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

Authors: Douadi Drihem
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We present characterizations of the Besov-type spacesBp,qs,τ and the Triebel-Lizorkin-type spaces Fp,qs,τ by differences. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0.


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## Body

## 1. Introduction

TheBp,qs,τ spaces and Fp,qs,τ spaces have been studied extensively in recent years. When τ=0 they coincide with the usual function spaces Bp,qs and Fp,qs, respectively, studied in detail by Triebel in [1–3]. When s∈ℝ, τ∈[0,∞) and 1≤p, q<∞ the Bp,qs,τ 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 Bp,qs,τ cover certain Campanato spaces, studied in [6, 7]. Later on, Drihem gave in [8] a characterization for Bp,qs,τ spaces by local means and maximal functions. For a complete treatment of Bp,qs,τ spaces and Fp,qs,τ 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 Ḃp,qs,τ and Ḟp,qs,τ (p≠∞), which generalize the homogeneous Besov-Triebel-Lizorkin spaces Ḃp,qs,τ, Ḟp,qs,τ and established the relation between Ḟp,qs,τ and Qα 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. Section2.1 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 Bp,qs,τ spaces and Fp,qs,τ spaces by differences.


## 2. Preliminaries

### 2.1. Notation and Conventions

As usual,ℝn 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,…,αn)∈ℕ0n, we write |α|=α1+⋯+αn and Dα=∂|α|/∂x1α1⋯∂xnαn. For v∈ℤ,let Bv be the ball of ℝn with radius 2-v and v+=max{v,0}. The Euclidean scalar product of x=(x1,…,xn) and y=(y1,…,yn) is given by x·y=x1y1+⋯+xnyn. We denote by |Ω| the n-dimensional Lebesgue measure of Ω⊆ℝn. For any measurable subset Ω⊆ℝn the Lebesgue space Lp(Ω), 0<p≤∞ consists of all measurable functions for which(2.1)‖f∣Lp(Ω)‖=(∫Ω|f(x)|pdx)1/p<∞,0<p<∞,‖f∣L∞(Ω)‖=ess-supx∈Ω|f(x)|<∞.By𝒮(ℝn) we denote the Schwartz space of all complex-valued, infinitely differentiable, and rapidly decreasing functions on ℝn and by 𝒮′(ℝn) the dual space of all tempered distributions on ℝn. We define the Fourier transform of a function f∈𝒮(ℝn) by(2.2)F(f)(ξ)=(2π)-n/2∫Rne-ix⋅ξf(x)dx.
Its inverse is denoted by ℱ-1f. Both ℱ and ℱ-1 are extended to the dual Schwartz space 𝒮′(ℝn) in the usual way.Letτ∈[0,∞) and p∈(0,∞]. Let Lτp(ℝn) be the collection of functions f∈Llocp(ℝn) such that(2.3)‖f∣Lτp(Rn)‖=sup⁡BJ1|BJ|τ(∫BJ|f(x)|pdx)1/p<∞,
where the supremum is taken over all J∈ℤ∖ℕ and all balls BJ of ℝn with radius 2-J. Obviously, when τ=0, then Lτp(ℝn)=Lp(ℝn). Furthermore,(2.4)Lτp(Rn)↪S′(Rn),
(see [9, page 46]).Ifs∈ℝ, 0<q≤∞ and J∈ℤ, then ℓq,J+s is the set of all sequences {fk}k≥J+ of complex numbers such that(2.5)‖{fk}k≥J+∣lq,J+s‖=(∑k≥J+2ksq|fk|q)1/q<∞,
with the obvious modification if q=∞. We recall that for any 0<θ≤1 and any J∈ℤ(2.6)(∑k≥J+|fk|)θ≤∑k≥J+|fk|θ,(2.7)(y+z)d≤max⁡(1,2d-1)(yd+zd),y,z≥0,d>0.Letf be an arbitrary function on ℝn and x, h∈ℝn. Then(2.8)Δhf(x)=f(x+h)-f(x),ΔhM+1f(x)=Δh(ΔhMf)(x),M∈N.
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 formula(2.9)ΔhMf(x)=∑j=0M(-1)jCjMf(x+(M-j)h),
where CjM are the binomial coefficients.Recall thatηj,N(x)=2jn(1+2j|x|)-N, for any x∈ℝn, j∈ℕ0 and N>0. By c we denote generic positive constants, which may have different values at different occurrences.


### 2.2. TheBp,qs,τ Spaces and Fp,qs,τ Spaces

In this subsection we present the Fourier analytical definition ofBp,qs,τ spaces, Fp,qs,τ spaces and recall their basic properties. We first need the concept of a smooth dyadic resolution of unity.Definition 2.1.
LetΨ be a function in 𝒮(ℝn) 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
(2.10)φj(x)=φ1(2-j+1x)forj=2,3,….
Then we have suppφj⊂{x∈ℝn:2j-1≤|x|≤3·2j-1}, φj(x)=1 for 3·2j-2≤|x|≤2j and Ψ(x)+∑j≥1φj(x)=1 for all x∈ℝn. The system of functions {φj} is called a smooth dyadic resolution of unity. We define the convolution operators Δj by the following:
(2.11)Δjf=F-1φj*f,j∈N,Δ0f=F-1Ψ*f,f∈S′(Rn).
Thus we obtain the Littlewood-Paley decomposition
(2.12)f=∑j≥0Δjf,
of all f∈𝒮′(ℝn)(convergence in𝒮′(ℝn)).TheBp,qs,τ spaces and Fp,qs,τ spaces are defined in the following way.Definition 2.2.
(i) Lets∈ℝ, τ∈[0,∞) and 0<p, q≤∞. The space Bp,qs,τ is the collection of all f∈𝒮′(ℝn) such that
(2.13)‖f∣Bp,qs,τ‖=sup⁡BJ1|BJ|τ(∑j≥J+2jsq‖Δjf∣Lp(BJ)‖q)1/q<∞,
where the supremum is taken over all J∈ℤ and all balls BJofℝn with radius 2-J.
(ii) Lets∈ℝ, τ∈[0,∞), 0<p<∞ and 0<q≤∞.The space Fp,qs,τ is the collection of all f∈𝒮′(ℝn) such that
(2.14)‖f∣Fp,qs,τ‖=sup⁡BJ1|BJ|τ‖(∑j≥J+2jsq|Δjf|q)1/q∣Lp(BJ)‖<∞,
where the supremum is taken over all J∈ℤ and all balls BJofℝn with radius 2-J.Remark 2.3.
The spacesBp,qs,τ  and  Fp,qs,τ  are independent of the particular choice of the smooth dyadic resolution of unity  {φj}  appearing in their definitions. They are quasi-Banach spaces (Banach spaces if  p≥1, q≥1).  In particular,
(2.15)Bp,qs,0=Bp,qs,Fp,qs,0=Fp,qs,
where Bp,qs and Fp,qs are the Besov spaces and Triebel-Lizorkin spaces respectively. If we replace the balls BJ 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 Δjf (j∈ℕ0) be the functions introduced in Definition 2.1. For any a>0, any x∈ℝn and any J∈ℤ we denote (Peetre’s maximal functions)(2.16)Δj,J*,af(x)=sup⁡y∈BJ|Δjf(y)|(1+2j|x-y|)aj∈N0,f∈S′(Rn).We now present a fundamental characterization ofBp,qs,τ spaces and Fp,qs,τ spaces.Theorem 2.4.
Lets∈ℝ, τ∈[0,∞), 0<p, q≤∞ and  a>n/p. Then
(2.17)‖f∣Bp,qs,τ‖*=sup⁡BJ1|BJ|τ(∑j≥J+2jsq‖Δj,J*,af∣Lp(BJ)‖q)1/q,
is an equivalent quasinorm in Bp,qs,τ.Theorem 2.5.
Lets∈ℝ, τ∈[0,∞), 0<p<∞, 0<q≤∞ and  a>n/min(p,q). Then ∥f∣Fp,qs,τ∥*=supBJ(1/|BJ|τ)∥(∑j≥J+2jsq|Δj,J*,af|q)1/q∣Lp(BJ)∥, is an equivalent quasinorm in Fp,qs,τ.Remark 2.6.
Theorem2.4 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  Δj,J*,af  can be replaced by  Δj*,af, where
(2.18)Δj*,af(x)=sup⁡y∈Rn|Δjf(y)|(1+2j|x-y|)aj∈N0,f∈S′(Rn).


