOLEV'S well-known mollification method, see S. L. SOBOLEV[4]. (For n = 1 this method is also known as STEKLOV'S mollification method.) It holds that 1101~11L, = 1. Whence, it follows that (g)h ELrdRn) C " ( R ~ ~and ) II(g)hllLp l l g I / L p * Since for functions of c$'(RrI)it holds II(g)h - gllLp(Rn)+ 0 if h 10, one obtains by the last estimate that the last relation is true for arbitrary functions g belonging t o

Lp(Rn).If g has a compact support then (g)h E C$'(RIJ.Setting g ( z ) = ak

it follows for D" = -, 0 p ( ( g ) h (2))=

s

ax;

Dzmh(z

R.

(3

- f(x),

5 k 5 m, by the definition of derivatives of distributions

- Y ) g ( Y ) dy

= (-1)Ia'

s Dimh(z -

Rri

g ( Y ) dy = oh * D"g. (7)

v (i1 f(x) ( - f ( x ) in WEj(RI,)if h 10. Hence, CF(RIJ,and so also S(R,), is dense

Using the preliminary considerations, whence it follows that C$'(Rn)3 tends to v

v

(3

-

in W;j(Rn)*

Step 2. It is not hard to see that for f E S(Rn)c D(AY)we have

where D ( A y ) is normed by ll*ll~pj or by Il*l&;,. By Theorem 1.13.1/2, (0, co) belongs to the resolvent set of Aj . Particularly, (Aj - E)"' is an isomorphic mapping from D ( A y ) onto Lp(Rn).Since X(Rn)is dense in W;j(RI,)and D(AY) = D((Aj- E ) m ) is closed, one obtains D(Ay) 3 W g j ( R I , )Here, . ( 8 ) holds for f E WEj(&,), too. On the other hand, using Fourier t,ransformations, i t follows that (Aj - E)IJ1is an isomorphic mapping from S(Rn) onto S(Rn). Completion shows that (Aj - E)" is also an isomorphic mapping from W;fj(Rn) onto LJR,). But then we have D(A7) = W g j ( R n )Formula . ( 5 )is a consequence of Remark 2.3.312. R e m a r k 1. The first step contains a complete description of SOBOLEV'S molli[4]) which is very important. We used a similar fication method (see S. L. SOBOLEV method in the first chapter several times.

2.5.1. Equivalent Norma and Translation Groups

R e m a r k 2 . I n the case considered here we have K m = 1.13.4/2. We sha.11

189

n

n D(A7). See Remark

and The norms II.()L;(Q6,A) and and 1.13.2.

11.11 L:,j0,6),~) have the same meaning as in Subsections 1.13.4

T h e o r e m . Let 0 < s < 00, 1 ~ - k , a n d i f 0 < 6 6 co,then

00.

If k and I are integers

where

A11 these norms are equivalent to 11 f 11 B;,f. Proof. The proof follows immediately from (2.4.2/16) with so = 0 and s1 = m > s

= O m , the last lemma, a,nd Theorem 1.13.6/1.

R e m a r k 3. Clearly, on the basis of the considerations to Theorem 1.13.6/1 (see particularly (1.13.6/9)),one obtains numerous further equivalent norms. For instance, ajf

C by C and - by Pf. Further, one can add in j=1 IalSk ax! (lo), (ll),or (12) norms of the form Ilf IIq or 11 f 11 with 0 5 o < s. This is a conse-

in (10) one can replace

I1

quence of (2.3.3/8) and (2.3.213). Those norms in which 1 is as small as possible are of special interest. For 0 < s < co,we set and

+ {s}, [s] integer, O 6 {s} < 1 , = [s]- + {s}-, [s]- integer, O < {s}+

= [s]

1.

The smallest number for I is I = 1 + [{a)']. Then we have k = [s]-. This shows that the spaces Bi,q(R,,),where s is an integer, hold a special position. R e m a r k 4. The spaces W;(RJ,0 < s < 00, 1 < p < 00, are defined in (2.3.1/7). For the Sobolev space W;(Rn)with s = 1, 2 , 3 , . . . , we obtained in Theorem 2.3.3(b) and Remark 2.3.312 a satisfactory description. Now, for the Slobodeckij spaces

190

2.5. Equivalent Norms in the Spaces Bi,JRJ

W;(R,,)= B;,p(Rn),0 < s

+ integer, it follows from the Theorem and Remark 3

R e m a r k 5. * By the last theorem, we obtained the usual definitions for the Slobodeckij spaces W”,R,), 0 < s integer, 1 < p < 00 (N. ARONSZAJK[l], L. N. SLOBODECKIJ[l], E. GAGLIARDO [l]) and for the Besov spaces Bi,q(Rn), 0 < s < 00, 1 < p < co, 1 5 q 5 co (0.V. BESOV [1, 21). Numerous further equivalent norms, partly considered in the following subsections, can be found in S. M. NIKOL’SKIJ [7] and M. H. TAIBLESON[1, I]. I n this connection we refer also t o P. L. BUTZER,H. BERENS[l]; P. L. BUTZER,R. J. NESSEL [l]; and P. I. hZORKIN [13]. Further references are given in Remark 2.3.3/3 and Remark 2.3.1/2.

+

2.6.2.

Equivalent Norms and Gauss-Weierstrass Semi-Groups

On the basis of the considerations on analytic semi-groups in Subsection 1.14.5, we obtain in an easy way further equivalent norms in the space B;,q(Rn). Lemma. Let 1 < p < co and n

[ W ( t )f] (x)= (4nt)-T

J

R”

lx-yp

e - 7 f ( y )d y , 0 < t < co,f

E LJR,).

If additionally W ( 0 )= E , then { W(t)}Ost< is an analytic semi-group i n the s Lp(Rn).I f A is the corresponding infinitesimal operator, then

Amf = A m / , D(Am)= W26n(R,,), m = 1 , 2 , . . . Proof. Step 1. By the well-known formula

(1)

p

(2)

if follows that

II W)II 6

1-

(4 )

For f E S(Rn),(1) is the uniquely determined bounded solution of the classical Cauchy problem of the heat conduction equation

The uniqueness of the solution shows that

W(t, + t z ) f = W ( t , ) [

W 2 )

fl,

0

s t,,

t2

<

03 *

Together with (4), the semi-group property follows from the last equation. With the aid of (3), one shows for f E S(Rn)in the usual way that

I l w ( t ) f - f l l L p + o if t l 0 .

(6)

2.5.2. Equivalent Norms and Gauss-Weierstrass Semi-Groups

191

By (4), it follows that (6) is true for f E LJR,), too. Hence, { W(t))ostt< a, is a strongly continuous semi-group with B = 0 in (1.13.1/1). Step 2. One obtains by ( 5 )and the definition of A that S(R,) c D(A) and Af = Af if f E S ( R , ) . Here we used A W ( t )f = W ( t )Af. By Theorem 2.3.3, it holds for f E S(R,) that

II(d - E ) fllr.,

=

l1F-V

+ 1512)Y I I L ,

- llfllJ";.

Since A is a closed operator, and since S(R,) is dense in v , ( R , , ) , it follows that D ( A ) 3 W;(R,) and Af = Af if f E W&,(R,).Conversely, let f e L p ( R I z ) Then . Fl(1 I[l2)-l F f belongs t o W;(R,,)and

+

-(A - E ) (F-'(l

+

1512)-'

Ff)

=

f.

Whence it follows D ( A ) = W8,(R,,).Further, one obtains that ( A - E ) m is an isomorphic mapping from D ( A m )onto L p ( R f f As ) . before, i t follows D(Anl)= Wim(R,). Step 3. By derivation it follows that the assumptions of Definition 1.14.5 are satisfied. Hence { W(t))Ost
If A = LJR,,), then LZ(A) = LI,(L,) has the same meaning as before, see Subsection 1.5.1. T h e o r e m . (a) Let 0 < s < 03, 1 - ,

llti,g i s an equivalent norm in the space B;,q(Rll).

[If

(b) Let - 00 < s < 00, 1 -, 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
AZmf= ( - l ) m A m f , Proof. Stepl. Since (1x12 + t

D(A2m) = WEm(Rn), m

2 ) - 7

=

1,2, . . .

(2)

is a harmonic function in the space Rn+l- {0},

+

--n+l

it follows by differentiation with respect to t that t(lsI2 ts) * is also a harmonic function in R,++, . But then P ( t )f with f E S(R,,) is also a harmonic function in R,++1. Since

I1t(1xI2+

IIL,

t2)+

=

1

n+l

(1x12

+ l)+IL+

one obtains in the usual way that u(x,t ) = P ( t )f , f Dirichlet's problem

aa,

du(s, t ) + -(2,t ) = 0 i f at2

(2,

(3) E S(Rn)) is

t ) E RA+l,

a bounded solution of

u(x,0) = f ( x ),

(4)

Step 2. By (3),one obtains [lP(t)[lLp+Lp 5 1. To derive the semi-group property we use formula (9) proved in Remark 1. For f E S(Rn)and for 0 -< t,, t, < m, it h o b P(tl) [P(t2)f] = F-lFP(tl) F-1FP(t2)f

=

= F-1 e-IWi e-l% Ff = P(tl

P-IFP(t,) F-1(e-IWFf)

+ t2)f .

(Here, we use (2.2.1/4).)Together with IIP(t)II 5 1 this yields the semi-groupproperty in Lp(Rn).One obtains the continuity in the same manner as in Lemma 2.5.2. Using d AG(t)f = -G(t)f , it follows from ( l ) by , differentiation with respect to t , that P(t)

at is an analytic semi-group in the sense of Definition 1.14.5, where B = 0 in (1.13.1/1). Step 3. For f E S(R,), one obtains by ( 4 )that

AZmf= (- 1)" Afr2f, S(Rn) c D ( A z m ) .

(5)

Further, it follows similarly to the third and the fourth step of the proof of Theorem 2.3.3 t h a t II(AZnl+ E ) fllL, = IIP-l(l + Itlam) FAILp llfllw;m.

-

Since S ( R n )is dense in WEm(Rn)and since A 2 m + E is a closed operator, it must be D(A2m) 3 Wim(R,). Here, ( 5 )holds for f E WEm(Rn).Let f E D(Azm)and 9 E S(R,). 13

Triebel, Interpolation

194

2.5. Equivalent Norms in the Spaces Bi,,(R,)

If one takes into consideration that A2mf may be obtained as a limit t of (l),then it follows in the sense of S'(R,) that (A2"f) (fp)= f(A2"p)

=

f((-l)md"rp)

=

[(-l)'nd'Jlf]

4 0 on the basis

(p).

Hence, A l n f E Lp(R,).But now one obtains in the same manner as above f E WE"'(R,). Theorem. (a) Let 0 < s < co, 1 s

11f 11 B:r is an equivalent norm in the space B;,JR,). Proof. Step 1. If m is an even number, then it follows from the last lemma that

(Lp(Rn),o(Am))B,q= Bi,q(Rn),

8

(7)

= me.

Now, for even numbers m, ( 6 ) is an immediate consequence of Theorem 1.14.5. Xtep 2. To prove (6) for arbitrary natural numbers m > s, it is sufficient t o show

For f that

E Lp(R,),it

follows from (1) (or the fact that P(t) is an analytic semi-group)

Hence, one can estimate the right-hand side of (8) from above by the left-hand side. We prove the conversion. If the right-hand side of (8) is finite, then there exists a sequence ti +

00

if j + co such that

/I

am+lP(tj)

atm+i

/I

LP

+ 0 if

j+

restriction of generality we may assume that for almost all x E R, we have

Without am+lP(tj)f(z)

00.

atm+l

+ 0 if j + co. (Otherwise one would choose a suitable sub-sequence.) For such an x

on0 obtains that

-a=m-m f atm

and

This proves (8).

I"

m

dz = - t

a J 1 +atl P ( t ) f

m+ 1

1 m

2.5.3. Equivalent Norms and Cauchy-Poisson Semi-Groups

195

R e m a r k 1. One can derive a formula analogous t o (2.5.218)if one uses

Here, c, has the same meaning as in ( 1 ) . Later on, we shall need this formula. So, t

we give a short proof here. For abbreviation we set cnF t

Since (1212

+

Hence,

is harmonic in R:+l, it follows that

n+1

t 2 ) T

g ( t , t ) = a,(&

e+It

+ a2(&e W .

+

Since g ( t , t ) is uniformely bounded with respect t o t, it follows a2(&= 0 if ( 0. By continuity one obtains u 2 ( t )= 0. Further, it holds g ( t , t ) = g(& 1) and a,(() = a(lE1).

Consequently, n a(lEl) E a. Since c,F

follows a = ( 2 4 - T .

(

t

+ t2)c)

(0) = (2n)-%, it

R e m a r k 2 . To prepare later applications, we investigate the fractional powers of the positive operator - A + E in the sense of Definition 1.15.1. For f E S(R,), it follows from ( 9 )that

Remark 2 . 2 4 4 yields that the right-hand side tends t o zero if t 1 0 (the numbers B = BLfrom Remark 2.2.414 tend t o zero if t J 0). Hence,

A f = -F-'1[1 F f , f

E

S(R,)c D ( A ) .

Using (2), whence i t follows that

Ajf = ( S+ce

ltlj

(1

i

l ) j F-lIEIjFf,

+ 1[I2)-T is a

f

E S(R,,) c

D(Aj), j

=

1,2,

...

(lob)

multiplier and since S(R,,)is dense in W#ln), it follows

W;(R,)c D ( A j ) .Further (lob)holds for f E W$(R,>), 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

-OO,~E&(&)-

Completion shows that this formula is valid for f E Lp(Rn),too. Hence, the assumptions of Theorem 1.15.3 are satisfied and one obtains for a > 0

o(Aa)

'[Lp(Rn)to(A")Ie

= [Lp(Rn),W;(Rn)le = Hi(Rn).

(13)

Here, m is a natural number with rn > a and Om = a. Further we used (2.4.2111). Clearly, (11) is a special case of (13). The formulas (10) and (11) are known. Using the Riesz transformation, one can represent Af from (10a) with the aid of singular integrals (and without Fourier transformations). We refer t o E. GORLICH [l], W. TREBELS [2], and W. TREBELS,U. WESTPHAL [l].

R e m a r k 3. * The investigation of boundary values of harmonic and analytic J.E. LITTLEWOOD [3]. The extension functions in the disk goes back t o G. H. HARDY, of these considerations t o harmonic functions in the half-space R:+l is due t o E. M. STEIN[2] and E. M. STEIN,G. WEISS[4].See also C. FEFFERMAN, E. M. STEIN [l]. A comprehensive treatment of the spaces B;,q(Rn) on the basis of formula (6) was [l]. Here, M. H. TAIBLESON used the papers written by given by M. H. TAIBLESON J. E. LITTLEWOOD [l, 21 and A. ZYOMUND[l]. G. H. HARDY,

2.5.4.

Equivalent Nornis and Approximation

The problem of approximation of non-smooth functions by smooth functions is very important in analysis and its applications. We remind of the approximation of continuous functions or of functions belonging t o Lp, defined in bounded domains , by polynomials or by trigonometrical functions. Here, the question of the rate of R,, of approximation is of great importance (theorems of Jackson type). In thi;r sense, one can consider the problem of approximation of functions of.L,(R,,), 1 < p < co, by smooth functions. An appropriate class of smooth functions (which are analogous to the trigonometric functions) are the entire analytic functions of exponential type. The problem of approximation of functions of Lp(R,) by entire analytic functions of exponential type is of great interest in the theory of the Besov spaces B$,,(R,). A systematic treatment and references can be found in S. M. NIEOL'SKIJ [7]. Our methods developed here are sufficient t o prove one of the main results of this theory.

197

2.5.4. Equivalent Norms and Approximation -

D e f i n i t i o n . Let 1 < p < 03 and 0 < t < 00. M p , t is the set of all functions f ( x l , . . . , x,,) belonging to Lp(R,,),which are restrictions of entire analytic functions f ( z l , . . . z,) of n complex variables of (spherical) exponential type t. An entire analytic function f(zl, ,2,) i s said to be a function of (spherical) exponential type t , if for any E > 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 <

00

and 1 O

I

*

i s a n equivalent norm in the space BiJR,,).

P r o o f . It is sufficient t o show that llfl]g;,g is a n equivalent norm in the space B8p,q(R,).If 1c is a real number then

(with the usual modification for q = co). Let {vj(x)}Fo E @,) be a system of functions considered in Example 2.3.1 such that (2.3.1/13) with vk instead of @k and with n

c = (2n)-F holds. Then

is a decomposition of f where

1

C vj * f

(j:o

E M p , l ,t

= 2.y+k. Whence it follows that

198

2.6. Duality Theory for the Spaces Bi,,(Rn) and Fi,g(Bn)

where c > 0 is independent of k. The last part of (4) can be proved in the following 2 - N + k . Then, using the multiplier theorem described i n Remark2.2.414, way. Let g EM,, one obtains that

This proves the right-hand side of (4). Together with (3) is follows that

(with the usual modification for q =

5 c‘ Replacing

00

2=1

00).

Now, Theorem 2.3.2 yields

I

2-bllfllB’,,, 5

C”llfllB:,;

C v, by v k + 1 in ( 5 ) ,one obtains with the aid of (3)the converse estimate.

00

j=k+l

Duality Theory for the Spaces Bi,q(Rn)and Fi,@(R,)

2.6.

Theorem 2.3.2 shows that S(R,) is dense hi B;,q(R,,),q < co,and dense in F;,&R,). I n agreement with the consideration in Subsection 1.11.2, Theorem 2.3.2(c) gives the possibility to interpret the dual spaces (Bi,q(R,J)r and (FiJR,))’ as subspaces of S’(R,). All the investigations of this section must be understood in this sense. Hence, f belongs t o (Bi,q(Rn))‘ c S’(R,) if and only if +ere exists a number c such that for all q.~E S(R,)

If(v)I 5 cllqllBi,7

similarly for (Fi,q(Rn))’.

The Dual Spaces of B’,q(R,) and H;(R,)

2.6.1.

We recall the Definition 2.3.2 of the spaces i i , m ( R , l ) . T h e o r e m . (a) Let -00 < s < 00 and 1 < < a.Then

(b) Let

-00

< s < 00, 1 5 q < 00 and 1 < p <

00.

Then

2.6.2. The Dual Spaces of F;,JBm)

199

(c) Let -co < s < co and 1 < p < co. Then

P r o o f . Step 1. Let f E (LISp(R,))‘.Then for

v E S(R,)

(4) Here we used Theorem 2.3.3 with F-1 instead of F (this is possible, since it holds (F-lv) ( t )= ( F v )( - E ) ) . By Theorem 2.3.4 and by (LJR,))’ = Lp,(R,,),it follows f E H;:(Rn)and

lfIl1,;

5 llfll(B;)J.

(5)

Step 2. Let f E H;:(R,). Analogously to (4), one obtains that

If(d1 5 lflH;; llvllB;. Whence it follows the inverse inequality of ( 5 ) and the proof of (1). Step 3. We prove (b). By (2.4.2/14), (1.11.2/3a), and the part (a) of the theorem, it follows for so c s < sl,0 < 8 c 1, and s = (1 - 8) so + Os, that

(BSp,q(Rn))‘= ((H$(Rn))’, (H;(Rn))’)o,q’= Bp:,q,(Rn)-]

Step 4 . I n a similar manner one obtains the proof of (c) by (1.11.213b). R e m a r k 1. * M. H. TAIBLESON [l, 111 proved (2) for 1 < q < 00 on the basis of interpolation theory. Direct proofs are given by P. I. LIZORKIN[6] and H. TRIEBEL [19]. (2) with q = 1 and (3) are proved by T. M. FLETT [l]. Further results on duality [2] and J. PEETRE [33]. theory can be found in T. M. FLETT R e m a r k 2. Let -co < s < co and 1 < p < co. Then the theorem shows that H;(R,) and B;,q(Rn)with 1 < q < 00 are reflexive Banach spaces. Since i i + , ( R , , ) is a strict subspace of Bi,w(Rn),see Remark 2.3.212, it follows by the theorem that B;,l(R,) and B;,w(Rn)are not reflexive Banach spaces. B u t then B;+(R,,) is also not reflexive. Otherwise B;,w(Rn),as a closed subspace of Bi,w(R,,),would be reflexive. (A closed subspace of a reflexive Banach space is also a reflexive Banach V. L. SMUL’JAN; space. This is an easy consequence of the theorem of W. F. EBERLEIN, see for instance K. YOSIDA[l], Appendix t o chapter V, 4.) 2.6.2.

The Dual Spaces of F’,@,)

The proof of Theorem 2.6.1 is based on the application of Theorem 1.11.2 and on the fact that B&(R,,) is the interpolation space of spaces whose dual spaces are known. For the spaces FrpP (R,) the situation is not so favourable. Application of Theorem 1.11.3 and of (2.4.2112) gives only a partial result, since (2.4.2112)describes only spaces Fi,q(R,)where q is contained in the interval between 2 and p .

200

2.7. The Holder Spaces C'(R,,)

Theorem. Let

-00

<s <

00,

1
c

00,

and 1

00.

Then

This theorem is due to H. TRIEBEL [19]. Later on, we shall be concerned above all with the spaces B;,q(Rn)and H;(Rn) (= F;,?(Rn)).So we omit the proof of the theorem, which is somewhat complicated. For detads we refer to the cited paper. Remark. The theorem shows that FAQ(Rn),-00 1 < q < co, are reflexive Banach spaces.

2.7.

< s < 00, 1 < p <

00,

The Holder Spaces C'(R,,)

Although this book is concerned above all with the Sobolev-Slobodeckij-BesovLebesgue spaces, we want to describe here shortly the Holder spaces. This has several reasons. 1. Beside the Sobolev-Slobodeckij-Besov-Lebesgue spaces the Holder spaces are the most important function spaces. They play a fundamental role in the theory of the ordinary and partial differential equations. 2. The Holder spaces are needed for a satisfactory formulation of the embedding theorems for the spaces B;,q(Rn)and F;,,(R,). 3. The methods developed in the previous sections are also applicable to the Holder spaces, they give far-reaching results (interpolation theorems, equivalent norms).

2.7.1.

Definition of the II6lder Spaces

.

If f(x)isacontinuousfunctiondefinedinR,,thenA~,hER,,,l= 1 , 2 , . . , A t = A h , has the same meaning as in Subsection 2.5.1. Definition. (a) If t 2 0 is a n integer then Ct(Rn)denotes the completion of S(Rn)i n the norm (b) I f 0 < t = [ t ] + { t } , [ t ] integer, 0 < { t } < 1, then

CYRn) = { f I f

E Crt1(Rn) 9

IlfIIcl < a},

2.7.2. Interpolation and Equivalent Norma

201

R e m a r k 1. It is easy to see that Ct(R,) and W(R,) are Banach spaces. For sake of simplicity, we shall write C(R,) = Co(R,).I n the next subsection, we shall show that Ct(Rn)= Vt(R,)if t is not a n integer. Hence the definition of W(R,) is only necessary if t is an integer. R e m a r k 2. The formulas (2) and (3) may be written in the form

R e m a r k 3. It is easy t o see that for t = 0 , 1 , 2 , . . ., Cg(R,,)is dense in Ct(R,). R e m a r k 4. References with respect t o the theory of Holder spaces can be found in the well-known book written by C. MIRANDA[l].

2.7.2.

Interpolation and Equivalent Norms

T h e o r e m 1. (a) Let 0

6

to < t,

< co and 0 < O < 1. Then

(Cto(R,),Ch(R,,))o,oo = V ( R , ) where t = (1 - 0) to

+ Ot,.

(1)

(b) Let 0 c t =I= integer. Then

Ct(R,) = V t ( R t , ) . Proof. Step 1. Analogously t o 2.5.1, there are considered in C(Rn)the n commutative strongly continuous semi-groups {G,(T)}~,,, m , j = 1, . . ., n, of isometric operators, [Qj(r) fl( 5 ) = f ( s ~* . - 3 - 1 Xj r , x j + 1 , * . ., x,), f E C(R,). (3) 3

3

+

If A, denotes the corresponding infinitesimal operator, then

If K m has the same meaning as in Definition 1.13.3, then IlfIlKm = I l f l l c m for f E S(R,,). Since S(R,) is dense in Cm(R,,), it follows K m nCm(R,). On the other hand let f E Km. Then the method in the first step of the proof of Lemma 2.5.1 shows that CT (RJ is dense inKm.Hence, K m = Cm(R,,).Using Theorem 1.13.6/1, one obtains that

(C(Rn),CnWn))o,m = vem(WWith the aid of the same theorem it follows that (2) is true.

Step 2. Theorem 1.13.6/2 and Theorem 1.10.2 yield (1). Beside A i we shall use A S , defined in Subsection 2.5.1.

(4)

202

2.8. Embedding Theorems for Different Metrics

T h e o r e m 2. Let t > 0. Further let k and 1 be integers with 0 5 k < t and 1 > t - k. Then for 0 < 6 5 co

v f ( R n )= {f I f

IIfIIgt)< a},

E Ck(Rn),

r = 1,2937

All these norms are equivalent to the norm I(f Ilut. P r o o f . The proof follows immediately by Theorem 1.13.6/1 and the above considerations. R e m a r k 1. On the basis of Theorem 1 and the above considerations, one can obtain further equivalent norms. For instance in ( 5 ) , (6), and (7), one may replace sup If(x)l by Ilfllck. Further, Theorem 1.13.4/1 shows that

SR”

Ilfll&

= SUP zsRn

I f ( 4 + la/-rn c

SUP

XEH,

lo”f(4

is an equivalent norm in the space C1n(Rn),m = 1 , 2 , . . . For 0 6 to < t, < co and Bt, $. integer, 0 < 8 < 1, it follows by Theorem 1 and (1.3.3/5) t,hat

t = (1 - @ t o

+

llfllct 5 cIIfIlt.;o“IIfII&

3

f E C‘1(Rn)-

(8)

Theorem 1.13.4/1 and (1.3.3/5) yield that (8) it also true for (1 - B ) t , + Bt, = t = integer. Formulas of such a type are of great importance in the theory of partial differential operators. Equivalent norms of the type ( 5 ) and (6) where only pure derivatives appear are only possible in spaces Ct(R,,)with t integer. See Remark 1.13.4/2,

+

R e m a r k 2. The results of this subsection are analogous to the considerations on See e.g. Theorem 2.5.1. the spaces Bi,q(R,,).

2.8.

Embedding Theorems for Different Metrics

Embedding theorems are of great importance for the theory of spaces of functions and distributions. There are two types of problems not independent of each other. 1. Given a space B;,q(R,) (or J’i,q(Rn), Hi(R,J, Wi(R,J).One asks for spaces B:,u(Rn) (or F:,u(Rn), H;(R,,),W:(R,), Ct(Rn))such that B;,q(Rn)c B;,JRn) (similarly for the

2.8.1. Embedding Theorem

203

other spaces). 2. If u(xl, . . ., xrl)is an element of B i J R J , then one investigates the , on an ( n - 1)-dimensional hyper“boundary value ” or “trace” u(xl,. . ., x , + ~0) plane and one mks whether this trace belongs to spaces of the type Bi,,(R,,-l).One tries t o find a comprehensive characterization of such traces. Generalizing this problem one considers also traces on 1-dimensional hyperplanes, 1 < n. Clearly, there is a close connection between these two problems. The obtained results are very important for applications in the theory of ordinary and partial differential equations. We shall return t o these questions later on. This section is concerned with problem 1. Problem 2 will be considered in the following section.

2.8.1.

Embedding Theorem

T h e o r e m . (a). Let co > q

2 p > 1,

1

6 r 5 co, and -a < t 5 s < 00 with

F&(Rn) c F&r(Rn). (c) Let 1 < p -C 00, and t 2 0. Then B$‘ (d) Let 1 < p

(R,)

(3)

c~(R,).

< 00, 0 < t

(4)

+ integer, and 1 5 r 6 co. Then

B ~ : ~ ( R c~~)( R , ) . (e) Let 1 g o of, the type described in Example 2.3.1. Additionally we assume (Fyk)

(E) = 1

for

1 Let f E Bilr(Rn)and - = 1 0

5 E supp Fy,, k

= 0, 1 , 2 ,

...

1 1 -+. Then one obtains by Theorem

P

4

(7) 1.18.9/1

Ilf * v k l l L , = Ilf * v k * V k l l L , 5 l l y k l l L o Ilf * vkllG* If y ( x )is the generating function in the sense of Example 2.3.1, and if one takes into consideration yk(x) = 2 k n y ( 2 k x ) for k = 1 , 2 , . . ., then k=l,2,

...

2.8. Embedding Theorems for Different Metrics

204

Now, ( 2 ) is a consequence of l l f l l B ~ , , ' Scll{f

S

* v k } IIIk(Lr)

cWf

* Pk}IIi:+"

(;-f)cLP,S

C"llfIIB;,,*

Step 2. To prove (3) we need a preliminary consideration. Let 1 < r, q < CO. If the SyStemS {Q)k(z)}&, and {?#k(z))gO have the same meaning as in the last step, then the matrices (Kk,j(X)}-m< k , j < m and {Kk,j(X)}-m< k , j

-

FKk,k(Z) =

=

F&,k(Z)

lzlt

2-"(Py,) (z), k

=

1 , 2 , . . .,

k

=

1,2,.

(x), -2"lzI-t(Fyk) =0

Kk,j(x) = K k , j ( z )

. .,

otherwise

satisfy the assumptions of Theorem 2.2.4(b). Together with Remark 2.2.412 whence it follows

-

and a converse estimate if one uses (Kk,j(x)}-m< k , j < Whence it follows

IlrlF;,,

- Ilf *

P)OllL,

+

Step 3. To prove (3) we w e n

F-1lsl-x

= c,(z1

1 :k

-1

1

Ip-"14tF(f * vdlP]F

1 1 1 < x < co, - + - =

--n X',

x

xr

!I

IL,

.

1.

(8)

(9)

The validity of (9) one may show in the following way. For 1 < x < 2 it holds n

l ~ lE L1(Bn) - ~ + L2(Rn)and

consequence of

[~-1/(h)]

hence P 1 l x / - % E

(+)

( 5 ) = A - " [ F - ~ / ( ~ ) I if

+ L2(R,).But o
then, (9) is a

co.

The same argument is true if one replaces F-l by F. One obtains (9) for 2 < x < oc, by FF-l = E. Finally, (9) for x = 2 follows by n

1xl-T;

n 121-2

if

x

t 2.

206

2.8.1. Embedding Theorem

and

Theorem 1.18.9/3 yields

l(

k=l

F ( f * ~k)llr)'ll

lF-l[lXlc

Together with (8) whence it follows that

llfllF;,r I cllfllF;,r*

f

Lt

5c

I/(2 k-1

IF-'[lxl'

F(f * Vk)llr)'il

r,

ES(%).

Since S(R,) is dense in F&(R,), one obtains ( 3 ) .

Step 4. We prove (4)f o r t = 0 , 1 , 2 , . . .Let {P)k]$o and { Y k } & be the same systems as in the first step where additionally it is supposed that

5 FQ)/((E)= (274-5.

n

m

k=O

Then for f

E B$+'(R,)) and

loci 5 t, if follows by

m

(2.2.1/4)

m

Analogously t o the first step it holds that (Dayk)(x) = Zk"+lblk(D"y)( 2 b ) for k = 1 , 2 , .

1 1 . . For += 1 it follows that

P

c IIDaf *

P'

k=O

c W

00

Q)kllLm

k=O

IIDavkllL,*

I l f * Q)k\lL,

00

Hence, C Oaf* P)k converges in Lm(R,).(10) yields that this limit coincides with k=O

Oaf.Now, if one takes into consideration that S(R,) is dense in B 2 l t ( R , Jthen one obtains (4)for t = 0, 1, 2 , .

..

206

2.8. Embedding Theorems for Different Metrics

+

Step 5. We prove (5). (This contains (4) for t integer as a special case.) For . .,0 < 8 < 1, and t = Om $; integer, Theorem 2.4.l(a) and Theorem 2.7.211 yield m = 1,2,

.

B'+t(Rn) p,r

=

-+m

(B:(Rn), B ~ ! (Rn))e,r ~ c (co(Rn),Cm(Rn))e,r c (Co(Rn),Cm(Rn))e,m =

C'(Rn)*

Step 6 . One obtains formula (6) by (5) and Theorem 2.3.2(d). R e m a r k 1. We describe a number of simple conclusions of the theorem. Using the inclusion properties of Theorem 2.3.2 one obtains: (a) If co > 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 <

(11)

then

00,

F;,r(Rn) c Fi,e(Rn).

(12)

Further it is clear, t,hat one may exchange the spaces B;,,&Rn)and Fk,e(R,,)on the right-hand sides of (11) and (12). (b) If 1 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<, , , from 2.5.2 by the semi-group { P ( T ) } ~co~ from . < 2.5.3.The use of more general semigroups in this connection was conaidered in H. TRIEBEL [20].

2.9.

Direct and Inverse Embedding Theorems (Embedding on the Boundary)

The application of function spaces in the theory of ordinary and partial differential equations is based on the fact that the elements of some function spaces have boundary values. This section is concerned with such boundary values. The basis for our considerations are the abstract embedding theorems obtained in Section 1.8. While for the Sobolev spaces we shall derive rather simply a final result, there are needed additional considerations for the Lebesgue spaces and the Besov spaces. Further we shall introduce special Sobolev spaces with weights. This is helpful for the application of the abstract results in Section 1.8.But later on we shall be concerned with much more general spaces of such a type. So we restrict ourselves a t this moment to the facts needed here. 14

1 b e l d , Interpolation

210

2.9. Direct and Inverse Embeddmg Theorems (Embedding on the Boundary)

2.9.1.

Direct and Inverse Embedding Theorems for the Sobolev Spaces ( I = n 1)

-

If s = 1 , 2 , . . . and 1 < p < O D , then the Sobolev spaces W i ( R n )are normed in the sense of Theorem 2.3.3(b) and Remark 2.3.312 by and bv

For abbreviation, we set W;(Rn) = LJR,).

=-

D e f i n i t i o n . If R: = { x I x = ( x l , . . ., x,,) E Rll;xn 0}, then W;(RA), . . ., 1 < p < co,denotes the restriction of Wi(R,,)to R: , that meam

s = 1, 2,

W",R;)

=

{f I f

E Lp(RA),39 E

Wi(R,) with g ( x ) = f ( x ) for

x

E R,+},

(2)

where the infimum is taken over all g in the sense of (2). The mapping g + f is said to be a restriction operator. R e m a r k 1. Wi(RA)is a factor space of W i ( R n ) ,and hence Wi(RA)is ZL Banach space. L e m m a 1. The restriction operator from Wi(R,,)onto Wi(R:), where s = 0, 1, 2, .. . , 1 < p < O D , is a retraction in the sense of Definition 1.2.4. If N is a natural number then there exists a corresponding coretraction independent of s = 0, 1, . . . , N and 1 < I, < 00. Proof. We have t o show that there exists an operator S E L (Wi(RA),WC(R,,)) such that (Sf)( x ) = f ( x ) for x E R:. Functions f ( x ) ,infinitely differentiable in the closure of R: and vanishing for large values of 1x1, are dense in W;,(R,C).It is sufficient t o extend functions of such a type over the boundary of R:. Let 0 < ill < co. We set < I, < ... < if

xn 2 0,

Now i t is easy t o see that one can (uniquely) determine the coefficients aj such that

Whence it follows that S f ( x ) is an N-times continuously differentiable function in R,, vanishing for large values of 1x1. Now one obtains the assertion of the lemma by the explicit expression of ( S f )( x ) . R e m a r k 2. The method is due t o G. M. FICRTENGOL'C [l]. See also H. TRIEBEL [17], p.377. As a simple conclusion one obtains that Wi(R,+)may be normed by formulas ( l a ) and ( l b ) after replacing L, by L,(R:).

2.9.1. Embedding Theorems for the Sobolev Spaces ( E = n - 1)

L e m m a 2 . Lets = 1, 2 , . . . and 1 <

W”,R:,) =

1

v, (P,-,P

211

< co. T h e n , in the sense of Definition 1.8.1/1, 1

-

W p L - 1 ) ;P, -, LP(R,H))

P

(5)

Proof. Step 1. For smooth functions f(x)of the type considered in the proof of the ) llfll v, are equivalent t o each other. Here, one uses that last lemma, l l f l l k ; ( n ~and W”,R;t) may be normed in the sense of Remark 2. Step 2. Let f(x’, t ) E V,$where x‘ = ( x l , . ., x,-~) and t = x,. Then f(x’, t + d), 6 > 0, belongs also t o V sand it holds f(x’,t 6) -+ f(x’, t). We fix 6 and use Sobolev’s mollification method described in the first step of the proof of Lemma 2.5.1,where h < 6, with respect t o RA. Using the notation introduced there, we choose for 7f(x’, t + 6). convenience o ( x ) = o(l)(x’)o ( 2 ) ( t ) . Then it follows (f(x’,t The function ( f ( x ‘ ,t + S)),, , infinitely differentiable in R: , may be exiended by (4) to R,. Whence it follows (f(x‘,t + d)),, E Wi(R;). Since the considerations of the first step are also true for these functions, now one obtains (5). Theorem. L e t s = 1,2,.. ., 1 (RA)then

@ ( R 3 = ( f I f E K(R:,L %f = 0 ) . (7) Proof. Step 1. The product of two retxactions is again a retraction. Hcncc it follows by Lemma 1 that one may restrict oneself by the proof of (a) to the spaces Wi(I3:). Lemma 2.5.1, Theorem 1.13.2,and Theorem 1.12.1 yield that { Wi(R,,-l),LP(Rn-l)) is a quasilinearizable interpolation couple. Xow, (a) is a consequence of ( 5 ) , Theorem 1.8.5(a),and (2.4.2/16). Step 2. Clearly, it holds %f = 0 for f E k i ( R i ) .To prove the converse asscrtion we assume that f is the restriction of a function belonging to C,”(R,) and %f = 0.

a’f (x)= O(x;;J),j = 0, . . . , s. Now we consider for 0 < il < 1 But then it follows axi the infinitely differentiable real functions p.(t)defined on [0, co),for which

0 5 XA(t) 5 1,

xn(t) = 0 for 0

6 t < 1,

Xr(t) = 1 for

2A

t < CO, (8a)

212

2.9. Direct and Inverse Embedding Theorems (Embedding on the Boundary)

Step 3. Now let f E W“,RA) be an arbitrary function with ‘8f = 0. The function f(x)is odd-extended to R,, - R:, and the so obtained function is also denoted by f . Then f belongs to W“,R,,). This follows with the aid of partial integration from the identity

J

Daf.g?ax=(-i)lal

j

f-Dag,dx,

If,

li,,

lbllss, v E ~ r ( ~ , , ) .

(10)

(See also the following remark.) Now we apply Sobolev’s mollification method described in the first step of the proof of Lemma 2.5.1, where ~ ( x=) G(lxl), t o f. We obtain W;(R:) 3 f l l + f for h J, 0 and %fh = 0. Approximation of fh(z) by ~ ( zfh(x), ) where ~ ( xare ) restrictions of suitable functions belonging t o C r (R , ), leads to the situation considered in the second step. R e m a r k 3. The boundary values must be understood in the sense of the considerations of Subsection 1.8.5. If f(x)is the restriction of a function belonging to CF(R,,), then it follows that ‘8f coincides with the usual (classical) boundary values. Since 8 is a continuous operator from Wi(RA)into

n B;;$F(Rfl-l) one may obtain the

5-1

,

1

j=O

boundary values of an arbitrary function f E WP(RA)in the following way: If f k -+f where fh(x)are restrictions of functions belonging t o Cg(R,), then 1V:

8-

1

1

On the basis of this limit process, one may carry over integral identities, including boundary integrals, from smooth functions to functions belonging to W;(R,), resp. W ; (RA).The identity (10) is an example. R e m a r k 4. The theorem is of fundamental importance for applications of the theory of function spaces to partial differential equations. It shows that we obtained 1 in some sense a final result for the description of boundary values. Since s - j - P + integer, one may formulate part (a) in the following way: % is a retraction from

W;(R,f)(resp. W$(R,,)) to the product of Slobodeckij spaces

n

S-1

J=o

1

Wi-j-F(Rn-l). The

fact that 8 is a retraction includes the so-called “direct” embedding theorems as well as the so-called “inverse” embedding theorems. Here, “direct embedding theorem” means that ‘8 is a linear continuous mapping from W”,R:) onto

n J=o

W;-J-T(R,,-J. As “inverse embedding theorem” one denotes the statement

n

W;-J-F(Rn-l)into ?V;(R:).

.s- 1

1

that there exists a coretraction, that means a linear continuous mapping from s-1

J

=O

1

R e m a r k 5. It is the aim of the following subsections to extend the theorem t o Lebesgue spaces H”,(R,,) and to Besov spaces B;,,(R,,) (and hence also to Slobodeckij spaces W;(R,,)). But there are needed some preliminaries for the application of Theorem 1.8.5. Among other t.liings it will be helpful to introduce special Sobolev spaces with weights.

2.9.2. The Spaces W'~lz,b,@n) and Jt.6,:z;(R+,)

213

R e m a r k 6. One may consider the embedding of functions belonging to Wi(R,,) on I-dimensional hyper-planes, for instance R,= {x I x = ( q , . . . , xl,0, . . ., O ) } . We shall return to this question in Subsection 2.9.4. It seems to be meaningful t o use an iteration, from R, t o R,,-l, from R,.-l t o R,1-2,and so on. But the theorem shows that for such a procedure one needs embedding theorems for the Slobodeckij spaces. R e m a r k 7. The investigation of the boundary values of Sobolev spaces was one reason for the introduction of Slobodeckij spaces. References with respect to this question are given in Remark 2.3.1/2. Further references can be found in Remark 2.9.412.

To obtain final results of the type of Theorem 2.9.1 for the Lebesgue-Besov spaces (without weights), we must be concerned temporarily with special Sobolev spaces with weights. Later on, those spaces are considered extensively. But for sake of completeness and for preparation of the later chapters, we shall prove a little more than needed in Subsection 2.9.3. Essentially we shall use in Subsection 2.9.3 only t,he ideas developed in the third and in the fourth step of the proof of Theorem 1 of this subsection. D e f i n i t i o n 1. Let nz = 0, 1 , 2 , . . ., 1 < p < 03, and -1 < LX < 03. Then WEl.T,,lm(Rn) denotes the completion of X(R,,)in the norm

Wg,:(R;) is the restriction of W;12sle(R,l) to R;. W;,l,nlD(R,l) = Lp,lxnl@,,) and similarly for R;. R e m a r k 1. If one considers a space containing all functions f E Lp,l,,,la(R,)such l Z00,n l then ~ ~ Rit" )is easy t o see that this apace is complete. (Here, that I l f ~ ~ ~ ~ ~ < the derivations must be understood in the sense of D'(RA)and Df(R;)).But then is meaningful, too. WFX;(R,+)is constructed the above definition of WElxn,s(Rll) analogously to Definition 2.9.1. As a factor space, WF,x,$Rf,)is also complete. The different equivalent norms in WF(B,-l) (see for instance (2.9.111))show that one can replace Ilf 11 W~lz,l.(n,)by

Here,

5'

=

(x,,. . ., z~-~).

L e m m a . (a) The restriction operator from W;";lx,le(Rfl) onto W2,a(R;) i s a re" traction in the sense of Definition 1.2.4. If N is a natural number then there exists a corresponding coretraction independent of m = 0, 1 , 2 , . . ., N , 1 -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 <

oc),

1

P

P r o o f . Step 1. For s - - = integer, we prove the first part of (c). Let f E S(R,,).

P

For fixed X' E R,,-l, it follows by Theorem 2.8.l(c) that

If one takes into consideration q = 1, and if one uses (2.5.1/10),it follows by integration over LP(Rn-Jfrom the triangle inequality for integrals that

Since S(R,,)is densein B:,l(Rf,),(14)holds also for f E B;,l(R,,)(seealso Remark 2.9.1/3)' The Lemma yields that (14) is true for f E Bi,l(R:),too, if one replaces Bi,l(Bn)by

B;,l(R:). Step 2. We fix j

1 0, 1 , 2 , . . . and choose a natural number m > 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,<] 7

1 Now, Theorem 2.4212 yields that 93 for s > -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<. K. GOLOVKIN, S. V. USPEKSKIJ,etc. Traces of functions on manifolds (boundaries of domains) under

2.9.4. Embedding Theorems for Lebesgue-Besov Spaces (General Case)

226

weak assumptions are considered in G. N. JAKOVLEV [2] and S. V. USPENSKIJ [6]. The theorems of 2.9.1,2.9.3, and of this subsection contain also characterizations of the spaces &RA) and .&q(RA), and corresponding approximation properties of functions belonging t o H;(R,) and B;,JR,) where rrtf = 0. These results are proved [21], some special cases, however, are known before. A complete in H. TRIEBEL description of the case p = 2 and 1 = n - 1 can be found in J. L. LIONS,E. MAQENES 1 [2, I]. For p += 2 and 1 = n - 1 the non-singular case (that means s - - += integer)

P

for the W;-spaces is also considered by J. L. LIONS,E. MAGENES[l,V]. The corresponding investigations for the H;-spaces are due t o E. SHAMIR [l] and for the B;,pspaces are due t o E.MAQENES [l]. The density of CF(RA) in H;(R+,) and in W;(Ri) 1 for s < - was proved in J. L. LIONS, E. MAGENES[l, IV]. There is also shown that 1

p

1

WF(R+n)= WF(RA) holds. Finally, we point out that the embedding theorem (2.8.1/4) which was of special importance for the above considerations is also known. We refer to P. GRISVARD [4], p. 242, and 0. V. BESOV [3]. R e m a r k 3. Until this moment, we considered separately the embedding theorems for different metrics (Theorem 2.8.1) and the embedding theorems for different dimensions (Theorem 2.9.1, Theorem 2.9.3, and the theorem of this subsection). Clearly, one may combine these results. In this connection we mention a simple rule. n The number s - - is called the differential dimension of the space Bi,q(R,) and the

P space H;(R,) (resp. B;,#2;) and H;(RA)). The Holder spaces Q(RA) have the differential dimension t. For all the above embedding theorems the following rule is satisfied: The differential dimension is unchanged, the upper index is getting smaller. Here, for the embedding on the boundary, only the mapping f(x) + / ( X I , 0 ) is considered. If one considers the mapping f(x) + E , . f ( x ' , 0 ) , then one has t o modify the rule in an obvious manner. The rule is correct for the so-called limit exponents described in Remark 2.8.1/5. Using these results, one may immediately obtain further embedding theorems on the basis of Remark 2.8.1/1 and Remark 2.8.112. Using the rule, it follows in a n easy way by Theorem 2.8.1 and by the above theorem that &,f = f(x', 0) for 1 = 1, . . .,n - 1 is a continuous operator from B;,JR,)

n l into q,JRr), 0 q 2 p > 1 , 1 5 r P P

5 00,

(2)

and from H",R,,)

n l into B;,p(Rl), 0 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 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 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 <

H“,R,,) where 1 1 -_

P’

< U < -

P

< co. The first step sliows that Q is a continuous operator in

1 -_

.

P’

1

< u < -.

Let s = m

P

+u

where m = 0 , 1 , 2 , .

..

and

. By the first step, one obtains for f E C r ( R i )that

Theorem 2.3.3(a) and GD’f = DaGf yield

s clllllw;(R,+).

llGfllqRn)

By completion, it follows that G is a continuous operator from H>(Ri)into H>(R,,), 1 -1 < s - - + 0 , 1 , 2 , ...

P

1 Step3.Letm = 0 , 1 , 2 , . . ., - -1 < uo,u1 <-and sj = m + uj where j = 0, 1. P’ P Then G is a continuous operator from (B;(R;), fi;(R;))e,qinto 0

(a;(Rn),H;(Rn))e,q = Bi,q(Rn),

= (1

- 0) $0 + 081, 1 5 4 S

[sl---]1 - = m - 1 , the operator !R defined in (2.9.3/9) has the [ same structure in both cases. Denoting temporarily by a corresponding coretraction

Since s - 1 -=

G%

is a projection from H;j(Ri) (for instance of the form (2.9.2/12)), then P = E onto H$(Ri). Since projections are special retractions, it follows by Theorem 1.2.4, Theorem 2.9.3, and Theorem 2.10.1 that

(&(R:)) fi;(R:))e,q = Bi,,(R+,), 1 5 q <

00.

(2)

Whence it follows that G is a continuous operator from B;,q(R:) into Bi,q(R,,),

1 -1 < s - -

P

* 0 , 1 , 2 , . . ., 1

q

<

00.

Step 4. To prove that G is a coretraction, we set

where 0 < 1, < a, such that

. . . < lfi+l < co and x’ = (x,, . . ., x , , - ~ )Choosing . the coefficients

then ‘%y belongs t o H;(R:) and t o gPJRA) where

1 -<s
P‘

1 5 q 6 co. Whence it follows, with the aid of the last assertion of the first step, that 6 (after completion) is a retraction corresponding to G.

230

2.10. The Spaces H;(R:) and B:,*(R:)

R e m a r k 2.* J.L.LIoNs, E.MACENES [ l , I V ] proved that C 5 is a continuous 1 operator from W8,(R,+) into W;(R,), 0 s 5 1, s $: -. A direct proof for the case 1 13 0 s < - was given in Remark 1. We mentioned above that a complete treatment

s

s

P

of the important case p = 2 can be found in J. L. LIONS,E. MAGENES [2, I]. The mapping properties for t 5 with respect t o tha spaces Hi(RA) are considered in [3]. I n the first and in the second step of the proof we essentially followed R. SEELEY the treatment given in R. SEELEY[3]. 1 1 1 1 1 T h e o r e m 2 . ( a ) L e t 1 < p < 00, 1 sq < co,- + - = - + - = 1 , - < s
P

1 1 1 1 (b)Let l < p < c o , l s q < c o , - + - = - + - = 1 ,

P’

1,

P’

q

Then it holds in the sense of the theory of distributions in D’(RA)

Proof. Step 1. Let 1 s. For q E ( i i J R A ) ) ‘ and 9 E S(R,), we use formula (2.9.3/4) where the convergence of g ( v j ) and the independence of the choice of the system (yj)jzlfollow from the considerations below. It holds that 4-

I(W (941 5 1

s

llqll(;;,q(R;))* Ilx+(Pj llSll(E8P.‘l @:))‘

6 cllqll(~,*(,t:)), Theorem 2.6.1 yields gq g E B;!,,,(R:) and 1l911B:;

llv - G2vllB;,g(Rv,) llvL3;,gw,,).

