Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces by Differences

Abstract

We present characterizations of the Besov-type spaces 𝐵^{𝑠,𝜏𝑝,𝑞} and the Triebel-Lizorkin-type spaces 𝐹^{𝑠,𝜏𝑝,𝑞} by differences. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking 𝜏=0.

1. Introduction

The 𝐵^{𝑠,𝜏𝑝,𝑞} spaces and 𝐹^{𝑠,𝜏𝑝,𝑞} spaces have been studied extensively in recent years. When 𝜏=0 they coincide with the usual function spaces 𝐵^{𝑠𝑝,𝑞} and 𝐹^{𝑠𝑝,𝑞}, respectively, studied in detail by Triebel in [1–3]. When 𝑠∈ℝ, 𝜏∈[0,∞) and 1≤𝑝, 𝑞<∞ the 𝐵^{𝑠,𝜏𝑝,𝑞} spaces were first introduced by El Baraka in [4, 5]. In these papers, El Baraka investigated embeddings as well as Littlewood-Paley characterizations of Campanato spaces. El Baraka showed that the spaces 𝐵^{𝑠,𝜏𝑝,𝑞} cover certain Campanato spaces, studied in [6, 7]. Later on, Drihem gave in [8] a characterization for 𝐵^{𝑠,𝜏𝑝,𝑞} spaces by local means and maximal functions. For a complete treatment of 𝐵^{𝑠,𝜏𝑝,𝑞} spaces and 𝐹^{𝑠,𝜏𝑝,𝑞} spaces we refer the reader the work of Yuan et al. [9]. Yang and Yuan, in [10–12], have introduced the scales of homogeneous Besov-Triebel-Lizorkin-type spaces 𝐵̇^{𝑠,𝜏𝑝,𝑞} and 𝐹̇^{𝑠,𝜏𝑝,𝑞} (𝑝≠∞), which generalize the homogeneous Besov-Triebel-Lizorkin spaces 𝐵̇^{𝑠,𝜏𝑝,𝑞}, 𝐹̇^{𝑠,𝜏𝑝,𝑞} and established the relation between 𝐹̇^{𝑠,𝜏𝑝,𝑞} and 𝑄_𝛼 spaces. See also [13] for further results.

Our main purpose in this paper is to characterize these function spaces by differences. These results are a generalization of some results given in [17], and [9, Chapter 4, Section 4.3]. All these results generalize the existing classical results on Besov spaces and Triebel-Lizorkin spaces by taking 𝜏=0.

The paper is organized as follows. Section 2.1 collects fundamental notation and concepts and Section 2.2 covers results from the theory of these function spaces. Some necessary tools are given in Section 3. These results are used in Section 4 to obtain the characterization of 𝐵^{𝑠,𝜏𝑝,𝑞} spaces and 𝐹^{𝑠,𝜏𝑝,𝑞} spaces by differences.

2. Preliminaries

2.1. Notation and Conventions

As usual, ℝ^𝑛 the 𝑛-dimensional real Euclidean space, ℕ the collection of all natural numbers, and ℕ₀=ℕ∪{0}. The letter ℤ stands for the set of all integer numbers. For a multi-index 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛0, we write |𝛼|=𝛼₁+⋯+𝛼_𝑛 and 𝐷^𝛼=𝜕^|𝛼|/𝜕𝑥^𝛼1₁⋯𝜕𝑥^𝛼𝑛_𝑛. For 𝑣∈ℤ,let 𝐵_𝑣 be the ball of ℝ^𝑛 with radius 2^−𝑣 and 𝑣⁺=max{𝑣,0}. The Euclidean scalar product of 𝑥=(𝑥₁,…,𝑥_𝑛) and 𝑦=(𝑦₁,…,𝑦_𝑛) is given by 𝑥⋅𝑦=𝑥₁𝑦₁+⋯+𝑥_𝑛𝑦_𝑛. We denote by |Ω| the 𝑛-dimensional Lebesgue measure of Ω⊆ℝ^𝑛. For any measurable subset Ω⊆ℝ^𝑛 the Lebesgue space 𝐿^𝑝(Ω), 0<𝑝≤∞ consists of all measurable functions for which‖𝑓∣𝐿^𝑝(Ω‖)𝑓‖∣=𝐿_∞_Ω||𝑓(𝑥)||^𝑝𝑑𝑥^1/𝑝<∞,0<𝑝<∞,(Ω)‖=ess-sup_𝑥∈Ω||𝑓(𝑥)||<∞.(2.1)

By 𝒮(ℝ^𝑛) we denote the Schwartz space of all complex-valued, infinitely differentiable, and rapidly decreasing functions on ℝ^𝑛 and by 𝒮^(ℝ^𝑛) the dual space of all tempered distributions on ℝ^𝑛. We define the Fourier transform of a function 𝑓∈𝒮(ℝ^𝑛) byℱ(𝑓)(𝜉)=(2𝜋)^−𝑛/2_ℝ𝑛𝑒^{−𝑖𝑥⋅𝜉}𝑓(𝑥)𝑑𝑥.(2.2) Its inverse is denoted by ℱ⁻¹𝑓. Both ℱ and ℱ⁻¹ are extended to the dual Schwartz space 𝒮^(ℝ^𝑛) in the usual way.

Let 𝜏∈[0,∞) and 𝑝∈(0,∞]. Let 𝐿^𝑝𝜏(ℝ^𝑛) be the collection of functions 𝑓∈𝐿^𝑝loc(ℝ^𝑛) such that‖‖𝑓∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖=sup_𝐵𝐽1||𝐵_𝐽||^𝜏_𝐵𝐽||𝑓(𝑥)||^𝑝𝑑𝑥^1/𝑝<∞,(2.3) where the supremum is taken over all 𝐽∈ℤ⧵ℕ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽. Obviously, when 𝜏=0, then 𝐿^𝑝𝜏(ℝ^𝑛)=𝐿^𝑝(ℝ^𝑛). Furthermore,𝐿^𝑝𝜏(ℝ^𝑛)↪𝒮^(ℝ^𝑛),(2.4) (see [9, page 46]).

If 𝑠∈ℝ, 0<𝑞≤∞ and 𝐽∈ℤ, then ℓ^{𝑠𝑞,𝐽}₊ is the set of all sequences {𝑓_𝑘}_𝑘≥𝐽+ of complex numbers such that‖‖𝑓_𝑘_𝑘≥𝐽⁺∣ℓ^{𝑠𝑞,𝐽}₊‖‖=_𝑘≥𝐽+2^𝑘𝑠𝑞||𝑓_𝑘||^𝑞^1/𝑞<∞,(2.5) with the obvious modification if 𝑞=∞. We recall that for any 0<𝜃≤1 and any 𝐽∈ℤ_𝑘≥𝐽+||𝑓_𝑘||^𝜃≤_𝑘≥𝐽+||𝑓_𝑘||^𝜃,(2.6)(𝑦+𝑧)^𝑑≤max1,2^𝑑−1𝑦^𝑑+𝑧^𝑑,𝑦,𝑧≥0,𝑑>0.(2.7)

Let 𝑓 be an arbitrary function on ℝ^𝑛 and 𝑥, ℎ∈ℝ^𝑛. ThenΔ_ℎ𝑓(𝑥)=𝑓(𝑥+ℎ)−𝑓(𝑥),Δ^ℎ𝑀+1𝑓(𝑥)=Δ_ℎΔ^𝑀ℎ𝑓(𝑥),𝑀∈ℕ.(2.8) These are the well-known differences of functions which play an important role in the theory of function spaces. Using mathematical induction one can show the explicit formulaΔ^𝑀ℎ𝑓(𝑥)=^𝑀_𝑗=0(−1)^𝑗𝐶^𝑀𝑗𝑓(𝑥+(𝑀−𝑗)ℎ),(2.9) where 𝐶^𝑀𝑗 are the binomial coefficients.

Recall that 𝜂_𝑗,𝑁(𝑥)=2^𝑗𝑛(1+2^𝑗|𝑥|)^−𝑁, for any 𝑥∈ℝ^𝑛, 𝑗∈ℕ₀ and 𝑁>0. By 𝑐 we denote generic positive constants, which may have different values at different occurrences.

2.2. The 𝐵^{𝑠,𝜏𝑝,𝑞} Spaces and 𝐹^{𝑠,𝜏𝑝,𝑞} Spaces

In this subsection we present the Fourier analytical definition of 𝐵^{𝑠,𝜏𝑝,𝑞} spaces, 𝐹^{𝑠,𝜏𝑝,𝑞} spaces and recall their basic properties. We first need the concept of a smooth dyadic resolution of unity.

Definition 2.1. Let Ψ be a function in 𝒮(ℝ^𝑛) satisfying Ψ(𝑥)=1 for |𝑥|≤1 and Ψ(𝑥)=0 for |𝑥|≥3/2. We put 𝜑₀(𝑥)=Ψ(𝑥), 𝜑₁(𝑥)=Ψ(𝑥/2)−Ψ(𝑥) and 𝜑_𝑗(𝑥)=𝜑₁2^−𝑗+1𝑥for𝑗=2,3,….(2.10) Then we have supp𝜑_𝑗⊂{𝑥∈ℝ^𝑛∶2^𝑗−1≤|𝑥|≤3⋅2^𝑗−1}, 𝜑_𝑗(𝑥)=1 for 3⋅2^𝑗−2≤|𝑥|≤2^𝑗 and Ψ(𝑥)+∑_𝑗≥1𝜑_𝑗(𝑥)=1 for all 𝑥∈ℝ^𝑛. The system of functions {𝜑_𝑗} is called a smooth dyadic resolution of unity. We define the convolution operators Δ_𝑗 by the following: Δ_𝑗𝑓=ℱ⁻¹𝜑_𝑗∗𝑓,𝑗∈ℕ,Δ₀𝑓=ℱ⁻¹Ψ∗𝑓,𝑓∈𝒮^(ℝ^𝑛).(2.11) Thus we obtain the Littlewood-Paley decomposition 𝑓=_𝑗≥0Δ_𝑗𝑓,(2.12) of all 𝑓∈𝒮^(ℝ^𝑛)(convergencein𝒮^(ℝ^𝑛)).

