Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 384593, 27 pages

doi:10.1155/2012/384593

Research Article

New Herz Type Besov and Triebel-Lizorkin Spaces with Variable Exponents

Baohua Dong and Jingshi Xu

Department of Mathematics, Hainan Normal University, Haikou 571158, China

Received 19 October 2012; Accepted 3 December 2012

Academic Editor: Baoxiang Wang

Copyright © 2012 Baohua Dong and Jingshi Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The authors establish the boundedness of vector-valued Hardy-Littlewood maximal operator in Herz spaces with variable exponents. Then new Herz type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Finally, characterizations of these new spaces by maximal functions are given.

1. Introduction

Recent decades, variable exponent function spaces have attracted many attention. In fact, since the variable Lebesgue and Sobolev spaces were systemically studied by Kováik and Rákosník in [1], there are many spaces introduced, such as Bessel potential spaces with variable exponent, Besov and Trieble-Lizorkin spaces with variable exponents, and Morrey spaces with variable exponents; see [2–10] and reference therein. When the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces, many results in classic harmonic analysis and function theory are also hold for the variable exponent case; see [11–13]. All mentioned spaces have many applications in differential equations, here we refer the readers to the survey [14].

Meanwhile, there are another type function spaces, Herz type spaces, have attracted many authors' interests for the last three decades. We recommend the readers to the monograph [15] and references therein. Indeed, many results in classic Lebesgue spaces have been generalized to Herz type spaces; see [16–20]. Herz type Besov and Triebel-Lizorkin spaces have been developed; see [21–24]. In 2010, Izuki in [25] introduced Herz space with variable exponent and and proved the boundedness of sublinear operators on them. Then, in [26] Izuki obtained the boundedness of commutators on Herz spaces with variable exponent. In [27] Izuki established the boundedness of vector-valued sublinear operators on Herz spaces with variable exponent and and obtained equivalent norms and wavelet characterization of Herz-Sobolev spaces with variable exponent. In [28], Shi and the second author of this paper introduced Herz type Besov and Triebel-Lizorkin spaces with variable exponent, and and and , and obtained their equivalent quasi-norms. Then, in [29], they gave a direct characterization of those spaces and its application. In 2012, Almeida and Drihem considered Herz spaces and in [30], where the exponent is variable as well, and gave boundedness results for a wide class of classical operators acting on such spaces.

Motivated by the previous papers, the goal of this paper is to introduce new Herz type Besov and Triebel-Lizorkin spaces with variable exponents. The structure of the paper is as follows. In Section 2 we will establish the boundedness of vector-valued Hardy-Littlewood maximal operator in spaces and . In Section 3 we give characterizations of these Herz type Besov and Triebel-Lizorkin spaces with variable exponents by maximal functions.

2. Vector-Valued Estimates

In this section we will establish vector-valued estimates of Hardy-Littlewood maximal operator in Herz spaces with variable exponents. Before stating our result, we recall some definitions and notations. First, we give some convention. Throughout this paper denotes the Lebesgue measure and the characteristics function for a measurable set . We also 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.

Definition 2.1. Let be a measurable set in with . Let be a measurable function. Denote

where , and

Then is a Banach space with the norm .

Let be the collection of all locally integrable functions on . Given a function , the Hardy-Littlewood maximal operator is defined by

where . We also use the following notation: and . The set consists of all satisfying and .   is the set of satisfying the condition that is bounded on . It is well known that if then is equivalent to , where is the conjugate exponent to , that means ; see [12]. For more about the set , one can see [11, 12, 31, 32]. Moreover, we define to be the set of measurable functions on with the range in such that and . Given , one can define the space as in Definition 2.1. This is equivalent to defining it to be the set of all functions such that , where and . We also define a quasi-norm on this space by .

Lemma 2.2 (see [25]). Let . Then there exist depending only on and such that for balls in and all measurable subsets ,

Lemma 2.3 (see [25]). Let . Then there exists a positive constant such that for balls in ,

Lemma 2.4 (see [11]). Let and , then there exists a positive constant such that for all sequences of locally integrable functions on ,

For giving the definition of the Herz spaces with variable exponent, let us introduce the following notations. Let .  .  .

denotes the set of all nonnegative integers. For , we denote if and .