## 2.1. Notation and Conventions

As usual,ℝn 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,…,αn)∈ℕ0n, we write |α|=α1+⋯+αn and Dα=∂|α|/∂x1α1⋯∂xnαn. For v∈ℤ,let Bv be the ball of ℝn with radius 2-v and v+=max{v,0}. The Euclidean scalar product of x=(x1,…,xn) and y=(y1,…,yn) is given by x·y=x1y1+⋯+xnyn. We denote by |Ω| the n-dimensional Lebesgue measure of Ω⊆ℝn. For any measurable subset Ω⊆ℝn the Lebesgue space Lp(Ω), 0<p≤∞ consists of all measurable functions for which(2.1)‖f∣Lp(Ω)‖=(∫Ω|f(x)|pdx)1/p<∞,0<p<∞,‖f∣L∞(Ω)‖=ess-supx∈Ω|f(x)|<∞.By𝒮(ℝn) we denote the Schwartz space of all complex-valued, infinitely differentiable, and rapidly decreasing functions on ℝn and by 𝒮′(ℝn) the dual space of all tempered distributions on ℝn. We define the Fourier transform of a function f∈𝒮(ℝn) by(2.2)F(f)(ξ)=(2π)-n/2∫Rne-ix⋅ξf(x)dx.
Its inverse is denoted by ℱ-1f. Both ℱ and ℱ-1 are extended to the dual Schwartz space 𝒮′(ℝn) in the usual way.Letτ∈[0,∞) and p∈(0,∞]. Let Lτp(ℝn) be the collection of functions f∈Llocp(ℝn) such that(2.3)‖f∣Lτp(Rn)‖=sup⁡BJ1|BJ|τ(∫BJ|f(x)|pdx)1/p<∞,
where the supremum is taken over all J∈ℤ∖ℕ and all balls BJ of ℝn with radius 2-J. Obviously, when τ=0, then Lτp(ℝn)=Lp(ℝn). Furthermore,(2.4)Lτp(Rn)↪S′(Rn),
(see [9, page 46]).Ifs∈ℝ, 0<q≤∞ and J∈ℤ, then ℓq,J+s is the set of all sequences {fk}k≥J+ of complex numbers such that(2.5)‖{fk}k≥J+∣lq,J+s‖=(∑k≥J+2ksq|fk|q)1/q<∞,
with the obvious modification if q=∞. We recall that for any 0<θ≤1 and any J∈ℤ(2.6)(∑k≥J+|fk|)θ≤∑k≥J+|fk|θ,(2.7)(y+z)d≤max⁡(1,2d-1)(yd+zd),y,z≥0,d>0.Letf be an arbitrary function on ℝn and x, h∈ℝn. Then(2.8)Δhf(x)=f(x+h)-f(x),ΔhM+1f(x)=Δh(ΔhMf)(x),M∈N.
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 formula(2.9)ΔhMf(x)=∑j=0M(-1)jCjMf(x+(M-j)h),
where CjM are the binomial coefficients.Recall thatηj,N(x)=2jn(1+2j|x|)-N, for any x∈ℝn, j∈ℕ0 and N>0. By c we denote generic positive constants, which may have different values at different occurrences.


