Homogeneity Property of Besov and Triebel-Lizorkin Spaces

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.

1. Introduction

The present note deals with classical Besov spaces 𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛) and Triebel-Lizorkin spaces 𝐅^{𝑠𝑝,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 as𝑇_𝜆∶𝑓⟶𝑓(𝜆⋅).(1.1) 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≤∞,‖‖𝑓(𝜆⋅)∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖∼𝜆^{𝑠−(𝑛/𝑝)}‖‖𝑓∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖,(1.2) for all 0<𝜆≤1 and all𝑓∈𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)withsupp𝑓⊂{𝑥∈ℝ^𝑛∶|𝑥|≤𝜆}.(1.3) The same property holds true for the spaces 𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛). This extends and completes [6], where corresponding results for the spaces 𝐵^{𝑠𝑝,𝑞}(ℝ^𝑛), defined via Fourier-analytic tools, were established, which coincide with our spaces 𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛) if 𝑠>𝑚𝑎𝑥(0,𝑛(1/𝑝−1)). Concerning the corresponding F-spaces 𝐹^{𝑠𝑝,𝑞}(ℝ^𝑛), 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 Section 2. Then, we focus on equivalent quasinorms 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 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 ℕ₀=ℕ∪{0}. Let ℝ^𝑛 be Euclidean 𝑛-space, 𝑛∈ℕ, ℂ the complex plane. The set of multi-indices 𝛽=(𝛽₁,…,𝛽_𝑛), 𝛽_𝑖∈ℕ₀, 𝑖=1,…,𝑛, is denoted by ℕ^𝑛0, with |𝛽|=𝛽₁+⋯+𝛽_𝑛, as usual. We use the symbol “≲” in𝑎_𝑘≲𝑏_𝑘or𝜑(𝑥)≲𝜓(𝑥)(2.1) always to mean that there is a positive number 𝑐₁ such that𝑎_𝑘≤𝑐₁𝑏_𝑘or𝜑(𝑥)≤𝑐₁𝜓(𝑥)(2.2) for all admitted values of the discrete variable 𝑘 or the continuous variable 𝑥, where (𝑎_𝑘)_𝑘, (𝑏_𝑘)_𝑘 are nonnegative sequences and 𝜑, 𝜓 are nonnegative functions. We use the equivalence “~” in𝑎_𝑘∼𝑏_𝑘or𝜑(𝑥)∼𝜓(𝑥)(2.3) for𝑎_𝑘≲𝑏_𝑘,𝑏_𝑘≲𝑎_𝑘or𝜑(𝑥)≲𝜓(𝑥),𝜓(𝑥)≲𝜑(𝑥).(2.4) 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, quasinormed by‖‖𝑓∣𝐿_𝑝(ℝ^𝑛)‖‖∶=_ℝ𝑛||𝑓(𝑥)||^𝑝d𝑥^1/𝑝(2.5) with the appropriate modification if 𝑝=∞. Moreover, let Ω denote a domain in ℝ^𝑛. Then, 𝐿_𝑝(Ω) is the collection of all complex-valued Lebesgue measurable functions in Ω such that‖‖𝑓∣𝐿_𝑝(Ω)‖‖∶=_Ω||𝑓(𝑥)||^𝑝d𝑥^1/𝑝(2.6) (with the usual modification if 𝑝=∞) is finite.

Furthermore, 𝐵_𝑅 stands for an open ball with radius 𝑅>0 around the origin,𝐵_𝑅={𝑥∈ℝ^𝑛∶|𝑥|<𝑅}.(2.7) Let 𝑄_𝑗,𝑚 with 𝑗∈ℕ₀ and 𝑚∈ℤ^𝑛 denote a cube in ℝ^𝑛 with sides parallel to the axes of coordinates, centered at 2^−𝑗𝑚, 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.

2.1. Function Spaces Defined via Differences

If 𝑓 is an arbitrary function on ℝ^𝑛, ℎ∈ℝ^𝑛, and 𝑟∈ℕ, thenΔ^1ℎ𝑓(𝑥)=𝑓(𝑥+ℎ)−𝑓(𝑥),Δ^ℎ𝑟+1𝑓(𝑥)=Δ^1ℎΔ^𝑟ℎ𝑓(𝑥)(2.8) are the usual iterated differences. Given a function 𝑓∈𝐿_𝑝(ℝ^𝑛), the 𝑟-th modulus of smoothness is defined by𝑑𝜔_𝑟(𝑓,𝑡)_𝑝=sup_{||ℎ||≤𝑡}‖‖Δ^𝑟ℎ𝑓∣𝐿_𝑝(ℝ^𝑛)‖‖,𝑡>0,0<𝑝≤∞,_{𝑟𝑡,𝑝}𝑓(𝑥)=𝑡^−𝑛_|ℎ|≤𝑡||Δ^𝑟ℎ𝑓(𝑥)||^𝑝dℎ^1/𝑝,𝑡>0,0<𝑝<∞,(2.9) denotes its ball means.

