Scholarly article on topic 'Homogeneity Property of Besov and Triebel-Lizorkin Spaces'

Homogeneity Property of Besov and Triebel-Lizorkin Spaces Academic research paper on "Mathematics"

CC BY
0
0
Share paper
OECD Field of science
Keywords
{""}

Academic research paper on topic "Homogeneity Property of Besov and Triebel-Lizorkin Spaces"

Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 281085,17 pages doi:10.1155/2012/281085

Research Article

Homogeneity Property of Besov and Triebel-Lizorkin Spaces

Cornelia Schneider1 and Jan Vybiral2

1 Applied Mathematics III, University Erlangen-Nuremberg, Cauerstrafie 11, 91058 Erlangen, Germany

2 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrafie 69, 4040 Linz, Austria

Correspondence should be addressed to Jan Vybiral, jan.vybiral@oeaw.ac.at

Received 9 November 2011; Accepted 10 January 2012

Academic Editor: Hans Triebel

Copyright © 2012 C. Schneider and J. Vybiral. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the classical Besov and Triebel-Lizorkin spaces defined via differences and prove a homogeneity property for functions with bounded support in the frame of these spaces. As the proof is based on compact embeddings between the studied function spaces, we present also some results on the entropy numbers of these embeddings. Moreover, we derive some applications in terms of pointwise multipliers.

1. Introduction

The present note deals with classical Besov spaces Bp,q(m") and Triebel-Lizorkin spaces Fp (rn) defined via differences, briefly denoted as B- and F-spaces in the sequel. We study

Fs ' pq

the properties of the dilation operator, which is defined for every X> 0 as

TX : f f (A).

The norms of these operators on Besov and Triebel-Lizorkin spaces were studied already in [1] and [2, Sections 2.3.1 and 2.3.2] with complements given in [3-5].

We prove the so-called homogeneity property, showing that, for s > 0 and 0 <p, q <<x>,

A-) | B^(R»)|| ~ As-(n/p ||f |

for all 0 <1 < 1 and all

f e BsPrCj(Rn) with supp f c (x e rn : |x| < 1}. (1.3)

The same property holds true for the spaces Fpq(rn). This extends and completes [6], where corresponding results for the spaces RSq(rn), defined via Fourier-analytic tools, were established, which coincide with our spaces Bp,q(rn) if s > max(0,n(l/p - 1)). Concerning the corresponding F-spaces Fpq(rn), the same homogeneity property had already been established in [7, Corollary 5.l6, page 66].

Our results yield immediate applications in terms of pointwise multipliers. Furthermore, we remark that the homogeneity property is closely related with questions concerning refined localization, nonsmooth atoms, local polynomial approximation, and scaling properties. This is out of our scope for the time being. But we use this property in the forthcoming paper [8] in connection with nonsmooth atomic decompositions in function spaces.

Our proof of (1.2) is based on compactness of embeddings between the function spaces under investigation. Therefore, we use this opportunity to present some closely related results on entropy numbers of such embeddings.

This paper is organized as follows. We start with the necessary definitions and the results about entropy numbers in Section 2. Then, we focus on equivalent quasinorms for the elements of certain subspaces of Bp,q(rn) and Fp,q(rn), respectively, from which the homogeneity property will follow almost immediately in Section 3. The last section states some applications in terms of pointwise multipliers.

2. Preliminaries

We use standard notation. Let n be the collection of all natural numbers, and let no = n u {0}. Let rn be Euclidean n-space, n e n, c the complex plane. The set of multi-indices ( = ((l, ...,(n), (i e n0, i = 1,...,n, is denoted by nn, with |(| = (l + ••• + (n, as usual. We use the symbol "<" in

< bk or q(x) < f(x) (2.1)

always to mean that there is a positive number cl such that

ak < clbk or <p(x) < (x) (2.2)

for all admitted values of the discrete variable k or the continuous variable x, where (ak)k, (bk)k are nonnegative sequences and f are nonnegative functions. We use the equivalence in

ak ~ bk or <p(x) ~ f(x) (2.3)

ak < bk, bk < ak or ^(x) < f (x), f (x) < ^(x). (2.4)

If a e r, then a+ := max(a, 0) and [a] denotes the integer part of a.