E B;,,.(R,,).

s

,(fp) cllgll ($#;))’

E2v)llB;,g(n;)

Since Gq is an extension of q, if follows *

(6)

1 Without changes the considerations are also true for g E (Bi,q(Ri))’if - - < 8 1 13’ < - . One concludes analogously for the spaces of type H .

P

Xtep2. Lct 1 < p < co, 1

1 1 s q < co,< s < co a n d s - - =+ 1 , 2 , . . . Further ?, 2,

2.10.3. Lift Property

23 1

let g E B;?,,.(R;).We choose an element f E B;F,,.(R,) such that g is the restriction of f . If C5 has the meaning of (l),and if v E C,j”(R,t),then it follows by Theorem 1 and Theorem 2.6.1 that

Is(v)I = If(@v)I5 llfllB-:

llGPllB;4(Iin)

P .P ,(R,)

Whence it follows g

E ( @ , q ( R i ) ) Construction ’. of

the infimum with respect to f yields 1 1 the reversion to (6). Similarly, one concludes for < s < - and for the H spaces. P P R e m a r k 3. * We mentioned above that the description of the spaces H i ( R i ) and Bi,q(R:),where s < 0, as dual spaces of spaces with positive upper index is helpful for applications to differential operators. Similar considerations are possible if one replaces R i by Q where Q i s a domain in R,. We return to this point in Section 4.8. Spaces over domains with negative upper index are introduced by J. LERAY[l], L. SCHWARTZ [2], and P. D. LAX [l] for the case of Hilbert spaces. The further [l]. Further, in this connecdevelopment of this theory is due to Ju. M. BEREZANSKIJ [3], the book by J. L. LIONS, E. MAGEtion we mention the book by L. HORMANDER NES [2, I] and the survey paper by L. R. VOLEVIE,B. P. PANEJACH [l]. The extension of this investigation to the spaces Wi(RA),where s < 0 and p 2, goes back to J. L. LIONS,E. MAGENES[l].

+

2.10.3.

Lift Property

In this subsection, statements for the spaces i;,,(R;) and the spaces &,(RZ)analogous to Theorem 2.3.4 are proved. Definition. Let - 03 < s < 03, 1 < p < 00, and 1 q 5 00. One sets

4 , q ( R i )= {f

I f E B;,q(Rn),

& ( ~ i=) { f I f E H;(Rn),

SUPP f

SUPPf c

= El

7

(la) (1b)

R e m a r k 1. The spaces &,q(R;) (and f i p ( R i ) )are considered not only as spaces over R i but also as closed subspaces of Bi,q(Rn) -(and H”,R,)). Similarly, one may define &,q(R;) and e P ( R ; )where R; = R, - R i . Clearly, and

Bi,qW H;(R:)

= =

B;,q(Rn)I&,q(RL)

H;(R,)@,(R;).

We show that C$(Ri) is dense in &p,q(Ri)as well as in &,(R’,) where - 00 < s < 00, 1 < p < co, 1 5 q < co. If N is a natural number, then it holds f ( z + h) + f ( z ) in W f ( R , ) if h + 0. Translations of such a type and multiplication with functions belonging to CF(Rn)are continuous operators in W f ( R I JBy . Theorem 2.6.1, similar . properties (inclusively f ( z + h) + f ( z ) if h + 0) hold in the space W i N ( R n )Inter-

232

2.10. The Spaces Hi(R:) and Bi,,(R:)

polation shows that this is also true for the spaces BA,(R,,) and HB,(R,,), - 00 < s < 00, 1 < p < co,1 5 q < co.Now, with the aid of Theorem 2.3.2, one obtains the desired property. Theorem. (a) Let -a < 0 , s < 03, 1 < p < 00, and 1 q co. Then Jsf = P1(ixn + (1 + 1x’I2)+)”P f , (2) where x’ = (xl, . .,x,,-~), is an isomorphic mapping from B;,,(R;) onto BZr,d(R:) and from %(R;) onto @;S(R;). 1 1 (b) Let 1 < p < 00, 1 5 q < 00, - < s < 00 and s - 1 , 2 , . . . Then

-

.

IP

~ J R ;= ) B;,,(R;)

1,

-

+

Bp(~;) = &(R;).

and

(3)

1 1 (c) L e t 1 < p < co, 1 5 q < c o , a n d - - 1 < s < - . T h e n 2) 2,

e , , ( R ; ) = B;,,(R;) and &(R;) = H;(Ri). (4) Proof. Step 1. The proof of part (a) is based on a modification of the well-known Paley-Wiener theorem. If one replaces ( 1 + lx12)T in (2.3.4/1) and in (2.3.4/2) by (ix,, (1 1 ~ ’ 1 ~ ) f ) ~ then , one may replace I , in Theorem2.3.4 by J , . Now, for the proof of part (a), it is sufficient - to show that supp J,f c R+,holds for every function f E S’(R,,) with supp f c R,+ . This is equivalent to the proof of 6

+ +

+

SUPPP(~x,, (1 +

-

Ix’~~)*)”

F-lv c R, -

= R,,

for v E CF(R,,) with supp q~ c R; = R,, - R ; . Now let type. Then one may extend

- Ri

v be a function of

such a

-8

withfixed x’ E R,,-,analyticaUyin the half-planeIm -c 5 0. Here E > 0 is a suitable poaitive number such that supp v c {x 1 x,, 5 -&}. Denoting this function by (P-lv)(x‘,t),then I(F-lv) (x’,t)l 5 c ea1m7. For 5, > 0 and N 2 0, it follows by Cauchy’s theorem that

[P(ixn + (1

+ IX’I~)’~

F-lvI (6’3 En)

(6)

-iN+m

= (2n)-+

J J

-iN-m

R,-l

e-i<s’.c)-iTtn(i-c

+ (1 + 1x’1~)*)”(~-1qI)

(51,

z) ax’at.

If N + 03 then one obtains the desired assertion. Xtep 2. Part (b)and part (c)follow from the last part of Remark 1, Theorem 2.10.211, and Theorem 1.2.4. 1 Remark 2. The question arises whether (3) remains valid if s - - is a natural

P

.

number. In Remark 43.213, we give a negative answer for the spaces B i J R ; )

where 1 < p , q < co. At that place we also obtain explicit norms for the spaces Bi,,(R,+),see also Section 4.4. Furthermore, we shall show in Remark 2.10.5/2 that the spaces Bi,,(RZ) and &p,q(R;),resp. the spaces Hi(RA) and k P ( R ; ) ,do not coincide, 1 < p < co, 1 < q < co,and -co < s < - 1

+ -.1

P R e m a r k 3. As mentioned in the first step of the proof, J , gives also an isomorphic mapping from Bp,q(R,,)onto B;;;(R,,), and from H;(R,,)onto H;--"(R,,). Furthermore, similarly to Remark 1 it holds that

As an immediate consequence one obtains: J , is a n isomorphic mapping from Ba,,q(R;)onto q;gd(R;), and from H;(R;) onto Hg-*(R;) where - 00 < CT,s < 00, 1 < p < co, and 1 5 q 5 00. Considering

38f = F-'(i.,, - (1 + Is'l"+)"Ff

instead of J,f, then one obtains the same assertion for the spaces B;,,(R;) and H:(R;).

2.10.4.

Interpolation of the Spaces hi(R:),ki(R:), &*(It:),

and i;,JR:)

Theorem 2.10.3 shows that the spaces B;,,(R;) and g;,,(R;), resp. &(RL) and gP(RL) 1 coincide, 1 5 q < co and 0 < s - - integer. Hence, it will be sufficient to prove

r,

+

a general interpolation theorem for the spaces gP,,(R,+) and e P ( R ; ) . T h e o r e m 1. Under the corresponding hypotheses for the parameters the form& (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) remain valid after 'replacing B : . (R,,)by"B:.(RA)and H : (R,,)by 5:(RA).Further it holds that

@;,p:)* %q,(R;))B,q = &,,(R3

+

(1)

9

--oo < ~ 0 < ~00, ~ a. 1 81, o < e < 1, s = (1 - e p , e8,, 1 < < 1 I qo,q1,q 5 a. Proof. We prove (1). The proof of all the other formulas is completely analogous. Theorem 2.10.3(a) yields that we may assume without loss of generality that 1 1 - < so, s1 < 00 and sj - - integer for j = 0, 1. Let additionally qo < co and 1,

+

I,

+

< 00. Then it follows by Theorem 1.17.1/1, Theorem 2.10.2/1, and Theorem 2.10.3(b)that(2) (B:qo(R,+)? &,,(R;))fhq = B;,q(R;)*

q1

Now, Theorem 1.2.4 shows that G from Theorem 2.10.2/1 is also for q = co a coretraction from &p,q(R;)into B;,,(R,,).Hence, one may extend (2) t o the values qo = 00 and/or q1 = 03.

234

2.10. The Spaces Hi(R:) and B;,*(R:)

R e m a r k . On the bask of Theorem 1.2.4 and the previous results, one may derive immediately further interpolation theorems. Let j = 0, 1 , 2 , . . . be given. For 1 1 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<

* B;,#W,

1 -l+-,l
P

qmJ * H p ; ) ,

(4)

00,1
2.10.3 and Remark 2.10.3/2.

2.11.

Structure Theory

In this section, the isomorphic structure of the spaces B;,,(R,,), H;(Rn),B;,q(Ri), H“,Ri),Bi,q(R:),and I&(R:) is considered. As a consequence, one obtains that many of these spaces have unconditional Schauder bases.

236

2.11. Structure Theory

2.11.1.

Preliminaries (Properties of the Spaces L, and 4 )

In this subsection, we collect some known facts on the geometric properties of the spaces 1, and L, . D e f i n i t i o n 1. A subset {bj},tl of a complex separable Banach space B is said to be a Schauder basis if everyelement b E B maybe represented in a unique way as convergent m series b = C Bjbj, B j complex J -1

(topological basis). A Schauder basis is said to be unconditional if any rearrangement vf the elements of {bj}Fl is also a Schauder basis. P. ENFLO [l] and S. KWAPIEI~ [l] have shown that there exist separable Banach spaces without Schauder basis. Treatments on bases in Banach spaces can be found in J. LINDENSTRAUSS, L. TZAFRIRI[2] and M. M. DAY[l]. T h e o r e m 1. (a) Let 1 < p < oc), and let L? c R, be a (bounded or unbounded) domain. Then Lp(L?)(with respect to the Lebesgue measure) is isomorphic to Lp((O,1)). In L,@) there exists an unconditional Schader basis independent of p (for instance, a suitable system of Haarfunctions). (b) For 1 5 q < oc) and 1 5 p < co, there exist in lq(lp)an unconditional Schauder basis (here lq(lp)must be understood in the sense of (1.18.1/1) where Aj = 1,). R e m a r k 1. Part (b) of the theorem is easy to show. A proof of the isomorphism JA.B. RUof the spaces L,(L?)and L,( ( 0 , l ) )can be found in M. A. KRASNOSEL’SKIJ, TICKIJ [l], 9 12 (the proof given there is also applicable t o unbounded domains). A proof that a system of functions of Haar type is a Schauder basis in Lp((O,1)) is given in M. A. KRASNOSEL’SKIJ, P. P. ZABREJKO,E. I. PUSTYL’NIK,P. E. SOBOLEVSKIJ [l], 1.5. See also Remark 4.9.412. It is, however, deep t o show that such a basis is unconditional. We refer t o E. M. STEIK [4] and V. I?. GAPO~KIN [l]. Investigations on unconditional bases in L,((O, 1)) can be found also in Z. A. ANTURIJA [ l ] and S. V. BOBKAREV [l]. I n Subsection 4.9.4, it will be shown that some systems of functions of Haar type are Schauder bases in the spaces B;,,(Q) and 1 1 H”,Q),l < p < 0 0 , l < q < c/3, -1 + - - 5 s - . R e r e Q i s a c u b e i n R , . 2,

I,

T h e o r e m 2. Let 1 < po < 03 and 1 < p , < co. (a) L,,((O, 1)) and LP1((O, 1)) are isomorphic if and only if po = pl. (b) lp0 and l,, are isomorphic if and only if po = p l . (c) LpO((O, 1)) and lP1 are isomorphic if and only if p o = p , = 2. R e m a r k 2. A proof of Theorem 2 is given in the classical book by S. BANACH [l], Chapter 12. D e f i n i t i o n 2. An infinite-dimensional Banach space B is said to be a prime Banach space if every infinite-dimensional complemented subspace of B is isomorphic to B. T h e o r e m 3. The spaces I,, 1 5 p < 00, are prime Banach spaces. R e m a r k 3. This theorem is due t o A. PEE.CZY&SKI [I]. There is also a proof that co is a prime Banach space, too. J. LINDENSTRAUSS [ l ] has shown that 1, is also a prime Banach space. Lp((O,1)) with p + 2 is not a prime Banach space, since Z,

2.11.2. Structure of the Spaces Hi(R,) and B;,JR")

237

is a complemented subspace which is not isomorphic to L,((O, 1)) (Theorem 2). In the fourth step of the proof of Theorem 2.11.2, we construct a n example of a complemented subspace of Lp(( 0 , l ) )isomorphic t o I , .

R e m a r k 4. * In connection with geometric properties of Banach spaces, in partic[l], M. M. DAY ular of the classical Banach spaces L, and I,, we refer to S. BANACH [l], J. LINDENSTRAUSS [2,3], J. LIXDENSTRAUSS, L. TZAFRIRI[2], S. G. KREJN [l], and V. D. MILMAN[l]. There further references are given.

2.11.2.

Structure of the Spaces HJR,) and BiJRJ

We start with preliminary considerations which are of self-contained interest. Let and

q/>= { x 1 IzjI

Qo = qo,

Qk

2k for j = 1, . . ., n}, k = 0, 1 , 2 , . = q,(

- qk-l

. .,

for k = 1 , 2 , . . .

By Lemma 2.2.4, the characteristic function x k ( x ) of Qk is a multiplier in LP(Rn), 1 < p < 00. The operator obtained by continuous extension in LJR,) of F-'xkFp, pl E S(R,,),is also denoted by F-lxkF. L e m m a . If 0 < s < co, 1 < p < 00, and 1 notations of Definition 2.3.1/1, we have

B;,q(Rn) = {f

Here,

I f E L,(Rn); IIf IIEi,,

=

5q6

CQ,

then, in the sense of the

II{F-'XjFf) II~:(L~) <

Ilfllz;, is a n equivalent norm in the space BAq(RrI).

CQ}.

(1)

Proof. A homogeneity argument shows that there exists a number c independent of j = 0, 1 , 2 , . . . such that

IIF-lXjFfllLp 6 cIIfIlLp, f E Lp(Rtt)* (2) Let {plj}~,, E @ be a system of functions in the sense of Definition 2.3.1/2. Then (2)is also valid for Fplj instead of x j . It is assumed that (2.3.1/13)holds with plkinstead of @ k . Now it is not hard t o see that the norms l l f l l ~ ; , , from (1) and from (2.3.2/1) are equivalent t o each other. R e m a r k 1. Equivalent norms of the type (1) are well-known, see for instance S. M. NIKOL'SKIJ[7], p. 374. T h e o r e m . (a) Let 1 < p < 00 and -00 < s < 00. Then Hg(R,) is isomorphic to Lp((O,1)), and there exists a n unconditional Schauder basis. ( b )Let 1 < p < 00, 1 6 q 5 00, and - 00 < s < 00. Then B;,q(Rr,)is isomorphic to W p ) .

(c) Let 1 < p < co, 1 5 q < 00 and -CQ < s < co. Then B;,q(Rr,)i s a separable Banach space and there exists a n unconditional Schauder basis. (d) Let 1 < p < 00 and - 00 < s < 00. Then Bg,,(R,,) i s not a separable Banach space. Proof. Step 1. Theorem 2.3.3 yields that H;(Rn) is isomorphic to Lp(R,). Now, (a) is a consequence of Theorem 2.11.1/1. Since I , is not separable, one obtains (c)

238

2.11. Structure Theory

and (d) by (b) and Theorem 2.11.1/1. Hence, Theorem 2.3.4 shows that it is sufficient t o prove (b) for s > 0. Step 2. Let P, = F-lxjF, j = 0 , 1 , . . . We show that

P,Pk = dj,,
00,

in Lp(R,). Let f E Lp(Rn)and S(R,) 3 f l the multiplier property of xj yield =

(pjf)

and

+f LP

03

v E S(R,,). Parseval's

formula and

J fFxjF-lv ax

lim (Pjfi)(v) =

I-

and

(31

4

6

(pkpjf)(v)= lim (F'-lxjFfl? F-'xkF@)L2(Rn) 1- m

=

1- w

d.ik(xjPf1> F@)L.,(R,)= ajk(Pjf)

*

Whence it follows (3). In particular, (3) shows that Pi is a projection in L,>(R!J. The will be denoted by R(Pj).We want t o show that Af = {2'1Pjf)zo range of Pj in Lp(Rn) is an isomorphic mapping from B;,q(Rn)onto

1

03

C

j-0

8 R(P,) c Zq(Lp(Rn)), 1
5 q 5 co, 0 < s < co. By ( l ) ,it is sufficient to prove that R ( A )3 C

Let

m

{fj}p0E C

1

j-0

=

to

j-0

8 R(P,). Since

1

j$2+fj

the series f =

W

C

2-'jfj

j=O

LP

5

C

j-0

@

< co, R(qi).

'1I

,C

~ ~ { ~ j ~ ~ 2-#J, ~ l q ( ~ p ~ I-N

converges in Lp(R,J.By ( 3 ) and Pjfj = f j , it follows Pl,f

2-skfk. Now, (1) yields f E B&(R,) and Af m

W

=

{fj}T0.Hence, B&(R,#)is isomorphic

8 R(Pj)considered as a subspacc of Zq(Lp(R,J).

Step 3. To consider metric operator G I ,

m

C

8 R(P,), we introduce in L,,(R,i),1 < p

In

and show

(Glf) (2)

< co, the iso-

j-O

. ., -j + 1, -j + 2 , . . .

= 2 P f ( 2 1 ~ ) ,I = 0, + 1 , + 2 , .

Pl+jGi = GlPj, j

=

1 , 2 , . . ., 1 =

v E S(R,J.Then one obtains (5)by ( S + j G l f )(v) = J (FGif)xj+lF-lY dz

Let f E S(R,) and

Rm

(4)

(5)

239

2.11.2. Structure of the Spaces Hi(R,) and Bi,,(Rm)

..

(5) yields that is a n isometric mapping from R (P ,) onto R (P l+ l) ,1 = 1, 2 , . Hence, for the proof that Bi,q(Rn)is isomorphic t o Zq(Zp), it will be sufficient to show that R(Po)c Lp(Rn)and R ( P l ) c LJR,,) are isomorphic t o l p . I n the fourth step, we shall show that R(Po)is isomorphic to l p . Since R ( P l )is isomorphic t o a comple, follows by Theorem 2.11.1/3 that mented infinite-dimensional subspace of R(Po ) it R ( P l )is also isomorphic to I , . Step 4. To finish the proof of (b), we must show that R(Po)is isomorphic to l p . Let f = (kl,. . ., k,,), kj are integers, and fn = (kin, . . .,k,,n).We want to show that

K f = {f(fn)}r, f

E &Po) >

(6)

is an isomorphic mapping from R(Po)c Lp(Rn)onto l p , 1 n

00.

We choose a

function p ES ( R,)such that ( F v )(5) = (2n)-yin a neighbourhood of Qo. Extending the definition of K in a n obvious way, it follows that and

Ilg(~, * f)Il~, S IIK(P, * /)II~,6

119 * fllf,m

S cllfII~, for f ~Lco(Rn)

I c ~ ~ (-f~ n) I/(Z)I I dx 5

Ii,

f

C I I ~ I I ~for ,

f E Ll(Rn)*

If one takes into consideration K(g, * f ) = Kf for f E R(Po),then Theorem 1.18.6/1 and (1.18.3/9) yield that K is a linear and continuous mapping from R(Po)into l p . Now we consider the operator S, S{gf}

=

Ff = cxo(x)1e - i ( h , s ) g f ,

f,

f

a t the first moment defined only on the finite sequences of I,,; (h, x> = n

(7) n

1 kjxj,

J -1

f = ( k l , . . ., klJ has the same meaning as above. By the explicit expression of F-lx0, see the proof of Lemma 2.2.4, it follows that f belongs t o Lp(Rn),1 f(n.f), x E Qo,

(8)

shows the possibility of such a choice of c. Using Parseval's formula, the multiplier property of xo(x),the continuity of K , and (8), it follows for v E S(Rn)that

240

2.11. Structure Theory

Since p E S(R,,)is an arbitrary function, one obtains that Il~{gdIILp5

Cll{~~>llI,.

S is extended by continuity t o the whole space 1,. By (6), (7)) and ( 8 ) , it follows SK = E and K S = E. Whence it follows that S E L(l,, R(P,))is the inverse operator t o K E L(R(P,),l p ) . Hence, R(P,) and 1, are isomorphic. R e m a r k 2. I n the proof of the theorem, we followed the treatment given in H.TRIEBEL[23]. The proof that R(P,) is isomorphic t o 1, is essentially due t o J. PEETRE [21]. R e m a r k 3. We mention an interesting special case. It follows immediately from part (b) of the theorem that B;,,(R,,) is isomorphic to 1,) 1 < p < co. Particularly, the Slobodeckij spaces W;(R,J,where 0 < s + integer, are isomorphic t o 1,) while the Sotolev spaces W”,R,) = H”,R,,), s = 0 , 1 , 2 , . . . are isomorphic t o Lp((O,1)). Hence, these spaces belong t o different isomorphism classes. 2.11.3.

Structure of the Spaces H,”(RL),fi,”(R:), Bi,g(R,+), and i i J R L )

1 Theorem 2.10.3 shows that for 0 < s - - $: integer and q < co the spaces H;(RA)

P

coincide with the spaces fi,(R:) and the spaces B;,,(Ri) coincide with the spaces

%,,(RA). T h e o r e m . (a) Let -co < s < co and 1 < p < 00. Then @,(R;) and Hi(RA) are separable spaces, isomorphic to L,((O, 1)))with an unconditional Schauder basis. ( b ) L e t -co < s < c o a n d l < p < co. T h e n f o r l s q < 00 thespacesk,.,(RA)and for 1 < q 6 00 the spaces B;,,(Ri) are isomorphic to lq(lp).For 1 5 q < co,@,,,(RA) and, for 1 < q < co,Bi,,(R:) are separable spaces; they have unconditional Schauder bases. For q = 00, Bi,,(RA)is not a separable space. Proof. Step 1. Theorem 2.10.3 yields that fi,(R;) is isomorphic t o L,(Ri) and hence by Theorem 2.11.1/1 also isomorphic to L,((O, 1)). Whence, together with Theorem 2.10.5/1, it follows part (a).

Step 2. We prove (b) for the spaces kP,,(RA).By Theorem 2.10.3, we may assume 1 without loss of generality that 0 < s < -. Then by Theorem 2.10.3, we have P %,,(R3 = B;,,(R:). if x,, 2 0 , ( S f ) ( x ) = f(x’ if x,, < 0 , x’ = ( x l , . . ., x , ~ - J , f(x’, -x,J

(

is a special extension operator from Lp(RA)into Lp(R,,)and from Wi(R;)into Wk(R,). S is a coretraction corresponding to the restriction operator from LJR,,) onto L,(RA) and from Wi(R,)onto Wi(R:),see Lemma2.9.1/1. Now, Theorem 2.10.1 and Theorem 1.2.4 yield that S is an isomorphic mapping from B;,,(R;), 0 < s < 1, 1 < p < 03, 1 q < co,onto a complemented subspace of B;,,(R,,). It is easy t o see that I,

( I f ) (4 = 3 ( f ( 4 +

f

W

9

-xn)),

f E B;,q(Rn)

9

(2)

-

2.11.3. Structure of the Spaces Hi(R:), Hi(R:), Bi,,(R:), and&,,(R:)

241

1

is a projection onto this subspace which will be denoted by IBgJR,). The operators P i , A , and GI have the same meaning as in the second and in the third step of the proof of Theorem 2.11.2. For f E S(Rn),it holds P,If = IPjf. Extension by continuity shows PjI = IP, . In particular, PjI are projections from Lp(R,)onto I R ( P j )= R(IPj).Further, it follows that W

A(IBi,q(Rn))=

C j=O

8 R(IPj) c lq(Lp(Rn))-

Finally, GLis an isometric mapping from R(IP,) onto R(IPL+l). Since R(IP,) and R(IP,) are complemented subspaces of R(P,) and R(P,), respectively, and since by

the third and the fourth step of the proof of Theorem 2.11.2 these spaces are isomorphic to l , , the desired assertion for the spaces e P q ( R :is ) a consequence of Theorem 2.11.1/3. Whence, together with Theorem 2.10.511 and Lemma 1.11.1, it follows the Part (b). R e m a r k 1. The method of the proof of the second step may be generalized. Let

1;2 = {x I z = (q,. . ., zn) E R,;

s

q > 0 for j = 1 , . . ., m)

s

where 1 s m n. Further let 1 < p < 03, 1 q 5 03, and 0 < s < 1. Then B;,@) is defined by restriction of BgJR,,) to Q. Replacing S from (1) by (Sf) (x)= f(lzil9 and I from (2) by

---

7

k m l ?zm+1,

* *

-

9

zn)

then one may repeat the considerations of the second step. It follows that BiJQ) is isomorphic t o lq(lp). R e m a r k 2. If q is restricted to 1 < q < a,Theorems 2.9.3 and 2.10.3 and the above theorem show that the isomorphism structure of all the spaces Bi,#ZA), &JRA), &JRA), H i ( R i ) ,@ ( R i ) , and &i(RA) is known with exception of 1

HF+m (Ri) and

m = 1,2,.

..

(3)

With the aid of other methods one may determine the structure of these spaces, too. We return t o that point in Remark 4.9.417.

2.12.

Diversity of the Spaces B;,*(Rn)and

Hi(&)

By Definition 2.3.1/1, it follows immediately Bi,2(Rn)= H;(R,), -00 < s < 00. It will be shown that, with exception of this trivial case, the B-spaces and the H-spaces never coincide. 16 Triebel, Interpolation

242

2.12. Diversity of the Spaces B;,f(Rn) and Hi(&,)

Theorem. (a) Let -a < so, s, < co; 1 < p o , p1 < co, and 1 5 qo, q1 5 co. Then it holds B2,91(Rn) = B2,,q,(Rr1) if and only if so = s,, po = pl, and go = yl. (b) Let -co < so, s, < 00 and 1 < p o , p1 < 03. Then it holds H"6.(R,J= I<;l(Rrl)if and only if so = s, and po = p , . (c) Let - co < so, s, < 00 ; 1 < p , , p , < 00, and 1 5 qo 5 03. Then it holds B;po,q,(Rn) = H:,(R,) if and only if so = s1 and po = p1 = qo = 2. P r o o f . Step 1. Let B;;o,90(R,,) = B2,q,(R,l).Then Theorem 2.3.4 yields B;,$a(R,,) B;T,"l(R,,),-co < E < co. By Theorem 2.4.l(a) i t follows by interpolation B;:,po(R,,)= B ~ , , J R r 1Since ). B~o,po(Rn) is isomorphic t o lpo (see Theorem 2.11.2)) and since l,, is a complemented subspace of Z,,o(l,l,),Theorem 2.11.2 and Theorem 2.11.1/3 yield p 1 = po = p . We suppose so $: sl.Then the inclusion properties of Theorem 2.3.2 and of Theorem 2.3.4 show that LJR,,) = WF(R,,)where rn is an arbitrary large natural number. Now, the embedding theorems, for instance Theorem 2.8.1,give a contradiction. Hence, so = sl. Finally, one obtains by Definition 2.3.1/1 that qo = q l . =

po

Step 2. Let H>o(R,,)= H;,(R,J. Theorem 2.11.2(a) and Theorem Z.ll.l/Z(a) yield =

pl. Further, one shows in the same manner as in the first step that so = sl.

Step 3. Let B~o,,,o(R,l) = HS;I(R,). Using (2.4.2/14) it follows analogously to the = B2,p,(R,1).Now, part (a) yields so = s1 = s first step by interpolation B20,po(R,l) 1 and po = p1 = p . By Theorem 2.3.4, we may assume s > -. Suppose B;,,,(R,,) s-

= H",R,,).Then byTheorem2.9.3, we have B,,,:

1 -

S--

1

I,

= Bp,pP (R,,-J.Part

( a )shows

qo = p . Now, since B;,JR,) and H;(R,) for p $: 2 are not isomorphic, part (c) is a

consequence of Theorem 2.11.2.

Remark.*) The proof uses essentially the structure of the spaces B;,JR,#)and I$(&). Some statements of the theorem are proved earlier with the aid of other methods. So, 0. V. BESOV[ 2 ] has shown that B;,q(Rn)$: Hi(R,) holds for 1 < p < 00 and q + 2. K. I<. GOLOVKIN[3] has given a proof for B;,2(Rrl) $. H;(R,,)for 1 < p < co and p $: 2. Further we refer to M. H. TAIBLESON [ l , I, particularly Lemma 221. *) After finishing the manuscript the author had a very interesting disrussion with Prof. -4. PELCZYBSKI(during the meeting in Halle, May 1974) on the isomorphism properties of the spaces Zg(Zp). A. PEWZYPSSKI sketched a proof of the following theorem: Let 1 5 q,, q1 5 co and 1 < p , , p 1 < 03. Then the spaces Zfo(Zp,) and Zfl(Zpl) are isomorphic if and only if q, = q1 and JJ, = p , . His proof based on the lemma: Let 1 5 q < co and 1 < p < co.Let X be a closed subspace of Z,(Z,). Then either X is isomorphic to a subspace of ,Z or X contains a subspace isomorphic to If. With the aid of the results on subspaces of L,((O,l ) ) , 1 < r < 2, (see J. LINDENSTRAUSS, L. TZAFRIRI [2], Theorem 11.3. 14) and a duality argument, it follows by the last lemma: Let 1 5 q 5 co and 1 < p , T < co. Then the spaces Zf(Zp) and L,((O, 1)) are isomorphic only in the trivial case p = q = r = 2. Using Theorem 2.11.2 one obtains: (a) Let -co < s,s, < 00 and 1 < p , , p 1 < 00. Then the spaces H:(R,) and Hi:(R.) are isomorphic if and only if p o = yl. (b) Let - co < so, s1 < co; 1 < p o , p1 < az~., and 1 5 q,,, q1 5 co. Then the spaces B2,fo(R,,)and l?&l(B,,) are isomorphic if and only If po = p 1 and go = q l . (c) Let - co < so, s1 < 03 ; 1 < p , , d, < co, and 1 5 qo 5 03. Then the spaces B:,f,(R,,) and HZ(R.) are isomorphic if and only if 6 0 = qo = p1 = 2. Similar results hold for the spaces defined over R,+,see Theorem 2.11.3.

243

2.13.1. Definitions

2.13.

Anisotropic Spaces

I n the spaces B&(R,,) and F;,q(R,l)and their special cases H;(R,,) and W;(R,,),there Such spaces are said to be is no distinction between the different directions in R,,. isotropic spces. They are helpful for investigations on elliptic and hyperbolic differena2u

tial operators like -- A u or Amu where nb is a natural number. It is plausible at2 ri that for considerations on semi-elliptic differential expressions like C ( - l ) l ' l k x

d2"'klL

-, where ml,are natural numbers,

I' = 1

one needs spaces in which different di-

rections are treated differently. Spaces of such a type are said to be anisotropic spces. One of the main aims of the book by S. M. NIKOL'SKIJ [7] is the systematic treatment of anisotropic spaces of Lebesgue-Besov type. Probably, a great part of the investigations in this chapter may be carried over to the anisotropic case. But we resbrict ourselves to some remarks.

2.13.1.

Definitions

Using Theorem 2.2.4,it follows that one may replace in (2.3.1/1)-(2.3.1/5) the number 2 by a number c > 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,<00 for k = l , ...,n, 1 < p < 0 0 , and 16qs00.Then one replaces M , , j = 1, 2 , . . . , i n (2.3.1/1) by

M J. = E J.+ 1 - E .I-1'

and M , = E l . Hence one uses ellipsoids, where the main axes are determined by the numbers s k , instead of balls. One sets

with the usual modification for q = 00. Similarly, one defines the spaces FgL(Rn). Analogously, one may determine the spaces BgL(Rn) and F:L(Rn) where ( 8 ) = (81, . . ., s,,), - 00 < sj < 0. But there arise some difficulties if some of the numbers sj in (s) = ( d l , . . . , s , ~ )are negative and the others are positive. Analogous16*

244

2.13. Anisotropic Spaces

s

where 0 5 k, < s,, 1, > s,. - k,, k, and I, are integers, r = 1, . . ., n, 0 < 6 03. All the norms are equivalent t o each other. S. M. NIKOL'SKIJ [7] proved in his book numerous direct and inverse embedding theorems on the boundary, embedding theorems for different metrics and theorems on equivalent norms. But it does not exist a systematic treatment on the interpolation properties of these spaces. In the next subsection we describe an example. A further generalization of these spaces, [7], can be obtained if one replaces BZi(R,,)in also considered in S. M. NIKOL'SKIJ (1) by B::;(R,o *

2.13.2.

Interpolation Theorem

The previous considerations are sufficient t o prove a simple but important interpolation theorem for anisotropic function spaces. T h e o r e m . Let (s) = (sl,. . ., sn), sj > 0 , l 5 a,q 5 CQ, 1 < p < 03,andO
~gm,~

wl,

Proof. Using the notations of (2.13.1/2) and of Subsection 2.5.1, then it follows (Lp(Rri),wF$(Rn))o,,q= BS($(Rn),

sj

= m.0. J

,

0 < s, < m i .

The considerations of Subsection 2.5.1, Theorem 1.12.2, and Theorem 1.13.2 (see also Theorem 1.14.4/2) show t,hat

R e m a r k 1. The proof shows that the abstract interpolation theorems of the first chapter (in particular Theorem 1.12.2 and Theorem 1.14.4/2) are general enough t o be it basis for a large part of the interpolation theory for anisotropic spaces. R e m a r k 2. A special case of the theorem, namely sj = Amj, where mj are natural [4]. There are also further results in numbers, 0 < A < 1 , is proved in P. GRISVARD this direction. The above theorem was formulated in H. TRIEBEL[27]. I n the quoted paper, it is the basis for investigations on the &-entropyfor compact embedding operators in anisotropic function spaces defined in bounded domains. See also Remark 4.10.3/3.

3.

LEBESGUE-BESOV SPACES WITH WEIGHTS IN DOMAINS

3.1.

Introduction

The last chapter was concerned with the theory of the Lebesgue-Besov spaces (and as special cases also with the theory of the Sobolev-Slobodeckij spaces) in R,,and in EL. This chapter and the following one deal with Lebesgue-Besov spaces with and without weights in domains. For some questions it is helpful to consider spaces without weights as special cases of spaces with weights. The systematic treatment of spaces with weights began a t the end of the fifties, see the survey paper by L. D. KuDRJAVCEV [l], and has been developed since that time to a comprehensive part of the theory of function spaces. The main field of application are the degenerate (elliptic) differential operators. This makes clear that a large part of the papers is concerned with Sobolev spaces with weights where the weights are powers of distances to manifolds (in particular boundaries of domains). But now also more general weights are considered. Further, some papers are also concerned with SlobodeckijBesov spaces with weights. The main problems are the following: 1. Density of smooth functions in the spaces under consideration, 2. Embedding theorems for different metrics (of the type described in Section 2 . Q 3. Direct and inverse embedding theorems on the boundary (of the type described in Section 2.9). There are only very few papers which are concerned with the interpolation theory of these spaces : P. GRISVARD[2], B. HANOUZET [l,21, B. HELFFER [l], C. GOUDJO[l], A. FAVINI [S, 71, and H. TRIEBEL [lo, 1411, 18, 22, 25, 281. In this chapter we consider two large classes of Sobolev-Slobodeckij-Lebesgue-Besov spaces with weights and a further, smaller class in Section 3.9. It is the main purpose to describe the interpolation properties of these spaces. Further there are proved embedding theorems and structure properties. A short description of some aspects of the theory of weighted P. I. LIZORKIN, spaces can be found in 0. V. BESOV,V. P. IL’IN,L. D. KUDWAVCEV, S. M. NIKOL’SKIJ [l]. See also S. L. SOBOLEV [8]. Further references and a short description of some problems are given in Section 3.10. This chapter is based on the [22, 10,1411,27]. Recently, A. A. AVANTAGGIATI [l] has given papers by H. TRIEBEL a comprehensive survey of the up-to-date situation of the theory of weighted function spaces and applications to degenerate elliptic differential equations. There can be found more than 150 references.

3.2.

Definitions and Fundamental Properties

As mentioned in the introduction, we shall be concerned with two comparatively large classes of function spaces with weights. The main aim of this chapter is the description of interpolation properties as well as the preparation of the later applica-

246

3.2. Definitions and Fundamental Properties

tions to (degenerate) elliptic differential operators. Since the two classes of functions are not independent of each other, we shall give a parallel development of the theory. I n this section some spaces are defined and a number of itnportant properties is proved. Further spaces are defined in connection with thgdescription of interpolation theory. Further we refer to Section 3.9, where a new class of functions is considered.

3.2.1.

The Spaces WF(f2; a)

As before, R, is the n-dimensional real Euclidean space. If 52 is an (open) domain in R,, then its boundary is denoted by aQ; aQ = 0 - 9. D e f i n i t i o n 1. An unbounded domain Q c R,, is said to be a domain of cone-type if there exists a one-side-unbounded open cone K with the vertex in the origin of the system of coordinates, such that for any x E 0 it holds x + K c 52. R e m a r k 1. After a suitable rotation K may be represented in the form n- 1

( x I x E R , , ;~ x ~ < c x ~ ;> xO , , j=l

where c is a suitable positive number. R,, and RA = { x I x E R,,, x,, > 0 } are special domains of cone-type. D e f i n i t i o n 2 . A bounded domain 52 c R,, is said to be a C”1-domain, where m = 1 , 2 , . . . or ni = co, if there exists a finite number of open balls K J ,j = 1, . . . , N , N

such that U K ,

3

J=1

a52 and KJ n a 9

+ 0 , and

if there exist m-fold differentiable real

vector-functions f ( J ) ( x = ) ( f i J ) ( x ).,. ., f!:)(x)) defined in E, such that y = f ( J ) ( zis ) a one-to-one mapping from Kl onto a bounded domain in R,,, where 8 9 n KJ is a part of the hyper-plane { y I y E R,,; y, = 0 } and 52 n Ii/ is a simply connected domain in the half-space RA. Further it is assumed that

R e m a r k 2. I n this chapter we shall be concerned with three types of domains: 1. Arbitrary domains, 2. Domains of cone-type, 3. C”-domains. But generally, the statements for function spaces defined over bounded Cm-domainsare also true for domains with weaker smoothness assumptions, e.g. for Cm-domains, where m is a number which one has to determine in dependence on the special situation. But we shall not describe explicitly weakenings of such a type later on. Further, in the next chapter, there is considered a fourth class of domains, bounded domain,. of cone-type, see Definition 4.2.3. R e m a r k 3. It is possible to define unbounded Cnl-domains. But, however, there [5]. are additional difficulties. We refer to F. E. BROWDER If the distance of a point x E 52 t,o the boundary aQ is denoted by d ( x ) ,then we set

52a Here 1 > 0.

=

{ x I x ED; d ( z ) < A}.

3.2.1. The Spaces W;(B; a)

247

D e f i n i t i o n 3. (a) Let 52 be an unbounded donmin of cone-type and let I< be a corresponding cone. A positive and continicous function a(x) defined in 52 is said to be a weightfunction of type 1 if there exist positive numbers 1,c , and C such that a ( x ) 5 c min a ( y ) for x € 5 2 - f2ZA and Iy-al5rl

+ Aly) 5 a(z + &y)

for z E 352, y E A, IyI where for any element y E K with IyI = 1 we have a(z

{XI

+ py;zEaQ;o<

=

1,0 <

l1 5 1, 5 C

(3)

< c)

52~~. (4) (b) Let 52 be a n unbounded domain of cone-type and let K be a corresponding cone. A positive and continuous function a ( x ) defined in 52 is said to be a weight-function of type 2 if there exist positive numbers 1,c , and C such that we have (2)) (4), and

+

x =z

,U

+

1,y) 2 a(z 1,y) for z E a52, y E I{, (yI = 1 , 0 < 1, S 1, 5 C. (3') (c) Let 52 be a bounded C"-domain. A positive and continuous function a ( x ) defined in 52 is said to be a wight-function of type 3 if there exist positive numbers 1,cl, and c,, and a positive and monotonically increasing ( = not decreasing) function e ( t ) defined in the interval (0, 1)such that

a(z

Cl@(d(X))

5 44 5 cze(d(x))

for x E QA.

(5)

(d)Let 52 be a bounded Cm-domain.A positive and continuous function a ( x )defined in SZ is said to be a weight-function of type 4, if there exist positive numbers 1,cl, and c,, and a positive and monotonically decreasing ( = not increasing) function e(t) defined in the interwall (0,1)such that (5) holds. R e m a r k 4. Formula (2) is a weak smoothness assumption and a t the same time it growth condition. A qualitative growth of a(x) like IxIx or like exlsl is in agreement with (2). But if a@)grows faster than an exponential function, e.g. like elzJx,x > 1, then (2) is not satisfied in general. (3), (3')) and ( 5 ) describe monotony properties of the weight function a ( x ) near the boundary. Important examples for weight functions of type 3 (and also of type 1 for a large class of unbounded domains of conetype) are a ( x ) = dx(x),0 5 x < co. Similarly a ( x ) = d"(x) with -co < x 5 0 are examples of weight functions of type 4 (and also of type 2 for a large class of unbounded domains of cone-type). As usual, the set of all complex-valued distributions over 52 is denoted by D'(l2). AS before, CF(52)is the set of all complex-valued infinitely differentiable functions defined over 52 with compact support. D e f i n i t ' i o n 4. Let 52 be an unbounded donaain of cone-type [or a bounded Cmdomain]. Further let a(x) be a weight function of type 1 or of type 2 [or of type 3 or of type 41. Then for rn = 0, 1, 2, . . . and 1 < p < co one sets

W?(Q;4

(Sobolev spaces with weights). The completioiz of C?(Q) in W r ( Q ; o )i s denoted by J'fF(52;a). W:(52, a) = Lp(52;a). For a ( x ) = 1 one writes W;(Q), resp. J'fy(52).

248

3.2. Definitions and Fundamental Properties

R e m a r k 5. * For a(x) = 1, one obtains the usual Sobolev spaces without weights. Sobolev spaces with weights are considered by many authors. References are given in Section 3.10. At the moment, we restrict ourselves t o the quotation of those papers which mark the beginning of the theory of function spaces with weights. The case Q = Ri and a(z) = x: was considered by L. D. KUDRJAVCEV[l], S. V. USPENSKIJ [2], and P. GRISVARD[2] (see also Subsection 2.9.2). Sobolev spaces in bounded s The spaces W;(Ri, (1 1 ~ 1 ~ ) ~ ) domains where a(x) =d”(z) are treated by J . N E ~ A[l]. are introduced by L.D.KUDRJAVCEV [2]. During the meantime these spaces are generalized by few authors, see Subsection 3.10.3. The above definition was given in H. TRIEBEL[22, I].

+

3.2.2.

Properties of the Spaces W:(sd; a)

If Q is a bounded domain, then Cm(Q) denotes the set of ell complex-valued infinitely differentiable functions defined in Q, whose derivatives (inclusively the function itself) can be extended continuously to 0.If Q is an unbounded domain then cm(Q) is defined analogously where additionally it is assumed that the functions vanish for large values of 1x1,x E Q. Similarly, the sets are defined, where m. = 0, 1,2,. .. T h e o r e m . (a) All the spaces described in Definition 3.2.114 are Banach spaces. (b) If 9 is an unbounded domain of wne-type, and if a(x) i s a weight function of type 1, then isw(&?) i s dense in WF(Q;a). (c) If9 is a bounded C”-domain, and if a(x) is a weight function of type 3, then Cm(Q) is dense in WF(Q;0 ) . Proof. Step 1. It follows by standard-arguments that WF(Q;a) is a Banach space. Here, as usual, one has to identify functions different only on a set of measure zero. B u t then it is clear that @F(Q;a) is also a Banach space. Step 2. To prove (b), we fix y E K where (y( = 1 and set cot = { z I z E R , ; z = z - t y ; z E Q } , 0 < t < CO.

ern@)

Here K is a cone belonging to Q. cot is also a n unbounded domain of cone-type. It is sufficient t o approximate functions f ( x ) E W;(Qn; a) vanishing for large values of 1x1,x E Q, by functions belonging t o ew(Q). If f ( x ) is such a function, then it is not hard t o see that f ( x + 2ty) is a n element of WF(0,).Let E > 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 <

<

(5)

00.

R e m a r k 1. 9 is an arbitrary domain, bounded or unbounded, without any smoothness assumptions with respect to the boundary. For any domain there exist by functions e ( x )with the required properties: We start with a representation of non-empty bounded domains c o ( j j , j = N , N + 1, . . . , such that

a

m

a) w ( i + l ) , u

l=n.

(,)(jj

= Q,

d ( a w ( j ) a, t o ( j + l ) )2 c2-J

(6)

for j = N , N + 1, . . . Here c is a suitable positive number; d ( a o ( j ) ,a o ( J + l )is ) the . is easy to see that representations of distance of the boundaries of w i j ) and u J ( j + l )It such a type always exist. Sett’ing ~ ( x=) 2 j for z ~ w ( j + l ) w ( i ) , j = N , N + 1 , . . .,

~ ( x=) 2N-l

for x

EW(.~),

and using Sobolev’s niollification method (first step of the proof of Lemma 2.5.1) in a neighbourhood of i3wc.i) where the mollification radius is hi = c‘2-j, then one obtains a function e ( x )with the desired properties. Here c‘ is a sufficiently small positive [lo], p. 117. Condition ( 2 ) means that e ( x ) tends to number. See also H. TRIEBEL infinity, if x approaches to the boundary or if x tends to infinity. Condition (1) is

3.2.3. The Spaces B;,JS; Q”; Q’) and H i ( 8 ; @;

essential. If Q is a bounded domain theii 2-l} i = N , N is a representation of the above typc. It holds that d ( a w ( / ) ,a o ( J + ’ ) ) = 2-5-1. UJ(JJ

= {X I 2 E Q, d(x) >

e’)

251

+ 1, . . .

Then the described construction of e(x) yields that there exist two positive numbers c1 and c2 such that cld(z)

s e-’(x) 5 c,d(z).

Hence, e-l(x) is only a mollification of the distance function d(x).If one has a bounded C”-domain then one ran set p-l(x) = d(x)near the boundary. It is not very hard to then one may choose describe further examples. If Q = R,,, e(z) = (1

+ Ix12))” or

e(z) =

e(1+121*)‘,

11 > 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 0. Further let ,u and v be two real numbers such (b) Let 1 (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 + 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/<},& is a Cauchy sequence in B ; , # 2 ; $'; e') or H i @ ; @; @), respectively, and if w is a bounded domain with 6 c 9,then one obtains that fk

f and

qvjfis

___*

fj;*p(R")

(HpL))

vjf

if k

+

co .

(5)

Whence the completeness of B;,#; e"; e') or H;(SZ; p p ; @), respectively, follows in the usual way. Step 3. We show that C Z ( 9 ) is dense in B i J Q ; e"; e') where q < co. Let f E B i J Q ; @; e'). Then by Lemma 1 it follows for M > K M+2

Ilf - fk c Y/
3.2. Definitions and Fundamental Properties

254

Theorem 2.3.2 yields that f

'11 + 2

C

j=K

y j may be approximated in B;,,(O; ep;e') by func-

tions belonging to C$ (52). One concludes analogously for Hi(52; ep; e').

L e m m a 2. Let 52 be an arbitrary domain, let e ( x ) be a function in the sense of Deand il be real numbers where finition 3.2.311 and let {yj(x)}j"rnE IY.Further let p , ,LA, 1 < p < co and 0 < il < 1. Then there exists for any multi-index 01 a positive number c = c ( a ) such that for all j = N , N + 1 , . . . and for all y E Qj

If

01

= (0,

. . . , 0) then one may replace y J ( z )in (6) by 1.

Proof. We choose a number co > 0 such t81iat

.{ I I x for y

E

52! and j

=

s

- YI

c02-J}

'J-1

52]tl

N,N

+ 1, . . . Then the left-hand side of (6) can be estimated by

+c

1

IV(@//)(z) D"(y~,z))l/~ Ix - ylp-"-'p d x .

I L -YI(C&J

Here z = x(x) x by

+ (1 - x ( x ) )y where 0 5 x ( x ) s

1. One obtains the desired estimate

lV(@'/I(z) D"y,(z))(/'5 ~Dp+Jp+J/~l"l.

T h e o r e m 2 . Let 52 be a n arbitrary domain, let p(x) be a function in the sense of Definition 3.2.3/1. Further let 1 < p < 00, s 2 0, and v 2 ,LA + sp. Then IlfIl:v;cn;Qw) =

for s = 0 , 1 , 2 , . . . and

[i.(,&XS

+

e'(x) lD"f(x)l~' p"(x) If(x)l"

llfll7vp;Q~;Q~)

for 0 < s = [s] J q Q ;9 ; e*).

(7b)

+ {s}, [s] integer; 0 < { s ) < 1 : are

Proof. Step 1. Let f

E

W",Q; @;

(7a)

equiwa2ent norm.? in the space

e'). For .s = 0, 1, 2 , . . . oneobtainsimmediately

l l f l l % ~ ( ~ ; ~S; e~ I~l f)l I t ~ ~ ( ~ ; e * ; e - , .

(8)

3.2.4. Properties of the Spaces B;,,(B; $; QY) and H#2; f ;e’)

255

To prove (8) also for s $. integer, we use a system of functions {yj(~)}i”,N E Y having the property (3.2.3/8).Then Lemma 2 and (2.5.1/15)yield

m

With the aid of Remark 2.4.2/6 one obtains for

s

2 ~ ’ + ~ ( ~ l ~ ~ l l ~ ” ( y , fC?”’llY,fIl )Il/tp

I \I

--P

1011

=

[9]

that (.)

