# Homogeneity Property of Besov and Triebel-Lizorkin Spaces

Authors: Cornelia Schneider; Jan Vybíral
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

The present note deals with classical Besov spacesBp,qs(ℝn) and Triebel-Lizorkin spaces Fp,qs(ℝn) defined via differences, briefly denoted as B- and F-spaces in the sequel. We study the properties of the dilation operator, which is defined for every λ>0 as(1.1)Tλ:f⟶f(λ⋅).
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-calledhomogeneity property, showing that, for s>0 and 0<p,q≤∞,(1.2)‖f(λ⋅)∣Bp,qs(Rn)‖~λs-(n/p)‖f∣Bp,qs(Rn)‖,
for all 0<λ≤1 and all(1.3)f∈Bp,qs(Rn)withsupp⁡f⊂{x∈Rn:|x|≤λ}.
The same property holds true for the spaces Fp,qs(ℝn). This extends and completes [6], where corresponding results for the spaces Bp,qs(ℝn), defined via Fourier-analytic tools, were established, which coincide with our spaces Bp,qs(ℝn) if s>max(0,n(1/p-1)). Concerning the corresponding F-spaces Fp,qs(ℝn), the same homogeneity property had already been established in [7, Corollary  5.16, 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 Section2. Then, we focus on equivalent quasinorms for the elements of certain subspaces of Bp,qs(ℝn) and Fp,qs(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. Letℕ be the collection of all natural numbers, and let ℕ0=ℕ∪{0}. Let ℝn be Euclidean n-space, n∈ℕ, ℂ the complex plane. The set of multi-indices β=(β1,…,βn), βi∈ℕ0, i=1,…,n, is denoted by ℕ0n, with |β|=β1+⋯+βn, as usual. We use the symbol “≲” in(2.1)ak≲bkorφ(x)≲ψ(x)
always to mean that there is a positive number c1 such that(2.2)ak≤c1bkorφ(x)≤c1ψ(x)
for all admitted values of the discrete variable k or the continuous variable x, where (ak)k, (bk)k are nonnegative sequences and φ, ψ are nonnegative functions. We use the equivalence “~” in(2.3)ak~bkorφ(x)~ψ(x)
for(2.4)ak≲bk,bk≲akorφ(x)≲ψ(x),ψ(x)≲φ(x).
If a∈ℝ, then a+:=max(a,0) and [a] denotes the integer part of a.Given two (quasi-) Banach spacesX and Y, we write X↪Y if X⊂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≤∞, stands for the usual quasi-Banach space with respect to the Lebesgue measure, quasinormed by(2.5)‖f∣Lp(Rn)‖:=(∫Rn|f(x)|pdx)1/p
with the appropriate modification if p=∞. Moreover, let Ω denote a domain in ℝn. Then, Lp(Ω) is the collection of all complex-valued Lebesgue measurable functions in Ω such that(2.6)‖f∣Lp(Ω)‖:=(∫Ω|f(x)|pdx)1/p
(with the usual modification if p=∞) is finite.Furthermore,BR stands for an open ball with radius R>0 around the origin,(2.7)BR={x∈Rn:|x|<R}.
Let Qj,m with j∈ℕ0 and m∈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, χj,m stands for the characteristic function of Qj,m.


### 2.1. Function Spaces Defined via Differences

Iff is an arbitrary function on ℝn, h∈ℝn, and r∈ℕ, then(2.8)(Δh1f)(x)=f(x+h)-f(x),(Δhr+1f)(x)=Δh1(Δhrf)(x)
are the usual iterated differences. Given a function f∈Lp(ℝn), the r-th modulus of smoothness is defined by(2.9)ωr(f,t)p=sup⁡|h|≤t‖Δhrf∣Lp(Rn)‖,t>0,0<p≤∞,dt,prf(x)=(t-n∫|h|≤t|(Δhrf)(x)|pdh)1/p,t>0,0<p<∞,
denotes its ball means.Definition 2.1.
(i) Let0<p,q≤∞,s>0, and r∈ℕ such that r>s. Then, the Besov space Bp,qs(ℝn) contains all f∈Lp(ℝn) such that
(2.10)‖f∣Bp,qs(Rn)‖r=‖f∣Lp(Rn)‖+(∫01t-sqωr(f,t)pqdtt)1/q
(with the usual modification if q=∞) is finite.
(ii) Let0<p<∞,0<q≤∞,s>0, and r∈ℕ such that r>s. Then, Fp,qs(ℝn) is the collection of all f∈Lp(ℝn) such that
(2.11)‖f∣Fp,qs(Rn)‖r=‖f∣Lp(Rn)‖+‖(∫01t-sqdt,prf(⋅)qdtt)1/q∣Lp(Rn)‖
(with the usual modification if q=∞) is finite.Remark 2.2.
These are theclassical BesovandTriebel-Lizorkin spaces,in particular, when 1≤p,q≤∞(p<∞for the F-spaces)and s>0.We will sometimes write Ap,qs(ℝn) when both scales of spaces Bp,qs(ℝn) and Fp,qs(ℝn) are concerned simultaneously.
Concerning the spacesBp,qs(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 spacesFp,qs(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 Definition2.1 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 ∫01 can be replaced by ∫0∞ resulting again in equivalent quasinorms, (cf. [14, Section  2]).
