Homogeneity property of Besov and Triebel-Lizorkin spaces

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.


Introduction
The present note deals with classical Besov spaces B s p,q (R n ) and Triebel-Lizorkin spaces F s p,q (R n ) 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 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 > 0 and 0 < p, q ≤ ∞, f (λ·)|B s p,q (R n ) ∼ λ s− n p f |B s p,q (R n ) , (0.1) for all 0 < λ ≤ 1 and all f ∈ B s p,q (R n ) with supp f ⊂ {x ∈ R n : |x| ≤ λ}.
The same property holds true for the spaces F s p,q (R n ). This extends and completes [CLT07], where corresponding results for the spaces B s p,q (R n ), defined via Fourier-analytic tools, were established, which coincide with our spaces B s p,q (R n ) if s > max 0, n 1 p − 1 . Concerning the corresponding F-spaces F s p,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 that the homogeneity property is closely related with questions concerning refined localization, non-smooth 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 [SV11] in connection with non-smooth atomic decompositions 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 such embeddings. This note is organized as follows. We start with the necessary definitions and the results about entropy numbers in Section 1. Then we focus on equivalent quasi-norms for the elements of certain subspaces of B s p,q (R n ) and F s p,q (R n ), respectively, from which the homogeneity property will follow almost immediately in Section 2. The last section states some applications in terms of pointwise multipliers. 0 The second author acknowledges the financial support provided by the START-award "Sparse Approximation and Optimization in High Dimensions" of the Fonds zur Förderung der wissenschaftlichen Forschung (FWF, Austrian Science Foundation).

Preliminaries
We use standard notation. Let N be the collection of all natural numbers and let N 0 = N ∪ {0}. Let R n be Euclidean n-space, n ∈ N, C the complex plane. The set of multi-indices β = (β 1 , . . . , β n ), β i ∈ N 0 , i = 1, . . . , n, is denoted by N n 0 , with |β| = β 1 + · · · + β n , as usual. We use the symbol ' ' in always to mean that there is a positive number c 1 such that for all admitted values of the discrete variable k or the continuous variable x, where (a k ) k , (b k ) k are non-negative sequences and ϕ, ψ are non-negative functions. We use the equivalence '∼' in 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 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. L p (R n ), with 0 < p ≤ ∞, stands for the usual quasi-Banach space with respect to the Lebesgue measure, quasi-normed by Let Q j,m with j ∈ N 0 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 cube in R n concentric with Q and with side length r times the side length of Q. Furthermore, χ j,m stands for the 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 are the usual iterated differences. Given a function f ∈ L p (R n ) the r-th modulus of smoothness is defined by denotes its ball means.
Definition 1.1. (i) Let 0 < p, q ≤ ∞, s > 0, and r ∈ N such that r > s. Then the Besov space Remark 1.2. 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 s p,q (R n ) when both scales of spaces B s p,q (R n ) and F s p,q (R n ) are concerned simultaneously. Concerning the spaces B s p,q (R n ), the study for all admitted s, p and q goes back to [SO78], we also refer to [BS88,Ch. 5,Def. 4.3] and [DL93, Ch. 2, §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 F s p,q (R n ) has been described in detail in [Tri83] for those spaces which can also be considered as subspaces of S ′ (R n ). Otherwise one finds in [Tri06, Section 9. The spaces are quasi-Banach spaces (Banach spaces if p, q ≥ 1). Note that we deal with subspaces of L p (R n ), in particular, for s > 0 and 0 < q ≤ ∞, we have the embeddings where 0 < p ≤ ∞ (p < ∞ for F-spaces). Furthermore, the B-spaces are closely linked with the Triebel-Lizorkin spaces via 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., 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 s p,q (R n ) using subatomic decompositions, which introduces B s p,q (R n ) as those f ∈ L p (R n ) which can be represented as s > 0, 0 < p, q ≤ ∞ (with the usual modification if p = ∞ and/or q = ∞), ̺ ≥ 0, and k β j,m (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 B s p,q (R n ) the following is known.
Proposition 1.3. Let 0 < p, q ≤ ∞, s > 0, k ∈ N with k > s, and let h ∈ C k (R n ). Then is a linear and bounded operator from B s p,q (R n ) into itself. The proof relies on atomic decompositions of the spaces B s p,q (R n ), cf. [Sch10a, Prop. 2.5]. We will generalize this result in Section 3 as an application of our homogeneity property.