T

&,‘; (2J”lly,fllLp)

P

5 C22p’I/vJfIl&i + c~2pJ+S”llllt)lfII~p* (9) Now, if one takes into consideration v 2 p + sp, then (8) is a consequence of the

last two estimates. Xtep 2. To prove the reversion to (8) we start with f E W”,(sz; @; e’) and assume s integer. Using Lemma 2 and (2.5.1/15) (where one may restrict oneself in (2.5.1/11) t o the summation over la( = [s]) one obtains

IlfII%i(n

,ep; ev)

256

3.2. Definitions and Fundamental Properties

m

Analogously to (9),it follows for 0 < t < s that 2 j ~ + p l g - l 1 j l l y ~5 f l lc~(2Jfi(lyjfll ~ bi)t'8 ( 2 J f i + j ~ a l l y ~ f l l ~ ~ ) ( b - ~ ) / a

5 ~2jfillvjf115,; + C(E) 2jfi+jspIIvjfllPLp

(11)

> 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 ~ ~ ~ ~ ~ % ;<~ co Q belongs ; ~ ~ ; t~o ~W;(Q; ) ,@; e’). If this is ensured, then one may replace lIfIIt;:(n;p’;e”) in (14) by l fI ; (Q;P’;pY) Although this assertion does not play any

qw),

z(n;er;pY) +

-

role later on, a clarification of this question would be of interest. Q v )(7b), onemay alsoconsidernormsofthe Remark2.Besidenorms ~ \ j ~ \ $ p ; p ~from form

llfll ibi(n;ep;e’)

+

integer. Later on, in the development of the interpolation theory for the spaces W?(Q; G) defined in 3.2.1, we shall be concerned with spaces of such a type, see Theorem 3.3.1. Assuming v 2 p + s p and f E W @ ; e”; e’), then the second step of the proof of Theorem 2 yields

s

Ilf II w;(n;ew;ev)6 cllf II ki(n;e’ 17

Tricbel, lnterpolntion

;eu)

.

258

3.2. Definitions and Fundamental Properties

Similarly to Theorem 2, the question arises whether 11f 11 &;(Q; ew is an equivalent norm in W i ( Q ; ep; e'). We describe a counterexample. Let Q = (0, l), 1 < p < 03, 1 ,u = p, 0 < I < -, and p + I p 5 Y < p + 1. Further let e-'(x) = d ( z ) for ;Qv)

P

x E (0, E ) u (1 - E , l), 0 < E < -). By Theorem 3, we have f ( x ) , = e-'(z): W i ( ( 0 ,l),p, e"). On the other hand, \ ~ f ~ ~ & ; ( ~ ~ Q w does ; e v ) not exist.

E

R e m a r k 3. With exception of Theorem 3, we followed in this subsection essentially the treatment given in H. TRIEBEL[22,II].

3.2.5.

Equivalent Norms and Coinpact Embedding in Wr(sZ) [Part I]

By Definition 32.114, the spaces q(Q) are special cases of the spaces W:(SZ; a) with a(x) = 1. Later on, we shall be concerned extensively with equivalent norms, compact embeddings and interpolation properties for the spaces WF(Q). But for the considerations in the following subsections it is useful t o prove here some simple results of such type. Theorem. Let Q be a bounded C"-domain.

(a) If 1 m,, then the embedding from WF(SZ)into W T ( Q )is compact. (c) If 1 < p < co and m = 0, 1, 2 , . ., then the restriction operator from WT(R,,) onto WT(SZ)is a retraction in the sense of Definition 1.2.4. If M is a natural number, then there exists a corresponding coretraction independent of 1 < p < co and m = 0 , 1 , ..., M . Proof. Step 1. Theorem 3.2.2(c)yields that Ern@)is dense in W;(Q). Hence, for the proof of (c), it will be sufficient t o extend a function f E ( ? ~ ( Q )to a function belonging $0 WT(R,,).If the balls K j have the meaning of Definition 32.112, and if y j ( x )E C,"(Kj) is a resolution of unity (see the third step of the proof of Theorem 3.2.2),then it is sufficient t o extend fyj . After application of the transformation y = f G ) ( x )from Definition 3.2.112, one may use the operator S from the proof of Lemma 2.9.1/1. Returning t o the original coordinates, multiplying the function extended over 8 0 with a function y E C$(U K j u SZ), identical 1 in a neighbourhood of Q, then one obtains an admissible coretraction.

...,

.

Step 2. The theorem of ARZELLASCOLIyields that a set of bounded functions in cl(Q) is a precompact set in c(Q).Then it follows from the first step and from (2.8.1/16) that the embedding operator from WF(Q) into LJQ) is compact for sufficiently large values of m. By the first step, Theorem 1.2.4, and (2.4.2/11),one obtains that WY(I.2)is an interpolation space with respect to W;*(Q),m2 > m,, and Lp(Q). Now, (b) is a consequence of Theorem 1.16.4/2.

with v < ,LI + sp

Wi(Q;Q P ; p’)

3.2.6. The Spaces

259

Step 3. To prove (a), we assume that Ilfllyv;cn, is not an equivalent norm. Then there exists a sequence (f,(x)}T1c W P ( 0 )such that 1 C lID”fjllLp= 1 and IlfjllL, + C I I ~ ” f j I I L p< (2) 15 la15 111- 1

7-

IcI = rn

Then it follows that { f j ( x ) } ~isl bounded in Wy(SZ) and hence, by part (b), precompact in W;-l(SZ). The last part of (2) shows that {fj(x)}Flis precompact in WT(sL), too. Without loss of generality, we may assume that f j +f . Then it w,” follows by the second part of (2) that f = 0. This is a contradiction t o the first part of (2). R e m a r k . As mentioned above, we shall obtain esscntially more general results, later on. We do not give references a t the moment.

3.2.6.

The Spaces Wi(sd;ep;e’) with v

< p + sp

The investigations for the spaces w”,(O; e”; e”) in Subsectioii 3.2.4. were based essentially on the hypothesis v 2 p + sp. Now we want t o consider the spaces W“,SZ; e”; e’) with v < p + sp. For this purpose we assume that SZ is a bounded C”-domain and that we have e-l(x) N d ( x ) *) near the boundary in the sense of Remark 3.2.311. It will be shown that there are a number of essential differences in comparison with the spaces W;(SZ; @,p”), v 2 p + sp. Further we shall see that these spaces are related to the spaces W;(SZ, e”) from Subsection 3.2.1. L e m m a 1. Let SZ be a bounded C”-domain. Purther let e(x) be a weight function in the sense of Definition 3.2.311 with e-l(x) d ( x ) near the boundary. (a) Let 1 < p < co, na = 1 , 2 , . ., -co < p < co and p mnp += 1 kp, where Ic = 0, . . , , m - 1. Then there exist a positive number c and a domain w where 6 c SZ, such that J eP+llz+) I f ( x ) l P ax c j e’(x) lofif(x)IP + c J I f ( x ) l P d~ (1) R n la1

.

-

+

c

=Ill

+

(zx

0

for all elements f E cm(SZ) for which the left-hand side of (1) i s finite. (Iia this m e the right-hand side is also finite). 1 (b) Let 1 < p < 00, 0 < 1 < 1, and il += - . Then there exist a positive number c and P a domain 0) where 6 t SZ, such that

c”@)

for all elements f E for which the left-hand side of (2) i s finite. Proof. Step 1. For f E ~ ~ ( S Z ) t,he , left-hand sicle of (1) is finite if and only if

DYf(x)lel>= 0 for 0 5 IyI 5 m *) 17 *

-

‘1.

+ [’?,

But then the right-hand side of (1)

means that there exist two positive numbers c1 and c2 such that c1 d ( r ) 5

6ct d(z).

@-1(r)

260

3.2. Definitions and Fundamental Properties

is also finite, and one can approximate such a function by functions belonging to C r ( i 2 )inthe sense of (1). Hence, for the proof of (l),we may euppose f(x) E Cr(i2).

Using the notations of the third step of the proof of Theorem 3.2.2, and applying the transformation of coordinates y = f ( . i ) ( x )from Definition 3.2.112, then we have yj(x) f ( z ) = g,(y) E C r ( R : ) ,j = 1, . . ., N . Ifp mp > 1, then we choose a number x such that p mp > - x p p > 1. Writing y = (y’, yn), then we have

+

+

+

and

m

+

If p mp < 1, then we choose a number x such that p Then it holds

+ mp <

-xp

+ p < 1.

and (analogously t o the above consideration)

Returning to the original coordinates X, and taking into consideration f(x) =

N

C yj(x) f(x) for x E 9,it follows that

j=O

R

@‘+“””(x)If(r)lpdx

5

c

\

Ij

(la1C 1lD”f(x)IP+ If(x)lp)dx.

e ” + ( m - l ) p (5)

=

Since $+(rJL-l)/’(x) 5 E&‘+”~/ (x)near the boundary, one obtains (1) for m = 1. Iteration and application of Theorem 3.2.5(a) with respect t o a suitable bounded C’”-domain w ,where 6 c SZ, yield the desired result. 8tep 2. To prove (2), we use again the method of local coordinates. If g(y) = gj(y) has the same meaning as in the first step, and if one extends this function by g(y’, yn) = g(y‘, -yn) for yn < 0, then g(y) belongs to Wi(R,,) and hence also to Wi(R,,). Further it is sufficient to prove

3.2.6. The Spaces W i ( Q ;pp;

e’) with v

ip

+ sp

1

If - < 1 < 1, then the left-hand side of ( 5 ) is finite only if g(x‘, 0)

P

261

=

0. Then

Theorem 2.9.3 and the method of approximation introduced there show that one may

(

restrict oneself in any case 0 < 1 < 1 , 1 $. - to functions g(x) E C$(R,J vanishing Pl ) in a neighbourhood of {x I x,, = O } . Now we suppose that there exists a positive number c such that for all u(t)E Cz((0, a)) m

m m

Extending u(t) by u(t) = u(- t ) to ( - co,0 ) , putting u ( t ) = g(x’,t ) in ( 6 ) ,and integrating over x’ E Rrl-l,then it follows that

1

l%-Ap

R”

1 9 ( 4 I p dz

1

m

c

h-a 11g(x’,Xn) - dx’,xn

dh

+ h)Il&n)T + cllgII&t,)*

(7)

0

I n bhe notation of Subsection 2.5.1 the right-hand side of (7) is the p-th power of Since the norm in the interpolation space (LP(R,J,Wp,n(Rn))A,p.

(&(Rn), J+’i(%))A,p

c

(Lp(Rn),wi,n(Rn))A,p 9

(5) is a consequence of (7) and (2.5.1/15). Step 3. We prove (6). Let t

~ ( t= ) ~ ( t)

t

(wit) - r w ( t ) we have

u ( t ) = v(t)

(8) yields

s

-

~ (dtt ) =-

t

0

$)’

= u‘(t) -

I”t

00

t-AI+(t)lP

‘s

‘s

BY t

t

dt

( ~ ( t) ~ ( tdt. ) )

0

u ( t )d t

t

0

+v(t) = u ’ ( t ) t

at

(9)

v(t)-.

5c

t

m

\

t-”p-l

0

0

jlu(t) - u(t)l!’d t dt 0

s c s 1 lu(t + t) m m

u(t)lp

(t

+~)-~p-ldtdt

0 0

2c

m m

\ 1Iu(t + z) - u(t)l/’t-’~)-ldt d t .

0 0

262

3.2. Definitions and Fundamental Properties m

+

Using (3) or (4), where p mp = Ap, n = 1, and g,(t) = consequence of (9) and (10). R e m a r k 1. (3) and (4) are modifications of Hardy’s well-known inequality m

J t-“lf(t)p at 5 0

1

00, 0

(+y

- 11

c 3

+= 1, f E C,”((O,

m

J t-“+”lf’(t)pat, 0

co)). See G. H. HARDY, J. E. LITTLEWOOD, G. P 6 L Y A

[l], Theorem 330. It is possible to prove that

- 11)

(lb

cannot be improved. Fur-

ther, one can show that both the sides of (11) are equal if and only if f(t) = 0. For p = 2 we shall need (11) later on. We give a short proof here for this case. Let cr > 1 and let f be a real function. Then m

j- t-“f2(t) at 0

m

= J t-“ 0

t

J ( f Z ( 2 ) ) ’ at 0

m

0

One obtains (11) by application of Holder’s inequality. If cr < 1 , then one starts with m

f2M

=

1

- (f2(t))’dt t

and concludes similarly. Finally, one may consider also complex-valued functions f(t). The proof shows that (11) with p = 2 and cr < 1 is also valid for functions f ( t ) differentiable in [0, 00) and vanishing for large values of t . R e m a r k 2 . The third step of the above proof is due t,oP. GRISVARD 121, Lemma 4.1. Definition. Let Q c R, be a bounded C”-domain. Furtlm let e(x) be a weight function in the sense of Definition 3.2.3/1 with @-1(x) d ( x )near the boundary (Remark 3.2.3/1). Let 1 < p < co, s 2 0 and v < ,u + sp. If s is a n integer, then one sets N

“wll,(SZ;e

”; @’) =

IIfIIwi(n;ep;ev) =

1f I f

[l

ELlpOc (Q);

(,z,$

e”(4lD”f(4l” + e’(4 li(x)l”) d X ] ” P < 0 0 ) . (W

3.2.6. The Spaces Wi(f2;

If s

=

e”; e’)

with

Y

+ sp

263

[s] + {s}, where [s]is an integer and 0 < {s} < 1, then one sets

le””W D a f ( 4 - @P’P(Y)

Ix - yln+(s)/,

o”f(Y)I”

dxdy]”

+ IlfIIm[;(n;p;ev)<

00

I

*

(12b)

(As before for s = 0, it is assumed that p = v, and W;(Q; e”; e”) = LJQ; p”).) The completion of Cm(52)nWi(52; e”; e’) in W$2; e”; e’) is denoted by W;(Q;@‘; e’). The completion of Cg(52) in W@; e”; e’) is denoted by @p(52; el; Q’).

R e m a r k 3 . To justify the last part of the definition, one has t o show that W”,Q;e”; e’) is a Banach space. It is sufficient t o prove the completeness. Let {f,}jTl be a Cauchy sequence. Then there exists a function f E Lkc(52)such that for any domain o with 6 c 52 fj

+f

and

oafi+ D”f

for

la1

= [s]

in L p ( o ) .

Hence, uaing standard methods for the proof of completeness of function spaces (see [17], p. 28), it follows that W;(52;@; e’) is a Banach space. e.g. H. TRIEBEL R e m a r k 4. We do not discuss the question here, under which conditions it holds W;(Q;e”; @’) = Wi(J-2;e”; @’).

For considerations of such a type the above definition does not seem to fit well. For such investigations it is more useful t o define Wi(J-2; Q”; e’) as the set of all functions f ELlpOC(Q) such that I I f l l ~ ( n ; P ” ; P ” ) < a,where has the meaning of (3.2.4/14). If one defines the spaces Wt(52;Q”; e’) with v 2 ,u + sp in this way, then it follows, by Theorem 3.2.413 and Theorem 3.2.411, that even Cr(52) is dense in Wi(J-2;e”; e’). We do not also give a detailed discussion under which conditions we have W p 2 ;e”; e’) = +p; e”; @’), v < ,u sp.

IIfI/+w:Q;e’;PY) +

But in the following lemma we consider two important special cases. In particular, it will be shown that, for any number E > 0 and for any number 1 0 such that p + sp > v > ,u sp - E and

+

For v >= ,u + sp we write for sake of simplicity Wi(52;e”; e’) = ki(52;Q”; p”), see Theorem 3.2.411. L e m m a 2. Let 52 c Rn be a bounded Cw-domain.k’urther let e ( x ) be a weight function in the sense of Definition 3.2.311 with e-l(x) d ( x ) near the boundary. (a) I t holds Wb,(Sz; 1 ; e’) = Wdp(52; 1; e‘) (13) N

264

3.2. Definitions and Fundamental Properties

if and only if either 0 5 s

5 -,1 P

1 or-<s<03 (b) Let m

v z s p - p

and

P

=

v arbitrary,

1 , 2 , . . . and 1 ,u 2 (m - 1 ) p + 1 or -co
5 s 6 -. 1

Then it follows by Theorem 2.9.3(d) and by the

P

method of local coordinates that Cr(52) is dense in Wi(D) = W,(Sz; 1 ; 1). Then the approximation method from the fourth step of the proof of Theorem 2.9.3 shows that Cr(D) is also dense in W $ ? ;1 ; @”), -co < v < co. 1 Step 2. Let - < s < 03. Theorem 2.9.3 and the method of local coordinates show that f

1,

E

c”(s2)belongs to kp(D)= kp(D;1 ; 1) if and only if

n

if and only if

D’flsn

=

0 for IyI

=

0,.. .,

(15), (17), and the approximation method of the proof of Theorem 2.9.3 show that for

[+] 2 - f][s

C r ( Q ) is dense in W;(D; 1; e’). Now we suppose that (18) is untrue. We consider functions f belonging to W“,D; 1 ; e’) with OafIan = 0 for

[

:I

IayI < s - -

.

If C r ( Q )is dense in W;(D; 1 ; e”),then Lemma 1 yields that f can be approximated also in W”,D) by functions belonging to C g ( 9 ) With . the aid of the method of local coordinates it follows that a corresponding approximation property holds if one replaces D by R,+.Now we extend the transformedfunctions g odd or even over {x I x,,= 0 } such that

ax[.-.a,[

’1-g 7-1

is an even-extended function. Then one obtains a corresponding

approximation in W i ( R i ) .This is

it

contradiction to Theorem 2.9.3. This proves (a).

3.2.6. The Spaces W#2;

e”;

e’) with v

+ sp

265

Step 3. Let m = 1 , 2 , . . ., 1 < p < 03, and p 5 0. The method of local coordinates, Lemma 1, Theorem 3.2.5, and Theorem 2.9.212 show, that C,jO(Q) is dense in W;(Q; e”; e’) if and only if p 5 -mp 1. For p 2 0 the set Cg(Q)is dense in WT(Q;ep;e’) if and only i f f E cW(f2)n WT(f2; p8; e’) has the consequenceDYflaQ=O Iyl 5 m - 1. But this is the same as p 2 ( m - 1)p + 1. for 0

+

-

Theorem. Let l2 c Rn be a bounded C“-domain. Let e(x) be a weight function in the sense of Definition 3.2.3/1 with e-l(x) d ( x ) near the boundary (Remark 3.2.311). Further let 1 < p < 03, 0 < s = [s] + {s}, where [s] is an integer and 0 5 (s} < 1, v < p sp. I f 1 {s} 8 - and p + sp 4 1 + kp where k = 0, . ., [s] - 1 , (19)

+

.

P

then

*p(sz; e”; e’)

=

Wi(Q;e”;

(20)

@”+SP).

Proof. By Theorem 3.2.412 and Theorem 3.2.413, it is sufficient t o prove that there exists a positive number c such that for all f E Cg(0)

m

Here w is a suitable domain, where 6 c 0.(If s is an integer, then one has to modify.) Formula (21) is a consequence of both the parts of Lemma 1 and Theorem 3.2.5. R e m a r k 5. For the validity of (20), condition (19) is not only sufficient, but also 1 necessary. Let {s} = -. To show that (20) is untrue, it is sufficient to prove that

e

-q#]

P

an element of @;(l2; e”; e’). Let e(x) = d(z) near the boundary. The method of local coordinates and Theorem 2.9 3 yield that e-[”(x)belongs to kp(Q), see also the second step of the proof of Lemma 2. Using v < p + sp and the approxP

(5) is

imation method of the proof of Theorem 2.9.3, then we have e P (x) 1 E @p(Q; p@;e’). Let {s} - and v < p + sp. Then H. TRIEBEL[22,II] has shown -&-[.I

+

P

that(2O)istrueifandonlyifp + s p + 1 + k p w i t h k = O , . . . , [ s ] - 1 . W e d o n o t go into detail here and refer t o the cited paper (the assumption v = 0 used there is not necessary if one takes into consideration Lemma l(a)). If s is an integer, then A. KUFNER[l], where a characterthis follows also from the second part of J.KADLEC, ization of the corresponding spaces for singular values of the parameters is given. J. KADLEC, A. KUFNER [l]. See also 0. V. BESOV, 1 R e m a r k 6. Lemma 1 shows that for fixed s, p , and p with {s) - all the spaces

+

+P additionally p + sp $: 1 + k p

@,(f2; ep;e’) with - co < v < sp p coincide. If with k = 0, . . ., [s] - 1, then these spaces are equal t o W i ( 0 ;e”; ep+sll).Further, ~ ( x=) $(x) is for p 0 a weight function of type 3 and for p 2‘0a weight function

266

3.3. Interpolation Theory for the Spaces W r ( S ; e )

of type 4 in the sense of Definition 3.2.113. Definition 3.2.114, Lemma l(a), and the last theorem yield @(Q; @") = WF(D; @;

if

m = 1 , 2 ,...,

(22)

,"+nq,

1
and p + l - k p

for k = l ,

..., m .

Hence, for an important class of weight functions, we obtained a connection between the two types of spaces considered here which will be of interest later on. Finally we mention the special case p = - m p which leads for p = 2 to the spaces f i n @ ) considered in J. L. LIONS, E. MAGENES[a, I], Chapter 2, 6.3. See also (14).

3.3.

Interpolation Theory for the Spaces W;(sd; u)

This section is concerned with the interpolation theory for the spaces WF(D;o) and W;(D; o ) of Definition 3.2.114. We restrict ourselves to the real method. Although the results are rather general, there is a number of unsolved problems. So, the spaces (WTl(Q;o), WTs(Q;O))e,q are only determined for weight functions of type 1 and type 3 while the spaces (@T(Q; o), Fbp(D;u ) ) ~are , ~only determined for weight functions of type 2 and type 4 (see Definition 3.2.1/3). But later, on the basis of Remark 3.2.6/6, we shall be concerned with (J@(Q; d-@1(x)), $F(Q;& P ~ ( X ) ) ) ~ , + where D is a bounded Cm-domain,d ( x ) denotes the distance to the boundary, and -00 < p l , p2 < 00. (See Theorem 3.4.3 and Remark 3.4.3/2.) For D = R,+, U(X) = a$, m, = 0 , and m2 = 1, P. GRISVARD[2] treated also the cases not con[22, I]. sidered here. We shall follow the treatment given in H. TRIEBEL

3.3.1.

The Spaces Wr(sd;a) with Weight Functions of Type 1

Lemma. Let D c R, be a n unbounded domain of cone-type (Definition 3.2.1/1). Let K be the corresponding cone. Further let vj E K , j = 1, . . ., n, with lvjl = 1, be n linearly independent vectors. Let o(x)be a weight function of type 1. Then G j ( t ) ,t 2 0,

[Gj(t)U ] ( x ) = U ( Z

+ vjt),

j = 1, . . ., m, u E L J Q ; c),

are strongly continuous semi-groups of operators with the infinitesimal operators

(1)

Aj,

Proof. Step 1. The semi-group property is clear. By (3.2.1/2) and (3.2./13), it follows IlGj(t) u I I L p ( R ; o )S ~ l + ~ l l u I I ~ ~ ( ~ ; o ) *) All derivationg-must be understood in the sense of D'(Q).

3.3.1. The Spaces W,"(S;u) with Weight Functions of Type 1

267

where c > 0 is a suitable number. One obtains the continuity in the same manner as in the second step of the proof of Theorem 3.2.2. Step2. Let u E D ( A j )and v E C r ( Q ) .Using

where 0 < t

s t,(p), it follows for t 1 0 , in the sense of the theory of distributions,

au Hence Aju = -E L J Q ; a). Now, for the proof of (2), one has to show that each avj dU function u E L,,(Q;a) such that -E LJQ; a) belongs t o D(Aj). It ,holds

cm(Q)

dVj

c D ( A j ) .On the other band, it follows, analogously t o the second step of the proof of Theorem 3.2.2, that is dense in

c"(52)

Whence it follows the desired result. We shall use the notations A,, and Af, from 2.5.1. If the vectors vj have the same meaning as in the last lemma, then one sets

I

Ma = h I h E K ;h Here0 < 6 <

c ajvj 11

=

j=1

with

aj

> 0 and

Ihl

I

56 .

(3)

00.

Theorem. Let 52 c R,, be a n unbounded domain of cone-type. Let a ( . ) be a weight function of type 1. Further let m, and m2 be integers, co > m2 > m, 2 0,1 s-k,andifO<6< 00,then

s

Bi,,(Q.;0)= (WYCQ; 4,W F ( Q ;d ) o , q r =

1.2. where

For q =

00

= {u I u E W",Q; 0 ) ; llull$!l,a)<

one must replace

equivalent norms in B&(Q; a).

(

Ma

'

l-\qf$-

by;;;

00)s

1.1.)

(4)

All these norms are

268

3.3. Interpolation Theory for the Spaces W r ( Q ; a)

Proof. The semi-groups CJj(t),j = 1, . . ., n, are commutative t o each other. If Km has the same meaning as in Definition 1.13.3, then it follows, by the above lemma, t,hat

=

and

aru

{ulw

E ~ p ( ~ ; o ) , r =

0 , . . .,m, j = 1 , . . .,n)

(7)

K m = WT(SZ;0 ) .

Now the theorem is a consequence of Theorem 1.13.6/2 and a simple transformation of coordinates. R e m a r k 1. B;,$2; a) is independent of the choice of m, and m2.This justifies the notation. R e m a r k 2. The theorem is analogous to Theorem 2.5.1. This is also true for the proof. If one wants to make the number l in the theorem as sma,ll as possible, then I n contrast t o Theorem 2.5.1, one has t o choose k = [s]- and 1 = 1 + [{s}']. one cannot set 6 = co,generally. The previous considerations show that for such a statement it is necessary that IICli(t)II is uniformly bounded. But for the important special case a@)= 1 it is easy t o see that [lCJj(t)ll5 1. Hence in that case one may also choose 6 = co in the above theorem.

3.3.2.

The Spaces

kT(Sa;a) with Weight Functions of Type 2

Lemma. Let SZ c R, be a n unbounded domain of cone-type and let a(x) be a weight function of type 2. If K and vj , j = 1, . . ., n, have the same meaning as in Lemma 3.3.1, and if one extends the functions of L,,(Q;a) outside of Q by zero, then Gj(t),t 2 0 ,

[G,(t) u] (2)= u ( x - v.it), j = 1, . . ., n , u EL^ (Q; a), (1) have the form are strongly continuous semi-group. Their infinitesimal operators au

A.u =

--.

D(Aj)i s the completion

av.; of CF(Q)in

au

the space

I.

{uI u E LJQ; a),-€L,,(SZ;a) (3) avja Proof. Using (3.2.1/3') instead of (3.2.1/3),then one obtains the boundedness of llGj(t)[l,the strong continuity and the semi-group property in the same manner a,s in it follows, similarly to the the first step of the proof of Lemma 3.3.1. For u E D(d;), second step of the proof of Lemma 3.3.1, that au AjU = - € Lp(SZ; a) avj Further, it holds that, Gj(t)u ED(A,;)and IlA,j8;(t) u - ~ , U ~ ~ L-,, 0, ~ifQ t~J, ~0 .~

3.3.2. The Spaces WF(Q;a) with Weight Functions of Type 2

269

The function G,(t) u vanishes near the- boundary. It is not hard to see that such a function can be approximated in D(A,) by funstions belonging t o Cg(Q). On the other hand, all functions of Cr(.Q) belong to D(A.,).Whence it follows the lemma.

To make easier the formulation of the following theorem, we introduce some abbreviations. Q1has the same meaning as in (3.2.1/1), Ma was defined in (3.3.1/3). Analogously t o Theorem 3.3.1, one sets for 1 0 , and s < 1 = integer R

(5)

Theorem. Let Q c R , be an unbounded domain of cone-type and let a(x) be a weight function of type 2 . Further let m, and m, be integers, co > m2 > m1 2 0, 1 s - k , a n d i f O < 6 < co,then

(@T(sz; a), W ~ ( Sa)>e,q ; = {u I u E *:(Q; r

=

IIUIITJ;;~) < co>,

(6)

r~),

1 . 2 . where

*

+

~ ~ u l ~(2) ( k , l ,= d ) IIuIILp(Q;b)

c

[(IpU[IB*&k(R;u),d

la1 5 k

+ IIDaUII(8-k,p,q,o)l'

(8)

All these norms are equivalent norms in the space (W?(Q; a), W F ( m ;u ) ) ~ , ~ . Proof. If Km has the same meaning as in Definition 1.13.3 with respect to the semigroups dj(t),then it follows from the proof of the lemma that completion of C r ( 9 )in the space =

Kll~

n

n

n

j=l

D ( @ ) is the

n D(Aj")described in (3.3.1/7), and that

j=l

@;(Q; a).

Theorem 1.13.612, Theorem 1.13.4/1, and a transformation of coordinates yield that (I@; a), @@; C J ) ) ~ is , ~ the set of all elements u E @(Q; a) such that

or similar expressions with q =

03

c

la1 5 k

instead of

c n

i=l

one has to modify in the-usual way.) Setting

sz(/,, = {x I z E sz, 5 - ?> # 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 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 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 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

+ sp

279

2 ,u + sp such that (4)

B;,JQ; e” ; e’) = Bk,JQ; ex; e”)* (5) Proof. Step 1 . We prove (2).Using the previous notations, it follows from (2.8.1/19) and the monotony of the 1,-spaces

If t > 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 ,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
a [m-

-1P

(3)

[m- a + l

av

p

r,, - a+l1 is a retraction from WT(Q; d’(x)) onto

1

8 ’ B;;

a+l

j=O

.

-’(aQ).Further it holds

@;(a; a&@))= { f I f E WyPm(Q;a”(%)), %f = O } .

(b) If

OL

(4)

2 mp - 1 , then

W;(Q; d”(x))= Q(Q; d”(x)). (5) Proof. One can choose the functions f ( j ) ( x )from the above definition such that the normal direction with respect to a 9 is mapped in the y,-direction. Further it follows that (yjf)(f(j)-l(y))belongs to the space

from (2.9.213). With the aid of the method of local coordinates, it follows in the same way as in the first step of the proof of Theorem 2.9.211 that 8 is a linear and a+l [!)I--]a+l . B L P T v J (aQ). In the continuous mapping from W T ( Q ; d ” ( x ) )into j=O

same manner one obtains by Theorem 2.9.212 that % is a retraction, and that (4) and ( 5 ) hold.

282

3.6. Direct. and Inverse Embedding Theorems (Embedding on the Boundary)

R e m a r k 2. It was explained in Subsection 2.9.2 how to understand the boundary values, namely with the aid of interpolation theory in the sense of Theorem 1.8.5(a). For f E cw(Q), it is clear that the so defined boundary values the usual boundary values. Further it holds (in any case)

5 a < co, it was proved in Theorem 3.2.2(c) that Cm(Q) is dense in WT(Q;d"(z)). Now, let f E Wr(SZ;d"(x)) and let ew(Q) 3 ti -+ f in WT(J2;d u ( x ) ) . Then by formula (6) it follows that Rf can be represented as a limit of Bfj in For 0

[-3- (aSZ). By the results of 0. V. BESOV,A. KUFNER[l], it foln

a+l

B,, p j=O lows that Ca(SZ) is also dense in W r ( Q ;d"(x))where -1 < u < 0. Hence one may carry over the above considerations to this case, too.

R e m a r k 3. I n the formulation of the theorem it is not necessary to distinguiah the normal direction. Let v(z)E R,, Iv(x)l = 1, be vectors defined in a neighbourhood of a Q , whose components are m-fold differentiable functions. If v(x)is a non-tangential vector in any point on x E aSZ, then the theorem is also valid. This follows from ) that the vectors the proof of the theorem, if one chooses the functions f ( J ) ( x such v ( x ) are mapped in the y,-direction. R e m a r k 4 . The theorem shows that the space B;,,(aQ), 0 < s < 0 0 , l < p < a, can also be normed in the following way. Choosing a natural number m and a number

-1 < a < co such that

m

-i- - s, then P

I n Remark 3.6.311, we shall see that the spaces B;,,(aSZ) can be normed in a similar way,O < s < co, 1 < p < a , l = < q 5 00. R e m a r k 5. References will be given in 3.10.1.

3.6.2.

Direct and Inverse Embedding Theorems for the Spaces B;JR,+; x;)

The half space RA is an unbounded domain of cone-type in the sense of Definition 3.2.1/1. Further, a(x) = x: where 0 =< cx < co is a weight function of type 1 in the sense of Definition 3.2.113. The spaces B i J R ; ; x:) are described explicitly in Theorem 3.3.1 as interpolation spaces of the spaces WT(R,+; G).I n contrast to the last subsection, we do not consider the case -1 < a < 0, since we did not develop an interpolation theory for these spaces. Previously we mentioned the paper by P. GRISVARD[2] where the case -1 < u < 0 is treated (at least partly).

As before, one sets x'

=

(xl,. . ., x,-J.

253

3.6.2. Embedding Theorems for the Spaces B;,,(B:; <)

Theorem. Let 1 < p < T h e n '8,

CO,

1

sq

i s a retraction from BL,,(R,+; xz) onto

CO,

[.-+In

0

I,

5 a < 00, and Or+l < S < C O .

a+1

(Rn-l).

BP,* P

j=O

Proof. Step 1. Theorem 3.2.2(b) yields that Cm(RA)is dense in W';(R:;s:). Whence it follows that these spaces coincide with the spaces @i,lz;(Ri) from Definition 2.9.212. Fixing a and choosing two sufficiently large natural numbers m 1 and m2, then it follows from the proof of Theorem 2.9.212 (as well as from the third ~ ) h(x')is a continand the fourth step of the proof of Theorem 2.9.2/1)that ~ ( x ,P(xn) a+l ,,,,- -

uous mapping from B,, P (Rn-l)into W;'(R,f; x:), r = 1,2. Interpolation yields that for sufficiently large values of s the mapping x(x,J P ( q J h(x') is continuous s-

from B,,,, O1

a+l -

(R,,-.l)into Bi,,(RA; 2:). Kow let s. be an arbitrary number such that

P

< S. Further let m be a sufficiently large natural number. Using (2.9.3/15) P and the considerations of that place, as well as the explicit norms of Bi,,(R;; x:) from Theorem 3.3.1, then one obtains that x(xJ P(xn)h(x') is a continuous mapping +

s-

from B , ,

a+l -

P

(R,L-l)into B;,,(Ri ;

c)for alls such that -I- < s. By similar considerP OL

ation, whence it follows that the operator G{ho,. . ., h,} from (2.9.2/12), where

, is a continuous mapping from

[.--+I-n j=O

B;,,(R:; $) such that

B,,

(Rn-J into

P

Step 2. To prove that !)is I a continuous mapping from Bk,,(R;; 2:) into

[+I-

n

j=O

B,,

v ( t ) = 1 for 0

o+l

(R,l--l),we need preliminaries. Let v ( t ) E cm((O,GO)) such that

p

t

6

1. We set 9

T f = FJ(x,,)

2

S . . . j"

1

/(xi

+ xrtri, + xn:n~zt 22

*

*

- 9

xnzn) dr1.

.

*

dzn*

1

Taking into consideration that a 2 0,then it follows that T is a linear and continuous mapping from WT(R:; x:) into itself, nz = 1 , 2 , . . ., 1 < < CO. This is true for T m ,too. If f E C"(R,+),then differentiation and partial integration yields IITfII V $ ( R ~ ; Z ~5+ ~c l)l f I I L p ( R : ; r f ) *

284

3.6 Direct and Inverse Embedding Theorems (Embeddingon the Boundary)

By iteration we have

11 Trnf11 W,"(R; ;zz+mp)5 I l f 11 L p ( R i ;.z,")

*

After extension by continuity, Tnl is a continuous mapping from Lp(RA;x:) into W';"(RA; e + " P ) . But then, Trnis also a continuous mapping from

Here 0 < s < m. The explicit form of the K-functional (Subsection 1.3.1, and Subsection 1.3.2) and (1.18.5/2)yield (Wy(RA;e + r n p ) , W;(R,t; x:)) c WF(R:;X : + ( ~ - ' ) P 1(5) -9P rn

Step 3. After the preliminaries in the second step, it is not very hard to finish the it holds (T'flf)( X I , 0 ) = f(x', 0). But then it follows from (3), proof. For f 6Crn(R;t)

(4),( 5 ) , and Theorem 2.9.2/2 that %,of, s

=

oL

f(x', 0) for s > --

+

(P

u+i

--

is a continuous

Whence it follows that '8 is a continmapping from Bi,p(RA;xz) into Bp,p p (R,L-l).

[.-+In

uous mapping from B;,p(RA;x:) into

'

u+l

BP,+, p

j=O

(R,L-l).This shows, to-

gether with the first step, that % is a retraction. Step 4. One obtains the corresponding assertion for the spaces B;,,(R,C;x:) by interpolation of the spaces Bi?(Ri; x:) where E is a sufficiently small positive number. See Theorem 1.2.4, Theorem 1.10.2, and Theorem 3.3.1. Here, in the case a + 1 , one has to omit the top order derivatives < s+&--

[.- I-

[

+P

'1-

e).

in the expression for '8 corresponding t o the spaces BiT;(Ri;

R e m a r k 1. Similar considerations as in the second and the third step can be found in P. GRISVARD[2]. R e m a r k 2. Let f$,(RA; question arises whether

g ) be

the completion of CF(R:) in B;,,(R;; xz). The

&(R:,; x:) = { f I f E B;>,&R:; x i ) ; 8f = 0). (6) Similar assertions are contained in the Theorems 2.9.211, 2.9212, 2.9.3, and 3.6.1. Let f E Bi,,(R,+;x:) such that %f = 0 and 1 K } . We choose Cm(RA)3 f j -+ f in B;,,(R,f;x:) if j -+ co. If G has the same meaning as in the first step, then it follows (in a notation easy to understand) for a suitable function ~ ( xE )Cm(R:)that Cm(R:)3 gj = [ f j - E(%jj)]X(X) -+ f , 8 g j = 0 . This yields that i t is sufficient to consider functions f eCm(R,,)such that 8f = 0. Now, let f be such a function. Further let s > ___ . We choose two non-negative @

P

integersrn,andm,such that s = (1-8) m1 +Om,, 0 < 8 < 1. Setting%=

285

3.6.3. Embedding Theorems for the Spaces B;,,(Q; du(z))

then it follows by (2.9.217) (with the notations introduced there), (1.3.3/5), and Theorem 3.3.1 that

Ilf(x) - x A ( x ~ ~(x)I B;, (R: 5 cllf(x) -

Ifs--

+

P

f(x)II

-

C1a-sp+xp+p+l

e

1-8 V(R;;z:)

- xA(xn) f ( x ) I I w ~ ( ~ ~ ; ~ , " )

, 0<1<1.

(7)

is not an integer, then it follows that the right-hand aide of (7) con-

verges t o zero if 1 10. Whence it follows that (6) holds for 1 and s - IY integer. Similarly one proves that

P

P

00,

+

1

s q < co,

(8) B;,@:; G)= B;,,@;; %3 holds for 1 < p < 03, 1 S q < co, and 0 < s <-a . For the singular cases u + l P s - -= 0, 1 , 2 , . . . the situation is essentially more complicated. Assuming P u + l that B;,,(R,+;xz) are reflexive spaces for 1 < p < co, 1 < q < co and s ~

=

P

0 , 1 , 2 , . . . (for u = 0 this is ensured by Theorem 2.11.3), then one can apply the

- 1 , 2 , ..., above method. One would obtain that (6) also holds for s - a+ and (8) also for s = a ~

+

,1

00,

1

03.

P

Another possibility to prove

P (6) and (8) for s - -= 0 , 1 , 2 , . . ., 1 < p < co, 1 < q < P u + l

00,

is the further

development of the methods in H. TRIEBEL [21].

3.6.3.

Direct and Inverse Embedding Theorems for the Spaces B;,q(Sa;d"(x))

Let ll c R, be a bounded C"-domain. Further let d ( x ) be the distance of a point x E SZ t o the boundary all. For oc 2 0 the spaces B;,,(SZ; d"(x)) are determined in

Theorem 3.3.3 as interpolation spaces. As mentioned in the introduction to Subsection 3.6.2, it is meaningful to consider also the case - 1 < u < 0. Since we did not develop an interpolation theory for these spaces, we shall restrict ourselves to the case 0 6 u < 00. Further we shall use the notations of Subsections 3.6.1. Theorem. Let Q c R,,be a bounded C"-domain. Further let co > a 2 0, 1 -

i-, and 1 V

sq2

00.

Then 93,

[,7-?]-

is a retraction from B;,,(SZ; d"(x))onto

n

j=O

b-

B,,

a+l . -1

p

(all).

286

3.6. Direct and Inverse Embedding Theorems (Embedding on the Boundary)

Proof. Using the method of local coordinates in the same manner as in the proof of Theorem 3.6.1, then one obtains the theorem on the basis of Theorem 3.6.2. R e m a r k 1. By the theorem, one may describe equivalent norms in the spaces

B;,S(aS)without the explicit use of the balls K j , the functions f ” ( z ) and yj(x)from

Definition 3.6.1. See also Remark 3.6.1/4. By the above theorem, it follows immediatelyfors>O,l
If I f

1

B;,,(aQ)

=

E LJaQ), 3g

Ilflb;,can)

= inf llglls~~%. glan=f

E B;:T

(S)such that f

I

= glaaj,

1

(2)

Here B”pq(S)= B$,,(LR;do@)). R e m a r k 2. It follows by Theorem 3.3.3, Theorem 3.2.2(c), and Theorem 1.6.2, that C”(S) is dense in BA,(SZ;d”(x)),where s > 0, 1 < p < 00, 1 5 q < 00. Then one may carry over Remark 3.6.112 concerning the boundary values. R e m a r k 3. The method of local coordinates permits also to carry over Remark 3.6.212. We denote by &,q(Q;d * ( z ) ) the completion of C g (Q) in Bi,q(Q;d”(s)). Then : C%+1 integer we have (a) For 1 < < 03, 1 5 q < oc), s > -, and s - -

P -&,@; d ” ( 4 ) = {f I f E B ; & & d”(z)), %f

+

+

P

=

0) *

(3)

( b ) F o r l < P < o c ) , l s Q < 0 0 , a n c l O < s <we ~ have +

&,,(S; d”(z)) = Bi,,(Q;d ” ( z ) ) .

P

One may also carry over the remarks to the singular cases 3.6.4.

.9

a+l

(4)

- -= 0, 1, 2 , . . . P

Direct and Inverse Embedding Theorenis for the Spaces J+’F(Rn; a), w:(R,’; a), B,S,g(Rn;a), and Bi,Q(R,’;a)

In this subsection we consider Q

= R,, , which is an unbounded domain of cone-type, and a weight function a(z) of type 1 in the sense of Definition 3.2.1/3. In this case, where exist that means that a(x) is a positive continuous function defined in R,,, two positive numbers c and 1 such that

~ ( z5) c min a(y), x E R,,. Iu-zl 21

(1) is a weak smoothness assumption and a growth condit,ion. By (1) there exist four positive numbers c1 , c,) x l , and x 2 such that, cle-qIzl

a@)

5 c2e+l.

(2)

The spaces Wr(R,,;a) and Bi,,(R,; a) are clet,eriiiined in Definition 3.2.1/4 and Theorem 3.3.1, respectively. I n the case under consideration, one may replace ill,, in (3.3.1/6) and (3.3.1/6) by K , = {x I 1x1 < S}. To obtain a full description o f the

3.6.4. The Spaces W:(R,; a), W:(R:; a), Bi,(R,; a), and BiJR:; a)

287

be11:~viour of functions belmgiig t o WF(R,;a) and Bi,q(Rn; a) near {x I x , ~= 01, o introduce corresponding spaces over R,+.

7 .

Definition. Let a(x) be a weight function of the described type. (a) W r ( R i ;a), where m = 0 , 1, 2 , . . and 1 0 , 1 < p < BiJR,; a ) to R:.

00,

and 1

sq

co, denotes the restriction of

R e m a r k 1. As usual, we set

sfi;,&R,;o)

R e m a r k 2. The function a(x) need not be a weight function of type 1 in the sense of Definition 3.2.1/3 for the unbounded domain of cone-type RL. But if one supposes that a(x) is a weight function of type 1 in R;, then one obtains by the following theorem that the spaces WT(RL;a)and Bi,q(R;t;a)coincide with the descriptions in Definition 3.2.1/4 and Theorem 3.3.1. See also Remark 3. Theorem 1. (a) Let m = 0 , 1 , 2 , . . ., and 1 < p <

03.

Then for f

E

WF(R,+; a)

(b) Let 1 < p < 00, 1 5 q 5 co, and 0 < 8 < 1. If m, and m, are two integers such that 0 m, < m, < 00, and if s = (1 - 8) m, + Om,, then

-

Bi,q(Rn';a) = ( WT1(R:; a): WF1(R:;a))o,q (6) The spaces Bi,q(R:;a ) can be normed by (3.3.1/5) and (3.3.1/6) after replacing l2 = R,+by M , = { h I Ihl 5 S} A R,+. Proof. Step 1 . Let X ( t ) be an infinitely differentiable function defined on R, such that x ( t ) = 1 for - 1 t < co and X ( t ) = 0 for - 00 < t < - 2 . Using the extension method of Lemma 2.9.1/1 where one has to replace (Sf)(2)by ~(x,)(Sf)(x),then one obtains part (a) of the theorem. At the same moment it follows that the restriction a) is a retraction. Then (4) is the conseoperator from Wr(R,,;a) onto WF(R,+; quence of the definition of the spaces B;,q(R,,;a) and of Theorem 1.2.4.

s

Step 2. The second step of the proof of Theorem 3.2.2 yields that W T ( R i ; u ) coincides with the space of (3.2.1/6)(one has t o extend Definition 3.2.1/4 t o the spacea considered here). Now one can repeat the consideration of Subsection 3.3.1, where one sets vj = (0,. . ., 0, 1, 0, . . ., 0 ) in Lemma 3.3.1. Whence it follows (3.3.1/5) and (3.3.1/6)where SZ = R: and M6 = { h I (hl 5 S} A R,+. R e m a r k 3. If a(x) is a weight function of type 1 in R:, then one obtains by the

288

3.6. Direct and Inverse Embedding Theorems (Embeddmgon the Boundary)

second step of the proof and by Theorem 3.3.1 that the spaces WT(R,+;a) and

Bi,,(R,+;a) coincide with the previously defined spaces. We set 2’ = (zl, . .,x , , - ~and ) a(x’) = a(%’, 0). T h e o r e m 2 . (a) Let m = 1 , 2 , . and 1 < p < 00. Then 8,

.

..

is a retraction from WF(R,,;a),resp. WF(Ri;a), onto

If F@(R,+;a) is the completion

of

C ~ ( R ;i n ) Y ( R , + ; a), then

{ f I f E WF(R,+; a), ’$If= O } . 1 (b)Let 1 c p c 00,lsq 5 0 0 , a n d s > -.Then 8,

6’T(R,+;a)

=

P

is a retraction from BO,JR,; a), resp. B;,,(R,+;a), onto

[.-$I-

S - - - J1

j=O

I f Bi,,(R,+;a) denotes the completion of C$’(R,+) i n B i J R ; ; a), then for 1 < q < 1

&&C; 4

= {f

If

E B;,,(R;; a),

W

2,

(c) Let 1

(10)

= 0).

If s - - is not an integer, then (10) i s also valid for q

=

00

1

1

00,

5 q < 00, and 0 c s < A . Then CF(R;) i s dense i n

1

B;,,(R,+;a). If 1 < q <

00

and 1 < p <

2,

00,

1

then C$(R,+)is dense i n B:,(R,+; a).

Proof. Step 1.Let Rn-l = u K j be a suitable covering of R,,-l with congruent balls W

j=l

of sufficiently small radius. Further let yj(z’)E CT ( K j ) such that 0

5 y j ( z ‘ )5

,xy j ( z ’ )= 1, IDYyj(z’)l 5 c y W

1,

J=1

if i

=

1, 2 , .

..

(11)

(Outside of K j , the functions y j ( z ’ ) are extended by zero.) We prove the first parts of (a) and (b) with p = q. Since the product of two retractions is again a retraction, one may restrict oneself to the spaces WF(Ri;a) and BZ,,(R;;a) (see also the first step of the proof of Theorem 1). Further, one may restrict oneself to the investigation of functions f ( s )where f ( z ) = 0 for q12 17 > 0. The norms of the spaces Bi,p(Rn-l;a) and Bi,p(R;;a) are described a t the beginning of this subsection and in Theorem 1. Choosing the number 6 appearing there and the above number 17 sufficiently small,

3.7. Structure Theory

IIUllB~JRm-,. P 0 -)

289

m

C cjIIYj(z’)U ( ~ ’ ) I I ~ ; , ~ ( R ~ .- ~ )

(14)

j=

N

Here cj are suitable numbers, for instance cj = 8(xj), where xj is the centre of the ball K j . Now Theorem 2.9.3 yields that ‘8 is a linear and continuous mapping from

n

in-1

,,,-L-j

W r ( R ; ; a )into B , , p (Rn-l;a). Similarly one concludes for B;,,(R,+;a) 1 j=O with s > -. Now let xj(x‘) E C$(Kj) such that x j ( x ’ ) = 1 in a neighbourhood of P supp yj . (Here one sets xj(z’) = 0 outside of Kj .) If Q has the same meaning as in (2.9.2/12), then it follows from the third step of the proof of Theorem 2.9.3 by a suitable choice of the functions %,(xi)that 6 ,

C x j ( s ’ ) Q {yjho j=1

-

00

*9

vjhx} = Q {hot *

* *

9

hx}

(15)

is a coretraction corresponding to 8 in the cases considered here. We used the formulas (12), (13), and (14). Hence, % is a retraction. Step 2. Theorem 1.2.4 and the interpolation theorems for the spaces under coneider-

[.-$I-n

*- 1 -i a) onto (Rn-l; a). ation yield that % ia a retraction from B$,q(R,+; Bp,qp (See also the fourth step of the proof of Theorem 3.6.2.) j = O Step 3. Let q < a.Then C”(R:) is dense in B$,q(R;;a) and dense in F ( R , + a). ; Then the explicit formulas for the norms of these spaces, the method developed in the third step of the proof of Theorem 2.9.211 (see also (15)),and the corresponding assertions of Theorem 2.9.3 yield (S), (lo),and part (c) of the theorem. Remark 4. One may carry over immediately further assertions of Theorem 2.9.3 to the case consideredhere. So,itfollows from Theorem 2.9.3(c) that (10) is untrue if 1 q = 1 ands - - = 1,2,. ., P

3.7.

