Homogeneity property of Besov and Triebel-Lizorkin spaces
aa r X i v : . [ m a t h . F A ] D ec Homogeneity property of Besov and Triebel-Lizorkin spaces
Cornelia Schneider and Jan VybíralAugust 15, 2018
Abstract
We consider the classical Besov and Triebel-Lizorkin spaces defined via differences and prove ahomogeneity property for functions with bounded support in the frame of these spaces. As the proofis based on compact embeddings between the studied function spaces we present also some resultson the entropy numbers of these embeddings. Moreover, we derive some applications in terms ofpointwise multipliers.
Keywords and Phrases: homogeneity, Besov spaces, Triebel-Lizorkin spaces, differences, pointwisemultipliers, entropy numbers.
Introduction
The present note deals with classical Besov spaces B sp,q ( R n ) and Triebel-Lizorkin spaces F sp,q ( R n ) definedvia differences, briefly denoted as B- and F-spaces in the sequel. We study the properties of the dilationoperator, which is defined for every λ > T λ : f → f ( λ · ) . The norms of these operators on Besov and Triebel-Lizorkin spaces were studied already in [Bo83] and[ET96, Sections 2.3.1 and 2.3.2] with complements given in [Vyb08], [Sch09a], and [SV09].We prove the so-called homogeneity property , showing that for s > < p, q ≤ ∞ , k f ( λ · ) | B sp,q ( R n ) k ∼ λ s − np k f | B sp,q ( R n ) k , (0.1)for all 0 < λ ≤ f ∈ B sp,q ( R n ) with supp f ⊂ { x ∈ R n : | x | ≤ λ } . The same property holds true for the spaces F sp,q ( R n ). This extends and completes [CLT07], wherecorresponding results for the spaces B sp,q ( R n ), defined via Fourier-analytic tools, were established, whichcoincide with our spaces B sp,q ( R n ) if s > max (cid:16) , n (cid:16) p − (cid:17)(cid:17) . Concerning the corresponding F-spaces F sp,q ( R n ), the same homogeneity property had already been established in [Tri01, Cor. 5.16, p. 66].Our results yield immediate applications in terms of pointwise multipliers. Furthermore, we remark thatthe homogeneity property is closely related with questions concerning refined localization, non-smoothatoms, local polynomial approximation, and scaling properties. This is out of our scope for the timebeing. But we use this property in the forthcoming paper [SV11] in connection with non-smooth atomicdecompositions in function spaces.Our proof of (0.1) 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 suchembeddings.This note is organized as follows. We start with the necessary definitions and the results about entropynumbers in Section 1. Then we focus on equivalent quasi-norms for the elements of certain subspaces of B sp,q ( R n ) and F sp,q ( R n ), respectively, from which the homogeneity property will follow almost immediatelyin Section 2. The last section states some applications in terms of pointwise multipliers. The second author acknowledges the financial support provided by the START-award “Sparse Approximation andOptimization in High Dimensions” of the Fonds zur Förderung der wissenschaftlichen Forschung (FWF, Austrian ScienceFoundation). Preliminaries
