Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent

Abstract

We introduce new Besov and Triebel-Lizorkin spaces with variable integrable exponent, which are different from those introduced by the second author early. Then we characterize these spaces by the boundedness of the local Hardy-Littlewood maximal operator on variable exponent Lebesgue space. Finally the completeness and the lifting property of these spaces are also given.

1. Introduction

Variable exponent function spaces have attracted many attentions because of their applications in some aspects, such as partial differential equations with nonstandard growth [1], electrorheological fluids [2], and image restoration [3–5]. In fact, since the variable Lebesgue and Sobolev spaces were systemically studied by Kováčik and Rákosník in [6], there are many spaces introduced, such as, Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, and Hardy spaces with variable exponent; see [7–20] and references therein. When the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces, many results in classical harmonic analysis and function theory hold for the variable exponent case; see [21–23].

Let be a measurable function. Denote by the space of all measurable functions on such that for some with the norm Then is a Banach space with the norm .

We will use the following notations: and . The set consists of all satisfying and . Moreover, we define to be the set of measurable functions on with the range in such that . Given , one can define the space as above. This is equivalent to defining it to be the set of all functions such that , where and . We also define a quasinorm on this space by .

Let be a locally integrable function on ; the local variant of the Hardy-Littlewood maximal operator is given by for some constant . We denote the set of such that is bounded on . In 2013 Danelia et al. gave characterizations of , a vector-estimate for the local Hardy-Littlewood maximal operator if , and a Littlewood-Paley square-function characterization of the variable exponent Lebesgue spaces when belongs to in [24].

In 2001 Rychkov used the boundedness of the local Hardy-Littlewood maximal operator to prove a stronger result of the Peetre type for spaces and and gave the lifting property for these spaces in [25].

Motivated by the previous papers, the goal of this paper is to introduce new Besov and Triebel-Lizorkin spaces with variable exponent. To state our result, we need some notations.

Throughout this paper denotes the Lebesgue measure for a measurable set denotes the set of all nonnegative integers. Let and be the dual space of . For , and is the largest integer less than or equal to .

Given a function on , let denote the maximal number such that has vanishing moments up to order . In other words, for all multiindices with . If no moments of vanish, then put . A pair of functions is called satisfying the condition, if and .

Take a function satisfying condition, which is possible for any . (Indeed, the assumption is void for and is satisfied automatically for . For , any with Fourier transform near the origin will do the job.) The notation was introduced by Schott in [26]. More precisely, let it be the set of all for which the estimate is valid with some constants , . It is evidently that includes temperate distributions .

Now, we give the definition of Besov spaces and Triebel-Lizorkin spaces with variable exponent.

Definition 1. Let ,, , as above, , and for .(i)The Besov space with variable exponent is the set of with where (ii)For , the Triebel-Lizorkin space with variable exponent is the set of with where

The key point is to prove that different choices of in Definition 1 do not really change the spaces, leading to equivalent quasinorms. For that has been proved by the second author in 2008, see [19]. To go on, we recall variant Peetre-type maximal functions which was introduced by Rychkov in [25]. Let where

Now it is the position to state our main result.

Theorem 2. Let , , and with such that . Suppose that , and the pairs , and satisfy the condition. Then there are positive constants , , and such that for each , , and all one has Since for any , one immediately gets a consequence of Theorem 2.

Corollary 3. The spaces and with , , and with such that are independent of the particular choice of the function in Definition 1. The quasinorms arising for different are equivalent.

The proof of Theorem 2 will be given in Section 2. In Section 3 we study the completeness and the lifting property of these spaces by using Theorem 2. We will use the notation if there exists a constant such that . If and we will write . Finally we claim that is always a positive constant but it may change from line to line. Other notations will be explained when we meet them.

2. Proof of Theorem 2

We will use the idea of [25] by Rychkov to prove Theorem 2. First we need some lemmas.

Lemma 4 (see [25, Theorem 1.6]). Let a function have nonzero integral, and let . Then for any there exist two functions , , such that has vanishing moments up to order and where and for .

Before the next lemma we denote a special convolution operator which is given by

Lemma 5 (see [25, Lemma 2.10]). Let , , , and and , . Then there is a constant depending only on , , , , such that for all and each , , one has

Lemma 6 (see [24, Corollary 3.2]). Let and ; then there exists a positive constant such that for all sequences of locally integrable functions on

Lemma 7 (see [23, Lemma ]). Let be a real or complex vector space and be a semimodular on . Then and are equivalent. If is continuous, then also and are equivalent, as are and .

Lemma 8. Let and . Then there exists a positive constant such that for where is a positive constant and are locally integrable functions on .