Given two (quasi-) Banach spaces X and Y, we write X Y if X c Y and the natural embedding of X in Y is continuous. All unimportant positive constants will be denoted by c, occasionally with subscripts. For convenience, let both dx and | ■ | stand for the (n-dimensional) Lebesgue measure in the sequel. Lp(rn), with 0 < p < to, stands for the usual quasi-Banach space with respect to the Lebesgue measure, quasinormed by

||/ | Lp(rn)|| :=([ \f (x)|pdx)^ 02.5)

vr» /

with the appropriate modification if p = to. Moreover, let Q denote a domain in rn. Then, Lp(Q) is the collection of all complex-valued Lebesgue measurable functions in Q such that

||/ | Lp(Q)|| := (jj/(x)|Pd^ (2.6)

(with the usual modification if p = to) is finite.

Furthermore, Br stands for an open ball with radius R> 0 around the origin,

Br = {x e rn : |x| <R}. (2.7)

Let Qj,m with j e n0 and m e Zn denote a cube in rn with sides parallel to the axes of coordinates, centered at 2-jm, and with side length 2-j+1. For a cube Q in rn and r > 0, we denote by rQ the cube in rn concentric with Q and with side length r times the side length of Q. Furthermore, Xjm stands for the characteristic function of Qj,m.

2.1. Function Spaces Defined via Differences

If f is an arbitrary function on rn, h e rn, and r e n, then

(Ah/ (x) = f (x + h) - f (x), (A-1/) (x) = Ah (Ahf)(x) (2.8)

are the usual iterated differences. Given a function f e Lp (rn), the r-th modulus of smoothness is defined by

Wr (f,t)p = sup||Ahf | Lp (rn )||, t> 0, 0 <p < to,

p \h\<t

/ f \1/p (2.9)

dlpf(x) = ( t~n \ h<i\(Arhf) (x)\pd^ , t> 0, 0 <p< to,

denotes its ball means.

Definition 2.1. (i) Let 0 < p,q < to, s > 0, and r e N such that r > s. Then, the Besov space

Bp,q(rn) contains all f e Lp(r") such that

| = 11/ | lp(rn)|| + ( i rsqwr(f,typ j

(2.10)

(with the usual modification if q = œ) is finite.

(ii) Let 0 <p < œ, 0 < q < œ, s > 0, and r e n such that r > s. Then, Fpq(rn ) is the collection of all f e Lp(r") such that

| Fp/q(r")|r = ||f | Lp (rn)||

t-sqd[P/(-)q y) q ^ (r-

| Lp (rn )

(2.11)

(with the usual modification if q = to) is finite.

Remark 2.2. These are the classical Besov and Triebel-Lizorkin spaces, in particular, when 1 < p,q < to ( p < to for the F-spaces) and s > 0. We will sometimes write Ap q(rn) when both scales of spaces Bp,q(rn) and Fp,q(rn) are concerned simultaneously.

Concerning the spaces Bp q(Rn), the study for all admitted s, p, and q goes back to [9], we also refer to [10, Chapter 5, Definition 4.3] and [11, Chapter 2, Section 10]. There are as well many older references in the literature devoted to the cases p,q > 1.

The approach by differences for the spaces Fp,q(rn) has been described in detail in [12] for those spaces which can also be considered as subspaces of S'(rn). Otherwise, one finds in [13, Section 9.2.2, pp. 386-390] the necessary explanations and references to the relevant literature.

The spaces in Definition 2.1 are independent of r, meaning that different values of r > s result in norms which are equivalent. This justifies our omission of r in the sequel. Moreover, the integrals J0 can be replaced by J0°° resulting again in equivalent quasinorms, (cf. [14, Section 2]).

