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

We present characterizations of the Besov-type spaces 𝐵𝑠,𝜏𝑝,𝑞 and the Triebel-Lizorkin-type spaces 𝐹𝑠,𝜏𝑝,𝑞 by differences. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking 𝜏=0.


Introduction
The B s,τ p,q spaces and F s,τ p,q spaces have been studied extensively in recent years.When τ 0 they coincide with the usual function spaces B s p,q and F s p,q , respectively, studied in detail by Triebel in 1-3 .When s ∈ R, τ ∈ 0, ∞ and 1 ≤ p, q < ∞ the B s,τ p,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,τ p,q cover certain Campanato spaces, studied in 6, 7 .Later on, Drihem gave in 8 a characterization for B s,τ p,q spaces by local means and maximal functions.For a complete treatment of B s,τ p,q spaces and F s,τ p,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 Ḃs,τ p,q and Ḟs,τ p,q p / ∞ , which generalize the homogeneous Besov-Triebel-Lizorkin spaces Ḃs,τ p,q , Ḟs,τ p,q and established the relation between Ḟs,τ p,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,τ p,q spaces and F s,τ p,q spaces by differences.

Notation and Conventions
As usual, R n the n-dimensional real Euclidean space, N the collection of all natural numbers, and N 0 N ∪ {0}.The letter Z stands for the set of all integer numbers.For a multi-index α α 1 , . . ., α n ∈ N n 0 , we write |α| 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 x 1 , . . ., x n and y y 1 , . . ., y n is given by x • y x 1 y 1 • • • x n y 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

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 by Its inverse is denoted by F −1 f.Both F and F −1 are extended to the dual Schwartz space S R n in the usual way.
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 If s ∈ R, 0 < q ≤ ∞ and J ∈ Z, then s q,J is the set of all sequences {f k } k≥J of complex numbers such that with the obvious modification if q ∞.We recall that for any 0 < θ ≤ 1 and any Let f be an arbitrary function on R n and x, h ∈ R n .Then 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 where C M j are the binomial coefficients.Recall that η j,N x 2 jn 1 2 j |x| −N , for any x ∈ R n , j ∈ N 0 and N > 0. By c we denote generic positive constants, which may have different values at different occurrences.

The B s,τ
p,q Spaces and F s,τ p,q Spaces In this subsection we present the Fourier analytical definition of B s,τ p,q spaces, F s,τ p,q spaces and recall their basic properties.We first need the concept of a smooth dyadic resolution of unity.

2.10
Then we have supp The system of functions {ϕ j } is called a smooth dyadic resolution of unity.We define the convolution operators Δ j by the following:

Journal of Function Spaces and Applications
Thus we obtain the Littlewood-Paley decomposition The B s,τ p,q spaces and F s,τ p,q spaces are defined in the following way.
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,τ p,q is the collection of all f ∈ S R n such that 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,τ p,q and F s,τ p,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,0 p,q B s p,q , F s,0 p,q F s p,q , 2.15 where B s p,q and F s p,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 j f j ∈ N 0 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

2.16
We now present a fundamental characterization of B s,τ p,q spaces and F s,τ p,q spaces.Theorem 2.4.Let s ∈ R, τ ∈ 0, ∞ , 0 < p, q ≤ ∞ and a > n/p.Then is an equivalent quasinorm in B s,τ p,q .

Some Technical Lemmas
To prove our results, we need some technical lemmas.The following lemma for Δ j f, in place of Δ * ,a j f, is given in 14, pages 87-89 for the B s,τ p,q spaces and 1 ≤ p < ∞ .Further results, can be found in 12, Lemma 2.4 .Lemma 3.1.Let Δ j f 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,τ p,q ).Then there is a constant c > 0, independent of j, such that for any Here one uses A s,τ p,q to denote either B s,τ p,q or F s,τ p,q .
Proof.Let ψ, ψ 0 ∈ S R n be two functions such that Fψ 1 and Fψ 0 1 on supp ϕ 1 and supp ϕ 0 respectively.Then Since ϕ ∈ S R n , the right-hand side is bounded by cη j−1,N * |Δ j f| y , for any N > 0. Hence we get for all f ∈ S R n and any Using the same method given in 9, Proposition 2.6 we obtain for any The proof is completed.
for any ball B J of R n with radius 2 −J and any function From the definition of Δ M h f we have Take the L p B J -norm to estimate 3.6 form above by The lemma is proved.Remark 3.3.Let M, A, τ, and p be as in Lemma 3.2.Let J ∈ N 0 .By the embedding L p B 0 → L p B J there is a constant c > 0, independent of J and A, such that for any ball B J of R n with radius 2 −J and any function