The 𝐵^{𝑠,𝜏𝑝,𝑞} spaces and 𝐹^{𝑠,𝜏𝑝,𝑞} spaces are defined in the following way.

Definition 2.2. (i) Let 𝑠∈ℝ, 𝜏∈[0,∞) and 0<𝑝, 𝑞≤∞. The space 𝐵^{𝑠,𝜏𝑝,𝑞} is the collection of all 𝑓∈𝒮^(ℝ^𝑛) such that ‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖=sup_𝐵𝐽1||𝐵_𝐽||^𝜏⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞‖‖Δ_𝑗𝑓∣𝐿^𝑝𝐵_𝐽‖‖^𝑞⎞⎟⎟⎠^1/𝑞<∞,(2.13) where the supremum is taken over all 𝐽∈ℤ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽.
(ii) Let 𝑠∈ℝ, 𝜏∈[0,∞), 0<𝑝<∞ and 0<𝑞≤∞. The space 𝐹^{𝑠,𝜏𝑝,𝑞} is the collection of all 𝑓∈𝒮^(ℝ^𝑛) such that ‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖=sup_𝐵𝐽1||𝐵_𝐽||^𝜏‖‖‖‖‖⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞||Δ_𝑗𝑓||^𝑞⎞⎟⎟⎠^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖‖<∞,(2.14) where the supremum is taken over all 𝐽∈ℤ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽.

Remark 2.3. The spaces  𝐵^{𝑠,𝜏𝑝,𝑞}  and  𝐹^{𝑠,𝜏𝑝,𝑞}  are independent of the particular choice of the smooth dyadic resolution of unity  {𝜑_𝑗}  appearing in their definitions. They are quasi-Banach spaces (Banach spaces if  𝑝≥1, 𝑞≥1).  In particular, 𝐵^{𝑠,0𝑝,𝑞}=𝐵^{𝑠𝑝,𝑞},𝐹^{𝑠,0𝑝,𝑞}=𝐹^{𝑠𝑝,𝑞},(2.15) where 𝐵^{𝑠𝑝,𝑞} and 𝐹^{𝑠𝑝,𝑞} are the Besov spaces and Triebel-Lizorkin spaces respectively. If we replace the balls 𝐵_𝐽 by dyadic cubes 𝑃 (with side length 2^−𝐽) we obtain equivalent norms.

The full treatment of both scales of spaces can be found in [9]. Let Δ_𝑗𝑓 (𝑗∈ℕ₀) be the functions introduced in Definition 2.1. For any 𝑎>0, any 𝑥∈ℝ^𝑛 and any 𝐽∈ℤ we denote (Peetre’s maximal functions)Δ^{∗,𝑎𝑗,𝐽}𝑓(𝑥)=sup_{𝑦∈𝐵𝐽}||Δ_𝑗𝑓(𝑦)||1+2^𝑗||𝑥−𝑦||^𝑎𝑗∈ℕ₀,𝑓∈𝒮^(ℝ^𝑛).(2.16)

We now present a fundamental characterization of 𝐵^{𝑠,𝜏𝑝,𝑞} spaces and 𝐹^{𝑠,𝜏𝑝,𝑞} spaces.

Theorem 2.4. Let 𝑠∈ℝ, 𝜏∈[0,∞), 0<𝑝, 𝑞≤∞ and 𝑎>𝑛/𝑝. Then ‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖^∗=sup_𝐵𝐽1||𝐵_𝐽||^𝜏⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞‖‖Δ^{∗,𝑎𝑗,𝐽}𝑓∣𝐿^𝑝𝐵_𝐽‖‖^𝑞⎞⎟⎟⎠^1/𝑞,(2.17) is an equivalent quasinorm in 𝐵^{𝑠,𝜏𝑝,𝑞}.

Theorem 2.5. Let 𝑠∈ℝ, 𝜏∈[0,∞), 0<𝑝<∞, 0<𝑞≤∞ and 𝑎>𝑛/min(𝑝,𝑞). Then ‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖^∗=sup_𝐵𝐽(1/|𝐵_𝐽|^𝜏)‖(∑_𝑗≥𝐽+2^𝑗𝑠𝑞|Δ^{∗,𝑎𝑗,𝐽}𝑓|^𝑞)^1/𝑞∣𝐿^𝑝(𝐵_𝐽)‖, is an equivalent quasinorm in 𝐹^{𝑠,𝜏𝑝,𝑞}.

Remark 2.6. Theorem 2.4 for  0<𝑝, 𝑞<∞  is given in [8, Theorem 4.5]. For Theorem 2.5 see [12, Theorem 1.1]. In addition if  𝑎>𝑛max(1/𝑝,𝜏), then in Theorem 2.4  Δ^{∗,𝑎𝑗,𝐽}𝑓  can be replaced by  Δ^𝑗∗,𝑎𝑓, where Δ^𝑗∗,𝑎𝑓(𝑥)=sup_{𝑦∈ℝ𝑛}||Δ_𝑗𝑓(𝑦)||1+2^𝑗||𝑥−𝑦||^𝑎𝑗∈ℕ₀,𝑓∈𝒮^(ℝ^𝑛).(2.18)

3. Some Technical Lemmas

To prove our results, we need some technical lemmas. The following lemma for Δ_𝑗𝑓, in place of Δ^𝑗∗,𝑎𝑓, is given in [14, pages 87–89] (for the 𝐵^{𝑠,𝜏𝑝,𝑞} spaces and 1≤𝑝<∞). Further results, can be found in [12, Lemma 2.4].

Lemma 3.1. Let Δ_𝑗𝑓 be as in Definition 2.1 and let 𝑠∈ℝ,  𝑎>0,  𝜏∈[0,∞) and 0<𝑝, 𝑞≤∞(0<𝑝<∞ for the space 𝐹^{𝑠,𝜏𝑝,𝑞}). Then there is a constant 𝑐>0, independent of 𝑗, such that for any 𝑓∈𝒮^(ℝ^𝑛)‖‖Δ^𝑗∗,𝑎𝑓‖‖_∞≤𝑐2^{𝑗(𝑛/𝑝−𝑠−𝑛𝜏)}‖‖𝑓∣𝐴^{𝑠,𝜏𝑝,𝑞}‖‖,𝑗∈ℕ₀.(3.1) Here one uses 𝐴^{𝑠,𝜏𝑝,𝑞} to denote either 𝐵^{𝑠,𝜏𝑝,𝑞} or 𝐹^{𝑠,𝜏𝑝,𝑞}.

Proof. Let 𝜓,𝜓₀∈𝒮(ℝ^𝑛) be two functions such that ℱ𝜓=1 and ℱ𝜓₀=1 on supp𝜑₁ and supp𝜑₀ respectively. Then ||Δ_𝑗𝑓(𝑦)||=||𝜓_𝑗∗Δ_𝑗𝑓(𝑦)||,𝑦∈ℝ^𝑛,(3.2) with 𝜓_𝑗=2^{(𝑗−1)𝑛}𝜓(2^𝑗−1) if 𝑗∈ℕ. Since 𝜑∈𝒮(ℝ^𝑛), the right-hand side is bounded by 𝑐𝜂_{𝑗−1,𝑁}∗|Δ_𝑗𝑓|(𝑦), for any 𝑁>0. Hence we get for all 𝑓∈𝒮^(ℝ^𝑛) and any 𝑥∈ℝ^𝑛Δ^𝑗∗,𝑎𝑓(𝑥)≤𝑐sup_{𝑦∈ℝ𝑛}𝜂_{𝑗−1,𝑁}∗||Δ_𝑗𝑓||(𝑦).(3.3) Using the same method given in [9,Proposition 2.6] we obtain for any 𝑦∈ℝ^𝑛𝜂_{𝑗−1,𝑁}∗||Δ_𝑗𝑓||(𝑦)≤𝑐2^{𝑗(𝑛/𝑝−𝑠−𝑛𝜏)}‖‖𝑓∣𝐴^{𝑠,𝜏𝑝,𝑞}‖‖.(3.4) The proof is completed.

Lemma 3.2. Let 𝑀∈ℕ, 𝐽∈ℤ⧵ℕ, 𝐴>0, 𝜏∈[0,∞) and 1≤𝑝≤∞. Then there is a constant 𝑐>0, independent of 𝐽 and 𝐴, such that ‖‖‖_|ℎ|≤𝐴||Δ^𝑀ℎ𝑓(⋅)||𝑑ℎ∣𝐿^𝑝𝐵_𝐽‖‖‖≤𝑐𝐴^𝑛||𝐵_𝐽||^𝜏‖‖𝑓∣𝐿^𝑝𝜏‖‖,(3.5) for any ball 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽 and any function 𝑓 such that ‖𝑓∣𝐿^𝑝𝜏‖<∞.