The spaces are quasi-Banach spaces (Banach spaces if p,q > 1). Note that we deal with subspaces of Lp(rn), in particular, for s > 0 and 0 < q <to, we have the embeddings

Ap,q(rn) ^ Lp(rn), (2.12)

where 0 <p <to (p < to for F-spaces). Furthermore, the B-spaces are closely linked with the Triebel-Lizorkin spaces via

Bp,min(p,q) (rn) ^ F^) ^ Bp,max(p,q) (^ (2.13)

(cf. [15, Proposition 1.19 (i)]). The classical scale of Besov spaces contains many well-known function spaces. For example, if p = q = to, one recovers the Holder-Zygmund spaces Cs(rn), that is,

B* (rn)= Cs(rn), s > 0. (2.14)

Recent results by Hedberg and Netrusov [16] on atomic decompositions, and by Triebel [13, Section 9.2] on the reproducing formula provide an equivalent characterization of Besov spaces Bpq(rn) using subatomic decompositions, which introduces Bpq(rn) as those f e Lp(rn) which can be represented as

f (x) = XX X jmjm(x), x e rn, (2.15)

j=0 meZn

with coefficients 1 = m e c : ¡5 e nn, j e n0, m e zn} belonging to some appropriate sequence space bp,Qq defined as

bspq := {l : ||1 | b^ < to}, (2.16)

/ to / p \q/p\1/q

||l | b^H = sup2Q5 (jr2?(s-n/p)q( £ \l?m\p) ) , (2.17)

11 11 ¡eNn =0 \meZ» ' ' / /

s > 0, 0 <p, q < to (with the usual modification if p = to and/or q = to), q > 0, and kj5m(x) are certain standardized building blocks (which are universal). This subatomic characterization will turn out to be quite useful when studying entropy numbers.

In terms of pointwise multipliers in Bp,q(rn), the following is known. Proposition 2.3. Let 0 <p,q <to,s > 0,k e n with k> s, and let h e Ck(rn). Then,

f hf (2.18)

is a linear and bounded operator from Bp,q(rn) into itself.

The proof relies on atomic decompositions of the spaces Bp q(rn), (cf. [17, Proposition 2.5]). We will generalize this result in Section 4 as an application of our homogeneity property.

2.2. Function Spaces on Domains

Let Q be a domain in m". We define spaces Apq(Q) by restriction of the corresponding spaces on m", that is, Aspq(Q) g|o = f. Furthermore,

on r", that is, Apw(Q) is the collection of all f e Lp (Q) such that there is a g e Aspq(rn) with

| Ap/q(Q)ll = infllg | Ap (r" )II, (2.19)

where the infimum is taken over all g e Ap,q(rn) such that the restriction g|q to Q coincides in Lp (Q) with f.

In particular, the subatomic characterization for the spaces Bp,q(rn) from Remark 2.2 carries over. For further details on this subject, we refer to [18, Section 2.1].

Embeddings results between the spaces Bp,q(rn) hold also for the spaces Bp,q(Q), since they are defined by restriction of the corresponding spaces on rn. Furthermore, these results can be improved, if we assume Q c rn to be bounded.

Proposition 2.4. Let 0 < s2 < si < to, 0 < p\,p2,q\,q2 < to, and Q c rn be bounded. If

6+ = s1 - s2 - d( — - —^ > 0, (2.20)

\p1 pi) +

one has the embedding

Bp1 ,q1 (Q) ^ B* (Q). (2.21)

Proof. If p1 < p2, the embedding follows from [19, Theorem 1.15], since the spaces on Q are defined by restriction of their counterparts on rn. Therefore, it remains to show that, for p1 > p2, we have the embedding

B^q(Q) ^ B^(Q). (2.22) Let f e D(rn) with support in the compact set Q1 and

f(x) = 1 if x e Q c Q1. (2.23) Then, for f e Bp!/q2 (Q), there exists g e Bp^(rn) with

g|q = f' \\f I Bplq2(Q)|| ~ ||g I BpU(rn)||. (2.24)

We calculate

||f I BpU(Q)|| <||fg I Bgq(rn)||

<|| fg I Bp!,q2(rn)|| (2.25) < Cf ||g I B^(rn)|| ~||f | Bp?q(Q)||.

The last inequality in (2.25) follows from Proposition 2.3. In the 2nd step, we used (2.10) together with the fact that

|K(fg) I Lp2 (rn)|| < Cq11| Ah(fg) I Lp1 (rn) ||, p1 >p2, (2.26)

which follows from Holder's inequality since supp fg c Q1 is compact.

2.3. Entropy Numbers

In order to prove the homogeneity results later on, we have to rely on the compactness of embeddings between B-spaces, Bpq(Q), and F-spaces, Fpq(Q), respectively. This will be established with the help of entropy numbers. We briefly introduce the concept and collect some properties afterwards.