We use standard notation. Let N be the collection of all natural numbers and let N = N ∪ { } . Let R n be Euclidean n -space, n ∈ N , C the complex plane. The set of multi-indices β = ( β , . . . , β n ), β i ∈ N , i = 1 , . . . , n , is denoted by N n , with | β | = β + · · · + β n , as usual. We use the symbol ’ . ’ in a k . b k or ϕ ( x ) . ψ ( x )always to mean that there is a positive number c such that a k ≤ c b k or ϕ ( x ) ≤ c ψ ( x )for all admitted values of the discrete variable k or the continuous variable x , where ( a k ) k , ( b k ) k arenon-negative sequences and ϕ , ψ are non-negative functions. We use the equivalence ‘ ∼ ’ in a k ∼ b k or ϕ ( x ) ∼ ψ ( x )for a k . b k and b k . a k or ϕ ( x ) . ψ ( x ) and ψ ( x ) . ϕ ( x ) . If a ∈ 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 ⊂ Y and the natural embeddingof X in Y is continuous. All unimportant positive constants will be denoted by c , occasionally withsubscripts. For convenience, let both d x and | · | stand for the ( n -dimensional) Lebesgue measure in thesequel. L p ( R n ), with 0 < p ≤ ∞ , stands for the usual quasi-Banach space with respect to the Lebesguemeasure, quasi-normed by k f | L p ( R n ) k := (cid:18)Z R n | f ( x ) | p d x (cid:19) p with the appropriate modification if p = ∞ . Moreover, let Ω denote a domain in R n . Then L p (Ω) is thecollection of all complex-valued Lebesgue measurable functions in Ω such that k f | L p (Ω) k := (cid:18)Z Ω | f ( x ) | p d x (cid:19) p (with the usual modification if p = ∞ ) is finite.Furthermore, B R stands for an open ball with radius R > B R = { x ∈ R n : | x | < R } . (1.1)Let Q j,m with j ∈ N and m ∈ Z n denote a cube in R n with sides parallel to the axes of coordinates,centered at 2 − j m , and with side length 2 − j +1 . For a cube Q in R n and r >
0, we denote by rQ the cubein R n concentric with Q and with side length r times the side length of Q . Furthermore, χ j,m stands forthe characteristic function of Q j,m . Function spaces defined via differences If f is an arbitrary function on R n , h ∈ R n and r ∈ N , then(∆ h f )( x ) = f ( x + h ) − f ( x ) and (∆ r +1 h f )( x ) = ∆ h (∆ rh f )( x )are the usual iterated differences. Given a function f ∈ L p ( R n ) the r -th modulus of smoothness is definedby ω r ( f, t ) p = sup | h |≤ t k ∆ rh f | L p ( R n ) k , t > , < p ≤ ∞ , (1.2)and d rt,p f ( x ) = t − n Z | h |≤ t | (∆ rh f )( x ) | p d h ! /p , t > , < p < ∞ , (1.3)denotes its ball means . 2 efinition . (i) Let 0 < p, q ≤ ∞ , s >
0, and r ∈ N such that r > s . Then the Besov space B sp,q ( R n )contains all f ∈ L p ( R n ) such that k f | B sp,q ( R n ) k r = k f | L p ( R n ) k + (cid:18)Z t − sq ω r ( f, t ) qp d tt (cid:19) /q (1.4)(with the usual modification if q = ∞ ) is finite. (ii) Let 0 < p < ∞ , 0 < q ≤ ∞ , s >
0, and r ∈ N such that r > s . Then F sp,q ( R n ) is the collection ofall f ∈ L p ( R n ) such that k f | F sp,q ( R n ) k r = k f | L p ( R n ) k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z t − sq d rt,p f ( · ) q d tt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (1.5)(with the usual modification if q = ∞ ) is finite. Remark . These are the classical Besov and
Triebel-Lizorkin spaces , in particular, when 1 ≤ p, q ≤ ∞ ( p < ∞ for the F-spaces) and s >