Proof. Since 1≤𝑝≤∞, the left-hand side is bounded by _|ℎ|≤𝐴‖‖Δ^𝑀ℎ𝑓∣𝐿^𝑝𝐵_𝐽‖‖𝑑ℎ.(3.6) From the definition of Δ^𝑀ℎ𝑓 we have ||Δ^𝑀ℎ𝑓(𝑥)||≤^𝑀_𝑚₌₀𝐶^𝑀𝑚||𝑓(𝑥+(𝑀−𝑚)ℎ)||.(3.7) Take the 𝐿^𝑝(𝐵_𝐽)-norm to estimate (3.6) form above by ^𝑀_𝑚=0𝐶^𝑀𝑚_|ℎ|≤𝐴‖‖𝑓∣𝐿^𝑝𝐵^𝐽‖‖𝑑ℎ,(3.8) where if 𝑥₀ the centre of 𝐵_𝐽 then 𝑥₀+(𝑀−𝑚)ℎ is the centre of 𝐵_𝐽. Using the fact that |𝐵_𝐽|=|𝐵_𝐽| to estimate (3.8) from above by 𝑐𝐴^𝑛|𝐵_𝐽|^𝜏‖𝑓∣𝐿^𝑝𝜏‖.
The lemma is proved.

Remark 3.3. Let 𝑀, 𝐴, 𝜏, and 𝑝 be as in Lemma 3.2. Let 𝐽∈ℕ₀. By the embedding 𝐿^𝑝(𝐵₀)↪𝐿^𝑝(𝐵_𝐽) there is a constant 𝑐>0, independent of 𝐽 and 𝐴, such that ‖‖‖_|ℎ|≤𝐴||Δ^𝑀ℎ𝑓(⋅)||𝑑ℎ∣𝐿^𝑝𝐵_𝐽‖‖‖≤𝑐𝐴^𝑛‖‖𝑓∣𝐿^𝑝𝜏‖‖,(3.9) for any ball 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽 and any function 𝑓 such that ‖𝑓∣𝐿^𝑝𝜏‖<∞.
For 𝑠>0, 𝑀∈ℕ, 𝜏∈[0,∞), 1≤𝑝<∞ and 0<𝑞≤∞, we set ‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀=‖‖𝑓∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖+sup_𝐵𝐽1||𝐵_𝐽||^𝜏‖‖‖‖₂^+−𝐽+1₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}||Δ^𝑀ℎ𝑓(⋅)||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖.(3.10)

Here the supremum is taken over all 𝐽∈ℤ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽.

Lemma 3.4. Let 𝑠>0, 𝑀∈ℕ, 𝐽∈ℤ, 𝜏∈[0,∞), 1≤𝑝<∞ and 0<𝑞≤∞. Then there is a constant 𝑐>0, independent of 𝐽, such that ‖‖‖‖‖⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞_|𝑣|≤1||Δ^𝑀2_−𝑗𝑣𝑓(⋅)||𝑑𝑣^𝑞⎞⎟⎟⎠^1/𝑞‖‖‖‖‖⎛⎜⎜⎝∣𝐿^𝑝𝐵_𝐽‖‖‖‖‖≤𝑐||𝐵_𝐽||^𝜏‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀,(3.11)_𝑗≥𝐽+2^𝑗𝑠𝑞_|𝑣|>1||Δ^𝑀2_−𝑗𝑣𝑓(⋅)𝜔(𝑣)||𝑑𝑣^𝑞⎞⎟⎟⎠^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖‖≤𝑐||𝐵_𝐽||^𝜏‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀,(3.12) for any ball 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽, any 𝜔∈𝒮(ℝ^𝑛) and any function 𝑓 such that ‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖_𝑀<∞.