Let X and Y be quasi-Banach spaces, and let T : X ^ Y be a bounded linear operator. If additionally, T is continuous, we write T e L(X,Y). Let UX = {x e X : ||x | X|| < 1} denote the unit ball in the quasi-Banach space X. An operator T is called compact if, for any given e> 0 we can cover the image of the unit ball UX with finitely many balls in Y of radius e.

Definition 2.5. Let X,Y be quasi-Banach spaces, and let T e L(X,Y). Then, for all k e n, the kth dyadic entropy number ek(T) of T is defined by

ek(T) = inf|e > 0 : T(Ux) c 1J (y + eUY) for some yx,...,y1k-1 e Y1, (2.27)

I j=1 J

where UX and UY denote the unit balls in X and Y, respectively.

These numbers have various elementary properties which are summarized in the following lemma.

Lemma 2.6. Let X,Y, and Z be quasi-Banach spaces, and let S,T e L(X,Y) and R e L(Y,Z).

(i) (Monotonicity) ||T|| > e1(T) > e2(T) > ••• > 0. Moreover, ||T|| = e1(T), provided that Y is a Banach space.

(ii) (Additivity) If Y is a p-Banach space (0 <p < 1), then, for all j, k e n,

ep+k-1(S + T) < ej(S) + epk(T). (2.28)

(iii) (Multiplicativity) For all j, k e n,

ej+k-1(RT) < ej (R)ek(T). (2.29)

(iv) (Compactness) T is compact if and only if

lim ek(T)= 0. (2.30)

Remark 2.7. As for the general theory, we refer to [20-22]. Further information on the subject is also covered by the more recent books [2, 23].

Some problems about entropy numbers of compact embeddings for function spaces can be transferred to corresponding questions in related sequence spaces. Let n > 0 and {Mj }jeN0 be a sequence of natural numbers satisfying

Mj ~ 2jn, j e n0. (2.31)

Concerning entropy numbers for the respective sequence spaces bp^(Mj), which are defined as the sequence spaces bpQ in (2.17) with the sum over m e zn replaced by a sum over m = 1,..., Mj, the following result was proved in [24, Proposition 3.4].

Proposition 2.8. Let d> 0, 0 < o\,o2 < to, and 0 < q1,q2 < to. Furthermore, let q1 > q2 > 0,

0 <p1 < pi <to, 6 = d - 02 - n(----M > 0. (2.32)

Then the identity map

id : b™ (Mj) b%'t (Mj) (2.33)

is compact, where Mj is restricted by (2.31).

The next theorem provides a sharp result for entropy numbers of the identity operator related to the sequence spaces bp^ (Mj).

Theorem 2.9. Let n> 0, 0 < s1,s2 < to, and 0 < q1,q2 < to. Furthermore, let q1 > q2 > 0,

0 <p1 < pi <to, 6 = s1 - s2 - n(----^^ > 0. (2.34)

For the entropy numbers ek of the compact operator

id : (Mj) (Mj), (2.35)

one has

ek(id) ~ fc-ö/«+i/P2-i/pir k e n. (2.36)

Remark 2.10. The proof of Theorem 2.9 follows from [25, Theorem 9.2]. Using the notation from this book, we have

bpiQ (Mj) = c [2%, (2j(si-n/pi)^M0], i = 1,2. (2.37)

Recall the embedding assertions for Besov spaces Bp q(Q) from Proposition 2.4. We will give an upper bound for the corresponding entropy numbers of these embeddings. For our purposes, it will be sufficient to assume Q = Br.

Theorem 2.11. Let

0 < s2 < s1 < go, 0 <p1rp2 0 < q1fq2 < to,

Ä /11

o+ = s1 - s2 - n[---

(2.38)

Journal of Function Spaces and Applications Then, the embedding

id: Bp; q (Q) Bpq (Q) (2.39)

is compact, and, for the related entropy numbers, one computes

ek(id) < k-(s1-s2)/n, k e n. (2.40)

Proof.