Structure Theory

In Section 2.11, we considered the structure of the spaces H$(R,), B;JR,) etc. Now we complete these results determining the structure of the spaces B;,p(Q;Q’; Q’) and H i @ ; $‘; e’) of Definition 3.2.312; v 2 u , + s p . Beside Theorem 2.11.113, we [l]. need here a further result due to A. PELCZY~SKI Lemma. Suppose that for the given B a m h space A there exists a number q, where 1 5 q co,such that Zq(A)is isomorphic to A . If B i s a second Banach space,and if there ezist a retraction from A onto B and a retraction from B MLto A , then B is isomorphic to A . 19

Tricbel, Interpolation

290

3.8. Embeddmg Operators and Width Numbers

P r o o f . Temporarily we denote the isomorphism of two spaces by “-”. By the considerations in Subsection 1.2.4, it follows in the sense of direct decompositions Then

A=AO@Al, A

- &(A)

N

B=Bo@Bl,

lq(A0) 0 lq(A1) Z q ( 4

-

Ao-B,

lq(B)a3 l q ( 4

0 lq(Al)8 B I @ A

-

-

Bo-A.

lq(A) 8 B1

0 Jq(4 @B N

Bo 0 B1 = B .

R e m a r k . The above lemma is sufficient for our purposes. More general results ~ SExamples KI of Banach spaces A of the above type may be found in A. P E ~ C Z YEl]. are Lp((O,1)) and Z p , 1 6 p < 03. Theorem 2.11.2 and Theorem 2.11.3 yield that B;,q(R,,),Hi(R,,),B;,q(R3,H;(R:), q , q ( R 3 9and @,(R3, 1 1 ; in the case of the spaces Pp,&R;),one has t o assume additionally q < 00.) For the proof of the following theorem we need a further statement. If w c R, is a bounded domain with smooth boundary, then we shall prove in Subsection 4.9.3 that H i ( w ) ,0 < s < 00, 1 < p < 00, is isomorphic to L p ( w ) ,and hence isomorphic t o L,((O, 1)).Here H;(w) is the restriction of H;(R,) t o w . See also Theorem 2.11.2(a). Theorem. Let the hypotheses of Definition 3.2.312 be satisfied. Then H ; ( Q ;f ; e’), 1 0 , l = p + s p , is isomorphic to l p . I n p r t i c u h r , if s zs an integer, then W;(Q; ep; e’) is isomorphic to Lp((O,l)),and if s is not an integer, then the space W;(Q; e”; p’) i s isomorphic to l p . s s

>= 0,

Proof. Step 1 . Using the first step of the proof of Theorem 3.4.2 and the considerations in Subsection 1.2.4, then it follows easily that H i @ ; $; @“ is isomorphic t o B complemented subspace of Z,(I€i(R,))and B;,,(Q; @‘; e’) is isomorphic t o a compleTheorem 2.11.2(b)and Theorem 2.11.1/3 yield that mented subspace of Zp(Bi,p!Rn)). B i , p ( Q ;e”; e’) is isomorphic to l p . Step 2. It follows from the first step and from Theorem 2.11.2(a) that there exists a retraction from Lp((O,1)) onto H;(Q; e”; e’). The lemma and the above remark show that one still needs, for the proof of the theorem, a retraction from H i ( Q ;e”; Q’) onto Lp(( 0 , l ) ) . Let w be a n open ball in R, such that 6 c Q. We want to show that the restriction operator from H ; ( Q ; e”; e’) onto H i ( w ) is a retraction. This is a consequence of Lemma 2.9.3 and the method of local coordinates. Together with the lemma and the above remarks, one obtains the desired result.

3.8.

Embedding Operators and Width Numbers

~n Section 1.16, width ideals and entropy ideals were introduced. They are useful for a qualitative description of compact operators. It is meaningful to use these geometric characteristics for the qualitative description of compact embedding operators in Lebesgue-Besov spaces with and without weights. Later, in Section 4.10, we shall

3.8.1. Equivalent Norm8 and Compact Embedding in Wr(l2)[Part 111

291

extensively treat compact embedding operators in spaces without weights. This section is concerned with the determination of width numbers and approximation numbers of embedding operators for the spaces of type W i ( Q ; @; @'+*p). Here Q c R,, is a bounded Cm-domain. Further, ~ ( xis ) a weight function in the sense of Remark 3.2.311 such that e-l(x) d ( x ) near the boundary. The investigations in Subsection 3.2.6, in particular Theorem 3.2.6, show that this is an important class of Sobolev-Slobodeckij spaces with weights. With the aid of the method of piecewise approximation by polynomials, M. 8. BIRMAN, M. Z. SOLOYJAR [l,21 determined the &-entropyand the widths of embedding operators in Sobolev-Slobodeckij spaces. A. EL KOLLI [2,3] extended these investigations t o spaces W$?; Q"; $'), 1 < p < CQ, s = 1 , 2 , . . . , in the sense of Definition 3.2.6. I n Subsection 3.8.2, we shall prove the results obtained by A. EL KOLLI. Using these assertions, the interpolation properties of the spaces Wi(l2; p"; Q @ + ~ P ) , and the theorems of Section 1.16, we shall obtain in Subsection 3.8.3 a satisfactory characterization of the embedding operators with the aid of widths. Essentially we shall follow the treatment given in H. TRIEBEL[27].

-

3.8.1.

Equivalent Norms and Compact Embedding in Wr(sd) [Part II]

This subsection is the continuation of Subsection 3.2.5. For the investigation of compact embedding operators in spaces with weights we need some simple properties of Sobolev spaces without weights over bounded domains. We shall be extensively concerned with Lebesgue-Besov-Sobolev-Slobodeckijspaces over domains, later on. So we restrict ourselves, a t the moment, t o the proof of those assertions which are needed in the following two subsections.

If Q = Q is a cube in R,, then WT(Q),1 < p < CQ, m = 1 , 2 , . . .,is the restriction of W;"(R,)to Q. It is easy t o see that Theorem 3.2.5 is also true for l2 = Q.

L e m m a 1. Let Q c R,, be either a bounded Cw-domain or a cube. Let 1 0 such that for all f E WF(Q)

** Ilf IILg(Q) 6 cllf II w;(n).

(2)

We assume that one cannot find such a number c. Then, similarly to the third step of the proof of Theorem 3.2.5, there exists a precompact sequence {fj}Flsuch that

Assuming without loss of generality that fj -+ f, then it follows from the first w; part of (3) that I(fjll,,,(f2) = 1 , and from the second part that f = 0. This proves (2). 19*

3.8. Embedding Operators and Width Numbers

292

For the later considerations we need a lemma on the geometry of Banach spaces, which we formulate without proof. L e m m a 2. Let Bktl be a ( k + l)-dimensional subspace of a Banach space B, k = 0, 1 , 2 , . . . If s k + , is the embedding operator from Bk+l into B , then (4) dk(sk+i9 Bk+i B ) = 1 . R e m a r k 1. The lemma is a modification of a known theorem of M. G. KREJN, M. A. KRASNOSEL’SKIJ, D. P. MIL’MAN [l]. A proof can also be found in G. G. LORENTZ [Z]and in T. KATO[3], IV, 2, Lemma 2.3. If {aj}$l and {bj}Zl are two sequences of positive numbers, then aj bj means that there exist two positive numbers c1 and cs such that 9

-

.

claj 5 bj 5 czaj, j = 1 , 2 , . . T h e o r e m . Let SZ c Rn be either a bounded C”-domain or a cube. Further let 1 < p < 00 and m = 1 , 2 , . . . Then for the embedding operator I from WF(Q) into LJQ) it holds that rn sj(I, Y ( Q ) , L ~ ( Q ) ) dj(I, WF(Q),L ~ ( Q ) j-“ (5a)

-

and

--

111

-

, j = 1 , 2 , . . ., &(I, W;(SZ), Lp(SZ))5 c j where c is a suitable positive number. Proof. Step 1. The extension method (see Theorem 3.2.5(c)) and Theorem 1.16.1/1 yield that one may restrict oneself t o the case 52 = Q = (210 < xk < 1 , k = I , . . . , n } . Further, it follows by Lemma 1.16.112 that it is sufficient to prove sj 6 c j - -ni

- -in n

and

d j 2 c l j ” , G I > 0. Step 2. If q is a cube whose length of the side is 1, and if f ( x ) E WF(q)such that

[ ~ p f d= x o for

4

o s I ~sI m - 1 ,

then it follows by Lemma 1 that

If q is a cube whose lenght of the side is 2-k, k = 0, 1 , 2 , . . ., and if f the property (B), then one obtains by the last inequality that

E

WF(q)with

where c is a constant independent of f and k. Step 3. We divide the cube Q into 2kncubes q l , whose lenghts of the sides are 2 - k . Iff E WF(Q),Ilfllw,m 5 1, then one determines in ql a polynomial Pl(x) of degree m - 1, such that (6) holds with f - Pl instead of f.The coefficients of Pl(x)depend linearly on f ( x ) . (7)yields

3.8.2. Compact Embedding in W F ( 0 ;&';

293

8+nTl')

If L denotes the number of linearly independent polynomials of degree m - 1, then one obtains that SL.Bh(I;

and

Sj(I;

WT(Q),L,(Q))5 c2-kn'

W?(Q),LJQ))6 ci

- -ni

(8)

.

(9)

m. -_

, we start with the same decomposition of Q as in the Step 4. To prove dj 2 c'j last step and choose functions yl(x)E CF(ql)such that

-

llvzllq(qd = 1 and l l ~ l l l L p ( q ~ ) Fk". (The constants of estimates in the last formula are independent of k.) If

is a sequence of complex numbers such that

2kn

C IoclIp

1=1

=

at-

{oc1}1=1

1, and if one sets

2kn

Pl(4 = 1C WPz(") =1

9

-

then it follows that

I I V I l 1 Y ~ ( Q )= 1 and llvIILp(Q) rk". Hence one obtains by Lemma 2 that dakn-i(I; WF(Q),L,(Q)) 2 ~2-k", c > 0 . (10) Together with (9), whence it follows the theorem. R e m a r k 2. For the later considerations, we need a sharper version of formula (5). Let $2, = [5 121 - 1 < 151 < 21 (11)

The considerations of the third and the fourth step yield that one must replace (8) (and similarly (10)) by S L . ~ ~ - ~ U ~ WF(Q,), - I ) ( I ; LJQJ) 6 ~2-km, where c is independent of 1 and k. Whence it follows that n -1 sj(l;

-

-

w;(Q,),L,(Q~)) d j ( I ; w ; ( Q ~L,(QJ) ), j-

111

211)1"

7

(12)

where '' is independent of 1. It holds a similar estimate for d j . R e m a r k 3. As before mentioned, we shall generalize the above theorem essentially, later on. See Subsection 4.10.1 and Subsection 4.10.2. There are also given references. N"

3.8.2.

Compact Embedding in Wp(sd;e"; e"'"''')

-

I n this subsection, Q c R, is a bounded Cm-domain.Further, let ~ ( xbe ) a weight function in the sense of Remark 3.2.3/1 such t,hat p - l ( s ) d ( x ) near the boundary. The spaces WT(Q; Q"; ~ f i + ~ l and p ) L,(Q; @') are introduced in Definition 3.2.312. See also Theorem 3.2.412. Embedding operators are denoted by I . T h e o r e m . Let 1 p + mp, then there does not exist an embedding of W?(Q; p'; efi+"p) into LJQ ; @'I.

3.8. Embedding Operators and JVidth Numbers

294

(b) If Y = ,u + mp, then the embedding operator from W,“(Q; e”; e p + r r ’ l l ) into Lp(Q;e’) is continuous, but not compact. mp < Y < p (c) I f p mp, then n

+

+

(d) If v = p

+ -,m p n

then

d j ( I ; WF(Q;e”; p (e) I f -a < v < p

+ -,mnp

+ n q ,

LJQ;

@V))

s

Sj(.

m

. .) 5 c

then

(f) The widths & ( I ; WT(Q; e”; L, (0; @)) cun be estimutedfrom above by the right-hand sides of ( l ) ,(2), and (3), multiplied with suitable constants. P r o o f . Step 1 . Setting ~1

= {X

1%

E Q;

2-1 < d ( z ) < 2-l+l), 1 = N , N

+ 1, . . .,

(4)

then one may construct in an easy way a sequence of functions {pl(x)}& belonging to C$(sZ) such that SUPP V l

= 62,

114)rll~~(n;e”;e’+”P) P =

1,

I/P)111L,~o;e~, P N

2z”-fi-mp).

(5)

Whence it follows (a) and (b). Step 2. Let v 0.

(6)

Further, it is possible t o construct R 2‘(n-1)functions pl of type ( 5 ) with pairwisedisjoint supports. Here K is a suitable positive integer. Then Lemma 3.8.112 and the method of the fourth step of the proof of Theorem 3.8.1 yield S&(n-1)-1(.

. .) 2

WF(Q; @”;

d~21(”-lLl(I; -1

@p+mp),

L J Q; @”))

m p i. p -v

-

2 - c2 p , where c is a suitable positive number. Whence it follows that Sj(.

. .) 2 d-,(I; W?(Q; e”; @ p + m p ) , Lp(Q; e v , )

2 cj

--~r+ m p p(n-l)

v

.

(7)

Hence for the proof of the theorem, one has still t o show that the left-hand sides of (l),(2), and (3) can be estimated from above by the right-hand sides. Step 3. We consider a model case, identifying 2 !. with the unit ball. If Q is a n arbitrary bounded C”-domain, then one has t o modify the investigations. Let f E WT(f2; $‘; e@+‘”p)such that llfll 1. Multiplication with 21 transforms

s

3.8.2. Compact Embedding in W;(Q; e”; d’”)

ojz

from (4) into

Dl from

(3.8.1/11); 1 = 1, 2, 3,

295

... . We set wo =

Using (3.8.1/12)and returning t o the original coordinates, then it follows for suitable functions f&) belonging to jl-dimensional subspaces of Lp(wl)that

J If(4 - f Z ( 4 l e’(4 ~ ax

Here we used that, by the construction of Subsection 3.8.1, one has on the right-hand side of (8) only D”f with la1 = m. The function f l ( x )depends linearly on f . Further it holds

j”

If(z)lJ’e’(x)ax

s

Q

R

- U0~’

Now we use (8) for 1

J If(x)lPe”+”‘P(z) ax s c2-.41@-v+rnp).(9)

C2-Jl(fi-”+lnp)

JI

=

0, 1 , 2 , . . ., M - 1 , and choose

By summation, it follows from (8) and (9) that there exist functions g M ( 5 ) belonging to a

M-1

1Ilf

C jr

( L O

-dimensional subspace of L,(S; ev(s)),such that

- gM~~Lp(12;ey) 5 c2

-_ M

(~r+nlp-v)

The functions gM depend linearly on f . If v < p

This relation and (11) yield (3). For v M-1

j= C j l - M 2 and

M ( w+ P -

=p

+ -,my, n

(11)

then one obtains by (10)

+9 it holds n

v)%

z=o

- - i2

M(rnp + p - v ) A mp

.

1% 1 Together with (11) whence it follows ( 2 ) . For v > p M-

C h - 2

M(inp+p - v)&

In/’

+M ( v -

p-

+mp we have n

2)“ n rnp = 2.1/(n-1).

1=0

Together with (11) whence it follows ( 1 ) . Step 4. Now, (f) is a consequence of Lemma 1.16.1/2. R e m a r k 1. Theorem 3.2.6 shows that the spaces WF(D; e p ; position. I n particular, for p 1 - 1p (1 = 1, . . . , m)

+

Wr(l-2;@; @”) = WT(D; @ p ; @ ’ + m p) .

e”+mP)

hold a special

296

3.8. Embedding Operators and Width Numbers

Together with (3.2.4/13a), whence it follows that W r ( S ;@; @ + m p ) are the spaces considered in A. ELKOLLI[2,3]. With respect to the widths dj the theorem coincides essentially with the results due to A. EL KOLLI[2,3]. Further, A. EL KOLLI [2,3] has shown that (l),(2), and (3) are also valid if one replaces W:(sZ; e”;eP+”&p) with p = 1 - Zp (2 = 1,. . ., m ) by @(Q; e’; @). Remark 2. I n the above considerations, one may replace the spaces W F ( S ;ep; eP+”’p) by the more general spaces Wy(sZ;e”; ex) with x 2 ,u + mp. I n this way, one may obtain more general results. But they have not the same final character as the above theorem.

3.8.3.

Compact Embedding in W;(sd; g”; Q’”+’~)

In this subsection, we generalize the results of Subsection 3.8.2 essentially. l2 c R,,

-

is a g h a bounded C”-domain. Further, let e(x) be a weight function in the sense of Remark 3.2.3/1 such that e-l(x) d(x) near the boundary. The spaces W i ( S ;ep; pp+sp), 0 5 s < a,and Lp(O; e”) are determined in Definition 3.2.312. See also Theorem 3.2.412. Embedding operators are denoted by I. Theorem. Let p, v , p , q, and a be real numbers such that 1 - -n 2, !I’ n n (a) If -L< - s +- -, then there does not exist an embedding from ! I ? , q W p 2 ; e’; @ P + S P ) into Lq(S;e’). n n P V - - -, then the embedding from Wi(f2; ep;ec+8p) into (b) If - ! I = -s P q Lq(sZ;e’) is c o n t i n w , but not compact.

: 1.

+

(f) If p = q, then one may r e p h e ,,5“ in (1) and (3) by “”’. Proof. 8 k p 1. The sets 01 have the same meaning as in (3.8.2/4). If y ( x )E Cc(R,) such that supp v c then it follows that

2

:I

I 1x1 < -

and if one sets y l ( x )= rp(21(s - X J ) where x1€ 0 1 ,

-c2

zm-

41 +L!! p

p

, m

= 0, 1,2,.

..

(4)

3.8.3. Compact Embedding in W:(sZ; p";

f+'p)

297

One may assume that supp v 1 c w l holds. Formula (4), interpolation of WF(Q; ep;@'+mp) and L,(Q;@) on the basis of Theorem 3.4.2, and (1.3.3/5) yield In

lp

llPiIlw~(n;ew;pw+*~p, = c21"-7+7, < 1=1,2

Setting 1yi = c'2

-ls+

-- In p

lp

p

vZ,then one obtains by

)..., o < s < c Q .

(5)

(4) and ( 5 ) that

Hence, it follows (a). 8tep2. Now we consider the cases (b)-(e), where we assume without loss of V

P

n

n

generality - - - 2 0. Let A = - - and Q P P q 8 2 so 2 A, and one obtains by Theorem 3.2.4/3(a) and by Theorem 3.6.1 that

3

W t ( 0 ;e";

(7)

@N+SP).

Hence, for the caaes (b)-(e) the embedding from W;(Q; e"; ep+'P) into L,(Q; e') is continuous. (6) yields that the embedding is not compact for s = so. This proves (b). Step 3. If 0 < 8 < 1, and if one again sets 1 = 1-8 -=a P 1

Q

p =-((I q

e

+-,q

n

P

n -and

Q

8=A(l-O)+se=se+n

- e) v + ep),

then one obtains by Theorem 3.4.2 that W$Q; @; $zu) and H:(Q; to 2L = + I p K(e; q ; q H;(Q; 9 ; p + S p ) ) .

),

q(n;

Forp - v >

--sqn

it holds

sq it follows that For p - v = - n

-P - -v-- - -8 O

Q

n

+ -1- a

sq > p - v > -sq that and for - n

(8)

1 q'

F ;fi;')belong

298

3.9. The Spaces W ; , ~ ( B J

Further it holds (Theorem 3.5.1). Now for p = q, Theorem 3.8.2, Theorem 1.16.311, and the second step yield (l),(2), and (3).Here one may replace WS,in the formulation of the theorem v by H i . Taking into consideration B e = " -and using Theorem 1.16.311, a q then one obtains (l),(2), and (3) for q > 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,
+ 08,

and

p = (1

- 0) p1 + 0p2.

Then, if s i s not an integer, (w;,p,(Rn), ~ ~ , p , ( R pn = ) )w;,p(Rn)* ~ P r o o f . The spaces G;: from Subsection 3.4.1 are identified with WF(R,), k Further, one replaces llfllq in (3.4.1/1)by

=

(1) 1,2.

If vk = p,,. - skp, k = 1 , 2 , then all the considerations in the proof of Lemma 3.4.1 remain valid. [(3.4.1/2) is satisfied, the hypotheses vk 2 pk used there are not necessary now]. The theorem follows by a modification of the proof of Theorem 3.4.2. R e m a r k . Introducing the more general spaces h>,,(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<1

P

P

[ e l 8-L onto n w p , p

-k

k =O

(c) If 0

s

(Rn-1).

1

- , the72

P

G”,,(RA) = w ; , p w . (3) 1 1 P r o o f . Step 1. Let- < s < co and { s } - . Remark 1 and the following P P considerations show that one may restrict oneself in the proof of (a) t o the spaces wi,JRn). The system Z of Definition 3.9.112 will be denoted temporarily by Z,,, to indicate the dimension n. Let {c,(x)}go E 2,. Then {&(x’, O)}$o E Znd1.One may assume that the functions cj(x) depend only on x‘ in a neighbourhood of xn = 0. Let f(x) E w8p,+(RI,),y = 2-jx, and f(x) = f(2Jy)= Ry). Then it follows by Theorem 3.9.1 that

+

304

3.9. The Spaces w;&)

The exponents of 2 coincide in (4) and (5).Then Theorem 2.9.3yields that ‘8 is a coninto tinuous mapping from iodp,t(Rn)

1 Step 2. Let again - < 8 < 03 and

+ -1 . The transformation y = 2-jx maps K j P IyI < 4}, i = 1,2, . . . Further, let (Cj(z)}$oE 2,

(8)

P 1 from Definition 3.9.1/2 into {y I -< 2 be the system of functions from the first step. Let xo(x) E C$(R,) such that 1 x0(z)= 1 for 121 5 2 and let xl(z) E C$(R,) such that xl(z) = 1 for - 5 121 5 4 2 1 and xl(x) = 0 for 1x1 5 -. If E and S have tho same meaning as in (2.9.2/12) and 4 (2.9.1/4), then for one sets

h&’)

1 E w8-7 - k

(Rn-l), k = 0, . . ,,

P7t

where the above notations are used. Now with the aid of (4) and (5)it follows that is a continuws operator from

36

[a-

n$1 w

j-0

* - l - k ~p ,

(Rn-l) ~ into w;,+(R,) such

6

that

= E. Herb we used that G is an extension operator in the sense of Theorem 2.9.3. (See the third step of the proof of Theorem 2.9.3.) Hence ‘8 is a retraction. This proves (a). Step 3. Part (b) and part (c) of the theorem are consequences of Theorem 3.9.1 and Theorem 2.9.3. R e m a r k 2. Using the spaces h;,JR,) and b;,,+(R,,),resp. the corresponding spaces over R:, introduced in Remark 3.9.1/3, one may generalize the theorem essentially. In particular, after replacing

one may prove the part (a) of the theorem also for the case {s} = L.

P

R e m a r k 3. On the pattern of Theorem 2.9.4 and on the basis of the theorem, one may determine traces of functions belonging t o w;,,(R,,) on R Lwhere O < 1 < n - 1. R e m a r k 4. * For s = 1,2, . . ., the theorem was proved in B. HANOUZET [3,4]. See also L. D. KUDRJAVCEV [2,3], Ju. S. NIKOL‘SKIJ [2,3], T. S. PIGOLKINA [2,3].

3.9.4. Structure Theory

3.9.4.

306

Structure Theory

With the aid of the methods of Section 3.7, we shall determine the structure of the spaces w;,+(R,,). T h e o r e m . Let 1 < p < 00 and -GO
+

Proof. The first step of the proof of Theorem 3.9.3 yields that S , is a linear and continuous mapping from w;,JR,) into Zp( W;(R,,)). If the functions xo(x) and x l ( x ) have the same meaning as in the second step of the proof of Theorem 3.9.3 then R ,

is a linear and continuous mapping from Zp( W;(R,,)) into w;,JR,,). Since RS = E , the mapping R is a retraction from Zp( W;(R,,)) onto w;,JR,,). Now, all the other considerations are the same as in the proof of Theorem 3.7.

3.10.

Complements

I n contrast to the second chapter, hitherto we have given only few references in this chapter. There are several reasons. First, we considered mainly the classes of spaces introduced by the author. Secondly, in the theory of the Sobolev-SlobodeckijBesov spaces with weights there are many possible modifications of the types of the considered weights and of the methods how these weights are put in the corresponding norms. See Definition 3.2.114, Definition 3.2.312, Theorem 3.2.412, Remark 3.2.412, Definition 3.2.6, and Definition 3.9.111. Although these classes of spaces are rather general, of course, they do not give a complete description of the theory of function spaces with weights. In particular, if one considers not only isotropic, but also anisotropic spaces with weights then one obtains many new possible modifications. I n this section we describe very briefly a number of types of spaces more or less closely related t o the previously considered spaces. Essentially, we restrict ourselves t o references. Some of this papers are also concerned with anisotropic spaces, but we shall not point out this explicitly. A survey on the results obtained in the theory of function spaces with weights by Soviet mathematicians can be found in : 1 . Investigations on the theory of differentiable functions of several variables and their applications, Trudy mat. inst. Steklova akad. nauk SSSR, 106, Moscow 1969*),

306

3.10. Complements

2. Embedding theorems and their applications, "Nauka ", Moscow 1970*). Further we refer to 5 10 of the paper by 0. V. BESOV,V. P. IL'IN, L. D. KUDRJAVCEV, P. I. LIZORKIN, S. M. NLKOL'SKIJ [l]. 3.10.1.

Spaces with Weights g(x)

r~

d"(r)

Let 52 c Rn be a (bounded or unbounded) domain. Further let d(z) be the diitance of a point x E f2to the boundary 632. From a historical point of view, a t first Sobolev(Slobodeckij-Besov) spaces are considered, whose weights have tho form d"(z), - 00 < x < oc), or can be estimated from below and from above by d"(z). Examples are the spaces WF(52;CT) from Definition 3.2.1/4, if a(z) = d"(z) (see also 3.6.1 and 2.9.2), and the spaces B i J R ; ; and Bi,q(52;d"(x)) from 3.6.2 and 3.6.3. The first [l]. systematic treatment of spaces of such a type was given in L. D. KUDRJAVCEV The development of this theory at the beginning of the sixties is characterized by papers written by P. I. LIZORKIN[l],V. P. IL'IN[2], S. V. USPENSEIJ[2], J. N E ~ A S [l], and P. GRISVARD [2]. The trace method of interpolation theory leads also to spaces of such a type. See Subsections 1.8.1 and 2.9.2. In this connection we refer to [l].There can be found interpolation the paper by J. L. LIONS[3]. See also C. GOUDJO theorems for spaces of such a type. The main purpose of the quoted papers is to obtain embedding theorems for different metrics, as well as direct and inverse embedding theorems on manifolds of lower dimensions. In the meantime, there are also considered ankotropic spaces, where the weights have the form xi (for 52 = R,+)

c)

111

and I-IxTj (for 52 = {z I F~> 01). SLe A. D. DSABRAILOV [2,4,5], A. D. BERIEV[l], j=1

S. V. USPENSRIJ[4,5], R. D. KULOV[l,21 and Ju. S. NIKOL'SKIJ [4]. Further references can be found in the cited papers. Further we refer to § 10 in the paper written by 0. V. BESOV,V. P. IL'IN, L. D. KUDRJAVCEV, P. I. LIZORRIP,S. M. NIROL'SIUJ [l]. 3.10.2.

Spaces with Mixed Norms

Mixed norms with and without weights are considered by several authors. If 1 < pj < C O , = ~ 1 , . . . , n , then

IltIlPu... Pn 7

is said to be a mixed norm. See A. BENEDEK, R. PANZONE [l], V. I. BURENKOV [l], A. CH. GUDIEV[l], and JA.S. BUQROV[1,2]. On this basis one may introduce Sobolev-Slobodeckij-Besovspaces with weights and mixed norms, and investigate their properties. We refer to R. D. KULOV[l, 21 and A. S. DSAFAROV, g. F. MAIKEDOV El]. There are also given further references. Interpolation theorems for spaces of such a type can be found in A. BENEDEK, R. PANZONE [l].

3.10.4. The Spaces of

I. A. KWRIJANOV

307

The Spaces of L. D. Kudrjavcev; Generalizations

3.10.3.

We pointed out in Remark 3.9.1/1 that the spaces w;,+(R,)of Definition 3.9.1/1 are closelyrelated to spaces introduced by L. D. KUDRJAVOEV [2,3]. If e(z)= (1 + lzI8)+, s = 1 , 2 , . . ., and if 1 < p < 00, then L. D. KUDRJAVCEV [2,3] considers spaces with the norm (1) II F’pW II L,(R.) + IIf II L+) .

c

lal=s

Here cc) is a compact set in R, , e.g. the unit ball. Similarly, spaces over R i are introduced. The main aim is the investigation of embedding theorems and of traces on hyper-planes. For this purpose, one needs corresponding spaces where s is not an integer. Using Hardy’s inequality (3.2.6/3) or (3.2.6/4),respectively, and the equivalent norms in Theorem 3.2.5, then it follows that (1)and Ilfllw;,(B,, are equivalent norms on C$’(R,) for p - sp + n - 1 -1, -1 - p , . . ., -1 - (s - 1)p; that means p k p - n for k = 1, . . .,s. Spaces of such a type, partly with more general weight functions and in more general domains, are considered by several authors. [l, 2,3], J u . S. NIKOL‘SKIJ[2,3], Ju. V. RYBALOV We refer to T.S. PIGOLKINA [l, 21, L. D. KUDRJAVCEV [4], and B. HANOUZET [2,3,4]. Ju. V. RYBALOV is concerned with the problem under which conditions the embedding between spaces of such a type is continuous or compact, respectively. I n this connection we refer also to R. A. ADAMS[4].

+

+

3.10.4.

The Spaces of I. A. Kiprijanov

A special class of Sobolev-Slobodeckij spaces with weights is treated in I. A. KIPRIJANOV [1,4]. The main idea may be described as follows. The Fourier transform is one of the most important tools for the investigation of the spaces Hi@,,).One can ask whether there exist other integral transforms qualified for treating other 1 classes of function spaces. If JY(t), v 2 --, is the usual Bessel function, then 2 a

-

&4x)=j” Jsz

f ( z )dz,

JY(@

0

00

f ( 4= J JG

JY(s&(Q)

ds

0

denote the direct and the inverse Bessel transform, respectively. The spaces corresponding to this transform may be defined as the completion of C$(R:) in the norm

R:

. . ., y 2 0.

R,+

One obtains more general spaces if one combines several (one-dimensional) Bessel transforms and a k-dimensional Fourier transform. Further, one may also define spaces of such a type in the case that s is not aninteger. A treatment of the theory of such spaces can be found in the quoted papers by I. A. KIFRIJANOV.There are embedding theorems, extension theorems etc. Further [l]. V we refer to B. M. B O G A ~ s = 1, 2,

20

*

308

3.10. Complements

3.10.5.

The Spaces of A. Kufner

Let Q c R, be a bounded domain and let r ( x ) be the distance of a point x E SZ t o a given fixed point on the boundary dQ. A. KUFNER[l] introduced Sobolev spaces with weights whose norms are given by

1 < p < 00, - 00 < a < 00, rn = 1 , 2 , . . . Probably, one may consider these spaces in analogy t o the spaces W;(Qn; e) from Definition 3.2.114. I n particular, one may develop an interpolation theory for these spaces on this basis.

3.10.6.

Axiomatics of a Class of Spaces with Weights

In Definition 3.2.311 and Definition 3.2.312 the weight function e ( x ) and the spaces H i @ ; e”; e’), Bi,q(SZ;e”; e”),and W”,SZ; Q ” ; e’) are introduced. The investigations of these spaces show that the special properties of the spaces H;(R,), B & ( R , ) ,and W;(R,) are used not very much. Hence it seems t o be meaningful to mtroduce axiomatically a larger class of spaces containing the spaces H;(R,) and Bi,q(R,Jas special cases. Such a class in the set of all Banach spaces G having the following properties : (a) X(R,)c G c S’(R,), S(R,,) is dense in G . (b) If f ( x ) E a, then f(Ax) belongs also t o G , (here f(h) must be understood in the sense of the theory of distributions), 1 0 is a complex number. Further it holds

+

Ilf(h)I\C

5 cAD\lfllG for 1 < 1 K

where c > 0 and D E ( - 00, co)are suitable numbers. (c) If f E G and q E C$(R,) then qf belongs also to G. If q lDyqWI5 C ( Y ) for 0 IIyi <

E

C$(R,) and if

00,

then there exists a positive number C = C ( c ( y ) )such that

IIqfIlc 5

Cllfllc.

(d) There exists a number c > 0 such that for all h > 0 and all f E

and

II(f)/iIlG

5

EG

cllfllG-

Here, ( f ) h has the same meaning as in (2.5.116) (in the sense of the theory of distribu. tions). Hi(R,) and Bi,q(R,,)are contained in this class. If G, and G, are two Banach spaces satisfying the above conditions, then, by a suitable choice of p and v, one may replace the spaces Hi(&) and Lp(Rn)(resp. Bi,q(R,) and Lp(Rn))in Definition 3.2.312 by G, and G,. We do not go into detail here. The definition of the Banach spaces G and the consideration of the corresponding spaces with weights can be found in W. FECHNER [l].

4.

LEBESGUE-BESOV SPACES WITHOUT WEIGHTS IN DOMAINS

4.1.

Introduction

Thia chapter is concerned with the theory of the Lebesgue-Besov spaces without weights defined over domains (and hence as special cases also with the theory of the Sobolev-Slobodeckij spaces without weights defined over domains). The aim of the investigations is clear by the last tw-o chapters. General remarks can be found in the introductions 2.1 and 3.1. Although the Lebesgue-Besov spaces without weights are contained in the corresponding spaces with weights, it is useful to develop the theory of these spaces in a separate chapter. Firstly, this is justified from a historical point of view. Secondly, there is a number of important results which cannot be obtained by specialization from the theory of the spaces with weights. I n particular, in the following treatment this holds for extension theorems and structure theorems, as well as for certain problems related t o compact embedding and equivalent norms. On the other hand, there are some important parts of the theory for the spaces without weights which can be obtained in an easy way from corresponding investigations of chapter 2 and chapter 3. This is the case for the interpolation theory, the theory of embedding theorems, and some parts of the structure theory and the theory of equivalent norms. As in the last two chapters, we restrict ourselves t o isotropic spaces, that means to spaces where is no distinction between the directions in R, . Historical remarks and references have been given in the second chapter. So we restrict ourselves in this chapter t o references which contain specific investigations on Lebesgue-Besov spaces defined over domains.

4.2.

Definitions, Extension Theorems

The Lebesgue-Besov spaces (and hence as special cases also the Sobolev-Slobodeckij spaces) over domains are defined as restrictions of the corresponding spaces over R,,. This coincides with the investigations for the spaces defined over R,+in the second chapter, but not with the considerations in the third chapter. But the extension theorems of this section and the interpolation theory of Section 4.3 will show that by suitable specializations of the weight functions the spaces defined here are the same as the spaces of the third chapter.

310

4.2. Definitions, Extension Theorems

4.2.1.

Definitions

D e f i n i t i o n 1. Let SZ c R,, be an arbitrary (bounded or unbounded)domain. Further, let -a < s < co, 1 < p < a,and 1 5 q 5 co. Then B;,q(SZ)i s the restriction of B;,q(Rn) to Q,

Here glQ E D’(SZ) denotes the restriction of g E D’(R,) to SZ in the sense of the theory of distributions. Yurther, one sets H;(SZ) if s = 0, 1 , 2 ., W.,(Q) = {B;,+(Q) if 0 < s integer. R e m a r k 1. See Definition 2.3.1/1, Definition 2.9.1, and Definition 2.9.3. The Besov spaces B;,q(Q)and the Lebesgue spaces Hi(SZ) are Banach spaces, since they are factor spaces. Similarly t o the above notations, WS,(SZ)is called a Sobolev space (8 = 1 , 2 , . . .), resp. Slobodeckij space (s integer). Historical remarks and references can be found in Remark 2.3.112 and Remark 2.3.114. R e m a r k 2. Setting u ( x ) = 1 in Definition 3.2.114, then one obtains a Sobolev space without weight. The extension theorems in 4.2.2 and 4.2.3 will show that this space coincides with WF(SZ).See also Remark 4.2.2/3 and Remark 4.2.3/8. The spaces B&(Q; a) are defined in Theorem 3.3.1 and Theorem 3.3.3 by interpolation. We shall obtain similar statements for the spaces Bi,JQ) in Subsection 4.3.1. Hence, also these spaces coincide with the coreesponding spaces with weight, where a ( x ) = 1.

+

..

+

D e f i n i t i o n 2. Under the same hypotheses a s i n Definition 1, one denotes by Bi,q(Q), &@), and Wi(SZ) the completion of CF(L?) in B;,q(Q), H;(Q), and W”,SZ), respectively. R e m a r k 3. Later on, we shall define further subspaces of B;,JQ) and HS,(Q); see Definition 4.3.312 and Definition 4.3.2.

4.2.2.

First Extension Method

We shall describe two extension methods. The first one has the advantage that it is also applicable t o spaces B;,,(Q) and H i @ ) , where s < 0. For the second one, we shall need only weak smoothness assumptions for the boundaries of the considered domains. L e m m a . Let (w,,SZ,) and (02, Q2) be two couples of bounded C”-domains in R,, N = 1 , 2 , . . ., such that W j t Qj for j = 1, 2. Further, let y = h ( x ) be a one-to-one muppin,g from a, onto D2, having a non-vanishing Jacobian, such that w 1i s mapped

4.2.2. First Extension Method

311

onto I&. Let the components of h ( x )be N-foldwntinuouslydifferentiable functions. Then ( K f )( Y ) = f(h-'(y)),

f ( 4E C r n ( W 1 ) ,

(1)

can be extended i n a unique way to an isomorphic mapping from HpN(w1)onto H j N ( o z ) , l
-

Proof. Step 1 . Let dl be a domain such that 0,c and 6,c 9,.A function g(x) E Cr(f&) will be extended by zero outside of fi,; similarly (Kg) ( y ) . The explicit 1 1 expression of the norm of HF(R,), Theorem 2.3.3, = 1, and Theorem P P 2.6.l(a)yield

fi,

+

5

(2)

~ ~ K g ~ ~ H ~ H cllgllH;N(R,) ( R ~ ) *

-

Here c is independent of g. Approximation by smooth functions and a limit process show that (2) holds for all g E H j N ( R n )where supp g c 9,.

Step 2. Let f E H p N ( w l ) .It follows that one obtains an equivalent norm in H j N ( w , ) if one restricts oneself in (4.2.1/2) to elements g E H i N ( R n )such that aupp g c fil. Now, the lemma is a consequence of (2), where one has to take into consideration that one may replace h by h-l in the above considerations. Theorem. Let 9 c R, be a bounded Cm-domain. If -co < s < co, 1 < p < 00, q co, then the restriction operator from B;JR,J onto B;,#2) and from H;(R,) onto H ; ( S ) i s a retraction. If N i s a natural number, then there exists a wrresponding coretraction independent of 1 < p < co,1 q 03 and Is1 c N .

an& 1

s s

Proof. Step 1 . With the aid of the method of local coordinates, described in the third step of the proof of Theorem 3.2.2, it follows by Lemma 2.9.3 and the constructions used there that a coretraction G from H f ( 9 ) into H f ( R n ) exists (see also the following second step).

Step 2. We want t o Show that the same mapping G is also a coretraction from H i N @ ) into HiN(R,,). Suppose that the balls Kj and the functions yj(x)and ~)i(x) have the same meaning as in the third step of the proof of Theorem 3.2.2. Since g -t yjg is a continuous mapping from H i N ( R , J into itself, i t follows that g + yjg is a180 a continuous mapping from H i N ( 9 )into itself. Hence, it is sufficient to extend y;g for g E H i N @ ) . Since Cg(Rn)h dense in H;"(R,), it follows that the restriction of C g ( R n ) t o 9 is dense in H i N @ ) . Hence, one may restrict oneself to functions of such a type. Applying the above lemma t o h ( x ) = f ( J ) ( zfrom ) Definition 3.2.1/2, using afterwards the method of Lemma 2.9.3, multiplying the extended distribution by q ; ( f ( J ) - l ( y ) and ) , using finally the lemma in the reverse direction, thenone obtains the desired extension operator. Step 3. Now the theorem is a consequence of the above steps, Theorem 1.2.4, (2.4.2/11), and (2.4.2/14). R e m a r k 1. The coretraction described in the theorem is sometimes called extension operator.

312

4.2. Definitions, Extension Theorems

Remark 2. Clearly, Lemma 2.9.3 is the baais of the last theorem. If one considers only spaces Bi,,(52) and H $ ? )where Is1 < N , then it is sufficient to suppose that 52 is a bounded CaY-domain.Further, it is possible to extend the theorem to special unbounded domains. Remark 3. If s = m = 1 , 2 , . . ., then it follows from the explicit form of the coretraction 6 and from Theorem 3.2.2(c),that Wr(52)= WF(52;o),where o(x) = 1, Definition 3.2.114.

4.2.3.

Second Extension Method

In this subsection, we describe a second extension method, applicable to spaces

B;,&2) and H $ ? )where s > 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 -

-&,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 <

c

00,

1

(2b)

00,

then Cr(SZ)is dense i n

8;,q(~ a )d ~ J Q c ) i+;(~). 1

5 q < w,- P

1 <s

< co,and

s

1

- - + integer, then

l?;,,(SZ)

(3)

P

&;(sz).

~ J S Z ) = 8;,,(~)and f i ; ( ~ = ) (4) Proof. Step 1. We start with preliminaries. We use the method of local coordinates, similarly t o the second step of the proof of Theorem 4.2.2. If yj has the former meaning, then it follows from the previous considerations and by Theorem 4.3.1/1, that g --f y,g is a continuous mapping from B;,4,(SZ)into itself; similarly for If;@). A corresponding assertion holds for the spaces B;,@) and @,(SZ). Step 2. Let f E and q < w. Then yif E Since Cr(B,,)is dense in B;,q(R,),one obtains that

e,,(SZ)

4,$2).

(The notations must be understood in the sense of the theory of distributions.) Choosing h in a suitable way, then (y,f)(x h ) has a compact support in SZ. Such distributions can be approximated in B;,q(R,) by functions belonging to C r (Q) ( eCr(R,)).Whence it follows that C$(SZ) is dense i118~,,(9). Similarly, one conNow one obtains (3) immediately from the definition. cludes for Pp(SZ). Step3. Lemma 4.2.2 is true for H:(col) = WF(co,), too. Then it follows by interpolation on the basis of Theorem 4.3.1/1, that one can extend Lemma 4.2.2 t o all spaces B;,,(w,) and Hi(col). Now, using the method of local coordinates, one obtains (2) from Theorem 2.9.3(d). In the same manner, it follows (4) from (3) and Theorem 2.10.3(b, c ) .

+

T h e o r e m 2 . Let 9 c R, be a bounded Cm-domuin. Then, after replacing B : .( R tT ) by i : . ( S Z ) , and H:(R,) by H":(SZ), the f o r m u b (2.4.1/3),(2.4.1/7), (2.4.1/8),(2.4.2/9), (2.4.2/10), and (2.4.2/14) are also valid under the corresponding hypotheses for the parameters. Further it holds that

(q,,.(Q), 8.;,q1(Q)),,, =

&,e)s*+ esx (Q) , (5) - w < so,sl c 00,s~ sl,o c e < i , i c p < a , i q o , q l , q Proof. Using the method of local coordinates, one can extend Theorem 2.10.413 to the domain SZ in a similar way as in the first and the third step of the last proof. But then the theorem is a consequence of the marked formulas (inclusively Theorem 2.4.212) and Theorem 1.2.4. R e m a r k 2. The theorem is similar to Theorem 2.10.4/1. Comparison with Theorem l(a, c) shows that one obtains by the last theorem a rather general interand fi;(SZ). On the other hand, one sees that polation theory for the spaces &,@)

+

s

4.3.2. The Spaces

-

i;,,(Q), B;,,(Q), k;(Q), and I?;(Q)

319

1 the spaces where s - -is

an integer have a special position. In this connection, 1 two problems arise. Firstly, one asks, whether one can obtain (4) also for s - 23 = 0, 1 , 2 , . . . with the aid of other methods. (2.10.5/4) and the method of local 1 coordinates yield that for -co < s < -1 -, 1 < p < co, and 1 < q < 00, it holds that P

P

+

g;,g(Q) iB;,&Q) and @&C?) 4 H;(Q). Secondly, explicit norms for the spaces Bi,,(Q)and DP,,(Q) as well as for the spaces H,"(Q) and fi,"(Q) are of interest. For the spaces B;,g(Q)and ~ p , g ( Q ) where s > 0 as well as for the Sobolev spaces W r ( Q ) where m = 1 , 2 , . . ., one may solve the

second problem in a satisfactory way. See Theorem 4.2.4 for the Sobolev spaces. Setting a(x) = 1 in Theorem 3.3.3, then one obtains with the aid of Theorem 4.2.4 and Theorem 4.3.1/1 equivalent norms in BE,,@) for s > 0. We return to this problem in Section 4.4 in more detail. Formula (4), Theorem 2, and Theorem 3.3.4, where a(x) = 1, .yield

s

s > 0, 1 < p < co, 1 q 6 00. Here 0 is defined in (3.2.1/1).If q = co,then one has to modify the last expression in the usual way. If d(x) denotes the distance of a point x E Q to the boundary, then, for 1 0, Remark 3.3.2 yields

Ilfllf;P.P (Q)

llfll :;,p(n)

s

+ d - 9 4 If(4l" dx. n

(7)

Hence, the question I?$,@) = &,g(Q) is identical with the problem whether there exists a number c > 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
00.

P

(10)

320

4.3. Interpolation Theory

This can be proved in the following way. Using the method of local coordinates, then one obtains by Theorem 2.9.3 that dk(x) belongs to &,,(52). On the other hand, it is not hard t o see that the left-hand side of (8) is not finite for f ( x ) = d k ( x ) .

R e m a r k 3. Using Theorem 3.3.1 and Theorem 3.3.2 where 52 = R: and o ( x ) = 1, then one obtains (6) and (10) for 52 = RA.This is a complement to (2.10.313). R e m a r k 4. * The spaces Jv;(52) ( = ffi(52)for s = 0 , 1 , 2 , . . .and = pp,p(SZ) for 0 < s #= integer) have been introduced in J. L. LIONS,E. MAUENES[l ; I I L V ] and E. MAGENES[l] in connection with the corresponding spaces if;(52). There are statements of the type of formula (4) as well as special cases of Theorem 2 . I n this connection, we refer also t o P. GRISVARD[6] and P. KRBE [4]. I n the last paper, there are considered angular domains.

4.3.3.

The Spaces

B;,q,(Bj)(sa)

and H;,(BjJ(sa)

I n the Subsection 4.3.1 and in the Subsection 4.3.2, we developed an interpolation theory for the spaces B;,$2) and H,"(Q), and for the spaces pP,,(52) and g;(SZ), respectively. The question arises, whether one may generalize these results by interpolating closed subspaces of B;,,(Q) and H;(SZ). I n Remark 1.17.1/4, we mentioned the difficulties arising by the Interpolation of subspaces. (One obtains a further counter-example in the sense of Remark 1.17.1/4 by Theorem 4.3.111, Theorem 4.3.211, Theorem 4.3.212, and (4.3.2/10).) For the theory of elliptic differential operators, subspaces of B;,q(SZ)and of Hd(52) between &,,(52) and B;,q(52), or between I?;@) and H,d(SZ),respectively, are of special interest, which are determined by differential operators. I n this subsection, we formulate an important result by P. GRISVARD[5,6,7] and R. SEELEY[2,3]. We omit the rather complicated proof. D e f i n i t i o n 1. Let SZ c R, be a bounded C"-domain. Further let

B j f ( x )=

C

bj,a(x)oaf, bj,a(x)E c"(aQ)

(1)

9

la15 mj

j = 1, . . . , k , be differential operators on aSZ, Then {Bj}ik_lis said to be a normal system if 0 5 m, < m2 < . . . < mk and if for any normal vector vz with respect to a52 and the p i n t x C bj,JX) V: 0 , j = 1 , . ., k. lal=mj

+

.

E aSZ

it holds t h d

R e m a r k 1. Normal system of boundary value operators play an important in the theory of elliptic differential equations. We shall return to this point later on. D e f i n i t i o n 2 . Let 52 c R, be a bounded C"-domain. Further, let B j be the differential operators described i n Definition 1. For s > 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

<8 -- . P

321

(5)

R e m a r k 2. The method of local coordinates and Theorem 2.9.3 show that the definition is meaningful. T h e o r e m . Let 9 c R,,be a bounded C”-domain. Further let {Bj}ik_lbe a normal system in the sense of Definition 1. Let m be a natural number such that m > 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
+

cN

ck(Q)

R e m a r k 2. The theorem and the definition yield that em(1(2) is dense in for k = 0, 1, 2, . . . Here cm(Q) is the set of all infinitely differentiable functions defined in where Q is a bounded C”-domain. Further it follows that these spaces are separable ones. A corresponding assertion for the spaces cl(Q)and Ct(R,), where t is not an integer, is not true. See Remark 4.9.416.

s,

4.6.2.

Interpolation and Equivalent Norms

T h e o r e m 1. Let Q t R,, be a bounded C”-domain. (a) If 0 5 to < t, < co and 0 < 8 < 1 then

(Cto(Q),Cfl(SZ)),,, (b) If 0 < t

+ integer, then

= W(Q)

where t

= (1

CyQ) = W(J-2).

- 8 ) to + Bt,.

(1) (2)

Proof. The proof is an immediate consequence of Theorem 4.5.1, Theorem 2.7.211, and Theorem 1.2.4. T h e o r e m 2. Let 9 c R, be a bounded C”-domain. (a) If m = 0 , 1 , 2 , . . . ) then

4.6. The Holder Spaces c'(S2)

326

are equivalent mrms in C.l(Q). (b) If has the same meaning as in (4.4.1/2),then for t > 0

a,,,,

are equivalent mrms in

c

%((a). Here k and 1 are integers such that 0 s k < t and 1 > t -k .

