1 < s < n=p. II - CiteSeerX

Conditions on composition operators which map a space of Triebel-Lizorkin type into a Sobolev space. The case 1 < s < n=p. II Winfried Sickel

Abstract.

Let G : R ! R be a continuous function. Denote by TG the corresponding composition operator which sends f to G(f ). Then we investigate necessary and sucient conditions on the parameters s; p; q; r and on the function G such that an inclusion like s (Rn ))  W m (Rn ) TG (Fp;q p s denotes a space of Triebel-Lizorkin type and W m denotes a Sobolev space, reis true. Here Fp;q p spectively. Necessary and sucient conditions for such an inclusion to hold will be given in cases G(t) = tk ; k 2 N , G(t) = jtj , G(t) = t jtj? ;  > 1, G 2 C 1 , and G a periodic C 1 -function. 1991 Mathematics Subject Classi cation: 46E35, 47H30. Running title: Composition operators 1

0

1 Introduction We are interested in composition mappings. Therefore, let G : R ! R be some continuous function. Then the associate composition operator TG is given by f ?! G(f ). The properties of TG like boundedness, continuity etc. depend strongly on the domain of de nition, cf. e.g. Appell and Zabrejko [AZ], Bourdaud [Bo 3], Sickel [Si 3, Si 6], and Runst and Sickel [RS, Chapter 5]. Even in case of Sobolev spaces Wpm (Rn ) the behavior of these simple operators is not completely well understood. Dahlberg [Da] proved

TG (Wpm (Rn )) = f G(f ) : f 2 Wpm(Rn ) g  Wpm(Rn ); 1 < p < 1; 2  m < np

(1)

implies G(t) = ct for some c 2 R. Hence, there is no nonlinear composition operator which acts on Wpm (Rn ) if 2  m < n=p. Our aim here consists in an investigation of necessary and sucient conditions on G connected with the embedding

TG (Wpr (Rn ))  Wpm(Rn ); m  r ; 1

(2)

in dependence on m; r; p; and n. In fact, we shall deal with the more general problem obtained s (Rn ). by replacing Wpr (Rn ) by spaces of Triebel-Lizorkin type Fp;q s (Rn ) represent certain generalizations of Sobolev spaces. They include several The spaces Fp;q classical types of function spaces:

 Fp;m (Rn ) = Wpm(Rn ) if 1 < p < 1; m = 1; 2; : : :, and Wpm(Rn ) denote the 2

Sobolev spaces,

s (Rn ) = W s(Rn ) if 1  p < 1; 0 < s 6= integer, and W s denote the  Fp;p p p

Slobodeckij spaces (known to include the trace classes of Sobolev spaces),

 Fp;s (Rn ) = Hps(Rn ) if 1 < p < 1, and Hps(Rn ) denote the Bessel potential 2

spaces.

These classes have in common that they are well-adapted to elliptic partial dierential equas (Rn ) tions, cf. Triebel [Tr 1]. The degeneracy result stated in (1) extends to those spaces Fp;q as long as 1 + p1 < s < np ; 1 < p < 1; (3)

cf. Bourdaud [Bo 3], Runst and Sickel [RS, 5.3.1] or [Si 5, Si 7]. Since for nontrivial G the mapping TG can not be into (if (3) holds) one could ask for an optimal image space. This space has to be larger than the original one, e.g. de ned by weaker regularity conditions. So it makes sense, given s; m; p; q; and n and having the embeddings s ,! W m at hand to ask for properties of G implied by Fp;q p s (Rn ))  W m (Rn ) : TG (Fp;q p

(4)

It turns out that (4) implies on the one side a condition on s and m (if it is not satis ed then we have a similar degeneracy result as in (1)) and on the other hand that G must belong to a certain weighted Sobolev space. This supplements and improves some earlier results of the author, cf. [Si 5] (for that reason we called this paper part II). The more general problem to replace also Wpm (Rn ) by spaces of fractional order will be discussed in [Si 7], but cf. also [Si 5] and [RS, 5.3]. Let us point out that the description of TG (Wpm (Rn )) necessarily leads to spaces characterized by fractional order of regularity, cf. [Si 7] and [RS, Chapter 5]. So also this more general problem turns out to be of interest. As we hope these investigations will be useful in connection with a discussion of the existence and regularity properties of solutions of nonlinear partial dierential equations, where the nonlinearity comes from a composition, cf. Amann [Am], Cazenave and 2