Function spaces on domains Ω
Let Ω be a domain in R n . We define spaces A s p,q (Ω) by restriction of the corresponding spaces on R n , i.e. A s p,q (Ω) is the collection of all f ∈ L p (Ω) such that there is a g ∈ A s p,q (R n ) with g Ω = f . Furthermore, where the infimum is taken over all g ∈ A s p,q (R n ) such that the restriction g Ω to Ω coincides in L p (Ω) with f .
In particular, the subatomic characterization for the spaces B s p,q (R n ) from Remark 1.2 carries over. For further details on this subject we refer to [Sch11b, Sect 2.1].
Embeddings results between the spaces B s p,q (R n ) hold also for the spaces B s p,q (Ω), since they are defined by restriction of the corresponding spaces on R n . Furthermore, these results can be improved, if we assume Ω ⊂ R n to be bounded.

Entropy numbers
In order to prove the homogeneity results later on we have to rely on the compactness of embeddings between B-spaces, B s p,q (Ω), and F-spaces, F s p,q (Ω), 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 T : X → Y be a bounded linear operator. If additionally, T is continuous we write T ∈ L(X, Y ). Let U X = {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 U X with finitely many balls in Y of radius ε.
Definition 1.5. Let X, Y be quasi-Banach spaces and let T ∈ L(X, Y ). Then for all k ∈ N, the kth dyadic entropy number e k (T ) of T is defined by 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 1.6. Let X, Y and Z be quasi-Banach spaces, let S, T ∈ L(X, Y ) and R ∈ L(Y, Z). Remark 1.7. As for the general theory we refer to [EE87], [Pie87] and [Kön86]. Further information on the 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 to corresponding questions in related sequence spaces. Let n > 0 and {M j } j∈N0 be a sequence of natural numbers satisfying (1.14) Concerning entropy numbers for the respective sequence spaces b s,̺ p,q (M j ), which are defined as the sequence 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 following result was proved in [Sch11a, Prop. 3.4] Proposition 1.8. Let d > 0, 0 < σ 1 , σ 2 < ∞, and 0 < q 1 , q 2 ≤ ∞. Furthermore, let ̺ 1 > ̺ 2 ≥ 0, 0 < p 1 ≤ p 2 ≤ ∞ and δ = σ 1 − σ 2 − n 1 Then the identity map 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 the sequence spaces b s,̺ p,q (M j ).
Theorem 1.9. Let n > 0, 0 < s 1 , s 2 < ∞, and 0 < q 1 , q 2 ≤ ∞. Furthermore, let ̺ 1 > ̺ 2 ≥ 0, For the entropy numbers e k of the compact operator Remark 1.10. The proof of Theorem 1.9 follows from [Tri97, Th. 9.2]. Using the notation from this book Recall the embedding assertions for Besov spaces B s p,q (Ω) from Proposition 1.4. We will give an upper bound for the corresponding entropy numbers of these embeddings. For our purposes it will be sufficient to assume Ω = B R .
Since Ω ⊂ R n is bounded we have This coincides with (1.14). We introduce the (nonlinear) operator S, where g is given by (1.22). Recall that the expansion is not unique but this does not matter. It follows that S is a bounded map since Next we construct the linear map T , It follows that T is a linear (since the subatomic approach provides an expansion of functions via universal building blocks) and bounded map, We complement the three bounded maps Ex , S, T by the identity operator Hence, taking finally Re Ω we obtain f by (1.21), 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 B s1 p1,q1 (Ω) is mapped by S • Ex into a bounded set in b s1,̺1 p1,q1 (M j ). Since the identity operator id from (1.23) is compact, this bounded set is mapped into a pre-compact set in b s2,̺2 p2,q2 (M j ), which can be covered by 2 k balls of radius ce k (id) with e k (id) ≤ ck − δ n + 1 This follows from Theorem 1.9, where we used p 2 ≥ p 1 . Applying the two linear and bounded maps T and Re Ω afterwards does not change this covering assertion -using Lemma 1.6(iii) and ignoring constants for the time being. Hence, we arrive at a covering of the unit ball in B s1 p1,q1 (Ω) by 2 k balls of radius ce k (id) in B s2 p2,q2 (Ω). Inserting δ = s 1 − s 2 − n 1 p 1 − 1 p 2 in the exponent we finally obtain the desired estimate e k (id) ≤ ck − s 1 −s 2 n , k ∈ N.