## 2.2. TheBp,qs,τ Spaces and Fp,qs,τ Spaces

In this subsection we present the Fourier analytical definition ofBp,qs,τ spaces, Fp,qs,τ spaces and recall their basic properties. We first need the concept of a smooth dyadic resolution of unity.Definition 2.1.
LetΨ be a function in 𝒮(ℝn) 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
(2.10)φj(x)=φ1(2-j+1x)forj=2,3,….
Then we have suppφj⊂{x∈ℝn:2j-1≤|x|≤3·2j-1}, φj(x)=1 for 3·2j-2≤|x|≤2j and Ψ(x)+∑j≥1φj(x)=1 for all x∈ℝn. The system of functions {φj} is called a smooth dyadic resolution of unity. We define the convolution operators Δj by the following:
(2.11)Δjf=F-1φj*f,j∈N,Δ0f=F-1Ψ*f,f∈S′(Rn).
Thus we obtain the Littlewood-Paley decomposition
(2.12)f=∑j≥0Δjf,
of all f∈𝒮′(ℝn)(convergence in𝒮′(ℝn)).TheBp,qs,τ spaces and Fp,qs,τ spaces are defined in the following way.Definition 2.2.
(i) Lets∈ℝ, τ∈[0,∞) and 0<p, q≤∞. The space Bp,qs,τ is the collection of all f∈𝒮′(ℝn) such that
(2.13)‖f∣Bp,qs,τ‖=sup⁡BJ1|BJ|τ(∑j≥J+2jsq‖Δjf∣Lp(BJ)‖q)1/q<∞,
where the supremum is taken over all J∈ℤ and all balls BJofℝn with radius 2-J.
(ii) Lets∈ℝ, τ∈[0,∞), 0<p<∞ and 0<q≤∞.The space Fp,qs,τ is the collection of all f∈𝒮′(ℝn) such that
(2.14)‖f∣Fp,qs,τ‖=sup⁡BJ1|BJ|τ‖(∑j≥J+2jsq|Δjf|q)1/q∣Lp(BJ)‖<∞,
where the supremum is taken over all J∈ℤ and all balls BJofℝn with radius 2-J.Remark 2.3.
The spacesBp,qs,τ  and  Fp,qs,τ  are independent of the particular choice of the smooth dyadic resolution of unity  {φj}  appearing in their definitions. They are quasi-Banach spaces (Banach spaces if  p≥1, q≥1).  In particular,
(2.15)Bp,qs,0=Bp,qs,Fp,qs,0=Fp,qs,
where Bp,qs and Fp,qs are the Besov spaces and Triebel-Lizorkin spaces respectively. If we replace the balls BJ 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 Δjf (j∈ℕ0) be the functions introduced in Definition 2.1. For any a>0, any x∈ℝn and any J∈ℤ we denote (Peetre’s maximal functions)(2.16)Δj,J*,af(x)=sup⁡y∈BJ|Δjf(y)|(1+2j|x-y|)aj∈N0,f∈S′(Rn).We now present a fundamental characterization ofBp,qs,τ spaces and Fp,qs,τ spaces.Theorem 2.4.
Lets∈ℝ, τ∈[0,∞), 0<p, q≤∞ and  a>n/p. Then
(2.17)‖f∣Bp,qs,τ‖*=sup⁡BJ1|BJ|τ(∑j≥J+2jsq‖Δj,J*,af∣Lp(BJ)‖q)1/q,
is an equivalent quasinorm in Bp,qs,τ.Theorem 2.5.
Lets∈ℝ, τ∈[0,∞), 0<p<∞, 0<q≤∞ and  a>n/min(p,q). Then ∥f∣Fp,qs,τ∥*=supBJ(1/|BJ|τ)∥(∑j≥J+2jsq|Δj,J*,af|q)1/q∣Lp(BJ)∥, is an equivalent quasinorm in Fp,qs,τ.Remark 2.6.
Theorem2.4 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  Δj,J*,af  can be replaced by  Δj*,af, where
(2.18)Δj*,af(x)=sup⁡y∈Rn|Δjf(y)|(1+2j|x-y|)aj∈N0,f∈S′(Rn).