One can replace in (6) by laljk Ihl 5 6,where 0 < 6 < 00.

c

la1 = k

. Further, one can replace h E R,,in ( 5 )and ( 6 ) by

Proof. Step 1 . The construction of the coretraction and Theorem 4.5.1 yield that (3) is an equivalent norm in Cm(Q). I n the same manner as in Theorem 3.2.5, one proves that (4) is also an equivalent norm in Cm(Q). Step 2. The proofs of the Theorems 3.3.1 and 3.3.3 may be carried over without changes t o the C-spaces. But then it follows from the modified Theorem 3.3.3 in the same manner as in the proof of Theorem 4.4.1 that (5) and (6) (inclusively the marked modifications) are equivalent norms in W(Q). R e m a r k 1. The mentioned modification of Theorem 3.3.3 gives also the possibility t o describe equivalent norms in the spaces (Cto(Q),Ct1(Q)),, where 1 5 q < co. See Theorem 4.4.1. Since the method is clear, we do not go into detail here. R e m a r k 2. For 0 2 to < t, < oc,, 0 < 0 < 1, and t = (1 - @ t o O t l , Theorem 4.5.1 and (2.7.2/8) yield

+

Ilfll&n)

e s CllfIl~~.(Q, Ilfll~h(Q, 1-8

3

f

E C"(Q).

(7)

Inequalities of such a type are very important in the theory of elliptic differential operators. See for instance C. MIRANDA[l]. R e m a r k 3. If t is not an integer, then W(Q) = C'(Q), Theorem l(b). Then it follows by (6) that

This is the usual definition of the norm in the Holder space replace sup If(4l = Ilfllcotn) by l l f l I C t w 2 ) ~

Here one may

cl(Q).

XED

R e m a r k 4. Similarly to (4.2.4/3)and to (5), one may ask whether there one can

4.G.1. Embedding Theorems for Arbitrary Domains

replace

101

c

=m

SUP lo”f(4lin ( 4 ) by

sen

c SUP 11

j = 1 ref2

1

a’”f(z)

1.

327

J. BOMAN [l] has shown that

this is not true. See also Remark 1.13.412. For the later considerations it is useful to extend Definition 4.5.1 to arbitrary domains. D e f i n i t i o n . Let Q c R, be an arbitrary domain. Then one sets

for 0

+ { t ] ,[t]is an integer, 0 < {t} < 1. Further, one sets cl(52)= { f I f ( z ) is [t]-fold continuously differentiable in a and

= [t]

*

IlfIlCl(Q, < .o>. R e m a r k 5. If Q is a bounded C”-domain, then the definition coincides with the above results. With the aid of the usual methods, it follows that ct(S)is a Banach space in any case. R e m a r k 6. P. L. BUTZER, H. JOHNEN [ l ]considered Holder spaces on manifolds with the aid of interpolation methods;

Embedding Theorems for Different Metrics, Inclusion Properties

4.6.

Definition 4.2.1/1 gives the possibility t o carry over embedding theorems and inclusion properties from the Lebesgue-Besov spaces Hi(R,,) and Bi,q(Rn)to the corresponding spaces defined over arbitrary domains. For bounded domains one obtains further embedding theorems. Embedding Theorems for Arbitrary Domains

4.6.1.

The spaces ct(52)have the same meaning as in Definition 4.5.2. Theorem. Let 52 c R, be an arbitrary domain. (a)Let-oo<sO,l < p < 0 0 , a n d l ~ q , ~ q , ~ 0 0 . T h e n

q 3 Q= ) B ; , m = Bi,ql(Q)= q,q,(Q) = q,,(Q) = B;;:,;;“Q).

(b) Let

-00

(c) Let

00

<s <

00

and 1 q 2 p > 1,l 5 r I 00, - 00 < t 6 s < n s---2t--. P -

(1)

Then

00, and

(3)

328

4.6. Embedding Theorems for Different Metrics, Inclusion Properties

If additional 0 < t (f) Let 1 < p <

00

+ integer, then ( 6 ) i s also valid for s = t + -.n2,

and t 2 0. Then

B;F(Q) c C t ( Q ) . (7) P r o o f . (1) and (2) are consequences of (2.3.2/3)and (2.3.3/8),and Definition 4.2.1/1. All the other assertions of the theorem follow from Theorem 2.8.1, Remark 2.8.1/1, Remark 2.8.1/2, and (2.8.1/18). R e m a r k 1. If 0 5 t 5 s < 03 and 00 > q 2 p > 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 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 <

i s a retraction from H;(SZ) onto

00.

Then '8)

[.-$I-

n

j=O

s-L-j

(80). Further, it holds

Bp,p

B;(Q)= { f 1 f E H p - 2 ) ;'8f = 0 ) . (b) Let 1 < p < co, 1 traction from

q

B;,,(Q) onto

&,,(Q) =

5

n

[s-$1j=O

{f I f

03,

and-

1

V

<s <

E B;,&.n); %f

(a0). For

B;,,(SZ) into

(d) If 1 < p <

0 0 , l< q

and dense i n Hj(SZ).If 1

00

it holds 1 -

1

r)

a ["-7 'If

1

-, P s

is a continuous map-

a'V[.-$]

1

< co, and - co < s

< p < co and - co c

(3)

- - i s not an integer. It holds BLl(Q)

- - = 0, 1 , 2 , . . . Then P LJaSZ). s

1
= O} *

1

(c) Let 1 < p < co and

Then %, defined i n (1)) i s a re-

00.

s-L-j

Bp,q

Formula (3) with q = 1 i s true if and only if s

ping from

(2)

then

Ce(SZ) i s dense in BK,(Q)

1

< -,then Cp(SZ)i s dense i n B;,,(SZ). P

Proof. Part (d) coincides with Theorem 4.3.2/1(a). Using the method of local coordinates (see the third step of the proof of Theorem 4.3.2/1),then the other parts of the theorem follow from Theorem 2.9.3. R e m a r k 1. See Theorem 2.9.3, Theorem 3.6.1, and Theorem 3.6.3. R e m a r k 2. The theorem is also valid if v = v(y), y E aSZ, are non-tangential vectors whose components are Cm-functionson aQ.

4.7.2.

Direct and Inverse Embedding Theorems (General Case)

The transformation of Theorem 2.9.4 gives rise t o some difficulties. We restrict ourselves to the description of the situation, however, we do not go into detail. Let M be an I-dimensional C"-manifold with M c Q, I = 1, . . ., n - 1. It is assumed that M is open in the sense of I-dimensional local coordinates on M . Further, let

4.7.2. Direct and Inverse Embedding Theorems (General Case)

332

M , be an 1-dimensionalC"-manifold, R,c ill,such that the boundary aM, = N o- M , (with respect to M ) is an (1 - 1)-dimensional C"-manifold. It is assumed that M , can be covered in M by a finite number of systems of local coordinates. Similarly to Definition 3.6.1, one may define the spaces B;,,(M,), 0 < s < 00, 1 < p < co, and 1 5 q 5 co. Let vk(x), k = 1 , . . ., n - 1, be n - 1 linearly independent vectors in R, spanning the normal plane in the point x E M . Suppose that the components of vk(x) are Cm-functionsdefined on ill.Similarly t o the above notations, we write

Theorem. Let D c R, be a bounded C"-domain. Further, let M and M , be the described C" -manifolds, n-1 (a) Let 1 < p < co and < s < co. Then 8, P ~

is a retruction from

H;(D) onto la1 5

(b) Let 1 < p

n

[+I-

< co, 1 5 q 5 co, and-

a retraction from B;,JD) onto

n-l

B ~ ~ ~ ~ ' ( M , ) . n-1

P

<s <

00.

Then 8,defined in ( l ) , is

Proof. Using local coordinates on M , the theorem can be obtained from Theorem 2.9.4. R e m a r k 1. The other parts of Theorem 2.9.4 may be also carried over. I n the problem of approximation of functions where %f = 0 in the sense of Theorem 2.9.4, one has t o take into consideration whether the intersection of M , and aD is empty or not. R e m a r k 2. References for the theory of embedding theorems can be found in Remark 2.9.412.

4.8.

Duality Theory

The spaces B;,@) and H$Q) have been obtained in Definition 4.2.1/1 by restriction of corresponding spaces over Rn. For spaces with s < 0, however, there is another possibility of definition, used by J. L. LIONS,E. MAGENES[l, 111; 2, I] in connection with the theory of elliptic differential operators. For instance, starting with the spaces H;(D), where s > 0 and 1 < p c co,in the sense of Definition 4.2.1/1, then

332

4.8. Duality Theory

they set H;?(Q) = (g;(SZ))’. Here the dual spaces are constructed similarly to Section 2.6 and Subsection 2.10.2 in the sense of the dual couple {CF(SZ)= D(SZ), D’(SZ)}. I n Theorem 4.8.2, we shall see that these two possibilities of definition coincide, with the exception of singular cases.

4.8.1.

The Dual Spaces of i;,@(O) and fi;(J;))

Theorem. Let SZ c R,, be an arbitrary (bounded or unbounded) domain. Further, let -co<s
Then

P

P’

9

9’

(e,,(Q))’ B;s,.(Q) =

and

(fi;(G?))’ = H;.”(SZ).

(2)

Proof. One obtains (2) similarly to (2.10.6/2). R e m a r k . The theorem is a generalization of Theorem 2.10.5/1.

4.8.2.

The Dual Spaces of Bi,Jsd) and H J O )

T h e o r e m . Let

SZ c R,, be a bounded C”-domain. q <

(a) Let 1 < p < co, 1 (4.8.1/1)i s valid. Then

(&,,(Q))’

=

1

a,-< s < P

03,

P

(&;(SZ))’

H;?(Q). (1) 1 and -1 + - < s < Suppose that (4.8.1/1) is

Bp’,,(SZ) and

(b) Let 1 < p < co, 1 < q <

1 and s - - i1 , 2 , . . . Suppose that

00,

valid. Then

(BK,(SZ))’ = B;?,,@) and

=

-.P

1

2,

(H;(Q))’

=

H;f(Q).

(2)

1

( c ) Let 1 < p < co, 1 < q < co,and -a < s < -. Suppose that (4.8.1/1)i s valid. 2,

Proof. Step 1 . Part (a) and part (b) are consequences of Theorem 4.8.1 and Theorem 4.3.2/1. Step 2 . Bp,,(Q) is a closed subspace of Bi,,(R,). Theorem 2.11.2 yields that B;,,(R,,) is a reflexive Banach space for 1 < p , q < 03. Closed subspaces of reflexive spaces are also reflexive spaces. This follows from the Hahn-Banach theorem and the theV. L. SMUL‘JAN (see K. YOSIDA [l], Appendix to Chapter 5 , orem by w. F. EBERLEIN,

333

4.9.1. Elliptic Boundary Value Problem

4). Similarly, one concludes for fii(SZ). Now, (3) follows from Theorem 4.7.l(d) and Theorem 4.8.1. R e m a r k . A duality theory for general Sobolev-Besov spaces in domains can also be found in T. MURAMATU[5, 61.

4.9.

Structure Theory

This section is the counterpart t o Section 2.11. We shall see that one may carry over some parts of the results obtained there. But we shall need new methods, since the method of local coordinates does not fit for problems of such a type. There are needed some deep results of the theory of general regular elliptic boundary value problems, which we shall describe in the first subsection without proof. But some important results of this section are independent of these tools (e.g. W i ( Q ) lp for 0 < s += integer, or the existence of Schauder bases in BA,(Q) and in H;(Q)).

-

4.9.1.

Elliptic Boundary Value Problems

Regular elliptic boundary value problems are considered in the fifth chapter. We need here some of the results of this theory (only partly derived in the fifth chapter), however, t o develop the structure theory for the Lebesgue-Beaov spaces H;(Q) and B;,&?) (and so also for the Sobolev-Slobodeckij spaces W;(Q)). The theorems are formulated without proofs. Concerning the references, we restrict ourselves to few papers. More references, comments, and historical remarks can be found in the fifth chapter. D e f i n i t i o n . Let Q c R,, be a bounded C"-domain. (a) The differential operator A ,

i s said to be properly elliptic, if C a J x ) tU= a(x, 5 ) 1u1= 2m

+ 0 *)

for all R,, 3 5 += 0 and

x

ED,

(2)

5 E R,, and R, the polynomial a ( x , 5 + tr])in the complex variable t has exactly m roots ti = tk+(x,5, q), k = 1, . . .,m, with positive imaginary part. Here a J x ) are complexand if for any x E 0 and for any coupk of linearly independent vectors

q

E

valued infinitely differentiable functions in 0.One sets a+(x,t,797) =

n 111

k- 1

(t -

ti).

(3)

( b ) The system {Bj}Fl from (4.3.3/1) i s said to be complemented (with respect to A ) if, for any x E aSZ, the normal vector v x , and any tangential vector [, += 0 in the p i n t x *) As before, one sets 't =

t? . . . 5:-

for

5

=

( E , , . . ., t,,)E R , and a

= (al, . . ., a,,).

334

4.9. Structure Theory

with respect to a 0 , the polynomials

+ t v x ) = I C-nij

bj(x7 5,

a1

bj,a(x)(5,

+

t~z)”

in t are linearly independent modulo a+(x,5, vz t). R e m a r k 1. Comments and references to this definition are given in Chapter 5, see 5.2.1. The assertion on the roots of a ( x ; 5 + t y ) is called the “root-condition)). For n 2 3, it is a consequence of (2), see Remark 5.2.111. Theorem. Let 0 c R, be a bounded C”-domain and A be a p r o p r l y elliptic differential operator. Let {Bj}Y:l be a normal system (Definition 4.3.3/1) with m,,, < 2m, that is complemented (with respect to A ) .I t i s additionally assumed that for all R,, 3 6 0 and for all x E 0 )

)

+

+

and that for all x E a 0 , the normal vector v,, all tangential vectors 5, 0, and all t E ( - co,01 the above defined polyuomials bj(x;5, + tv,) are linearly independent modulo

...

/.=l

(T

- t:(t)). IIere, tk+(t)are the roots with positive imaginary part of the

polynomial

-

+

a@, 6, t V . 1 ) t . For sufficiently large positive nuinbeis Q 2 eo the operator A + QE gives a n isomorphic mapping from W a ; ~ ($2) ~ l (Befinition 4.3.312) onto W i ( 0 ) , 1 < p < co, k = 0, 1, 2, . . . Here, Po = e 0 ( A ,B,, . . . , B,J is independent of p and k. (b) Considering A, (a)

)

D ( A J = ivi:y~~1 ($21, (5) A,,f = A f , as an unbounded operator in L,(Q), then A , + QE is for Q 2 eo a positive operator in the seme of Definition 1.14.1, 1 < p < 00. (Here, eo has the same meaning as in part (a).) If AS, where z is a complex number, are the fractional powers of A, in the sewe of Section 1.15, then A;, -co < t < co, are bounded operators in L p ( 0 ) and , there exist numbers c > 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,}-,,,<, is a two-side scale, then {B,)O(z,c< ,is a one-side scale. l < , , - 03 < T < 00. Clearly, it is sufficient to The same holds for { B t + T ) o ( ~ ~where prove the scale property for large values of N . R e m a r k 2. For fixed 1 < p < co and fixed 1 5 q 5 co, Theorem 2.3.4 yields <s< is a two-side scale. The same holds for {Pi,q(R,J}-m< 8 < , , that {B;,,(RrL)}-, where 1 < p , q < 03, and in particular for {H;(RrL)}.-m < $ <, . It follows by Theorem 2.10.3that{~p,q(R~)}-m,s,m for 1 < p < 03 and 1 5 q 5 oo,and{&(R~))-,,,,, for 1 < p < co are two-side scales. I n all these cases, it is possible t o choose operators A!" independent of N . For the later considerations, however, it is meaningful to use the above definition. 71

T h e o r e m . Let J2 c R, be a bounded C " - d m i n . (a) I f < p c 00 and 1 5 q 5 03 are fix&, then { H ; ( {B~,q(12)},,<s< , and {4,q(SZ))o<s< are one-side scales. (b) If 1 ,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
D ( ( A , + eE)')

(4.3.2/4) yields &(Q) then it follows that

D ( ( A+ ~

=

=

i$m~

(Q) *)

1

for m - 1 + - < 2m0

P

1

5 m + - . (3) P

f i F ( Q ) . Using again Theorem 1.15.3 and Theorem 4.3.2/2,

e q e ) = iiFme(~) for o 5 8 5 +.

Hence one obtains by Theorem 1.15.2 that {&p(Q)}05s<m is a one-side scale. The operator A,, where p = 2, is self-adjoint in L,(Q). But then it follows by Example 1.15.1(b) that ( A , + eE)' is also self-adjoint. Now one obtains with the aid of Theorem 4.8.1 that (A,, + PIC)', 0 I -8 I*, is an isomorphic mapping from H i @ ) 1

1

P

P'

onto H;-2m'(Q), where - + - = 1 and 0

>= s 2 s - 2m6 >=

-m. The construc-

tion of the fractional powers in Subsection 1.15.1 yields that ( A , = (Ap,+ eE)'f for f E C$(Q). Whence it follows that

{H;(sz) for

-CO

< s < 0,

&,(Q)

for

+ eE)'f

o s s < m}

is a two-side scale. Step 3. Part (a) is a conseq~enceof Theorem 4.3.1/1, Theorem 4.3.212, and the interpolation property. To prove part (b) for the B-spaces, we use H;(Q) = &,(Q) 1

1

P'

P

for - - < s < - ,Theorem 4.3.2/1. By interpolation (Theorem 4.3.1/1 and Theorem 4.3.2/2) it follows that 1 < p < 00,l 5 q Bi,q(Q)= P,,,(Q),

s

00,

1

1

- 7< s < -. P P

Now (b) for the B-spaces is a consequence of the second step, Theorem 4.3.1/1, and Theorem 4.3 2/2. R e m a r k 3. The proof shows that we constructed for every number N another set of operators {AtN))in the sense of the above definition. On the other hand, it follows by the construction that the operators AtN) are independent of 1 < p < co and 1 q 03. This property will be of great importance, later on.

s s

R e m a r k 4. The theorem and the proof go essentially back t o H. TRIEBEL[23]. We refer also t o i.A. ALIMOV [I]. *) We do not write additional indices to indicate the choosen value of

k (k = 0 or k = m).

4.9.3. Isomorphism Ropertiea

4.9.3.

337

Isomorphism Properties

Theorem. Let 9 c R,, be a bounded C”-domain. Further,let 1 < p < co. (a) The spaces H;(SZ), where - m < s <

Z?;(SZ)

and

where k

g;(S),where 0

00,

s<

00

,

1 +m +- 1 < s < 00, 2,

are isomorphic to Lp((O,1)). Here m = 1 , 2 , . . . and k = 0, 1, . . . (b) The spaces B;,p(SZ),&,,(9), and B;,,(SZ), where -co < s < 03, are isomorphic to l p . Proof. Xtep 1. Theorem 2.11.1/1(a) yields that H;(SZ) = f$(SZ) = LJQ) is isomorphic to Lp((O,1)). Now, one obtains the assertions for the spaces H;(SZ) andH;(SZ) 1 from Theorem 4.9.2. Theorem 4.3.2/1 yields = HP(S) for s 2 0 and s - o

nit- 1

=+ integer. Finally, the cases H p 1

iT(SZ)

1

%;(a)

(a), where

m = 0,1,2,.

. ., are

2,

obtained by

(4.9.2/3)and = HT(SZ). Step 2. Similarly t o Theorem 4.7.1, it follows that (1) is a complemented subspace of HJSZ). Hence (1) is isomorphic t o a complemented subspace of Lp((O,1)). Let o be a bounded C”-domain, where i3 c SZ. Then it follows with the aid of the former extension methods that H ; ( w ) is isomorphic to a complemented subspace of &JS) and isomorphic t o a complemented subspace of (1). Since H ; ( w ) is isomorphic to Lp(( 0 , l ) )and since Lp(( 0 , l ) )is a space in the sense of Lemma 3.7, it follows by the same lemma that (1) is isomorphic t o Lp((O,1)). Step 3. Bi,p(SZ)and &p,p(s2) are isomorphic t o complemented subspaces of Bi,p(R,,). This is a consequence of Theorem 4.2.2, the proof of Theorem 4.3.212, and Theorem 1.2.4. Now it follows by Theorem 2.11.2(b) and Theorem 2.11.1/3 that Bi,p(Q)and B;,p(SZ)are isomorphic t o l p . The considerations to Theorem 4.7.1 show that Bi,p(Q)is a complemented subspace of B;,p(SZ)(clearly, one may restrict oneself 1 to- < s < m). Using Theorem 2.11.1/3, it follows that ii,p(s2) is isomorphic to Zp.

P

R e m a r k 1. The theorem is the counterpart to Theorem 2.11.2 and Theorem 2.11.3 (see also Theorem 3.7 and Theorem 3.9.4). In contrast t o these results, the above theorem does not give a determination of the structure of the spaces BiJSZ) for p + q. Probably, for bounded domains SZ the spaces B;,@) and Zq(Zp) are not isomorphic. But a clarification of this problem seems to be open. I n the next subsection, we shall prove the existence of Schauder bases in these spaces if q < co. 22

Triehel, Intrrpolation

338

4.9. Structure Theory

On the other hand, if Q is an unbounded domain of cone-type, then the spaces Bg,,(Q), where 1 < p < 00, 1 6 q co, and 0 < s < 1, are isomorphic t o /,(Ip). This can be proved in the following way. Let Q be an unbounded domain of cone-type. Then there exist n linearly independent vectors vl, . . ., v,, and a point y E Q such that y + M, where

s

M

=

(

XIXER,,,

x

=

5 ap,,

j=1

aj >

01,

is contained in and all the points of y + M have a distance t o dQ larger than a suitable positive number E . By Remark 4.2.318 and Theorem 2.11.2(b), there exists a retraction from lq!lp)onto B;,,(Q). On the other hand, it follows from the properties of M that the restriction operator from B;,,(Q) onto Bi,,(M) is also a retraction. But with the aid of an affine transformation of coordinates, one obtains by Remark 2.11.3/1 that Bb,,(M) is isomorphic t o lq(lp).Since lq(lp) is a space in the sense of Lemma 3.7, the same lemma yields that B;,,(Q) is isomorphic to lq(Zp).

a,

R e m a r k 2. The theorem yields that the Sobolev spaces W i ( Q ) ,s = 0, 1 , 2 , . . ., are isomorphic to Lp((O,l)),and the Slobodeckij spaces W;(Q),0 < s += integer, are isomorphic t o lp . Hence, these spaces belong to different isomorphism classes. R e m a r k 3. The second step of the proof shows that any complemented subspace k;(Q)c V c Hi(f2) is isomorphic to Lp((O,1)). Similarly, any infinitely dimensional complemented subspace V of B;,JQ) is isomorphic t o lp

V of Hi(52) such that

.

R e m a r k 4. The question arises whether one can weaken the assumptions for the domain 52. Clearly, it is sufficient if 52 is a bounded CN-domain, where N is a natural number, sufficiently large in dependence on s. If 52 is a bounded domain of conetype, then it follows by Theorem 4.2.3 that Bi,p(Q)for 0 < s < co (and hence also Wi(52)for 0 < s =# integer) is isomorphic to l p . The same theorem yields that H$2) for 0 5 s < co and for domains of such a type is isomorphic t o a complemented subspace of Lp((O,1)). We shall return t o this problem in Theorem 4.9.4/2.

4.9.4.

Sehauder Bases

T h e o r e m 1. Let 52 c B, be bounded C ” - d o m i n . (a) All the spaces of Theorem 4.9.3 have an unconditional Schauder basis.

(b) The spaces B;,,(Q) where 1 < p < co, 1 < q <

00,

and -1

have a common Schauder basis (a system of functions of Haar type). (c) Let 1 < p < co. The spaces

B;,q(52),where - co < s < co and 1 5 q < 00 , Pp,q(Q), where 0 < s < 00 and 1 q < 0 0 ,

s

&&2),

where 0 < s <

have (individual) Schauder h e s .

00

and 1 < q < co

1 1 +c s < -,

P

2,

4.9.4.Schauder Bases

339

Proof. Step 1 . Part (a) is an immediate consequence of Theorem 4.9.3 and Theorem 2.11.1/1. Step 2 . Temporary, we assume that the space BI,,(Q) with 1 < p < CQ,1 q < 03, 1 and 0 < o < - has a Schauder basis. We prove part (c) under this hypothesis.

P

Since for the above parameters it holds B;,,(Q) = B;,,(Q), Theorem 4.3.211, one obtains the desired assertion for the spaces B;,,(Q) and @p,q(Q) from Theorem 4.9.2 and Theorem 4.3.2/1.It follows from (4.3.214)and Theorem 4.7.l(d)that one still m+-

1

has to prove the statement for the spaces BP,, (Q) with m = 1,2,. . . and 1 < q < 03. For this purpose we use the operator from the second step of the proof of Theorem 4.9.2.For two suitable numbers 8, and 8, such that 1 1 m-

1

it holds that

D((A,

+< 2m8, < m + - < 2m0, < 2 m P ?,

+ @E)O')= H;y2,)(Q),

Interpolating these spaces by (*, .)a,,, is an isomorphic mapping from

=

0,1.

then Theorem 1.17.1/1yields that (AP + @E)O

1

< -, by a suitable choice of a. Whence it follows that the P where 0 < s < 03 and 1 < q < 03 have also a Schauder basis.

onto B;,,(Q) where 0 < s spaces &&2)

I

Step 3. To prove that B;,,(Q) where 1 < p <

03,

1

5 q < 00,

has a Schauder basis, we need some preliminaries. I n the cube Q,

Q = { x I x E R , , ,<~ xj < l,j = 1,2,.. .,n}, we consider the decompositions Q =

.

<

5. I(--)

(1)

N

U Q,, where

r=l

+

mj -(=)

1

and 0 < s < -, ?J

+

2k-1

for j = 1,.. . , l ,

for j = z

+ 1,. . . , n .

(2)

For fixed k = 1,2,. . and I = 0,1,.. ., n (with the usual interpretation for 1 = 0 and 1 = n ) , there are running mj = 0, . . ., 2k - 1 (resp. 2k-1- 1). (The sign of equality in (2)must be chosen in a suitable way.) First we show that the step functions whose steps me determined by the cubes Q, are dense in Wi(Q),1 < p < 00, 1 0 s< Since Q is a bounded domain of cone-type, it follows by Theorem 4.4.212

-.P

22*

340

4.9. Structure Theory

One obtains by Definition 4.2.1/1 and Theorem 2.3.2 that it is sufficient to approximate functions f(x), which are continuously differentiable in &. Let f(x) be such a function. If Z is a decomposition in the above sense, and if one denotes the centres of the cubes Q, by x ( ~ )then , one sets fz(x) = / ( x ( ~ ) for ) 2 E Q,.

It holds that

s

I f ( 4 - f z ( 4 - (/(!I) - fz(Y))IP dxdy 15

- yp+v

QxQ

Since the measure

c IQ, N

j=1

x Qjl tends t o zero if

k + 03 in (2), the first summand

on the right-hand side tends also t o zero. Using Definition 4.5.2, and denoting by d j ( x )the distance of x E Qj to aQj , then one can estimate the second summand on the right-hand side of (4) by

N I~ I I f l l ~ j &z )llQj=l ~ l l f *lPl ~ q g ) 9

k 2 k&). Whence it follows that the step functions of the above type are dense in W;(Q), 0

s

1

< -.

2,

Step 4. We need a second preliminary consideration. If all the symbols have the same meaning as in the last step, then we show that Pz, (Pzf) (4 = - / f ( y ) dy IQjl

is a projection in WP(Q), 1 < p <

from

For 0 < s < I it, holds t h a t

P

00,

for

5 EQj,

02s<

-.2,1

(5) For the space Lp(Q),this follows

4.9.4. Schauder Bases

341

For fixed j the first sum is to take over all r such that Q j n 0, 0 and r ==! j. The second sum contains all the other pairs of j and r. To estimate the first sum, we fix j and msume without loss of generality

jf(x) dx = 0, where

Q

a,

U

=

Setting a, = (Pzf)(x)for x E Qr , then

OrnGj

*

Qr. 0

Here the constants are independent of the decomposition. By the same argument as in the proof of Lemma 3.8.1/1, one obtains that

Using (8), then a homogeneity-argument yields

where the constant c is independent of the decomposition. Putting (11) in (9), and summing over j , then one obtains that N

c c‘

j=i r

* * *

S

P cllfIIWi(Q)*

(12)

QjXQr

To estimate the second summand in (7),we use a.I - ar

=

1QjI-l

IQrI-’

J (/(XI

Setting bj,r = inf Ix - yI one obtains that XEQj YOQI

- f(Y)) dx dY

9

342

4.9. Structure Theory

Now it follows by (7), (12), (13), and (6), that P z is a projection in WJQ),and that

5

IIP;.IIrvpd(q,+n~;co,

(14)

c,

holds where the constant c is independent of the decomposition Z. Step 5 . Now it is not hard to see that the spaces BiJQ), where 1 < p < co,

1

q <

1

and 0 < s < - , have a common Schauder basis. For this purpose, all

03,

P

the decompositions (2) are numbered, Z,, Z,, . . . , such that one passes from Zj to Z ,, by bisecting exactly one of the elements Q,. Then, P, and Pz,+, - PZ, are one-dimensional projections (the ranges are one-dimensional). It follows by 1 interpolation that (14) is also true for BS,,,(Q),1 < p < 00, 0 < s < - , 1 5 4 < co. Further,

P cl(Q)is dense also in these spaces. But then it follows from the third step

f

a,

= pzlf

+ 1C- 1

( P Z , +-~ Pzj) f

9

f

E

B;,q(Q).

The generating functions of the projection spaces of Pzj and Pz,+l - Pz,are orthogonal in L,(Q).Whence it follows easily that these functions are a Schauder basis in B i J Q ) (system of functions of Haar type). Step 6 . I n the same manner, one proves the existence of a common Schauder basis in the spaces BZ,,(Q -

1 < p < co, 1

At

U

Q r i ) , where Q r i are some of the cubes of the type (2),

1 q < co, 0 < s < - (cube with cube-shaped holes). One con1-1

P

cludes similarly if D is a ball with ball-shaped holes. Now, a bounded C”-domain can be transformed in this standard domain. Whence it follows that there exists a 1 common Schauder basis in all the spaces BZ,,(D),where 1 < p < co, 0 < s < - ,

P

lSq
Step 7. The operators Pz (resp. the modified operators in the sense of the last step) are self-adjoint in L,(Q)(resp. L,(D)). Now, Theorem 4.8.1 and Theorem 4.3.2/1 yield 1 that(14)hold~alsoforBS,,~(D),~herel < p < c o , l < q < c o , a n d - l +-<s
P

Since cl(D)is dense also in these spaces, it follows in the previous manner that the above system of functions is it Schauder basis in these spaces, too. A further interpolation gives also the spaces BE,$)). R e m a r k 1. The theorem is a generalization of a corresponding result in

H. TRIEBEL[23]. The steps 3-6 are carried over from this paper.

R e m a r k 2. The fifth step contains also a proof of the existence of a Schauder basis in Lp((O,1)). See Theorem Z.ll.l/l(a) and Remark 2.11.1/1. R e m a r k 3. The spaces Bi,,(Q)play a special role with respect to their boundary behaviour; see Theorem 4.7.1. This is the reason why the above proof is not applicable 1 1 to the spaces B,,,(Q) where s = k + -, k = 0, 1, 2, . . . If 0 < .s - - is not an

P

P

integer, then these spaces coincide with the spaces pp,,(SZ), Theorem 4.3.2/1.

4.9.4. Schauder Bases

343

R e m a r k 4. The proof shows that the step functions are a dense subset in all the 1 spaces Hj(SZ) and B;,,(SZ) where 1 < p < co, 1 q < co, and -co < s < -.

s

I,

On the other hand, Theorem 4.7.1 yields that step functions (with exception of constant functions) does not belong to HJSZ) and B:,q(Q),where 1 < p < CO, 1 1 5 q 5 00, and- < s < 00. P The question arises whether one can prove the existence of Schauder basis under weaker assumptions for the domain SZ. For this purpose we need a lemma, which we formulate without proof. Lemma. A complemented subspace of L,,((O,1)) has a ~%hauderbasis, 1 < p < a. R e m a r k 5 . The lemma follows from the theory of the 9,,-spaces, which are A. PELCZYI~KI [l]. Lp((O,1)) is an gP-space. introduced by J. LINDENSTRAUSS, J. LINDENSTRAUSS, H. P. ROSENTHAL[l] have shown that each complemented subspace of an g,,-space is either isomorphic t o a Rilbert space or is again an 9,,space. Finally, W. B. JOHNSON, H. P. ROSENTHAL, M. ZIPPIN [l] proved that each separable SP-space has a Schauder basis. Whence the lemma follows. T h e o r e m 2. Let Q c R, be a bounded domain of cone-type. If 1 < p < co and 0 < s < co then the spaces B;,,(SZ) and H;(Q) (and hence also W#2)) haveaSchauder basis. Proof. The proof of the theorem is an immediate consequence of the above lemma and Remark 4.9.314.

R e m a r k 6. * The existence of Schauder bases in Sobolev spaces WF(Q),1 < p < CO, . . . , where SZ c R, is a bounded domain with smooth boundary, was , JOHN, J. N ~ 6 a s[l]. Further, Z. CIESIELSKI,J. DOMSTA [l] proved by S. F U ~ I K0. described explicit constructions for such bases in cube-shaped domains Q. These bases are also Schauder bases in C m ( Q ) , m = 1 , 2 , . . . Concerning the existence of [1,2], Z. CIESIELSKI [2], and Schauder bases in CnL(Q)see also S. SCHONEFELD J. RYLL [I]. If follows from Lemma 12 in B. S. MITJAGIN [3] that Cm(R,) in the in the sense of Definition 4.5.1 are isomorphic sense of Definition 2.7.1 and to Cm(Q).The same holds for cF(SZ),the set of all functions of Cml(SZ)vanishing a t the boundary inclusively all derivatives up to the order m. Hence all these spaces have in= Schauder bases. On the other hand, the spaces c'(SZ)and Ct(R,), where 0 < t =I teger, have not a Schauder basis, since these spaces are not separable spaces. Z. CIESIELSKI [I] showed that these spaces are isomorphic to 1,. m = 1,2,

cm(Q)

R e m a r k 7. I n the considerations in this section, we used essentially that the domains SZ are bounded. The investigations in S. AGMON[2] and R. SEELEY[l] are valid only in this case (in contrast t o D. FUJIWARA [ l , 2, 31). Let D = RA. Then i t follows by Remark 2.11.3/2 that one still has to determine the isomorphic structure of the spaces in (2.11.3/3)in this case. I n the same manner as in the second m+-

step of the proof of Theorem 4.9.3, one shows that H P 1

1

P

(R:)isisomorphic toLp((0,l)).

By Theorem 2.9.3 it follows that Bg+',(R:), where m = 1 , 2 , . . ., 1 -+m 1

< p , q < 00, is

a complemented subspace of BLq (RA) and hence by Theorem 2.11.3(b)isomorphic

344

4.10. Qualitative Properties of Embedding Operators

t o a complemented subspace of

On the other hand,

Z&). 1

-+m

is a complemented subspace of B;q 1

(R,+).This is a consequence of the extension

method. Since BLq ({z I z E R,; xn > l}) is isomorphic to E (1 ) (Theorem 2.11.3(b)), 3

m

+ .:,

and since Lemma 3.7 is applicable to Zq(Zp), it follows that B;q 1 < q < 03, m = 1, 2, . . ., is isomorphic to Zq(l,,).

4.10.

( R i ) ,1 < p < 03,

Qualitative Properties of Embedding Operators

The width numbers for embedding operators in Sobolev-Slobodeckij spaces with weights are estimated in Section 3.8. In this section embedding operators in Lebesgue-Besov spaces (and hence as special cases also in Sobolev-Slobodeckij spaces) without weights in domains are considered. We are concerned with approximation numbers, width numbers and &-entropy.Although there remain open a number of problems, one obtains far-reaching results on the basis of the interpolation theorems of Section 1.16, the scale properties of Theorem 4.9.2, and the duality theory of Section 4.8.

4.10.1.

Embedding Operators in Bounded Domains of Cone-Type (Approximation Numbers, Width Numbers)

The width numbers d j and d j , and the approximation numbers sj have the same meaning as in Definition 1.16.1/2. Theorem. Let IR c R, be a bounded domain of cone-type. 1 (a) Let 1 < < 00, 0 6 s < -, and s < t < 03. Then it holds for the embedding

operator I that

P

sj(I; wL(Q), W;(Q)

-

t-a

dj(1; wp(Q), w;(Q)) jy,

(1)

N

t-8

d j ( 1 ; Wt,(Q), W;(Q)) 6 cj-", (2) where c is a suitable positive namber. Here j = 1, 2, . One can r e p h e W;@) with s > 0 in (1) anal (2) by Bi,p(Q) with 1 5 e 5 p . n n 1 Further, let (b) Let 1 < p , q < 00, 0 < s <-, s < t < 03, and s - - < t 13 P P 1 5 u, e 6 00. Then

..

-.

4.10.1. Embedding Operators in Bounded Domains of Cone-Tspe

d j ( I ; Bi,cr(Q),B;,,(Q)) I cj-" if q 2 P Replacing B;JQ) by LP(Q),then (3) remains valid if s = O . 1 n (c) Let 1 < q 5 p < co; 0 I s < - ; s < t < co and s - - < t ?,

t-8

1

1

?,

345

(3b)

--.nq

Then

& ( I ; W;(Q), W;(Q)) 5 sj(. . .) 5 C? n q P . (4) If 8 > 0 then one can replace Wi(Q) by BiJQ), where 1 5 p co. Proof. Step 1 . Theorem 4.2.3 and Theorem 1.16.111 yield that one can restrict oneself to the case Q = Q = {x I 0 < xk < 1, k = 1, . . .,n } . We prove (1) and (2). By Lemma 1.16.1/2, it is sufficient to show that .--t-

5 cj"

aj

t-s

and dj

t -8

2 c ' j n wherec' > 0.

t-8

we use the same ideas as in the second and in the Step 2. To prove si cj-" third step of the proof of Theorem 3.8.1. As in the second step of the proof of Theorem 3.2.6, it follows by Theorem 4.3.112 and Theorem 1.16.412 that the embedding from Wb(Q)into Wi(Q)is compact. Now, similarly t o (3.8.1/1),oneobtainsfor t =k integer

and a corresponding expression for t = integer. Now, let f E Wp(Q)such that ~ ~ / ~ ~5w1.; If ~ the Q ) cubes qLhave the same meaning as in the third step of the proof of Theorem 3.8.1, then one determines in q1a polynomial PI of degree [t]- such that

J

DP(f - PI) dx

=

0 for 0

5

I [t]-.

QI

The coefficients of Pz(x) depend linearly on f. Now, W i ( q l )is normed similarly t o (5), where the last sum is replaced by IlfIIL,,(ql). Then it follows that

where c > 0 is independent of k . (For t = integer one has t o modify in the usual way.) Setting P(x)= Pl(x)for x E q l , 1 = 1, . . . , 2 h , then

s

QxQ

I f ( 4 - P(X)- f(Y) 1 2

+ P(Y)lP & d y

- yp+sp

346

4.10. Qualitative Properties of Embedding Operators

If d,(x) denotes the distance of a point x mate the second sum by

c Jlf(x) - P l ( x ) pd;"(x)

E q1 to

the boundary a q l , then one can esti-

2kn

c

(8)

ax.

1 = i QI

It follows from the proof of Lemma 3.2.6/1(b)that (3.2.6/2)is also valid for Q = qc and e(x) = dil(x). Applying (3.2.6/2)to f - P 1 ,then one can omit (in the sense of ( 5 ) )the second summand on the right-hand side of (3.2.6/2). Whence it follows that one can estimate (8) by the first summand on the right-hand side of (7).Now, (6) yields

SL.zkn(1;

Wk(Q), W;(Q)) 5 C2-k(t-s).

Here L is the number of the linearly independent polynomials of degree [ t ] - . Now, one obtains that t-s

j = 1,2,.. .

s j ( I ; W',(Q), W;(Q))5 c j - 7 ,

(9)

Step 3. Let t = 1 , 2 , . . . Similarly to the fourth step of the proof of Theorem 3.8.1, we show that t-s

d j ( I ; W",Q), Wg(Q))2 c j - - i t , j = 1,2,.. ., c > 0. For this purpose, we construct functions y l ( x )E C 2 ( q l )such that

-

J

ll~~llwr;~Ql) = 1, l l ~ ~ l l ~ .2-'c(f-s), ~ ( ~ ~ )and

y&) dx

0.

(11)

Q1

(All the symbols have the same meaning as in the second step.) Let quence of complex numbers such that

=

(10)

2 h

1 IcxlIp

1=1

=

{ ~ r ~ ) lbe = ~a 2kn

. 2kn

1, and let y ( x ) =

Then llyllw;(Q)= 1 holds. The considerations of the second step yield

Whence it follows (lo),similarly to (3.8.1/10).

Step 4 . Hitherto, we proved (1)and ( 2 ) for t

=

1, 2 , . . . Let again 0

C

2=1

se-

alyl(z).

1

5 s < -. P

Further, let m = 1 , 2 , . . . and N > m. Setting A , = B = W;(sZ), A , = W;(Q) and A = WF(Q),then (10)for arbitrary t > s is a consequence of Theorem 1.16.3/1(b). This proves (1)and ( 2 ) . 1 Step 5. Let 1 < p < co, 0 so < s < s, < t , and s1 < -. Then i t follows by

s

P

Theorem 4.3.1/2and (1.3.3/5) that

IlfllB;@(Q, 5 CllflIl:(Q)

1-8

e

ll/ll,,.p)

3

f

E

Wi(Qn)

7

(13)

where s = (1 - 0) so + Osl and 1 5 e 5 00. Assuming that P ( x ) has the same meaning as in the second step, and replacing f in (13)by f - P, then one obtains in the same manner as above

4.10.1 Embedding Operators in Bounded Domains of Cone-Type

347

Whence it follows that (14) is also true for d, or dJ instead of 8,. Since Bi,,(Q)c Wi(Q) for 1 2 e p , one obtains ( 1 ) for B;,,(Q), 1 5 Q 5 p , instead of W;(Q). Th‘IS proves

s

(8).

Step 6 . We prove (3) and (4). The explicit norms for the Besov spaces described in Theorem 4.4.212 yield

B;*,,(Q)= B;,,,(Q)

for

1 <

q1

s Po <

(15)

*

s

Hence, for the proof of (3), one can restrict oneself t o q p . For p = q, (4)is contained in (2), while one obtains (3) with p = q by part (a) and Theorem 1.16.3/1(a), similarly to the fourth stcp. Let q < p . Theorem 4.6.1 and (4.6.1/8)yield t-LL+?

B;,,(Q) c B,,,,Q

(Q),

W#2)

n

c

n

W:,g+p(Q).

Now one obtains (3) and (4) from the above results, (1.16.1/26), and a corresponding formula for d J . R e m a r k 1. * Ford,, ( 1 ) has been proved by A. E L KOLLI[l, 31. The inclusion properties (4.6.1/1) and (4.6.1/2) show that one can replace in (3) Bi,,(Q) by and/or Bi,,(Q)by Wi(f2). Further, (3) remains true for s = 0, if one replaces B;&2) by L,(Q). This case, with W@2) instead of B i , J Q ) ,was treated by M. 8. BIRMAN, M. Z. SOLOMJAK [l,21. I n these papers, the technique of piecewise approximation by polynomials, which played an important role in the proof of the above theorem and in the proof of Theorem 3.8.1, is also developed. (A. EL KOLLIuses also this method.) Further references can be found in Subsection 4.10.2.

wq(Q)

R e m a r k 2. Formula (4) may be improved by a stronger use of Theorem 4.6.1. But we do not go into detail, since we shall obtain essentially sharper results in the following subsection. R e m a r k 3. In connection with this theorem there arise a number of problems. 1 (a) The restriction 0 s < - is based on the method. I n the next subsection, we

P

shall see that (3) and (4) are valid for bounded C”-domains and for - 03 < s n and s - - < t

P

n

<$

t

< co

- - . (b) I n contrast to ( l ) ,the formulas ( Z ) , (3), and (4) contain

only estimates, but not asymptotic assertions. For bounded Cw-domains we shall see in the following subsection that one may replace 5 in (2) by -. The corresponding problem for (3) and (4) seems to be more complicated. We shall return to this question in Remark 4.10.2/3. R e m a r k 4. The Theorem is the counterpart to Theorem 3.8.3. As in the first step of the proof of that theorem, it follows that there does not exists a continuous embedding from W i ( Q ) into L,,(Q) for 0 < t < co, 1 t n P 4 (4.6.1/8) yields that, for - - = t - - , there exists a continuous embedding from

n

P

P

348

4.10. Qualitative Properties of Embedding Operators

Wi(Q) into LJQ). Using again the functions of the first step of the proof of Theorem 3.8.3, one obtains that the embedding is not compact in this case. With the aid of 1 (12), one can prove corresponding results for W;(Q), 0 < s < - , instead of LJQ).

r,

Further results of such a type can be obtained by Theorem 4.6.1, Theorem 4.3.112, and the consideration in 1.16.4. If Q is a bounded C"-domain, then one can also use Theorem 4.9.2. We do not go into detail. R e m a r k 5. I n Remark 1.16.1/3, we mentioned the axiomatic theory of general [7, 81. I n connection with (1) it would be of approximation numbers by A. PIETSCH interest t o prove (1) for the isomorphism numbers, mentioned there. This would have the consequence, that (1) is true for all general approximation numbers (in the framework of t h k theory).

4.10.2.

Embedding Operators in Bounded C"-Domains (Approximation Numbers, Width Numbers)

Theorem. Let Q c R, be a bounded Cm-domain. (a) Let l < p , q < c o , 1

n n and s - - < t - - . P q

-co<s
6 (T, e 5 co. Then

. .)

d , ( I ; B ~ , ~ ( B;,JQ)) Q), 5 s,(.

1

t-s

1

2 c j - T + r - T if q

Further, let

p,

t 6

s

d,(I; Bi,&2), B;,#2)) 6 sj(. . .) cj" if q 2 P These formulasremain valid after replacing dj(. . .) by d j ( . .). (b) Let 1 < p <

00,

-co < s < t < co, 1 5

d,(I; BL,,(Q), B;,,(Q))

N

s,(. . .)

Here one can replace B;,JQ) by H;(Q). (c) Let 1 < p < 00, - m < s < t < 00, p d j ( I ; B;,,(Q), B;,,(Q))

.

(T

-

5

and 1

s e 5 p . Then

t-a

j-".

2 5 (T

00,

and 1 5

00,

e

s co. Then

t-s N

j-7.

Here one can replace Bb,,(Q) by H i @ ) . n (d)Letl
d , ( I ; B:,@), Cs(Q))

5

d j ( 1 ; Bi,,(Q),C S ( Q ) )

5 cj-"+Q.

t-8

1

s cj-n+q, I-8

1

Proof. Step 1. Let 1 < p < co and 0 < s < t <

-.1 P

Since the embedding operator

349

4.10.2. Embedding Operators in Bounded Ca,-Domains

I is self-adjoint in the sense of the duality theory of Section 4.8, it follows by Theorem 4.10.l(c), Lemma 1.16.1/3, Theorem 4.8.1, and Theorem 4.3.2/1 that t-8

s,(I; BpSp,(Q),Bpf,p,(Q))5 c j - 7 ,

1

5 e < 03.

(6)

Suppose that the polynomials P l ( x )and the cubes q1 have the same meaning as in the second step of the proof of Theorem 4.10.1 (or as in the proof of Theorem 3.8.1, respectively). Setting Q, = q, , then the operator P from tbe second step of the proof of Theorem 4.10.1 coincides with the operator Pz from (4.9.4/5).Pz is self-adjoint. Now, the second step of the proof of Lemma 1.16.1/3 shows that one may use this operator in the approximation of the embedding operator I(B;?,,,(Q),B;! @)) in the sense of (6). Similarly to the fifth step of the proof of Theorem 4.10.1, ig follows s j ( I ; BP?,~,(Q), B$,,,,(Q)) 6 cj-“,

1-8

1

5e6

1

03,

5 or 5

(7)

00.

Here, e = 00 is an admissible value, too. Starting with (7)and repeating the above considerations, one obtains that

Step 2. Theorem 4.3.2/1 yields B;,,(Q)

=

B;,,(f2),

1

03,

0

1

5 s <-, I,

1

5

@

< 00.

(9)

Now it follows from the scale properties of Theorem 4.9.2 and Remark 4.9.2/3 that 1 (8) remains valid for 0 5 s < t < s + - . Similarly, one obtains with the aid of (9) I,

1-s s j ( I ;Bb,,(Q), Bp,,(Q)) 5 cj n , 1

se<

00,

1

5 r~ < 00.

Now, using Theorem 4.8.1, then it follows that (8) also holds for t - 1

< t 5 0 and 1 < e

s

s

(10) 1

+-< s P

1
-

2,

.

But this is a consequence of (4.10.1/1).Using (4.10.1/1) with s = 0, then it follows that (2) is also true for H i ( Q ) instead of B;,,(SZ).

350

4.10. Qualitative Properties of Embedding Operators

Step 5 . Using Theorem 1.16.3/2, then it follows that for the proof of (3) it is sufficient t o prove (11) with dJ instead of d J . This formula is a consequence of Lemma 1.16.1/3, formula (2),and Theorem 4.8.1 (in relation with Theorem 4.3.2/1). Similarly, it follows that one may replace B;,,(SZ) by H;(Q). Step 6. One obtains (4) and (5) from part (a) with e = 1, (4.6.1/7), (1.16.1/25), and corresponding formulas for dJ and sj . R e m a r k 1. The inclusion properties (4.6.1/1) and (4.6.1/2) show that one can replace in (1) (and the corresponding formulas for d J ) , (2), (3), (4),and (5) B:,,(SZ) (resp. BL,,(Q))by H;(Q) (resp. H>(SZ)),and B;,$2) by H;(Q). Oneobtains thesharpest assertions contained in (1)) (4), and (5) if CT = co and e = 1. We note some interesting special cases. (1)-(5) yield : q 1 ; W i ( Q ) ,W i ( Q ) )5

(a) for 0

5 s < t < co, 1 < q 5 p <

SJ(.

00,

. .) 5 c j

1 JJ

(12)

?L

1L

P

! I

and s - - < t - -,

formula for dj instead of d., ,

and a corresponding

t -a

d J ( I ; W',(S), W ; ( Q ) ) 5 s j ( . . .) 6 c j - y (13) s < t < 03, 1 < p 2 q < co, and a corresponding formula for dJ instead of

(b) for 0

5

for0

5 s < t < 0 3 , l< p < 00, t-* 1 . .) 5 cj-'E +-q _

d j ( I ; W i ( Q ) ,Cu(Q)) for 1 < p < co and 0

4

1

4

l-Y

s

sj(.