0. We shall sometimes write A sp,q ( R n ) when both scales of spaces B sp,q ( R n ) and F sp,q ( R n ) are concerned simultaneously.Concerning the spaces B sp,q ( R n ), the study for all admitted s , p and q goes back to [SO78], we also referto [BS88, Ch. 5, Def. 4.3] and [DL93, Ch. 2, §10]. There are as well many older references in theliterature devoted to the cases p, q ≥ F sp,q ( R n ) has been described in detail in [Tri83] for thosespaces which can also be considered as subspaces of S ′ ( R n ). Otherwise one finds in [Tri06, Section 9.2.2,pp. 386–390] the necessary explanations and references to the relevant literature.Definition 1.1 is independent of r , meaning that different values of r > s result in norms which areequivalent. This justifies our omission of r in the sequel. Moreover, the integrals R can be replaced by R ∞ resulting again in equivalent quasi-norms, cf. [DP88, Sect. 2].The spaces are quasi-Banach spaces (Banach spaces if p, q ≥ L p ( R n ), in particular, for s > < q ≤ ∞ , we have the embeddings A sp,q ( R n ) ֒ → L p ( R n ) , where 0 < p ≤ ∞ ( p < ∞ for F-spaces). Furthermore, the B-spaces are closely linked with the Triebel-Lizorkin spaces via B sp, min( p,q ) ( R n ) ֒ → F sp,q ( R n ) ֒ → B sp, max( p,q ) ( R n ) , (1.6)cf. [Sch09b, Prop. 1.19(i)]. The classical scale of Besov spaces contains many well-known function spaces.For example, if p = q = ∞ , one recovers the Hölder-Zygmund spaces C s ( R n ), i.e., B s ∞ , ∞ ( R n ) = C s ( R n ) , s > . (1.7)Recent results by Hedberg, Netrusov [HN07] on atomic decompositions and by
Triebel [Tri06,Sect. 9.2] on the reproducing formula provide an equivalent characterization of Besov spaces B sp,q ( R n )using subatomic decompositions , which introduces B sp,q ( R n ) as those f ∈ L p ( R n ) which can be representedas f ( x ) = X β ∈ N n ∞ X j =0 X m ∈ Z n λ βj,m k βj,m ( x ) , x ∈ R n , with coefficients λ = { λ βj,m ∈ C : β ∈ N n , j ∈ N , m ∈ Z n } belonging to some appropriate sequence space b s,̺p,q defined as b s,̺p,q := (cid:8) λ : k λ | b s,̺p,q k < ∞ (cid:9) (1.8)where k λ | b s,̺p,q k = sup β ∈ N n ̺ | β | ∞ X j =0 j ( s − n/p ) q X m ∈ Z n | λ βj,m | p ! q/p /q , (1.9) s >
0, 0 < p, q ≤ ∞ (with the usual modification if p = ∞ and/or q = ∞ ), ̺ ≥
0, and k βj,m ( x ) are certainstandardized building blocks (which are universal). This subatomic characterization will turn out to bequite useful when studying entropy numbers.In terms of pointwise multipliers in B sp,q ( R n ) the following is known.3 roposition . Let 0 < p, q ≤ ∞ , s > k ∈ N with k > s , and let h ∈ C k ( R n ). Then f −→ hf is a linear and bounded operator from B sp,q ( R n ) into itself.The proof relies on atomic decompositions of the spaces B sp,q ( R n ), cf. [Sch10a, Prop. 2.5]. We willgeneralize this result in Section 3 as an application of our homogeneity property. Function spaces on domains
ΩLet Ω be a domain in R n . We define spaces A sp,q (Ω) by restriction of the corresponding spaces on R n , i.e. A sp,q (Ω) is the collection of all f ∈ L p (Ω) such that there is a g ∈ A sp,q ( R n ) with g (cid:12)(cid:12) Ω = f . Furthermore, k f | A sp,q (Ω) k = inf k g | A sp,q ( R n ) k , where the infimum is taken over all g ∈ A sp,q ( R n ) such that the restriction g (cid:12)(cid:12) Ω to Ω coincides in L p (Ω)with f .In particular, the subatomic characterization for the spaces B sp,q ( R n ) from Remark 1.2 carries over. Forfurther details on this subject we refer to [Sch11b, Sect 2.1].Embeddings results between the spaces B sp,q ( R n ) hold also for the spaces B sp,q (Ω), since they are definedby restriction of the corresponding spaces on R n . Furthermore, these results can be improved, if weassume Ω ⊂ R n to be bounded. Proposition . Let 0 < s < s < ∞ , 0 < p , p , q , q ≤ ∞ , and Ω ⊂ R n be bounded. If δ + = s − s − d (cid:18) p − p (cid:19) + > , (1.10)we have the embedding B s p ,q (Ω) ֒ → B s p ,q (Ω) . (1.11)P r o o f : If p ≤ p the embedding follows from [HS09, Th. 1.15], since the spaces on Ω are definedby restriction of their counterparts on R n . Therefore it remains to show that for p > p we have theembedding B s p ,q (Ω) ֒ → B s p ,q (Ω) . (1.12)Let ψ ∈ D ( R n ) with support in the compact set Ω and ψ ( x ) = 1 if x ∈ ¯Ω ⊂ Ω . Then for f ∈ B s p ,q (Ω), there exists g ∈ B s p ,q ( R n ) with g (cid:12)(cid:12) Ω = f and k f | B s p ,q (Ω) k ∼ k g | B s p ,q ( R n ) k . We calculate k f | B s p ,q (Ω) k ≤ k ψg | B s p ,q ( R n ) k≤ k ψg | B s p ,q ( R n ) k≤ c ψ k g | B s p ,q ( R n ) k ∼ k f | B s p ,q (Ω) k . (1.13)The last inequality in (1.13) follows from Proposition 1.3. In the 2nd step we used (1.4) together withthe fact that k ∆ rh ( ψg ) | L p ( R n ) k ≤ c Ω k ∆ rh ( ψg ) | L p ( R n ) k , p > p , which follows from Hölder’s inequality since supp ψg ⊂ Ω is compact.4 ntropy numbers In order to prove the homogeneity results later on we have to rely on the compactness of embeddingsbetween B-spaces, B sp,q (Ω), and F-spaces, F sp,q (Ω), respectively. This will be established with the helpof entropy numbers. We briefly introduce the concept and collect some properties afterwards.Let X and Y be quasi-Banach spaces and T : X → Y be a bounded linear operator. If additionally, T is continuous we write T ∈ L ( X, Y ). Let U X = { x ∈ X : k x | X k ≤ } denote the unit ball in thequasi-Banach space X . An operator T is called compact if for any given ε > U X with finitely many balls in Y of radius ε . Definition . Let
X, Y be quasi-Banach spaces and let T ∈ L ( X, Y ). Then for all k ∈ N , the k th dyadicentropy number e k ( T ) of T is defined by e k ( T ) = inf ε > T ( U X ) ⊂ k − [ j =1 ( y j + εU Y ) for some y , . . . , y k − ∈ Y , where U X and U Y denote the unit balls in X and Y , respectively.These numbers have various elementary properties which are summarized in the following lemma. Lemma . Let
X, Y and Z be quasi-Banach spaces, let S, T ∈ L ( X, Y ) and R ∈ L ( Y, Z ). (i) ( Monotonicity ) k T k ≥ e ( T ) ≥ e ( T ) ≥ · · · ≥
0. Moreover, k T k = e ( T ), provided that Y is aBanach space. (ii) ( Additivity ) If Y is a p -Banach space (0 < p ≤ j, k ∈ N e pj + k − ( S + T ) ≤ e pj ( S ) + e pk ( T ) . (iii) ( Multiplicativity ) For all j, k ∈ N e j + k − ( RT ) ≤ e j ( R ) e k ( T ) . (iv) ( Compactness ) T is compact if, and only if,lim k →∞ e k ( T ) = 0 . Remark . As for the general theory we refer to [EE87], [Pie87] and [Kön86]. Further information onthe subject is also covered by the more recent books [ET96] and [CS90].Some problems about entropy numbers of compact embeddings for function spaces can be transferred tocorresponding questions in related sequence spaces. Let n > { M j } j ∈ N be a sequence of naturalnumbers satisfying M j ∼ jn , j ∈ N . (1.14)Concerning entropy numbers for the respective sequence spaces b s,̺p,q ( M j ), which are defined as the se-quence spaces b s,̺p,q in (1.9) with the sum over m ∈ Z n replaced by a sum over m = 1 , . . . , M j , the followingresult was proved in [Sch11a, Prop. 3.4] Proposition . Let d >
