#### Abstract

We present characterizations of the Besov-type spaces and the Triebel-Lizorkin-type spaces by differences. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking .

#### 1. Introduction

The spaces and spaces have been studied extensively in recent years. When they coincide with the usual function spaces and , respectively, studied in detail by Triebel in [1–3]. When , and , the 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 cover certain Campanato spaces, studied in [6, 7]. Later on, Drihem gave in [8] a characterization for spaces by local means and maximal functions. For a complete treatment of spaces and 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 and (), which generalize the homogeneous Besov-Triebel-Lizorkin spaces , and established the relation between and 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 .

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 spaces and spaces by differences.

#### 2. Preliminaries

##### 2.1. Notation and Conventions

As usual, the -dimensional real Euclidean space, the collection of all natural numbers, and . The letter stands for the set of all integer numbers. For a multi-index , we write and . For ,let be the ball of with radius and . The Euclidean scalar product of and is given by . We denote by the -dimensional Lebesgue measure of . For any measurable subset the Lebesgue space , consists of all measurable functions for which

By we denote the Schwartz space of all complex-valued, infinitely differentiable, and rapidly decreasing functions on and by the dual space of all tempered distributions on . We define the Fourier transform of a function by Its inverse is denoted by . Both and are extended to the dual Schwartz space in the usual way.

Let and . Let be the collection of functions such that where the supremum is taken over all and all balls of with radius . Obviously, when , then . Furthermore, (see [9, page 46]).

If , and , then is the set of all sequences of complex numbers such that with the obvious modification if . We recall that for any and any

Let be an arbitrary function on and , . 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 are the binomial coefficients.

Recall that , for any , and . By we denote generic positive constants, which may have different values at different occurrences.

##### 2.2. The Spaces and Spaces

In this subsection we present the Fourier analytical definition of spaces, spaces and recall their basic properties. We first need the concept of a smooth dyadic resolution of unity.

*Definition 2.1. *Let be a function in satisfying for and for *. *We put , and
Then we have , for and for all . The system of functions is called a smooth dyadic resolution of unity. We define the convolution operators by the following:
Thus we obtain the Littlewood-Paley decomposition
of all .

The spaces and spaces are defined in the following way.

*Definition 2.2. *(i) Let , and , *.* The space is the collection of all such that
where the supremum is taken over all and all balls * of * with radius .

(ii) Let , , and *. *The space is the collection of all such that
where the supremum is taken over all and all balls * of * with radius .

*Remark 2.3. *The spaces and are independent of the particular choice of the smooth dyadic resolution of unity appearing in their definitions. They are quasi-Banach spaces Banach spaces if , *.* In particular,
where and are the Besov spaces and Triebel-Lizorkin spaces respectively. If we replace the balls by dyadic cubes (with side length ) we obtain equivalent norms.

The full treatment of both scales of spaces can be found in [9]. Let () be the functions introduced in Definition 2.1. For any , any and any we denote (Peetre’s maximal functions)

We now present a fundamental characterization of spaces and spaces.

Theorem 2.4. *Let , , , and . Then
**
is an equivalent quasinorm in .*

Theorem 2.5. *Let , , , and . Then , is an equivalent quasinorm in .*

*Remark 2.6. *Theorem 2.4 for , is given in [8, Theorem 4.5]. For Theorem 2.5 see [12, Theorem 1.1]. In addition if *,* then in Theorem 2.4 can be replaced by , where

#### 3. Some Technical Lemmas

To prove our results, we need some technical lemmas. The following lemma for , in place of , is given in [14, pages 87–89] (for the spaces and ). Further results, can be found in [12, Lemma 2.4].

Lemma 3.1. *Let be as in Definition 2.1 and let , , and , for the space ). Then there is a constant , independent of , such that for any **
Here one uses to denote either or .*

*Proof. *Let be two functions such that and on and respectively. Then
with if . Since , the right-hand side is bounded by , for any . Hence we get for all and any
Using the same method given in [9,Proposition 2.6] we obtain for any
The proof is completed.

Lemma 3.2. *Let , , , and . Then there is a constant , independent of and , such that
**
for any ball of with radius and any function such that .*

*Proof. *Since , the left-hand side is bounded by
From the definition of we have
Take the -norm to estimate (3.6) form above by
where if the centre of then is the centre of . Using the fact that to estimate (3.8) from above by .

The lemma is proved.