## 3. Some Technical Lemmas

To prove our results, we need some technical lemmas. The following lemma forΔjf, in place of Δj*,af, is given in [14, pages 87–89] (for the Bp,qs,τ spaces and 1≤p<∞). Further results, can be found in [12, Lemma 2.4].Lemma 3.1.
LetΔjf be as in Definition 2.1 and let  s∈ℝ,  a>0,  τ∈[0,∞) and  0<p, q≤∞(0<p<∞ for the space  Fp,qs,τ). Then there is a constant  c>0, independent of  j, such that for any  f∈𝒮′(ℝn)(3.1)‖Δj*,af‖∞≤c2j(n/p-s-nτ)‖f∣Ap,qs,τ‖,j∈N0.
Here one uses Ap,qs,τ to denote either Bp,qs,τ or Fp,qs,τ.Proof.
Letψ,ψ0∈𝒮(ℝn) be two functions such that ℱψ=1 and ℱψ0=1 on suppφ1 and suppφ0 respectively. Then
(3.2)|Δjf(y)|=|ψj*Δjf(y)|,y∈Rn,
with ψj=2(j-1)nψ(2j-1) if j∈ℕ. Since φ∈𝒮(ℝn), the right-hand side is bounded by cηj-1,N*|Δjf|(y), for any N>0. Hence we get for all f∈𝒮′(ℝn) and any x∈ℝn(3.3)Δj*,af(x)≤csup⁡y∈Rnηj-1,N*|Δjf|(y).
Using the same method given in [9,Proposition 2.6] we obtain for any y∈ℝn(3.4)ηj-1,N*|Δjf|(y)≤c2j(n/p-s-nτ)‖f∣Ap,qs,τ‖.
The proof is completed.Lemma 3.2.
LetM∈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
(3.5)‖∫|h|≤A|ΔhMf(⋅)|dh∣Lp(BJ)‖≤cAn|BJ|τ‖f∣Lτp‖,
for any ball  BJ of  ℝn with radius  2-J and any function f such that ∥f∣Lτp∥<∞.Proof.
Since1≤p≤∞, the left-hand side is bounded by
(3.6)∫|h|≤A‖ΔhMf∣Lp(BJ)‖dh.
From the definition of ΔhMf we have
(3.7)|ΔhMf(x)|≤∑m=0MCmM|f(x+(M-m)h)|.
Take the Lp(BJ)-norm to estimate (3.6) form above by
(3.8)∑m=0MCmM∫|h|≤A‖f∣Lp(B̃J)‖dh,
where if x0 the centre of BJ then x0+(M-m)h is the centre of B̃J. Using the fact that |B̃J|=|BJ| to estimate (3.8) from above by cAn|BJ|τ∥f∣Lτp∥.
The lemma is proved.Remark 3.3.
LetM,A,τ,and p be as in Lemma 3.2. Let J∈ℕ0. By the embedding Lp(B0)↪Lp(BJ) there is a constant c>0, independent of J and A, such that
(3.9)‖∫|h|≤A|ΔhMf(⋅)|dh∣Lp(BJ)‖≤cAn‖f∣Lτp‖,
for any ball BJ of ℝn with radius 2-J and any function f such that ∥f∣Lτp∥<∞.
Fors>0, M∈ℕ, τ∈[0,∞), 1≤p<∞ and 0<q≤∞, we set
(3.10)‖f∣Fp,qs,τ‖M=‖f∣Lτp(Rn)‖+sup⁡BJ1|BJ|τ‖(∫02-J++1t-sqsup⁡|h|≤t|ΔhMf(⋅)|qdtt)1/q∣Lp(BJ)‖.Here the supremum is taken over allJ∈ℤ and all balls BJ of ℝn with radius 2-J.Lemma 3.4.
Lets>0, M∈ℕ, J∈ℤ, τ∈[0,∞), 1≤p<∞ and  0<q≤∞. Then there is a constant  c>0, independent of  J, such that
(3.11)‖(∑j≥J+2jsq(∫|v|≤1|Δ2-jvMf(⋅)|dv)q)1/q∣Lp(BJ)‖≤c|BJ|τ‖f∣Fp,qs,τ‖M,(3.12)‖(∑j≥J+2jsq(∫|v|>1|Δ2-jvMf(⋅)ω(v)|dv)q)1/q∣Lp(BJ)‖≤c|BJ|τ‖f∣Fp,qs,τ‖M,
for any ball  BJ of  ℝn with radius  2-J, any ω∈𝒮(ℝn) and any function f such that ∥f∣Fp,qs,τ∥M<∞.Proof.
LetP be a dyadic cube with side length 2-J. This result, for P in place of BJ, 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 anyx∈ℝn and any i∈ℤ(3.13)(∑v≥i2(s+n)vq(∫|h|≤2-v|ΔhMf(x)|dh)q)1/q≤c(∫02-i+1t-(s+n)q(∫|h|≤t|ΔhMf(x)|dh)qdtt)1/q.
Here we will prove that the left-hand side of (3.12) is bounded by
(3.14)‖f∣Lτp(Rn)‖+sup⁡BJ1|BJ|τ‖(∫02-J+t-sqsup⁡|h|≤t|ΔhMf(⋅)|qdtt)1/q∣Lp(BJ)‖=‖f∣Fp,qs,τ‖M′.
Obviously, ∥f∣Fp,qs,τ∥M′≤∥f∣Fp,qs,τ∥M. We write
(3.15)2js∫|v|>1|Δ2-jvMf(x)ω(v)|dv=∑k=0∞2js∫2k<|v|≤2k+1|Δ2-jvMf(x)ω(v)|dv≤c∑k=0∞2(s+n)j-Nk∫2k-j<|h|≤2k-j+1|ΔhMf(x)|dh,
where N>0 is at our disposal and we have used the properties of the function ω,
(3.16)|ω(x)|≤c(1+|x|)-N,
for any x∈ℝn and any N>0. Now the right-hand side of (3.15) in ℓq,J+s-norm is bounded by (with σ=min(1,q))
(3.17)c(∑k=0∞2-Nσk(∑j≥J+2(s+n)jq(∫2k-j<|h|≤2k-j+1|ΔhMf(x)|dh)q)σ/q)1/σ=(∑k=0J+-1⋯+∑k≥J+⋯)1/σ=(IJ(x)+IIJ(x))1/σ≤21/σ-1((IJ(x))1/σ+(IIJ(x))1/σ),
by (2.7). Here we put ∑k=0J+-1⋯=0 if J+=0. Take the Lp(BJ)-norm we obtain that the left-hand side of (3.12) is bounded by