Proof. Let 𝑃 be a dyadic cube with side length 2^−𝐽. This result, for 𝑃 in place of 𝐵_𝐽, is already known, see [9, Lemmas 4.3, 4.4]. By simple modifications of their arguments we will give another proof of (3.12). The proof is given only when 0<𝑞<∞. The case 𝑞=∞ is similar.
Before proving this result we note that for any 𝑥∈ℝ^𝑛 and any 𝑖∈ℤ_𝑣≥𝑖2^{(𝑠+𝑛)𝑣𝑞}_{|ℎ|≤2−𝑣}||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^𝑞^1/𝑞≤𝑐^2−𝑖+1₀𝑡^{−(𝑠+𝑛)𝑞}_|ℎ|≤𝑡||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^𝑞𝑑𝑡𝑡^1/𝑞.(3.13) Here we will prove that the left-hand side of (3.12) is bounded by ‖‖𝑓∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖+sup_𝐵𝐽1||𝐵_𝐽||^𝜏‖‖‖‖₂^+−𝐽₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}||Δ^𝑀ℎ𝑓(⋅)||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖=‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀.(3.14) Obviously, ‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖^𝑀≤‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖_𝑀. We write 2^𝑗𝑠_|𝑣|>1||Δ^𝑀2_−𝑗𝑣𝑓(𝑥)𝜔(𝑣)||𝑑𝑣=^∞_𝑘=02^𝑗𝑠_{2𝑘<|𝑣|≤2𝑘+1}≤𝑐_∞||Δ^𝑀2_−𝑗𝑣𝑓(𝑥)𝜔(𝑣)||𝑑𝑣_𝑘=02^{(𝑠+𝑛)𝑗−𝑁𝑘}_{2𝑘−𝑗<|ℎ|≤2𝑘−𝑗+1}||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ,(3.15) where 𝑁>0 is at our disposal and we have used the properties of the function 𝜔, ||𝜔(𝑥)||≤𝑐(1+|𝑥|)^−𝑁,(3.16) for any 𝑥∈ℝ^𝑛 and any 𝑁>0. Now the right-hand side of (3.15) in ℓ^{𝑠𝑞,𝐽}₊-norm is bounded by (with 𝜎=min(1,𝑞)) 𝑐⎛⎜⎜⎝^∞_𝑘=02^{−𝑁𝜎𝑘}⎛⎜⎜⎝_𝑗≥𝐽+2^{(𝑠+𝑛)𝑗𝑞}_{2𝑘−𝑗<|ℎ|≤2𝑘−𝑗+1}||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^𝑞⎞⎟⎟⎠^𝜎/𝑞⎞⎟⎟⎠^1/𝜎=⎛⎜⎜⎝^𝐽+−1_𝑘=0⋯+_𝑘≥𝐽+⋯⎞⎟⎟⎠^1/𝜎=𝐼_𝐽(𝑥)+𝐼𝐼_𝐽(𝑥)^1/𝜎≤2^1/𝜎−1𝐼^𝐽(𝑥)^1/𝜎+𝐼𝐼_𝐽(𝑥)^1/𝜎,(3.17) by (2.7). Here we put ∑^{𝐽+−1𝑘=0}⋯=0 if 𝐽⁺=0. Take the 𝐿^𝑝(𝐵_𝐽)-norm we obtain that the left-hand side of (3.12) is bounded by 2^1/𝜎−1‖‖𝐼_𝐽^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖+2^1/𝜎−1‖‖𝐼𝐼_𝐽^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖.(3.18) Let us estimate (𝐼_𝐽)^1/𝜎 in 𝐿^𝑝(𝐵_𝐽)-norm. After a change of variable 𝑗−𝑘−1=𝑣, we get for any 𝑥∈ℝ^𝑛 (here 𝐽⁺=𝐽 and 𝑘<𝐽) 𝐼_𝐽(𝑥)≤𝑐^𝐽+−1_𝑘=02^{(𝑠+𝑛−𝑁)𝜎𝑘}_{𝑣≥𝐽+−𝑘−1}2^{(𝑠+𝑛)𝑣𝑞}_{|ℎ|≤2−𝑣}||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^𝑞^𝜎/𝑞=^𝐽+−2_𝑘=0⋯+^𝐽+−1_𝑘=_𝐽+−1=𝑀₁⋯(𝑥)+𝑀₂(𝑥).(3.19) Here we put ∑^{𝐽+−2𝑘=0}⋯=0 if 𝐽⁺≤1. We have 𝑀₁(𝑥)≤𝑐^𝐽+−2_𝑘=02^{(𝑠+𝑛−𝑁)𝜎𝑘}^{2−𝐽+𝑘+2}₀𝑡^{−(𝑠+𝑛)𝑞}_|ℎ|≤𝑡||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^𝑞𝑑𝑡𝑡^𝜎/𝑞.(3.20) Since 𝐿^𝑝/𝜎(𝐵_𝐽) is a normed spaces and 𝐵_𝐽⊂𝐵_{𝐽−𝑘−2}, the right-hand side in 𝐿^𝑝/𝜎(𝐵_𝐽)-norm can be estimated from above by 𝑐^𝐽+−2_𝑘=02^{(𝑠+𝑛−𝑁)𝜎𝑘}‖‖‖‖₂^{−𝐽+𝑘+2}₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}||Δ^𝑀ℎ𝑓(⋅)||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿^𝑝𝐵_{𝐽−𝑘−2}‖‖‖‖^𝜎≤𝑐||𝐵_𝐽||^{𝐽𝜏𝜎+−2}_𝑘=02^{(𝑠+𝑛−𝑁+𝑛𝜏)𝜎𝑘}‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀^𝜎.(3.21) We choose 𝑁>2(𝑠+𝑛)+𝜏𝑛. This yields that the last expression is bounded by 𝑐||𝐵_𝐽||^𝜏𝜎‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀^𝜎,(3.22) where 𝑐>0 is independent of 𝐽. Now (𝑀₂)^1/𝜎 in 𝐿^𝑝(𝐵_𝐽)-norm is bounded by 𝑐2^{(𝑠+𝑛−𝑁)𝐽+}‖‖‖‖_𝑣≥02^{(𝑠+𝑛)𝑣𝑞}_{|ℎ|≤2−𝑣}||Δ^𝑀ℎ𝑓(⋅)||𝑑ℎ^𝑞^1/𝑞≤𝑐2_{(𝑠+𝑛−𝑁)𝐽+}∣𝐿^𝑝𝐵_𝐽‖‖‖‖‖‖‖‖_𝑣≥12^{(𝑠+𝑛)𝑣𝑞}_{|ℎ|≤2−𝑣}||Δ^𝑀ℎ𝑓(⋅)||𝑑ℎ^𝑞^1/𝑞+𝑐2_{(𝑠+𝑛−𝑁)𝐽+}∣𝐿^𝑝𝐵_𝐽‖‖‖‖‖‖‖_|ℎ|≤1||Δ^𝑀ℎ𝑓(⋅)||𝑑ℎ∣𝐿^𝑝𝐵_𝐽‖‖‖,(3.23) where we have used (2.7). Using the embedding 𝐿^𝑝(𝐵₀)↪𝐿^𝑝(𝐵_𝐽) and Remark 3.3, we obtain ‖‖𝑀₂∣𝐿^𝑝/𝜎𝐵_𝐽‖‖=‖‖𝑀₂^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖^𝜎≤𝑐2^{(𝑠+𝑛−𝑁)𝜎𝐽+}‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀^𝜎=𝑐2^{(𝑠+𝑛−𝑁+𝑛𝜏)𝜎𝐽+}||𝐵_𝐽||^𝜏𝜎‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀^𝜎≤𝑐||𝐵_𝐽||^𝜏𝜎‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀^𝜎,(3.24) because of 𝑁>𝑠+𝑛+𝑛𝜏. Therefore, ‖‖𝐼_𝐽^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖^𝜎≤=‖‖‖‖𝐼𝑀_𝐽1∣𝐿^𝑝/𝜎𝐵_𝐽‖‖≤𝑐||𝐵_𝐽∣𝐿^𝑝/𝜎𝐵_𝐽‖‖+‖‖𝑀₂∣𝐿^𝑝/𝜎𝐵_𝐽‖‖||^𝜏𝜎‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀^𝜎.(3.25) Now let us estimate (𝐼𝐼_𝐽)^1/𝜎 in 𝐿^𝑝(𝐵_𝐽)-norm. We write 𝐼𝐼_𝐽(⋅)=𝑐_𝑘≥𝐽+2^{−𝑁𝜎𝑘}⎛⎜⎜⎝^𝑘+𝐽++1_𝑗_=𝐽+⋯+^∞_𝑗=𝑘_+𝐽++2⋯⎞⎟⎟⎠^𝜎/𝑞≤𝑐_𝑘≥𝐽+2^{−𝑁𝜎𝑘}⎛⎜⎜⎝⎛⎜⎜⎝^𝑘+𝐽++1_𝑗_=𝐽+⋯⎞⎟⎟⎠^𝜎/𝑞+⎛⎜⎜⎝^∞_𝑗=𝑘_+𝐽++2⋯⎞⎟⎟⎠^𝜎/𝑞=𝑐_𝑘≥𝐽+⎞⎟⎟⎠2^{−𝑁𝜎𝑘}𝑆¹_𝑘(⋅)+𝑆²_𝑘(⋅).(3.26) After a change of variable 𝑗−𝑘−1=𝑣, we get 𝑆¹_𝑘(𝑥)^1/𝜎≤𝑐2^{(𝑠+𝑛)𝑘}⎛⎜⎜⎝^𝐽+_𝑣=𝐽_+−𝑘−12^{(𝑠+𝑛)𝑣𝑞}_{|ℎ|≤2−𝑣}||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^𝑞⎞⎟⎟⎠^1/𝑞≤𝑐2^{(𝑠+𝑛)𝑘}^{2+𝑘−𝐽+2}₂^+−𝐽𝑡^{−(𝑠+𝑛)𝑞}_|ℎ|≤𝑡||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^𝑞𝑑𝑡𝑡^1/𝑞≤𝑐2^{(𝑠+𝑛)𝑘}_{|ℎ|≤2+𝑘−𝐽+2}||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^{2+𝑘−𝐽+2}_2+−𝐽𝑡^{−(𝑠+𝑛)𝑞}𝑑𝑡𝑡^1/𝑞≤𝑐2^{(𝑠+𝑛)𝐽++(𝑠+𝑛)𝑘}_{|ℎ|≤2+𝑘−𝐽+2}||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ.(3.27) Therefore there exists a constant 𝑐>0 independent of 𝐽 and 𝑘 such that ‖‖𝑆¹_𝑘^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖≤𝑐2^{(𝑠+𝑛)𝐽++(𝑠+𝑛)𝑘}‖‖‖_{|ℎ|≤2+𝑘−𝐽+2}≤𝑐2_{(𝑠+𝜏𝑛)𝐽++(𝑠+2𝑛)𝑘}Δ^𝑀ℎ𝑓(⋅)𝑑ℎ∣𝐿^𝑝𝐵_𝐽‖‖‖≤𝑐2_{(2𝑠+2𝑛+𝜏𝑛)𝑘}||𝐵_𝐽||^𝜏‖‖𝑓∣𝐿^𝑝𝜏‖‖||𝐵_𝐽||^𝜏‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀,(3.28) by Lemma 3.2 (combined with Remark 3.3) and the fact that 𝑘≥𝐽⁺. Now let us estimate (𝑆²_𝑘)^1/𝜎in 𝐿^𝑝(𝐵_𝐽)-norm. We have 𝑆²_𝑘(𝑥)^1/𝜎≤𝑐2^{(𝑛+𝑠)𝑘}^∞_𝑣=𝐽++12^{(𝑠+𝑛)𝑣𝑞}_{|ℎ|≤2−𝑣}||Δ^𝑀ℎ𝑓(𝑥)||𝑑ℎ^𝑞^1/𝑞≤𝑐2^{(𝑛+𝑠)𝑘}^2+−𝐽₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}||Δ^𝑀ℎ𝑓(𝑥)||^𝑞𝑑𝑡𝑡^1/𝑞.(3.29) Therefore, ‖‖𝑆²_𝑘^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖≤𝑐2^{(𝑛+𝑠)𝑘}‖‖‖‖₂^+−𝐽₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}||Δ^𝑀ℎ𝑓(⋅)||^𝑞𝑑𝑡𝑡^1/𝑞≤𝑐2_{(𝑛+𝑠)𝑘}∣𝐿^𝑝𝐵_𝐽‖‖‖‖||𝐵_𝐽||^𝜏‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀.(3.30) Consequently, ‖‖𝐼𝐼_𝐽^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖^𝜎=≤‖‖𝐼𝐼_{𝐽𝑘≥𝐽+}∣𝐿^𝑝/𝜎𝐵_𝐽‖‖≤𝑐2^{−𝑁𝜎𝑘}_𝑘≥𝐽+‖‖𝑆^1𝑘^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖^𝜎+‖‖𝑆²_𝑘^1/𝜎∣𝐿^𝑝𝐵_𝐽‖‖^𝜎2^{(2𝑠+2𝑛+𝜏𝑛−𝑁)𝜎𝑘}||𝐵_𝐽||^𝜏𝜎‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀^𝜎≤𝑐||𝐵_𝐽||^𝜏𝜎‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^𝑀^𝜎,(3.31) since 𝑁>2(𝑠+𝑛)+𝜏𝑛. This finishes the proof of Lemma 3.4.

For 𝑠>0, 𝑀∈ℕ, 𝜏∈[0,∞), 1≤𝑝≤∞, and 0<𝑞≤∞, we set‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀=‖‖𝑓∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖+sup_𝐵𝐽1||𝐵_𝐽||^𝜏^2+−𝐽+1₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}‖‖Δ^𝑀ℎ𝑓∣𝐿^𝑝𝐵_𝐽‖‖^𝑞𝑑𝑡𝑡^1/𝑞.(3.32) Here the supremum is taken over all 𝐽∈ℤ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽. Similar arguments yield.

Lemma 3.5. Let 𝑠>0, 𝑀∈ℕ, 𝐽∈ℤ, 𝜏∈[0,∞), 1≤𝑝≤∞ and 0<𝑞≤∞. Then there is a constant 𝑐>0, independent of 𝐽, such that ⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞_|𝑣|≤1‖‖Δ^𝑀2_−𝑗𝑣𝑓∣𝐿^𝑝𝐵_𝐽‖‖𝑑𝑣^𝑞⎞⎟⎟⎠^1/𝑞⎛⎜⎜⎝_𝑗≥𝐽+≤𝑐||𝐵_𝐽||^𝜏‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀,2^𝑗𝑠𝑞_|𝑣|>1‖‖Δ^𝑀2_−𝑗𝑣𝑓∣𝐿^𝑝𝐵_𝐽‖‖||𝜔(𝑣)||𝑑𝑣^𝑞⎞⎟⎟⎠^1/𝑞≤𝑐||𝐵_𝐽||^𝜏‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀,(3.33) for any ball 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽, any 𝜔∈𝒮(ℝ^𝑛) and any function 𝑓 such that ‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖_𝑀<∞.

Now we recall the following lemma which is useful for us.

Lemma 3.6. Let 0<𝑎<1, 𝐽∈ℤ and 0<𝑞≤∞. Let {𝜀_𝑘} be a sequences of positive real numbers, such that ‖‖𝜀_𝑘_𝑘≥𝐽⁺∣ℓ^0𝑞,𝐽₊‖‖=𝐼<∞.(3.34) The sequences {𝛿_𝑘∶𝛿_𝑘=∑^{𝑘𝑗=𝐽}₊𝑎^𝑘−𝑗𝜀_𝑗}_𝑘≥𝐽⁺𝑎𝑛𝑑{𝜂_𝑘∶𝜂_𝑘=∑^∞𝑗=𝑘𝑎^𝑗−𝑘𝜀_𝑗}_𝑘≥𝐽⁺, are in ℓ^0𝑞,𝐽₊ with ‖‖𝛿_𝑘_𝑘≥𝐽⁺∣ℓ^0𝑞,𝐽₊‖‖+‖‖𝜂_𝑘_𝑘≥𝐽⁺∣ℓ^0𝑞,𝐽₊‖‖≤𝑐𝐼,(3.35)𝑐depends only on 𝑎 and 𝑞.

