Research Article | Open Access
Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces by Differences
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 .
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  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. . 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  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 , 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.
2.1. 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.
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 .
3. Some Technical Lemmas
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 .
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
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
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