0, 0 < σ , σ < ∞ , and 0 < q , q ≤ ∞ . Furthermore, let ̺ > ̺ ≥ < p ≤ p ≤ ∞ and δ = σ − σ − n (cid:18) p − p (cid:19) > . (1.15)Then the identity map id : b σ ,̺ p ,q ( M j ) → b σ ,̺ p ,q ( M j ) (1.16)is compact, where M j is restricted by (1.14).The next theorem provides a sharp result for entropy numbers of the identity operator related to thesequence spaces b s,̺p,q ( M j ). 5 heorem . Let n >
0, 0 < s , s < ∞ , and 0 < q , q ≤ ∞ . Furthermore, let ̺ > ̺ ≥ < p ≤ p ≤ ∞ and δ = s − s − n (cid:18) p − p (cid:19) > . (1.17)For the entropy numbers e k of the compact operatorid : b s ,̺ p ,q ( M j ) → b s ,̺ p ,q ( M j ) (1.18)we have e k (id) ∼ k − δn + p − p , k ∈ N . Remark . The proof of Theorem 1.9 follows from [Tri97, Th. 9.2]. Using the notation from this bookwe have b s i ,̺ i p i ,q i ( M j ) = ℓ ∞ h ̺ i ℓ q i (cid:16) j ( s i − npi ) ℓ M j p i (cid:17)i , i = 1 , . Recall the embedding assertions for Besov spaces B sp,q (Ω) from Proposition 1.4. We will give an upperbound for the corresponding entropy numbers of these embeddings. For our purposes it will be sufficientto assume Ω = B R . Theorem . Let 0 < s < s < ∞ , < p , p ≤ ∞ , < q , q ≤ ∞ , and δ + = s − s − n (cid:18) p − p (cid:19) + > . Then the embedding id : B s p ,q (Ω) → B s p ,q (Ω) (1.19)is compact and for the related entropy numbers we compute e k (id) . k − s − s n , k ∈ N . (1.20) Proof.
Step 1: Let p ≥ p , δ + = δ , and let f ∈ B s p ,q (Ω), then by [DS93, Th. 6.1] there is a (nonlinear)bounded extension operator g = Ex f such that Re Ω g = g (cid:12)(cid:12)(cid:12) Ω = f (1.21)and k g | B s p ,q ( R n ) k ≤ c k f | B s p ,q (Ω) k . We may assume that g is zero outside a fixed neighbourhood Λ of Ω. Using the subatomic approach for B s p ,q ( R n ), cf. Remark 1.2, we can find an optimal decomposition of g , i.e., g ( x ) = X β ∈ N n ∞ X j =0 X m ∈ Z n λ βj,m k βj,m ( x ) , k g | B s p ,q ( R n ) k ∼ k λ | b s ,̺ p ,q k (1.22)with ̺ > M j for fixed j ∈ N be the number of cubes Q j,m such that rQ j,m ∩ Ω = ∅ . Since Ω ⊂ R n is bounded we have M j ∼ jn , j ∈ N . This coincides with (1.14). We introduce the (nonlinear) operator S , S : B s p ,q ( R n ) → b s ,̺ p ,q ( M j )by Sg = λ, λ = n λ βj,m : β ∈ N n , j ∈ N , m ∈ Z n , rQ j,m ∩ Ω = ∅ o , g is given by (1.22). Recall that the expansion is not unique but this does not matter. It followsthat S is a bounded map since k S k = sup g =0 k λ | b s ,̺ p ,q ( M j ) kk g | B s p ,q ( R n ) k ≤ c. Next we construct the linear map T , T : b s ,̺ p ,q ( M j ) → B s p ,q ( R n ) , given by T λ = X β ∈ N n ∞ X j =0 M j X m =1 λ βj,m k βj,m ( x ) . It follows that T is a linear (since the subatomic approach provides an expansion of functions via universalbuilding blocks) and bounded map, k T k = sup λ =0 k T λ | B s p ,q ( R n ) kk λ | b s ,̺ p ,q ( M j ) k ≤ c. We complement the three bounded maps Ex , S , T by the identity operatorid : b s ,̺ p ,q ( M j ) → b s ,̺ p ,q ( M j ) with ̺ > ̺ , (1.23)which is compact by Proposition 1.8 and the restriction operatorRe Ω : B s p ,q ( R n ) → B s p ,q (Ω) , which is continuous. From the constructions it follows thatid (cid:0) B s p ,q (Ω) → B s p ,q (Ω) (cid:1) = Re Ω ◦ T ◦ id ◦ S ◦ Ex . (1.24)Hence, taking finally Re Ω we obtain f by (1.21), where we started from. In particular, due to the factthat we used the subatomic approach, the final outcome is independent of ambiguities in the nonlinearconstructions Ex and S . The unit ball in B s p ,q (Ω) is mapped by S ◦ Ex into a bounded set in b s ,̺ p ,q ( M j ) . Since the identity operator id from (1.23) is compact, this bounded set is mapped