The spaces are quasi-Banach spaces (Banach spaces ifp,q≥1). Note that we deal with subspaces of Lp(ℝn), in particular, for s>0 and 0<q≤∞, we have the embeddings
(2.12)Ap,qs(Rn)↪Lp(Rn),
where 0<p≤∞ (p<∞ for F-spaces). Furthermore, the B-spaces are closely linked with the Triebel-Lizorkin spaces via
(2.13)Bp,min⁡(p,q)s(Rn)↪Fp,qs(Rn)↪Bp,max⁡(p,q)s(Rn),
(cf. [15, Proposition  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 𝒞s(ℝn), that is,
(2.14)B∞,∞s(Rn)=Cs(Rn),s>0.
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 Bp,qs(ℝn) using subatomic decompositions, which introduces Bp,qs(ℝn) as those f∈Lp(ℝn) which can be represented as
(2.15)f(x)=∑β∈N0n∑j=0∞∑m∈Znλj,mβkj,mβ(x),x∈Rn,
with coefficients λ={λj,mβ∈ℂ:β∈ℕ0n,j∈ℕ0,m∈ℤn} belonging to some appropriate sequence space bp,qs,ϱ defined as
(2.16)bp,qs,ϱ:={λ:‖λ∣bp,qs,ϱ‖<∞},
where
(2.17)‖λ∣bp,qs,ϱ‖=sup⁡β∈N0n2ϱ|β|(∑j=0∞2j(s-n/p)q(∑m∈Zn|λj,mβ|p)q/p)1/q,s>0,0<p,q≤∞  (with the usual modification if p=∞ and/or q=∞), ϱ≥0, and kj,mβ(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 inBp,qs(ℝn), the following is known.Proposition 2.3.
Let0<p,q≤∞,s>0,k∈ℕ with  k>s, and let  h∈Ck(ℝn). Then,
(2.18)f⟶hf
is a linear and bounded operator from Bp,qs(ℝn) into itself.The proof relies on atomic decompositions of the spacesBp,qs(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

LetΩ be a domain in ℝn. We define spaces Ap,qs(Ω) by restriction of the corresponding spaces on ℝn, that is, Ap,qs(Ω) is the collection of all f∈Lp(Ω) such that there is a g∈Ap,qs(ℝn) with g|Ω=f. Furthermore,(2.19)‖f∣Ap,qs(Ω)‖=inf⁡‖g∣Ap,qs(Rn)‖,
where the infimum is taken over all g∈Ap,qs(ℝn) such that the restriction g|Ω to Ω coincides in Lp(Ω) with f.In particular, the subatomic characterization for the spacesBp,qs(ℝn) from Remark 2.2 carries over. For further details on this subject, we refer to [18, Section  2.1].Embeddings results between the spacesBp,qs(ℝn) hold also for the spaces Bp,qs(Ω), since they are defined by restriction of the corresponding spaces on ℝn. Furthermore, these results can be improved, if we assume Ω⊂ℝn to be bounded.Proposition 2.4.
Let0<s2<s1<∞,  0<p1,p2,q1,q2≤∞, and  Ω⊂ℝn be bounded. If
(2.20)δ+=s1-s2-d(1p1-1p2)+>0,
one has the embedding
(2.21)Bp1,q1s1(Ω)↪Bp2,q2s2(Ω).Proof.
Ifp1≤p2, the embedding follows from [19, Theorem  1.15], since the spaces on Ω are defined by restriction of their counterparts on ℝn. Therefore, it remains to show that, for p1>p2, we have the embedding
(2.22)Bp1,q2s2(Ω)↪Bp2,q2s2(Ω).
Let ψ∈D(ℝn) with support in the compact set Ω1 and
(2.23)ψ(x)=1ifx∈Ω̅⊂Ω1.
Then, for f∈Bp1,q2s2(Ω), there exists g∈Bp1,q2s2(ℝn) with
(2.24)g|Ω=f,‖f∣Bp1,q2s2(Ω)‖~‖g∣Bp1,q2s2(Rn)‖.
We calculate
(2.25)‖f∣Bp2,q2s2(Ω)‖≤‖ψg∣Bp2,q2s2(Rn)‖≤‖ψg∣Bp1,q2s2(Rn)‖≤cψ‖g∣Bp1,q2s2(Rn)‖~‖f∣Bp1,q2s2(Ω)‖.
The last inequality in (2.25) follows from Proposition 2.3. In the 2nd step, we used (2.10) together with the fact that
(2.26)‖Δhr(ψg)∣Lp2(Rn)‖≤cΩ1‖Δhr(ψg)∣Lp1(Rn)‖,p1>p2,
which follows from Hölder’s inequality since suppψg⊂Ω1 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,Bp,qs(Ω), and F-spaces, Fp,qs(Ω), respectively. This will be established with the help of entropy numbers. We briefly introduce the concept and collect some properties afterwards.LetX and Y be quasi-Banach spaces, and let T:X→Y be a bounded linear operator. If additionally, T is continuous, we write T∈L(X,Y). Let UX={x∈X:∥x∣X∥≤1} denote the unit ball in the quasi-Banach space X. An operator T is called compact if, for any given ε>0 we can cover the image of the unit ball UX with finitely many balls in Y of radius ε.Definition 2.5.