Definition 2.1. (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 ‖‖𝑓∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖_𝑟=‖‖𝑓∣𝐿_𝑝(ℝ^𝑛)‖‖+¹⁰𝑡^−𝑠𝑞𝜔_𝑟(𝑓,𝑡)^𝑞𝑝d𝑡𝑡^1/𝑞(2.10) (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 of all 𝑓∈𝐿_𝑝(ℝ^𝑛) such that ‖‖𝑓∣𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖_𝑟=‖‖𝑓∣𝐿_𝑝(ℝ^𝑛)‖‖+‖‖‖‖¹⁰𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}𝑓(⋅)^𝑞d𝑡𝑡^1/𝑞∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖(2.11) (with the usual modification if 𝑞=∞) is finite.

Remark 2.2. These are the classical Besov and Triebel-Lizorkin spaces, in particular, when 1≤𝑝,𝑞≤∞ ( 𝑝<∞ for the F-spaces) and 𝑠>0. We will 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 [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 𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛) has been described in detail in [12] for those spaces which can also be considered as subspaces of 𝒮^(ℝ^𝑛). 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 ∫¹⁰ can be replaced by ∫^∞0 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 𝐿_𝑝(ℝ^𝑛), in particular, for 𝑠>0 and 0<𝑞≤∞, we have the embeddings 𝐀^{𝑠𝑝,𝑞}(ℝ^𝑛)↪𝐿_𝑝(ℝ^𝑛),(2.12) 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),(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=∞, one recovers the Hölder-Zygmund spaces 𝒞^𝑠(ℝ^𝑛), that is, 𝐁^𝑠∞,∞(ℝ^𝑛)=𝒞^𝑠(ℝ^𝑛),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 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 represented as 𝑓(𝑥)=_{𝛽∈ℕ𝑛0}^∞_𝑗=0_{𝑚∈ℤ𝑛}𝜆^{𝛽𝑗,𝑚}𝑘^{𝛽𝑗,𝑚}(𝑥),𝑥∈ℝ^𝑛,(2.15) with coefficients 𝜆={𝜆^{𝛽𝑗,𝑚}∈ℂ∶𝛽∈ℕ^𝑛0,𝑗∈ℕ₀,𝑚∈ℤ^𝑛} belonging to some appropriate sequence space 𝑏^{𝑠,𝜚𝑝,𝑞} defined as 𝑏^{𝑠,𝜚𝑝,𝑞}∶=𝜆∶‖‖𝜆∣𝑏^{𝑠,𝜚𝑝,𝑞}‖‖<∞,(2.16) where ‖‖𝜆∣𝑏^{𝑠,𝜚𝑝,𝑞}‖‖=sup_{𝛽∈ℕ𝑛0}2^𝜚|𝛽|⎛⎜⎜⎝^∞_𝑗=02^{𝑗(𝑠−𝑛/𝑝)𝑞}_{𝑚∈ℤ𝑛}||𝜆^{𝛽𝑗,𝑚}||^𝑝^𝑞/𝑝⎞⎟⎟⎠^1/𝑞,(2.17)𝑠>0,0<𝑝,𝑞≤∞ (with the usual modification if 𝑝=∞ and/or 𝑞=∞), 𝜚≥0, and 𝑘^{𝛽𝑗,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^{𝑠𝑝,𝑞}(ℝ^𝑛), the following is known.

Proposition 2.3. Let 0<𝑝,𝑞≤∞,𝑠>0,𝑘∈ℕ with 𝑘>𝑠, and let ℎ∈𝐶^𝑘(ℝ^𝑛). Then, 𝑓⟶ℎ𝑓(2.18) 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. [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 Ω be a domain in ℝ^𝑛. We define spaces 𝐀^{𝑠𝑝,𝑞}(Ω) by restriction of the corresponding spaces on ℝ^𝑛, that is, 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|_Ω=𝑓. Furthermore,‖‖𝑓∣𝐀^{𝑠𝑝,𝑞}(Ω)‖‖=inf‖‖𝑔∣𝐀^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖,(2.19) where the infimum is taken over all 𝑔∈𝐀^{𝑠𝑝,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^{𝑠𝑝,𝑞}(ℝ^𝑛) from Remark 2.2 carries over. For further details on this subject, we refer to [18, Section  2.1].