into a pre-compact setin b s ,̺ p ,q ( M j ) , which can be covered by 2 k balls of radius ce k (id) with e k (id) ≤ ck − δn + p − p , k ∈ N . This follows from Theorem 1.9, where we used p ≥ p . Applying the two linear and bounded maps T andRe Ω afterwards does not change this covering assertion – using Lemma 1.6(iii) and ignoring constants forthe time being. Hence, we arrive at a covering of the unit ball in B s p ,q (Ω) by 2 k balls of radius ce k (id)in B s p ,q (Ω). Inserting δ = s − s − n (cid:18) p − p (cid:19) in the exponent we finally obtain the desired estimate e k (id) ≤ ck − s − s n , k ∈ N . Step 2: Let p > p . Since by Proposition 1.4, B s p ,q (Ω) ⊂ B s p ,q (Ω) , we see that B s p ,q (Ω) ⊂ B s p ,q (Ω) ⊂ B s p ,q (Ω) , and therefore (1.20) is a consequence of Step 1 applied to p = p . This completes the proof for the upperbound. 7 emark . By (1.6) and the above definitions we have B sp, min( p,q ) (Ω) ֒ → F sp,q (Ω) ֒ → B sp, max( p,q ) (Ω) . (1.25)In other words, any assertion about entropy numbers for B-spaces where the parameter q does not playany role applies also to the related F-spaces.Therefore, using Lemma 1.6(iv) and Theorem 1.11 we deduce compactness of the corresponding embed-dings related to B- and F-spaces under investigation. Our first aim is to prove the following characterization.
Proposition . Let 0 < p, q ≤ ∞ , s > R > k f | B sp,q ( R n ) k ∼ (cid:18)Z ∞ t − sq ω r ( f, t ) qp dtt (cid:19) /q for all f ∈ B sp,q ( R n ) with supp f ⊂ B R . Proof.
We shall need, that B sp,q ( B R ) embeds compactly into L p ( B R ). This follows at once from thefact that B sp,q ( B R ) is compactly embedded into B s − εp,q ( B R ), cf. Remark 1.12, and B s − εp,q ( B R ) ֒ → L p ( B R ),which is trivial.We argue similarly to [CLT07]. We have to prove, that k f | L p ( R n ) k . (cid:18)Z ∞ t − sq ω r ( f, t ) qp dtt (cid:19) /q for every f ∈ B sp,q ( R n ) with supp f ⊂ B R . Let us assume, that this is not true. Then we find a sequence( f j ) ∞ j =1 ⊂ B sp,q ( R n ), such that k f j | L p ( R n ) k = 1 and (cid:18)Z ∞ t − sq ω r ( f j , t ) qp dtt (cid:19) /q ≤ j , (2.1)i.e., we obtain that k f j | B sp,q ( R n ) k is bounded. The trivial estimates k f j | L p ( R n ) k = k f j | L p ( B R ) k and k f j | B sp,q ( B R ) k ≤ k f j | B sp,q ( R n ) k imply that this is true also for k f j | B sp,q ( B R ) k . Due to the compactness of B sp,q ( B R ) ֒ → L p ( B R ), we mayassume, that f j → f in L p ( B R ) with k f | L p ( B R ) k = 1. Using the subadditivity of ω ( · , t ) p , we obtain that (cid:18)Z ∞ t − sq ω r ( f j − f j ′ , t ) qp dtt (cid:19) /q ≤ j + 1 j ′ . Together with the estimate k f j − f j ′ | L p ( R n ) k →
0, this implies that ( f j ) ∞ j =1 is a Cauchy sequence in B sp,q ( R n ), i.e. f j → g in B sp,q ( R n ). Obviously, f = g follows.The subadditivity of ω ( · , t ) p used to the sum ( f − f j ) + f j implies finally, that (cid:18)Z ∞ t − sq ω r ( f, t ) qp dtt (cid:19) /q = 0 . As ω r ( f, t ) is a non-decreasing function of t , this implies that ω r ( f, t ) = 0 for all 0 < t < ∞ and finally k ∆ rh f | L p ( R n ) k = 0 for all h ∈ R n . By standard arguments, this is satisfied only if f is a polynomial oforder at most r . Due to its bounded support, we conclude, that f = 0, which is a contradiction with k f | L p ( R n ) k = 1 . With the help of this proposition, the proof of homogeneity quickly follows.