(3.18)21/σ-1‖(IJ)1/σ∣Lp(BJ)‖+21/σ-1‖(IIJ)1/σ∣Lp(BJ)‖.
Let us estimate (IJ)1/σ in Lp(BJ)-norm. After a change of variable j-k-1=v, we get for any x∈ℝn (here J+=J and k<J)
(3.19)IJ(x)≤c∑k=0J+-12(s+n-N)σk(∑v≥J+-k-12(s+n)vq(∫|h|≤2-v|ΔhMf(x)|dh)q)σ/q=∑k=0J+-2⋯+∑k=J+-1J+-1⋯=M1(x)+M2(x).
Here we put ∑k=0J+-2⋯=0 if J+≤1. We have
(3.20)M1(x)≤c∑k=0J+-22(s+n-N)σk(∫02-J+k+2t-(s+n)q(∫|h|≤t|ΔhMf(x)|dh)qdtt)σ/q.
Since Lp/σ(BJ) is a normed spaces and BJ⊂BJ-k-2, the right-hand side in Lp/σ(BJ)-norm can be estimated from above by
(3.21)c∑k=0J+-22(s+n-N)σk‖(∫02-J+k+2t-sqsup⁡|h|≤t|ΔhMf(⋅)|qdtt)1/q∣Lp(BJ-k-2)‖σ≤c|BJ|τσ∑k=0J+-22(s+n-N+nτ)σk(‖f∣Fp,qs,τ‖M′)σ.
We choose N>2(s+n)+τn. This yields that the last expression is bounded by
(3.22)c|BJ|τσ(‖f∣Fp,qs,τ‖M′)σ,
where c>0 is independent of J. Now (M2)1/σ in Lp(BJ)-norm is bounded by
(3.23)c2(s+n-N)J+‖(∑v≥02(s+n)vq(∫|h|≤2-v|ΔhMf(⋅)|dh)q)1/q∣Lp(BJ)‖≤c2(s+n-N)J+‖(∑v≥12(s+n)vq(∫|h|≤2-v|ΔhMf(⋅)|dh)q)1/q∣Lp(BJ)‖+c2(s+n-N)J+‖∫|h|≤1|ΔhMf(⋅)|dh∣Lp(BJ)‖,
where we have used (2.7). Using the embedding Lp(B0)↪Lp(BJ) and Remark 3.3, we obtain
(3.24)‖M2∣Lp/σ(BJ)‖=‖(M2)1/σ∣Lp(BJ)‖σ≤c2(s+n-N)σJ+(‖f∣Fp,qs,τ‖M′)σ=c2(s+n-N+nτ)σJ+|BJ|τσ(‖f∣Fp,qs,τ‖M′)σ≤c|BJ|τσ(‖f∣Fp,qs,τ‖M′)σ,
because of N>s+n+nτ. Therefore,
(3.25)‖(IJ)1/σ∣Lp(BJ)‖σ=‖IJ∣Lp/σ(BJ)‖≤‖M1∣Lp/σ(BJ)‖+‖M2∣Lp/σ(BJ)‖≤c|BJ|τσ(‖f∣Fp,qs,τ‖M′)σ.
Now let us estimate (IIJ)1/σ in Lp(BJ)-norm. We write
(3.26)IIJ(⋅)=c∑k≥J+2-Nσk(∑j=J+k+J++1⋯+∑j=k+J++2∞⋯)σ/q≤c∑k≥J+2-Nσk((∑j=J+k+J++1⋯)σ/q+(∑j=k+J++2∞⋯)σ/q)=c∑k≥J+2-Nσk(Sk1(⋅)+Sk2(⋅)).
After a change of variable j-k-1=v, we get
(3.27)(Sk1(x))1/σ≤c2(s+n)k(∑v=J+-k-1J+2(s+n)vq(∫|h|≤2-v|ΔhMf(x)|dh)q)1/q≤c2(s+n)k(∫2-J+2k-J++2t-(s+n)q(∫|h|≤t|ΔhMf(x)|dh)qdtt)1/q≤c2(s+n)k∫|h|≤2k-J++2|ΔhMf(x)|dh(∫2-J+2k-J++2t-(s+n)qdtt)1/q≤c2(s+n)J++(s+n)k∫|h|≤2k-J++2|ΔhMf(x)|dh.
Therefore there exists a constant c>0 independent of J and k such that
(3.28)‖(Sk1)1/σ∣Lp(BJ)‖≤c2(s+n)J++(s+n)k‖∫|h|≤2k-J++2ΔhMf(⋅)dh∣Lp(BJ)‖≤c2(s+τn)J++(s+2n)k|BJ|τ‖f∣Lτp‖≤c2(2s+2n+τn)k|BJ|τ‖f∣Fp,qs,τ‖M′,
by Lemma 3.2 (combined with Remark 3.3) and the fact that k≥J+. Now let us estimate (Sk2)1/σin Lp(BJ)-norm. We have
(3.29)(Sk2(x))1/σ≤c2(n+s)k(∑v=J++1∞2(s+n)vq(∫|h|≤2-v|ΔhMf(x)|dh)q)1/q≤c2(n+s)k(∫02-J+t-sqsup⁡|h|≤t|ΔhMf(x)|qdtt)1/q.
Therefore,
(3.30)‖(Sk2)1/σ∣Lp(BJ)‖≤c2(n+s)k‖(∫02-J+t-sqsup⁡|h|≤t|ΔhMf(⋅)|qdtt)1/q∣Lp(BJ)‖≤c2(n+s)k|BJ|τ‖f∣Fp,qs,τ‖M′.
Consequently,
(3.31)‖(IIJ)1/σ∣Lp(BJ)‖σ=‖IIJ∣Lp/σ(BJ)‖≤∑k≥J+2-Nσk(‖(Sk1)1/σ∣Lp(BJ)‖σ+‖(Sk2)1/σ∣Lp(BJ)‖σ)≤c∑k≥J+2(2s+2n+τn-N)σk|BJ|τσ(‖f∣Fp,qs,τ‖M′)σ≤c|BJ|τσ(‖f∣Fp,qs,τ‖M′)σ,
since N>2(s+n)+τn. This finishes the proof of Lemma 3.4.Fors>0, M∈ℕ, τ∈[0,∞), 1≤p≤∞, and 0<q≤∞, we set(3.32)‖f∣Bp,qs,τ‖M=‖f∣Lτp(Rn)‖+sup⁡BJ1|BJ|τ(∫02-J++1t-sqsup⁡|h|≤t‖ΔhMf∣Lp(BJ)‖qdtt)1/q.
Here the supremum is taken over all J∈ℤ and all balls BJ of ℝn with radius 2-J. Similar arguments yield.Lemma 3.5.
Lets>0, M∈ℕ, J∈ℤ, τ∈[0,∞), 1≤p≤∞ and  0<q≤∞. Then there is a constant  c>0, independent of  J, such that
(3.33)(∑j≥J+2jsq(∫|v|≤1‖Δ2-jvMf∣Lp(BJ)‖dv)q)1/q≤c|BJ|τ‖f∣Bp,qs,τ‖M,(∑j≥J+2jsq(∫|v|>1‖Δ2-jvMf∣Lp(BJ)‖|ω(v)|dv)q)1/q≤c|BJ|τ‖f∣Bp,qs,τ‖M,
for any ball  BJ of  ℝn with radius  2-J, any ω∈𝒮(ℝn) and any function f such that ∥f∣Bp,qs,τ∥M<∞.Now we recall the following lemma which is useful for us.Lemma 3.6.
Let0<a<1, J∈ℤ and  0<q≤∞. Let  {εk} be a sequences of positive real numbers, such that
(3.34)‖{εk}k≥J+∣lq,J+0‖=I<∞.
The sequences  {δk:δk=∑j=J+kak-jεj}k≥J+and{ηk:ηk=∑j=k∞aj-kεj}k≥J+, are in  ℓq,J+0 with
(3.35)‖{δk}k≥J+∣lq,J+0‖+‖{ηk}k≥J+∣lq,J+0‖≤cI,cdepends only on  a and  q.