LetX,Y be quasi-Banach spaces, and let T∈L(X,Y).Then, for all k∈ℕ, the kth dyadic entropy number ek(T)  of T is defined by
(2.27)ek(T)=inf⁡{ε>0:T(UX)⊂⋃j=12k-1(yj+εUY)forsomey1,…,y2k-1∈Y},
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.
LetX,Y, and  Z be quasi-Banach spaces, and let  S,T∈L(X,Y) and  R∈L(Y,Z). (i)
(Monotonicity)∥T∥≥e1(T)≥e2(T)≥⋯≥0. Moreover, ∥T∥=e1(T), provided that Y is a Banach space.(ii)
(Additivity) IfY is a p-Banach space (0<p≤1), then, for all j,k∈ℕ,
(2.28)ej+k-1p(S+T)≤ejp(S)+ekp(T).(iii)
(Multiplicativity) For allj,k∈ℕ,
(2.29)ej+k-1(RT)≤ej(R)ek(T).(iv)
(Compactness)T is compact if and only if
(2.30)lim⁡k→∞ek(T)=0.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. Letn>0 and {Mj}j∈ℕ0 be a sequence of natural numbers satisfying
(2.31)Mj~2jn,j∈N0.
Concerning entropy numbers for the respective sequence spaces bp,qs,ϱ(Mj), which are defined as the sequence spaces bp,qs,ϱ in (2.17) with the sum over m∈ℤn replaced by a sum over m=1,…,Mj, the following result was proved in [24, Proposition  3.4].Proposition 2.8.
Letd>0,  0<σ1,σ2<∞, and  0<q1,q2≤∞.  Furthermore, let ϱ1>ϱ2≥0,
(2.32)0<p1≤p2≤∞,δ=σ1-σ2-n(1p1-1p2)>0.
Then the identity map
(2.33)id:bp1,q1σ1,ϱ1(Mj)⟶bp2,q2σ2,ϱ2(Mj)
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 spacesbp,qs,ϱ(Mj).Theorem 2.9.
Letn>0,  0<s1,s2<∞, and  0<q1,q2≤∞. Furthermore, let  ϱ1>ϱ2≥0,
(2.34)0<p1≤p2≤∞,δ=s1-s2-n(1p1-1p2)>0.
For the entropy numbers ek of the compact operator
(2.35)id:bp1,q1s1,ϱ1(Mj)⟶bp2,q2s2,ϱ2(Mj),
one has
(2.36)ek(id)∼k-δ/n+1/p2-1/p1,k∈N.Remark 2.10.
The proof of Theorem2.9 follows from [25, Theorem  9.2]. Using the notation from this book, we have
(2.37)bpi,qisi,ϱi(Mj)=l∞[2ϱilqi(2j(si-n/pi)lpiMj)],i=1,2.
Recall the embedding assertions for Besov spaces Bp,qs(Ω) 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 Ω=BR.Theorem 2.11.
Let(2.38)0<s2<s1<∞,0<p1,p2≤∞,0<q1,q2≤∞,δ+=s1-s2-n(1p1-1p2)+>0.
Then, the embedding
(2.39)id:Bp1,q1s1(Ω)⟶Bp2,q2s2(Ω)
is compact, and, for the related entropy numbers, one computes
(2.40)ek(id)≲k-(s1-s2)/n,k∈N.Proof.
Step 1.
Letp2≥p1, δ+=δ, and let f∈Bp1,q1s1(Ω), then, by [26, Theorem  6.1], there is a (nonlinear) bounded extension operator
(2.41)g=ExfsuchthatReΩg=g|Ω=f,(2.42)‖g∣Bp1,q1s1(Rn)‖≤c‖f∣Bp1,q1s1(Ω)‖.
We may assume that g is zero outside a fixed neighbourhood Λ of Ω. Using the subatomic approach for Bp1,q1s1(ℝn), cf. Remark 2.2, we can find an optimal decomposition of g, that is,
(2.43)g(x)=∑β∈N0n∑j=0∞∑m∈Znλj,mβkj,mβ(x),‖g∣Bp1,q1s1(Rn)‖~‖λ∣bp1,q1s1,ϱ1‖
with ϱ1>0 large.
LetMj for fixed j∈ℕ0 be the number of cubes Qj,m such that
(2.44)rQj,m∩Ω≠∅.
Since Ω⊂ℝn is bounded, we have
(2.45)Mj~2jn,j∈N0.
This coincides with (2.31). We introduce the (nonlinear) operator S,
(2.46)S:Bp1,q1s1(Rn)⟶bp1,q1s1,ϱ1(Mj)
by
(2.47)Sg=λ,λ={λj,mβ:β∈N0n,j∈N0,m∈Zn,rQj,m∩Ω≠∅},
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
(2.48)‖S‖=sup⁡g≠0‖λ∣bp1,q1s1,ϱ1(Mj)‖‖g∣Bp1,q1s1(Rn)‖≤c.
Next we construct the linear map T,
(2.49)T:bp2,q2s2,ϱ2(Mj)⟶Bp2,q2s2(Rn),
given by
(2.50)Tλ=∑β∈N0n∑j=0∞∑m=1Mjλj,mβkj,mβ(x).