Weissler [CW], and Johnsen and Runst [JR] for rst steps in this direction . This note is organized as follows. In Section 2 we collect the needed material about funcs (Rn ))  tion spaces of Triebel-Lizorkin type. First consequences of embeddings like TG (Fp;q s (Rn ) or TG (C 1 (Rn ))  F s (Rn ) for the regularity of G will be derived in Section 3. The Fp;q p;q 0 main results of this paper are contained in Subsections 4.1 and 4.2. Whereas we shall deal with necessary conditions in the rst one, the Subsection 4.2 is devoted to the investigation of sucient conditions. In Subsection 4.3 we apply this to the mappings f ! f m ; m 2 N ; m  2, f ! jf j; and f ! f jf j?1 ;  > 1. Further, in Subsection 4.4 we state necessary and sucient conditions in cases G 2 C01 and G a periodic C 1 -function. Finally, in Subsection 4.5 we present a detailed comparison of the quality of composition operators induced by polynomials and induced by C01-functions. It turns out that depending on s; p; and n we may have the situation that the mappings induced by polynomials of low order are better behaved than composition operators induced by an arbitrary non-trivial C01-function and vice versa. All the time we restrict us to the case s < n=p. For sucient conditions in case that s 62 (1; n=p) we refer to the literature, cf. Bourdaud [Bo 3], Bourdaud and Kateb [BK 3] Marschall [Mar], Runst [Ru], Triebel [Tr 2], Yamazaki [Ya], and [Si 2, Si 3, RS]. Composition operators on subspaces of Bessel potential spaces Hps with 1 < s < n=p are investigated in Adams and Frazier [AF 1, AF 2]. Colin [Co] has treated s < n=p in case of Slobodeckij spaces. Finally, an investigation of the case 1 < s < 1 + 1=p has been given in Bourdaud and Kateb [BK 3]. Acknowledgement. The author would like to express his gratitude to the referee for the most helpful remarks. The referee suggested an essential simpli cation of the proof of Proposition 2 and indicated how to extend Theorem 6 to the general case stated in Theorem 4.

2 Some information about spaces of Besov-Triebel-Lizorkin type In general, all functions, distributions, etc. are de ned on the Euclidean n-space Rn . If there is no danger of confusion we will not indicate this. By N we denote the set of all natural numbers, by N 0 the same set including 0. To introduce spaces of Triebel-Lizorkin type we make use of the characterization by dierences. Recall, if f (x) is an arbitrary function and h 2 Rn we put m   X m (?1)` f ( x + (m ? `)h ); m
h f (x) = `=0 `

3

m = 1; 2; : : : :

De nition 1 Let s > 0; 1  q  1 and 1  p < 1. Let M be a nonnegative integer such that M  s < M + 1. Then s Fp;q

=



f 2 Lp : f real-valued; s k = k f jL k + k k f jFp;q p

Z

1

t?sq ( 1

tn

0

Z jhj 0) or f 2 Lp If G is a continuous function, then the composition G(f ) for f 2 Fp;q is well-de ned. In cases we are interested in the mapping TG must have some boundedness properties. This will be made clear by the following proposition.

Proposition 1 Let G : R ! R be a continuous function. Let 1  u < 1 and 1  v  1. Let r > 0 be a positive real number and let m be a natural number. Further we assume 0 < s < n=p. Suppose either s )  Fr TG (Fp;q u;v

or

s )  Wm TG (Fp;q 1

or

s )  Lu : TG (Fp;q

(5)

Let Bt (z ) be the open ball in Rn with radius t > 0 and centre z . Then the restriction of TG s (Bt (z )) is a bounded operator. to Fp;q