Embeddings results between the spaces 𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛) hold also for the spaces 𝐁^{𝑠𝑝,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.

Proposition 2.4. Let 0<𝑠₂<𝑠₁<∞, 0<𝑝₁,𝑝₂,𝑞₁,𝑞₂≤∞, and Ω⊂ℝ^𝑛 be bounded. If 𝛿₊=𝑠₁−𝑠₂−𝑑1𝑝₁−1𝑝₂₊>0,(2.20) one has the embedding 𝐁^𝑠1_𝑝1,𝑞1(Ω)↪𝐁^𝑠2_𝑝2,𝑞2(Ω).(2.21)

Proof. If 𝑝₁≤𝑝₂, the embedding follows from [19, Theorem  1.15], since the spaces on Ω are defined by restriction of their counterparts on ℝ^𝑛. Therefore, it remains to show that, for 𝑝₁>𝑝₂, we have the embedding 𝐁^𝑠2_𝑝1,𝑞2(Ω)↪𝐁^𝑠2_𝑝2,𝑞2(Ω).(2.22) Let 𝜓∈𝐷(ℝ^𝑛) with support in the compact set Ω₁ and 𝜓(𝑥)=1if𝑥∈Ω⊂Ω₁.(2.23) Then, for 𝑓∈𝐁^𝑠2_𝑝1,𝑞2(Ω), there exists 𝑔∈𝐁^𝑠2_𝑝1,𝑞2(ℝ^𝑛) with 𝑔||_Ω=𝑓,‖‖𝑓∣𝐁^𝑠2_𝑝1,𝑞2(Ω)‖‖∼‖‖𝑔∣𝐁^𝑠2_𝑝1,𝑞2(ℝ^𝑛)‖‖.(2.24) We calculate ‖‖𝑓∣𝐁^𝑠2_𝑝2,𝑞2(Ω)‖‖≤≤‖‖‖‖𝜓𝜓𝑔𝑔∣∣𝐁𝐁^𝑠2_{𝑝2,𝑞2𝑠2𝑝1,𝑞2}(ℝ^𝑛)‖‖≤𝑐_𝜓(ℝ^𝑛)‖‖‖‖𝑔∣𝐁^𝑠2_𝑝1,𝑞2(ℝ^𝑛)‖‖∼‖‖𝑓∣𝐁^𝑠2_𝑝1,𝑞2(Ω)‖‖.(2.25) The last inequality in (2.25) follows from Proposition 2.3. In the 2nd step, we used (2.10) together with the fact that ‖‖Δ^𝑟ℎ(𝜓𝑔)∣𝐿_𝑝2(ℝ^𝑛)‖‖≤𝑐_Ω1‖‖Δ^𝑟ℎ(𝜓𝑔)∣𝐿_𝑝1(ℝ^𝑛)‖‖,𝑝₁>𝑝₂,(2.26) 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, 𝐁^{𝑠𝑝,𝑞}(Ω), and F-spaces, 𝐅^{𝑠𝑝,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 let 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 2.5. 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 𝑒_𝑘(𝑇)=inf⎧⎪⎨⎪⎩𝜀>0∶𝑇𝑈_𝑋⊂^2𝑘−1_𝑗=1𝑦_𝑗+𝜀𝑈_𝑌forsome𝑦₁,…,𝑦₂^𝑘−1∈𝑌⎫⎪⎬⎪⎭,(2.27) 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 2.6. Let 𝑋,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) If 𝑌 is a 𝑝-Banach space (0<𝑝≤1), then, for all 𝑗,𝑘∈ℕ, 𝑒^{𝑝𝑗+𝑘−1}(𝑆+𝑇)≤𝑒^𝑝𝑗(𝑆)+𝑒^𝑝𝑘(𝑇).(2.28)(iii)(Multiplicativity) For all 𝑗,𝑘∈ℕ, 𝑒_{𝑗+𝑘−1}(𝑅𝑇)≤𝑒_𝑗(𝑅)𝑒_𝑘(𝑇).(2.29)(iv)(Compactness) 𝑇 is compact if and only if lim_𝑘→∞𝑒_𝑘(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 {𝑀_𝑗}_𝑗∈ℕ0 be a sequence of natural numbers satisfying 𝑀_𝑗∼2^𝑗𝑛,𝑗∈ℕ₀.(2.31) Concerning entropy numbers for the respective sequence spaces 𝑏^{𝑠,𝜚𝑝,𝑞}(𝑀_𝑗), which are defined as the sequence spaces 𝑏^{𝑠,𝜚𝑝,𝑞} in (2.17) with the sum over 𝑚∈ℤ^𝑛 replaced by a sum over 𝑚=1,…,𝑀_𝑗, the following result was proved in [24, Proposition  3.4].