4. Characterizations with Differences

We are able to state the main results of this paper.

Theorem 4.1. Let 1≤𝑝<∞, 0<𝑞≤∞, 𝜏∈[0,∞)  and  𝑀∈ℕ. Assume 𝑛min(𝑝,𝑞)<𝑠<𝑀,0≤𝜏<1𝑝(4.1) or 𝑛min(𝑝,𝑞)<𝑠<𝑀−𝑛𝜏+𝑛𝑝,𝜏≥1𝑝.(4.2) Then ‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖_𝑀 is an equivalent quasinorm in 𝐹^{𝑠,𝜏𝑝,𝑞}.

Theorem 4.2. Let 1≤𝑝≤∞, 0<𝑞≤∞, 𝜏∈[0,∞) and 𝑀∈ℕ. Assume 0<𝑠<𝑀,0≤𝜏<1𝑝(4.3) or 0<𝑠<𝑀−𝑛𝜏+𝑛𝑝,𝜏≥1𝑝.(4.4) Then ‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖_𝑀 is an equivalent quasinorm in 𝐵^{𝑠,𝜏𝑝,𝑞}.

Remark 4.3. Theorems 4.1 and 4.2 for  0≤𝜏<𝑠/𝑛+1/𝑝  and  0≤𝜏<𝑠/𝑛+1/𝑝−max(1/min(𝑝,𝑞)−1,0),  respectively, are given in [9] Theorems 4.6 and 4.7, respectively.