Proof The assumption (5) means that G(f ) are Lebesgue measurable functions for all

admissible functions f . s )  F r and To prove the assertion we assume the contrary. That means we suppose TG (Fp;q u;v s (Bt (z )) (the proof in the other cases is the same). Without loss of TG is not bounded on Fp;q s generality we may take z = 0. This implies the existence of a sequence ffj g1 j =1  Fp;q (Bt (0)) such that s (B (0))k  C r k  2jn=u : k fj jFp;q and k G(fj )jFu;v t s (Bt (0)) into F s (Rn ). By E we denote a linear and continuous extension operator from Fp;q p;q Without loss of generality we may assume supp Efj  Bt+1 (0). Therefore, see e.g. [Tr 1, 3.3.4] or Seeger [Se]. De ne

f (x) =

1 X

j =j0

(Efj )(2j (x ? zj )) ;

zj = (j; 0; : : : ; 0) ;

where j0 (t) is chosen in such a way that

supp (Efj )(2j (
? zj )) \ supp (Efj+`)(2j+` (
? zj+` )) = ; ;

` = 1 ; j  j0 (t) :

s and s < n=p it follows Thanks to well-known properties of the dilation operator on Fp;q s k k f jFp;q

1 X

j =j0

n

s kc 2j (s? p ) k Efj jFp;q

5

1 X

j =j0

n

s (B (0))k < 1 ; 2j (s? p ) k fj jFp;q t

(6)

r the homogeneous counterpart of F r , de ned as the set of cf. [Tr 1, 3.4.1]. Denote by F_u;v u;v all tempered distributions f (modulo polynomials) such that r k=k k f jF_u;v

 Z1

t?rv ( 1

tn

0

Z

jhj  g =

1 [

`=1

(f t : G(f`(t; 0; : : : ; 0)) >  g \ [?`; `]) :

This shows that G itself must be Lebesgue measurable. In addition the local integrability of G(f`) with respect to Rn implies the local integrability of G with respect to R. Step 2. Let I = [a; b] be some nite interval. For M 2 N 0 ; M + 1 > s  M , s > 0 it follows that s (I )k k G jFp;q

k

= k G jLp (I )k + 1 1 t?sq a ? x b ? x j [ M +1 ; M +1 ] \ [?t; t] j 0



Z

Z [

a?x b?x M +1 ; M +1 ]\[?t;t]

q dt  =q

j
Mh G(x)j dh t +1

1

(14)

jLp(I )k

s (I ), cf. [Tr 3, (usual modi cation if q = 1) may be taken as an equivalent quasi-norm on Fp;q 5.9]. Again we test TG with the simple functions f` 2 C01 . In case `=2 > M + 1 we obtain from (11)

Z

s (Rn )kp  c 1 > k G(f`) jFp;q jxi j`=2

i=1;:::;n 0Z 0 Z  MX  M + 1 1 ? sq @ @ j (?1)j G(x  t jhi j p the assertion (i) is proved. Step 2. The statement (ii) follows from

Ap ;p  c

Z

j

2

sup 2?j p j jG(y)jp dy  c j 2N0 ?2

Z

Lemma 1 results in the following corollary. 11

j G(y) jp % p(y) dy :

2

Corollary 1 Let G : R ! R be continuous. Let  0. (i) In Proposition 2 we may replace A ;p < 1 by G m 2 Lp (R; % ) for all  > p. (ii) In Proposition 3 we may replace A ;t < 1 by G m 2 Lt (R; % ) for all  > t. (

)

(

)

r ,! Lt Corollary 1(ii) yields also the following. Because of the embedding Ft;q

Corollary 2 Let 1  t < 1 and 1  v  1. Suppose 0 < r  s < np ;

s )  Fr TG (Fp;q t;v

then G belongs to Lt (R; % ) for some  0.

Remark 8 Corollary 2 re ects the dierence between the cases s < n=p and s > n=p. In the

latter case there can not be boundedness conditions on G, because the underlying function spaces contain only bounded functions, cf. [Tr 1, 2.7.2].

A general bound for the regularity of the composition We put



Pm = p : p(t) =

R

m X `=0

a`



t` ;

a` 2 R; ` = 0; : : : ; m ; m 2 N 0 :

If G 62 Pm?1 , then we have ?ww jG(m) (y)jp dy  c > 0 for w suciently large. In this case Proposition 2 gives the following bound for the maximal regularity of the composition.