Proposition 2.8. Let 𝑑>0, 0<𝜎₁,𝜎₂<∞, and 0<𝑞₁,𝑞₂≤∞. Furthermore, let 𝜚₁>𝜚₂≥0, 0<𝑝₁≤𝑝₂≤∞,𝛿=𝜎₁−𝜎₂−𝑛1𝑝₁−1𝑝₂>0.(2.32) Then the identity map id∶𝑏^𝜎1,𝜚1_𝑝1,𝑞1𝑀_𝑗⟶𝑏^𝜎2,𝜚2_𝑝2,𝑞2𝑀_𝑗(2.33) is compact, where 𝑀_𝑗 is restricted by (2.31).

The next theorem provides a sharp result for entropy numbers of the identity operator related to the sequence spaces 𝑏^{𝑠,𝜚𝑝,𝑞}(𝑀_𝑗).

Theorem 2.9. Let 𝑛>0, 0<𝑠₁,𝑠₂<∞, and 0<𝑞₁,𝑞₂≤∞. Furthermore, let 𝜚₁>𝜚₂≥0, 0<𝑝₁≤𝑝₂≤∞,𝛿=𝑠₁−𝑠₂−𝑛1𝑝₁−1𝑝₂>0.(2.34) For the entropy numbers 𝑒_𝑘 of the compact operator id∶𝑏^𝑠1,𝜚1_𝑝1,𝑞1𝑀_𝑗⟶𝑏^𝑠2,𝜚2_𝑝2,𝑞2𝑀_𝑗,(2.35) one has 𝑒_𝑘(id)∼𝑘^{−𝛿/𝑛+1/𝑝2−1/𝑝1},𝑘∈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 𝑏^{𝑠𝑖,𝜚𝑖}_{𝑝𝑖,𝑞𝑖}𝑀_𝑗=ℓ_∞2^𝜚𝑖ℓ_𝑞𝑖2^{𝑗(𝑠𝑖−𝑛/𝑝𝑖)}ℓ^𝑀𝑗_𝑝𝑖,𝑖=1,2.(2.37) Recall the embedding assertions for Besov spaces 𝐁^{𝑠𝑝,𝑞}(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=𝐵_𝑅.

Theorem 2.11. Let 0<𝑠₂<𝑠₁𝛿<∞,0<𝑝₁,𝑝₂≤∞,0<𝑞₁,𝑞₂≤∞,₊=𝑠₁−𝑠₂−𝑛1𝑝₁−1𝑝₂₊>0.(2.38) Then, the embedding id∶𝐁^𝑠1_𝑝1,𝑞1(Ω)⟶𝐁^𝑠2_𝑝2,𝑞2(Ω)(2.39) is compact, and, for the related entropy numbers, one computes 𝑒_𝑘(id)≲𝑘^{−(𝑠1−𝑠2)/𝑛},𝑘∈ℕ.(2.40)