Definition 2.5. Let and with .

(i) The homogeneous Herz space is defined by

 where

(ii) The nonhomogeneous Herz space is defined by

 where

Definition 2.6. A function is called locally -Hölder continuous on if there exists such that

for all . If for all , then is called - Hölder continuous at origin. is called satisfying the -Hölder decay condition if there exist and a constant such that

for all .

By , we denote the discrete Lebesgue space equipped with the usual quasi-norm. Let us denote

for sequences of measurable functions (with the usual modification when .)

Proposition 2.7. Let and . If satisfies -Hölder decay condition at infinity, then

Additionally, if is -Hölder continuous at the origin, then

We have the boundedness of vector-valued Hardy-Littlewood maximal operator in Herz spaces with variable exponents, which generalizes the results in [27, 33].

Theorem 2.8. Let with and be -Hölder continuous, both at the origin and at infinity, such that , where are constants satisfying (2.4), , then there is a constant independent of sequences of locally integrable functions on such that

Proof. Here we only prove the result for homogeneous Herz space, for nonhomogeneous Herz space it can be proved by similar way. We only consider sequences of locally measurable functions on such that .

Firstly, by Proposition 2.7 we have

To continue, we let . Now we estimate by Minkowski inequality as follows:

By the same way we consider ,

So we get

with .

Secondly, we divide the proof into three steps.

Step  1. We consider . By Lemma 2.4 we have

Similarly, we have

Thus

Step  2. We consider . We recall that the Hardy-Littlewood maximal operator satisfies the size condition

for all with compact support and a.e .

Therefore, for for all , by the size condition and the generalized Minkowski inequality, we obtain

By Hölder's inequality we get

On the other hand, by using Lemmas 2.2 and 2.3, we have

We put (2.27) into (2.26) and get

here .

For , we can make (2.28) further calculation by Hölder's inequality

For , we have

Similarly, we have

here .

For , we have

For , we have

Thus

Step  3. We consider .

For for all , by size condition and generalized Minkowski inequality, we have

By Hölder's inequality we get

On the other hand, by using Lemmas 2.2 and 2.3, we have

We put (2.37) into (2.36) and get

here .

Similar to Step  2, we can make (2.38) further calculation and obtain

Similarly, we have

here . By the same argument followed (2.28), we have

Thus

We get what we want by putting , , and into (2.20).

We remark that there is an analogue of Theorem 2.8 for sublinear operators.

Theorem 2.9. Let and be -Hölder continuous, both at the origin and at infinity, such that , where are constants satisfying (2.4), Suppose that is a sublinear and bounded operator on satisfying size condition

for all with compact support and a.e. , and for any , there exists a positive constant such that for all sequences of locally integrable functions on ,

Then

for all sequences of locally integrable functions on , where is independent of .

3. The Quasi-Norm Characterizations

Let be the Schwartz space on be its dual space on . For ,   denotes its Fourier transform, and denotes its inverse Fourier transform. Take with and

Now define and set for all . Let . Then is a resolution of unity, that means for all .

Now, we introduce the Herz-type Besov spaces and Triebel-Lizorkin spaces with variable exponents as following.

Definition 3.1. Let be a resolution of unity as above, and with .

(i) The set

is named to the Herz-type Besov space with variable exponents and denoted by . The quasi-norm of in this space is denoted by

(ii) For , the set

is named to the Herz-type Triebel-Lizorkin space with variable exponents and denoted by . The quasi-norm of in this space is denoted by

Here and are the spaces of all sequences of measurable functions on with finite quasi-norms

The spaces and are defined similarly by replacing the norm with . To make these spaces definite, we need to show them independent of the choice of the resolution of unity . To this aim we need more notation.

Let , integer be such that

Here (3.7) and (3.8) are Tauberian conditions, while (3.9) expresses moment conditions on .

Let us recall the classical Peetre maximal operator introduced in [34]. Given a sequence of function , a tempered distribution and a positive number , we define the system of maximal functions

Since makes sense pointwise everything is well defined. We will often use dilates of a fixed function , where might be given by a separate function. Also continuous dilates are needed. Let . We define by

Now we have equivalent quasi-norms for these new spaces.