Corollary 3 Let 0  m  s < n=p. Let G be continuous and suppose G 62 Pm? . Then s )  W m implies TG(Fp;q p n + ( n ? s) (26) m  p n ?p sp+ 1 : 1

1

p

Remark 9 The condition G 62 Pm? is essential, at least if m  2. We shall stress this 1

point in Subsection 4.3. Moreover, as we shall show in Subsection 4.2, the bound in (26) is s )  Wm unimprovable if 2  m < n=p. There is a large class of functions G such that TG (Fp;q p holds with s; m and p as above. Remark 10 Of course, m  s and (26) interact. A simple calculation gives n + 1 ( n ? s) p p p n ?s+1 p

s

()

1 + p1  s :

(27)

This shows that (26) represents a restriction only if s > 1+1=p. Specializing s = m we obtain: if G 62 Pm?1 , 0 < m < n=p and TG (Wpm )  Wpm hold, then necessarily m  1 + 1=p follows. 12

The mapping properties of f ! f k are well-known, see the next subsection. That allows to complement this assertion: if G is a nonlinear function, 0 < m < n=p and TG (Wpm )  Wpm hold, then necessarily m  1 + 1=p follows. With other words: if 1 + 1=p < m < n=p and TG(Wpm )  Wpm, then G is a linear function. This interesting fact has been observed rst time by Dahlberg [Da]. d(s) Fig.1 Remark 11 Of some interest turns out to be also the following observation. We study the dierence . .... .... .. ..

d(s) = s ?

........................... ......... .... ....... .... ..... .... ..... .... ..... . . .... . ... .... . . . . ... .... . ... . . ... .... . . . .. .... .. . . . ... .... . . . ... .... ... . . . ... .... . ... . ... . ... . . . ... .... . . ... ... .... . . ... .... . ... . ... .... . . .. .... ... . . .. ... . . . ... ... . ... . . . ... ... . . ... ... .... . . .. ....

n + 1 ( n ? s) p p p n ?s+1 p

for xed n and p (n > p). Obviously, d(1+1=p) = d(n=p) = 0 and d(s) > 0 if 1 + 1=p < s < n=p.

1 + p1

s0

n p

............. .........

s

Moreover, the function d(s) (cf. Fig. 1, p = 2; n = 10) is concave on this interval, hence, it attains a maximal value d(s0 ) there. We have

r

s0 = np + 1 ? n ?p 1

and

d(s0 ) =

r n ? 1



p?1 p + ? 1 p p(n ? 1) 2

This shows that d(s) has a bound depending on p and n, but no a priori bound for xed p and independent of n. To nish this subsection we collect all our ndings with respect to necessary conditions.

Theorem 3 Suppose 0  m  s < n=p. Let G : R ! R be an arbitrary function if m  1 s )  Wm and a continuous function if m = 0, but not an element of Pm? . Then TG (Fp;q p 1

implies:

 m

n + 1 ( n ? s) p p p n ?s+1 ; p

 G(0) = 0 ;

 G 2 Wpm;`oc(R) ; 

sup w? w1

R w

?w

1=p jG m) (y)jp dy < 1

with =

(

13

n + 1 ( n ? s) ? m ( n ? s + 1) p p p p : n ?s p

Remark 12 We shall give some comments to the last condition in Theorem 3. The rst

condition implies that must be nonnegative. Fix s; p; and n. If m decreases, then the bound for increases, hence the condition 4 becomes less restrictive. Vice versa if m increases, then the bound for decreases and consequently, the condition 4 becomes more restrictive. Now we x m; p; and n. If s " n=p, then the condition 4 becomes less restrictive. To consider the converse, that means s # 0, we x m = 0. Then the bound decreases, hence the condition becomes more and more restrictive. This behaviour is in a good correspondence with s ,! F s?"; " > 0 and the conditions one could derive immediately from the embeddings Fp;q p;q m m ? ` Wp ,! Wp ; ` 2 N . With other words: Theorem 3 describes the impression we already had from the embedding relations in a quantitative way.

4.2 Sucient conditions on G Here we have a partial inverse of Theorem 3. We have to subdivide our investigations into the three cases m  2, m = 1, and m = 0.

Theorem 4 Suppose 1 < p < 1 and 2  m  s < n=p. Let G(0) = 0. If sup w?

