#### Abstract

The authors introduce Herz-Morrey-Hardy spaces with variable exponents and establish the characterization of these spaces in terms of atom. Applying the characterization, the authors obtain the boundedness of some singular integral operators on these spaces.

#### 1. Introduction

The Herz spaces go back to Beurling and Herz; see [1, 2]. Firstly, they attracted a lot of authors’ attention because they could be used to characterize Fourier multipliers for Hardy spaces; see [3]. Then, in 1989 Chen and Lau in [4] and García-Cuerva in [5] introduced now called nonhomogeuous Herz type Hardy spaces. They found that these Herz type Hardy spaces have a decomposition via central atoms. After that, Lu et al. considered homogeuous Herz type Hardy spaces and also obtained a central atomic decomposition for them. Since then Herz type spaces have been studied extensively; see monograph [6] for details. Meanwhile, in the last three decades, the interest of the study for variable exponent spaces has been increasing year by year. Variable exponent spaces have many applications: in electrorheological fluid [7], in differential equations [8] and references therein, and in image restoration [9–11], for instance. Indeed, many spaces with variable exponents appeared, such as: Lebesgue spaces, Sobolev spaces and Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, Campanato spaces with variable exponent, and Hardy spaces with variable exponent; see [12–23] and references therein. Moreover, the atomic, molecular, and wavelet decompositions of variable exponent Besov and Triebel-Lizorkin spaces were given in [13, 14, 20, 21, 24]. The duality and reflexivity of spaces and were discussed in [25]. The atomic and molecular decompositions of Hardy spaces with variable exponent and their applications for the boundedness of singular integral operators were obtained in [22, 26].

Recently, as a generalization of Lebesgue spaces with variable exponent, Herz spaces with variable exponents are introduced. In fact, in 2010 Izuki proved the boundedness of sublinear operators on Herz space with variable exponents and in [27]. In 2012, Almeida and Drihem obtained boundedness results for a wide class of classical operators on Herz spaces and in [28]. Shi and the first author in [29] considered Herz type Besov and Triebel-Lizorkin spaces with one variable exponent. Then Dong and first author in [30] established the boundedness of vector-valued Hardy-Littlewood maximal operator in spaces and and gave characterizations of Herz type Besov and Triebel-Lizorkin spaces with variable exponents by maximal functions. In [31], Wang and Liu introduced a certain Herz type Hardy spaces with variable exponent. In 2013, Samko introduced Herz spaces with three variable exponents and obtained the boundedness of Hardy-Littlewood maximal operator on them. In [32–34], the boundedness of singular integrals and their commutators of BMO functions are discussed in Herz Morrey spaces with variable exponents. The Herz-Morrey spaces with constants were considered in [35, 36]; however, there is no theory of Herz-Morrey type Hardy spaces. In this paper we fill the gap and introduce Herz-Morrey-Hardy spaces with variable exponents.

The outline of the paper is as follows. In the rest of the section we will recall some definitions and notions. In Section 2, we will define the Herz-Morrey-Hardy spaces with variable exponents and and give their atomic characterization. In Section 3, we obtain that certain singular integral operators are bounded from Herz-Morrey-Hardy spaces with variable exponents into Herz-Morrey spaces with variable exponents as an application of the atomic characterization.

Throughout this paper denotes the Lebesgue measure and the characteristic function for a measurable set . For a multi-index , we denote . 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 1. *Let be a measurable set in with . Let be a measurable function. Denotewhere , 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 and what follows . 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 satisfies the following global log-Hölder continuous then ; see [37–42].

*Definition 2. *Let be a real-valued function on . If there exists such that, for all , ,then is said local log-Hölder continuous on .

If there exists , such that for all ,then is said log-Hölder continuous at origin.

If there exist and a constant such that for all then is said log-Hölder continuous at infinity.

If is both local log-Hölder continuous and log-Hölder continuous at infinity, then is said global log-Hölder continuous.

The sets of log-Hölder continuous functions, log-Hölder continuous functions at origin, log-Hölder continuous functions at infinity, global log-Hölder continuous are denoted by , , , and , respectively.

We denote by the conjugate exponent to , which means . It is also well known that is equivalent to ; see [39].

For simplicity, we denote by . We will use the following results.

Lemma 3 (see [43]). *Let . If and , then is integrable on and**where .*

Lemma 4 (see [27]). *Let . Then there exist , , and a positive constant depending only on and such that for all balls in and all measurable subsets ,*

Lemma 5 (see [27]). *Let . Then there exists a positive constant such that, for any ball in ,*

To give the definition of Herz-Morrey spaces with variable exponents, let us introduce the following notations. Let , , , , and . The symbol denotes the set of all nonnegative integers. For , we denote if and .

*Definition 6. *Let , , and . Let be a bounded real-valued measurable function on . The homogeneous Herz-Morrey space and nonhomogeneous Herz-Morrey space are defined, respectively, bywhereHere there is the usual modification when .