Proof. Step 1. Let 𝑝₂≥𝑝₁, 𝛿₊=𝛿, and let 𝑓∈𝐁^𝑠1_𝑝1,𝑞1(Ω), then, by [26, Theorem  6.1], there is a (nonlinear) bounded extension operator 𝑔‖‖𝑔=∣E𝐁x𝑓_{𝑠1𝑝1,𝑞1}suchthatRe_Ω𝑔=𝑔||_Ω=𝑓,(2.41)(ℝ^𝑛)‖‖≤𝑐‖‖𝑓∣𝐁^𝑠1_𝑝1,𝑞1(Ω)‖‖.(2.42) We may assume that 𝑔 is zero outside a fixed neighbourhood Λ of Ω. Using the subatomic approach for 𝐁^𝑠1_𝑝1,𝑞1(ℝ^𝑛), cf. Remark 2.2, we can find an optimal decomposition of 𝑔, that is, 𝑔(𝑥)=_{𝛽∈ℕ𝑛0}^∞_𝑗=0_{𝑚∈ℤ𝑛}𝜆^{𝛽𝑗,𝑚}𝑘^{𝛽𝑗,𝑚}(𝑥),‖‖𝑔∣𝐁^𝑠1_𝑝1,𝑞1(ℝ^𝑛)‖‖∼‖‖𝜆∣𝑏^𝑠1,𝜚1_𝑝1,𝑞1‖‖(2.43) with 𝜚₁>0 large.
Let 𝑀_𝑗 for fixed 𝑗∈ℕ₀ be the number of cubes 𝑄_𝑗,𝑚 such that 𝑟𝑄_𝑗,𝑚∩Ω≠∅.(2.44) Since Ω⊂ℝ^𝑛 is bounded, we have 𝑀_𝑗∼2^𝑗𝑛,𝑗∈ℕ₀.(2.45) This coincides with (2.31). We introduce the (nonlinear) operator 𝑆, 𝑆∶𝐁^𝑠1_𝑝1,𝑞1(ℝ^𝑛)⟶𝑏^𝑠1,𝜚1_𝑝1,𝑞1𝑀_𝑗(2.46) by 𝑆𝑔=𝜆,𝜆=𝜆^{𝛽𝑗,𝑚}∶𝛽∈ℕ^𝑛0,𝑗∈ℕ₀,𝑚∈ℤ^𝑛,𝑟𝑄_𝑗,𝑚∩Ω≠∅,(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 ‖𝑆‖=sup_𝑔≠0‖‖𝜆∣𝑏^𝑠1,𝜚1_𝑝1,𝑞1𝑀_𝑗‖‖‖‖𝑔∣𝐁^𝑠1_𝑝1,𝑞1(ℝ^𝑛)‖‖≤𝑐.(2.48) Next we construct the linear map 𝑇, 𝑇∶𝑏^𝑠2,𝜚2_𝑝2,𝑞2𝑀_𝑗⟶𝐁^𝑠2_𝑝2,𝑞2(ℝ^𝑛),(2.49) given by 𝑇𝜆=_{𝛽∈ℕ𝑛0}^∞_𝑀𝑗=0^𝑗_𝑚=1𝜆^{𝛽𝑗,𝑚}𝑘^{𝛽𝑗,𝑚}(𝑥).(2.50) It follows that 𝑇 is a linear (since the subatomic approach provides an expansion of functions via universal building blocks) and bounded map, ‖𝑇‖=sup_𝜆≠0‖‖𝑇𝜆∣𝐁^𝑠2_𝑝2,𝑞2(ℝ^𝑛)‖‖‖‖𝜆∣𝑏^𝑠2,𝜚2_𝑝2,𝑞2𝑀_𝑗‖‖≤𝑐.(2.51) We complement the three bounded maps Ex,𝑆, 𝑇 by the identity operator id∶𝑏^𝑠1,𝜚1_𝑝1,𝑞1𝑀_𝑗⟶𝑏^𝑠2,𝜚2_𝑝2,𝑞2𝑀_𝑗with𝜚₁>𝜚₂,(2.52) which is compact by Proposition 2.8 and the restriction operator Re_Ω∶𝐁^𝑠2_𝑝2,𝑞2(ℝ^𝑛)⟶𝐁^𝑠2_𝑝2,𝑞2(Ω),(2.53) which is continuous. From the constructions, it follows that id𝐁^𝑠1_𝑝1,𝑞1(Ω)⟶𝐁^𝑠2_𝑝2,𝑞2(Ω)=Re_Ω∘𝑇∘id∘𝑆∘Ex.(2.54) Hence, taking finally Re_Ω, 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 𝐁^𝑠1_𝑝1,𝑞1(Ω) is mapped by 𝑆∘Exinto a bounded set in 𝑏^𝑠1,𝜚1_𝑝1,𝑞1𝑀_𝑗.(2.55) Since the identity operator id from (2.52) is compact, this bounded set is mapped into a precompact set in 𝑏^𝑠2,𝜚2_𝑝2,𝑞2𝑀_𝑗,(2.56) which can be covered by 2^𝑘 balls of radius 𝑐𝑒_𝑘(id) with 𝑒_𝑘(id)≤𝑐𝑘^{−𝛿/𝑛+1/𝑝2−1/𝑝1},𝑘∈ℕ.(2.57) This follows from Theorem 2.9, where we used 𝑝₂≥𝑝₁. Applying the two linear and bounded maps 𝑇 and Re_Ω 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 𝐁^𝑠1_𝑝1,𝑞1(Ω) by 2^𝑘 balls of radius 𝑐𝑒_𝑘(id) in 𝐁^𝑠2_𝑝2,𝑞2(Ω). Inserting 𝛿=𝑠₁−𝑠₂−𝑛1𝑝₁−1𝑝₂(2.58) in the exponent, we finally obtain the desired estimate 𝑒_𝑘(id)≤𝑐𝑘^{−(𝑠1−𝑠2)/𝑛},𝑘∈ℕ.(2.59)Step 2. Let 𝑝₁>𝑝₂. Since, by Proposition 2.4, 𝐁^𝑠2_𝑝1,𝑞2(Ω)⊂𝐁^𝑠2_𝑝2,𝑞2(Ω),(2.60) we see that 𝐁^𝑠1_𝑝1,𝑞1(Ω)⊂𝐁^𝑠2_𝑝1,𝑞2(Ω)⊂𝐁^𝑠2_𝑝2,𝑞2(Ω),(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 𝐁^{𝑠𝑝,min(𝑝,𝑞)}(Ω)↪𝐅^{𝑠𝑝,𝑞}(Ω)↪𝐁^{𝑠𝑝,max(𝑝,𝑞)}(Ω).(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≤∞, 𝑠>0, and let 𝑅>0 be a real number. Then, ‖‖𝑓∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖∼^∞0𝑡^−𝑠𝑞𝜔_𝑟(𝑓,𝑡)^𝑞𝑝𝑑𝑡𝑡^1/𝑞(3.1) for all 𝑓∈𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛) with supp𝑓⊂𝐵_𝑅.