It follows that T is a linear (since the subatomic approach provides an expansion of functions via universal building blocks) and bounded map,
(2.51)‖T‖=sup⁡λ≠0‖Tλ∣Bp2,q2s2(Rn)‖‖λ∣bp2,q2s2,ϱ2(Mj)‖≤c.
We complement the three bounded maps Ex,S, T by the identity operator
(2.52)id:bp1,q1s1,ϱ1(Mj)⟶bp2,q2s2,ϱ2(Mj)withϱ1>ϱ2,
which is compact by Proposition 2.8 and the restriction operator
(2.53)ReΩ:Bp2,q2s2(Rn)⟶Bp2,q2s2(Ω),
which is continuous. From the constructions, it follows that
(2.54)id(Bp1,q1s1(Ω)⟶Bp2,q2s2(Ω))=ReΩ∘T∘id∘S∘Ex.
Hence, taking finally ReΩ, 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 Exand S. The unit ball in Bp1,q1s1(Ω) is mapped by S∘Exinto a bounded set in
(2.55)bp1,q1s1,ϱ1(Mj).
Since the identity operator id from (2.52) is compact, this bounded set is mapped into a precompact set in
(2.56)bp2,q2s2,ϱ2(Mj),
which can be covered by 2k balls of radius cek(id) with
(2.57)ek(id)≤ck-δ/n+1/p2-1/p1,k∈N.
This follows from Theorem 2.9, where we used p2≥p1. Applying the two linear and bounded maps T and ReΩ 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,q1s1(Ω) by 2k balls of radius cek(id) in Bp2,q2s2(Ω). Inserting
(2.58)δ=s1-s2-n(1p1-1p2)
in the exponent, we finally obtain the desired estimate
(2.59)ek(id)≤ck-(s1-s2)/n,k∈N.Step 2.
Letp1>p2. Since, by Proposition 2.4,
(2.60)Bp1,q2s2(Ω)⊂Bp2,q2s2(Ω),
we see that
(2.61)Bp1,q1s1(Ω)⊂Bp1,q2s2(Ω)⊂Bp2,q2s2(Ω),
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
(2.62)Bp,min⁡(p,q)s(Ω)↪Fp,qs(Ω)↪Bp,max⁡(p,q)s(Ω).
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 Lemma2.6 (iv) and Theorem 2.11, we deduce compactness of the corresponding embeddings related to B- and F-spaces under investigation.


## 2.1. Function Spaces Defined via Differences

Iff is an arbitrary function on ℝn, h∈ℝn, and r∈ℕ, then(2.8)(Δh1f)(x)=f(x+h)-f(x),(Δhr+1f)(x)=Δh1(Δhrf)(x)
are the usual iterated differences. Given a function f∈Lp(ℝn), the r-th modulus of smoothness is defined by(2.9)ωr(f,t)p=sup⁡|h|≤t‖Δhrf∣Lp(Rn)‖,t>0,0<p≤∞,dt,prf(x)=(t-n∫|h|≤t|(Δhrf)(x)|pdh)1/p,t>0,0<p<∞,
denotes its ball means.Definition 2.1.
(i) Let0<p,q≤∞,s>0, and r∈ℕ such that r>s. Then, the Besov space Bp,qs(ℝn) contains all f∈Lp(ℝn) such that
(2.10)‖f∣Bp,qs(Rn)‖r=‖f∣Lp(Rn)‖+(∫01t-sqωr(f,t)pqdtt)1/q
(with the usual modification if q=∞) is finite.
(ii) Let0<p<∞,0<q≤∞,s>0, and r∈ℕ such that r>s. Then, Fp,qs(ℝn) is the collection of all f∈Lp(ℝn) such that
(2.11)‖f∣Fp,qs(Rn)‖r=‖f∣Lp(Rn)‖+‖(∫01t-sqdt,prf(⋅)qdtt)1/q∣Lp(Rn)‖
(with the usual modification if q=∞) is finite.Remark 2.2.
These are theclassical BesovandTriebel-Lizorkin spaces,in particular, when 1≤p,q≤∞(p<∞for the F-spaces)and s>0.We will sometimes write Ap,qs(ℝn) when both scales of spaces Bp,qs(ℝn) and Fp,qs(ℝn) are concerned simultaneously.
Concerning the spacesBp,qs(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 spacesFp,qs(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 Definition2.1 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 ∫01 can be replaced by ∫0∞ resulting again in equivalent quasinorms, (cf. [14, Section  2]).
The spaces are quasi-Banach spaces (Banach spaces ifp,q≥1). Note that we deal with subspaces of Lp(ℝn), in particular, for s>0 and 0<q≤∞, we have the embeddings
(2.12)Ap,qs(Rn)↪Lp(Rn),
where 0<p≤∞ (p<∞ for F-spaces). Furthermore, the B-spaces are closely linked with the Triebel-Lizorkin spaces via
(2.13)Bp,min⁡(p,q)s(Rn)↪Fp,qs(Rn)↪Bp,max⁡(p,q)s(Rn),
(cf. [15, Proposition  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 𝒞s(ℝn), that is,
(2.14)B∞,∞s(Rn)=Cs(Rn),s>0.