## 4. Characterizations with Differences

We are able to state the main results of this paper.Theorem 4.1.
Let1≤p<∞, 0<q≤∞, τ∈[0,∞)  and  M∈ℕ. Assume
(4.1)nmin⁡(p,q)<s<M,0≤τ<1p
or
(4.2)nmin⁡(p,q)<s<M-nτ+np,τ≥1p.
Then ∥f∣Fp,qs,τ∥M is an equivalent quasinorm in Fp,qs,τ.Theorem 4.2.
Let1≤p≤∞,  0<q≤∞, τ∈[0,∞) and  M∈ℕ. Assume
(4.3)0<s<M,0≤τ<1p
or
(4.4)0<s<M-nτ+np,τ≥1p.
Then ∥f∣Bp,qs,τ∥M is an equivalent quasinorm in Bp,qs,τ.Remark 4.3.
Theorems4.1 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 Theorem4.1.
LetBJ be any ball centered at x0∈ℝn and of radius 2-J, J∈ℤ. We will do the proof in three steps. Step 1.
We have withs>0,
(4.5)∑j≥0|Δjf|=∑j≥02-js2js|Δjf|≤csup⁡j∈N02js|Δjf|≤c(∑j≥02jsq|Δjf|q)1/q.
Let f∈Fp,qs,τ. Then,
(4.6)‖f∣Lτp(Rn)‖≤‖∑j≥0|Δjf|∣Lτp(Rn)‖≤c‖(∑j≥02jsq|Δjf|q)1/q∣Lτp(Rn)‖≤c‖f∣Fp,qs,τ‖.Step 2.
For anyx∈BJ we put
(4.7)HJ(x)=∫02-J++1t-sqsup⁡|h|≤t|ΔhMf(x)|qdtt.
After a change of variable t=2-y, we get
(4.8)HJ(x)=ln⁡2∫J+-1+∞2ysqsup⁡|h|≤2-y|ΔhMf(x)|qdy.
Then
(4.9)HJ(x)≤c∑k≥J+2ksqsup⁡|h|≤2-k+1|ΔhMf(x)|q.Let ψ,ψ0∈𝒮(ℝn) 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∈BJ, j∈ℕ0, and |h|≤2-k+1(4.10)|Δh1Δjf(x)|=|Δh1(ψj*Δjf)(x)|≤2-ksup⁡|x-y|≤c2-k∑|α|=1|Dαψj*Δjf(y)|,
with some positive constant c, independent of j and k, and ψj(·)=2(j-1)nψ(2j-1·) for j=1,2,…. By induction on M, we show that
(4.11)|ΔhMΔjf(x)|≤2-kMsup⁡|x-y|≤c2-k∑|α|=M|Dαψj*Δjf(y)|.
We see that if |α|=M and a>0(4.12)|Dαψj*Δjf(y)|=2(j-1)n|∫RnDα(ψ(2j-1(y-z)))Δjf(z)dz|≤2(j-1)(n+M)∫Rn|(Dαψ)(2j-1(y-z))||Δjf(z)|dz.
Suppose that 0≤j≤J+-1. The right-hand side in (4.12) may be estimated as follows:
(4.13)c2j(n+M)Δj*,af(y)∫Rn|(Dαψ)(2j-1(y-z))|(1+2j|y-z|)adz≤c2jMΔj*,af(y).
Then we obtain for any x∈BJ, |h|≤2-k+1 and any k≥J+(4.14)|ΔhMΔjf(x)|≤c2(j-k)Msup⁡|x-y|≤c2-kΔj*,af(y)≤c2(j-k)M(1+2j-k)asup⁡|x-y|≤c2-kΔj*,af(y)(1+2j|x-y|)a≤c2(j-k)MΔj*,af(x),
if 0≤j≤J+-1.
Suppose now thatJ+≤j≤k. By our assumption on x and k we have
(4.15)|y-x0|≤|x-x0|+|x-y|<2-J+c2-k≤(c+1)2-J,
which implies that y is located in some ball B̃J, where B̃J={y∈ℝn:|y-x0|<(c+1)2-J}. Writing the integral in (4.12) as follows
(4.16)∫B̃J-1⋯dz+∑i≥0∫B̃J-i-2∖B̃J-i-1⋯dz=Ij,J(y)+∑i≥0IIj,J-i(y).
We recall that
(4.17)Δj,l*,af(y)=sup⁡z∈B̃l|Δjf(z)|(1+2j|y-z|)a,
for any j∈ℕ0, f∈𝒮′(ℝn) and any l∈ℤ. We have
(4.18)Ij,J(y)≤Δj,J-1*,af(y)∫B̃J-1|(Dαψ)(2j-1(y-z))|(1+2j|y-z|)adz≤c2-jnΔj,J-1*,af(y)≤c2-jnΔj,J-2*,af(y).
Let us estimate IIj,J-i. Since ψ∈𝒮(ℝn), we have
(4.19)|Dαψ(x)|≤c(1+|x|)-2N,
for any x∈ℝn and any N>0. Then for any N large enough, IIj,J-i(y) does not exceed
(4.20)cΔj,J-i-2*,af(y)∫B̃J-i-2∖B̃J-i-1(1+2j-1|y-z|)-2N+adz≤c2-iNΔj,J-i-2*,af(y)∫Rn(1+2j-1|y-z|)-N+adz≤c2-iN-jnΔj,J-i-2*,af(y),
where we have used 2j-1|y-z|>(c+1)2j-J+i-1≥(c+1)2i-1. Therefore,
(4.21)|Dαψj*Δjf(y)|≤c2jM∑i≥02-iNΔj,J-i-2*,af(y).
Then we obtain for any x∈BJ any |h|≤2-k+1 and any J+≤j≤k(4.22)|ΔhMΔjf(x)|≤c2(j-k)M∑i≥02-iNsup⁡|x-y|≤c2-kΔj,J-i-2*,af(y)≤c2(j-k)M(1+2j-k)a∑i≥02-iNsup⁡|x-y|≤c2-kΔj,J-i-2*,af(y)(1+2j|x-y|)a.
Consequently, for any J+≤j≤k there is a constant c>0 independent of J, j, and k such that
(4.23)|ΔhMΔjf(x)|≤c2(j-k)M∑i≥02-iNΔj,J-i-2*,af(x).
Finally for j≥k+1 we have for x∈BJ and |h|≤2-k+1(4.24)|ΔhMΔjf(x)|≤∑m=0MCmM|Δjf(x+(M-m)h)|≤2Msup⁡|x-y|≤C2-k|Δjf(y)|≤2Msup⁡|x-y|≤C2-k|Δjf(y)|(1+2j|x-y|)a(1+2j|x-y|)a.
We remark also that by our assumption on x and k we have
(4.25)|y-x0|≤|x-x0|+|x-y|<2-J+C2-k≤(C+1)2-J,
and this implies that y is located in some ball B̃J, where B̃J={y∈ℝn:|y-x0|<(C+1)2-J}. Then,
(4.26)|ΔhMΔjf(x)|≤c2(j-k)aΔj,J*,af(x),
if j≥k+1, where Δj,J*,af is given in (4.17) (with B̃J a ball centered at x0 and of radius (C+1)2-J).
We write,(4.27)ΔhMf(x)=∑j≥0ΔhMΔjf(x)=∑j=0J+-1⋯+∑j=J+k⋯+∑j≥k+1⋯.
Here we put ∑j=0J+-1⋯=0 if J+=0. Let us estimate each term in ℓq,J+s-norm. We have by (4.14) and Lemma 3.1(4.28)∑j=0J+-1|ΔhMΔjf(x)|≤∑j=0J+-12(j-k)MΔj*,af(x)≤c∑j=0J+-12(j-k)M+j(n/p-s-nτ)‖f∣Fp,qs,τ‖≤c2-kM∑j=0J+-12j(M+n/p-s-nτ)‖f∣Fp,qs,τ‖≤c2J(M+n/p-s-nτ)-kM‖f∣Fp,qs,τ‖,