Z w

w1

with

=

?w

j Gm(y) jp dy

 =p 1

n + 1 ( n ? s) ? m( n ? s + 1) p p p p n ?s p

1, a second one in s p  t in any case, cf. (50). Again we make use of (33). To continue we have to replace Fp;q by Wtm . The inequalities corresponding to Wtm ,! Lp0 (?`) and Wtm ,! Wpkk are: ( nt ? m) ( ? `)  pn  nt ( ? `) 0

and

k ? m + nt  pn  nt : k

These inequalities yield ` X n n  n ( ? `) + n ` = n  ; n n ( t ? m) + m  p = p + t t t 0 k=1 pk

which means that

`

1

k G ` (f ) @@xf1 : : : @@xf` jLpk  c k f jWtmk ( )

(53)

1

1

is valid if t is restricted by (50). We continue with (38). The embeddings

Wtm ,! Lp1 ((?m)p+1) ;

Wtm ,! Wp12 (mp?2) ;

and

Wtm ,! Wp23

take place if the following inequalities are satis ed: (( ? m)p + 1) ( nt ? m)  pn  (( ? m)p + 1) nt ; 1 (mp ? 2) ( nt ? m + 1)  pn  (mp ? 2) nt ; 2

and

2 + nt ? m  pn  nt : 3

Summing up we obtain

( nt ? m) p + mp  pn + pn + pn  nt p : 1 2 3 Again the restrictions for t given in (50) are guaranteeing that we may choose p1 ; p2 ; and p3 such that (1=p1 ) + (1=p2 ) + (1=p3 ) = 1. Hence

Z

@f (x))m jp dx  c k f jW m kp j G m (f (x)) ( @x t (

)

1

(54)

as long as t satis es (50) and m  2. Moreover, (50) guarantees also Wtm ,! Lp . Hence

Z

j G(f (x)) jp dx  c k f jLp kp  c k f jWtmkp :

(55)

To nish the proof it will be sucient to establish the counterpart of what we did in Step 5 of the proof of Theorem 4. We give some comments. To prove limj !1 G(fj ) = G(f ); f 2 Wtm we need to modify the choice of v only. First observe, that t  p in any case. If we choose v 20

such that t  vp  n=( nt ? m), then by assumption k f ? fj jLvp k ! 0 if j ! 1. By this we get a = 1 ? pt  v10  1 ? np ( nt ? m) = b : (56) Again we need to have f 2 Lpv0 (?1) which is equivalent to

c = np ( ? 1) ( nt ? m)  v10  ( ? 1) pt = d :

(57)

Our assumption (50) yields a  d and b  c. Consequently, the intersection of (57) and (56) is nonempty. This proves limj !1 G(fj ) = G(f ) in Lp. So, (51) follows from the Fatou property of Wpm . The counterpart of Theorem 5 in the cases m = 0 and m = 1 is treated e.g. in Appell and Zabrejko [AZ].

Proposition 5 ([AZ, Chapter 9]) Suppose 1  p < 1 and p < n. Suppose G(0) = 0. If G is locally Lipschitz continuous and satis es

j G0(y) j  AG j y j ;

y 2 R;

(58)

with some  0, then TG (Wt1 )  Wp1 holds for all t such that

np ( + 1)  t  p ( + 1) : n + p Moreover, there exists some constant c such that

k G(f ) jWpmk  c AG k f jWtmk

(59)

+1

(60)

holds with c independent of G and f .

4.3 Properties of the mapping f ! jf j;  > 0 An interesting family of mappings is induced by G (y) = jyj ;  > 0: Our aim is to demonstrate the results of Subsections 4.1 and 4.2 using this example. Observe,

G 2 Lt (R; % ) Now let us suppose

()

s )L ; TG (Fp;q t

21

1 + t < :

t  1:

Proposition 3 yields

n

  n ?t s : p

Theorem 4 proves the suciency of the latter condition.

Proposition 6 Let  > 0 and let 0 < s < n=p. Then the following assertions are equivalent: (i)

s )  Lt , TG (Fp;q

(ii)

n `oc TG (L`oc r )  Lt with r = np ?s ;

(iii)

  np ?t s :

n

Proof The equivalence of (i) and (ii) is explained above. Next observe that `oc TG (L`oc r )  Lt

()

t  r :

s ,! Lr holds if and only if Hence we have (i) =) (ii). Further, it is well-known that Fp;q p  r  n=((n=p) ? s), cf. e.g. [ST]. This yields (ii) =) (i).

We continue with a detailed investigation of the case  > 1. Therefore we have to do some preliminary work.

Lemma 2 Let  > 1.