Step 1. Let p2 > p1, 6+ = 6, and let f e Bp1 q(Q), then, by [26, Theorem 6.1], there is a (nonlinear) bounded extension operator

g = Exf such that ReQg = g|Q = f, (2.41)

||g | B^(rn)|| < cllf | B^(Q)II. (2.42)

We may assume that g is zero outside a fixed neighbourhood A of Q. Using the subatomic

^>1 q(

approach for Bpq (m"), cf. Remark 2.2, we can find an optimal decomposition of g, that is,

gx = zz z ^,mkUx)< ||g i bspu: (m")|| ~ iia i bsp\q\\ (2.43)

ßeN" j=0 meZ"

with q 1 > 0 large.

Let Mj for fixed j e no be the number of cubes Qjrm such that

rQjrm n Q = 0. (2.44)

Since Q c rn is bounded, we have

Mj ~ 2jn, j e n0. (2.45) This coincides with (2.31). We introduce the (nonlinear) operator S,

S : Bp1q (rn) bp\% (Mj) (2.46)

Sg = X, X = [x[m : p e Nn, j e n0, m e zn, Qm n Q/ 0}, (2.47)

where g is given by (2.43). Recall that the expansion is not unique, but this does not matter. It follows that S is a bounded map since

||i | bp* (Mj)|| ||S|| = sup ||| ^ ])| < c. (2.48)

g/p ||g | Bp1,q1 (rn)|| "

Next we construct the linear map T,

T : b%% (Mj) Bp22,q2 (rn), (2.49)

given by

^ = XX Zjmkj,m(x)- (2.50)

^eNJ;=0 m=1

It follows that T is a linear (since the subatomic approach provides an expansion of functions via universal building blocks) and bounded map,

ITl | Bgq (R")|L l|T 11 = sup N, , s M ^ c- (2.51)

i / 0 ||i | bpqtM

We complement the three bounded maps Ex,S, T by the identity operator

id : bp]* (Mj) bp* (Mj) with Q1 > Q2, (2.52)

which is compact by Proposition 2.8 and the restriction operator

ReQ : Bsplq2 (Rn) Bg^(Q), (2.53)

which is continuous. From the constructions, it follows that

id (Bp1,q1 (Q) Bp^,q2 (Q)) = ReQ o T o id o S o Ex. (2.54)

Hence, taking finally ReQ, we obtain f by (2.41), where we started from. In particular, due to the fact that we used the subatomic approach, the final outcome is independent of ambiguities in the nonlinear constructions Ex and S. The unit ball in Bp11,q1 (Q) is mapped by S o Ex into a bounded set in

C1( Mj).

(2.55)

Since the identity operator id from (2.52) is compact, this bounded set is mapped into a precompact set in

d (M;), (2.56)

which can be covered by 2k balls of radius cek (id) with

ek(id) < cr6/n+1/p2-1/p1, k e n. (2.57)

This follows from Theorem 2.9, where we used p2 > p1. Applying the two linear and bounded maps T and ReQ afterwards does not change this covering assertion—using Lemma 2.6 (iii) and ignoring constants for the time being. Hence, we arrive at a covering of the unit ball in Bp1 (Q) by 2k balls of radius cek(id) in Bp^(Q). Inserting

6 = S1 - S2 - n( — - —^ (2.58)

1 2 p1 p2

in the exponent, we finally obtain the desired estimate

ek (id) < ck-(s1-s2)/n, k e n. (2.59) Step 2. Let p1 > p2. Since, by Proposition 2.4,

Bp1q(Q) c BpM2 (Q), (2.60)

we see that

Bp^ (Q) c Bp/i?2 (Q) c BpM2 (Q), (2.61)

and, therefore, (2.40) is a consequence of Step 1 applied to p1 = p2. This completes the proof for the upper bound.

Remark 2.12. By (2.13) and the above definitions, we have

^mrn^) (Q) ^ Fp,q(Q) ^ B;,max(pq) (Q). (2.62)

In other words, any assertion about entropy numbers for B-spaces where the parameter q does not play any role applies also to the related F-spaces.

Therefore, using Lemma 2.6 (iv) and Theorem 2.11, we deduce compactness of the corresponding embeddings related to B- and F-spaces under investigation.

3. Homogeneity

Our first aim is to prove the following characterization.