Proof. We will need that 𝐁^{𝑠𝑝,𝑞}(𝐵_𝑅) embeds compactly into 𝐿_𝑝(𝐵_𝑅). This follows at once from the fact that 𝐁^{𝑠𝑝,𝑞}(𝐵_𝑅) is compactly embedded into 𝐁^{𝑠−𝜀𝑝,𝑞}(𝐵_𝑅), cf. Remark 2.12, and 𝐁^{𝑠−𝜀𝑝,𝑞}(𝐵_𝑅)↪𝐿_𝑝(𝐵_𝑅), which is trivial.We argue similarly to [6]. We have to prove that ‖‖𝑓∣𝐿_𝑝(ℝ^𝑛)‖‖≲^∞0𝑡^−𝑠𝑞𝜔_𝑟(𝑓,𝑡)^𝑞𝑝𝑑𝑡𝑡^1/𝑞(3.2) for every 𝑓∈𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛) with supp𝑓⊂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 ‖‖𝑓_𝑗∣𝐿_𝑝(ℝ^𝑛)‖‖=1,^∞0𝑡^−𝑠𝑞𝜔_𝑟𝑓_𝑗,𝑡^𝑞𝑝𝑑𝑡𝑡^1/𝑞≤1𝑗,(3.3) that is, we obtain that ‖𝑓_𝑗∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖ is bounded. The trivial estimates ‖‖𝑓_𝑗∣𝐿_𝑝(ℝ^𝑛)‖‖=‖‖𝑓_𝑗∣𝐿_𝑝𝐵_𝑅‖‖,‖‖𝑓_𝑗∣𝐁^{𝑠𝑝,𝑞}𝐵_𝑅‖‖≤‖‖𝑓_𝑗∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖(3.4) imply that this is true also for ‖𝑓_𝑗∣𝐁^{𝑠𝑝,𝑞}(𝐵_𝑅)‖. Due to the compactness of 𝐁^{𝑠𝑝,𝑞}(𝐵_𝑅)↪𝐿_𝑝(𝐵_𝑅), we may assume, that 𝑓_𝑗→𝑓 in 𝐿_𝑝(𝐵_𝑅) with ‖𝑓|𝐿_𝑝(𝐵_𝑅)‖=1. Using the subadditivity of 𝜔(⋅,𝑡)_𝑝, we obtain that ^∞0𝑡^−𝑠𝑞𝜔_𝑟𝑓^𝑗−𝑓_𝑗^′,𝑡^𝑞𝑝𝑑𝑡𝑡^1/𝑞≤1𝑗+𝑗1^.(3.5) Together with the estimate ‖𝑓_𝑗−𝑓_𝑗∣𝐿_𝑝(ℝ^𝑛)‖→0, this implies that (𝑓_𝑗)^∞𝑗=1 is a Cauchy sequence in 𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛), that is, 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 ^∞0𝑡^−𝑠𝑞𝜔_𝑟(𝑓,𝑡)^𝑞𝑝𝑑𝑡𝑡^1/𝑞=0.(3.6) As 𝜔_𝑟(𝑓,𝑡) is a nondecreasing function of 𝑡, this implies that 𝜔_𝑟(𝑓,𝑡)=0 for all 0<𝑡<∞ and finally ‖Δ^𝑟ℎ𝑓∣𝐿_𝑝(ℝ^𝑛)‖=0 for all ℎ∈ℝ^𝑛. 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∣𝐿_𝑝(ℝ^𝑛)‖=1.