(i) We have

G (t) = jtj 2 Wpm;`oc(R)

()

 > m ? 1p :

(61)

1 p

(62)

(ii) Let m be a non-negative integer satisfying m <  + 1=p. Then there exist two positive constants c1 and c2 such that

c1

1 w?m+ p



Z w

?w

jGm) (y)jp dy (

 =p 1

 c w?m 2

+

holds for all w  1.

Proof The statements (i) and (ii) follow by explicit calculations. Now we are in position to apply our results from the preceding subsections.

Theorem 6 Let  > 1 be a real number but not an even natural number. Let 1 < p < 1. (i) Suppose 0 < s < n=p and m 2 N . Then the following assertions are equivalent: 0

22





jf j

: f

s 2 Fp;q

 m 1 increases the singularity in the origin and hence, leads to some loss in the regularity. That proves the sharpness of t0 . 23

Finally, we consider the possibility t > p. Of course, one can not use local arguments. We study the composition operator on smooth functions, having some decay near in nity. Let

g (x) = (1 ? (x)) jxj? ;

 > 0:

A simple calculation gives

g 2 Wtm

 > 1t

()

and

g 2 Wpm

()

 > 1p :

Take  such that p < (1=) < t; then g belongs to Wpm but g does not belong to Wpm . Step 4. Suciency in (ii). In case m  2 the result follows from Theorem 5, if m = 1 one has to apply Proposition 5 and if m = 0, then the assertion becomes obvious. This completes the proof.

Remark 16 It is not hard to see that Theorem 6 remains true if we replace f ! jf j by f ! f jf j? . 1

We excluded the case of  to be an even natural number. If  = 2k for some natural number k, then the condition m <  + 1=p disappears. The same is true if we replace f ! jf j by f ! f k with k 2 N ; k  2.

Theorem 7 Let k 2 N ; k  2 and let 1 < p < 1. (i) Let 0 < s < n=p and m 2 N . Then the following assertions are equivalent:



s  f k : f 2 Fp;q



0

 Wpm ;

 m  s ? (k ? 1)( np ? s) . (ii) Suppose m 2 N 0 and m < n=p. Then the following assertions are equivalent:

 



fk

: f

nk m(?1)+ np

2 Wtm



 Wpm ;

 t  pk.

Proof For a proof see [Si 1, Si 4] or [RS, 5.3.2].

24

4.4 Composition operators induced by either C 1-functions or periodic C 1functions 0

The aim of this subsection consists in a further discussion of the results derived in Section 3, Subsection 4.1 and Subsection 4.2, now concentrated on C01 -functions or periodic C 1functions. C01 -functions G

Suppose G 2 C01(R). Then 0 < sup

Z w

w1

?w

j G m) (t) jp dt (

 =p 1

< 1:

So, the conditions implied by Theorem 3 are

G(0) = 0

and

m

n + 1 ( n ? s) p p p n ?s+1 : p

For the nontrivial case m  2 suciency is stated in Theorem 4. This leads to the following quite satisfactory answer.

Theorem 8 Let 1 < p < 1 and let 2  m  s < n=p. Suppose G 2 C 1(R) and G 6 0. s )  W m holds if and only if Then TG (Fp;q p n + ( n ? s) G(0) = 0 and m  p n ?p sp+ 1 : p 0

1

Periodic C 1-functions G Let G 6 0 be a smooth periodic function. Then there exist two positive constants c1 and c2 such that Zw c1 w  j G(m) (t) jp dt  c2 w ?w

holds for all w  1. So, the conditions implied by Theorem 3 are

G(0) = 0

n p

m  n ?s+1 : p

and

Note if 0 < s < n=p, then n p

n ?s+1 p

()

1:

Theorem 9 Let 1 < p < 1 and let 2  m  s < n=p. Suppose G is periodic, G 2 C 1(R), s )  W m holds if and only if and G 6 0. Then TG (Fp;q p G(0) = 0

n p

m  n ?s+1 : p

and

Remark 17 The quantity n=(n ? sp + p) has similar properties as the quantity de ned in (26), cf. Fig. 2 and Fig. 3 (p = 2; n = 10). In Fig. 2 we plotted the identity mapping against the quantities n + 1 ( n ? s) n p p p  g(s) = n ? s + 1 and g (s) = n ? ps + 1 : p

p

. .... .... .. ..