Theorem . Let 0 < λ ≤ f ∈ B sp,q ( R n ) with supp f ⊂ B λ . Then k f ( λ · ) | B sp,q ( R n ) k ∼ λ s − n/p k f | B sp,q ( R n ) k (2.2)with constants of equivalence independent of λ and f .8 roof. We know from Proposition 2.1 that k f ( λ · ) | B sp,q ( R n ) k ∼ (cid:18)Z ∞ t − sq ω r ( f ( λ · ) , t ) qp dtt (cid:19) /q , as supp f ( λ · ) ⊂ B . Using ∆ rh ( f ( λ · ))( x ) = (∆ rλh f )( λx ), we get ω r ( f ( λ · ) , t ) p = sup | h |≤ t k ∆ rh ( f ( λ · )) k p = sup | h |≤ t k (∆ rλh f )( λ · ) k p = λ − n/p sup | h |≤ t k (∆ rλh f )( · ) k p = λ − n/p sup | λh |≤ λt k (∆ rλh f )( · ) k p = λ − n/p ω r ( f, λt ) p , which finally implies (cid:18)Z ∞ t − sq ω r ( f ( λ · ) , t ) qp dtt (cid:19) /q = λ − n/p (cid:18)Z ∞ t − sq ω r ( f, λt ) qp dtt (cid:19) /q = λ s − n/p (cid:18)Z ∞ t − sq ω r ( f, t ) qp dtt (cid:19) /q ∼ λ s − n/p k f | B sp,q ( R n ) k . The homogeneity property for Triebel-Lizorkin spaces F sp,q ( R n ) follows similarly. Proposition . Let 0 < p < ∞ , < q ≤ ∞ , s > R > k f | F sp,q ( R n ) k ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rt,p f ( · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) for all f ∈ F sp,q ( R n ) with supp f ⊂ B R . Proof.
