#### Abstract

We introduce Triebel-Lizorkin and Besov spaces by Calderón’s reproducing formula on product spaces of homogeneous type. We also obtain smooth atomic and molecular decompositions for these spaces.

#### 1. Introduction and Main Results

Atomic and molecular decompositions are significant tools in studying function spaces and operators in harmonic analysis. The atomic decomposition for Hardy spaces on was first introduced by Coifman in [1] and was extended to by Latter in [2]. Molecules in Hardy spaces were also introduced by Coifman in [3]. Coifman and Weiss in [4] extended molecules to more general setting in Hardy spaces.

The atomic decomposition of product Hardy spaces was established by Chang and Fefferman in [5, 6]. Recently, general atomic decomposition on product Hardy spaces was constructed by Han et al. in [7] and Han et al. in [8].

The smooth atomic decomposition for the Triebel-Lizorkin spaces on first is considered by Frazier and Jawerth in [9] and later extended to spaces of homogeneous type by Han and Sawyer in [10]; for similar results, see [11]. Smooth molecules of the Triebel-Lizorkin spaces on bidisc were given by Wang in [12].

In [13], Hardy spaces associated with different homogeneities were constructed. Wu in [14] introduced related Triebel-Lizorkin and Besov spaces similarly. In [7, 15], the authors studied the Hardy spaces on product spaces of homogeneous type and showed some properties of these spaces. A natural question arises: whether we introduce the Triebel-Lizorkin and Besov spaces on product spaces of homogeneous type and study some properties of these spaces.

The purpose of this paper is to answer the question. Namely, we introduce the Triebel-Lizorkin and Besov spaces on product spaces of homogeneous type. A theory of atomic and molecular decompositions for these spaces is also presented.

We begin with some necessary definitions and notation on spaces of homogeneous type.

A quasimetric on a set is a function : satisfying that (i) if and only if , (ii) for all , and (iii) there exists a constant such that for all , , and Any quasimetric defines a topology, for which the balls with all and all form a basis.

We now state definitions of spaces of homogeneous type.

*Definition 1. *A space of homogeneous type is a set with a quasimetric and a nonnegative Borel regular measure on such that , and there exists a positive constant such that
for all and all , where is assumed to be defined on a -algebra which contains all Borel sets and all balls .

We suppose that and for all . Further, suppose there exist constant and regularity exponent such that, for all and all , and .

Let us recall the definition of an approximation to the identity on spaces of homogeneous type.

*Definition 2 (see [7]). *Let be a space of homogeneous type as in Definition 1 and constant satisfying (1). A sequence of linear operators is said to be an approximation to the identity of order if there exists such that, for all , , , and , , the kernel of , is a function from into satisfying (i) ; (ii) for ; (iii) for ; (iv) + for and ; (v) .Moreover, a sequence of linear operators is said to be an approximation to the identity of older having compact support if there exists a constant such that, for all and all , and , , the kernel of is a function from into satisfying and(vi) if and ;(vii);(viii);(ix).

We present test functions on spaces of homogeneous type before we give test functions on product spaces of homogeneous type .

*Definition 3 (see [15]). *For fixed , let , , and , where is the regularity exponent on . A function defined on is said to be a test function of type , if, for all , satisfies the following conditions:(i);
(ii) for .

If is a test function of type , we write and the norm of is defined by

We denote by the class of with for fixed . Set if with . It is easy to see that with an equivalent norm for all and . Furthermore, we can check that is a Banach space with respect to the norm in . Let be the completion of the space in with . If , we then define .

We define the distribution space by all linear functionals from to with the property that there exists such that for all .

Let for be two spaces of homogeneous type as in Definition 1 and satisfies (1) with replaced by for . We define spaces of test functions on product space of spaces of homogeneous type.

*Definition 4 (see [15]). *Let , and , where is the regularity exponent on for . A function defined on is said to be a test function of type if, for any fixed , , as a function of variable of , is a test function of on . Similarly, for any fixed , , as a function of variable of , is a test function of on . Moreover the following conditions are satisfied:(i);(ii) for all with ;(iii)properties (i)-(ii) also hold with and interchanged.

If is a test function of type , we write
and the norm of is defined by

Similarly, we denote by the class of with for fixed . Set if with . It is easy to see that with an equivalent norm for all and . Furthermore, we can check that is a Banach space with respect to the norm in .

Let be the completion of the space in with for . If , we then define .

Also, we define the distribution space by all linear functionals from to with the property that there exists a constant such that for all .

We give Calderón’s reproducing formulas on product spaces of homogeneous type.

