Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 275791, 18 pages

http://dx.doi.org/10.1155/2012/275791

## Characterizations of Multiparameter Besov and Triebel-Lizorkin Spaces Associated with Flag Singular Integrals

Department of Mathematics, China University of Mining and Technology (Beijing), Beijing 100083, China

Received 25 February 2012; Accepted 9 April 2012

Academic Editor: Yongsheng S. Han

Copyright © 2012 Xinfeng Wu and Zongguang Liu. 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

We introduce the inhomogeneous multiparameter Besov and Triebel-Lizorkin spaces associated with flag singular integrals via the Littlewood-Paley-Stein theory. We establish difference characterizations and Peetre's maximal function characterizations of these spaces.

#### 1. Introduction and Main Results

The flag singular integral operators were first introduced by Müller, Ricci, and Stein when they studied the Marcinkiewicz multiplier on the Heisenberg groups in [1]. To study the -complex on certain CR submanifolds of , in 2001, Nagel et al. [2] studied a class of product singular integrals with flag kernel. They proved, among other things, the boundedness of flag singular integrals. More recently, Nagel et al. in [3, 4] have generalized these results to a more general setting, namely, homogeneous group. For other related results, see [5, 6].

For , Han and Lu [7] developed Hardy spaces with respect to the flag multiparameter structure via the discrete Littlewood-Paley-Stein analysis and discrete Calderón’s identity and proved the and boundedness for flag singular integral operators. The duality of was also established. More recently, Ding et al. studied the homogeneous Besov spaces and Triebel-Lizorkin spaces associated with flag singular integrals in [8] and proved the boundedness of flag singular integrals on these spaces. Similar results can also be found in [9].

The aim of this paper is to give the new difference characterization as well as Petree’s maximal function characterization of multiparameter Besov and Triebel-Lizorkin spaces associated with flag singular integrals, which reflect that the Besov spaces and Triebel-Lizorkin spaces have *flag multiparameter structure*. These characterizations are established for the inhomogeneous Besov and Triebel-Lizorkin spaces, but the argument goes through with only minor alterations in the homogeneous ones introduced in [8].

In order to describe more precisely questions and results studied in this paper, we begin with basic notations and notions. Let with

and let with

Let , and

then, for , the following Calderón’s reproducing formula holds:

where the series converges in .

A Schwartz function is said to be a *product test function* in if it satisfies

for all multi-indices .

*Definition 1.1. *A function defined on is said to be a test function in if there exists a function such that
and the seminorm of is defined by
where denote the seminorm in . Denote by the dual of .

Choose a Schwartz function on such thatNote that with Fourier transform is supported in . Define the operator by .

For , by taking the Fourier transform,

where the series converges in norm.

Now, we introduce the definition of inhomogeneous Besov spaces and Triebel-Lizorkin spaces associated with flag singular integrals.

*Definition 1.2. *Let and . The inhomogeneous *Triebel-Lizorkin space associated with flag singular integrals * is defined to be the collection of all such that
and the inhomogeneous *Besov space associated with flag singular integrals * is defined to be the collection of all such that

Throughout this paper, we always work on for some fixed and use to denote , similarly for , and so forth. We would like to point out that the multiparameter structures are involved in the definitions of and . The following result shows that the definition of the Besov spaces and Triebel-Lizorkin spaces is independent of the choice of ; thus, the Besov spaces and the Triebel-Lizorkin spaces are well defined.

Theorem 1.3. *If satisfies the same conditions as , and is defined similar to (1.8) with replaced by , then for and and ,
*

*Remark 1.4. *As the classical case, it is not hard to show that and are norms of and , respectively. Moreover, and are complete with respect to these norms and hence are Banach spaces. We omit the details.

Throughout this paper, we use the notations and . We introduce the following *flag multi-parameter Peetre maximal functions* (with respect to ). For and , define

For , define

An index is said to be *admissible* if

We point out again that the flag multiparameter structure is involved in the definition of Peetre’s maximal functions. The maximal function characterizations of Besov spaces and Triebel-Lizorkin spaces are as follows.