Proposition 3.1. Let 0 <p,q <to, s> 0, and let R> 0 be a real number. Then,

n / rto dt \ 1/q

| Bp q(rn) 11 rsqWr(f,t)qpj) (3.1)

for all f e Bp q(rn) with supp f c BR.

Proof. We will need that Bpq(BR) embeds compactly into Lp(BR). This follows at once from the fact that Bpq(BR) is compactly embedded into Bp-q (BR), cf. Remark 2.12, and Bp-q (BR) Lp(Br), which is trivial.We argue similarly to [6]. We have to prove that

aTO dt \1/q

3 t-sqf)qpdi) (3.2)

for every f e Bpq(rn) with supp f c BR. Let us assume that this is not true. Then, we find a sequence (fj ^ c Bp (rn), such that

aTO dA 1/q 1

^-sqWr(fj,t)qpd) < j, (3.3) that is, we obtain that ||fj | Bpq(rn)|| is bounded. The trivial estimates

||fj | Lp(rn)|| = ||fj | Lp(Br)||, ||fj | Bp,q(BR)|| < ||fj | Bpq(rn)|| (3.4)

imply that this is true also for ||fj | Bp (BR)||. Due to the compactness of Bp (BR) Lp(BR),

f] I "pq\BKJ N Lilc ^^a^uiCM ui upqy

fj - f

w(-, t)p, we obtain that

we may assume, that f] — f in Lp(BR) with Nf |Lp(BR)N = 1. Using the subadditivity of

(JT rsq^fj - fj<di)1/q < 1+j- (35)

Together with the estimate ||fj - fji | Lp(rn)| ^ 0, this implies that (fj);c=1 is a Cauchy sequence in Bpq(rn), that is, fj ^ g in Bpq(rn). Obviously, f = g follows.

The subadditivity of w(-, t)p used to the sum (f - fj) + fj implies finally that

aTO dt \1/q t-sq^ (f,t)qpd) = 0. (3.6)

As wr (f, t) is a nondecreasing function of t, this implies that wr (f, t) = 0 for all 0 < t < œ and finally ||Af | Lp(m")N = 0 for all h e m". By standard arguments, this is satisfied only if f is

a polynomial of order at most r. Due to its bounded support, we conclude that f = 0, which is a contradiction with ||f | Lp(rn) || = 1. □

With the help of this proposition, the proof of homogeneity quickly follows.

Theorem 3.2. Let 0 < X < 1 and f e Bspq(Rn) with supp f c Bx. Then,

X-) | Bp q(rn)J ~ Xs-n/plf | Bp q(rn)1 (3.7)

with constants of equivalence independent of 1 and f. Proof. We know from Proposition 3.1 that

n /cœ dt \1/q

A-) | B;/q(R")|| t-sqWrf (A-),f)PTJ ,

as supp f (A-) c Bi. Using Ah (f (A-))(x) = (Arkhf)(Ax), we get

(f (^ = sup|| Ah(f (A-))||p = sup||(AAhf)(A-)||p = r"/psup||(AAhf)(-)||p

|h|<t |h|<t | h | <t

= A-n/p sup ||(AAhf )(')||p = ^^r (JA),

|Ah|<At

which finally implies

aœ dt \1/q /cœ dt \1/q

rsqMr(f(A-),t);d) = A-/p(jo t-sqWrU,At)ld)

aœ jf \ 1/q

t-sqwrf,t)qvj) ~ As—n/p 11 f | Bp/q(R")||.

(3.10)

The homogeneity property for Triebel-Lizorkin spaces Fp,q(R") follows similarly. Proposition 3.3. Let 0 <p < œ, 0 <q <œ, s > 0, and let R> 0 be a real number. Then,

| Fp,q(rn)|| ~

for all f e Fp q(rn) with supp f c BR. Proof. We have to prove that

aœ dt\1/q ^ rsqd[pf (yd) | LP(Rn)

(3.11)

||f | Lp(rn)|

aœ dt \1/q t sqdrt/pf (o^y) | LP(r")

(3.12)

for every f e Fp (rn) with supp f c Br. Let us assume again that this is not true. Then, we

FS co"

find a sequence (fjc Fp (r") such that

llf I Lpy

aœ dt \1/q 3 t-sqdlpfj(^dt) | Lp(r")