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

Theorem 3.2. Let 0<𝜆≤1 and 𝑓∈𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛) with supp𝑓⊂𝐵_𝜆. Then, ‖‖𝑓(𝜆⋅)∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖∼𝜆^{𝑠−𝑛/𝑝}‖‖𝑓∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖(3.7) with constants of equivalence independent of 𝜆 and 𝑓.

Proof. We know from Proposition 3.1 that ‖‖𝑓(𝜆⋅)∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖∼^∞0𝑡^−𝑠𝑞𝜔_𝑟(𝑓(𝜆⋅),𝑡)^𝑞𝑝𝑑𝑡𝑡^1/𝑞,(3.8) as supp𝑓(𝜆⋅)⊂𝐵₁. Using Δ^𝑟ℎ(𝑓(𝜆⋅))(𝑥)=(Δ^𝑟𝜆ℎ𝑓)(𝜆𝑥), we get 𝜔_𝑟(𝑓(𝜆⋅),𝑡)_𝑝=sup_{||ℎ||≤𝑡}‖‖Δ^𝑟ℎ(𝑓(𝜆⋅))‖‖_𝑝=sup_{||ℎ||≤𝑡}‖‖Δ^𝑟𝜆ℎ𝑓(𝜆⋅)‖‖_𝑝=𝜆^−𝑛/𝑝sup_{||ℎ||≤𝑡}‖‖Δ^𝑟𝜆ℎ𝑓(⋅)‖‖_𝑝=𝜆^−𝑛/𝑝sup_{||||𝜆ℎ≤𝜆𝑡}‖‖Δ^𝑟𝜆ℎ𝑓(⋅)‖‖_𝑝=𝜆^−𝑛/𝑝𝜔_𝑟(𝑓,𝜆𝑡)_𝑝,(3.9) which finally implies ^∞0𝑡^−𝑠𝑞𝜔_𝑟(𝑓(𝜆⋅),𝑡)^𝑞𝑝𝑑𝑡𝑡^1/𝑞=𝜆^−𝑛/𝑝^∞0𝑡^−𝑠𝑞𝜔_𝑟(𝑓,𝜆𝑡)^𝑞𝑝𝑑𝑡𝑡^1/𝑞=𝜆^{𝑠−𝑛/𝑝}^∞0𝑡^−𝑠𝑞𝜔_𝑟(𝑓,𝑡)^𝑞𝑝𝑑𝑡𝑡^1/𝑞∼𝜆^{𝑠−𝑛/𝑝}‖‖𝑓∣𝐁^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖.(3.10)

The homogeneity property for Triebel-Lizorkin spaces 𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛) follows similarly.

Proposition 3.3. Let 0<𝑝<∞,0<𝑞≤∞,𝑠>0, and let 𝑅>0 be a real number. Then, ‖‖𝑓∣𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖∼‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}𝑓(⋅)^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖(3.11) for all 𝑓∈𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛) with supp𝑓⊂𝐵_𝑅.

Proof. We have to prove that ‖‖𝑓∣𝐿_𝑝(ℝ^𝑛)‖‖≲‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}𝑓(⋅)^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖(3.12) for every 𝑓∈𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛) with supp𝑓⊂𝐵_𝑅. Let us assume again that this is not true. Then, we find a sequence (𝑓_𝑗)^∞𝑗=1⊂𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛) such that ‖‖𝑓_𝑗∣𝐿_𝑝(ℝ^𝑛)‖‖=1,‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}𝑓_𝑗(⋅)^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖≤1𝑗,(3.13) which in turn implies that ‖𝑓_𝑗∣𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛)‖ is bounded. Again, the same is true also for ‖𝑓_𝑗|𝐅^{𝑠𝑝,𝑞}(𝐵_𝑅)‖. Due to 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 ‖𝑓∣𝐿_𝑝(𝐵_𝑅)‖=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 ‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}𝑓(⋅)^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖=0(3.14) or, equivalently, ^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}𝑓(𝑥)^𝑞𝑑𝑡𝑡=0(3.15) for almost every 𝑥∈ℝ^𝑛. Hence, 𝑑^{𝑟𝑡,𝑝}𝑓(𝑥)=0 for almost all 𝑥∈ℝ^𝑛 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<𝜆≤1 and 𝑓∈𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛) with supp𝑓⊂𝐵_𝜆. Then, ‖‖𝑓(𝜆⋅)∣𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖∼𝜆^{𝑠−𝑛/𝑝}‖‖𝑓∣𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖(3.16) with constants of equivalence independent of 𝜆 and 𝑓.