Theorem 1.5. *Let and . If is admissible, then for , one has*(i)*,
*(ii)*.** Here and in what follows, one uses the following notation:
*

In order to state our result for flag singular integrals, we need to recall some definitions given in [2]. Following closely from [2], we begin with the definitions of a class of distributions on an Euclidean space . A -normalized bump function on a space is a function supported on the unit ball with norm bounded by 1. As pointed out in [2], the definitions given below are independent of the choices of , and thus we will simply refer to “normalized bump function’’ without specifying .

*Definition 1.6. *A flag kernel on is a distribution on which coincides with a function away from the coordinate subspaces , where and satisfies(1) (Differential inequalities) for any multi-indices and ,
for all with ,(2) (Cancellation condition)
for all multi-index and every normalized bump function on and every ,
for every multi-index and every normalized bump function on and every ,
for every normalized bump function on and every and .

The boundedness of flag singular integrals on these inhomogeneous Besov spaces and Triebel-Lizorkin spaces is given by the following theorem, whose proof is quite similar to that in homogeneous case in [8]. We omit the proof here.

Theorem 1.7. *Suppose that is a flag singular integral defined on with the flag kernel , then, for and , is bounded on and on .*

As in the classical inhomogeneous Besov spaces and Triebel-Lizorkin spaces, we will give the difference characterization for and . However, the new feature is that the differences of functions are associated with the “flag.’’ More precisely, for and , we define the *first flag difference *(*associated the flag *) in by

where is the difference operator on , and is the difference operator on . For and , the *th flag difference operator * can be defined inductively by

Theorem 1.8. *If , , , and , where denotes the greatest integer function, one defines
**
then .*

As mentioned before, by slightly modifying the proof, we can prove difference characterizations and Peetre’s maximal function characterizations of homogeneous Besov and Triebel-Lizorkin spaces, introduced in [8]. We leave the details to the interested reader.

The following of the paper is organized as follows. In Section 2, we give some lemmas. The proof of Theorems 1.3 and 1.5 is presented in Section 3. Section 4 is devoted to the proof of Theorem 1.8.

#### 2. Some Lemmas

In this section, we present some lemmas, which will be used in the proofs of the theorems.

##### 2.1. Inhomogeneous Calderón’s Reproducing Formula in

Lemma 2.1. *The inhomogeneous Calderón’s reproducing formula holds
**
where the series converges in , and .*

We point out that in [8] the homogeneous Calderón’s reproducing formula was provided. Note that the convergence of these two kind of producing formulas are different. See [8] for homogeneous case.

*Proof. *For any , then by definition, there exists such that . We need to show that for all ,
tends to in the topology of as . We only consider the case when in the summation in (2.2) since the other case can be dealt with in the same way. Denote, in this case, the expression (2.2) by . By Fourier inversion,
Let so that
Since , Lebesgue’s dominated convergence theorem yields

On the other hand, using the cancellation conditions of , and the smoothness of , we can get

Now, (2.5) together with the estimate (2.6) implies that, for ,
Applying this to (here denotes any multi-index in ) and noting that , we obtain
This proves the convergence of series in (2.1) in . The convergence in follows by a standard duality argument. The convergence in can be proved similar to the product case, see (10, Theorem 1.1).

##### 2.2. Almost Orthogonality Estimates

The following lemma is the almost orthogonality estimates, which will be frequently used. See [7] for a proof.

Lemma 2.2. *Let . Given any positive integers and , there exists a constant such that
**
where are defined as in Section 1.*

##### 2.3. Maximal Function Estimates

The maximal function estimates are given as follows.

Lemma 2.3. *For , and for any and , there exists a constant depending only on and , but independent on , such that
*

*Proof. *By the almost orthogonality estimate in Lemma 2.2, for any and , we have the pointwise estimate
where . This proves (2.10).

*Remark 2.4. *Since the almost orthogonality estimates hold with or is replaced by , repeating the same argument as (2.11), we see that the estimate (2.10) is still valid if or is replaced by .

Denote by the strong maximal operator defined by

where the supremum is taken over all open rectangles in that contain the point .