*Remark 3.3. *Let *, **, **, *and be as in Lemma 3.2. Let *.* By the embedding there is a constant , independent of and , such that
for any ball of with radius and any function such that .

For , , , and , we set

Here the supremum is taken over all and all balls of with radius .

Lemma 3.4. *Let , , , , and . Then there is a constant , independent of , such that
**
for any ball of with radius , any and any function such that .*

*Proof. *Let be a dyadic cube with side length . This result, for in place of , 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 . The case is similar.

Before proving this result we note that for any and any
Here we will prove that the left-hand side of (3.12) is bounded by
Obviously, . We write
where is at our disposal and we have used the properties of the function ,
for any and any . Now the right-hand side of (3.15) in -norm is bounded by (with )
by (2.7). Here we put if . Take the -norm we obtain that the left-hand side of (3.12) is bounded by
Let us estimate in -norm. After a change of variable , we get for any (here and )
Here we put if . We have
Since is a normed spaces and , the right-hand side in -norm can be estimated from above by
We choose . This yields that the last expression is bounded by
where is independent of . Now in -norm is bounded by
where we have used (2.7). Using the embedding and Remark 3.3, we obtain
because of . Therefore,
Now let us estimate in -norm. We write
After a change of variable , we get
Therefore there exists a constant independent of and such that
by Lemma 3.2 (combined with Remark 3.3) and the fact that . Now let us estimate in -norm. We have
Therefore,
Consequently,
since . This finishes the proof of Lemma 3.4.

For , , , , and , we set Here the supremum is taken over all and all balls of with radius . Similar arguments yield.

Lemma 3.5. *Let , , , , and . Then there is a constant , independent of , such that
**
for any ball of with radius , any and any function such that .*

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

Lemma 3.6. *Let , and . Let be a sequences of positive real numbers, such that
**
The sequences , are in with
**depends only on and .*

#### 4. Characterizations with Differences

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

Theorem 4.1. *Let , , and . Assume
**
or
**
Then is an equivalent quasinorm in .*

Theorem 4.2. *Let , , and . Assume
**
or
**
Then is an equivalent quasinorm in .*

*Remark 4.3. *Theorems 4.1 and 4.2 for and , respectively, are given in [9] Theorems 4.6 and 4.7, respectively.

*Proof of Theorem 4.1. *Let be any ball centered at and of radius , . We will do the proof in three steps. *Step 1. *We have with ,
Let . Then,
*Step 2. * For any we put
After a change of variable , we get
Then
Let be two functions such that and on supp and supp respectively. Using the mean value theorem we obtain for any , , and
with some positive constant , independent of and , and for . By induction on , we show that
We see that if and
Suppose that . The right-hand side in (4.12) may be estimated as follows:
Then we obtain for any , and any
if .

Suppose now that . By our assumption on and we have
which implies that is located in some ball , where . Writing the integral in (4.12) as follows
We recall that
for any , and any . We have
Let us estimate . Since , we have
for any and any . Then for any large enough, does not exceed
where we have used . Therefore,
Then we obtain for any any and any
Consequently, for any there is a constant independent of , , and such that
Finally for we have for and
We remark also that by our assumption on and we have
and this implies that is located in some ball , where . Then,
if , where is given in (4.17) (with a ball centered at and of radius .

We write,
Here we put if . Let us estimate each term in -norm. We have by (4.14) and Lemma 3.1
where the last inequality can be obtained by our assumption on and . The last expression in -norm does not exceed
since . Therefore,
The second term can be estimated by (with )
Since again , then we can apply Lemma 3.6 to estimate the last expression by
Since is a normed space, so (4.31) in -norm is dominated by
Using the embedding and the fact that to estimate this expression from above by
where the first inequality is obtained by Theorem 2.5 and the second inequality follows by taking . Taking , then using again Lemma 3.6 to estimate (4.32) by
This expression, by Theorem 2.5, in -norm is bounded by . Hence we have for any and any ball of with radius
with some positive constant independent of . From this it follows that
for any .*Step 3. *Let 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
The function satisfies for and for . Then, taking and for , we obtain that is a smooth dyadic resolution of unity. This yields that
is a norm equivalent in (see Remark 2.3). Let us prove that
for any ball of with radius . First the left-hand side contains only when . Then
where is a ball centered at and of radius . Hence
where we have used the fact that . Moreover, it holds for and