Theorem 3.2. Let for with , with and be -Hölder continuous, both at the origin and at infinity, such that , where are constants satisfying (2.4) for . And let . Further, let belong to be given by (3.7), (3.8), and (3.9).

(i) For , then the space can be characterized by

where

Furthermore, are equivalent.

(ii) If , then for the space can be characterized by

where

Furthermore, are equivalent.

Theorem 3.2 is the generalization of Theorem 1 in [28] for the Herz type space only one variable exponent . For readers' convenience, we give the proof for -parts in outline. The proof of -parts is similar and a bit simpler. The idea of the proof goes back to [35]. In fact, we will use the argument in [35]. To go on, we need some lemmas.

Lemma 3.3 (see [36]). Let ,

Then for any there is a constant so that

where for all .

Lemma 3.4 (see [36]). Let . For any sequence of nonnegative numbers denote

Then

holds, where is constant only depend on .

Lemma 3.5. Let , and with . For any sequence of nonnegative measurable functions on denote

Then

hold with some constants and .

Proof. By Lemma 3.4, (3.25) follows immediately from (3.23). Now we prove (3.26).

Let . Here we can divide it into two cases.

Case  1 ( ). Since is a norm, we have

By Lemma 3.4, we obtain (3.26).

Case  2. If , take . Then

By Lemma 3.4, we obtain

In the last inequality, we used (3.26) has been proved for space . Raising to power , we obtain (3.26).

Lemma 3.6 (Theorem 6 in [37]). Let is a resolution of unity and let . Then there exist functions satisfy

such that

where the functions are defined via and .

Proof of Theorem 3.2. We divide the total proof into four steps. First, we prove the equivalence of (3.15) and (3.16). The next step is to build the bridge between (3.16) and (3.18) and to change from the system to a system . The equivalence of (3.18) and (3.19) goes parallel to (3.15) and (3.16). Indeed, Definition 3.1 can be seen as a special case of (3.19). Finally, we prove (3.19) is equivalent to the rest.

Step  1. We are going to prove that for every

From Lemmas 3.3 and 3.6, we have that, see [35], for , there exists positive constant for ,

Take now and put . We obtain for

Now we apply Lemma 3.5 in which yields

Next, using Theorem 2.8, we can obtain

Hence, we obtain

This proves . Since the reverse inequality is trivial, this finishes Step  1.

Step  2. Let be functions satisfying (3.25).

First, we are going to prove for all

Again from Lemmas 3.6 and 3.3, we have that, see [35], let , there exists a positive constant for any ,

for all and all .

Suppose first that . Then we take on both sides , which gives

Applying Lemma 3.5 we obtain that

which gives the desired result.

In case we argue as follows. The quantity is no longer a norm. This gives

Notice that the right-hand side is nothing more than a convolution of the sequences

Now we apply the -norm to both sides and get for all

We take both sides to power and apply the -norm. This gives (3.38).

Similarly, we obtain for all

Step  3. Choosing in Step  1 and omitting the integration over we see immediately

Step  4. What remains is to show that (3.19) is equivalent to the rest.

First, let us prove that for any

From [35], for , there exists a positive constant for any

If then we have

Thus we have

Now we use the majorant property of the Hardy-Littlewood maximal operator in [38] and continue estimating

An index shift on the right-hand side gives

Then with analogous arguments as after (3.39), we obtain (3.47).

Second, we prove . Since for all

Therefore we can obtain what we want. The proof is complete.

Remark 3.7. One can define homogenous spaces and . Then there is an analogue of Theorem 3.2 for these spaces. We omit the detail here and leave it for interested readers.

Acknowledgments

The authors would like to thank the referee for his carefully reading and suggestions which made the presentation more readable. J. Xu is supported by the National Natural Science Foundation of China (Grant no. 11071064 and 11226167) and the Natural Science Foundation of Hainan Province (no. 111006) and the Scientific Research Foundation of Hainan Province Education Bureau (no. Hjkj2011-19).

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