(3.13)

which in turn implies that ||fj | Fpq(rn)|| is bounded. Again, the same is true also for ||fj|Fpq(BR)||. Due to the compactness of Fpq(rn) ^ Lp(rn), we may assume that fj ^ f in Lp(Br) with ||f | Lp(BR)|| = 1. A straightforward calculation shows again that (fj)TO=1 is a Cauchy sequence in Fp (rn) and, therefore, fj ^ f also in Fp (rn). Finally, we obtain

t-sqd[pf (-)qj) | lp(rn)

(3.14)

or, equivalently,

t-sqdU (x)q- = 0

(3.15)

for almost every x e rn. Hence, drtpf (x) = 0 for almost all x e rn and almost all t > 0. By standard arguments, it follows that f must be almost everywhere equal to a polynomial of order smaller than r. Together with the bounded support of f, we obtain that f must be equal to zero almost everywhere. □

Theorem 3.4. Let 0 <1 < 1 and f e Fp (rn) with suppf c Bx. Then,

a-) | fp,q(rn)ll ~ \s-"/p\lf | fp,q(r")|

(3.16)

with constants of equivalence independent of A and f. Proof. We know from Proposition 3.3 that

AO | F;

p,q(r"

ot-sqd{p(f (A-))(-)q—j i Lp(rn)

(3.17)

as supp f (A-) c B1. Using Ah (f (A-))(x) = (ArAhf)(Ax), we get using the substitution h = Ah,

dlpf (A-))(x) ^ t"^^^ |A^f (A-)(x)|pd^ ^t"^^^ I (AAhf)(Ax)|

(At)" ) ~h<Xi I ( Af (Ax)|pdhJ = d\trp{f) (Ax),

(3.18)

Journal of Function Spaces and Applications which finally implies

aœ dt\1/q / dt\1/q rsqdlp(f (l-))(-)qTj | Lp(rn) = y t-sqdruf (l-)qTj | Lp(rn)

aœ dt \1/q t-sqd\pf (\-)qT) | Lp(rn)

aœ dt \1/q t-sqdr,pf (^d j | Lp(rn)

_ js-n/p

Js-n/p\\f | Fp,q(rn)i

(3.19) □

4. Pointwise Multipliers

We briefly sketch an application of the above homogeneity results in terms of pointwise multipliers. A locally integrable function y in M" is called a pointwise multiplier in A^R") if '

f ~ Vf (4-1)

maps the considered space into itself. For further details on the subject, we refer to [27, pp. 201-206] and [28, Chapter 4]. Our aim is to generalize Proposition 2.3 as a direct consequence of Theorems 3.2 and 3.4. Again let Bx be the balls introduced in (2.7).

Corollary 4.1. Let s > 0,0 <p,q <<x>, and 0 < X < l.Lety be a function having classical derivatives in B2X up to order 1 + [s] with

\DYy(x) \ < aJ-

< 1 + [s], x e B2j,

for some constant a> 0. Then, y is a pointwise multiplier in Bpq(Bx),

\\f | Bp,q(Bj) \\ < c\\f | Bp,q(Bj) \\ , (4.3)

where c is independent of f e Bpq(Bx) and of 1 (but depends on a).

Proof. By Proposition 2.3, the function ^(1-) is a pointwise multiplier in Bpq(B1). Then, (4.3) is a consequence of (3.7),

\\f | Bp,q(Bj)\\ ~ j-(s-n/p)\\yf (J.) | Bp,q(Bl)\\

IL i )

< j-(s-n/p)Nf(J.) | BSpJBi)\\ ~ \\f | BSpJBx)

Remark 4.2. In terms of Triebel-Lizorkin spaces Fp,q(m"), we obtain corresponding results (assuming p < œ) with the additional restriction on the smoothness parameter s that

s>n( , 1 . - M. (4.5)

min p, q p

This follows from the fact that the analogue of Proposition 2.3 for F-spaces is established using an atomic characterization of the spaces Fp q(M") which is only true if we impose (4.5), (cf. [13, Proposition 9.14]).

Acknowledgment

Jan Vybiral acknowledges the financial support provided by the START award "Sparse Approximation and Optimization in High Dimensions" of the Fonds zur Forderung der wissenschaftlichen Forschung (FWF, Austrian Science Foundation).