n p

Fig.2

..... .......... ........ .. ....... ........ ........ ........ ......... . . . . . . . . . . . . ... ........ ........ ....... ........ ........ ....... ....... ........ .... ... ....... ............ ........ . . . . . . . . . . . . ......... ...... ..... ..... ... ........ ...... ..... ....... ...... ...... ........ ........ ....... ..... ....... ......... .......... . . . . . . . . . . . . . . .. ..... ... ....... ........ ....... ....... .......... .............. ....... ........... ......... ....... ............... ......... ........ ................... ......... . .............................................................. . . . . . . . . . . . . . . . . ........................................................ ......... ................. ...... . .................. ......... .......................... ................................... ............................... . . . . .

g (s)

1

1

. .... .... .. ..

1+

........................................... ..... ......... .... ........ . ....... ............................... ....... ....... ...... ... ............ .... . . . . . . . . . . ..... ... . ..... .... ... ...... . . . . . . . .... ... ...... ..... .... ... . . . . . . . ... .. ... ............ ..... ..... ...... ........ ........ ...... ..... ........ . . . . . . . ..... ... .... ..... . ...... . . . . . . ....... ..... ..... . . . . . . ...... ... .... . . . .... . . . . ... .... ..... . . . . . . ... . ..... ..... . .. . . . . . ... .... . ....

d (s)

g  (s)

n p

1 p

Fig.3

d(s)

............ ..........

s

1

Similar as there one could study the dierence

1+

n p

1 p

............ .........

s

n

d (s) = s ? n ? ps + 1 p for xed n and p (n > p). Obviously, d (1) = d (n=p) = 0 and d (s) > 0 if 1 < s < n=p. Moreover, the function d (s) is concave on this interval, hence, it attains a maximal value d (s0 ) there. We have

r s0 = np + 1 ? np

and

d(s0 ) =

r n



2

p ?1 :

This shows that also d (s) has a bound depending on p and n, but no a priori bound for xed p and independent of n. Remark 18 Both, Theorem 8 and Theorem 9 improve some earlier results of the author, cf. [Si 2, Si 3, Si 5, Si 6] or [RS, 5.3.6]

A comparison of composition operators induced by polynomials and by C 1functions, respectively 0

We x k; p, and n and investigate the following functions:

fk (s) = s ? (k ? 1) ( np ? s) ; k  2 ; 26

and

g(s) =

n + 1 ( n ? s) p p p n ?s+1 ; p

(64)

cf. Theorem 6 and Theorem 8. Because of Remark 10 we concentrate on the case 1 + p1  s  np . While fk (s) is linear the function g(s) is nonlinear and convex. Moreover, we have

fk ( np ) = g( np ) = np

and

g(1 + p1 ) = 1 + 1p :

(65)

The function fk ? g has a second zero (beside s1 = np ) in point

?1: s2;k = np + 1 ? nkp

It holds

s2;k < np

k < n ?p 1

()

and

(66)

fk (s2;k ) = k + p1 :

(67)

A typical situation is described in Fig. 4 (k = 2; 3; p = 2; n = 10). Moreover, in any case we have s2;k > 1 + 1=p. This shows if kp < n ? 1, then there is some region where the mapping f ! f k behaves better than any mapping of type f ! G(f ); G 2 C01(R); G 6 0. .... ..... .. ..

More exactly, the loss of smoothness of the mapping f ! f k is less than the loss of smoothness under the mapping f ! G(f ); G 2 C01(R); G 6 0. For this reason we have the following re nement of Dahlberg's result, cf. [Da].

Fig.4

n p

f2 (..s)

... ........ ... ........... .... ........... ... ..................... ....... ....... . .. .... ........ ..... ........ ..... ....... ..... .......... ..... ........... . . . ..... ..... .... ................. ..... .... ...... .... ..... ........ .... ................... ....... . . .. . . .. ........ ... .......... .. .... .. ............... .... .. .............. ........ .. .... .. ............... .... .... . .. ............................ ............. .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... ....... . ... ..... .......................... . ...... ............................................... ..... .... ..... ..... ..... ..... ..... .... .... . . . . . . . . ... ... . ..... . .... ..... .... . ..... .... . ..... .... . ..... ... . .... .....

g (s)

f3 (s)

1+

1 p

n p

s2;2

............. ..........