Proposition 7. *Let , , and . If , then*

Proposition 7 is the generalization of Herz spaces with variable exponents in [28], and it was used in [33, 34].

Lemma 8. *Let , , and . Let be bounded and log-Hölder continuous both at the origin and at infinity such that , where , are constants in Lemma 4. Suppose that is a sublinear and bounded operator on satisfying size condition**for all with compact support and a.e. . Then there exists a positive constant such that**for any function belongs to and , respectively.*

Lemma 8 is the generalization of Herz spaces with variable exponents in [28]. For a proof, see [33].

#### 2. The Atomic Characterization

In this section, we will introduce Herz-Morrey-Hardy spaces with variable exponents and . To do this, we need to recall some notations. denotes the Schwartz space of all rapidly decreasing infinitely differentiable functions on , and denotes the dual space of . Let be the grand maximal function of defined bywhere and and is the nontangential maximal operator defined by

The grand maximal operator was firstly introduced by Fefferman and Stein in [44] to study classical Hardy spaces. For classical Hardy spaces, one can also see [45–47]. Nakai and Sawano generalized them to variable exponent case in [22].

*Definition 9. *Let , , , , and . The homogeneous Herz-Morrey-Hardy space with variable exponents and nonhomogeneous Herz-Morrey-Hardy space with variable exponents are defined, respectively, by

*Remark 10. *If and , these spaces were considered by Wang and Liu in [31]. If and are constant and , these are the classical Herz type Hardy spaces; see [6].

Let for and for . Then there exists such that for all . Therefore, by [46, Proposition in Page 57], there exists such that for all . This means that satisfies the size condition in Lemma 8. By Lemma 8, if and , then

Thus we are interested in the case , . In this case, we will establish a characterization of the spaces and in terms of central atom. For we denote by the largest integer less than or equal to .

*Definition 11. *Let and be log-Hölder continuous both at the origin and infinity, and nonnegative integer ; here , if , and , if , and as in Lemma 4.(i)A function on is called a central -atom, if it satisfies supp ; (2) ; (3) , .(ii)A function on is called a central -atom of restricted type, if it satisfies supp , ; (2) ; (3) , .

*Remark 12. *If and are constant, then taking we recover the classical case in [6].

Theorem 13. *Let , , , and be log-Hölder continuous both at the origin and infinity, , , , and as in Lemma 4.*(i)* if and only if in the sense of , where each is a central -atom with support contained in and . Moreover,* *where the infimum is taken over all above decompositions of .*(ii)* if and only if in the sense of, where each is a central -atom of restricted type with support contained in and . Moreover* *where the infimum is taken over all above decompositions of .*

*Proof. *We only prove (i). The proof of (ii) is similar. We use the ideas in [6]. To prove the necessity, we choose such that , , and . For , let , . For each , set , . It is obvious that and in . Let be a radial smooth function such that with , for . Let for and Observe that and for . Obviously, , . LetThen for . For each , we denote by the class of all the real polynomials with the degree less than . Let be the unique polynomial satisfyingWriteFor the term , let and , where and is a constant which will be chosen later. Note that , .

Now we estimate . To do this, let be the orthogonal polynomials restricted to with respect to the weight , which are obtained from by the Gram-Schmidt method, which meanswhere for , otherwise 0.

It is easy to see that for . On the other hand, from we infer that Thus, we deduce a.e. That is, almost everywhere for . Therefore for . By the generalized Hölder inequality we haveBy Lemma 5 we have Choose ; then and each is a central -atom with support contained in . Here and below we abuse and it is well defined in Definition 11. Thus,Now we estimate . By the condition of and Proposition 7 we consider it in two cases.*Case 1* (). Consider *Case 2* (). ConsiderHence, It remains to estimate . Let be the dual basis of with respect to the weight on , that is, Similar to the method of [48], letWe writewhereand is a constant which will be chosen later. Note that By a computation we have Since it follows that Take . It is easy to show that each is a central -atom with support contained in , andwhere is a constant independent of , , , and . Moreover, we haveUsing the same argument as before for , we obtainTherefore, Thus, we obtain thatwhere each is a central -atom with support contained in , is independent of andwhere is independent of and .

Sinceby the Banach-Alaoglu theorem we obtain a subsequence of converging in the weak topology of to some . It is easy to verify that is a central -atom supported on . Next, since another application of the Banach-Alaoglu theorem yields a subsequence of which converges weak in to a central -atom with support in . Furthermore, Similarly, there exists a subsequence of which converges weak in to some , and is a central -atom supported on . Repeating the above procedure for each , we can find a subsequence of converging weak in to some which is a central -atom supported on . By using the diagonal method we obtain a subsequence of such that, for each , in the weak topology of and therefore in .

Now we only need to prove that in the sense of . For each , note that . Using the same argument in [48], we have Recall that . If , then by Lemmas 3 and 4 we haveIf , let such that ; then by Lemmas 4 and 3 again we have LetThenwhich implies thatThis establishes the identity we wanted.

To prove the sufficiency, for convenience, we denote . Firstly we havewhere Now we have