Lemma 5 (see [7, 15]). *Suppose that . Let , let be an approximation to the identity of order on spaces of homogeneous type , and let for . Then there are families of linear operators and on such that, for all with ,
**
where the series converges in the norm of both the spaces with . Moreover, for and all , , the kernel of , satisfies conditions (i) and (ii) in Definition 2 with replaced by any and
**, the kernel of , satisfies conditions and in Definition 2 with replaced by for and (8).*

We also need the following result, which gives an analogue of the grid Euclidean dyadic cubes on a space of homogeneous type.

Lemma 6 (see [15]). *Let be a space of homogeneous type as in Definition 1. Then there exists a collection of open subsets, where is some index set, and constants such that*(i)* for each fixed and , if ;*(ii)*for any with , either or ;*(iii)*for each and each , there is a unique such that ;*(iv)*;*(v)*each contains some ball , where .*

We think of as being a dyadic cube with diameter and center . When and , we denote by , where , the set of all cubes , where is a fixed large positive integer, and a point in .

Throughout the paper, we use to denote positive constants, whose value may change from one occurrence to the next. We denote by that there exists a constant independent of the main parameters such that . is the characteristic function of . Let for any . We also denote , are the Hardy-Littlewood maximal function on spaces of homogeneous type , , respectively.

We introduce Triebel-Lizorkin space and Besov space on product spaces of homogeneous type.

*Definition 7. *Suppose that , , and let be the same as in Lemma 5. Let and , . Then Triebel-Lizorkin space is defined by
The Besov space is defined by

In order to check that the definitions for and are independent of the choice of approximations to the identity, we recall almost orthogonality estimate.

Lemma 8 (see [15]). *Let . Suppose that and are two approximations to the identity on spaces of homogeneous type, and , . Then for , there exists a positive constant depending only on such that , the kernel of , satisfies the following estimate:
*

By Calderón’s reproducing formula and almost orthogonality estimate, we can get the following product-type Plancherel-Pôlya inequalities. The proofs of the following theorems are similar to Proposition 4.1 in [10]. Here we omit the details.

Theorem 9. *Suppose that and are the same as in Lemma 8 for . Let , and for . Then there exists a constant such that for all with , for , and then
*

*Remark 10. *Let , , , and be as in Lemma 5. Then the kernel of has compact support, but not of . Since (11) holds for and satisfying only the smoothness condition for the second variable, we conclude that Theorem 9 holds with being replaced by .

#### 2. Smooth Atomic Decomposition

In this section, smooth atomic decomposition is presented. We first give definitions of the smooth atoms for and .

*Definition 11. *Suppose collections of open subset and satisfying the conditions in Lemma 6. Let and , and a function defined on is said to be a -smooth atom for if(i)
where is the center of for ;(ii) (iii)

Now, we also define certain spaces of sequences of indexed by dyadic rectangles in which will character the coefficients in our decomposition of and . Suppose that , with for , , . Let be as in Definition 11. Let and be the collection of sequences such that

Smooth atomic decompositions for and can be stated as follows.

Theorem 12. *Let be as in Definition 11. Suppose that , , , with for .*(i)*Then there exist sequence and -smooth atoms such that
* *with convergence in , and
* *Similarly, there exist sequence and -smooth atoms such that
* *with convergence in and
*(ii)*Conversely, if are -smooth atoms and
* *then
*

*Proof. *We first prove part (i). Let be as in Definition 11. Suppose ; by Lemma 5, we have
where
Similar to [7, 10], we can prove the convergence of the above series converges in and with and for . We now show that satisfies conditions in Definition 11. Here we only show that satisfies condition (iii) and we first prove that satisfies (15). By the size condition of and for , we have

We now verify (16); here, we only consider the case and similarly we estimate the case ; we get
which is a desired estimate. The proof of (17) is similar to that of (16) by symmetry. Here we omit the details.

We now estimate (18). If and , then
which is the desired result. Similarly we can consider another three cases.

Then, by Hölder’s inequality and Remark 10, we obtain

The proof of the case goes by the analogous argument to the case that , but using the vector-valued maximal inequality. Indeed, suppose that ; as in the proof of , we have
where and are as in (27) and (28), respectively. From [7, 10], we can also similarly obtain the series converges in the norm of and in with and for . To show the conclusion, using Fefferman-Stein vector-valued maximal inequality and Remark 10, we have

To prove part (ii), we need the discrete version of the Hardy-Littlewood maximal function estimate on one single factor , which is an analogue to a result of [9]. Also see [10].

Lemma 13.* Suppose that ** is a space of homogeneous type as in Definition 1 Let ** and **. Fix ** with ** and for any **where ** is the center of dyadic cube ** and ** depends only on **. *

We now turn to proof of part (ii). In this part, we denote . Suppose that for is as in Lemma 5 and is an -smooth atom. For , we consider it by four cases. We first estimate the case . In this case, applying the cancellation conditions of and ,

To estimate the case , , we only estimate , and afterwards we can consider the case similarly. In this case, we have