Proof. By homogeneity, it suffices to consider the case Let , where is the set of all unit dyadic cubes in . Then it is easy to get Since , by Minkowski’s inequality and Hölder’s inequality, Since and , the latter factor is uniformly bounded in . We take the th power of the above inequality and integrate it. We get It is easy to know that for and also that By these two observations and (21), we have Applying Lemmas 6 and 7 we obtain Using Lemma 7 again we obtain Thus we have This finishes the proof.

We give a notation of norm in which will be used in the following context:

Lemma 9 (see [27, Lemma 2]). Let , . For any sequence of nonnegative numbers denote Then holds, where is constant and only depends on , .

Lemma 10 (see [19, Lemma 3]). Let , , and . For any sequence of nonnegative measurable functions on , denote Then hold with some constants and .

Proof of Theorem 2. By Lemma 4, take , , with large enough so that (13) is true. It follows that where Because of the elementary inequality we have the following fact: To estimate , note that which follows easily from the moment conditions on and . Furthermore, is supported in the ball , in which By the last two estimates, We put this estimate in (36) and see that if we choose and take into account , then we arrive at with some . It is easy to see that, in the right side of (40), we have essentially the convolution with the sequence , which is of course a bounded operator on any , . Now by Lemmas 9 and 10 for and we easily obtain In other words, we reduce matters to prove (11) and (12) with , . Below we do it only for (11); the argument for (12) is similar.
Let and . By Lemma 5 and a discrete version of the Hardy inequality which we apply with and , we have Note that and . Let ; by Lemmas 6 and 8, the operators and are all bounded on . Hence the desired estimate (11) with , , follows.
This finishes the proof.

3. Some Applications

In this section, we will consider the completeness, the lifting property, and the related quasinorms of these spaces introduced in previous section.

Theorem 11. Let , , and with such that . Then the quasinormed spaces and are quasi-Banach spaces.

Proof. We only give the proof for and for ; it can be proved by the similar way. We use the similar argument in [25]. Let and with supp . We set and in the left side of (11). Analyzing the proof of Theorem 2 shows that only finite numbers of derivatives of the kernels are involved in the estimates, and therefore we know where is a constant and depends on , , , , but not on and .
It is easy to know
We take on both sides of the last inequality and get By (44) and (46) we have where , are constants and depend on , , , , but not on and .
Then we know that the following estimate is valid for all and with some constants , which may depend on , , , , but not on and . Thus we obtain that is continuously embedded in .
Now we conclude the proof of the theorem in a normal way. If a sequence of distributions is Cauchy sequence in , then by (48) it converges “pointwise.” By the completeness of , the sequence has a limit in . Again by (48), we have , since Cauchy sequences are bounded. Finally, by Lebesgue’s theorem on dominated convergence it is easily seen that in .
This finishes the proof.

In next context, we study the action of the Bessel potential operators in our Besov and Triebel-Lizorkin spaces with variable exponent. More precisely, we consider the following -dilated version: where denotes the identity operator.

For this operator acts by the rule , where It is well known that if , then and has the representation (see Stein’s book [28] for these matters), from which it follows rather easily that with is away from the origin and with an absolute constant . By the identity we see that for the distribution agrees in with a function, which again satisfies (52).

By the same argument in page 170 of [25] we know that the convolution can be defined as an element of for any , provided that . The next theorem states explicitly where it acts.

Theorem 12 (the lifting property). Let with such that . Then there is a constant so that for all , , and every positive one has

Proof. The idea of the proof comes from [25]. We use again (13) with and having vanishing moments up to large order . By an argument similar to that one used above to define on Besov and Triebel-Lizorkin spaces with variable exponent, one can establish the identity
From [25] by choosing sufficiently large, we have Now by using Theorem 2 with and , it follows easily that if belongs to or , then is in or , respectively. Then the condition on becomes with .
The fact that the maps in (54) are actually onto follows from the identity .
This finishes the proof.

It follows from Theorem 12 that is an equivalent quasinorm on for small and analogously for . The next theorem gives a version of this result for involving “pure” derivatives.

Theorem 13. Let , , and with such that . Then for any

Proof. For brevity, we only give the outline of the proof for Triebel-Lizorkin space with variable exponent; for Besov space with variable exponent it can be proved by similar way. The “” inequality follows immediately from Definition 1 by partial integration and invoking Theorem 2.
The “” inequality for even follows from Theorem 12. To obtain it for odd, it suffices to consider the case . In view of Theorem 12, it is sufficient to prove the estimate From [25] again, we have All this leads to the following counterpart of (56): where, for , has vanishing moments up to order and satisfy From (58), it is easily deduced by virtue of Theorem 12.
This finishes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for his carefully reading which made the presentation more readable. Jingshi Xu is supported by the National Natural Science Foundation of China (Grant nos. 11071064, 11361020, and 11226167) and the Natural Science Foundation of Hainan Province (no. 113004).