3.10
Here the supremum is taken over all J ∈ Z and all balls B J of R n with radius 2 −J .
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,τ p,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 ∈ Here we will prove that the left-hand side of 3.12 is bounded by

3.14
Obviously, f | F s,τ p,q M ≤ f | F s,τ p,q M .We write where N > 0 is at our disposal and we have used the properties of the function ω, for any x ∈ R n and any N > 0. Now the right-hand side of 3.15 in s q,J -norm is bounded by with σ min 1, 3.17 by 2.7 .Here we put Take the L p B J -norm we obtain that the left-hand side of 3.12 is bounded by

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

3.19
Journal of Function Spaces and Applications 9 Here we put

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

3.21
We choose N > 2 s n τn.This yields that the last expression is bounded by where c > 0 is independent of J. Now M 2 1/σ in L p B J -norm is bounded by

3.23
where we have used 2.7 .Using the embedding L p B 0 → L p B J and Remark 3.3, we obtain

3.24
because of N > s n nτ.Therefore, p,q M σ .

3.25
Now let us estimate II J 1/σ in L p B J -norm.We write

3.26
After a change of variable j − k − 1 v, we get

3.27
Therefore there exists a constant c > 0 independent of J and k such that 1/σ in L p B J -norm.We have Therefore,

3.31
since N > 2 s n τn.This finishes the proof of Lemma 3.4.
Here the supremum is taken over all J ∈ Z and all balls B J of R n with radius 2 −J .Similar arguments yield.

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,τ p,q M < ∞.
Now we recall the following lemma which is useful for us.

3.34
The sequences {δ k : c depends only on a and q.

Characterizations with Differences
We are able to state the main results of this paper.
Then f | B s,τ p,q M is an equivalent quasinorm in B s,τ p,q .
Proof of Theorem 4.1.Let B J be any ball centered at x 0 ∈ R n and of radius 2 −J , J ∈ Z.We will do the proof in three steps.
Step 1.We have with s > 0, Let f ∈ F s,τ p,q .Then, 4.6 Step 2. For any x ∈ B J we put After a change of variable t 2 −y , we get Then 4.9 Let ψ, ψ 0 ∈ S R n be two functions such that Fψ 1 and Fψ 0 1 on suppϕ 1 and suppΨ respectively.Using the mean value theorem we obtain for any x ∈ B J , j ∈ N 0 , and 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 We see that if |α| M and a > 0

4.12
Suppose that 0 ≤ j ≤ J − 1.The right-hand side in 4.12 may be estimated as follows:

4.13
Then we obtain for any x ∈ B J , |h| ≤ 2 −k 1 and any k ≥ J Suppose now that J ≤ j ≤ k.By our assumption on x and k we have which implies that y is located in some ball B J , where Writing the integral in 4.12 as follows We recall that 4.17 for any j ∈ N 0 , f ∈ S R n and any l ∈ Z.We have

4.18
Let us estimate II j,J−i .Since ψ ∈ S R n , we have for any x ∈ R n and any N > 0. Then for any N large enough, II j,J−i y does not exceed where we have used 2 j−1 |y − z| > c 1 2 j−J i−1 ≥ c 1 2 i−1 .Therefore,

4.21
Then we obtain for any x ∈ B J any |h| ≤ 2 −k 1 and any J ≤ j ≤ k

4.22
Consequently, for any J ≤ j ≤ k there is a constant c > 0 independent of J, j, and k such that

Journal of Function Spaces and Applications
Finally for j ≥ k 1 we have for x ∈ B J and |h| ≤ 2 −k 1