"

n

< t - -, and a corresponding formula for dJ instead of q

R e m a r k 2. * The first result of this type is due t o A. N. KOLMOGOROV [l]. He proved (14) for the widths d, and for the parameters s = 0, t = 1 , 2 , . . . , p = 2, and the domain SZ = (0, 1). These investigations are extended by many authors, where the width numbers dl and dJ are determined or estimated for embedding operators in Sobolev-Slobodeckij spaces W;(Q),s 2 0, 1 5 p < co, and in Holder asymptotic spaces e'(SZ).With the aid of the methods given by A. N. KOLMOOOROV, formulas for dJ from (14) with p = 2,O 5 s < t < co, s and t integers, are obtained [l,2,3]. (In Remark 1.16.1/1 we mentioned that the numbers d,, by J. W. JEROME d j , and sJ are equal in separable Hilbert spaces.) S . B. BABAD~ANOV, V. M. TICHOMIROV [l] proved (14) for SZ = (0, l), s = 0, and t = 1 with respect t o d , and d l . Formula (14) for d, and sl , where 52 = (0, l ) ,and t and s are integers, can be found in H. WALEK[2]. Further, V. M. TICHOMIROV [ l , 21 has proved that

-

-

t

d,(I; 6 ( S Z ) , @(Q)) O(Q),@(Q)) j-". (16) This formula is not contained in the above theorem (and its special cases) See also [2]. Extensions of the asymptotic formulas (16) for dJ to generalized G. G. LORENTZ [l], G. G.LORENTZ[2], and Hiilder spaces can be found in V.M.TICHOMIROV

35 1

4.10.2. Embedding Operators in Bounded Cm-Domains

J. W. JEROME, L. L. SCHUMAKER [l]. Formula (14) for the widths dj was proved by A. EL KOLLI[l, 31. Special cases of the general formula ( 2 ) for dj go back to H.TRIEBEL [15,27]. R e m a r k 3. * The considerations of Remark 2 are wncerned with the asymptotic behaviour of the width numbers for the embedding of spaces of the same type (Lpspaces, C-spaces). To obtain corresponding asymptotic formulas for the embedding of spaces of different types is essentially more complicated. Although there exists a number of partial results, a final clarification of this problem seems to be open. All the assertions of the theorem concerned with the general case ((l),(4), and ( 5 ) , as well as (12), (13), and (15)) contain only estimates from above. (12) and (13) with s = 0 are proved in &I. 8. BIRMAY,M. z. SOLOMJAK [I, 21 (see Remark 4.10.1/1). (15) for s = 0 and with respect to the width d.; can be found in M. Z. SOLOMJAK, V. M. TICHOMIROV [l]. See also J. W'. JEROME, L. L. SCHUMAKER [l]. Generalizations of these results are proved in H. TRIEBEL[15,27]. The paper written by M. Z. SOLOMJAK, V. M. TICHOMIROV [l] contains also the asymptotic formula d i ( I ; W p 2 ) ,c"(Q))

- p+7, t

1

00

> p 2 2,

t

n

- -> 0. P

I n the same paper, estimates of dJ in the case 1 < p < 2 are derived. By (1.16.1/6), one may replace Eo(Q) in (17) by L,(Q). Then one obtains as a special case

-

1

d / ( I ;W ; ( ( O ,111, L,((O, 1))) j-F. On the other hand, R. S. ISMAGILOV [21 proved that

(18)

3

wi((O> 'I), L m ( ( o ,

'))) 2 'i-"**) (19) These two formulas show that the widths dj and dJ can have an essentially different behaviour. One can derive some conclusions from the result found by R. 8. ISMAOILOV. SinceL,((O, 1)) is continuously embedded in Lp((O,l)),1 - + - , that means 03 > p > 10. (20) 'J(';

5

2

1

,

Using Theorem 1.16.1/2 and (14) for p = 2 and n = 1, then i t follows that d J ( I ;Hk((0, I)), L,((O, 1 ) ) ) 1

.A+-

ci

a

5

for t

1

2 1.

The embedding from H: P ( ( 0 ,1)) into Lp((0, 1)) is continuous, 2 from the interpolation couple

(21)

6p <

CO. Starting

and using Theorem 1.16.3/1 and (19), then one obtains a similar formula for-

1

2

1

-P

< t < 1. I n both cases, one gets an improvement of (12) for p > 10. Using the scale *) E. D. GLUSKIN[l] proved that

c0((O, 1)))

d,(I; IY:((O, l ) ) ,

N

if-'if I ? 2.

4.10. Qualitative Properties of Embedding Operators

352

properties of Theorem 4.9.2 (in relation with the interpolation properties of the spaces

H t and Theorem 1.16.1/2), then one may replace (21) by -(&a)+-

a

5 d j ( I ; H:((O, I)),H;((O, 1 ) ) ) 6 cj (22) for -co < s < t - 1 < m.*) (A corresponding formula holds for 0 < t - s 6 1.) Using Lemma 1.16.1/3 and the duality theorems of Section 4.8 in the same manner as above, then one obtains that .-(t-8)+-

for - 01)

a

d J ( C H i ( ( 0 , 111, H W , 1))) 6 C? (23) < s < t - 1 < co. Comparison with (12) where one has to replace d, by

d j shows that (23) is an improvement for 1 < p

1

one obtains also a n improvement for 1 < q < -. Using the theorems of Sublo 9 section 1.16.1, the duality theorems of Section 4.8 and the scale properties of Theorem 4.9.2, then one can obtain further results. For instance, starting with the interpolation couple { L p ,W i } ,then one can prove formulas of the type (21) and (22) with H i ( ( 0 , 1)) instead of Hk((0, l)),2 6 q < p < co. Similarly for (23). This shows that also in these cases the right-hand side of (12) can be improved. We omit explicit formulas, aince one cannot expect to obtain in this way exact asymptotic growth for d, and d j . But we want t o mention some other papers concerned with problems [l, 21 proved, that considered here. R. S. ISMAGILOV 1 1 1 6 q S p 6 2 ; - - - < t = integer. P P For the approximation numbers s j , this result is extended by R. SCHOLZ [l, 21 t o the n-dimensional case and to 1 6 q 5 p 6 a.For the special case (24),formula (12) is an asymptotic formula, IR = (0, l ) , n = 1, t is an integer, s = 0. Using again the above methods (Theorem 1.16.3/1(b), Theorem 4.9.2), then one may generalize (24) essentially. It holds that

\

q 1 ;H i ( ( 0 , l)),H i ( ( 0 , 1)))

-

1

1

j-(f-s)+T-T,

l < q S p 6 2 , --co<s s - - . Here one can replace H i ( ( 0 , l ) ) by Bi,o((O,1)) where 16 cr 5 00. 1

q

P

Using the duality theorems of Section 4.8 and Theorem 1.16.3/2(b)in the same manner as above, then (25) yields

dj(l;H;((oy1)), H;((O, 1))) *) kE((O,l)),where e >

--1

(I'

and p

-i

-( t - s ;+---1 Q

1 p,

2 5 q 6 1, < a ,

(26)

1 -+ integer, is a closed subspace of H:((O, 1 ) ) (Theorem U

4.3.2/1) with finite codimension. Hence, for these values, one can replace H in (21), (22), and (33) by l?.

4.10.2. Embedding Operators in Bounded Cw-Domains

t

1

1

- - > 8 - - . Here one canreplaceHi((0, 1)) by B&((O, 1)) where 1 q P

353

e 5 co.

The question arises whether (24) remains valid if p (and q ) become larger than 2. R. S. ISMAGILOV [2] proved the somewhat weaker result,

s,(I; Wt,((O, l)),L,((O, 1))) 1

1

q

P

1

N

1

j-t+T-T, 2

5q2p 5

00 )

t > - - - . See also (21). Ju. I. MAKOVOZ[l] showed that

wi((o?

'1))

wi((O> 'I)?

'))) j-'? (27) 1 5 p 5 q 6 00. On the basis of the above results, one may generalize this result essentially. It holds t h a t dJ(l;

Lp((o,

dJ(l;

Lp((o,

-

d,(I; H;W, I)),H;((O, 1))) d l ( C H;((O, 1)))H;((O, 1))) Pt-), (28) - co < s < t < 00, and 1 < p 5 q < 00. See (13) and (1 b) (and the corresponding formula for d J ) . (Whence it follows also an asymptotic formula for the corresponding [l],where numbers sJ .) Finally, we mention the paper written by N. P. KORNEJ~UK the widths d, are determined for the embedding operators from (generalized) Holder spaces into L p . R e m a r k 4. * I n Theorem 1.16.1/1, we proved that the width ideals K, and G,, and the approximation ideals S, are &-ideals. One can generalize the previous considerations asking whether the embedding operators between Lebesgue-Besov spaces (and hence also between Sobolev-Slobodeckij spaces) belong t o given Q-ideals. Of course, in the first place one will be interested in concrete &-ideals which can be described in an easy way. In the next subsection, we shall be concerned with the important class of entropy ideals. Here we quote some papers containing results of such a type. The first results on the qualitative behaviour of embedding operators is dur to K. MAURIN[l]. He proved that for bounded domains 52 the embedding operator from Wg(52) into L a ( 0 )is a Hilbert-Schmidt operator, provided that Tl

m > - holds. The Hilbert-Schmidt operators are the operators belonging 2 to G2 described in Subsection 1.19.7. This result was generalized by B. GRAMSCH [l], J. KADLEC,V. B. KOROTKOV [l], PHAM THE LAI [2,4] and H. TRIEBEL[4], determining the conditions under which the embedding operators from Wi(52)into Wi(52) belong t o the classes 6, and Gp,qof Subsection 1.19.7. The obtained results are contained in (14) with = 2. I n the paper by J. KADLEC,V. B. KOROTKOV [l], anisotropic spaces are also considered. A generalization of the above results (in the framework of Hilbert space theory) t o Sobolev-Besov spaces with weights can be found in H. TRIEBEL[12]. I n the paper by C. CLARK[l], conditions are given under which the embedding operator from Whn(Q) into Wi(S), where 52 is an unbounded domain, is a Hilbert-Schmidt operator. Corresponding results with respect to the E. P. CATclasses SPand Gp,mare contained in the papers written by P. J. ARANDA, TANEO [l] and J.-M. AUDRIN,PHAM THE LAI [l]. The classes G p and Gp,qcan be extended from Hilbert spaces t o Banach spaces such that one obtains Q-ideals. See [l]. In the papers Subsection 1.19.8, A. PIETSCH,H. TRIEBEL[l], and D. FREITAG [l], and H. KONIG [l] by A. PIETSCH, H. TRIEBEL[l], H. TRIEBEL[S], D. FREITAG

integer

23

=

Tnebel, Interpolation

354

4.10. Qualitative Properties of Embedding Operators

conditions are formulated under which embedding operators between (general) Lebesgue-Besov spaces belong t o these Q-ideals. A further important class of compact operators is the Q-ideal N , of all nuclear operators. An assertion whether embedding operators between Sobolev-Slobodeckij spaces belong to N , can be found in H. TRIEBEL[l].A sharper version of these results is due to A. PIETSCH [6]. A. PIETSCH, He proved that for bounded domains with smooth boundary the embedding operator from W&Q) into L,,(Q)is nuclear, provided that

t>n

l
if

If t < n 1 + - - ( Q

for co > q z p > 1 o r i f t < n f o r 1 < q $ p < 0 0 , then lP ) the corresponding embedding operators are not nuclear. It is of interest that there is a difference between the distinction of the cases in (29) on the one hand and in (12), (13), (26), (28)on the other hand. A. PIETSCH [6] is also concerned in this connection with the Q-ideals N , of r-nuclear operators and P, of absolutely-r-summing opera[l]. tors. See also s. V. K~SLJAKOV

R e m a r k 5. * With very few exceptions all the papers quoted in the last remarks are concerned with Lebesgue-Besov (Sobolev-Slobodeckij) spaces in bounded domains. For unbounded domains, there arise new problems. Let 9 c R, be a domain containing a n infinite set of pairwise disjunctive congruent balls K j .Then it is clear that the embedding between Lebesgue-Besov spaces (if there is any) cannot be compact. (To understand this, one has only to consider a family of functions v, E C g ( K i ) , yj(x) 0, generated by translation form a standard function.) Further, there exist unbounded domains 9 for which the embedding from WF(S), m = 1,2,..., l < p < 00, into LJQ) is compact but the embedding from W;(Q) into L p ( 9 )is not compact. The investigations on the compactness of the embedding from W r ( 9 )into L,(S) (and more genera1 into L q ( 9 ) )for unbounded domains begin with the paper by C. CLARK [2]. These results are extended by C. CLARK,R. A. ADAMS,and J. FOURNIER. In particular, we refer to R. A. ADAMS [1,3]. Necessary and sufficient conditions with respect to the domain 9 for the compactness of the embedding from W F ( 9 )into L,(S) are given by R. A. ADAMS[2]. Investigations on the compactness J. FOURNIER of the embedding from W r ( 9 )into L J 9 ) can be found in R. A. ADAMS, [l, 21. Necessary and sufficient conditions for the compactness of the embedding n n from H;(Q) into L q ( 9 )s, - - > - - , q 2 p , are given by M. S. BEKGER,

+

I)

u 1

M. SCHECHTER [l], Theorem 2.8. Qualitative characterizations of embedding operators in unbounded domains by approximation numbers, width numbers and so on in the same extent as for bounded domains seem to be a n open problem. For t,he case p = 2, we refer t o the just mentioned papers by C. CLARK [l], P. J. ARANDA, E. P. CATTANEO[l], and J.-M. AUDRIN,PHAM THE LAI [l]. Closely related to these considerations there is the question whether embedding operators in bounded domains with non-smooth boundaries are compact. We refer t o V. G. MAZ’JA[l] and the papers quoted theye.

4.10.3. Embedding Operators in Bounded Domains (e-Ent,ropy)

4.10.3.

355

Embedding Operators in Bounded Domains (r-Entropy)

Besides the width numbers, approximation numbers, and Q-ideals mentioned in Remark 4.10.2/4, the e-entropy is also usable for the qualitative characterization of embedding operators. The e-entropy was introduced in Definition 1.16.1/2. As before embedding operators are denoted by I. F'urther, we shall formulate the results only for bounded Cm-domains, although some statements are &o true for bounded domains of cone-type. It is not the aim of this subsection to give a systematic survey on the known results. We shall formulate two fundamental results due to M. 8. BIRMAN, M. Z. SOLOMJAK [l, 21 and G. F. CLEMENTS[Z] without proof, which are the basis for the further considerations. Lemma. Let SZ c R, be a bounded Co"-domain. n

@ ) I f<1p , q < c o , t - - - > q

n

--,aidt>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 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 

4

n

--. P

Let l 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 <

co,t

-

n

-- I1

(7)

E 1-8,

--> 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 <

Besov spaces

Further, similarly to (2.4.2/14),one can define “periodic”

00.

W&

(3) 1 q 00. I n this way, it seems t o be possible t o develop a systematic theory of these spaces in the extent of Chapter 2. The theory of equivalent norms in the spaces B”pq,,,can be found in 0. V. BESOV[2] and M. K. POTAPOV [l] (at least for the one-dimensional case). If one wants to carry over the methods of Chapter 2, then one has to replace the niultiplier theorems of Subsection 2.2.4 by the multiplier theorem of J. MARCINKIEWICZ [2] and S. G. MICHLIN [3, Appendix] for multiple Fourier series. If B;,q.n =

0

< 8 < 1,

-00

3

H&)e,q

< s0,s1 <

00,

3

s = (1 - 8 ) s o

+ 8sl, 1 ,

B;*q(c?) 1

and 0 < s < -. Thisrelation is useful for some prob-

P

lems (e.g. for the proof of the existence of Schauder bases in B;,q(Q)),see H. TRIEBEL [23].

4.11.2.

Spaces with Dominating Mixed Derivatives

I n Theorem 4.2.4 and Remark 4.2.411, we were concerned with the problem whether derivatives Dpf can be estimated in Lp-norms by given derivatives or polynomials of derivat’ives, (4.2.415).One may generalize this problem essentially considering spaces having dominating mixed derivatives. Let 1 = (Z1, . . ., l,,) where lj are natural numbers. S. M. NIKOL‘SKIJ [6] introduced spaces 8;W of Sobolev type, where the norms are det,ermined by Ilf llSilV

=

C

I(, = e,zj

llD”fllL9(R,,),

k

=

(ki,.

-

ku)-

Here 0, has only the two values 0 or 1. Similarly, one may define spaces of Lebesgue type or Besov type. The theory of these spaces is developed by P. I. LIZORKIN, S. M. NIKOL’SKIJ[l], A. D. D~ABRAILOV [l,31, T. I. AMANOV [2], A. P. UNINSKIJ [l], 0. V. BESOV[7], 0. V. BESOV,A. D. DBABRAILOV [ l ] and other mathematicians. An interpolation theory for these spaces can be found in P. GRISVARD[4]. Further, the theory of interpolation-n-tuples sketched in Subsection 1.19.10 is useful for the development of an interpolation theory for these spaces. We refer to A. YOSHIKAWA [2] and G. SPARR [l].

4.11.4. The Spaces of L. HOBMANDER and L. R. VOLEVI~, B. P. PANEJACH

4.11.3.

359

Spaces of Abstract Functions

Let A be a Banach space and 9 c R,, be a bounded domain. On the measurable subsets of Q, there are considered additive set functions v having values in A . If 1 1 1 < p < co and - + -T = 1, then @ J A , Q) denotes the set of all elements q such

P

that

P

(u

1144II L p 4 ,

Here the supremum is to take over all (real-valued or complex-valued) simple functions. (A function is said to be a simple function if it is a finite step function LO = C G,;c~,(x) where c, are constants and x ( 2 ) are the characteristic functions of *j measurable subsets Qj of Q.) !Pp(A,Q) is a subset of QP(A,Q) containing all elements = D%p which are continuous by translations with respect t o the above norm. E Y J A , 52) is said t o be the distribution derivative of v E @JA, Q) in the sense of vector-valued distributions if for all test functions w

J D“w d,v

R

= (-

1

l)lal w(2) d,va. R

!P?(A, Q) is the set of all functions v E @ J A , Q) whose distribution derivations of the order m belong t o !Pp(A,Q). I n this way, one receives the abstract counterpart t o the Sobolev spaces WF(Q). The spaces Y F ( A ,Q) are introduced by S. L. SOBOLEV [5,6, 71. In the papers by V. B. KOROTKOV [l,2 , 3 , 4 , 5 ] , Ju. I. GIL’DERMAN [l, 2,3], and other mathematicians, the abstract counterparts of the Slobodeckij spaces and other spaces considered before are introduced. There can also be found embedding theorems for these spaces. We do not go into detail here. An interpolation theory for these spaces does not seem t o exist.

4.11.4.

The Spaces of L. Hiirmander and L. R. Volavic, B. P. Panejach

In (2.3.3/1), we described equivalent norms for the spaces H;(RJ. L. R. VOLEVIO, B. P. PAXEJACH [l] (Appendix 2) generalized this approach t o the spaces H;(R,,) and introduced the spaces

HprJ

=

(f I f E s’(Rn),llfllH;

=

IIp-1p(4PflILp < a> 9

(1)

1 < p < w. Here, p ( x ) is a weight function satisfying some additional conditions. The rase 11 = 2 is of special interest. In this case, it is sufficient t o assume that p(x) is a positive continuous function and that there exist two positive numbers C and N such that p(z

+ ?A 2 (1 + ClxOJ P(Y)

(21

360

4.11. Complements

for all x E R,, and y E R,L.The spaces H$(R,,) coincide with spaces introduced by L. HORMANDER [3]. Further, L. HORMANDER [3] sets (3) Bg(Rn) = {f I f E S’(RnI7 IIfIIB; = IIP(~)F f I l ~ p< a>. L. HORMANDER and also L. R. VOLEVI~,B. P. PANEJACH define corresponding spaces over domains. An interpolation theory for the Hiirmander spaces can be found [5,6], see also A. FAVINI [5]. in M. SCHECHTER

4.11.5.

Spaces with Variable Order of Differentiat,ion

Let @(X) =

+ c(x)

be a real-valued function; s is constant, a(x) belongs t o S(R,). Consider the pseudodifferential operator Ae ,

(nef)(x)= b-l[(l+ IW)+(F~) d I ( x ) , f E s’(R,). For smooth functions (e.g. f E S(R,))this can be written in the form e(4

(A,/)(2)= (27zl-T J ” e i < ~ ~+~ )1 (~l 1 I1

fL

(F/) (6)a t .

2 ) T

Then A. UNTERBERGER [l, 21 and B. BEAUZAMY [1, 21 are concerned with the spaces

H$)(Rn)= {f I f E S ‘ ( R ~ ) , E Lp(Rn)} where 1 < p < co.Clearly, for a(s) = 0 one obtains the spaces Hj(Rn)from (2.3.3/1). But generally, the order of differentiation expressed by e(x)is variable from point t o introduced also spaces of Besov type with variable order of differpoint. B. BEAUZAMY entiation.

‘

5.

REGULAR ELLIPTIC DIFFERENTIAL OPERATORS

5.1.

Introduction

I n the previous chapters, the interpolation theory and the theory of the LebesgueBesov spaces have been developed. In the following chapters, the obtained results will be used for investigations on elliptic differential operators. The main aim is the description of mapping properties between function spaces. While the Chapters 6 and 7 are concerned with degenerate elliptic differential operators, this chapter deals with general regular elliptic differential operators in bounded domains. Section 5.2 contains the needed definitions and some consequences, mostly of preliminary character. I n Section 5.3, a proof of the a-priori-estimate for general regular elliptic differential operators in the framework of L,-theory is given. This is one of the main results of this chapter. Section 5.4 is concerned with the L,-theory for regular elliptic differential operators, both homogeneous and non-homogeneous boundary value problems are treated. Using interpolation theory, we shall obtain in Section 5.5 more general conclusions. Beside the a-priori-estimates, the theorems of this section are the main results of the chapter. I n Section 5.6, some special problems are considered, partly of importance for the later investigations. The final Section 5.7 completes the results of Sections 5.4 and 5.5. Among others, there are considered regular elliptic differential operators in Holder spaces and in Sobolev-Besov spaces with weights.

5.2.

Regular Elliptic Differential Operators

This section contains the basic notations for regular elliptic differential operators and the corresponding boundary value problems. Furthermore, there are proved some conclusions of the definitions needed in the further considerations.

5.2.1.

Definitions

Although some of the needed definitions are given in Subsection 4.9.1, we shall formulate them here again. D e f i n i t i o n 1. Let SZ be either RL or a bounded Cm-domain i n R,,.The differential operator A , Au = C a,(x) D a u , (1) 5 2111

1 0 1

i s said to be properly elliptic if a ( x , 5 ) = C a,(x) P =k 0 for all R,, 3 la1 =

h i

6 =# 0,

and all x

E

0, ( 2 )

362

5.2. Regular Elliptic Differential Operators

and i f for all couples 5 E R, and q E R, of linearly independent vectors and for all x E il the polynomial a ( x , + zq)i n the complex variable t has e m t l y m roots 7: = tlc+(x;5 ; q ) with positive imaginary part (and so also e m t l y m roots ti with negative imaginary part).Here, a,(x) are complex-valued functions, infinitely differentiable i n 0 ;i n the m e Q = R:, a, are constants. Let a + @ ;5

111

Ill

, t)~= I ~7(t - ti), I<= 1

a - ( x ; 5 , q , t) =

I7 (t - t i ) . k=l

R e m a r k 1. This definition coincides essentially with the first part of Definition 4.9.1. A differential operator A ,

Au

=

c aJx) Dau,

14S1

is said t o be elliptic (in Q) if

C a,(x)

la]=Z

+0

for all R,, 3 E $. 0 and all x E Q .

(4)

Here, Q is an arbitrary domain in R,. It is not assumed that 1 is an even number. But it is not hard t o see that 1 must be even, provided that n 2 3. To show this, we fix x E Q and consider two linearly independent vectors 5 E R,,and q E R,. Clearly, the polynomial in the complex variable L has no real roots. Let m be the number of the roots with positive imaginary parts. If 6 is fixed and q is variable (always linearly independent with respect t o t),then the roots are continuous functions of q. Hence, in such a procedure, the number of roots having positive imaginary part is always m . If n 2 3, thenq can be transformed into -7 in this way. Consequently, a,(x) (5 - Lq)”

c

lal=L

has also m roots with positive imaginary part. On the other hand, it is clear that this polynomial has exactly 1 - m roots with positive imaginary part, namely the roots of (5) with negative imaginary part. Hence, m = 1 - m , that means 1 = 2m is a n even number. For n = 2 such a statement is not true. A counter-example is the Cauchy-Riemann differential operator

au + %-., au ax, ax, which is elliptic in the above sense. The lest part of the definition, namely the assumption that the order of the elliptic differential operator is even, 2m, and that m is the exact number of roots of a ( x , 5 + tq)with positive imaginary part, is called the “root-condition”. The above considerations show that for n >= 3 an elliptic differential operator fullfils the root-condition automatically. Finally, we remark that an elliptic differential operator is said t o be uniformly elliptic i n Q if there exists a number c > 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 <

(5)

00.

Using the properties of a+(E',En) described in Lemma 1, then it is not hard to see that

(1

+ IE12)"'2 a+(F,t n ) v(IE'1)

(6) s+ni (1 It12)" satisfies the conditions of the Michlin-Hormander multiplier theorem formulated in

+

(

Remark 2.2.414. We notice that a i ( F )= lFlk a;

( L I ) has the needed differentiabi-

5' + 0. Let f E HF"(R,), (Ff)(t',En) = 0 for 15'1 < 1. Then Ff can be replaced in ( 5 )by ~(15'1) Ff. Now, the multiplier theorem lity properties, since a;(&')is analytic for

)

366

5.2. ltegular Elliptic Differential Operators

yields

To prove the reverse estimate, it will be sufficient to show that x+

“I

is also a multiplier. For 16’1 2 3, it follows from (1) that

+

+

a+([’,t n ) 2 c(15n1 15‘1)” 2 ~ ’ ( 1 l E 1 2 ~ ” , C’ > 0. Using again the differentiability properties of a:(p),one obtains that (7)is a multiplier. This proves the first assertion of the lemma. The second one can be obtained in exactly the same way. a d ( ~ 1(F, ) tn)= o for lt’l < 1. Then Lemma 3. (a)~ e f tE S(R,J,supp c

x,

-

supp F-la+([’,&,) Ff c R:. (b) Let f E S ( R , ) , supp f c supp F-l[a-(E’,

F,and (Ff) 5,)I-l Ff

([I,

[,J = 0 for

lt‘l < 1. Then

c R,.

Proof. We fix 5‘ E R,,-l where 16’1 2 1. Then it follows from (1)that a+(r,z) is an analytic function in the.lower half-plane {z I Im z 5 O}. Furthermore, la+(F,z)l 5 G I Z ~ ’ Now ~ . one can repeat the argumentation given in the first step of the proof of Theorem 2.10.3, where one must replace (iz + (1 + Idla)*)” by a+(E‘,t).T h i ~ proves part (a). Part (b)can be obtained in the same way. 6.2.3.

Regular Elliptic Problems in R,’

Throughout this subsection, A, AU = C aaD@u, Ial = 2 m

denotes a properly elliptic differential operator having only constant top-order coefficients. Further, let 9 = RA in the definitions in Subsection 5.2.1. Let {Bj)Fl, Bju = C b,i,llDdu, j = 1 , . ., m ,

.

tat = mj>

be differential operators having only constant top-order coefficients such that (A, B,, . . ., B,} is regular elliptic in R,+ (Definition 5.2.1/4). We shall prove two lemmas of preliminary character. We use the notations of the last subsection. Let 6 = (El, t 2 , . ., 0) = (p,0) and 11 = (0,. . ., 0 , l ) . Then the complementing condition, Definition 5.2.113, means that the polynomials in t, are linearly independent modulo a+([’,z). Here, b j , l ( p are ) homogeneous polynomials in 5‘ of degree mi - 1. Using the properties of a+(E’,t)described in Lemma 5.2.211, it follows m-1 ”9 C b j , i ( 6 ’ )7‘ = C bj,l(E’)2’ qj(E’, 7 ) a+([’,t) (2) 1=0

1=0

+

3

5.2.3. Regular Elliptic Problems in R:

367

where q,(g, t)is a polynomial in t if 5' is fixed and bj,l ( p )are analytic functions in t',5' 0, such that

+

bj,l(1p) = Ptj-'bj,i(5'), 1 > 0 . (3) We collect some properties of the functions bj,l(l') needed in the further considerations. L e m m a 1. Let (bj,I(['))j=l,...,, be the nzatrix of the functions described dove. 1 =o, .. . , I l l - 1 Then 0det (bj,l(t'))j=l ,..., 0 for E' (4) 1 =o,. . ., - 1 Let (c,,i(t')),=l,. .., be the corresponding inverse matrix, and let v(t)be an infinitely 1 = 0 , . . ., - 1 differentiable function in [O, co)such that ~ ( t =) 0 for 0 5 t 5 3 and v ( t ) = 1 for 1 t < 00. Then 111

111

+

+

111

+

cj,l(l') (1 15'12)*'m~-"~(15'I) is a Michlin-Hormander multiplier in L P ( R n - J ,1 < I, < 03 (Remark 2.2.414). Proof. (2), (5.2.1/11), and the complementing condition show that C d j b j , l ( P )= 0 , 1 = 0 , . . ., m - 1, 5'

j=l

(5)

+ 0,

has only the trivial solution d, = . . . = d , = 0. This is equivalent t o (4). Since bj,I(F)depends analytically on 5' 0, so also the determinant (4) and the elements C ~ , ~ of( Pthe ) inverse matrix. The usual calculation of c,,l(F) (for instance with the help of Cramer's rule) and (3) yield

+

*

Cj,J(AF)= 1-'nj+kj,J(5'), 1 > 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,

<

03},

(1)

5.3.1. The Spaces HF(BL)

369

where 5 = (tl,. . ., 5n-1,5,) = (P, 5,) E R,. Further, HF(R:) is the restriction of H;'(R,) to R:. Remark. For r = 0, we have H Y ( R , ) = H;(R,), Theorem 2.3.3(a).Consequently, H;O(R,') = H;(R,f), Definition 2.9.3. Clearly, HY(R,) and H;'(R:) are Banach spaces. The norm in HY(R,+)is given in the usual way, Lemma. Let A , Au =

C

a,D*u,

1111 = 2 m

be a properly elliptic differential operator having only constant top-order coefficients. Let 1 0 such that for all u E HF(R,f)

-

IlullHgr(R;t) 5 c ( I I A ~ ~ l H ~ - a m * v+ ( R ~~ ~) ~ ~ ~ f f ~ - l * r + l ( R ~ ) ) (2) Proof. Step 1 . First we prove some helpful inequalities. Let j and k be non-negative integers, k 2 j. We shall prove that there exists a number c > 0 such that for all u E HY(R,f)

Here a and r are given real numbers. The left-hand side of (3) can be estimated from above bv

Here we used that glRf = u has the consequence

The definition of Hg-ki'(R,) and the usual multiplier properties of the functions under consideration (Remark 2.2.414) show that (4)can be estimated from above by the right-hand side of (3). This proves (3). In the same way, one obtains that

c

lal=k

and

a+@,.

llFU!lHFr(Rf)

6 cI(UIIH g C k - ' B r + l ( R i )

(5)

..,O, k )

IIF-'(l

+ IEr12)"'z FUlIHg"(R:) = I I U \ I ~ ' + " ( R , f ) ,

(6)

where x is an arbitrary number.

Step 2. We prove (2). First we remark that the lifting properties for To,

+

j 0 h = F-'(it, - (1 1p12)+)uFh, - 0 0 < 0 < 0 0 , (7) described in Remark 2.10.313, hold also (using the same arguments) for the spaces H;'(R:), where r is fixed. In particular,

370

6.3. A-Priori-Estimates

Using the definition of

( 3 ) for k = 2m

- 1, and ( 6 )for x

=

1, i t follows that

Since A is elliptic, the coefficient a(o,..,,0,2m) cannot be zero. Hence,

+

aznlu -- cAu

aztt"

C

...,

c,LPu.

151 4m

a+(O,

0,b)

Putting this in ( 9 ) and using ( 5 ) for k = 2m, we get (2).

A-Priori-Estimates [Part I. R;, constant coefficients, Dirichlet problem]

6.3.2.

L e m m a . Let A ,

Au = la1C aaLPu, = 8m

(1)

be a properly elliptic differential operator having only constant top-order coefficients. Let 1 < p < co. Then there exists a positive number c swh that ~ O Tall u, SUPP u c R,+, ( F U )(P, En) = 0 . . .,E n - 1 , Eft) = ( P , 6,)E R,,, we have

41 E Hy(Rn),

E

=

(€1,

for

IE'I < 1 ,

(2)

(31 Proof. Clearly, f = Au E Him(Rn).Now we use the extension method described in Lemma 2.9.3. Since supp f c it follows that CIIAUllH;"'(R:).

IIUIIH:(R:)

(6f)(2)= f ( z ) IIEfllH;'"(R,)

2m+2

+ jI-:1 bjf(Zf,

(4)

-Ajzn),

5 CllfllH;'"(R;)

7

EflR: =

f*

(5)

Here, bj are suitable coefficients, determined in the proof of Lemma 2.9.3. Since ( F f )(E', it follows that

En) =

(W) (E',

Eft)

( F A u ) ( E f , E n ) = 0 for 15'1 < 1 s

= 0 for

15'1 < 1 .

(6) (7)

The first part of Lemma, 5.2.2/3 yields

z,

[F-la+(E', tn)F u ] ( d ,a,)= 0 for xfZ < 0. (8) Using supp (Au - Gf) c (6), and (7), then one obtains from the second part of Lemma 5.2.2/3 that [F-la+(E',En) F u ] ( x ' , ~ , = ) c[F-l(a-(E',

F A u ] (x',xn)

5.3.2. A-Priori-Estimates [Part I]

371

(7) and Lemma 5.2.212 yield

IIuL(IH r ( R , ) 5 Cll Gfllff;m(RH,)

*

Now, the inequality (3) is a consequence of (5). Theorem. Let A be the properly elliptic differential operator (1). Let 1 < p < ca and s = 0, 1, 2, . . . Then there exists a positive number c such that for all u E HP+*(R,f)

~1 E

HP+**'(R,,),

a& (x',O) = 0 ax;

for

i

= 0,.

. ., m

- 1,

( F u )(tf,En) = 0 for 15'1 < 1 . Here s = 0, 1 , 2 , . . ., and r is a non-negative number. We want t o show that

s

~ ~ ~ ~ ~ U ~ + * CIIAullIf;"'+'*'(R:) * r ( R ~ )

-

(13) (14)

For s = 0 and r = 0 this estimate coincides with (3), because the values of u ( x f ,x,,) for x,, < 0 do not play any role. We prove (14) for s = 0 and r 2 0. Let u be a function satisfying (12) and (13) for s = 0. We set g = P l ( l + Ic$'12)'/z Fu. Then it follows easily that g E H;(R,),

ajg (x',0) = 0 axi

for

j = 0, . . . , m - 1 ,

(Fg) ( t f t,n ) = 0 for IF1 < 1 Using (5.3.1/6) and (3), one obtains that

11 uI1

HFr(R;)

=

llgll H r ( R ; )

s cllAgllIf;"'(fZ:) =

=

cllF-'(l

+ 11f12)"2 FAuIIH;'"(R:)

cIIA~IIH;~.~(R:).

(15)

This proves (14) for s = 0 and r 2 0. We get the general case by induction. Let (14) be true for a fixed s (where s = 0 , 1 , 2 , . .) and all r 2 0. Let u be a function satisfying (12), (13), where s is replaced by s + 1. Then u belongs also t o Hpm+s,r+l(Rn). Lemma 5.3.1 yields

.

IIU"ffT+'+'''(R,+)

Applying (14) we get IIUIIfC+l+'*r(R:)

s c(

+ llU1lH;+lsr+'(R:)).

llAUllH;+l-mBr(Hn+)

5 c( \ ~ ~ ~ ~ ~ I I ~ + ' - m+ oIrl A( 'f~Z~ l~l I)~ ; ; . . ' + ' ( R i ) ) .

But the last summand can be estimated from above by IIAull,y;+1-m.r(R;). (This follows from (5.3.1/6) and (5.3.1/3).)This proves (14). Step 2. Now we drop the restriction (13). Let u be a function satisfying (12), where r = 0. Let rp(t) be an infinitely differentiable function in [0, 00) such that y ( t ) = 1 for 0 5 t 5 1, p(t) = 0 for 2 t < co. 44*

372

5.3. A-Priori-Estimates

We set

u ~ ( x= ) F-'v(IPl) Fu, u ~ ( x= ) F-l(l - ~ ( l t ' l )Fu. ) (16) By a simple application of the multiplier theorem (Remark 2.2.4/4), it follows that uo(x)and ul(z) satisfy (12) for r = 0. Because (13) is fulfilled for ul(x),we may apply (14) for r = 0 t o ul.Hence,

5

~ ~ ' % ~ ~ f f ~ + " (CIIA@i11fI;"+'(R:) R~)

=

CllF-'(l

- v(It'1))FAUIIH;"+'(R:).

Using the same argumentation as in the first step of the proof of Lemma5.3.1, it follows that ll'%IIHi+"(R~) 5 CIIAu~~Elp"'+'(R:). (17)

zD,

For estimating uo(x)we use the lifting property of formula (5.3.1/7), described a t the beginning of the second step of the proof of Lemma 5.3.1, IIuOIIH;+"(R:)

=

IIF-WIPI)F X + ~ U I I L ~ ( R : ) .

I I X + ~ U O I I L ~ (= R:)

Let v ( x ) = ( j 8 +m u) ( x ) for xn > 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<

03.

(b) The kernel N(A,) is independent of p and it holds

N(Ap) c c ? , { B j ) ( Q ) * (3) (c) I t holds the following alternative : either the resolvent set of A , is empty or the spectrum SAPof A , consists of isolated eigenvalues of finite algebraic multiplicity without any finite cluster p i n t . I n the second m e , S A P is independent of p, and the corresponding spaces of associated eigenvectors (and 0) are also independent of p , they are contained in ca”,{BjJ!Q). A finite number of associated eigenvectors belonging to different eigenvalues are lanearly independent. Proof. Step 1. Iterative application of Theorem 5.4.1 yields D(Ak,)c Wimk(Q), where k = 1 , 2 , . . . We remind of the estimate IluIIwp) EIIuIlw;(n) + cellullLp(n), 0 < s < t , (4)

s

where E > 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 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

s q 5 co,and s > 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 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,<}is a projection from H;(SZ) onto Hi,fB,)(Q)and from B;,q(Q)onto B;,q,lBj)(Q). 6.4.6.

The Operators ?In

Beside the homogeneous boundary value problems, Remark 5.4.413, we shall be concerned with non-homogeneous boundary value problems

AU

=

f,

Bjulan = y , , j = 1 , . . ., m .

(1)

It will be shown that one may reduce these problems to homogeneous problems. T h e o r e m . Let Q be a bounded C"-domain. Let A , B,, . . ., B , be regular elliptic, Definition 5.2.114. Let 1 < p < co,and s = 0 , 1, 2, . . . T h e n Zf),

%F)u

= {Au;B,u,

. . . , B,,,u},

(2)

387

5.4.5. The Operators 9fP m

considered as a m p p i n g from WF2"(Q) into W;(Q) x l-'IB~y~"''j-j=l

a @-operator. The kernel N(%g))is independent of p and s,

1

P

(aQ) is

iV(%!)) = N ( A p ) . Further, it holds codim R(%F)) = codim R ( A p )< co and there exists a projection from LJQ) x restriction to w;(Q)x

in

j=l

Ill

j=i

(3) (4) 1 p(aQ)

BE-"'j--

onto I?(%$')) whose

B ~ ~ - ~ ~p (aQ) ' ~ +is~a -projection ' onto R(%$)).

Proof. (3) is a consequence of Theorem 5.4.412. By the same theorem there exist N linearly independent functionals g l , . . ., g - , belonging t o (Lp(Q))',such that

R(Ap) = { f I f E L p ( Q ) , g j ( f ) = 0, j = 1, . . ., N } . By (5.4.4/9),one may choose gl, . . ., gn. in such a way that R(AF))= ( f I f E Wi(Q), g j ( f ) = 0, j = 1, * * *, N } , where s = 0,1, . . . , and that gj are also linear and continuous functionals on Wi(Q). Let m

1

{ f , v 1 , . ., vm}E W;(Q) x I1B~Y~+"-ll-"j(aQ) j=1

be given. If (5 has the meaning of Lemma 5.4.4, then (1) has a solution 'uE W ~ m m c ( J 2 ) if and only if there exists a function v E D ( A t ) )such that

A ~ )= v f - A G { V l ? .. .>pm>, (5) + G{cpl, . . ., qm}.(5) has a solution if and only if g j ( f ) - gj(AG{v,>* * vm}) = 0, j = 1, * * ., N . NOW,gj({vl, . . ., vm}) = gj(AG{y1,. . ., vnz}) is a linear and continuous functional namely u = v

- 9

ni

On j=i

B;y;,+

s-

1 -m . P ?(aQ), and

(gj,

g,) is a linear and continuous functional on

The N functionals (g, ,.gj), j = 1, . . ., N , are linearly independent. This proves (4). Taking into consideration that G and the functionals g j are independent of p and s, one obtains the last assertion of the theorem. R e m a r k 1. If I # then (3) and (4) yield that mapping from W;;C2"(Q)onto

%g) - AE is an isomorphic

(Here one must replace A by A - AE in the above considerations.) R e m a r k 2. I n connection with the above theorem and Theorem 5.4.412, there , arises the question whether one can characterize N ( A g ) )= N ( % t ) )( = N ( A p ) )R(AF)), 25*

388

5.4. L,-Theory in Sobolev Spaces

and R(%g”)) more precisely. A systematic treatment of this problem can be found in J. L. LIONS,E. MAGENES[2, I]. See also M. SCHECHTER [a], and J. L. LIONS,E. MAGENES [l,VI]. For this purpose, one needs the dual operators for A!) and %!). The results are similar t o the theorems of the Riesz-Schauder theory for compact operators in Banach spaces. I n the next subsection. we give some remarks in this direction.

5.4.6.

Complement,s

For sake of completeness, we add some remarks. Proofs of the cited results can be found in J. L. LIONS,E. MAGENES[2, I]. R e m a r k 1. In Subsection 5.4.2, we described an important result of the L,theory. To give a qualitative characterization of the differential operators Sj , T j , and Cj in (5.4.2/3)and (5.4.2/4)we use the notation of a Dirichlet system. A system of differential operators defined on the boundary of a bounded Cm-domain is said to be a Dirichlet system if it is a normal system, Definition 5.2.1/2, such that mj = j - 1 (perhaps after changing the numeration of the differential operators). Let SZ be a bounded C”-domain, let A be properly elliptic, Definition 5.2.111, and let B , , . . ., B , be a normal system, Definition 5.2.112, where mj are less than 2m. Then there exist differential operators S j , formula (5.4.2/3),j = 1, . . ., m, such that { B j , S.;):, is a Dirichlet system. Sj is not uniquely determined. For fixed Sj , there exist a second (uniquely determined) Dirichlet system {Cj , Tj}Pl, formulas (5.4.2/3) and (5.4.2/4),such that Green’s formula (5.4.2/5)is true. This result was first proved by N. ARONSZAJN, A. N. MILGRAM[l].The above formulation coincides essentially with Theorem 2.1, Chapter 2, by J. L. LIONS,E. MAGENES [2, I]. There can be found also a proof. R e m a r k 2. I n Subsection 5.4.2, we used the fact that A*, C,, . . ., C , is regular elliptic, provided that A , B , , . . ., B,, is regular elliptic. But one can prove a better result. Assume that A is properly elliptic and that B , , . . ., B , is a normal system such that all mi are less than 2m. Then it follows from Remark 1 that C,, . . .,C, is also a normal system. Now one can prove that {Bj}Cl satisfies the complementing condition with respect t o A , Definition 5.2.113, if and only if { C , } ~ satisfies , the complementing condition with respect t o A*. This important result is due to M. SCHECHTER [2] who gave a direct algebraic proof. Other proofs can be found in J. L. LIONS,E. MAGENES[2, I ] and Ju. M. BEREZANSKIJ [l]. R e m a r k 3. (5.4.2/6)was denoted as the adjoint (non-homogeneous) problem. If = 1, . . ., m, then one can avl-1 show that { C i } ~is~ also , a Dirichlet system. If A is a formally self-adjoint operator, then, in this special case, the adjoint problem coincides with the original problem (5.4.5/1).The boundary value problem is “self-adjoint”. a1-L

{Bj}j”ilis a Dirichlet system, for instance Bju = -, j

R e m a r k 4. With the aid of the adjoint problem, the corresponding operators A*(S) and and the Dirichlet’s systems described in Remark 1, one can characP terize N(AB))= N(%!)), R(A(,”)), and R(9$)). We refer again to J. L. LIONS,E. MAGENES [2, I].

5.5.1. Homogeneous Boundary Value Problems

389

R e m a r k 5. * The question arises whether the diverse assumptions are necessary for the theory developed in the last subsections. We add some remarks in this direc[2] proved that there exist properly elliptic differential operators, tion. M. SCHECHTER Definition 5.2.1/1, and normal systems {B,}Pl, Definition 5.2.112, where na, is less than 2m, such that {B,}Fl does not satisfy the complementing condition in Definition [2] that 5.2.1/3 (with respect to A ) . It follows from the considerations in J. PEETRE the a-priori-estimate (5.3.411) is true if and only if dim N(9lF))< co and R((U!)) is a closed subspace in Wi(sZ) x

n Bi:,2n7-n'J--(aQ). A. S. DYNIN[l,21 proved that 1

111

p

j=1

(5.3.4/1)for p = 2 and s = 0 , 1 , 2 , . . . is equivalent t o the assertion that A ,B, ,. . . .B,,, is a @-operis regular elliptic. Further, it was shown by A. S. DYNIN[l,21 that rator with (5.4.513) and (5.4.5/4)if and only if A , B, , . . . , B,,, is regular elliptic. A. DOUGLIS,L. NIConsiderations in this direction can also be found in S. AGMON, RENBERQ [I, I], 5 10. Finally, we refer t o a paper by 8 . AGMON [2]: A ray arg 1. = 8 in the complex plane, 0 8 < 2n,is said to be a direction of minimal growth of the resolvent if il = Q el0 belongs t o the resolvent set of A , for all sufficiently large positive values of e and if there exists a number c such that

II(Ap- ilE)-lII

5 cIAI-',

111 2

e > 0.

(1)

S. AGMON [2] formulated sufficient, and for p = 2 also necessary, conditions, ensuring that the resolvent set of A , is not empty and that arg 1 = 8 is a direction of minimal growth. For 8 = n tfhese conditions coincide with the hypotheses of Theorem 4.9.1.

5.5.

Boundary Value Problem [Part 11

I n (5.4.4/6)and (5.4.5/1)we described homogeneous and non-homogeneous boundary value problems. The theorems in the Subsections 5.4.4 and 5.4.5 may be considered as a theory for boundary value problems of such a type in Sobolev spaces (although the boundary values belong t o more general spaces). Using interpolation theory one can extend t,heseresults t o Lebesgue-Besov spaces without weights. I n Section5.7, we shall be concerned shortly with corresponding investigations in other functions spaces.

5.5.1.

Homogeneous Boundary Value Problems

It follows immediately from Theorem 5.4.412 that the spectrum of the operator A!' is independent of 1 < p < co and s = 0 , 1 , 2 , . . . We denote it simply by S A . Similarly, N ( A ) = N(A,,) = N ( A g ) )is the common kernel of all these operators. T h e o r e m . Let 1;2 be a bounded Cw-domain.Let A , B,,. . ., B,,, be regular elliptic, Definition 5.2.114.

390 (a)

5.5. Boundary Value Problems [Part I]

If 1 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 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 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,) <

(2 b)

(3)

00.

(b) I f 1 $ S A , then { A u - Au, B,u,

. . ., B,u} is an isomorphic m p p i n g from H;+2rn(9)onto H;(9) x

where s

n B::;rn

J=1

2 0, 1 0,1<2,
R e m a r k 1. * The application of interpolation theory to boundary value problems for elliptic differential operatorsis due to J. L. LIONS,E. MAGENES[l,particularlyIV] [4]. It is one of the main aims of the papers by J. L. LIONS, and M. SCHECHTER E. MAGENES [l] to extend these considerations to s < 0. For this purpose, new tools which are not developed here are needed. For p = 2, a systematic trestment also of these parts of the theory can be found in J. L. LIONS,E. MAGENES [2, 11. R e i n a r k 2. Using the methods developed by J. PEETRE [2], the above theorem yields immediately the a-priori-estimates

-

c I11

IIUllH;+aYn) IIAUIIH;(o) + llUllLP(Q, + lIB,ullB2m+*-m~-' PIP (an) /=1 for u E H F 2 " ( 9 ) , 1 0.

J=1

P,9

P

392

5.6. Distributionsof Eigenvalues, Associated Eigenvectors

5.6.

Distributions of Eigenvalues, Associated Eigenvectors, and Green Functions

This section deals with some special questions, partly of interest later on. We consider the distribution of eigenvalues of self-adjoint elliptic boundmy value problems, the problem of the density of the associated eigenvectors, and qualitative characterizations for Green functions.

5.6.1.

Dist,ributions of Eigenvalues, and Associated Eigenvectors in Hilbert

Spaces Let H be a complex separable Hilbert space. A self-adjoint operator A acting in H is said t o be an operator with pure point spectrum if its spectrum consists of isolated eigenvalues of finite multiplicity without any finite cluster point. By the theorem of F. RELLICH, a self-adjoint operator is a n operator with pure point spectrum if and only if the embedding operator from D ( A ) (equipped with the norm l l u l l D ( A ) = llAull + Ilull) into H is compact (see H. TRIEBEL [17], p. 277). For such a n operator A the eigenvalues (inclusively their multiplicities) are numbered by One sets

0

5 lilJ 5 11,1 5 . . . 5

6 lAj+ll

. . . + 00

for j

-,00.

