#### 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
*