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 Bp,qs(ℝn) using subatomic decompositions, which introduces Bp,qs(ℝn) as those f∈Lp(ℝn) which can be represented as
(2.15)f(x)=∑β∈N0n∑j=0∞∑m∈Znλj,mβkj,mβ(x),x∈Rn,
with coefficients λ={λj,mβ∈ℂ:β∈ℕ0n,j∈ℕ0,m∈ℤn} belonging to some appropriate sequence space bp,qs,ϱ defined as
(2.16)bp,qs,ϱ:={λ:‖λ∣bp,qs,ϱ‖<∞},
where
(2.17)‖λ∣bp,qs,ϱ‖=sup⁡β∈N0n2ϱ|β|(∑j=0∞2j(s-n/p)q(∑m∈Zn|λj,mβ|p)q/p)1/q,s>0,0<p,q≤∞  (with the usual modification if p=∞ and/or q=∞), ϱ≥0, and kj,mβ(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 inBp,qs(ℝn), the following is known.Proposition 2.3.
Let0<p,q≤∞,s>0,k∈ℕ with  k>s, and let  h∈Ck(ℝn). Then,
(2.18)f⟶hf
is a linear and bounded operator from Bp,qs(ℝn) into itself.The proof relies on atomic decompositions of the spacesBp,qs(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

LetΩ be a domain in ℝn. We define spaces Ap,qs(Ω) by restriction of the corresponding spaces on ℝn, that is, Ap,qs(Ω) is the collection of all f∈Lp(Ω) such that there is a g∈Ap,qs(ℝn) with g|Ω=f. Furthermore,(2.19)‖f∣Ap,qs(Ω)‖=inf⁡‖g∣Ap,qs(Rn)‖,
where the infimum is taken over all g∈Ap,qs(ℝn) such that the restriction g|Ω to Ω coincides in Lp(Ω) with f.In particular, the subatomic characterization for the spacesBp,qs(ℝn) from Remark 2.2 carries over. For further details on this subject, we refer to [18, Section  2.1].Embeddings results between the spacesBp,qs(ℝn) hold also for the spaces Bp,qs(Ω), since they are defined by restriction of the corresponding spaces on ℝn. Furthermore, these results can be improved, if we assume Ω⊂ℝn to be bounded.Proposition 2.4.
Let0<s2<s1<∞,  0<p1,p2,q1,q2≤∞, and  Ω⊂ℝn be bounded. If
(2.20)δ+=s1-s2-d(1p1-1p2)+>0,
one has the embedding
(2.21)Bp1,q1s1(Ω)↪Bp2,q2s2(Ω).Proof.
Ifp1≤p2, the embedding follows from [19, Theorem  1.15], since the spaces on Ω are defined by restriction of their counterparts on ℝn. Therefore, it remains to show that, for p1>p2, we have the embedding
(2.22)Bp1,q2s2(Ω)↪Bp2,q2s2(Ω).
Let ψ∈D(ℝn) with support in the compact set Ω1 and
(2.23)ψ(x)=1ifx∈Ω̅⊂Ω1.
Then, for f∈Bp1,q2s2(Ω), there exists g∈Bp1,q2s2(ℝn) with
(2.24)g|Ω=f,‖f∣Bp1,q2s2(Ω)‖~‖g∣Bp1,q2s2(Rn)‖.
We calculate
(2.25)‖f∣Bp2,q2s2(Ω)‖≤‖ψg∣Bp2,q2s2(Rn)‖≤‖ψg∣Bp1,q2s2(Rn)‖≤cψ‖g∣Bp1,q2s2(Rn)‖~‖f∣Bp1,q2s2(Ω)‖.
The last inequality in (2.25) follows from Proposition 2.3. In the 2nd step, we used (2.10) together with the fact that
(2.26)‖Δhr(ψg)∣Lp2(Rn)‖≤cΩ1‖Δhr(ψg)∣Lp1(Rn)‖,p1>p2,
which follows from Hölder’s inequality since suppψg⊂Ω1 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,Bp,qs(Ω), and F-spaces, Fp,qs(Ω), respectively. This will be established with the help of entropy numbers. We briefly introduce the concept and collect some properties afterwards.LetX and Y be quasi-Banach spaces, and let T:X→Y be a bounded linear operator. If additionally, T is continuous, we write T∈L(X,Y). Let UX={x∈X:∥x∣X∥≤1} denote the unit ball in the quasi-Banach space X. An operator T is called compact if, for any given ε>0 we can cover the image of the unit ball UX with finitely many balls in Y of radius ε.Definition 2.5.
LetX,Y be quasi-Banach spaces, and let T∈L(X,Y).Then, for all k∈ℕ, the kth dyadic entropy number ek(T)  of T is defined by
(2.27)ek(T)=inf⁡{ε>0:T(UX)⊂⋃j=12k-1(yj+εUY)forsomey1,…,y2k-1∈Y},
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.