Lemma 2.5. *Let , and , then for all and for all functions on whose Fourier transform is supported in the rectangle , one has the estimate
**
In particular, if with and , then for all ,
*

Lemma 2.5 can be proved as in the classical one-parameter case. We refer the reader to [11].

##### 2.4. An Embedding Result

The following lemma is an embedding result.

Lemma 2.6. *For and , one has the following continuous embedding:
*

*Proof. *For , by inhomogeneous Calderón’s reproducing formula (2.1),
where we have used Hölder’s inequality in the last inequality. This proves Lemma 2.6 for Besov spaces.

For the Triebel-Lizorkin spaces, by (2.1), the pointwise inequality , Hölder’s inequality, and Fefferman-Stein’s vector-valued inequality, we have
This ends the proof of Lemma 2.6.

#### 3. Proof of Theorems 1.3 and 1.5

We first prove Theorems 1.3 and 1.5 for Triebel-Lizorkin spaces by showing

The first inequality in (3.1) follows from the pointwise inequality

Next, for any admissible , fix . Since is admissible, we can choose such that , , and the thus inequality (2.14) holds. We apply Lemma 2.5 and the boundedness of to deduce

which gives the third inequality in (3.1).

Thus, to finish the proof of (3.1), it remains to verify the second inequality. For , is nonzero only when and . Thus, applying Calderón’s identity (2.1), Minkowski’s inequality, Remark 2.4, and Lemma 2.5, we deduce that

To finish the proof, it remains to show

By the inhomogeneous Calderón’s identity (2.1), we have

It follows that

We first estimate . By the support properties of and and Young’s inequality,

Next, we give the estimate for . For any ,where we have used Lemma 2.2 in the last inequality. Applying Minkowski’s inequality and Hölder’s inequality yields

where we have chosen as a small positive constant less than . Therefore, Combining the estimates (3.8) and (3.11), we obtain (3.5). This finishes the proof of (3.1), and hence, Theorems 1.3 and 1.5 for Triebel-Lizorkin spaces follow.

The proofs of Theorems 1.3 and 1.5 for Besov spaces are similar. By (3.2) and the maximal function estimate (2.14), . The conclusion of Theorem 1.5 for Besov spaces follows. By Calderón’s reproducing formula (2.1), Young’s inequality, the almost orthogonality estimate in Lemma 2.2, Hölder’s inequality, and Minkowski’s inequality, we have

as desired. This ends the proof of Theorem 1.3 for Besov case. Hence, the proofs of Theorems 1.3 and 1.5 are complete.

#### 4. Proof of Theorem 1.8

##### 4.1. Proof of Theorem 1.8 for Besov Space

By the moment conditions of ’s and ’s, we may write

By Minkowski’s inequality and Hölder’s inequality and noting that

we have

Hence,

where the last inequality follows from

This inequality together with the trivial inequality yields .

To prove the converse, by Lemma 2.6, it suffices to show thatBy the Calderón’s identity (2.1), we write

where the series on the right hand side converges in . Thus, by Minkowski’s inequality and Young’s inequality, we conclude that

It follows that the left hand side in (4.6) is dominated, up to a constant, by the sum of

For , we have

where we have used the estimate

To estimate , by the estimate

and Hölder’s inequality, we see that is majorized by

proving (4.6), where is a positive number such that . This concludes the proof of Theorem 1.8 for Besov spaces.

##### 4.2. Proof of Theorem 1.8 for Triebel-Lizorkin Space

Using the moment conditions on , we have

where . Therefore,

which, together with the obvious inequality , yields .

To show the converse, write

where the series converges in . It follows thatwhere and .

For , by Lemma 2.2,

Consequently,Therefore,

which together with the maximal function characterization of implies

As for , similar estimate to (4.19) yields

Therefore,

This estimate together with (4.21) and Lemma 2.6 yields . This ends the proof of Theorem 1.8.

#### Acknowledgments

The research was supported by NNSF of China (Grant nos. 11101423, 11171345) and the Fundamental Research Funds for the Central Universities of China (Grant no. 2009QS12). The authors would like to express their deep gratitude to the referee for his/her valuable comments and suggestions.

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