References

[1] G. Bourdaud, Sur les Operateurs Pseudo-Différentiels a Coefficients peu Réguliers, Habilitation thesis, Université de Paris-Sud, Paris, France, 1983.

[2] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, vol. 120 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1996.

[3] J. Vybiral, "Dilation operators and sampling numbers," Journal of Function Spaces and Applications, vol. 6, no. 1, pp. 17-46, 2008.

[4] C. Schneider, "On dilation operators in Besov spaces," Revista Matematica Complutense, vol. 22, no. 1, pp. 111-128, 2009.

[5] C. Schneider and J. Vybiral, "On dilation operators in Triebel-Lizorkin spaces," Functiones et Approximatio Commentarii Mathematici, vol. 41, no. part 2, pp. 139-162, 2009.

[6] A. M. Caetano, S. Lopes, and H. Triebel, "A homogeneity property for Besov spaces," Journal of Function Spaces and Applications, vol. 5, no. 2, pp. 123-132, 2007.

[7] H. Triebel, The Structure of Functions, vol. 97 of Monographs in Mathematics, Birkhauser, Basel, Switzerland, 2001.

[8] C. Schneider and J. Vybiral, "Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces," http://arxiv.org/abs/1201.2280v1.

[9] E. A. StoroZenko and P. Osval'd, "Jackson's theorem in the spaces LP(Rk), 0 < p < 1," Sibirskit Matematiceskit Zurnal, vol. 19, no. 4, pp. 888-901,1978.

[10] C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Boston, Mass, USA, 1988.

[11] R. A. DeVore and G. G. Lorentz, Constructive Approximation, vol. 303 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Berlin, Germany, 1993.

[12] H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhauser, Basel, Switzerland, 1983.

[13] H. Triebel, Theory of Function Spaces. III, vol. 100 of Monographs in Mathematics, Birkhauser, Basel, Switzerland, 2006.

[14] R. A. DeVore and V. A. Popov, "Interpolation of Besov spaces," Transactions of the American Mathematical Society, vol. 305, no. 1, pp. 397-414,1988.

[15] C. Schneider, "Spaces of Sobolev type with positive smoothness on R", embeddings and growth envelopes," Journal of Function Spaces and Applications, vol. 7, no. 3, pp. 251-288, 2009.

[16] L. I. Hedberg and Y. Netrusov, "An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation," Memoirs of the American Mathematical Society, vol. 188, no. 882, p. 97, 2007.

[17] C. Schneider, "Trace operators in Besov and Triebel-Lizorkin spaces," Zeitschrift für Analysis und ihre Anwendungen, vol. 29, no. 3, pp. 275-302, 2010.

[18] C. Schneider, "Traces in Besov and Triebel-Lizorkin spaces on domains," Mathematische Nachrichten, vol. 284, no. 5-6, pp. 572-586, 2011.

[19] D. D. Haroske and C. Schneider, "Besov spaces with positive smoothness on RK, embeddings and growth envelopes," Journal of Approximation Theory, vol. 161, no. 2, pp. 723-747, 2009.

[20] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, NY, USA, Oxford Science Publication, 1987.

[21] A. Pietsch, Eigenvalues and s-Numbers, vol. 13 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1987.

[22] H. Konig, Eigenvalue Distribution of Compact Operators, vol. 16 of Operator Theory: Advances and Applications, Birkhauser, Basel, Switzerland, 1986.

[23] B. Carl and I. Stephani, Entropy, Compactness and the Approximation of Operators, vol. 98 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1990.

[24] C. Schneider, "Trace operators on fractals, entropy and approximation numbers," Georgian Mathematical Journal, vol. 18, no. 3, pp. 549-575, 2011.

[25] H. Triebel, Fractals and Spectra, vol. 91 of Monographs in Mathematics, Birkhauser, Basel, Switzerland, 1997.

[26] R. A. DeVore and R. C. Sharpley, "Besov spaces on domains in Rd," Transactions of the American Mathematical Society, vol. 335, no. 2, pp. 843-864,1993.

[27] H. Triebel, Theory of Function Spaces. II, vol. 84 of Monographs in Mathematics, Birkhauser, Basel, Switzerland, 1992.

[28] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 1996.

Copyright of Journal of Function Spaces & Applications is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.