Proof of Theorem 4.1. Let 𝐵_𝐽 be any ball centered at 𝑥₀∈ℝ^𝑛 and of radius 2^−𝐽, 𝐽∈ℤ. We will do the proof in three steps. Step 1. We have with 𝑠>0, _𝑗≥0||Δ_𝑗𝑓||=≤𝑐_𝑗≥02^−𝑗𝑠_𝑗≥02^𝑗𝑠||Δ_𝑗𝑓||≤𝑐sup_𝑗∈ℕ02^𝑗𝑠||Δ_𝑗𝑓||2^𝑗𝑠𝑞||Δ_𝑗𝑓||^𝑞^1/𝑞.(4.5) Let 𝑓∈𝐹^{𝑠,𝜏𝑝,𝑞}. Then, ‖‖𝑓∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖≤‖‖‖‖_𝑗≥0||Δ_𝑗𝑓||∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖‖‖≤𝑐‖‖‖‖_𝑗≥02^𝑗𝑠𝑞||Δ_𝑗𝑓||^𝑞^1/𝑞≤𝑐‖‖𝑓∣𝐹_{𝑠,𝜏𝑝,𝑞}∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖‖‖‖‖.(4.6)Step 2. For any 𝑥∈𝐵_𝐽 we put 𝐻_𝐽(𝑥)=^2+−𝐽+1₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}||Δ^𝑀ℎ𝑓(𝑥)||^𝑞𝑑𝑡𝑡.(4.7) After a change of variable 𝑡=2^−𝑦, we get 𝐻_𝐽(𝑥)=ln2^𝐽+∞₊₋₁2^𝑦𝑠𝑞sup_{||ℎ||≤2−𝑦}||Δ^𝑀ℎ𝑓(𝑥)||^𝑞𝑑𝑦.(4.8) Then 𝐻_𝐽(𝑥)≤𝑐_𝑘≥𝐽+2^𝑘𝑠𝑞sup_{||ℎ||≤2−𝑘+1}||Δ^𝑀ℎ𝑓(𝑥)||^𝑞.(4.9)Let 𝜓,𝜓₀∈𝒮(ℝ^𝑛) be two functions such that ℱ𝜓=1 and ℱ𝜓₀=1 on supp𝜑₁ and suppΨ respectively. Using the mean value theorem we obtain for any 𝑥∈𝐵_𝐽, 𝑗∈ℕ₀, and |ℎ|≤2^−𝑘+1||Δ^1ℎΔ_𝑗𝑓(𝑥)||=≤||2Δ^1ℎ_−𝑘𝜓_𝑗∗Δ_𝑗𝑓(𝑥)||sup_{||||𝑥−𝑦≤𝑐2−𝑘|}_𝛼|=1||𝐷^𝛼𝜓_𝑗∗Δ_𝑗𝑓(𝑦)||,(4.10) with some positive constant 𝑐, independent of 𝑗 and 𝑘, and 𝜓_𝑗(⋅)=2^{(𝑗−1)𝑛}𝜓(2^𝑗−1⋅) for 𝑗=1,2,…. By induction on 𝑀, we show that ||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤2^−𝑘𝑀sup_{||||𝑥−𝑦≤𝑐2−𝑘|𝛼}_|=𝑀||𝐷^𝛼𝜓_𝑗∗Δ_𝑗𝑓(𝑦)||.(4.11) We see that if |𝛼|=𝑀 and 𝑎>0||𝐷^𝛼𝜓_𝑗∗Δ_𝑗𝑓(𝑦)||=2^{(𝑗−1)𝑛}||||_ℝ𝑛≤2_{(𝑗−1)(𝑛+𝑀)}𝐷^𝛼𝜓2^𝑗−1(𝑦−𝑧)Δ_𝑗𝑓(𝑧)𝑑𝑧||||_ℝ𝑛||(𝐷^𝛼𝜓)2^𝑗−1(𝑦−𝑧)||||Δ_𝑗𝑓(𝑧)||𝑑𝑧.(4.12) Suppose that 0≤𝑗≤𝐽⁺−1. The right-hand side in (4.12) may be estimated as follows: 𝑐2≤𝑐^{𝑗(𝑛+𝑀)}Δ2^𝑗∗,𝑎_𝑗𝑀𝑓(𝑦)_ℝ𝑛||(𝐷^𝛼𝜓)2^𝑗−1(𝑦−𝑧)||1+2^𝑗||𝑦−𝑧||^𝑎𝑑𝑧Δ^𝑗∗,𝑎𝑓(𝑦).(4.13) Then we obtain for any 𝑥∈𝐵_𝐽, |ℎ|≤2^−𝑘+1 and any 𝑘≥𝐽⁺||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤≤𝑐𝑐22^{(𝑗−𝑘)𝑀}_{(𝑗−𝑘)𝑀}sup_{||||𝑥−𝑦≤𝑐2−𝑘}Δ^𝑗∗,𝑎𝑓(𝑦)1+2^𝑗−𝑘^𝑎sup_{||||𝑥−𝑦≤𝑐2−𝑘}Δ^𝑗∗,𝑎𝑓(𝑦)1+2^𝑗||𝑥−𝑦||^𝑎≤𝑐2^{(𝑗−𝑘)𝑀}Δ^𝑗∗,𝑎𝑓(𝑥),(4.14) if 0≤𝑗≤𝐽⁺−1.
Suppose now that 𝐽⁺≤𝑗≤𝑘. By our assumption on 𝑥 and 𝑘 we have ||𝑦−𝑥₀||≤||𝑥−𝑥₀||+||𝑥−𝑦||<2^−𝐽+𝑐2^−𝑘≤(𝑐+1)2^−𝐽,(4.15) which implies that 𝑦 is located in some ball 𝐵_𝐽, where 𝐵_𝐽={𝑦∈ℝ^𝑛∶|𝑦−𝑥₀|<(𝑐+1)2^−𝐽}. Writing the integral in (4.12) as follows 𝐵_𝐽−1⋯𝑑𝑧+_𝑖≥0𝐵_{𝐽−𝑖−2⧵}𝐵_{𝐽−𝑖−1}⋯𝑑𝑧=𝐼_𝑗,𝐽(𝑦)+_𝑖≥0𝐼𝐼_{𝑗,𝐽−𝑖}(𝑦).(4.16) We recall that Δ^{∗,𝑎𝑗,𝑙}𝑓(𝑦)=sup_𝑧∈𝐵_𝑙||Δ_𝑗𝑓(𝑧)||1+2^𝑗||𝑦−𝑧||^𝑎,(4.17) for any 𝑗∈ℕ₀, 𝑓∈𝒮^(ℝ^𝑛) and any 𝑙∈ℤ. We have 𝐼_𝑗,𝐽(𝑦)≤≤Δ𝑐2^{∗,𝑎𝑗,𝐽−1}_−𝑗𝑛𝑓(𝑦)𝐵_𝐽−1||(𝐷^𝛼𝜓)2^𝑗−1(𝑦−𝑧)||1+2^𝑗||𝑦−𝑧||^𝑎𝑑𝑧Δ^{∗,𝑎𝑗,𝐽−1}𝑓(𝑦)≤𝑐2^−𝑗𝑛Δ^{∗,𝑎𝑗,𝐽−2}𝑓(𝑦).(4.18) Let us estimate 𝐼𝐼_{𝑗,𝐽−𝑖}. Since 𝜓∈𝒮(ℝ^𝑛), we have ||𝐷^𝛼𝜓(𝑥)||≤𝑐(1+|𝑥|)^−2𝑁,(4.19) for any 𝑥∈ℝ^𝑛 and any 𝑁>0. Then for any 𝑁 large enough, 𝐼𝐼_{𝑗,𝐽−𝑖}(𝑦) does not exceed 𝑐Δ^{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑦)𝐵_{𝐽−𝑖−2⧵}𝐵_{𝐽−𝑖−1}≤𝑐2_−𝑖𝑁1+2^𝑗−1||𝑦−𝑧||^{−2𝑁+𝑎}𝑑𝑧≤𝑐2Δ^{∗,𝑎𝑗,𝐽−𝑖−2}_{−𝑖𝑁−𝑗𝑛}𝑓(𝑦)_ℝ𝑛1+2^𝑗−1||𝑦−𝑧||^−𝑁+𝑎𝑑𝑧Δ^{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑦),(4.20) where we have used 2^𝑗−1|𝑦−𝑧|>(𝑐+1)2^{𝑗−𝐽+𝑖−1}≥(𝑐+1)2^𝑖−1. Therefore, ||𝐷^𝛼𝜓_𝑗∗Δ_𝑗𝑓(𝑦)||≤𝑐2^𝑗𝑀_𝑖≥02^−𝑖𝑁Δ^{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑦).(4.21) Then we obtain for any 𝑥∈𝐵_𝐽 any |ℎ|≤2^−𝑘+1 and any 𝐽⁺≤𝑗≤𝑘||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤𝑐2^{(𝑗−𝑘)𝑀}_𝑖≥0≤𝑐2_{(𝑗−𝑘)𝑀}2^−𝑖𝑁sup_{||||𝑥−𝑦≤𝑐2−𝑘}Δ^{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑦)1+2^𝑗−𝑘^𝑎_𝑖≥02^−𝑖𝑁sup_{||||𝑥−𝑦≤𝑐2−𝑘}Δ^{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑦)1+2^𝑗||𝑥−𝑦||^𝑎.(4.22) Consequently, for any 𝐽⁺≤𝑗≤𝑘 there is a constant 𝑐>0 independent of 𝐽, 𝑗, and 𝑘 such that ||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤𝑐2^{(𝑗−𝑘)𝑀}_𝑖≥02^−𝑖𝑁Δ^{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑥).(4.23) Finally for 𝑗≥𝑘+1 we have for 𝑥∈𝐵_𝐽 and |ℎ|≤2^−𝑘+1||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤^𝑀_𝑚₌₀≤2_𝑀𝐶^𝑀𝑚||Δ_𝑗𝑓(𝑥+(𝑀−𝑚)ℎ)||≤2_𝑀sup_{||||𝑥−𝑦≤𝐶2−𝑘}||Δ_𝑗𝑓(𝑦)||sup_{||||𝑥−𝑦≤𝐶2−𝑘}||Δ_𝑗𝑓(𝑦)||1+2^𝑗||𝑥−𝑦||^𝑎1+2^𝑗||𝑥−𝑦||^𝑎.(4.24) We remark also that by our assumption on 𝑥 and 𝑘 we have ||𝑦−𝑥₀||≤||𝑥−𝑥₀||+||𝑥−𝑦||<2^−𝐽+𝐶2^−𝑘≤(𝐶+1)2^−𝐽,(4.25) and this implies that 𝑦 is located in some ball 𝐵_𝐽, where 𝐵_𝐽={𝑦∈ℝ^𝑛∶|𝑦−𝑥₀|<(𝐶+1)2^−𝐽}. Then, ||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤𝑐2^{(𝑗−𝑘)𝑎}Δ^{∗,𝑎𝑗,𝐽}𝑓(𝑥),(4.26) if 𝑗≥𝑘+1, where Δ^{∗,𝑎𝑗,𝐽}𝑓 is given in (4.17) (with 𝐵_𝐽 a ball centered at 𝑥₀ and of radius (𝐶+1)2^−𝐽).
We write, Δ^𝑀ℎ𝑓(𝑥)==_𝑗≥0Δ^𝑀ℎΔ_𝑗𝑓(𝑥)_𝐽+−1_𝑗=0⋯+^𝑘_𝑗_=𝐽+⋯+_{𝑗≥𝑘+1}⋯.(4.27) Here we put ∑^{𝐽+−1𝑗=0}⋯=0 if 𝐽⁺=0. Let us estimate each term in ℓ^{𝑠𝑞,𝐽}₊-norm. We have by (4.14) and Lemma 3.1^𝐽⁺⁻¹_𝑗=0||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤^𝐽+−1_𝑗=0≤𝑐2^{(𝑗−𝑘)𝑀}_𝐽+−1Δ^𝑗∗,𝑎𝑓(𝑥)_𝑗=0≤𝑐22^{(𝑗−𝑘)𝑀+𝑗(𝑛/𝑝−𝑠−𝑛𝜏)}_{𝐽−𝑘𝑀+−1}‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖_𝑗=0≤𝑐22^{𝑗(𝑀+𝑛/𝑝−𝑠−𝑛𝜏)}_{𝐽(𝑀+𝑛/𝑝−𝑠−𝑛𝜏)−𝑘𝑀}‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖,(4.28) where the last inequality can be obtained by our assumption on 𝑠 and 𝜏. The last expression in ℓ^{𝑠𝑞,𝐽}₊-norm does not exceed 𝑐2^{𝐽𝑛(1/𝑝−𝜏)}‖‖‖2_(𝑘−𝐽⁺_{)(𝑠−𝑀)}_𝑘≥𝐽+∣ℓ^0𝑞,𝐽₊‖‖‖‖‖𝑓∣𝐹_{𝑠,𝜏𝑝,𝑞}‖‖≤𝑐2^{𝐽𝑛(1/𝑝−𝜏)}‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖,(4.29) since 𝑠<𝑀. Therefore, 𝐻_𝐽(𝑥)^1/𝑞≤+𝑐2𝑐⎛⎜⎜⎝^{𝐽𝑛(1/𝑝−𝜏)}_𝑘≥𝐽+‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖(4.30)⎛⎜⎜⎝_{𝑘𝑖≥0𝑗}_=𝐽+2^{(𝑗−𝑘)(𝑀−𝑠)+𝑠𝑗−𝑖𝑁}|||Δ_{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑥)|||⎞⎟⎟⎠^𝑞⎞⎟⎟⎠^1/𝑞+𝑐_𝑘≥𝐽+(4.31)_𝑗≥𝑘2^{(𝑗−𝑘)(𝑎−𝑠)+𝑠𝑗}|||Δ_{∗,𝑎𝑗,𝐽}𝑓(𝑥)|||^𝑞^1/𝑞.(4.32) The second term can be estimated by (with 𝜎=min(1,𝑞)) 𝑐⎛⎜⎜⎝_𝑖≥02^{−𝑖𝑁𝜎}⎛⎜⎜⎝_𝑘≥𝐽+⎛⎜⎜⎝^𝑘_𝑗_=𝐽+2^{(𝑗−𝑘)(𝑀−𝑠)+𝑠𝑗}|||Δ_{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑥)|||⎞⎟⎟⎠^𝑞⎞⎟⎟⎠^𝜎/𝑞⎞⎟⎟⎠^1/𝜎.(4.33) Since again 𝑠<𝑀, then we can apply Lemma 3.6 to estimate the last expression by 𝑐⎛⎜⎜⎝_𝑖≥02^{−𝑖𝑁𝜎}⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞|||Δ_{∗,𝑎𝑗,𝐽−𝑖−2}𝑓(𝑥)|||^𝑞⎞⎟⎟⎠^𝜎/𝑞⎞⎟⎟⎠^1/𝜎.(4.34) Since 𝐿^𝑝/𝜎(𝐵_𝐽) is a normed space, so (4.31) in 𝐿^𝑝(𝐵_𝐽)-norm is dominated by 𝑐⎛⎜⎜⎝_𝑖≥02^{−𝑖𝑁𝜎}‖‖‖‖‖⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞|||Δ_{∗,𝑎𝑗,𝐽−𝑖−2}𝑓|||^𝑞⎞⎟⎟⎠^𝜎/𝑞∣𝐿^𝑝/𝜎𝐵_𝐽‖‖‖‖‖⎞⎟⎟⎠^1/𝜎=𝑐⎛⎜⎜⎝_𝑖≥02^{−𝑖𝑁𝜎}‖‖‖‖‖⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞|||Δ_{∗,𝑎𝑗,𝐽−𝑖−2}𝑓|||^𝑞⎞⎟⎟⎠^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖‖^𝜎⎞⎟⎟⎠^1/𝜎.(4.35) Using the embedding 𝐿^𝑝(𝐵_{𝐽−𝑖−2})⊂𝐿^𝑝(𝐵_𝐽) and the fact that 𝐽⁺≥(𝐽−𝑖−2)⁺ to estimate this expression from above by 𝑐⎛⎜⎜⎝_𝑖≥02^{−𝑖𝑁𝜎}‖‖‖‖‖⎛⎜⎜⎝_{𝑗≥(𝐽−𝑖−2)+}2^𝑗𝑠𝑞|||Δ_{∗,𝑎𝑗,𝐽−𝑖−2}𝑓|||^𝑞⎞⎟⎟⎠^1/𝑞∣𝐿^𝑝𝐵^{𝐽−𝑖−2}‖‖‖‖‖^𝜎⎞⎟⎟⎠^1/𝜎≤𝑐||𝐵_𝐽||^𝜏_𝑖≥02^{𝑖(𝑛𝜏−𝑁)𝜎}^1/𝜎≤𝑐||𝐵_𝐽‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖||^𝜏‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖,(4.36) where the first inequality is obtained by Theorem 2.5 and the second inequality follows by taking 𝑁>𝑛𝜏. Taking 𝑎∈(𝑛/min(𝑝,𝑞),𝑠), then using again Lemma 3.6 to estimate (4.32) by 𝑐⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞|||Δ_{∗,𝑎𝑗,𝐽}𝑓(𝑥)|||^𝑞⎞⎟⎟⎠^1/𝑞.(4.37) This expression, by Theorem 2.5, in 𝐿^𝑝(𝐵_𝐽)-norm is bounded by 𝑐|𝐵_𝐽|^𝜏‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖. Hence we have for any 𝐽∈ℤ and any ball 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽1||𝐵_𝐽||^𝜏‖‖𝐻_𝐽^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖≤𝑐‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖,(4.38) with some positive constant 𝑐 independent of 𝐽. From this it follows that ‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀≤𝑐‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖,(4.39) for any 𝑓∈𝐹^{𝑠,𝜏𝑝,𝑞}.
Step 3. Let Ψ be the function introduced in Definition 2.1 and in addition radial symmetric. We make use of an observation made by Nikol’skij [15] (see also [16]). We put 𝜓(𝑥)=(−1)^{𝑀+1𝑀−1}_𝑖=0(−1)^𝑖𝐶^𝑀𝑖Ψ(𝑥(𝑀−𝑖)).(4.40) The function 𝜓 satisfies 𝜓(𝑥)=1 for |𝑥|≤1/𝑀 and 𝜓(𝑥)=0 for |𝑥|≥3/2. Then, taking 𝜑₀(𝑥)=𝜓(𝑥),𝜑₁(𝑥)=𝜓(𝑥/2)−𝜓(𝑥) and 𝜑_𝑗(𝑥)=𝜑₁(2^−𝑗+1𝑥) for 𝑗=2,3,…, we obtain that {𝜑_𝑗} is a smooth dyadic resolution of unity. This yields that sup_𝐵𝐽1||𝐵_𝐽||^𝜏‖‖‖‖‖⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞||Δ_𝑗𝑓||^𝑞⎞⎟⎟⎠^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖‖,(4.41) is a norm equivalent in 𝐹^{𝑠,𝜏𝑝,𝑞} (see Remark 2.3). Let us prove that 1||𝐵_𝐽||^𝜏‖‖‖‖‖⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞||Δ_𝑗𝑓||^𝑞⎞⎟⎟⎠^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖‖≤𝑐‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀,(4.42) for any ball 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽. First the left-hand side contains Δ₀𝑓 only when 𝐽⁺=0. Then ‖‖Δ₀𝑓∣𝐿^𝑝𝐵_𝐽‖‖≤_ℝ𝑛‖‖𝑓∣𝐿^𝑝𝐵_𝐽‖‖||ℱ⁻¹𝜓(𝑦)||𝑑𝑦,(4.43) where 𝐵_𝐽 is a ball centered at 𝑥₀+𝑦 and of radius 2^−𝐽. Hence ‖‖Δ₀𝑓∣𝐿^𝑝𝐵_𝐽‖‖≤≤𝑐𝑐||||𝐵𝐵_𝐽𝐽||^𝜏‖‖𝑓∣𝐿^𝑝𝜏‖‖||^𝜏‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀,(4.44) where we have used the fact that |𝐵_𝐽|=|𝐵_𝐽|. Moreover, it holds for 𝑥∈ℝ^𝑛 and 𝑗=1,2,…Δ_𝑗𝑓(𝑥)=(−1)^𝑀+1_ℝ𝑛Δ^𝑀2_−𝑗𝑣𝑓(𝑥)ℱ⁻¹Ψ(𝑣)𝑑𝑣,(4.45) with Ψ(⋅)=ℱ⁻¹Ψ(⋅)−2^−𝑛ℱ⁻¹Ψ(⋅/2) (see [17, Theorem 3.1]). Now, for 𝑗∈ℕ we write _ℝ𝑛||Δ=^𝑀2_−𝑗𝑣𝑓(𝑥)|||||ℱ⁻¹Ψ(𝑣)|||𝑑𝑣_|𝑣|≤1||Δ^𝑀2_−𝑗𝑣𝑓(𝑥)|||||ℱ⁻¹Ψ(𝑣)|||𝑑𝑣+(−1)^𝑀+1_|𝑣|>1||Δ^𝑀2_−𝑗𝑣𝑓(𝑥)|||||ℱ⁻¹Ψ(𝑣)|||𝑑𝑣.(4.46) Then the estimate (4.42) is an obvious consequence of (4.44) and Lemma 3.4. Therefore, ‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖≤𝑐‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀,(4.47) which completes the proof of Theorem 4.1.