Proof. We know from Proposition 3.3 that ‖‖𝑓(𝜆⋅)∣𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖∼‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}(𝑓(𝜆⋅))(⋅)^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖,(3.17) as supp𝑓(𝜆⋅)⊂𝐵₁. Using Δ^𝑟ℎ(𝑓(𝜆⋅))(𝑥)=(Δ^𝑟𝜆ℎ𝑓)(𝜆𝑥), we get using the substitution ℎ=𝜆ℎ, 𝑑^{𝑟𝑡,𝑝}(𝑓(𝜆⋅))(𝑥)=𝑡^−𝑛_|ℎ|≤𝑡||Δ^𝑟ℎ𝑓(𝜆⋅)(𝑥)||^𝑝𝑑ℎ^1/𝑝=𝑡^−𝑛_|ℎ|≤𝑡||Δ^𝑟𝜆ℎ𝑓(𝜆𝑥)||^𝑝𝑑ℎ^1/𝑝=(𝜆𝑡)^−𝑛_|ℎ_|≤𝜆𝑡|||Δ_𝑟ℎ𝑓(𝜆𝑥)|||^𝑝𝑑ℎ^1/𝑝=𝑑^{𝑟𝜆𝑡,𝑝}(𝑓)(𝜆𝑥),(3.18) which finally implies ‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}(𝑓(𝜆⋅))(⋅)^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖=‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝜆𝑡,𝑝}𝑓(𝜆⋅)^𝑞𝑑𝑡𝑡^1/𝑞=𝜆_𝑠∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}𝑓(𝜆⋅)^𝑞𝑑𝑡𝑡^1/𝑞=𝜆_{𝑠−𝑛/𝑝}∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖‖‖‖‖^∞0𝑡^−𝑠𝑞𝑑^{𝑟𝑡,𝑝}𝑓(⋅)^𝑞𝑑𝑡𝑡^1/𝑞∼𝜆_{𝑠−𝑛/𝑝}∣𝐿_𝑝(ℝ^𝑛)‖‖‖‖‖‖𝑓∣𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛)‖‖.(3.19)

4. Pointwise Multipliers

We briefly sketch an application of the above homogeneity results in terms of pointwise multipliers. A locally integrable function 𝜑 in ℝ^𝑛 is called a pointwise multiplier in 𝐀^{𝑠𝑝,𝑞}(ℝ^𝑛) if𝑓⟼𝜑𝑓(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 𝐵_𝜆 be the balls introduced in (2.7).

Corollary 4.1. Let 𝑠>0, 0<𝑝,𝑞≤∞, and 0<𝜆≤1. Let 𝜑 be a function having classical derivatives in 𝐵_2𝜆 up to order 1+[𝑠] with ||D^𝛾𝜑(𝑥)||≤𝑎𝜆^−|𝛾|,||𝛾||≤1+[𝑠],𝑥∈𝐵_2𝜆,(4.2) for some constant 𝑎>0. Then, 𝜑 is a pointwise multiplier in 𝐁^{𝑠𝑝,𝑞}(𝐵_𝜆), ‖‖𝜑𝑓∣𝐁^{𝑠𝑝,𝑞}𝐵_𝜆‖‖≤𝑐‖‖𝑓∣𝐁^{𝑠𝑝,𝑞}𝐵_𝜆‖‖,(4.3) where 𝑐 is independent of 𝑓∈𝐁^{𝑠𝑝,𝑞}(𝐵_𝜆) and of 𝜆 (but depends on 𝑎).

Proof. By Proposition 2.3, the function 𝜑(𝜆⋅) is a pointwise multiplier in 𝐁^{𝑠𝑝,𝑞}(𝐵₁). Then, (4.3) is a consequence of (3.7), ‖‖𝜑𝑓∣𝐁^{𝑠𝑝,𝑞}𝐵_𝜆‖‖≲∼𝜆𝜆^{−(𝑠−𝑛/𝑝)}_{−(𝑠−𝑛/𝑝)}‖‖𝜑𝑓(𝜆⋅)∣𝐁^{𝑠𝑝,𝑞}𝐵₁‖‖‖‖𝑓(𝜆⋅)∣𝐁^{𝑠𝑝,𝑞}𝐵₁‖‖∼‖‖𝑓∣𝐁^{𝑠𝑝,𝑞}𝐵_𝜆‖‖.(4.4)

Remark 4.2. 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1min(𝑝,𝑞)−1𝑝.(4.5) This follows from the fact that the analogue of Proposition 2.3 for F-spaces is established using an atomic characterization of the spaces 𝐅^{𝑠𝑝,𝑞}(ℝ^𝑛) which is only true if we impose (4.5), (cf. [13, Proposition  9.14]).

Acknowledgment

Jan Vybíral 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).