LetX,Y, and  Z be quasi-Banach spaces, and let  S,T∈L(X,Y) and  R∈L(Y,Z). (i)
(Monotonicity)∥T∥≥e1(T)≥e2(T)≥⋯≥0. Moreover, ∥T∥=e1(T), provided that Y is a Banach space.(ii)
(Additivity) IfY is a p-Banach space (0<p≤1), then, for all j,k∈ℕ,
(2.28)ej+k-1p(S+T)≤ejp(S)+ekp(T).(iii)
(Multiplicativity) For allj,k∈ℕ,
(2.29)ej+k-1(RT)≤ej(R)ek(T).(iv)
(Compactness)T is compact if and only if
(2.30)lim⁡k→∞ek(T)=0.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. Letn>0 and {Mj}j∈ℕ0 be a sequence of natural numbers satisfying
(2.31)Mj~2jn,j∈N0.
Concerning entropy numbers for the respective sequence spaces bp,qs,ϱ(Mj), which are defined as the sequence spaces bp,qs,ϱ in (2.17) with the sum over m∈ℤn replaced by a sum over m=1,…,Mj, the following result was proved in [24, Proposition  3.4].Proposition 2.8.
Letd>0,  0<σ1,σ2<∞, and  0<q1,q2≤∞.  Furthermore, let ϱ1>ϱ2≥0,
(2.32)0<p1≤p2≤∞,δ=σ1-σ2-n(1p1-1p2)>0.
Then the identity map
(2.33)id:bp1,q1σ1,ϱ1(Mj)⟶bp2,q2σ2,ϱ2(Mj)
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 spacesbp,qs,ϱ(Mj).Theorem 2.9.
Letn>0,  0<s1,s2<∞, and  0<q1,q2≤∞. Furthermore, let  ϱ1>ϱ2≥0,
(2.34)0<p1≤p2≤∞,δ=s1-s2-n(1p1-1p2)>0.
For the entropy numbers ek of the compact operator
(2.35)id:bp1,q1s1,ϱ1(Mj)⟶bp2,q2s2,ϱ2(Mj),
one has
(2.36)ek(id)∼k-δ/n+1/p2-1/p1,k∈N.Remark 2.10.
The proof of Theorem2.9 follows from [25, Theorem  9.2]. Using the notation from this book, we have
(2.37)bpi,qisi,ϱi(Mj)=l∞[2ϱilqi(2j(si-n/pi)lpiMj)],i=1,2.
Recall the embedding assertions for Besov spaces Bp,qs(Ω) 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 Ω=BR.Theorem 2.11.
Let(2.38)0<s2<s1<∞,0<p1,p2≤∞,0<q1,q2≤∞,δ+=s1-s2-n(1p1-1p2)+>0.
Then, the embedding
(2.39)id:Bp1,q1s1(Ω)⟶Bp2,q2s2(Ω)
is compact, and, for the related entropy numbers, one computes
(2.40)ek(id)≲k-(s1-s2)/n,k∈N.Proof.
Step 1.
Letp2≥p1, δ+=δ, and let f∈Bp1,q1s1(Ω), then, by [26, Theorem  6.1], there is a (nonlinear) bounded extension operator
(2.41)g=ExfsuchthatReΩg=g|Ω=f,(2.42)‖g∣Bp1,q1s1(Rn)‖≤c‖f∣Bp1,q1s1(Ω)‖.
We may assume that g is zero outside a fixed neighbourhood Λ of Ω. Using the subatomic approach for Bp1,q1s1(ℝn), cf. Remark 2.2, we can find an optimal decomposition of g, that is,
(2.43)g(x)=∑β∈N0n∑j=0∞∑m∈Znλj,mβkj,mβ(x),‖g∣Bp1,q1s1(Rn)‖~‖λ∣bp1,q1s1,ϱ1‖
with ϱ1>0 large.
LetMj for fixed j∈ℕ0 be the number of cubes Qj,m such that
(2.44)rQj,m∩Ω≠∅.
Since Ω⊂ℝn is bounded, we have
(2.45)Mj~2jn,j∈N0.
This coincides with (2.31). We introduce the (nonlinear) operator S,
(2.46)S:Bp1,q1s1(Rn)⟶bp1,q1s1,ϱ1(Mj)
by
(2.47)Sg=λ,λ={λj,mβ:β∈N0n,j∈N0,m∈Zn,rQj,m∩Ω≠∅},
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
(2.48)‖S‖=sup⁡g≠0‖λ∣bp1,q1s1,ϱ1(Mj)‖‖g∣Bp1,q1s1(Rn)‖≤c.
Next we construct the linear map T,
(2.49)T:bp2,q2s2,ϱ2(Mj)⟶Bp2,q2s2(Rn),
given by
(2.50)Tλ=∑β∈N0n∑j=0∞∑m=1Mjλj,mβkj,mβ(x).
It follows that T is a linear (since the subatomic approach provides an expansion of functions via universal building blocks) and bounded map,
(2.51)‖T‖=sup⁡λ≠0‖Tλ∣Bp2,q2s2(Rn)‖‖λ∣bp2,q2s2,ϱ2(Mj)‖≤c.
We complement the three bounded maps Ex,S, T by the identity operator
(2.52)id:bp1,q1s1,ϱ1(Mj)⟶bp2,q2s2,ϱ2(Mj)withϱ1>ϱ2,
which is compact by Proposition 2.8 and the restriction operator
(2.53)ReΩ:Bp2,q2s2(Rn)⟶Bp2,q2s2(Ω),
which is continuous. From the constructions, it follows that
(2.54)id(Bp1,q1s1(Ω)⟶Bp2,q2s2(Ω))=ReΩ∘T∘id∘S∘Ex.