(number of eigenvalues, which by modulus less than or equal t o 1).The approximation numbers sj are described in Definition 1.16.1/2. As before, embedding operators are denoted by I . x

T h e o r e m 1. Let A be a self-adjoint operator with pure p i n t spectrum. Further, let

> 0. Then the following assertions are equivalent to each other :

-"

(" means that the left-hand side can be estimated by the right-hand side from below and from above with the aid of constants indepndent of 1,resp. j). Proof. Step 1 . To prove the equivalence of (2) and (3), we may assume without in (2), one loss of generality that all the eigenvalues are simple. Putting 1 = obtains (3). Conversely, (2) is a consequence of (3). S t e p 2 . Assume, without loss of generality, that 0 is not an eigenvalue of A . Then D ( A ) may be normed by IIAulI. Whence i t follows that ID(&4)--+lf = A-lAD(A)-H,

5.0.1. Eigenvalues, and Associated Eigenvectors in Hilbert Spaces

393

where AD(.,)+"is an isometric mapping from D ( A ) onto H , and A-l is to be understood as a compact mapping from H into H . One obtains that s j ( 1 ; D ( A ) ,H ) = sj(A-l; H , H ) , j = 0, 1, 2,

. ..

(5) If hi, E D ( A ) are the orthonormal eigenelements for the eigenvalues A/(,then A-l has the spectral representation " 0 1 A-lh = C -(h,hrc)Hhk. k = 1 Ak

If I<

E L(H, H

such that

j+l 1=1

) such that dim R ( K )

s j, t,hen there

exist numbers p l ,

. . ., pj+l

J+1

Ip1I2 = 1 , and Kh = 0 for h =

Cpl?zl.By (1.16.1/4)

1=1

sj(A-l; H , H ) 2 i d Il(A-' - K ) hll K

=

inf K

j+l

C Ar21pi12 2 12j+11-'.

(1=1

To prove the converse inequality, we set

Kh We have

=

j

C &'(h, hk)H hk.

1; = 1

t ~ j ( A - lH; , H )

=

llj+ll-l.

(5) and (6) prove the equivalence of (3) and (4). R e m a r k 1. The formulas ( 5 )and (6) are of self-containedinterest. Weshallusethem later on. Further, it is not very hard t o see that dj = d j = si in separable Hilbert spaces (see Definition 1.16.1/2). A short proof can be found in H. TRIEBEL[15]. T h e o r e m 2. Let A and B be two self-adjoint operators in H such that

D ( B ) c D ( A ) and

llAhll

s IlBhll

for

hED(B).

Further, let A be an operator with pure point spectrum. Then B i s also an operator with pure point spectrum. If NA(2) and N H ( 2 are ) the corresponding functions for the distribution of the eigenvalues, formula (l),then NIj(2) 5 NA(2) for A 2 0 . (7) Proof. Step 1. The theorem of F. RELLICH(see for instance H. TRIEBEL[17], p. 277) yields that B is an operator with pure point spectrum. Step 2. Assume that there exists anumber 2 2 0 such that (7) is untrue. Then there exists an element h E D ( B ) c D ( A ) , I(h(l= 1 , with the €allowing properties: (a) h is a linear combination of eigenelements of B belonging to eigenvalues of B which by modulus less than or equal to 2 ; (b) h is orthogonal t o all eigenelements of A , for which the corresponding eigenvalues by modulus less than or equal to A. For such an element h the spectral representations of A and B yield IlAhll > 2, IlBhII 5 2 . This is a contradiction. R e m a r k 2. The above theorem is known as the Variation Principle of R. COURANT.

394

5.6. Distributions of Eigenvalues, Associated Eigenvectors

R e m a r k 3. If H , and H are two Hilbert spaces such that H , c H , then H , may be represented as the domain of definition of a n appropriate positive-definite operator

[l], $124, or J. L. LIONS,E. MAGENES[2, I], see for instance F. RIESZ,B. SZ.-NAGY Chapter 1,2.1. If there are two Hilbert spaces, H , c H , c H , then one can compare, with the aid of the above theorem, the distributions of the eigenvalues of the corresponding operators. We shall use this fact later on. L e m m a 1. Let H , and I€, be two Hilbert spaces. Let A k , k = 1 , 2 , be a semi-bounded (from below) self-adjoint operator in H,i having a pure p i n t spectrum. Let

+1

iv,,(il)

N

+ 1,

x,q

> 0, k = 1 , 2 .

Then B = A , Q E + E Q A , is a semi-bounded (from below) self-adjoint operator in A H , Q H , having a pure point spectrum. I t holds Nn(il) + 1 il"1+"2 + 1 . A Proof. The definition of B = A , Q E + E Q A , and H , Q H , yields that B is a semi-bounded (from below) self-adjoint operator with pure point spectrum (see j = 1 , 2 , . . .; k = 1 , 2 , are for instance Ju.M. BEREZANSKIJ[l], I, 3 2). If j = 1 , 2 , . . .; 1 = 1 , 2 , . . . , are the eigenvalues eigenvalues of A k r then A?) + of B (inclusively their multiplicities). We assume, without loss of generality, that A , and A , are positive-definite. Then one obtains that N&) = NA,(il - A?'). N

AT),

c

Whence it follows

y ) $ L

N U @ ) SNatl(il) NA,(il) On the other hand. we have

CAK'+XB.

This proves the lemma. The investigations below on associated eigenvectors are based on a result found M. G. KREJN[l], which we formulate without proof. For this by I. C. GOCHBERG, purpose, we need the classes G p , 1 p < 03, of compact operators in Hilbert spaces

s

described in 1.19.7. We recall that an element 0 =# v E

m

n D(A.i)of a linear operator

j=O

A acting in a Banach space is said t o be a n associated eigenvector if there exist a complex number f and natural number k such that ( A - LE)" v = 0. I n this case, il is an eigenvalue. The dimension of the space of all associated eigenvectors belonging to a given eigenvalue f (and the null vector) is said to be the algebraic multiplicity of the eigenvalue 1. T h e o r e m 3. Let A be a self-adjoint operator in a Hilbert s p c e H , having a pure p i n t spectrum. Let B be a linear operator in H such that D ( B ) 2 D ( A ) . If there exist a complex number 1 4 S A and a number 1 p < oc) such that B(A - AE)-l E Gp, then A + B with the domain of definition D ( A + B ) = D ( A ) is a closed operator. Its

s

395

5.6.2. Eigenvalues of Self-AdjointElliptic Differential Operators

spectrum consists of isolated eigenvalues of finite algebraic multiplicity, and the linear hull of its associated eigenvectors is dense in H . R e m a r k 4.A proof of this theorem can be found in I. C. GOOHBERG, M. G. KREJN [l], Chapter V, 9 10. The results obtained there are more general than the above theorem. The theorem does not depend on the choice of A 4 S A . This is a consequence of Hilbert's relation for resolvents, ( A - ,uE)-' - ( A - AE)-l = (,LA - A ) ( A - Ah')-' ( A - pE)-l, 1 4 S A ,U 4 S A . Further, it is not very hard to see that A + B is a closed operator in D ( A ) ,namely: I f K E L(H, H ) is an appropriate operator of finite rank, then it follows for u E D ( A ) J(Bul(= IIB(A - AE)-l ( A - AE) uI( - II[B(A- '-)&A I - K ] ( A - 1E)u I ~ + IIK(A - 1E)uII I

D ( A ) 3 uj + u, and Auj + Buj + v . Then ( 8 )and.(9) yield u E D ( A )and Au + Bu = v. Hence, A + B is a closed operator on D ( A ) . The following lemma will be useful for the later applications. L e m m a 2. Let A be a self-adjoint operator in the Hilbert s~ H having a pure p i n t spectrum. Assume that 0 is not a n eigenvalue. Let A-l E Gp for an appropriate p , 1 5 p < co. Let B be a linear operator in H such that D ( B ) 3 D ( A ) . If there exist positive wumbers c and e such that for all 1 > E > 0 and for all u E D ( A )

Let

P r o o f . Let ill be the eigenvalues of A such t8hatlA,( corresponding orthonormal eigenelements. We set'

Pjh

=

i 1

C - (h,h,) hl , h E H . 1=1 i l l

Choosing E = s;/('+Q)(A-l; H , H ) in

sj(BA-l; H , H ) Hence, BA-1 E Gp(l+e). 5.6.2.

5 1A21 5 . . . , and let hl be the

(lo), then ( 6 ) yields

5 IIB(A-l - Pj)ll

~$l(l+Q)(A-l; H, H).

Distributions of Eigenvalues of Self-Adjoint Elliptic Differential Operators

T h e o r e m . Let Q be a bounded Cw-domain. Let A , B,, . . ., B , be regular elliptic, Definition 5.2.114. Let the operator A , defined in (5.4.311) be self-adjoint in L2(Q) ( A 2i s considered as an unbounded operator in L2(Q)).Then

N(1) + 1

-

A"1Zm

+ 1.

(1)

396

5.6. Distributions of Eigenvaluea, Aasooiated Eigenvectors

Proof. (4.10.2/14) yields

-

s j ( l ; J+p(Q),L,(SZ)) j

--Ynr I)

.

This formula is also true after replacing W p ( Q ) by @i"(SZ).This is a consequence of the ideal properties of the numbers si , and the previously used fact that W p ( Q ) is isomorphic t o a complemented subspace of I@"(Lo),provided that w is a bounded C"-domain such that c w . (The isomorphism operator is given by the restriction operator from 1Vim(co) onto Wim(Q),while the corresponding inverse operator is an appropriate extension operator.) Now

kp(52)c D(A,) c W p ( J - 2 ) . Hence, (2) is also valid if one replaces Wim(Q)by D(A,). The theorem is a consequence of Theorem 5.6.1/1. R e m a r k 1. I n Theorem 5.4.4/1,we proved cAT,{~,l(SZ) = D ( A 2 ) .This fact, formula ( l ) ,and the methods developed in the eighth chapter yield that is isomorphic t o s, the nuclear space of rapidly decreasing sequences. R e m a r k 2. * Many papers are concerned with investigations on dittributions of eigenvalues for (regular and degenerate) elliptic differential operators. A treatment of the history of these topics as well as a short description of the most important methods can be found in C. CLARK[3]. There is also an extensive bibliography (120 items). So we omit here historical references. We restrict ourselves to the quotai'ion of some new papers which may be understood as a complement t o the bibliography given in C. CLARK [3]. The theory of the distribution of eigenvalues goes and T. CARLEMAN.It was developed extensively in back t o H. WEYL,R . COURANT, the fifties by L. GARDING,F. E. BROWDER, and (a bit later) S. AGMOK(references can be found in C. CLARK[3]). Now we mention some new papers. I n particular we refer t o the paper by S. AGMON[4]. Let A , be the operator of the theorem. Assume additionally that A , is semi-bounded and that a(x,5 ) from (5.2.1/2) is positive. Then it has been proved by S. AGMON[4] that

cT,{Bjl(Q)

N(1) =

[(zn)pJ ( Q

J' d 5 ) dX] ;l"l,m

a ( . r , t )< 1

+ O(A('"-");'"').

(3)

Here any number u, such that 0 < u < 3, is admissible. If A has constant coefficients, then any number u, such that 0 < a < 1, is admissible. (3) is not only a more precise description of the asymptotic law formulated in (l),but it gives a t the same time also an important remainder estimate. This result was generalized by K . MARUO, H.TANABE[l], H.TANABE[l], and K.MARUO[2] (weaking of the smoothness assumptions). We refer also t o M. NAGASE[l] and J. BRUNING[l]. A great part of the new investigations on spectral theory for elliptic differential operators is concerned with the following topics : (a) Remainder estimates in the sense of (3); (b) Weakening of the smoothness assumptions (for the coefficients and for the domains) ; (c) Investigation on t,he properties of the "spectral functions"; (d)Spectral functions and distributions of eigenvalues for degenerate elliptic differential operators. The last problem will be considered later on. As a complement t o the bibliography in C. CLARK [3], we refer to the following papers: G. I. BASS[l], R. BEALS[l, 2,3], M. 8. BIXMAN,

5.6.4. Green Functions of Elliptic Differential Operators

397

M. Z. SOLOMJAK [3,4,5], V. V. BORZOVEl], J. FLECKINUER, G. M~TMER [l], W. GROMES [l,21, G. GRUBB[l], L. HORMANDER [a, 51, A. N. KO~EVNIKOV [l], B.M.LEVITAN [l], G. M ~ T M E R [l], and V.N.TULOVSKIJ[2]. See also Remark 7.8.311 and Remark 7.8.312. 5.6.3.

Associated Eigenvectors of Elliptic Differential Operators

In this subsection, we describe a (comparatively simple) result which is an illustration of Theorem 5.6.113. By the same method, we shall obtain later essentially more general results in the framework of an L,-theory for strongly degenerate elliptic differential operators (Subsection 6.6.2).See also Subsections 7.5.1 and 7.6.4. Theorem. Let Q be a h & d Cm-dmnain. Let A , B , , . ., B , be regular elli(ptic, Definition 5.2.114. Assume that A is given by

.

where A , ,

A,u = Au, D(d2) = D(A,) = W,",;g,,(Q), (considered as an unbounded operator in L,(Q))i s self-adjoint. Then the spectrum of A , consists of isolated eigenvalues of finite algebraic multiplicity and the linear hull of the associated eigenvectors is dense in L,(Q).

Proof. The embedding from W , " ~ B ~into ~ ( QL,(Q) ) is compact. Whence it follows that 2, is an operator with pure point spectrum. Denote the second term on the righthand side of (1)by Bu. If il.!+ s;, , then B(L& is a continuous mapping from L,(f2)into W;(Q). Then (4.10.2114)yields that B(& - il.E)-l, considered as a mapping from L,(Q) into L,(Q), belongs to G,, where 00 > 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

=<

co.

<

00,-

l -1

P

Proof. Step 1. If s 2 0, then the theorem for the H-spaces coincides essentially 1 1 with Theorem 5.5.l(b). For 1 < p < 03 and = 1, we set P P A;,m = C ( - 1)Iu1D”(a,(z) u) ,

+

/a12 2 n i

Here, cj,,(x) and r i have the same meaning as in (54.214). Using Theorem 5.4.2 and its sketch of the proof given there, it follows that A’, . . ., is regular elliptic. The only difference in comparison with Theorem 5.4.2 is that we now avoid the ‘. conjugate-complex formulation” prefered there. Using this fact and (5.6.4/3), then it follows 0 I$ S,1;. Now by Theorem 5.4.412, it holds 0 I$ SA;. . Hence, by Theorem 5.5.l(b), the dual operator (A;,)‘ of A;, gives a n isomorphic mapping from L&2) onto (H;:i?,)(SZ))’. Since (A;,)’u = A u for u E CF(Q), it follows that A gives an isomorphic mapping from H;&)($2) onto LJQ) and from Lp($2) onto (H;:zj) ($2))’. 1 Theorem 1.11.2 and Theorem 4.3.3 yield that A for 0 < 2mO < - and for 1
c

c,

(a)

[(LP‘CQ)? H73,11(52))0,,*1’. Again by Theorem 4.3.3 (Definition 4.3.3/2), and Theorem 4.8.2 it follows that ( ~ P C Q ) (H;%,)(w’)o,q , =

Q,If

2mO

-

[ ( L p ’ ( Q ) , H;%,AQ))@, = ( B p , . q * , ( C , , ( Q ) ) ’ = (B;y:,cQ),’ 26 Triehel, Interpolation

(11

= B,;:mo(c.n,~ (2)

402

5.7. Boundary Value Problems [Part TI]

1 Now one obtains the desired assertion for the B-spaces, provided that - - 1 <s 0, we refer t o Theorem 5.5.l(b).The case s = 0 as well as the assertion for the parameter values q = 1 and q = 00 follow from Theorem 4.3.111and Theorem 4.3.3by interpolation. Step 2. The proof for the H-spaces follows the same line. Instead of Theorem 1.11.2, one has t o use Theorem 1.11.3. R e m a r k 1. The proof is based essentially on the formulas (1)and (2). Specializing the boundary value operators Bi and c,i,then one can extend the theorem to other negative values of s. For instance, for the Dirichlet boundary value problem for A , aj-lu that means Bju = -, i = 1, . . ., m, the boundary operators cj give also the avJ -1 1 1 Dirichlet problem, Remark 5.4.613. Let 0 < 2m8 < m + - 2me - p , *0,1,-.., P' ' m - 1, and 1 < q < co. Then it follows from Theorem 4.3.3 (Definition 4.3.3/2), Theorem 4.7.1,and Theorem 4.8.2that (3) [ ( L , , ( Q )H;)~;F~)(Q))O,~,]' , = (&y$(Q))' = BG:ye(Q). Applying again Theorem 4.3.3 and Theorem 4.3.111,it follows that A with the Dirichlet boundary conditions gives a n isomorphic mapping from H;i%;(SZ) onto s+2ni 1 H;(Q) and from B,,q,,B,,(Q)onto Bi,,(SZ), provided that 1 < p < 03, - m - 1 + 1 1 1 P < s < 03,s - 1, - - 2 , . . ., m,and 15 q 5 co.

+ -I ,

2,

-P

R e m a r k 2.* Theorems of the above type can be found in J. L. LIONS,E.MAGENES [l, particularly VI; 2,I ] and E.MAGENES[l]. 6.7.2.

Sobolev-Besov Spaces with Weights

One can extend the investigations on boundary value problems for regular elliptic operators t o some Sobolev-Besov spaces with weights. The considerations made P. GRISVARD [l] in Sobolev spaces with weights, which we shall by G. GEYMONAT, formulate without proof are the basis. We use the notations of Definition 3.2.114 and Theorem 3.3.3. Further, let d(x) be the distance of a point x E Q t o the boundary aQ. T h e o r e m 1. Let 9 be a bourded C " - d m i n . Let A , B,, . . ., B, be regular elliptic, Definition 5.2.114.Let 1 < p < 00 and -1 < OL < p - 1. f.l) For s = 0,1,2,. . . and u E WErn+*(Q; d"(x)) we have

I1u (I II?+'(Q;

(b) It ho2ds

-

dyq)

+

+ ,z rn

a+l

lI~ullw;(n;d.(z), ll~llLp(n;d~(z)) ll~,f~11B2m+'---A~ P can,. J=1 P!P

..

(u I U E W~m(Q;d"(x)), Au = 0 ; Bjulan = 0 for f = 1,. ,m} t

cm(Q).

(1)

(2) R e m a r k 1. Theorem 3.6.1yields that (1)is meaningful. By the same theorem it follows that the right-hand side of (1)may be estimated from above by the left-hand side. The main problem is the proof of the converse estimate. For this purpose,

5.7.2. Sobolev-Beaov Spaces with Weights

403

G. GEYMONAT, P. GRISVARD[l] generalized the methods by S. AGMON,A. DOUGLIS, L. NIRENBERG [l]. It follows from the introduction to Subsection 3.6.1, and from Theorem 3.6.1 that an extension to values a 4 ( - 1, p - 1) is not possible, in general. For a = 0 formula (1) coincides with (5.3.4/1). Remark 2. Consider, similarly to Theorem 5.4.412, the operator A!*"), AF'")u = A u , D ( A k ")) 2m+s

= W P , ~ ~ j ) ( Q , d a ( ~ ) ) = { ~ ~ ~ ~= W 0, ~ mj =+ 1, s ..., ( ~m,} .d (3) n(~));B

Then, by (2), (4) N(Ag?")= ) N(A,) c Czpj)(Q). This is the generalization of the first part of (5.4.418). Further, it follows from (1) and the results by J. PEETRE [2] that R(AF "1) is closed. In particular, using the case a = 0 , one obtains that A - ilE for il4 S A is an isomorphic mapping from D ( A P a ) ) onto W;(Q; d"(s)). Using Theorem 3.6.1, it follows that %P")u= {Au - Au; B 1 u ; .. .;B,,u}, (5) is an isomorphic mapping from WEm+s(Q; a"(%))onto

%Pa),

W;(Q; a"(%)) x

If il E SA, then

fi BZ;;'---"'j

a+l

j=l

(m).

Y

codimR(%;'")) = codimR(A2"))< 00 is independent of 1 < p < 00, s = 0 , 1 , 2 , . ., and -1 < a < p - 1. This result may be obtained in the same way as before (Theorem 5.4.5) where one has to take into consideration that e-(Q) is dense in W i ( Q ; d"(s)), Theorem 3.2.2(c) and Remark 3.2.212.

.

The last remark shows that one can develop the theory for operators A!*") and the considerations in Section 5.4 and Section 5.5. For sake of simplicity, we restrict ourselves to the case 0 $ S A . The spaces Bi,,(Q;a"(%)),bc 2 0, used in the following theorem, have been defined in Theorem 3.3.3. Theorem 2. Let Q be a bounded CO"-domuin.Let A , B , , . . ., B , be regular elliptic, Definition 5.2.114. Let 1 0, and 0 4 S A . 00, 0 Then Aw gives an isomorphic mazvping from

aka)similarly to

B;7;Gj)(Q;daW) = {u I u E BZ:"(Q; d"(s)); BiulaQ = 0 for j = 1, . . ., m } onto B;,,(Q; d"(x)), and { A u ;B,u; . . .; B,,u} gives an isomorphic mapping from B;TY(Q;a"(%)) onto B i , , ( Q ;d"(s)) x

fiB ~ ~ ~ s - n ' j(am). -a+l I)

j=l

Proof. Step 1. First, we remark that Lemma 5.4.4 may be extended to the spaces 2m+s W Zpl r,l (f~sj ) ( Qd"(z)), ; s = 0, 1 , 2 , . . ., and to the spaces B P , q , ( ~ j )a"($)), ( Q ; s > 0. This is a consequence of the proof of this lemma, and of the proof of Theorem 3.6.1 and 26*

404

5.7. Boundary Value Problems [Part 111

Theorem 3.6.3. Using this modified lemma, then Theorem 1.17.111 and Theorem 3.3.3 yield 2mik B2m + k0 ( wE{Bj)(D;da(z)), w p , ( B , ] ( Q ;d"(z))o,q = p , q , { B j } ( Q ; Now it follows by Remark 2 and Theorem 3.3.3 that Au is an isomorphic mapping from B ~ ; ' ~ ~ ,da(z)) l ( Q ;onto B;,*(D;d"(z)) where s > 0. #tep 2. Using the modified Lemma 5.4.4 described in the first step, then one may carry over the proof of Theorem 5.4.5. Then it follows by Theorem 3.6.3 that { A u ;B1u;. . .; B,,,,u}gives an isomorphic mapping from BiTB,"(O;da(x)) onto

R e m a r k 3. I n contrast to Theorem 1, we restricted ourselves in Theorem 2 to a 2 0. The reason is that we developed an interpolation theory only for these values of the parameter. R e m a r k 4. I n connection with Theorem 2, we refer also to C. GOUDJO[l] who obtained similar results, partly also for negative values of s in the sense of Theorem 5.7.1. See also D. FORTUNATO [l]. 5.7.3.

Holder Spaces

For sake of completeness, we shortly describe boundary value problems for regular elliptic operators in Holder spaces. If Q is a bounded C'"-domain, then one may introduce the spaces Ct(aQ),t 2 0 , similarly t o Definition 3.6.1. Here, one has t o use the spaces Ct(Rn-,), Definition 2.7.1. As the method is clear we do not go into detail.

Theorem. Let D be a bounded C"-domain. Let A , B,, . . ., B , be regular elliptic, Definition 5.2.114. (a) For o < t integer and u E C z n l + t ( Q )

+

m

+ integer, then Au gives an isomorphic mapping from

(b) If 0 4 S A and 0 < t

~ = oufor jl = 1, ~ . . .,~m> onto o ( D ) and { A u :B,u; . . .; BlIlu}gives an isomorphic mapping from C 2 m + t ( Q ) onto

C:;~;'(D)

CqD) x

=

{u I u E C2rrt+t(~); ~

1,'

j=l

C2n1+t-m,

caQ).

R e m a r k 1. (1) is proved in S. AGMON, A. DOUGLIS,L. NIRENBERG [l]. It is the main result of the theorem. Part (b) of the theorem may be obtained from (1) and the previous considerations. An extension of (1) to the values t =0, 1 , 2 , .. . is not possible. R e m a r k 2. * Estimates of type (1) are first proved for elliptic differential operators [l,21. The estimates of J. SCHAUDER are improved of second order by J. SCHAUDER and generalized by A. DOUGLIS,L. NIRENBERG[l], C. B. MOR.REY[2], and C. MrRANDA [l]. The above formulation of (1) is due to S. AGMON, A. DOUGLIS,L. NIRENBERG [l]. Further references can be found in C. ~ A N D [l]. A

6.

STRONGLY DEGENERATE OPERATORS

6.1.

Introduction

ELLIPTIC DIFFERENTIAL

Beside regular elliptic differential operators, degenerate elliptic differential operators have been extensively considered in the last years. There are many, very different, types of degeneration. On the one hand the domain 9 c R, may be unbounded, or on the other hand the coefficients of the differential operator may be singular if 1x1 + 00 or x + aQ, or they have singularities in 9. Further, it may be happen that the ellipticitiy condition (4.9.1/2) degenerates if x -+ a 9 . Clearly, there are many possibilities of variation, and one cannot expect to catch all these different types by a uniform theory. This chapter and the following one are concerned with some (comparatively comprehensive) classes. The class considered in this chapter is characterized in a very strong degeneration of the coefficients of the operators near the boundary (and at infinity). This has the consequence that the boundary does not play any role (in contrast to the regular elliptic differential operators of the last chapter). For instance, a formally self-adjoint operator of this type with the domain of definition C$'(sZ) is essentially self-adjoint in A&?).

6.2.

Definitions and Preliminaries

In this section, the classes of operators coiisidered here are defined. Further, some

related locally convex spaces are introduced. Finally, there are proved some simple properties.

6.2.1.

Definitions

I f Q c R,,, then, as usual, Cm(Q) denotes the set of all infinitely differentiable complex-valued functions defined on Q. Definition 1. Let Q be a n arbitrary domain in R,,.Further, let p(x) E Cw(Q) be a positive function such that : 1. For all multi-indices y , there exist positive numbers c, such that IDr@(X)I

cr@'+lqx) for all

2 E9.

(1)

406

6.2. Definitions and Preliminaries

2. For any positive number K , there exist numbers cK > 0 and rlc > 0 such that e(z) > K if d ( x ) 5 cK or if 1x1 ( d ( z )is the distance to the boundary). Then ~!5'@(~)(9) denotes the locally convex space

Se(r,(f2)

=

{f If

IIfIIt,a

E CoD(Q),

=

1 rI,, ( X E D )

(2)

e6(z)l o " f ( ~ ) < I

(3)

SUP X€Q

for all 1 = 0, 1, 2 , . . ., and a11 multi-indices a } . Remark 1. Comparison with Definition 3.2.3/1 shows that (3.2.3/1) is replaced by the sharper assumption (1). The functions e(z) considered in Remark 3.2.3/1 satisfy also the above assumptions. In particular, in every bounded domain there exist functions e ( x )such that e-l(x) coincides essentially with d ( x ) . Definition 2. Let f2 c R, be a n arbitrary domain, and let e(z) be a weight function in the sense of Definition 1. Further, let m be a natural number, and let ,u and v be real numbers such that v > p + 2m. One sets 1 x i = -(v(2m - I ) + p l ) , 1 = 0, 1 , . . ., 2 m . (4) 2m (a) The class %&(9; e ( x ) )consists of all differentia2 operators of the form

Here, ba(x)E C m ( 9 )are real functions, la1 = 21 with 1 = 0 , 1, . . ., m, where all their derivatives (inclusively the functions themselves) are bounded in 9. Further, it is assumed that there exists a positive number C such that for all 6 E R, and all x E 9

(ellipticity condition). Let as(x) E C " ( 9 ) , 0 5 IBI < 2wk, and Dyas(x) = o( e x l p l + 1 ~ (1x ) ) for all multi-indices y .

> 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
(3) for

o < 1y1 5 z + s,

(4)

408

6.2. Definitions and Preliminaries

then it follows that the second summand in (1)has also the desired form. This is also ( x;) ) . true if A is an element of i i ; ~ e~

6.2.3.

Properties of the Spaces S,(,)(sd)

A complete locally convex space where the topology is given by a countable set of semi-norms is said t o be a n (P)-space. T h e o r e m . (a) Se(,)(Q)from Definition 6.2.1/1 is an (F)-space.Corn@) i s a dense subset. (b) If there exists a number a > 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 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 --,

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---, 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
2m n

2m

Under the formulated conditions for z,whence it follows the convergence of (5) if

N+

00.

Remark. Both the theorems are the counterpart to Theorem 5.4.4. For some important special cases A E ‘91Ev(12;&)), differentiability properties for Green functions are considered by B. LANQEMANN [ l , 21.

7.

LEGENDRE AND TRICOMI DIFFERENTIAL OPERATORS

7.1.

Introduction

This chapter is concerned with ordinary degenerate elliptic differential operators in a bounded interval 9 = (a,b ) and with partial degenerate elliptic differential operators in bounded Cw-domains9.I n contrast to the last chapter, the coefficients of the differentia,lexpressions belong to Cm(9).If u(x, 6)for x E dz and E R,,has the same meaning as in (4.9.1/2),then differential expressions (4.9.1/1) where 0 < la(x,5)1 + 0

if

x +30,

0 =/=

5 ER, is fixed,

(1)

are considered. The way in which (1) is realized is of fundamental importance for the behaviour of the corresponding differential expression. We describe som emodel cases treated in this chapter. If D = ( - 1, l),then differential expressions of the type

= 0, . . ., 2 m - 1, are considered. Bu is a (non-symmetric) perturbation operator of lower order. If m = k = 1 and Bu = 0, then one obtains the classical Legendre differential operator. For this reason, we shall denote here all differential operators of such a type as (generalized)Legendre differential operators. In a bounded Cw-domainD c R,, degenerate differential operators of order 2m are considered. For the description of the situation, we assume at the moment m = 1 and set

Ic

Again, Bu is a (non-symmetric)perturbation operator, while ( ~ j , , ~ ( x ) )is~an ; ~Her..~~ mitian positive-definite matrix. If Aj(x),x E D,denote its eigenvalues, then 0 < A,(x)

5 A,(x) 5 . . . 5 &(x) < 0 0 .

(4)

The type of degeneration in the sense of (1) is determined by the behaviour of Ai(x) if x + a 9 . Clearly, there are many possibilities of variation. Of special interest’are the following two “extreme” cases: 1. If d(x) is the distance of a point x E SZ to the boundary, and if (yl, . . .,y,) are local coordinates in a neighbourhood V of z E a 0 such that V n aSZ is characterized by y, = 0 and the y,-direction coincides with the normal direction with respect to 8 9 , then the first case can be described by

430

7.2. Definitions

in V nQ.Again, Bu is a perturbation operator, (bj,k(x))$i**in-lis a n Hermitian positive-definite matrix such that n- 1

C

j , k =1

b.j,k(X)

L

c15I2,

(6)

where c is a suitable positive number independent of x . 2 . The second case can be described near the boundary by

Au=-

C a (d(x) 6,,lr(z) 'I

axj (6,,k(x))j;i-.>satisfies (6) with n instead of n - 1. In the first case the degeneration takes place only in the normal direction, while in the second case the differential expression degenerates uniformly in all directions of the coordinates. There are corresponding considerations for differential oper&ors of order 2m. Then one obtains generalizations of the above (generalized) Legendre differential operators and the classical Tricomi differential operators. Operators of such a type are called here (generalized) Tricomi differentia1 operators (of first and second type). The considerations of this chapter are carried out only in the framework of an L,-theory. An L,-theory in the extent of the last two chapters seems t o be unknown a t the moment. The main aim is the investigation of the self-adjoint and non-selfadjoint Legendre operators and Tricomi operators of first and second type. The nonself-adjoint operators are considered as perturbations of the self-adjoint ones (the assumptions are chosen in a suitable way). I n recent years, there are published comprehensive investigations on general Tricomi differential operators of arbitrary order with different types of degeneration near the boundary. There are obtained far-reaching reaults in the framework of the theory for boundary value problems in the sense of Chapter 5 (L,-theory). We do not [l,4,5], consider these problems here extensively and refer to N. SHIMAKURA M. I.V I ~ K ,V. V. G R U ~ I N[l], and P. BOLLEY, J. CAMUS [a, 6,7]. Some remarks can also be found in Subsection 7.9.1.

7.2.

j,k=l

Definitions

I n this section (general) Legendre differential operators and (general) Tricomi differential operators are defined. Some of the main ideas have been described in the introduction.

7.2.1.

Legendre Differential Operators

D e f i n i t i o n . Let -co < a < 6 < co, and Zet p ( x ) E @((a, 6 ) ) such that p ( z ) > 0 2 E (a,b ) and c o > l i m - =P(X) C,>O, co>lim- p ( x ) = cb > 0. (1) z i a 5 - a z t b b - X

for

43 1

7.2.2. Tricomi Differential Operators

If m = 1, 2 , . . . and k

=

0, 1, . . . , 2 m - 1, then one sets

where

D(Alll,,c) = D(BnL,k) = ??"((a,b ) )

-

if

k = m, ??L

+ 1, . . ., 2m - 1 ,

I u E C"((U,b ) ) ; u(')(u)= U'')(b) = 0

D(A,,,,k)= D(BllL,k) = {U for 1 = 0 , . . ., m - k - 1)

(4a)

(4b)

if k = 0, 1, . . ., m - 1 . Here bj(x) E Cm((a, b ) ) are complex-valued functions such that bi(Z) =

O(@+l+J+1(X)).

(5)

(0 must be understood in the sense of x a and x b ) . R e m a r k 1. Simple examples for p ( x ) are p ( x ) = ( x - a) (b - x ) and p ( z ) = C,(x - a ) for a < x < a + E , p ( x ) = C,,(b - x ) for b - E < x < b ,

(6)

b - a where 0 < E < -. The operators A,,,, and Bm,kwith their domains of definition 2 (4)are considered as unbounded operators in L,( (a,b ) ) . The restriction k 5 2m - 1 ensures that A m , k is a self-adjoint operator with pure point spectrum. For k 2 2m this is untrue. Further, it will be shown that also the cases k = 0, 1, . . m on the one hand and the cases k = m 1, . . . , 2 m - 1 on the other hand have essentially different behaviour. ( 5 ) is an additional assumption only for values j such that k - 2m + j + 1 > 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 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

+ #lu12) as,

2m

+ p. (7)

provided that one of the following two conditions i s satisfied: (a) p > -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

<2[+]

+ 1 = (mm+ 1

for m = 0(2), f o r m = l(2)

7.3.2. Properties of the Spaces WF((a,b ) , pa)

437

P r o o f . Step 1.We show that (9) holds, provided that (10) is satisfied. We may assume, without loss of generality, 0 in the norm

s

OL

+ 1. The completion of CF((a,b ) )

c2

gives the space W f ( ( a ,b ) , pa)) from Definition 3.2.1/4. Since oc < 2m, it follows from b

J’

lu12dx

5&

U

b

\

b--8

paIaJU(q2ds

+c J

lU12dZ,

a+6

(1

where E > 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 ~~~)
-

b

+ P-1142)dx J [I(P*U)’l2 + lulald z .

(1)

a

Theorem 3.2.4/1 yields that C$((a, b ) ) is dense in Wi((a,b), p , p - l ) . It follows by completion that (1) is true for all u E * ( ( a , b), p , p-l). By Theorem 3.2.4/3(b), Cnrubelongs t o Wi((a,b ) , p , p - l ) if u E Cm((a,6 ) ) . It follows by partial integration

II CntuII?v:((a,bmp-1) a

b

a

After reckoning of the second term, the factor in front of pm-lp’21u(m)!2 in the last integral equals

+ (” 2

2’

llL) =

m2

Now using (7.3.1/2) with x = m - 1 and with similarly to the above considerations that

-

1

-4 +->--. ~ ( 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(
(3)

(1

Since C g ( ( a ,b ) ) is a part of the range of C,, whence it follows (b).

7.4.

Self- Adjoint Legendre Differential Operators

This section is concerned with the theory of the Legendre differential operators A,,,, from Definition 7.2.1. We shall follow the treatment given in H. TRIEBEL [14,11.

7.4.1.

Self- Adjoint and Positive-Definite Operators

T h e o r e m . (a) The operator from Definition 7.2.1 is essentially self-adjoint i n L,( (a,b ) ) . The closure A,,,,/% i s an operator with pure p i n t spectrum. Its eigenvalues are rwn-negative. (b) is psitive-definite if and only if 2k m. Proof. Step 1 . By Definition 7.2.1, A,,,,, is a symmetric operator, and it holds 6

(A,,,,kU,u ) ~=,

j

$lU(n1)12

dX

2 0.

(1)

CL

Theorem 7.3.2/1 and the criterion of F. RELLICH(see for instance H. TRIEBEL [17], p. 277) yield that Friedrichs’s extension of is an operator with pure point spectrum. To prove that A,,,,/,is self-adjoint, it is now sufficient to verify that every element w with A:,,v = 0 belongs t o &4m,k). Step 2 . Let AZ,hv = 0. Then in the sense of the theory of distributions, it holds

0

= ( - 1 ) m ( @ ~ O ~ ~ ) ) ( ~and ~ ~ )

~ ( 1 1 ’ )

p,,1-1 =-

-

@ ’

where Pm-l is a polynomial of degree m 1. At first we consider the case k Since v belongs t o L,( (a,b ) ) , it follows by integration of (2)that =

O(p$-”’)), where p,, =

(2

- a ) (b - 2).

(2)

2 m.

442

7.4. Self -Adjoint Legendre Differential Operators

If2(k - m ) > 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-’<).Hence, v 3 0. Consequently, Am,/: is positive-

s

definite. Let 2k 2 m

+ 1. If

m

m-1

= l ( 2 ) , then mm-2 -k 5and 2

m - 2

-= 0. In both

If m = 0(2),then m - k 5 -and Am,kpo9 positive-definite. 2

cases,

n1-1 -

Am,k

= 0.

is not

Remark 1. The proof shows that in each caae the null space N ( A m , , ) of Am,,. consists of polynomials, the degree of which is less than or equal to m - 1 .

7.4.2. The Minimal Operator

A,,,*

443

Remark 2. The points a and b are symmetric in the differential operators Am,k. For the later considerations it will be useful to consider also more general operators. Let k, and kI,be two non-negative integers. Further, let

Crn((a,b ) )3 lim

z

!w > 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,/<) has always a solution. The lemma yields u E D(A!,a,k). Whence it follows that the operators Al,,k are essentially self-adjoint. Applying Lemma 7.3.1/1, one obtains by iteration that 2 2 2 (3) IIAk,kull L,((a,b)) -b llull L,((o,b,) llull W~'"'(ta,b),p"') ' In particular, D(AL,J = D(A;,,) is the completion of D(AL,k) in the space Wfrn((a, b), 9'Theorem ). 7.3.2/1 yields that D(A,",,) = D ( A $ , J coincides with the right-hand sides of (2).

446

7.4. Self-Adjoint Legendre Differential Operators

Step 2. If 1 = 0,2,4, . . ., then part (b) of the theorem is a consequence of the first step. For 1 = 1, one obtains the desired result from (7.4.1/1). Assuming that (b) is proved for 1 = O , l , . . . , n 1, and taking into consideration nk 2k - 2 < 2(nm + m) - 1, then it follows for u E D(A$,k)from h m m a 7.3.1/1 that

+

+

6

-

b

J

I + lUl2) dx

(pk+aklU(nm+'2rn) 2

a

-

2

llull W,"-'%.$),p"+9 Whence it follows part (b) of the theorem. Remark 1. The proof show8 that (2) does not only hold in set-theoretical sense, but ah0 in topological sense. As usual, the topology in Cm((a,b ) ) is given by the semi-nops sup lu(j)(z)l,j = 0 , 1 , 2 , . . ., and the topology in D(A$,,) is given .re(a,b)

j = 0,1, 2, . . . by the semi-norms IIAL,kUIIL,((&), Remark2.Thetheoremcannotbeextendedtothevaluesk = m 1 , . . .,2m-1. To show this, we assume that D(A&) and CC.((a, b ) ) are equal. We consider the function f(z) = (pC)(") where p,,(z) = (5 - a) (b - 5). By Remark 7.4.1/1, f ( z ) is orthogonal to the null space N ( i i m , k ) . Hence, there exists a solution of

+

( - 1)"

(?JrU("))(")

= f(5),

belonging to D(ii$,k) = Cm((a,b ) ) . This is a contradiction to k > m and ?JrUW

= (z- a)" ( b - 5)"

+ P"-l.

Whence it follows that D(A$,k) is strictly larger than c m ( ( ab)), , provided that

k > 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 0 real; /? = (PI, . . . ,/?n-l, BIZ) = (p’,B,) multi-index, where IBI = 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 <

multi-index, where x

1

< 03, I > o > -- red, 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 “<” instead of “5”.Then (1) is also true for x - q, provided that 7 > 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 by iterative application of (3.2.6/4). One obtains a corresponding estimate for It follows from the first step of the proof of Lemma 1 that one can reduce Lemma 2 to (12).

7.6.3.

Self- Adjoint Trieomi Differential Operators of First Type

T h e o r e m 1. Assume that Bnl,kfrom Definition 7.2.211 i s symmetric in L2(Q). i s an operator with pure p i n t (a) Then, Bm,ki s essentially self-adjoint, the cbsure spectrum, and D(Bm,k) is the cornpktion of D(Bm,k) in the norm

458

7.6. Tricomi Differential Operators

Here S has the same meaning as in 7.2.2, ( y l , . . . , y,,) = (y’, yn) are the corresponding local coordinates *), and d(x) is the distance of a point x E 52 to the boundary a52; y = (y‘, 0 ) and DY, mean that there are only derivatives with respect to y l , . . . , yl/n-l. (b) I/ additionally d e ( y ( J )= ) 0 in (7.2.2/3), then Bnl,kis bounded from below. Proof. Step 1 . We start with preliminaries. If Am,kis the operator from Definition 7.2.1 and if C is the essentially self-adjoint operator from Theorem 7.6.1, then we consider the operator G, GU =

0E

[Am,k

+ E 0 CI U ,

D(a) = D(A,,k) 63 D ( C ) ,

(2)

in the space

G

L2(S) = L2( (0, h ) ) L2@52). (3) Here (2) must be understood in the sense of the decompositon (3). (Adescription of the [l], I, 5 2.) Since methods for tensor products can be found in Ju. M. BEREZANSKIJ Am,kand c are essentially self-adjoint, it follows that a is also essentially self-adjoint. Since Gu is symmetric also on

I

u ~ c ~ ( S ) , - = 0 for j = 0 , . . . , m - k - 1 ay;l aiu as (for k 1m the boundary conditions are omitted), one obtains that D ( B ) is the completion of (4) in the norm

(See (7.4.3/1) and Theorem 7.6.1.) Here, b(x) is the distance of a point x E S t o the boundary as. Since An,,kand C are semi-bounded operators with pure point spectrum, it follows that 14 is also a semi-bounded operator with pure point spectrum.

Step 2. Assume that d,(y(j)) = 0 in (7.2.2/3).Let x0(x)and xl(x) be two real functions belonging t o Cm(52)such t h a t &)

+ &x)

=

1 in 52,

x0(x) E C$(Q), xo(x)= 1 in a neighbourhood of 52 - S. Taking into consideration dx = gj(y(J)) dyij’

. . . d&lI

dyn,

(7.6.1/9), and Gtding’s inequality (see for instance K. MAURIN [ 2 ] , Chapter 13.3), then one obtains for a suitable domain o,where 52 - S c w c 52, that ( B n i , k ~u)L..(Q) ,

L

= ( B n t , / i U , x&)L,(Q)

(B”l,kXOU, X 0 U ) L d Q )

*) See the footnote on p. 454.

+ (Bm,/cU, x?u)L,(s)

+ (Brn,kh.XP,X 1 4 L a ( S ) - EIIUlIL;(Q)

- C ( E ) 1141;2(Q)

7.6.3. Self-AdjointTricomi Differential Operators of First Type

459

Here c > 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./
+ 5c 181

2rn

1

da(x) P U ( ~ ) ,

where Bm,lL, is a n operator as treated in the second and in the third step. In S this representation has the form (7.2.213) with the properties mentioned there. Now, the theorem is the consequence of the above considerations, Lemma 7.6.212, and the well-known criterion by T. KATOon self-adjointness of operators (see for instance T. K.4TO [3], V, Theorem 4.1, or H. TRIEBEL [17], Theorem 17.8).

460

7.6. Tricomi Differential Operators

T h e o r e m 2. Let BN,,kbe the operator from Theorem 1, where dd(y(J)) = 0 in (7.2.213). Further, let k = 0, . . . , m. Then

D@:,J

co

=

n~@i,d

j=O

for i = 0 , 1 , 2 , . . .; 1 = 0,..., m - k - 1

I

.

(8)

(For nz = k the boundary conditions are omitted.) Further, D(Biz,k), where j = 1, 2, . . ., is the completion of D(B:,,) in the norm

c

Proof. The operators Am,&and from the first step of the proof of Theorem 1 are semi-bounded. They have a pure point spectrum. Whence it follows easily that A

D((?’) = D(A i , k ) @ LZ(8)n L,(8) $ D ( c j )

7

(10)

= 1 , 2 , . . . (7.6.114) and Theorem 7.4.3(b) yield that J I U J ( D ( G ~ ~ is equivalent to (5) with 2mj instead of 2m and 2kj instead of 2k. Using Theorem 5.2.2 and the decomposition method of the second step of the proof of Theorem 1, then one obtains that (9) is an equivalent norm on D ( B i , k ) (Iterative . application of Remark 5.2.213 yields , ~ ) to W,2mj*10c(Q).) Using (7.3.1/2)and Theorem that every function u E D ( B ~ ,belongs 4.2.4, it follows that

i

D ( B i , k )c Wi21J’-kli(S).

(11)

Now one obtains, by Theorem 4.6.1, D ( & J c C”(Q). Formula (9) and Theorem 3.6.1 yield that the boundary values in (8) must be understood in the sense of the embedding theorems. But then they can be understood also in the classical sense. R e m a r k 1. The set (8) can be determined more precisely. I n a special case, we shall return to this question in Theorem 7.6.512. R e m a r k 2. Theorem 4.6.1 and (11) show that (8) can be understood also in topological sense. Here, the topology in C”(Q) is generated by the semi-norms where all multi-indices y are admitted. The topology on D(B2,k) is sup (DYu(x)1, x€D

given by the semi-norms IIBjm,kUII&(Q), j = 0, 1, 2,

. ..

R e m a r k 3. Remark 7.4.312 yields that an extension of the theorem t o k = m

. . . , 2m - 1 is impossible.

+ 1,

R e m a r k 4. Analogous results for similar classes of operators are obtained by M. S. BAOUENDI, C. GOULAOUIC [8].

7.6.5. The Spaces c,q),(sZ)

7.6.4.

46 1

Non-Self- Adjoint Tricomi Differential Operators of First Type

T h e o r e m . The operator B,,,, from Definition 7.2.211 is closable in L,(sZ). D(Bm./J is the completion of D(B,,,,k) in the norm (7.6.311). The spectrum of Bln,k consists 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 L2(Q). Proof. Step 1. Let B i L , k t 6 = C ( - 1)Iu1 D*(b,(s) U ) la1 5 Bin

be the formally adjoint operator to B,,,,, . Further it holds in S n Kj ax = gj(y(J’) dy,“’ . . . dy?!, ayn in locd coordinates. Then (7.2.213) yields that BX,k has the same structure (7.2.213). Brn,,,= 3 BNl,k 3 BL,/
+

+

Bm,IC~ = B,,,~U CU. (1) Here, B,,+. is a symmetric Tricomi operator of first type, while C is a perturbation operator of order 2m - 1, satisfying the assumptions of the third term of (7.2.213). Step 2. The decomposition method of the second step of the proof of Theorem 7.6.311 as well as Lemma 7.6.212 and Theorem 7.6.311 yield IICullLr(Q)

5 &Ilini,kU1lL,(Q)

+ C&-PIIUIIL,(Q) 9

where p > 0 is a suitable number. One obtains by k method of local coordinates that

llull

w;(n)5 c(lIBm,kUIIL,(Q)+

-

(2)

E D(im,k),

5 2m - 1, (3.2.6/4), and the

(3)

~~u~~L,(Q))~

Since the embedding from Wi(sZ)into L2(Q)is compact, it follows by the theorem

-

[17], p. 277) that i m , k is a self-adjoint of F. RELLICH(see for instance H. TRIEBEL operator with pure point spectrum. Assume, without loss of generality, that 0 is

-

not an eigenvalue. (3), (5.4.1/5), and (4.10.2114) yield &,:k E @, if p > 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 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~<~ e~<, -. 2

(15)

1

Since every function v E Corn((,, b ) ) can be represented in the form v = purr where u E C"((a, b ) ) A LL2)((a, b ) ) , then Ilvllq&) and llwllhze coincide on C$((a, b ) ) . It follows from the lemma that C$( (a,b ) ) is dense in K1= D(&). Then C$( (a, b ) ) is 1 also dense inD(&) where 0 < 8 < -. The proof of the lemma yields that C,"((a,b)) 1 is also densein K", 0 s 1 inclusively s = - .Whence it follows (8)for 0 c s < 2l ) 2 R e m a r k 4. The theorem is due essentially t o H.TRIEBEL[14,II]. The proof 1 given there is not applicable to the singular values of the parameters s = - + 1 2 where 1 = 0, 1 , 2 , . . .

-.

(

R e m a r k 5. The proof shows the difficulties arising in the interpolation of function spaces with weights, which are out of the line of the considerations of Chapter 3. The theorcm is also useful for the construction of counter-examples in the sense of Remark 1.17.1/4: The theorem yields

( W W b), p z , I), L,((a, b)))*,2 = Wi((a, b ) , p , 1) = Wi((a,b ) , p , p ) * On the other hand, one obtains by Theorem 3.2.6 and Theorem 3.4.2 that

( f m a ,b), P2,I), L,((a, b)))*,2

=

(WE(@, %7 p z , p - 2 ) ,~ , ( ( ab,) ) * , 2

=

p-1).

It follows from Theorem 7.3.2/1 and from W22((a,b ) , pz,1)

=

q(a,b ) , p ,

Wi((a, b ) , p2)

that @,"((a,0), p 2 , 1) has a finite codimension with respect to W i ( ( a ,b ) , p2, 1). Assuming (in the sense of Remark 1.17.1/4), that the norms of Wi((a,b ) , p , p ) and IVl( (a,b ) , p ,p - l ) areequivalent on C$( (a,b ) ) ,then i t follows from Theorem 3.6.1 that v = 1 is a n element of Wi((a,b ) , p , p-'). This is a contradiction. Starting from this result one can construct the counter-example mentioned in Remark 1.17.1/4.

7.7.2.