where the last inequality can be obtained by our assumption on s and τ. The last expression in ℓq,J+s-norm does not exceed
(4.29)c2Jn(1/p-τ)‖{2(k-J+)(s-M)}k≥J+∣lq,J+0‖‖f∣Fp,qs,τ‖≤c2Jn(1/p-τ)‖f∣Fp,qs,τ‖,
since s<M. Therefore,
(4.30)(HJ(x))1/q≤c2Jn(1/p-τ)‖f∣Fp,qs,τ‖(4.31)+c(∑k≥J+(∑i≥0∑j=J+k2(j-k)(M-s)+sj-iN|Δj,J-i-2*,af(x)|)q)1/q(4.32)+c(∑k≥J+(∑j≥k2(j-k)(a-s)+sj|Δj,J*,af(x)|)q)1/q.
The second term can be estimated by (with σ=min(1,q))
(4.33)c(∑i≥02-iNσ(∑k≥J+(∑j=J+k2(j-k)(M-s)+sj|Δj,J-i-2*,af(x)|)q)σ/q)1/σ.
Since again s<M, then we can apply Lemma 3.6 to estimate the last expression by
(4.34)c(∑i≥02-iNσ(∑j≥J+2jsq|Δj,J-i-2*,af(x)|q)σ/q)1/σ.
Since Lp/σ(BJ) is a normed space, so (4.31) in Lp(BJ)-norm is dominated by
(4.35)c(∑i≥02-iNσ‖(∑j≥J+2jsq|Δj,J-i-2*,af|q)σ/q∣Lp/σ(BJ)‖)1/σ=c(∑i≥02-iNσ‖(∑j≥J+2jsq|Δj,J-i-2*,af|q)1/q∣Lp(BJ)‖σ)1/σ.
Using the embedding Lp(B̃J-i-2)⊂Lp(BJ) and the fact that J+≥(J-i-2)+ to estimate this expression from above by
(4.36)c(∑i≥02-iNσ‖(∑j≥(J-i-2)+2jsq|Δj,J-i-2*,af|q)1/q∣Lp(B̃J-i-2)‖σ)1/σ≤c|BJ|τ(∑i≥02i(nτ-N)σ)1/σ‖f∣Fp,qs,τ‖≤c|BJ|τ‖f∣Fp,qs,τ‖,
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
(4.37)c(∑j≥J+2jsq|Δj,J*,af(x)|q)1/q.
This expression, by Theorem 2.5, in Lp(BJ)-norm is bounded by c|BJ|τ∥f∣Fp,qs,τ∥. Hence we have for any J∈ℤ and any ball BJ of ℝn with radius 2-J(4.38)1|BJ|τ‖(HJ)1/q∣Lp(BJ)‖≤c‖f∣Fp,qs,τ‖,
with some positive constant c independent of J. From this it follows that
(4.39)‖f∣Fp,qs,τ‖M≤c‖f∣Fp,qs,τ‖,
for any f∈Fp,qs,τ.
Step 3.
LetΨ 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
(4.40)ψ(x)=(-1)M+1∑i=0M-1(-1)iCiMΨ(x(M-i)).
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 φj(x)=φ1(2-j+1x) for j=2,3,…, we obtain that {φj} is a smooth dyadic resolution of unity. This yields that
(4.41)sup⁡BJ1|BJ|τ‖(∑j≥J+2jsq|Δjf|q)1/q∣Lp(BJ)‖,
is a norm equivalent in Fp,qs,τ (see Remark 2.3). Let us prove that
(4.42)1|BJ|τ‖(∑j≥J+2jsq|Δjf|q)1/q∣Lp(BJ)‖≤c‖f∣Fp,qs,τ‖M,
for any ball BJ of ℝn with radius 2-J. First the left-hand side contains Δ0f only when J+=0. Then
(4.43)‖Δ0f∣Lp(BJ)‖≤∫Rn‖f∣Lp(B¯J)‖|F-1ψ(y)|dy,
where B¯J is a ball centered at x0+y and of radius 2-J. Hence
(4.44)‖Δ0f∣Lp(BJ)‖≤c|BJ|τ‖f∣Lτp‖≤c|BJ|τ‖f∣Fp,qs,τ‖M,
where we have used the fact that |B¯J|=|BJ|. Moreover, it holds for x∈ℝn and j=1,2,…(4.45)Δjf(x)=(-1)M+1∫RnΔ2-jvMf(x)F-1Ψ̃(v)dv,
with Ψ̃(·)=ℱ-1Ψ(·)-2-nℱ-1Ψ(·/2) (see [17, Theorem 3.1]). Now, for j∈ℕ we write
(4.46)∫Rn|Δ2-jvMf(x)||F-1Ψ̃(v)|dv=∫|v|≤1|Δ2-jvMf(x)||F-1Ψ̃(v)|dv+(-1)M+1∫|v|>1|Δ2-jvMf(x)||F-1Ψ̃(v)|dv.
Then the estimate (4.42) is an obvious consequence of (4.44) and Lemma 3.4. Therefore,
(4.47)‖f∣Fp,qs,τ‖≤c‖f∣Fp,qs,τ‖M,
which completes the proof of Theorem 4.1.Proof of Theorem4.2.
The first two steps closely follow the argument in [17, Theorem 3.1].Step 1.
Letf∈Bp,qs,τ. Since s>0, then we have
(4.48)‖f∣Lτp(Rn)‖≤∑j≥02-js2js‖Δjf∣Lτp(Rn)‖≤csup⁡j∈N02js‖Δjf∣Lτp(Rn)‖≤c‖f∣Bp,qs,τ‖.Step 2.
As in the proof of Theorem4.1 we have
(4.49)(∫02-J++1t-sqsup⁡|h|≤t‖ΔhMf∣Lp(BJ)‖qdtt)1/q≤c(∑k≥J+2ksqsup⁡|h|≤2-k+1‖ΔhMf∣Lp(BJ)‖q)1/q.
Let us estimate ΔhMΔjf. If j≤k, then as in the proof Theorem 4.1, we have
(4.50)|ΔhMΔjf(x)|≤c2(j-k)MΔj*,af(x),
for any x∈BJ, |h|≤2-k+1 and any a>0. If j>k we have for x∈ℝn and |h|≤2-k+1(4.51)|ΔhMΔjf(x)|≤∑m=0MCmM|Δjf(x+(M-m)h)|.
Hence we obtain for any j>k and any a>0(4.52)‖ΔhMΔjf∣Lp(BJ)‖≤∑m=0MCmM‖Δjf∣Lp(B̃J)‖≤∑m=0MCmM‖Δj*,af∣Lp(B̃J)‖,
where if x0 the centre of BJ then x0+(M-m)h is the centre of B̃J. We remark also that by our assumption on h and k we have
(4.53)|x-x0|≤|x-(x0+(M-m)h))|+(M-m)|h|<2-J+M2-k+1≤(2M+1)2-J,
for any x∈B̃J. We denote B̂J the ball in ℝn centred at x0 and of radius (2M+1)2-J. Since Lp(B̂J)⊂Lp(B̃J) and Lp(B̂J)⊂Lp(BJ), we get