Proof of Theorem 4.2. The first two steps closely follow the argument in [17, Theorem 3.1].Step 1. Let 𝑓∈𝐵^{𝑠,𝜏𝑝,𝑞}. Since 𝑠>0, then we have ‖‖𝑓∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖≤≤𝑐‖‖_𝑗≥0𝑓2∣𝐵^−𝑗𝑠2^𝑗𝑠_{𝑠,𝜏𝑝,𝑞}‖‖Δ_𝑗𝑓∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖≤𝑐sup_𝑗∈ℕ02^𝑗𝑠‖‖Δ_𝑗𝑓∣𝐿^𝑝𝜏(ℝ^𝑛)‖‖‖‖.(4.48)Step 2. As in the proof of Theorem 4.1 we have ^2+−𝐽+1₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}‖‖Δ^𝑀ℎ𝑓∣𝐿^𝑝𝐵_𝐽‖‖^𝑞𝑑𝑡𝑡^1/𝑞≤𝑐_𝑘≥𝐽+2^𝑘𝑠𝑞sup_{||ℎ||≤2−𝑘+1}‖‖Δ^𝑀ℎ𝑓∣𝐿^𝑝𝐵_𝐽‖‖^𝑞^1/𝑞.(4.49) Let us estimate Δ^𝑀ℎΔ_𝑗𝑓. If 𝑗≤𝑘, then as in the proof Theorem 4.1, we have ||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤𝑐2^{(𝑗−𝑘)𝑀}Δ^𝑗∗,𝑎𝑓(𝑥),(4.50) for any 𝑥∈𝐵_𝐽, |ℎ|≤2^−𝑘+1 and any 𝑎>0. If 𝑗>𝑘 we have for 𝑥∈ℝ^𝑛 and |ℎ|≤2^−𝑘+1||Δ^𝑀ℎΔ_𝑗𝑓(𝑥)||≤^𝑀_𝑚₌₀𝐶^𝑀𝑚||Δ_𝑗𝑓(𝑥+(𝑀−𝑚)ℎ)||.(4.51) Hence we obtain for any 𝑗>𝑘 and any 𝑎>0‖‖Δ^𝑀ℎΔ_𝑗𝑓∣𝐿^𝑝𝐵_𝐽‖‖≤^𝑀_𝑚₌₀≤_𝑀𝐶^𝑀𝑚‖‖Δ_𝑗𝑓∣𝐿^𝑝𝐵^𝐽‖‖_𝑚₌₀𝐶^𝑀𝑚‖‖Δ^𝑗∗,𝑎𝑓∣𝐿^𝑝𝐵^𝐽‖‖,(4.52) where if 𝑥₀ the centre of 𝐵_𝐽 then 𝑥₀+(𝑀−𝑚)ℎ is the centre of 𝐵_𝐽. We remark also that by our assumption on ℎ and 𝑘 we have ||𝑥−𝑥₀||≤||𝑥−𝑥₀+(𝑀−𝑚)ℎ||+(𝑀−𝑚)||ℎ||<2^−𝐽+𝑀2^−𝑘+1≤(2𝑀+1)2^−𝐽,(4.53) for any 𝑥∈𝐵_𝐽. We denote 𝐵_𝐽 the ball in ℝ^𝑛 centred at 𝑥₀ and of radius (2𝑀+1)2^−𝐽. Since 𝐿^𝑝(𝐵_𝐽)⊂𝐿^𝑝(𝐵_𝐽) and 𝐿^𝑝(𝐵_𝐽)⊂𝐿^𝑝(𝐵_𝐽), we get ‖‖Δ^𝑀ℎ𝑓∣𝐿^𝑝𝐵_𝐽‖‖≤𝑐^𝐽+−1_𝑗=02^{(𝑗−𝑘)𝑀}‖‖Δ^𝑗∗,𝑎𝑓∣𝐿^𝑝𝐵_𝐽‖‖+𝑐^𝑘_𝑗_=𝐽++𝑐_𝑗≥𝑘2^{(𝑗−𝑘)𝑀}‖‖Δ^𝑗∗,𝑎𝑓∣𝐿^𝑝𝐵^𝐽‖‖‖‖Δ^𝑗∗,𝑎𝑓∣𝐿^𝑝𝐵^𝐽‖‖.(4.54) Here we put ∑^{𝐽+−1𝑗=0}⋯=0 if 𝐽⁺=0. Lemma 3.1 gives ^𝐽⁺⁻¹_𝑗=02^{(𝑗−𝑘)𝑀}‖‖Δ^𝑗∗,𝑎𝑓∣𝐿^𝑝𝐵_𝐽‖‖≤𝑐^𝐽+−1_𝑗=0≤𝑐22^{(𝑗−𝑘)𝑀+𝑗(𝑛/𝑝−𝑠−𝑛𝜏)−𝐽𝑛/𝑝}_{𝐽(𝑀−𝑠−𝑛𝜏)−𝑘𝑀}‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖.(4.55) The second inequality follows by our assumption on 𝑠 and 𝜏. Since 0<𝑠<𝑀, then we can apply Lemma 3.6 to estimate (4.49) by 𝑐||𝐵_𝐽||^𝜏‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖‖‖‖2^{(𝑘−𝐽+)(𝑠−𝑀)}_𝑘≥𝐽+∣ℓ^0𝑞,𝐽₊‖‖‖+𝑐⎛⎜⎜⎝_𝑗≥𝐽+2^𝑗𝑠𝑞‖‖Δ^𝑗∗,𝑎𝑓∣𝐿^𝑝𝐵^𝐽‖‖^𝑞⎞⎟⎟⎠^1/𝑞≤𝑐||𝐵_𝐽||^𝜏‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖,(4.56) where we have used Theorem 2.4, combined with Remark 2.6, and the equation |𝐵_𝐽|=(2𝑀+1)^𝑛|𝐵_𝐽|.Step 3. First this step in [17, Theorem 3.1] contains a gap, but using the same arguments given in Step 3 in the proof of Theorem 4.1 (with Lemma 3.5 in place of Lemma 3.4), we can prove that ‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖≤𝑐‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖_𝑀.(4.57) This ends the proof of Theorem 4.2.