Hence, taking finally ReΩ, 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 Exand S. The unit ball in Bp1,q1s1(Ω) is mapped by S∘Exinto a bounded set in
(2.55)bp1,q1s1,ϱ1(Mj).
Since the identity operator id from (2.52) is compact, this bounded set is mapped into a precompact set in
(2.56)bp2,q2s2,ϱ2(Mj),
which can be covered by 2k balls of radius cek(id) with
(2.57)ek(id)≤ck-δ/n+1/p2-1/p1,k∈N.
This follows from Theorem 2.9, where we used p2≥p1. Applying the two linear and bounded maps T and ReΩ 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,q1s1(Ω) by 2k balls of radius cek(id) in Bp2,q2s2(Ω). Inserting
(2.58)δ=s1-s2-n(1p1-1p2)
in the exponent, we finally obtain the desired estimate
(2.59)ek(id)≤ck-(s1-s2)/n,k∈N.Step 2.
Letp1>p2. Since, by Proposition 2.4,
(2.60)Bp1,q2s2(Ω)⊂Bp2,q2s2(Ω),
we see that
(2.61)Bp1,q1s1(Ω)⊂Bp1,q2s2(Ω)⊂Bp2,q2s2(Ω),
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
(2.62)Bp,min⁡(p,q)s(Ω)↪Fp,qs(Ω)↪Bp,max⁡(p,q)s(Ω).
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 Lemma2.6 (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.
Let0<p,q≤∞,  s>0, and let  R>0 be a real number. Then,
(3.1)‖f∣Bp,qs(Rn)‖~(∫0∞t-sqωr(f,t)pqdtt)1/q
for all f∈Bp,qs(ℝn) with suppf⊂BR.Proof.
We will need thatBp,qs(BR) embeds compactly into Lp(BR). This follows at once from the fact that Bp,qs(BR) is compactly embedded into Bp,qs-ε(BR), cf. Remark 2.12, and Bp,qs-ε(BR)↪Lp(BR), which is trivial.We argue similarly to [6]. We have to prove that
(3.2)‖f∣Lp(Rn)‖≲(∫0∞t-sqωr(f,t)pqdtt)1/q
for every f∈Bp,qs(ℝn) with suppf⊂BR. Let us assume that this is not true. Then, we find a sequence (fj)j=1∞⊂Bp,qs(ℝn), such that
(3.3)‖fj∣Lp(Rn)‖=1,(∫0∞t-sqωr(fj,t)pqdtt)1/q≤1j,
that is, we obtain that ∥fj∣Bp,qs(ℝn)∥ is bounded. The trivial estimates
(3.4)‖fj∣Lp(Rn)‖=‖fj∣Lp(BR)‖,‖fj∣Bp,qs(BR)‖≤‖fj∣Bp,qs(Rn)‖
imply that this is true also for ∥fj∣Bp,qs(BR)∥. Due to the compactness of Bp,qs(BR)↪Lp(BR), we may assume, that fj→f in Lp(BR) with ∥f|Lp(BR)∥=1. Using the subadditivity of ω(·,t)p, we obtain that
(3.5)(∫0∞t-sqωr(fj-fj′,t)pqdtt)1/q≤1j+1j′.
Together with the estimate ∥fj-fj′∣Lp(ℝn)∥→0, this implies that (fj)j=1∞ is a Cauchy sequence in Bp,qs(ℝn), that is, fj→g in Bp,qs(ℝn). Obviously, f=g follows.
The subadditivity ofω(·,t)p used to the sum (f-fj)+fj implies finally that
(3.6)(∫0∞t-sqωr(f,t)pqdtt)1/q=0.
As ωr(f,t) is a nondecreasing function of t, this implies that ωr(f,t)=0 for all 0<t<∞ and finally ∥Δhrf∣Lp(ℝn)∥=0 for all h∈ℝn. 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.
Let0<λ≤1 and  f∈Bp,qs(ℝn) with suppf⊂Bλ. Then,
(3.7)‖f(λ⋅)∣Bp,qs(Rn)‖~λs-n/p‖f∣Bp,qs(Rn)‖
with constants of equivalence independent of λ and f.Proof.
We know from Proposition3.1 that
(3.8)‖f(λ⋅)∣Bp,qs(Rn)‖~(∫0∞t-sqωr(f(λ⋅),t)pqdtt)1/q,
as suppf(λ·)⊂B1. Using Δhr(f(λ·))(x)=(Δλhrf)(λx), we get
(3.9)ωr(f(λ⋅),t)p=sup⁡|h|≤t‖Δhr(f(λ⋅))‖p=sup⁡|h|≤t‖(Δλhrf)(λ⋅)‖p=λ-n/psup⁡|h|≤t‖(Δλhrf)(⋅)‖p=λ-n/psup⁡|λh|≤λt‖(Δλhrf)(⋅)‖p=λ-n/pωr(f,λt)p,
which finally implies
(3.10)(∫0∞t-sqωr(f(λ⋅),t)pqdtt)1/q=λ-n/p(∫0∞t-sqωr(f,λt)pqdtt)1/q=λs-n/p(∫0∞t-sqωr(f,t)pqdtt)1/q~λs-n/p‖f∣Bp,qs(Rn)‖.The homogeneity property for Triebel-Lizorkin spacesFp,qs(ℝn) follows similarly.Proposition 3.3.