4.24
We remark also that by our assumption on x and k we have and this implies that y is located in some ball B J , where Then, where Δ * ,a j,J f is given in 4.17 with B J a ball centered at x 0 and of radius C 1 2 −J .We write,

4.27
Here we put Let us estimate each term in s q,J -norm.We have by 4.14 and Lemma 3.1 where the last inequality can be obtained by our assumption on s and τ.The last expression in s q,J -norm does not exceed since s < M. Therefore, The second term can be estimated by with σ min 1,

4.33
Since again s < M, then we can apply Lemma 3.6 to estimate the last expression by

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 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 This expression, by Theorem 2.5, in L p B J -norm is bounded by c|B J | τ f | F s,τ p,q .Hence we have for any J ∈ Z and any ball with some positive constant c independent of J. From this it follows that for any f ∈ F s,τ p,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

4.40
The function ψ satisfies ψ x 1 for |x| ≤ 1/M and ψ x 0 for |x| ≥ 3/2.Then, taking ϕ 0 x ψ x , ϕ 1 x ψ x/2 − ψ x and ϕ j x ϕ 1 2 −j 1 x for j 2, 3, . .., we obtain that {ϕ j } is a smooth dyadic resolution of unity.This yields that sup 4.41 is a norm equivalent in F s,τ p,q see Remark 2.3 .Let us prove that for any ball B J of R n with radius 2 −J .First the left-hand side contains Δ 0 f only when J 0. Then where B J is a ball centered at x 0 y and of radius 2 −J .Hence

4.44
where we have used the fact that |B J | |B J |.Moreover, it holds for x ∈ R n and j 1, 2, . . .

4.46
Then the estimate 4.42 is an obvious consequence of 4.44 and Lemma 3.4.Therefore, 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,τ p,q .Since s > 0, then we have

4.48
Step 2. As in the proof of Theorem 4.1 we have Let us estimate Δ M h Δ j f.If j ≤ k, then as in the proof Theorem 4.1, we have

4.51
Hence we obtain for any j > k and any a > 0

4.52
where if x 0 the centre of B J then x 0 M − m h is the centre of B J .We remark also that by our assumption on h and k we have for any x ∈ B J .We denote B J the ball in R n centred at x 0 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

4.54
Here we put 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 where we have used Theorem 2.4, combined with Remark 2.6, and the equation 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 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 B J in the definition of B s,τ p,q and F s,τ p,q to a supremum taken with respect to balls B J of R n with radius 2 −J and J ∈ N 0 .Lemma 4.4.Let s ∈ R, τ ∈ 1/p, ∞ and 0 < q ≤ ∞.
Let 0 < p ≤ ∞.A tempered distribution f belongs to B s,τ p,q if and only if where the supremum is taken over all J ∈ N 0 and all balls B J of R n with radius 2 −J .Furthermore, the quasinorms f | B s,τ p,q # and f | B s,τ p,q are equivalent.Let 0 < p < ∞.A tempered distribution f belongs to F s,τ p,q if and only if where the supremum is taken over all J ∈ N 0 and all balls B J of R n with radius 2 −J .Furthermore, the quasinorms f | F s,τ p,q # and f | F s,τ p,q are equivalent.Proof.For each J ∈ Z and m m 1 , . . ., m n ∈ Z n , set

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

4.61
Here B l J is a ball of center 2 −J and Q l J 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. where the supremum is taken over all balls B 0 of R n with radius 1.
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,τ p,q .Since s > 0, then we have

4.65
Step 2. As in the proof of Theorem 4.1, there is a constant c > 0 independent of J such that 4.66 by Lemma 4.4.
Step 3. The left-hand side in 4.42 with J J contains Δ 0 f only when J 0. Then

4.67
We recall that for x ∈ R n and j 1, 2, . . .

4.68
As in the proof of Lemma 3.4, we can prove that 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.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,τ p,q and the Besov-type space B s,τ p,q is not interest.
with Remark 3.3 and the fact that k ≥ J .Now let us estimate S 2 k 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