We have to prove that k f | L p ( R n ) k . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rt,p f ( · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) for every f ∈ F sp,q ( R n ) with supp f ⊂ B R . Let us assume again, that this is not true. Then we find asequence ( f j ) ∞ j =1 ⊂ F sp,q ( R n ) such that k f j | L p ( R n ) k = 1 and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rt,p f j ( · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ j , which in turn implies, that k f j | F sp,q ( R n ) k is bounded. Again, the same is true also for k f j | F sp,q ( B R ) k . Dueto the compactness of F sp,q ( R n ) ֒ → L p ( R n ) we may assume, that f j → f in L p ( B R ) with k f | L p ( B R ) k = 1 . A straightforward calculation shows again that ( f j ) ∞ j =1 is a Cauchy sequence in F sp,q ( R n ) and, therefore, f j → f also in F sp,q ( R n ). Finally, we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rt,p f ( · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0or, equivalently, Z ∞ t − sq d rt,p f ( x ) q dtt = 0for almost every x ∈ R n . Hence, d rt,p f ( x ) = 0 for almost all x ∈ R n and almost all t >
0. By standardarguments it follows that f must be almost everywhere equal to a polynomial of order smaller then r .Together with the bounded support of f , we obtain that f must be equal to zero almost everywhere. Theorem . Let 0 < λ ≤ f ∈ F sp,q ( R n ) with supp f ⊂ B λ . Then k f ( λ · ) | F sp,q ( R n ) k ∼ λ s − n/p k f | F sp,q ( R n ) k (2.3)with constants of equivalence independent of λ and f .9 roof. We know from Proposition 2.3 that k f ( λ · ) | F sp,q ( R n ) k ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rt,p ( f ( λ · ))( · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , as supp f ( λ · ) ⊂ B . Using ∆ rh ( f ( λ · ))( x ) = (∆ rλh f )( λx ), we get using the substitution ˜ h = λhd rt,p ( f ( λ · ))( x ) = t − n Z | h |≤ t | ∆ rh f ( λ · )( x ) | p dh ! /p = t − n Z | h |≤ t | (∆ rλh f )( λx ) | p dh ! /p = ( λt ) − n Z | ˜ h |≤ λt | (∆ r ˜ h f )( λx ) | p d ˜ h ! /p = d rλt,p ( f )( λx ) , which finally implies (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rt,p ( f ( λ · ))( · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rλt,p f ( λ · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = λ s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rt,p f ( λ · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = λ s − n/p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18)Z ∞ t − sq d rt,p f ( · ) q dtt (cid:19) /q | L p ( R n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∼ λ s − n/p k f | F sp,q ( R n ) k . We briefly sketch an application of the above homogeneity results in terms of pointwise multipliers. Alocally integrable function ϕ in R n is called a pointwise multiplier in A sp,q ( R n ) if f ϕf maps the considered space into itself. For further details on the subject we refer to [Tri92, pp. 201-206]and [RS96, Ch. 4]. Our aim is to generalize Proposition 1.3 as a direct consequence of Theorems 2.2, 2.4.Again let B λ be the balls introduced in (1.1). Corollary . Let s >
0, 0 < p, q ≤ ∞ and 0 < λ ≤
1. Let ϕ be a function having classical derivativesin B λ up to order 1 + [ s ] with | D α ϕ ( x ) | ≤ aλ −| γ | , | γ | ≤ s ] , x ∈ B λ , for some constant a >
0. Then ϕ is a pointwise multiplier in B sp,q ( B λ ), k ϕf | B sp,q ( B λ ) k ≤ c k f | B sp,q ( B λ ) k , (3.1)where c is independent of f ∈ B sp,q ( B λ ) and of λ (but depends on a ). Proof.
By Proposition 1.3 the function ϕ ( λ · ) is a pointwise multiplier in B sp,q ( B ). Then (3.1) is aconsequence of (2.2), k ϕf | B sp,q ( B λ ) k ∼ λ − ( s − np ) k ϕf ( λ · ) | B sp,q ( B ) k . λ − ( s − np ) k f ( λ · ) | B sp,q ( B ) k ∼ k f | B sp,q ( B λ ) k . Remark . In terms of Triebel-Lizorkin spaces F sp,q ( R n ) we obtain corresponding results (assuming p < ∞ ) with the additional restriction on the smoothness parameter s that s > n (cid:18) p, q ) − p (cid:19) . (3.2)This follows from the fact that the analogue of Proposition 1.3 for F-spaces is established using an atomiccharacterization of the spaces F sp,q ( R n ) which is only true if we impose (3.2), cf. [Tri06, Prop. 9.14].10 eferences [BS88] C. Bennett and R. Sharpley. Interpolation of operators , volume 129 of
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Cornelia SchneiderApplied Mathematics IIIUniversity Erlangen–NurembergCauerstraße 1191058 ErlangenGermany [email protected]
Jan VybíralAustrian Academy of Sciences (RICAM)Altenbergerstraße 69A-4040 LinzAustria [email protected]@oeaw.ac.at