s

s )  W m . We put k = max fk 2 Corollary 4 Let 2  m  s < n=p. Suppose TG (Fp;q p N [ f?1g : kp < (n ? 1) g. (i) If k = 1, then it follows: m > g(s) implies G 2 P . (ii) If k  2, then it follows: m > max (g(s); f (s)) implies G 2 P . (iii) If k  2, then it follows: m > g(s) implies G 2 P . (iv) If k  2, then it follows: 0

0

0

1

0

2

0

0

1

2

8 < g(s) m>: f (s)

if s < s2;2 ; if s2;2  s < np :

2

implies G 2 P1 . (v) Let k0  3. Then it follows: m > max (g(s); fk (s)) implies G 2 Pk?1 ; 2  k  k0 .

27

(vi) Let m = 2. It follows: 2 > g(s) implies G 2 P1 .

Proof We sketch the proof. Most of the needed information are contained in Fig.4. Of inter-

est turns out to be the triangle with corners in the points ((1+1=p); (1+1=p)); (n=p; (1+1=p)), and (n=p; n=p). This area will be subdivided by the graph of g into two parts. The part above of the graph will be decomposed further by the lines fk ; k = 2; : : : ; k0 . The image space can have smoothness above of the graph of g if and only if G is a polynomial. From the position of the pair (m; s) in our triangle we can derive (by means of Theorem 7) its order. Note that 2 < n=p implies k0  1. If k0 = 1, then g(s) > f2 (s); s < n=p. Now we apply Corollary 3. Hence (i) follows. Assertions (ii)-(v) are consequences of the monotonicity of the fk with respect to k. Finally, (vi) is implied by f2 (s2;2 ) = 2 + 1=p, cf. Fig. 5.

. .... .... .. ..

n p

Fig.5

f2 (s)

...... ....... ..... .... ..... ... ..... .... .... .... ...... . . .... . . .. ... .... .... ........ ....... . ..... .... .... .... ..... ..... ..... .......... . . . . . . . . ...... ..... ..... ...... ..... .......... ... . ..... .... ....... ..... ......... .................... . . . . ........... . . . . . . . . . . . . ....... .. ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .............................................................. ..... ..... ..... ..... .... .... .... ..... ..... ..... ..... ..... . .............. ..... ...................... ..... ..................................... ..... ......................................... ..... . . . ..... . . . . ..... ..... ..... .... ..... ..... . . . ..... .... .....

g (s)

1+

2

1 p

1+

n p

1 p

............. ..........

s

Remark 19 The restriction to m  2 is essential. If m  1 there are simple conditions on s )  W m , cf. e.g. Proposition 4. G to guarantee TG(Fp;q p Remark 20 Dahlberg's statement itself (cf. (1)) becomes a consequence of (ii) (or also (iv)) because of s > max(g(s); f2 (s)) if 1 + 1=p < s < n=p.

A nal comment to f ! jf j;  > 1 Instead of fk we compare g(s) with f ;  > 1. In what follows we exclude the case  = 2k; k 2 N . Then Corollary 3 applies and gives the bound m  g(s) for the image space Wpm . The mapping properties of f ! jf j ;  > 1 are re ected by

f(s) = s ? ( ? 1) ( np ? s)

and

f (s) = min(f(s);  + 1p ) ;

(68)

cf. Theorem 6. As above it holds f (n=p) = n=p, the function f ? g has a second zero in the point s2; = (n=p) + 1 ? (n ? 1)=(p), s2; < n=p if and only if p < n ? 1, and f(s2; ) =  + p1 . Hence f (s) > g(s), only if f (s) >  + 1=p but never f (s) > g(s). 28

However, if s = s2; , then g(s) = f (s) = f (s) and this shows again that the restriction m  g(s) in Corollary 3 is best possible.

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[AZ]

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[JR]

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[RS]

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[Se]

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[Si 6] Sickel, W.: Composition operators acting on Sobolev spaces of fractional order { a survey on sucient and necessary conditions. In: Proc. Conf. \Function spaces, dierential operators, and nonlinear analysis"(ed. J.Rakosnik), Prometheus Publ. House, Prague 1996, 159-182 [Si 7] Sickel, W.: Necessary conditions on composition operators acting between Besov spaces. The case 1 < s < n=p. III. Forum Math. (to appear) [ST]

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[Zi]

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31