(4.54)‖ΔhMf∣Lp(BJ)‖≤c∑j=0J+-12(j-k)M‖Δj*,af∣Lp(BJ)‖+c∑j=J+k2(j-k)M‖Δj*,af∣Lp(B̂J)‖+c∑j≥k‖Δj*,af∣Lp(B̂J)‖.
Here we put ∑j=0J+-1⋯=0 if J+=0. Lemma 3.1 gives
(4.55)∑j=0J+-12(j-k)M‖Δj*,af∣Lp(BJ)‖≤c∑j=0J+-12(j-k)M+j(n/p-s-nτ)-Jn/p‖f∣Bp,qs,τ‖≤c2J(M-s-nτ)-kM‖f∣Bp,qs,τ‖.
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
(4.56)c|BJ|τ‖f∣Bp,qs,τ‖‖{2(k-J+)(s-M)}k≥J+∣lq,J+0‖+c(∑j≥J+2jsq‖Δj*,af∣Lp(B̂J)‖q)1/q≤c|BJ|τ‖f∣Bp,qs,τ‖,
where we have used Theorem 2.4, combined with Remark 2.6, and the equation |B̂J|=(2M+1)n|BJ|.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
(4.57)‖f∣Bp,qs,τ‖≤c‖f∣Bp,qs,τ‖M.
This ends the proof of Theorem 4.2.Finally we study, in addition, the caseτ∈[1/p,∞). Under this condition we can restrict supBJ in the definition of Bp,qs,τ and Fp,qs,τ to a supremum taken with respect to balls BJ of ℝn with radius 2-J and J∈ℕ0.Lemma 4.4.
Lets∈ℝ, τ∈[1/p,∞) and  0<q≤∞.
Let0<p≤∞. A tempered distribution f belongs to Bp,qs,τ if and only if
(4.58)‖f∣Bp,qs,τ‖#=sup⁡BJ1|BJ|τ(∑j≥J2jsq‖Δjf∣Lp(BJ)‖q)1/q<∞,
where the supremum is taken over all  J∈ℕ0 and all balls  BJ of  ℝn with radius  2-J. Furthermore, the quasinorms ∥f∣Bp,qs,τ∥# and ∥f∣Bp,qs,τ∥ are equivalent.
Let0<p<∞. A tempered distribution f belongs to Fp,qs,τ if and only if
(4.59)‖f∣Fp,qs,τ‖#=sup⁡BJ1|BJ|τ‖(∑j≥J2jsq|Δjf|q)1/q∣Lp(BJ)‖<∞,
where the supremum is taken over all  J∈ℕ0 and all balls  BJ of  ℝn with radius  2-J. Furthermore, the quasinorms ∥f∣Fp,qs,τ∥# and ∥f∣Fp,qs,τ∥ are equivalent.Proof.
For eachJ∈ℤ and m=(m1,…,mn)∈ℤn, set
(4.60)QJ,m={x=(x1,…,xn)∈Rn:mi≤2Jxi<(mi+1),i=1,2,…,n}.
This lemma for QJ,m in place of BJ 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
(4.61)QJ,m⊂∪l=1vBJl,BJ⊂∪l=1kQJl.
Here BJl is a ball of center 2-J and QJl 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 for1≤p≤∞(4.62)Mp(f)=sup⁡B0(∫B0|f(x)|pdx)1/p,
where the supremum is taken over all balls B0 of ℝn with radius 1.Theorem 4.5.
Let1≤p<∞, 0<q≤∞, τ∈[1/p,∞), M∈ℕ and  n/min(p,q)<s<M-nτ+n/p. Then
(4.63)‖f∣Fp,qs,τ‖M#=Mp(f)+sup⁡BJ1|BJ|τ‖(∫02-J+1t-sqsup⁡|h|≤t|ΔhMf(⋅)|qdtt)1/q∣Lp(BJ)‖
is an equivalent quasinorm in Fp,qs,τ. Here the supremum is taken over all J∈ℕ0 and all balls BJ of ℝn with radius 2-J.Theorem 4.6.
Let1≤p≤∞, 0<q≤∞, τ∈[1/p,∞), M∈ℕ and  0<s<M-nτ+n/p. Then
(4.64)‖f∣Bp,qs,τ‖M#=Mp(f)+sup⁡BJ1|BJ|τ(∫02-J+1t-sqsup⁡|h|≤t‖ΔhMf∣Lp(BJ)‖qdtt)1/q
is an equivalent quasinorm in Bp,qs,τ. Here the supremum is taken over all J∈ℕ0 and all balls BJ of ℝn with radius 2-J.Proof.
We will prove only Theorem4.5. The proof of Theorem 4.6 is similar. We employ the same notations given in the proof of Theorem 4.1.Step 1.
Letf∈Fp,qs,τ. Since s>0, then we have
(4.65)Mp(f)≤∑j≥02-js2jsMp(Δjf)≤csup⁡B0,j∈N02js‖Δjf∣Lp(B0)‖≤c‖f∣Fp,qs,τ‖#.
Step 2.
As in the proof of Theorem4.1, there is a constant c>0 independent of J such that
(4.66)1|BJ|τ‖(HJ)1/q∣Lp(BJ)‖≤c‖f∣Fp,qs,τ‖≤c‖f∣Fp,qs,τ‖#,
by Lemma 4.4.Step 3.
The left-hand side in (4.42) (with J+=J) contains Δ0f only when J=0. Then
(4.67)‖Δ0f∣Lp(B0)‖≤∫Rn‖f∣Lp(B¯0)‖|F-1ψ(y)|dy≤cMp(f).
We recall that for x∈ℝn and j=1,2,…(4.68)Δjf(x)=(-1)M+1∫RnΔ2-jvMf(x)F-1Ψ̃(v)dv=(-1)M+1∫|v|≤1Δ2-jvMf(x)F-1Ψ̃(v)dv+(-1)M+1∫|v|>1Δ2-jvMf(x)F-1Ψ̃(v)dv.
As in the proof of Lemma 3.4, we can prove that
(4.69)‖(∑j≥J2jsq(∫|v|≤1|Δ2-jvMf(⋅)|dv)q)1/q∣Lp(BJ)‖≤c|BJ|τ‖f∣Fp,qs,τ‖M#,‖(∑j≥J2jsq(∫|v|>1|Δ2-jvMf(⋅)ω(v)|dv)q)1/q∣Lp(BJ)‖≤c|BJ|τ‖f∣Fp,qs,τ‖M#,
for any J∈ℕ any ball BJ of ℝn with radius 2-J and any ω∈𝒮(ℝn). The proof is completed.Remark 4.7.
Recently, Yang and Yuan [18, Theorem 2] proved that
(4.70)Fp,qs,τ=F∞,∞s+n(τ-1/p),0<p<∞,s∈R,Bp,qs,τ=B∞,∞s+n(τ-1/p),0<p≤∞,s∈R,
if τ>1/p, 0<q≤∞ or if τ=1/p and q=∞. Under these conditions the study of the Triebel-Lizorkin-type space Fp,qs,τ and the Besov-type space Bp,qs,τ is not interest.



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*Source: 328908-2012-01-12.xml*