Let0<p<∞,0<q≤∞,s>0, and let  R>0 be a real number. Then,
(3.11)‖f∣Fp,qs(Rn)‖~‖(∫0∞t-sqdt,prf(⋅)qdtt)1/q∣Lp(Rn)‖
for all f∈Fp,qs(ℝn) with suppf⊂BR.Proof.
We have to prove that(3.12)‖f∣Lp(Rn)‖≲‖(∫0∞t-sqdt,prf(⋅)qdtt)1/q∣Lp(Rn)‖
for every f∈Fp,qs(ℝn) with suppf⊂BR. Let us assume again that this is not true. Then, we find a sequence (fj)j=1∞⊂Fp,qs(ℝn) such that
(3.13)‖fj∣Lp(Rn)‖=1,‖(∫0∞t-sqdt,prfj(⋅)qdtt)1/q∣Lp(Rn)‖≤1j,
which in turn implies that ∥fj∣Fp,qs(ℝn)∥ is bounded. Again, the same is true also for ∥fj|Fp,qs(BR)∥. Due to the compactness of Fp,qs(ℝn)↪Lp(ℝn), we may assume that fj→f in Lp(BR) with ∥f∣Lp(BR)∥=1. A straightforward calculation shows again that (fj)j=1∞ is a Cauchy sequence in Fp,qs(ℝn) and, therefore, fj→f also in Fp,qs(ℝn). Finally, we obtain
(3.14)‖(∫0∞t-sqdt,prf(⋅)qdtt)1/q∣Lp(Rn)‖=0
or, equivalently,
(3.15)∫0∞t-sqdt,prf(x)qdtt=0
for almost every x∈ℝn. Hence, dt,prf(x)=0 for almost all x∈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.
Let0<λ≤1 and f∈Fp,qs(ℝn) with  suppf⊂Bλ. Then,
(3.16)‖f(λ⋅)∣Fp,qs(Rn)‖~λs-n/p‖f∣Fp,qs(Rn)‖
with constants of equivalence independent of λ and f.Proof.
We know from Proposition3.3 that
(3.17)‖f(λ⋅)∣Fp,qs(Rn)‖~‖(∫0∞t-sqdt,pr(f(λ⋅))(⋅)qdtt)1/q∣Lp(Rn)‖,
as suppf(λ·)⊂B1. Using Δhr(f(λ·))(x)=(Δλhrf)(λx), we get using the substitution h̃=λh,
(3.18)dt,pr(f(λ⋅))(x)=(t-n∫|h|≤t|Δhrf(λ⋅)(x)|pdh)1/p=(t-n∫|h|≤t|(Δλhrf)(λx)|pdh)1/p=((λt)-n∫|h̃|≤λt|(Δh̃rf)(λx)|pdh̃)1/p=dλt,pr(f)(λx),
which finally implies
(3.19)‖(∫0∞t-sqdt,pr(f(λ⋅))(⋅)qdtt)1/q∣Lp(Rn)‖=‖(∫0∞t-sqdλt,prf(λ⋅)qdtt)1/q∣Lp(Rn)‖=λs‖(∫0∞t-sqdt,prf(λ⋅)qdtt)1/q∣Lp(Rn)‖=λs-n/p‖(∫0∞t-sqdt,prf(⋅)qdtt)1/q∣Lp(Rn)‖~λs-n/p‖f∣Fp,qs(Rn)‖.


## 4. Pointwise Multipliers

We briefly sketch an application of the above homogeneity results in terms of pointwise multipliers. A locally integrable functionφ in ℝn is called a pointwise multiplier in Ap,qs(ℝn) if(4.1)f⟼φf
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 Bλ be the balls introduced in (2.7).Corollary 4.1.
Lets>0,  0<p,q≤∞, and  0<λ≤1. Let  φ be a function having classical derivatives in  B2λ up to order  1+[s] with
(4.2)|Dγφ(x)|≤aλ-|γ|,|γ|≤1+[s],x∈B2λ,
for some constant a>0. Then, φ is a pointwise multiplier in Bp,qs(Bλ),
(4.3)‖φf∣Bp,qs(Bλ)‖≤c‖f∣Bp,qs(Bλ)‖,
where c is independent of f∈Bp,qs(Bλ) and of λ (but depends on a).Proof.
By Proposition2.3, the function φ(λ·) is a pointwise multiplier in Bp,qs(B1). Then, (4.3) is a consequence of (3.7),
(4.4)‖φf∣Bp,qs(Bλ)‖~λ-(s-n/p)‖φf(λ⋅)∣Bp,qs(B1)‖≲λ-(s-n/p)‖f(λ⋅)∣Bp,qs(B1)‖~‖f∣Bp,qs(Bλ)‖.Remark 4.2.
In terms of Triebel-Lizorkin spacesFp,qs(ℝn), we obtain corresponding results (assuming p<∞) with the additional restriction on the smoothness parameter s that
(4.5)s>n(1min⁡(p,q)-1p).
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,qs(ℝn) which is only true if we impose (4.5), (cf. [13, Proposition  9.14]).



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