Legendre Differential Operators (General Case)

The extension of Theorem 7.7.1 t o the Legendre differential operators An,,kfrom Theorem 7.4.1, givesrise to many difficulties. Theorem 7.4.3 and Remark 7.4.312 show that it is meaningful to restrict oneself to the eases Iz = 0, . . ., m. We formulate a partial result without proof. T h e o r e m . Let m = 3 , 4 . . . ., k = 2 , . . ., m - 1 and k even, k and m are relative 1 3 5 prime. Further, let s 2 0, 2ms += -, -, -, . . . and (4na - 2k) s 1 , 3 , 5 , . . . Then 2 2 2 D(AS,,J is the completion of D(A:,,<) (Theorem 7.4.3) in Wiff18((a, b ) , pzks,1).

+

30*

468

7.8. Distributions of Eigenvalues, Green Functions

R e m a r k . A proof of the theorem can be found in H. TRIEBEL [14,II]. The (rather unnatural) additional conditions are needed in the proof to avoid “singular” values 1 1 of the parameters. In the case m = k = 1, these are the values s = - + - where 4 2 1 = 0,1,2,...

7.7.3.

Tricomi Differential Operators of First Type

Let B,n,kbe the self-adjoint Tricomi differential operator of first type from Theorem 7.6.312. The domains of definition D ( & , k ) , s = 0, 1, 2, . . ., are determined in Theorem 7.6.312 by reduction t o the domains of definition D(88) from (7.6.3/10). This method may be carried over to fractional powers. (7.6.3110) is valid, too, if one replaces j by s 2 0. The domains of definition D(Am,k) are a t least partly determined in Theorem 7.7.1 and Theorem 7.7.2. With the aid of the method of local coordinates, it follows that D(C*)= Wim8(ds2).On this basis, one can determine the domains of definition D ( & J . (One has t o use the method from the proof of Theorem 7.6.312.) We do not go into detail here.

7.8.

Distributions of Eigenvalues, Green Functions

In this section, distributions of eigenvalues of Legendre differential operators and Tricomi differential operators are determined. The results are similar t o Theorem 5.4.2 (regular elliptic differentia1 operators) and t o Theorem 6.6.111 (strongly degenerate elliptic differential operators). On this basis, properties of Green functions are considered. 7.8.1.

Legendre Differential Operators

The used symbols have the same meaning as in Subsection 5.4.1. i s the self-adjoint operator from Theorem 7.4.1, then we have T h e o r e m 1. If 1

+ N(A)

-

1

1

+A X .

(1)

Proof. (7.4.2/3)yields that D(A,,J is a closed subspace of Wim((a,b ) , p“). One obtains from Theorem 7.3.2/1, Theorem 3.2.6, and (3.8.2/3) that s j ( 1 ; ~ i m ( ( a 61, , ~ z ’ c ) ,~ , ( ( aa))) ,

sj(I; @im((a,b), pm), ~ z ( ( ab))) , N

N

sj(I; Wgrn((a,b), pZk,f k - 4 m ) , &((a,a))) j-zm.

Now, (1) is a consequence of Theorem 5.4.1/1.

7.8.1. Legendre Differential Operators

469

R e m a r k 1. The question arises whether (1) can be extended t o an asymptotic formula with remainder estimate similar to (5.4.2/3). G . BERGER [l] proved that 1

N(I)=7c

b

--

1

ax

1

ilx +o(il=logil).

.J'..iX)

(2)

The method by G. BERGERie also applicable if k is not an integer, 0 < k < 2m. of

T h e o r e m 2. Let K s be the space from Definition 7.7.1, and let ;I be 720 eigenvalue the operator Al,l from Theorem 7.4.1. Then (& - AE)-1 can be represented i n the

A

A

G(%, y ) E K P 6 L , ( h b ) ) n &((a, b ) ) 6 K e ,

(4)

3 3 P r o o f . Step 1. We use the method from the proof of Theorem 5.4.4. If f j are the orthonormal eigenfunctions of & , then

if and only if 0 S

(Al,1

Q

< -.

- ilE)-l f

=

m

C

1 ~

j = 1 Aj

- il

(f, f j ) . &( ( a,b) f j

(5)

*

Here, we assume without loss of generality, that f j are real functions. Setting N

1

then one obtains similarly to (5.4.4/9) that

Theorem 1 , where m = k = 1, and Theorem 5.4.1/1 yield

ilj

+1

N

jz. If

e < -,2 3

then the right-hand side of (7) converges. Since D(A$) = Ii@ (Theorem 7.7.1), one obtains (3) and (4) similarly t o the second step of the proof of Theorem 5.4.4.

Step 2 . Assume that (3) and (4)hold for the right-hand side of (4)coincides with

H = D ([A,,, 0 E

3 e 2-. ByTheorem 2

7.7.1, the space on

+ E 0 A,,,]")

P

If {f,},?l denotes the system of orthonormal eigenfunctions of Al,l, then the system

((4+

+ l)-lfi(z)

m

fk(y))j,k-l

is compleke and orthonormal in H (after the choice

470

7.8. Distributions of Eigenvalues, Green Functions

of a suitable equivalent norm in H ) . Developing G(')(s, y) with respect t o this system and taking into consideration

R e m a r k 2. Using Theorem 7 . 7 2 , then one can prove a corresponding theorem for (Anl,k- AE)-l, provided that the assumptions of Theorem 7.7.2 are satisfied.

7.8.2.

Tricomi Differenticl Operators of First Type

T h e o r e m . Let Br,,,kbe the self-adjoint Tricomi differential operator of first type from Theorem 7.6.311. Then

1

+ N(A)

n

2%

+ 1.

(1) P r o o f . Step 1. It follows from the proof of Theorem 7.6.3/1 that one may assume, without loss of generality, that d d ( y ( j ) )= 0. If B is the closure of the operator G from (7.6.3/2),then Lemma 5.4.1/1, Theorem 7.8.1/1, and (7.6.113) yield N

n

1 + Nc(A) 1 + A=. (2) If xt and x; have the same meaning as in the second step of the proof of Theorem 7.6.3/1,then it follows (by a suitable choice of x;) from the considerations a t that place N

'J(X?I;D ( E , f l , k ) t &(G)) s,(I; D(G),&(s)). Now, one obtains from (2) and Theorem 5.4.1/1 that

Step 2 . Extension operators are denoted by S and restriction operators by is a ball such that W c 0, then I l V ~ " ( m ) -+L2(w) =

R.If

w

RL,(Q) L.Jm)ID(%,,,,k)+ L2(Q)SW;"'(w)+D(Em,k). -+

Then the ideal properties of the s-numbers ((l.lG.1/25)and (1.16.1/26)with s,, instead of all)and (4.10.2/14)yield 2rn

s j ( I ; D(Brn,k),&(GI) 2 ci-". (1) is a consequence of ( 5 ) ,(6), and Theorem 5.4.1/1.

(6)

7.8.3. Tricomi Differential Operators of Second Type

47 1

R e m a r k 1. We sketched in Subsection 7.7.3 how t o determine the domains of definition of fractional powers of B,n,k. Then one can derive differentiability properties of Green functions similar to Theorem 7.8.112. See also Theorem 5.4.2 and Theorem 5.4.4.

R e m a r k 2. * The question arises whether (1) can be reinforced in the sense of (5.4.2/3). For m = 1 and k = 0, 1, there are given asymptotic formulas for N ( I ) by M. GUILLEMOT-TEISSIER [a]. Her method is a reinforcemelit of the above considerations. Probably, there can be obtained asymptotic foriiiulas for N ( 1 ) also for the operators B m , kin this way. Further, 111. S. BaOUENDI, C. GOULAOUIC [8] considered operators of the type B1,l.They proved asymptotic formulas for N ( I ) in the sense of (5.4.2/3) without remainder estimates. R e m a r k 3. For generation of the spaces

cs(l2)-from Definition 7.6.5, we intro-

If to is a domain such that duced in Subsection 7.6.5 the self-adjoint operators B'm,k.

6 c Q, then it follows from (7.6.5/3) (and the extension method) that

Wp(o)c D(gm,k)c Wgm(Q). Then, using the consideration-of the proof of the above theorem, it follows that (1) is also true for the operators BmSk.

7.8.3.

Tricomi Differential Operators of Second Type

T h e o r e m . Let

holds and

A be the self-adjoint differential

+ N(A)

-

+ An-1 cl(l + A) 6 1 + N ( 1 ) 1

1

for n

operator from Theorem 7.6.611.

= 3,4,.

c2(1 + A log A)

..

Then it (1)

for n

=

1.

(2)

Here, c1 and c2 are positive constants. Proof. Step 1 . The definition of A , (7.3.1/2),and the method of local coordinates

yield

D(A+)=

Wi(Qn; b, 0)= W#2;

b).

By Theorem 3.6.1, C$'(Q) is dense in Wi(l2; 0). Then Theorem 3.8.2, Remark 3.8.2/1, and Theorem 5.4.1/1 prove (1) for n _2 3. Step 2. Let n = 2. In this case, one has t o use Remark 3.8.2/1 and (3.8.2/2). For the eigenvalues I j of A + E , one obtains by (5.4.115) and (5.4.1/6) that

472

7.8. Distributions of Eigenvalues, Green Functions t

Since -is a monotonically increasing function for large values of t, it follows from (3) log

t-= logt

il c

yields log t

N

log il.Hence,

cil log 1.

N(1)

(4)

This proves the right-hand side of (2). The left-hand side is a consequence of

ww)

D(A*),

Theorem 5.4.112 (in relation with Remark 5.4.1/3), Theorem 5.4.1/1, and (4.10.2/14). R e m a r k 1. * The theorem can be reinforced and generalized. The first considerations concerning the spectral properties of A were given by M. s. BAOUENDI, C. GOULAOUIC [2,3]. Reinforcements and generalizations are due t o N. SHIMAKURA [3,6], C. GOULAOUIC [4], V. N. TULOVSKIJ[l], P. GRISVARD [8],L. BOUTETDE MONVEL, P. GRISVARD[l], A. EL KOLLI [2,3], C. NORDIN[l], and PHAM THE LAI[a]. The results obtained in C. NORDIN[l] deal also with differential operators of the above type in Riemannian spaces. In particular, it is possible t o give a n asymptotic formula for N ( 1 )in the sense of (5.4.213) (without remainder estimate). For n = 2 one obtains 1

+ N(1)

-

1

+ illog1.

R e m a r k 2. * The above method permits important generalizations. If SZ c R, is a bounded C"-domain, then one can consider for 0 5 a < 2m the bilinear form

BJu,

V) =

J d"(x) C

ISI6m

I2

D@ugS, dx

(or modifications). If Ha denotes the closure of CC(s2) in the norm

IIUIIH,

= [B,(u,

U)

+ IIull~*~nJ*~

then, by Remark 5.4.113,thereexists a self-adjoint operator A, such that D ( A i ) = Ha. Here A, is a self-adjoint (semi-bounded) extension of

Now one obtains by Theorem 3.8.2, Remark 3.8.2/1, and Theorem 5.4.1/1 t h a t 1

+ N(1)

1

+ N(1)

-

-

n

1

+1

1

+ A==

5

n-1

for O

2m

5 a < -,n

2m for -< dc < 2m, n

(7)

7.9.1. Boundary Value Problems for Degenerate Elliptic Differential Operators

and a corresponding estimate for a

=

-.2m 1E

4 73

The formulas (7) and (8) are known. The

first results in this direction are due to I. A. SOLOMEBC) [l]. One can show that (7) and (8) and a corresponding formula for a

2m

= -rema.in valid n

if H, is the completion

[l] there are also of c”(Q)in the norm llullH,. I n the paper by I. A. SOLOMEQ~ asymptotic formulas for N ( 1 ) in the sense of (5.4.2/3) without remainder estimate. A reinforcement and a generalization of these results are given by I. L. VULIS, M. Z. SOLOMJAE [1,2] and I. L. VULIS[l]. Further, K. MARUO[l] and G. BERGER[l] improved (7) by an mymptotic formula with remainder estimate. We refer also t o M. G-EMOT-TEISSIER [5,6]. See also Theorem 6.6.1/2.

7.9.

Complements

In this section, there are sketched some results of the theory of degenerate singular

elliptic differential operators, which did not play any role in the previous considerations. Essentially, we restrict ourselves t o references.

7.9.1.

Boundary Value Problems for Degenerate Elliptic Differential Operators

The fifth chapter was concerned with general (regular) elliptic boundary value problems. Theorem 6.2.2 and the boundary value operators Bj a t that place were the main notions. On the other hand this chapter was concerned with very special boundary value problems for degenerate elliptic differential operators (see Definition 7.2.1, Definition 7.2.211, and Definition 7.2.212). In recent years, important progresses in the treatment of general boundary value problems for degenerate elliptic differential operators have been made. If SZ c R,,is a bounded C”-domain, then M. I . VI~ I K ,V. V. GRU~IN [l] considered differential operators which can be represented in local coordinates (see Subsection 7.2.2) by

Here 1, = lial 2 0, while ua(y(j)) arecontinuous coefficients in 8 (see Subsection 7.2.2). There are considered boundary value problems

L(z,D ) = f ,

(2)

in the framework of L,-theory. The main aim of the paper is the formulation of conditions under which (2), (3)is meaningful and under which (L(z, D), B,) is a @-operator between suitable function spaces (see Theorem 5.2.2 and Theorem 6.2.4). Closely related t o these investigations are the papers by N. SHIMAKURA [l,4,5] and P. BOLLEY, J. CAMUS[3,4,6] as well as the papers quoted at the end of the introduction 7.1,

474

7.9. Complements

in Remark 7.2.1/2, and in Remark 7.2.213. N.SHIMAKURA [4] and P. BOLLET, J. CAMUS[6] consider differential operators of the form

Here, y(J) are local coordinates in the sense of Subsection 7.2.2. Formula (4) is a representation in S, where S has the same meaning as in Subsection 7.2.2. k and rn are natural numbers, 1 5 k 5 m. P1Ii-“is a differential operator, the order of which is less than or equal to n~ - h. Clearly, (1) and (4) are closely related t o each other. J. CA4MUS[6] proved a-priori-estimates for the I n a comprehensive paper, P. BOLLEY, problem (3), (4),and investigated under which conditions ( L ( z ,D),B,)is a 0operator between suitable spaces.

7.9.2.

Tricomi Differential Operators, Analytic Functions, and Functions of Gevrey Classes

The last three chapters contain many assertions about differentiability properties of solutions of (regular and degenerate) elliptic differential operators. In 1900 D. HILBERT formulated in his famous lecture “Mathematische Probleme” the task to investigate under which conditions solutions of linear and non-linear differential equations are analytical. (The 19th problem. See “ Die Hilbertschen Probleme”, Ostwald’s Klassiker der exakten Wissenschaften 252, Akad. Verlagsgesellschaft Geest und Portig, Leipzig 1971. The book contains also surveys written by 0. A. OLEJKIK end A. G. SIGALOV on the recent situation of the investigations.) References concerning (linear) elliptic differential operators can be found in the just quoted book as well as in the third volume of J. L. LIONS,E. MAGENES[2]. Further, we refer to 0. A. OLEJNIK [l]. I n recent years, there are made a number of successful investigations of this problem for Tricomi differential operators. These questions are closely related t o the so-called Gevrey classes. (This is also valid for regular elliptic differential operators.) Let K c R, be a compact set. A function u belongs t o the Gevrey class Q J K ) , s 2 1, if u is a Cw-functionin a neighbourhood of K , and if there exists a constant L > 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 j . Using (8.2.1/2) whence it follows that D(A") is a Montel space. This proves (a). Step 3. Let A be an operator with pure point spectrum, let A, be its eigeiivalues (inclusively their multiplicities), 0

5 [All 6 l1,I 5 . . .)

and let {hJ}Eibe the corresponding complete system of orthonormal eigeiie1ement.s. Then

f

-+

{ ( f , 'J)lf};i

(4)

is an isomorphic mapping from D(A") onto the sequence space qil,) =

(5 I5

=

(i~ 61 l complex, ,

This is a consequence of the spectral resolution of A .

Step 4 . If D(A") is a nuclear space, then it follows from the first, step t)hat A is an operator with pure point spectrum. Consequcntly, D(.4 ") is isomorphic to from (5). If Ik,iand F , have the same meaning as in Subsection 8.2.1, then (8.2.1/5)holds. One obtains, similarly to (8.2.1/6),that Ile1Ib,=

(1

+ n:q*,

Ilflllbk,

= (1

+ n:")-*.

5.2.2. The Structure of the Spaces D ( A m )

11-79

Hence, Ikh.,j is nuclear if and only if

c (IA,l + z= m

Using

l)-(k-J)

1

Z(l

<

00.

1

+ I&l)-(k-j)

r= 1

(1

+ lArl)-(W),

then it follows that (6) is equivalent t o

1

+ 1111 2 CZ”,

I

=

1 , 2 , 3,

. . .,

(7)

where c and x are suitable positive numbers. (7) is equivalent t o

This proves (2). At the same time one obtains from the above considerations the reversion: If A is an operator with pure point spectrum and if (2) is valid, then D(A”) is a nuclear space.

Step 5 . D ( A ” ) is isomorphic to s if and only if s{~,}is isomorphic t o s. Denoting by I k , . , ( q ~ ~resp. } ) , F,(s(A~}), the operators, resp. spaces, from Subsection 8.2.1 with respect t o s { ~ then , it follows from ( 5 ) , (5.4.1/5),and (5.4.1/6)that

A corresponding relation holds for the space s, where the topology in s cam be generated by ( 5 ) , where AL = I . If sf^,) and s are isomorphic, then the approximation numbers from (9) can be cstimated from above and from below by the corresponding numbers sr(IA,,j,(s), Fk,(s),F , , ( s ) ) for the space s with a suitable choice of k’ and j‘. Hence, there exist positive numbers U, , 02, c1 and c2 such that CIP’ 5 1

+

s CZP,,

I

= 1, 2, 3,

. ..

(10)

Assuming that (10) is true, then s{nj} = s follows. In analogy t o (8),formula (3) is a consequence of (10). R e m a r k 1. The theorem is the basis forthe further considerations. If D ( A ” ) is a nuclear space, then the spaces Fj from Subsection 8.2.1 are Hilbert spaces. The generation of nuclear spaces by a sequence of Hilbert spaces is not restricted to spaces of the type D(A”). One can prove that every (general) nuclear space can be [3], generated in such a way. We do not go into detail here and refer t o A. PIETSCH 4.4.1. But this gives a motivation for the investigation of nuclear spaces with the aid of the theory of Hilbert spaces.

R e m a r k 2. * The spaces D ( A ” ) are introduced by B. S.MITJAGIX [l]. Part (a) of the theorem and a variant of part, (b) are proved by A. PIETSCH [l]. Part (a) goes back to H. TRIEBEL [ll]. An important motive for introclucing the spaces D(A”) is t,he question on the existence of bases in special nuclear spaces. Clearly, every nuclear space D ( A m )has a basis, namely the complete system of eigenelements of the corresponding operator A with pure point spectrum. I n this connection we refer t o B. S.MrTJAOIN [l] and W. WOJTY~SKI [l].

480

8.3. Structure of Nuclear Function Spaces

8.3.

Structure of Nuclear Function Spaces

Theorem 8.2.2(c) is the basis for the investigation of the structure of numerous nuclear function spaces. In the first subsection, we prove a comparatively simple but far-reaching criterion. The second and the third subsections are concerned with the structure of concrete nuclear function spaces.

8.3.1.

General Structure Theorem

T h e o r e m . Let Q c R, be an arbitrary domain, and let

Au =

c

la1 5 rn

a,(x)Wu,

A,

D(A)= C$($2),

(1)

be a symmetric operator i n L,(Q) with C"-coefficients a,(x). Further, it is assumed that there exists a self-adjoint extension A of A , and that D(A") is a nuclear ( F ) - s p c e .Then D(A") is isomorphic to the space s of rapidly decreasing sequences. P r o o f . Step 1 . Theorem 8.2.2(b) yields that A is an operator with pure point spectrum and that

N(I)

+1

c,I7a

(2)

holds, where c2 > 0 and t 2> 0. It follows from Theorem 8.2.2(c) that it is sufficient for the proof of the theorem t o verify 1

+ cl171 5 1 + N ( 1 )

(3)

for suitable numbers c1 > 0 and tl > 0. Step 2. The hypotheses yield

it?(,) c

D ( 2 )c D ( A ) .

Here, w is a n open ball such that 6 c $2. (The functions of l?g(w) are extended by zero outside of w.) Denoting extension operators by S, restriction operators by R, and embedding operators by I , then it follows that =

lGr(q-+~,(m)

R L , ( Q ) + L ID(A)-L,(Q) ,(~) SV+~(~)+D(.~) +

Whence, one obtains for the approximation numbers sj that

s j ( I ; $T(w), L,(o))

c s , ( l ; D ( A ) ,L2(Q)), c > 0 .

(5.4.1/5), (5.4.1/6), and (4.10.2/14) (the former extension methods show that this formula holds also for @r instead of W p ) yield

lAjl-' 2

rn

cj-",

c > 0, j 2 j o .

Whence i t follows (3). R e m a r k . A variant of this theorem was formulated by H. TRIEBEL [25] without proof. Hence, s p c e s D(A") of the above type are isomorphic to s if and only if (2) is satisfied.

8.3.3. The Spaces @(Q) and c"(Sa)

8.3.2.

The Spaces S,(x.(sd) and

48 1

c,"(sd)

T h e o r e m 1. Let D c R,, be a n arbitrary domain, and let be the ( F ) - s p m from Definition 6.2.111. Further, let e-"(x) E L,(D) where a 2 0 is a suitable number. Then SQ(z) (D) i s isomorphic to the space s of rapidly decreasing sequences. Proof. The theorem is a consequence of Theorem 8.2.2(c), Theorem 6.4.3(c), and Theorem 6.6.111. R e m a r k 1. It follows from Theorem 6.2.3 that the hypothesis e-"(z) E L ~ ( Q ) , where a 2 0 is a suitable number, is necessary if one wants to identify the spaces Se(zl(sZ) and D(AD") where A is a self-adjoint operator in L,(Q). T h e o r e m 2. (a) The Schwartz space S(R,) of rapidly decreasing infinitely differentiable functions is isomorphic to s. (b) If D c R, is a bounded domain, then the space c$(Q)from Lemma 6.2.3 is isomorphic to s. P r o o f . The theorem is a consequence of Theorem 1, Remark 6.2.312, and Lemma 6.2.3. R e m a r k 2. It is well-known that S(R,) is isomorphic to s. I n the proof that is isomorphic t o s there are not needed any smoothness assumptions for the [S, 11, 241. boundary. Theorem 1 and Theorem 2 are due t o H. TRIEBEL

c:(D)

8.3.3.

The Spaces

e,

(sd) and Ern (sd)

T h e o r e m . Let D c R, be a bounded C"-domain. Then the spaces c$(sZ) from Definition 7.6.5 and from Remark 7.6.511 are isomorphic to the space s of rapidly decreasing sequences. P r o o f . If 1 = j , then C,?$2) = c$(Q) holds. Hence, this case is contained in Theorem 8.3.212. If 1 < j , then the theorem is a consequence of Theorem 7.6.512, Theorem 7.8.2, Remark 7.8.213, and Theorem 8.2.2(c). and s was proved by H. TRIER e m a r k 1. * The isomorphism of the spaces cz(D) BEL [Ill. I n this paper, there are also other nuclear function spaces which are isomorphic t o s. The important special case cm(D)= Czo(D) was considered ealier by M. S. BAOUENDI, C. GOULAOUIC[3,2] and by H. TRIEBEL [7]. I n the above proof, C. GOUthere are used Tricomi differential operators of first type. M. S. BAOUENDI, LAOUIC [3,2] proved the isomorphism of cm(D) and s by the same method, but on the basis of Tricomi differential operators of second type : Namely, the isomorphism of cm(D)and s is a n immediate consequence of Theorem 8.2.2(c), Remark 7.6.614, C. GOULAOUIC [8] gave also a proof of the isoand Theorem 7.8.3. M. S. BAOUENDI, morphism of Cm(D)and s with the aid of (modified) Tricomi differential operators of C. GOUfirst type. I n the above theorem and also in the papers by M. S. BAOUENDI, LAOUIC, it is supposed that D is a bounded C'"-domain. Using other methods, M. ZERNER [ l ] proved that cm(D) and s are isomorphic if the boundary of the bounded domain D c R,,satisfies only a Lipschitz condition. Further, we refer to B. HAKOUZET [5] and M. S. BAOUEPJDI, C. GOULAOUIC, B. HASOUZET [I], where it is proved 31

Triebel, Interpolntion

482

8.3. Structure of Nuclear Function Spaces

that c”(l2)and s are isomorphic, provided that use modified Tricomi differential operators.

SZ is a domain of cuboid-type. They

R e m a r k 2. * The isomorphism between D ( A” ) and s is given by (8.2.2/4). At the (a,b ) ) , same time, the elements (IA,},?.~ are a basis in D(AO”).For special spaces, ern( c $ ( ( u ,b ) ) , S(R,), etc., there exist many explicit bases generating isomorphic mappings onto s. So, B. s. MITJAGIN [l] proved, that the 6ebyiev polynomials are a basis in C ” ( ( - l , 1)). The proof that the Legendre polynomials are a basis in em((- 1 , l ) ) is due to M. GUILLEMOT-TEISSIER El]. (This assertion follows from the above method and Theorem 7.4.3, if one takes into considerations that the Legendre polynomials are eigenfunctions of the classical Legendre differential operator.) The Hermite functions are a basis in S(R,), see B. S. MITJAGIN [l]. (This assertion follows also from the above considerations, since the Hermite functions are the eigenfunctions of the Hermite differential operator which is a special case of the operators treated in [3], 10.3, and in S. ROLEWICZ Chapter 6.) Further examples can be found in A. PIETSCH [I]. R e m a r k 3. A number of important nuclear spaces are not obtained by the above method. Examples are CF(Rl),Cm(R1)as well as the nuclear spaces of harmonic and [3].) analytic functions. (The definitions of these spaces can be found in A. PIETSCH B. S. MITJAGIN [l] proved that Cm(R1)is isomorphic t o ( s ) where ~ L = ( 1 , 2 , 3 , . . .). This structure result and Theorem 8.3.1 yield that Cm(R1)cannot be represented as D ( A ” )where A is a self-adjoint differential operator in L,(R,). Finally, we mention a [l]. I n this paper, there are considered conclusion of a result by G. V. ROZENBLJUM distributions of eigenvalues of self-adjoint polyharmonic differential operators A with Dirichlet boundary conditions in unbounded domains. G. V. ROZENBLJUM [l] proves that there exist domains SZ where A is an operator with pure point spectrum in L,(S) such that (in contrast t o the usual behaviour of eigenvalue distributions for differential operators)

N(1)

-

ecA, c

> 0,

holds. (The paper contains essentially sharper asymptotic formulas in the sense of (5.4.2/3) without remainder estimate.) Now, it follows from Theorem 8.3.1 and Theorem 8.2.2 t h a t D(A’”)is a Monte1 space but not a nuclear space. R e m a r k 4. I n connection with the above results, there arises the problem formulated by A. GROTHENDIECK [2] whether each nuclear (P)-space has a basis. As it was shown by N. M. ZOBIN,B. S. MITJAGIN[l] (see also B. S. MITJAGIN,N. M. ZOBIN [l]),the answer is negative.

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[I] 0

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E.;

33

Triehel, Interpolation

514

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[2]

nOIIepeW3HKH M HOXW TB B @yHKUEOHaJlbHblX IIpOCTpaHCTBaX

6 n u x e ~ ~ iYCIIeXH i. O n H O 3aMeYaH He

M aTeM . HayK

06

YCIIeXH MaT eM . HayK

PZ-MepHbIX

H TeOpHR HaHJIy'ILUHX IIpH-

15: 3 (1960), 81-120. IIOIIepeYHHKaX

MHOKeCTB

B

6aHaXOBbIX

IIpOCTpaHCTBaX.

20: 1 (1965), 227-230.

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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 ,

<

,

,

s.

BIRMAN, M. 291, 323, 347, 351, 355-357, 396,486 S. 192, 487 BOCHNER, S. V. 236,487 BO~KAREV, BOGAEEV, B. M. 307, 487 P. 430, 431, 433, 473, 474, 487 BOLLEY, BOMAN, J. 86, 316,487 V. V. 357, 397, 487 BORZOV, DE MONVEL, L. 472,487 BOUTET BR~ZIS, D. 149, 487 F. E. 149, 246, 378, 396, 397, 488 BROWDER, J. 396, 488 BRWNING, JA.S. 306,488 BUGROV, V. I. 170,306,323,329,488 BURENKOV, P. L. 6, 16, 23, 25, 29, 41, 62, 75, 81, BUTZER, 96, 98, 100, 127, 132, 143; 144, 150, 179, 190, 327, 485, 487489

A. P. 15, 21, 55, 57, 58, 61, 66, 67, CALDER~N, 72,117,129,134,145,146,151,160,161,169, 180, 181, 184, 186, 315,489 8. 171 CAMPANATO, CAMUS,J. 430,431, 433, 473, 474, 487 Z. A. 236, 489 CANTURIJA, CARLEMAN, T. 396 L. 166,489 CATTABRICA, E. P. 353, 354, 483 CATTANEO, K. 192,487 CHANDRASEKHARAN, CHERSI,F. 144,489 CIESIELSKI,Z . 343,489 C. 353, 354, 396, 489 CLARK, CLEMENTS,G. F. 355, 357,489 H. 0.454,489 CORDES, COTLAR,M. 136, 161,485,490 R. 151,393, 396,490 COURANT, A. A. 454, 490 CUMAK, CWIKEL,M. 129, 490 DAY,M. M. 236,237,490 DENY,J. 224, 490 DERRIDJ,M. 433,474, 490 N. 21, 144, 149, 490 DEUTSCH,

523

Author Index ~

DMITRIEV, V. J. 71, 143, 145,490 DOMSTA, J. 343,489 W. F. 144,490 DONOQHUE, DOUQLIS,A. 364, 368, 378, 389, 403, 404, 483, 490 DUNFORD, N. 5, 58, 59, 75, 127, 135, 151, 157, 384,409,490 DYNIN,A. S. 389,490 A. D. 306,358,486, 491 D~ABRAILOV, A. S. 306,491 D~AFAROV, EBERLEIN, W. F. 199, 332 R. E. 141,491 EDWARDS, S. D. 400, 504 EJDEL’HAN, ENFLO,P. 110, 236,491 FABES,E. B. 161, 378, 491 FARAUT, J. 454,491 FAVINI,A. 145, 146, 148, 149, 245, 278, 302, 360,491 FECHNER, W. 308,491 FEFFERMAN,~. 167,168,171,179, 196, 491,492 FEJQIN,V. I. 407,484 FICHTENQOL’C, G. M. 210, 492 FLECKINGER, J. 397,492 FLETT,T. M. 171, 192, 199,492 FOIA?, C. 144,492 FORTUNATO, D. 404,492 FOURNIER, J. 354,483 FREITAG, D. 148, 353, 492 FRIEDRICHS, K. 0. 151 FROLOV, N. N. 171,492 FUEIK,S. 343, 492 FUJIWARA, D. 103, 334, 343, 492 A. V. 433,492 FURSIKOV,

GAGLIARDO,E. 15, 16, 20, 21, 25, 28, 29, 69,

144, 151, 169, 183, 190, 484, 492 GAPAILLARD, C. 147, 493 V. F. 236, 493 GAPO~RIN, GARDINQ,L. 396 I. M. 108 GEL’FAND, I. V. 316,493 GEL’MAN, G. 397,402,403,493 GEYMONAT, GILBERT,J. E. 131, 493 GIL’DERMAN, Ju. I. 359, 493 GIRAUD,G. 160 I. M. 416,444,493 GLAZMAN, GLUSKIN,E. D. 351, 493 I. C. 107, 108, 394, 395, 400, 427, GOCHBERG, 493 GOLOVPIN, K.K. 85, 136, 151, 170, 180, 224, 242, 316, 493

~

GORLICH,E. 179, 196, 488, 494 C. 245, 278, 306, 404,494 GOODJO, GOULAOUIC, C. 21, 118, 131, 143, 149,433,460, 463, 471,472,474,481, 484, 485,494 GRAMSCH, B. 353,494 GRISVARD, P. 16, 41, 46, 51, 65, 66, 76, 93, 94, 148, 149, 171, 184, 186, 208, 217, 224, 225, 244, 245, 248, 262, 266, 284, 306, 320, 321, 358, 397,402, 403, 465,472,487,493,494 GROMES,W. 397,454,494 A. 477,482,494 GROTHENDIECK, GRUBB,G. 397,494 V. V. 430, 473, 517 GRU~IN, A. CH. 306,494 GUDIEV, M. 433, 472, 473, 482, GUILLEMOT-TEISSIER, 495

B. 242, 299, 304, 307, 433, 474, HANOUZET, 481,485, 487, 495 HARDY, G. H. 141, 151, 196, 207, 262, 495 F. 17, 138 HAUSDORFF, HAYAKAWA, K. 117, 118, 148, 495 G. W. 150,495 HEDSTROM, HEINIQ,H. P. 138,495 B. 245,495 HELRRER, G. 414,495 HELLWIQ, HERZ,C. S. 171, 495 HILBERT,D. 151, 160,474, 490 HILLE,E. 75, 76, 496 I. I. 180, 192,496 HIRSCHMAN, HOLMSTEDT, T. 29, 135, 144, 149, 496 L. 141, 152-154, 157, 160, 165, HORMANDER, 166, 171, 197, 231, 360, 364, 378, 381, 382, 397, 414,453, 459,496 HOVEL,H. W. 98,496 HUQHES, E. 321,496 IGARI,S. 166, 496 IL’IN, V.P. 151, 161, 170, 224, 226, 245, 306, 313, 316, 324, 329, 357,486,496 ISMAQILOV, R. S. 351, 352, 353, 496

G. N. 225, 249,497 JAKOVLEV, JAROSZEWSKA, M. 136,497 JEROME, J. W. 350, 351,357,496,497 JOHN,0. 343, 492 JOHNEN, H. 327, 488 R. 167, 171,497 JOHNSON, JOHNSON, W. B. 343, 497 KADLEC, J. 58, 249, 265, 353, 486,497, 501 A. I. 167,497 KAMZOLOV, KANTOROVI~, L. V. 141, 497

524

Author Index

KARAD~OV, G. E. 147, 149,497 KATO,T. 98, 143,292,421,459,497 KATRACHOV, V. V. 475,498 KAZARJAN, G. G. 316,498 KERZMAN, N. 136,485 KIPRIJANOV, I. A. 307,475,498 KISLJAKOV,S. V. 354,498 KNIEPERT,D. 407,498 KOHN,J. J. 475, 498 KOIZUMI, S. 136,498 KOLLI,A. EL 291, 296, 347, 351,472,498 KOLOMOQOROV, A. N. 108,350,357,498,499 KOMATSU, H. 93-95, 98, 100, 103, 107, 146, 208,499

KOMIJRA, T. 477, 499 KOMURA, Y. 477,499 KONDRA~OV, V. I. 207,226,499 KONIQ,H. 353,499 KORNEJCUK, N. P. 353,499 KOROTKOV, V. B. 353,359,497,499 KO~ELEV, A. I. 378,499 KOTHE,G. 476,499 KOVALENKO, I. A. 400,499 KO~EVNIKOV, A. N. 397,454,500 KRASNOSEL’SKIJ, M. A. 96, 98, 100, 236, 292, 499,500

KRASOVSKIJ, Jn. P. 400, 499 KRAYNEK, W. T. 136,500 K R E ~P., 135-137,149,161,166,167,320,485, 500

KREJN,M. G. 107,108,292,394,395,400,427, 493,500

KREJN,S. G. 15, 17,55,136,145-149,237,500 KRETSCHMER, H. 427,500 KUDRJAVCEV, L.D. 151, 170, 217, 245, 248, 298, 299, 304, 306, 307, 357, 486, 500

KUFNER, A. 58, 249, 265, 280, 282, 308, 475, 486,497, 501

K a o v , R. D. 306, 501 Ku~cov,N. P. 150, 501 KURATSUBO, S. 166,496 KWAPIE~~, S. 236, 501 LACROIX-SONRIER, M.-T. 143, 501 LANQEMANN, B. 407,428, 501 LARSEN, R. 167, 501 LAX,P. D. 231, 501 DE LEEUW, K. 86, 501 LERAY, J. 231, 501 LEVI,B. 151 LEVITAN, B. M. 397, 501 LINDENSTRAUSS, J. 120, 236, 237, 242, 343, 501,502

LIONS,J. L. 15, 16, 30, 33, 35, 37, 41, 44, 51,

55,61, 66, 69, 71, 75, 103, 117, 118, 120, 129, 143, 146, 149, 151, 180, 181, 183, 184, 224, 225, 227, 228, 230, 231, 266, 306, 317, 320, 331, 364, 378, 380-382, 388, 391, 394, 401, 402,474,490,492,502, 507 LITTLEWOOD, J . E . 141, 151, 196, 207, 282, 495 LITTMAN, W. 166, 167, 179, 502 LIZORKIN, P. I. 151, 161, 166, 167, 170, 179181, 190, 199, 224, 226, 245, 306, 367, 358, 486,503, 507 LOFSTROM, J. 6, 150, 167, 503 LOPATINSKIJ, JA.B. 364,503 LORENTZ, G. G. 108, 132, 145, 149, 292, 350, 356, 357,488, 503, 504, 515 LOSSIEVSKAJA, T. V. 400, 504 LOZANOVSKIJ, G. JA.129,504

~UAQENES, E. 16, 119, 120, 144, 151, 170, 180,

183, 224, 225, 227, 228, 230, 231, 266, 317, 320, 331, 364, 378, 380-382, 391, 394, 401, 402, 474, 502, 504 MAKOVOZ,Jn. I. 353,504 MAMEDOV,9. F. 306, 491 MANES,A. 149,504 MARCINKIEWICZ, J. 17, 132, 136, 137, 166,358, 504 MARUO, K. 396,473, 504. MATIJEUK, M. I. 400,504 MAURIN, K. 75,353,453,458,476,504 MAZ’JA,V. G. 316, 329, 354,493, 504 MCCARTHY,C. 166, 179,502 MCKEAN, H. P. 454, 505 M~RUCCI, C. 147, 505, 509 M~TIVIER, G. 397, 492, 505 MEYERS,N. 329, 505 MICHLIN, S. G. 152, 160, 161, 165, 166, 314, 358, 505 MILQRAM,A. N. 364, 388,484 MIL‘MAN, D. P. 292, 500 MIL’MAN, V. D. 237, 505 MINAKSHISUNDARAM,S. 454, 506 MIRANDA,C. 136, 201, 326, 404, 505 MIRKIL, H. 86, 501 MIRO~IN,N. V. 147, 505 MITJAQIN, B. S. 21, 108, 145, 168, 343, 479, 482, 505, 506, 518 MIYAZAKI, K. 148,506 MORREY, C. B. 207,404, 506 MOSTEFAI, A. 356, 506 MUCKENHOUPT,B. 141, 161, 506 MULLA,F. 170, 180, 181, 224,484

Author Index

MULLER-PFELFFER, E. 329,407,506 MURAMATU, T. 91, 94, 95, 141, 208, 313, 324, 329,333,506

MURTHY, M. K. V. 171,475,506 NAQASE,M. 396,506 NAJMARK, M. A. 444,506 NEEAS,J. 151, 224, 248, 306, 343, 492, 506 NERI, U. 161, 507 NEWEL,R. J. 190,488 NIHOL’SHIJ,Jn. S. 304, 306, 307, 507 NIKOL’SKIJ,S. M. 146, 151, 161, 166, 169, 170,

176, 181, 190, 196, 197, 207, 224, 237, 243,

244, 245, 306, 357, 358, 486, 503, 507 NIRENBERG, L. 151, 207, 364, 368, 378, 389, 403,404, 475, 483, 490, 498, 507

NORDIN,C. 472,507

OKIHIOLU, G. 0. 166, 507 OKLANDER, E. T. 16, 135, 136, 507 OLEJNIK,0. A. 474, 507 OLOFF,R. 147, 507 O’NEIL, R. 140, 141, 508 ORNSTEIN, D. 86, 316, 508 PANEJACH, B. P. 231, 359, 360, 517 PANZONE, R. 136, 306,485 PARASKA, V. I. 400, 508 PEETRE, J. 6, 15, 16, 21, 23,2830, 33, 35, 37,

38, 41, 44, 52, 53, 61, 66, 69, 71, 75, 81, 98, 115, 117, 118, 125, 129, 131, 134, 135, 140, 141, 143-146, 148-150, 161, 167, 170, 171, 176, 177, 183, 184, 186, 199, 207, 208, 240, 378, 382, 389, 391, 403, 502, 503, 508 PELCZY~SKI, A. 6,108,236,242,289,290,343, 502, 505, 509 PERSSON, A. 117, 509 PETUNIN, Jn. I. 16, 17, 119, 120, 145, 146, 148, 149, 500, 509 PHAM THELAI 118, 147, 353, 354, 472, 484, 493, 505,509 PHILLIPS, R. S. 75,496 RETSCH,A. 107-110, 119, 147, 348, 353, 354, 477, 479, 482, 509, 510 RQOLKIKA, T. S. 304, 307, 510 RLIHA, P. 207, 510 PLEIJEL, A. 454, 505 POINCARE, H. 160 POLETTI,31.P. 149, 504 P6LYA, G . 141, 262, 495 POTAPOV, M. K. 358,510 PRODI,G. 224, 510 PUSTYL’NIK, E. I. 96, 98, 100, 236, 500

525

RIESZ,F. 5, 143, 394, 398, 510 RIESZ,M. 16, 17, 135, 510 RIVIERE,N.M. 161, 166, 167, 171, 179, 378, 491, 502, 510

ROJTBERG, JA.A. 400,499 ROLEWICZ, S. 477, 482,510 ROSENTHAL, H. P. 343,497, 502 ROZENBLJUM, G. V. 482,510 RUTICKIJ,JA.B. 236,499 RYBALOV, Ju. V. 307,511 RYLL,J. 343, 511 SADOSKY, C. 161, 511 SAQHER, Y. 149, 171,491, 511 SAPIRO,Z. JA.364, 511 SCHATTEN, R. 107, 146, 147, 511 SCHAUDER, J. 404,511 SCHECHTER, M. 55, 58, 61, 144, 171, 186, 360, 364,378,382, 388, 389,391,486,511

SCHERER, K. S. 41, 71, 81, 98, 143, 150, 488, 511

SCHOLZ, R. 511 SCHONEFELD, S. 343,511 SCHUBERT, H. 19,22,23,512 SCHUMAKER, L. L. 351, 357,497 SCHWARTZ, J . T . 5, 56, 59, 75, 127, 135-137, 151. 153, 157, 160, 166, 384,409,490, 512

SCHWARTZ, L. 152, 197, 231, 512 SEDAEV, A. A. 21, 145, 512 SEELEY, R. 103, 227, 230, 320, 321, 334, 343, 465, 512

SEMENOV, E. M. 17, 21, 136, 145, 500, 512 SERRIN,J. 329, 505 SESTAKOV, V. A. 61, 512 SHAMIR, E. 225, 227, 323, 512 SHARPLEY, R. 145,512 SHIMAKURA, N. 334, 430, 431, 433, 472-474, 512

SHIMOGAKI, T. 145, 149, 504 SIGALOV, A. G. 474 SIMADER, C. G. 378, 513 SINGER,I. M. 454. 505 SLOBODECKIJ, L. Pi. 169, 190,224,378,513 SMITH,K. T. 169, 170, 316, 483,484.513 SMUL’JAN, V. L. 199, 332 SOBOLEV, S. L. 141, 146, 151, 169, 188, 207, 226, 313, 359, 513

SOBOLEVSKIJ, P. E. 96, 98, 100, 236, 400, 500, 513

SOLOMEBE, I. A. 413, 513 SOLOMJAH, M. Z. 110, 291, 323, 347, 351,355357, 397, 473, 486, 513, 517

SOLONNIKOV, V. A. 224, 313,496

526

Author Index

SPANNE, S. 171 SPARR,G. 21, 30, 115, 135, 145, 148, 149, 358, 509, 513 STAMPACCHIA, G. 171,475, 506, 513 STEIN,E.M. 17, 136, 138, 141, 151, 161, 167, 170, 171, 179, 180, 192, 196, 224, 226, 236, 314, 315, 492, 513, 514 E. A. 329, 513 STORO~ENRO, R. S. 141, 161, 179, 227,313,316, STRICHARTZ, 514 G. A. 454, 514 SUVOREENKOVA, P. 170, 180, 181, 224, 484 SZEPTYCKI, B. 5, 143, 394, 398, 510 SZ.-NAGY,

M. H. 171, 180, 181, 184, 190, 192, TAIBLESON, 196, 199, 207, 242, 514 H. 396,504,514 TANABE, L. 149, 514 TARTAR, V. 150, 509, 514 THOMEE, THORIN,G. 0. 16, 17, 135, 514 V. M. 108, 110, 350, 351, 357, TICHOMIROW, 484, 499, 505, 513, 515 TREBELS, W. 168, 179, 196,489, 515 TRICOMI,F. G. 160 TRIEBEL,H. 66, 81, 87, 91, 107, 108, 112, 115, 119, 125, 147-149, 153, 157, 165, 169, 170, 177, 186, 192, 199, 200, 207, 209, 210, 217, 225, 240, 244, 245, 249-251, 258, 263, 265, 266, 275, 277, 285. 291, 298, 299, 301, 302, 336, 351, 353, 354, 357, 358, 374, 392, 393, 400, 407, 414-417, 421, 423, 427, 431, 433, 441, 444, 454, 459, 461, 467, 468, 478481, 510, 515 TULOVSKIJ, V. N. 397,472, 516 L. 120, 236, 237, 242, 502 TZAFRIRI, UNINSKIJ,A. P. 358,516 A. 360, 516 UNTERBERQER,

USPENSKIJ,S. V. 151, 170, 180, 217, 224, 225 248, 306, 516

R. S. 150,495 VARGA, VILLAMARIN,A. F. 149, 517 VIBIK, M. I. 382, 430,473, 483, 517 V O L E V I ~L., R. 231,359, 360, 517 VULIS,I. L. 473, 517 WALEK,H. 103,517 WALSH,T. 141, 161, 517 WEBER,A. 329, 506 WEISS,G. 17, 136, 138, 141, 161, 196, 514 U. 41, 98, 100, 196, 485, 489, 496, WESTPHAL, 515, 517 WEYL,H. 396,444,517 R. L. 141, 161, 179, 506, 517 WHEEDEN, WIDDER,D. V. 192, 496 E. 416, 517 WIENHOLTZ, V. 143, 518 WILLIAMS, WLOKA, J. 171, 518 W. 479, 518 WOJTYNSKI,

YOSHIKAWA,A. 69, 103, 146, 149, 208, 358, 518 K. 143,518 YOSHINACA, YOSIDA,K. 5,58,75,76,96,152, 197, 199,215, 332, 476, 518 YOUNG, W. H. 17, 138 P. P. 96, 98, 100,236, 500 ZABREJKO, ZAFRAN,M. 171,518 ZERNER,M. 481, 518 ZIPPIN, M. 145, 343, 497, 518 ZOBIN, N. M. 482, 506, 518 ZUILY,C. 433, 474,490, 518 ZPGMUND,A. 17, 135, 136, 151, 160, 161, 196, 489. 518

SUBJECT INDEX

Analytic function, entire 197 approximation number 108 Bessel transform 307 bilinear forms 145 Category 19 Cauchy-Poisson semi-group 192 C”-domain 246 complex methods 85 convexity theorem 16, 17 - - (Rieaz-Thorin) 135 coretraction 22 Differential dimension 225 differential operator, properly elliptic 333, 361 Dirichlet’s problem for the Laplacian 65 discrete methods 40 domain of cone-type, bounded 314 - - - , unbounded 246 dual space of interpolation spaces 69, 72 - - of Z,(A) 68 Eigenvector, associated 383 elliptic 362 - , properly 333, 361 -, regular 363 -, uniformly 362 ellipticity rondition 406 embedding theorem, abstract 49, 50, 53 - -, direct 212, 224 - - , inverse 212, 224 €-entropy 108 extension method 310, 312 - operator 311 Fourier transform 152 - -, inverse 152 fractional power 99 (P)-space476 functor 19

Gauss-Weierstrass semi-group 190 Hardy’s inequality 262 Hardy-Littlewood-Sobolev inequality 141 Hausdorff-Young’s theorem 16, 17, 138 Ideal of operators 107 inequality, multiplicative 186 infinitesimal generator 76 operator 76 integral operator, singular 157 representation 312, 313 ,singular 160 interpolation couple 15, 18 - - ,quasilinearizable 52, 73, 74 interpolation functions 144 - functor20 - -, exact 21 - - of type f 21 - - of t y p e 2 1 - property 15, 20 - scales 145 - space 20

-

J-method 38 K-functional 23 K*-functional 28 K-method 23 Legendre differential operator 429, 431 L*-functional 28 lift property 180 limit exponent 207 L-method 28 Mean-method 30, 35 method of local coordinates 249 mixed norm 306 multiplier theorem 161, 166 - of type ( p , p) 166 multiplier-matrix of type (p, 2)) 165

528

Subject Index

Normal system (of differential operatore) 320 nuclear operator 477 - space 477 Operator, positive 91 - with pure point spectrum 392 @-operator382 9-ideal 107 Reiteration theorem 61, 62 resolution of unity 249 restriction operator 210, 213 retraction 22 root-condition 334 ’

Scale, one-side 335 - ,two-side 335 Schauder basis in L, 342 semi-group of operators 75 - - - ,analytic 95, 96 -8 - -, commutative 81 space, Besov 169, 181, 187, 190, 310 -, - ,in domaine 329 - , Bessel-potential cf. space, Lebesgue - ,Holder 200, 201, 324,326, 327 -, Lebesgue 169, 179, 310 - , - ,in domains 329

space, Liouville cf. space, Lebesgue -, Lorentz 127, 132 - ,Marcinkiewicz 132 - ,Monte1 476 - ,Nikol’skij 170 -, quasi-Banach 27, 107 -, Slobodeckij 169, 190, 310 -, Sobolev 169, 179,310 -, -, with weights 213 system, complemented 333, 363 -, Dirichlet 388 - ,normal 363 Theorem of Paley-Littlewood type 179 trace method 41 translation group 187 Tricomi differential operator 430, 432, 433 Variation principle 393 Weight function of type 1 247 - of type 2 247 - - of type 3 247 - - of type 4 247 width, Gelfand 108 - , Kolmogorov 108

-

Young’s inequality 139

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