Finally we study, in addition, the case 𝜏∈[1/𝑝,∞). Under this condition we can restrict sup_𝐵𝐽 in the definition of 𝐵^{𝑠,𝜏𝑝,𝑞} and 𝐹^{𝑠,𝜏𝑝,𝑞} to a supremum taken with respect to balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽 and 𝐽∈ℕ₀.

Lemma 4.4. Let 𝑠∈ℝ, 𝜏∈[1/𝑝,∞) and 0<𝑞≤∞.
Let 0<𝑝≤∞. A tempered distribution 𝑓 belongs to 𝐵^{𝑠,𝜏𝑝,𝑞} if and only if ‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖^#=sup_𝐵𝐽1||𝐵_𝐽||^𝜏_𝑗≥𝐽2^𝑗𝑠𝑞‖‖Δ_𝑗𝑓∣𝐿^𝑝𝐵_𝐽‖‖^𝑞^1/𝑞<∞,(4.58) where the supremum is taken over all 𝐽∈ℕ₀ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽. Furthermore, the quasinorms ‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖^# and ‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖ are equivalent.
Let 0<𝑝<∞. A tempered distribution 𝑓 belongs to 𝐹^{𝑠,𝜏𝑝,𝑞} if and only if ‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^#=sup_𝐵𝐽1||𝐵_𝐽||^𝜏‖‖‖‖_𝑗≥𝐽2^𝑗𝑠𝑞||Δ_𝑗𝑓||^𝑞^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖<∞,(4.59) where the supremum is taken over all 𝐽∈ℕ₀ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽. Furthermore, the quasinorms ‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖^# and ‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖ are equivalent.

Proof. For each 𝐽∈ℤ and 𝑚=(𝑚₁,…,𝑚_𝑛)∈ℤ^𝑛, set 𝑄_𝐽,𝑚=𝑥=𝑥₁,…,𝑥_𝑛∈ℝ^𝑛∶𝑚_𝑖≤2^𝐽𝑥_𝑖<𝑚_𝑖+1,𝑖=1,2,…,𝑛.(4.60) This lemma for 𝑄_𝐽,𝑚 in place of 𝐵_𝐽 is given in [9, Lemma 2.2]. By the properties of the dyadic cubes, there exists 𝑣,𝑘∈ℕ not depending on 𝐽 such that 𝑄_𝐽,𝑚⊂∪^𝑣𝑙=1𝐵^𝑙𝐽,𝐵_𝐽⊂∪^𝑘𝑙=1𝑄^𝑙𝐽.(4.61) Here 𝐵^𝑙𝐽 is a ball of center 2^−𝐽 and 𝑄^𝑙𝐽 is a dyadic cube of side length 2^−𝐽. The proof of this result is an obvious consequence of the previous embeddings, Lemma 2.2 of [9] and Remark 2.3.

Defining for 1≤𝑝≤∞𝑀_𝑝(𝑓)=sup_𝐵0_𝐵0||𝑓(𝑥)||^𝑝𝑑𝑥^1/𝑝,(4.62) where the supremum is taken over all balls 𝐵₀ of ℝ^𝑛 with radius 1.

Theorem 4.5. Let 1≤𝑝<∞, 0<𝑞≤∞, 𝜏∈[1/𝑝,∞), 𝑀∈ℕ and 𝑛/min(𝑝,𝑞)<𝑠<𝑀−𝑛𝜏+𝑛/𝑝. Then ‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^#𝑀=𝑀_𝑝(𝑓)+sup_𝐵𝐽1||𝐵_𝐽||^𝜏‖‖‖‖₂^−𝐽+1₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}||Δ^𝑀ℎ𝑓(⋅)||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖(4.63) is an equivalent quasinorm in 𝐹^{𝑠,𝜏𝑝,𝑞}. Here the supremum is taken over all 𝐽∈ℕ₀ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽.

Theorem 4.6. Let 1≤𝑝≤∞, 0<𝑞≤∞, 𝜏∈[1/𝑝,∞), 𝑀∈ℕ and 0<𝑠<𝑀−𝑛𝜏+𝑛/𝑝. Then ‖‖𝑓∣𝐵^{𝑠,𝜏𝑝,𝑞}‖‖^#𝑀=𝑀_𝑝(𝑓)+sup_𝐵𝐽1||𝐵_𝐽||^𝜏^2−𝐽+1₀𝑡^−𝑠𝑞sup_{||ℎ||≤𝑡}‖‖Δ^𝑀ℎ𝑓∣𝐿^𝑝𝐵_𝐽‖‖^𝑞𝑑𝑡𝑡^1/𝑞(4.64) is an equivalent quasinorm in 𝐵^{𝑠,𝜏𝑝,𝑞}. Here the supremum is taken over all 𝐽∈ℕ₀ and all balls 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽.

Proof. We will prove only Theorem 4.5. The proof of Theorem 4.6 is similar. We employ the same notations given in the proof of Theorem 4.1.Step 1. Let 𝑓∈𝐹^{𝑠,𝜏𝑝,𝑞}. Since 𝑠>0, then we have 𝑀_𝑝(𝑓)≤≤𝑐‖‖_𝑗≥0𝑓2∣𝐹^−𝑗𝑠2^𝑗𝑠_{𝑠,𝜏𝑝,𝑞}𝑀_𝑝Δ_𝑗𝑓≤𝑐sup_{𝐵0,𝑗∈ℕ0}2^𝑗𝑠‖‖Δ_𝑗𝑓∣𝐿^𝑝𝐵₀‖‖‖‖^#.(4.65)
Step 2. As in the proof of Theorem 4.1, there is a constant 𝑐>0 independent of 𝐽 such that 1||𝐵_𝐽||^𝜏‖‖𝐻_𝐽^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖≤≤𝑐𝑐‖‖‖‖𝑓𝑓∣∣𝐹𝐹^{𝑠,𝜏𝑝,𝑞}_{𝑠,𝜏𝑝,𝑞}‖‖‖‖^#,(4.66) by Lemma 4.4.Step 3. The left-hand side in (4.42) (with 𝐽⁺=𝐽) contains Δ₀𝑓 only when 𝐽=0. Then ‖‖Δ₀𝑓∣𝐿^𝑝𝐵₀‖‖≤_ℝ𝑛‖‖𝑓∣𝐿^𝑝≤𝑐𝑀_𝑝𝐵₀‖‖||ℱ⁻¹𝜓(𝑦)||𝑑𝑦(𝑓).(4.67) We recall that for 𝑥∈ℝ^𝑛 and 𝑗=1,2,…Δ_𝑗𝑓(𝑥)=(−1)^𝑀+1_ℝ𝑛=(−1)_𝑀+1Δ^𝑀2_−𝑗𝑣𝑓(𝑥)ℱ⁻¹Ψ(𝑣)𝑑𝑣_|𝑣|≤1Δ^𝑀2_−𝑗𝑣𝑓(𝑥)ℱ⁻¹Ψ(𝑣)𝑑𝑣+(−1)^𝑀+1_|𝑣|>1Δ^𝑀2_−𝑗𝑣𝑓(𝑥)ℱ⁻¹Ψ(𝑣)𝑑𝑣.(4.68) As in the proof of Lemma 3.4, we can prove that ‖‖‖‖_𝑗≥𝐽2^𝑗𝑠𝑞_|𝑣|≤1||Δ^𝑀2_−𝑗𝑣𝑓(⋅)||𝑑𝑣^𝑞^1/𝑞‖‖‖‖_𝑗≥𝐽∣𝐿^𝑝𝐵_𝐽‖‖‖‖≤𝑐||𝐵_𝐽||^𝜏‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^#𝑀,2^𝑗𝑠𝑞_|𝑣|>1||Δ^𝑀2_−𝑗𝑣𝑓(⋅)𝜔(𝑣)||𝑑𝑣^𝑞^1/𝑞∣𝐿^𝑝𝐵_𝐽‖‖‖‖≤𝑐||𝐵_𝐽||^𝜏‖‖𝑓∣𝐹^{𝑠,𝜏𝑝,𝑞}‖‖^#𝑀,(4.69) for any 𝐽∈ℕ any ball 𝐵_𝐽 of ℝ^𝑛 with radius 2^−𝐽 and any 𝜔∈𝒮(ℝ^𝑛). The proof is completed.

Remark 4.7. Recently, Yang and Yuan [18, Theorem 2] proved that 𝐹𝐵^{𝑠,𝜏𝑝,𝑞}=𝐹^{𝑠+𝑛(𝜏−1/𝑝)∞,∞},0<𝑝<∞,𝑠∈ℝ,_{𝑠,𝜏𝑝,𝑞}=𝐵^{𝑠+𝑛(𝜏−1/𝑝)∞,∞},0<𝑝≤∞,𝑠∈ℝ,(4.70) if 𝜏>1/𝑝, 0<𝑞≤∞ or if 𝜏=1/𝑝 and 𝑞=∞. Under these conditions the study of the Triebel-Lizorkin-type space 𝐹^{𝑠,𝜏𝑝,𝑞} and the Besov-type space 𝐵^{𝑠,𝜏𝑝,𝑞} is not interest.

Acknowledgments

The author would like to thank the referee for his very carefully reading and also his many careful and valuable remarks, which improve, the results and the presentation of this paper.