#### Abstract

We give a molecular characterization of the Hardy space associated with twisted convolution. As an application, we prove the boundedness of the local Riesz transform on the Hardy space.

#### 1. Introduction

In this paper, we consider the linear differential operators Together with the identity they generate a Lie algebra which is isomorphic to the dimensional Heisenberg algebra. The only nontrivial commutation relations are The operator defined by is nonnegative, self-adjoint, and elliptic. Therefore, it generates a diffusion semigroup . The operators in (1) generate a family of “twisted translations” on defined on measurable functions by The “twisted convolution” of two functions and on can now be defined as where . More about twisted convolution can be found in [1–3].

In [4], the authors defined the Hardy space associated with twisted convolution. They gave several characterizations of via maximal functions, the atomic decomposition, and the behavior of the local Riesz transform. As applications, the boundedness of Hömander multipliers on Hardy spaces is considered in [5]. The “twisted cancellation” and Weyl multipliers were introduced for the first time in [6]. Recently, Huang and Wang [7] defined the Hardy space associated with twisted convolution for . Huang gave the characterizations of the Hardy space associated with twisted convolution by the Lusin area integral function and Littlewood-Paley function in [8] and established the boundedness of the Weyl multiplier on the Hardy space associated with twisted convolution by these characterizations in [9]. The purpose of this paper is to give a molecular characterization for . As an application, we prove the boundedness of the local Riesz transform on the Hardy space .

We first give some basic notations about . Let denote the class of -functions on , supported on the ball such that and . For , let . Given , , and a tempered distribution , define the grand maximal function Then, the Hardy space can be defined by For any , define .

*Definition 1. *Let and . A function is a -atom for the Hardy space associated to a ball if (1);
(2);
(3).

We define the atomic Hardy space to be the set of all tempered distributions of the form (the sum converges in the topology of ), where are -atoms and .

The atomic quasinorm in is defined by where the infimum is taken over all decompositions and are -atoms.

The following result has been proved in [4, 7].

Proposition 2. *Let . Then, for a tempered distribution on , the following are equivalent:*(i)*;*(ii)*for some , , ;*(iii)*for some radial function , such that , we have
*(iv)* can be decomposed as , where are -atoms and .*

Corollary 3. *Let and . Then, with equivalent norms.*

In order to give the main result of this paper, we need the dual space of Hardy space .

*Definition 4. *Let ; a locally integrable function is said to be in the Campanato type space if there exists a constant such that, for every ball ,
The norm of is the least value of for which the above inequality holds.

The dual space of is the BMO type space (cf. [4]). Note that is identified with . Let denote the space of finite linear combinations of -atoms, which coincides with , the space of square integrable functions with compact support. By Proposition 2, is a dense subspace of . Set

Similar to the classical case in [10], we immediately obtain the following theorem which proves that is the dual space of for .

Theorem 5. *Let . Then*(a)*suppose ; then given by (11) extends to a bounded linear functional on and satisfies
*(b)*conversely, every bounded linear functional on can be realized as with and
*

*Remark 6. *We may define the space ,, , by
where . The norm of is the least value of for which the above inequality holds. Due to Theorem 5, is also identified with the dual space of . The proof is almost the same as that of Theorem 5. Thus, the space coincides with and .

*Definition 7. *Let , , and . Set , . A function is called a -molecule with the center if (1),
(2),
(3).

Then, we can obtain a molecular characterization of as follows.

Theorem 8. *Given as in Definition 7, then if and only if can be written as , where are -molecules and . The sum converges in norm and also in when . Moreover,
**
where the infimum is taken over all decompositions of into -molecules.*

Let be a -function on with compact support and such that on a neighborhood of zero. Define for .

We refer to the singular integral operators defined by left twisted convolution with these kernels as the local Riesz transforms. The terminology is motivated by the fact that they are essentially the operators which are formally defined as , , .

As an application of Theorem 8, we can prove the following.

Theorem 9. *The local Riesz transforms , are bounded on , where .*

*Remark 10. *When , Theorem 9 is proved by the connection between and Hardy space on the Heisenberg group (cf. Lemma 4.9 in [4]).

Throughout the paper, we will use to denote a positive constant, which is independent of main parameters and may be different at each occurrence. By , we mean that there exists a constant such that .

#### 2. Molecule Characterization of

In this section, we prove the main result of this paper. Firstly, we have the following lemma.

Lemma 11. *If is a -atom for supported in , then is a -molecule centered at and
**
where and is a positive constant that is independent of .*

*Proof. *Since
we get
Therefore,
This proves that is a molecule with center at .

The following lemma is the key step for the proof of Theorem 8.

Lemma 12. *If is a -molecule with center at , then and
**
where is independent of .*

*Proof. *If , let , , and . Denote , where is the characteristic function of .

Let
Then
Without loss of generality, we can assume that . Then
Therefore,
Let . Then
In the following, we will prove
In fact, by
we have
Since
we get
Therefore, (27) holds true. In particular, we have
For ,
where depends on . This proves that , where is a -atom supported on and .

Now, we prove that has atomic decomposition. For ,
Therefore,
Let
Then,
Thus, by Abel transform,
Following from (34), we obtain
Let and
Then, are -atoms, , and
Therefore,
holds pointwise, where are -atoms and are -atoms, and
When , it is easy to see that the sum in (42) converges in .

To prove for , we need to show that, for every ,
In fact, (44) implies that (42) holds in .

For any , there exists such that . If , then
If , then
Therefore, when ,
Thus,
Let ; the right side is . The left side is
This proves (42) and the case of for Lemma 12 is proved. Similarly, the case of can be proved as the case of . Lemma 12 is proved.

*Proof of Theorem 8. *Theorem 8 follows from Lemmas 11 and 12.

#### 3. The Boundedness of Local Riesz Transform on

In this section, we prove the boundedness of local Riesz transform on by using Theorem 8.

*Proof of Theorem 9. *By Theorem 8, it is sufficient to prove that, for any -atom , is a -molecule and the norm , where is independent of .

Assume that ; then
where the last equality is valid because is an atom supported on . This proves that satisfies moment condition.

Denote . Then, we have
Let . Then,
For ,
For , since
we get
Let . Then
Therefore,
This completes the proof of Theorem 9.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research is supported by the National Natural Science Foundation of China (11001002 and 10901018), the Beijing Natural Science Foundation (1142005), and the Program for New Century Excellent Talents in University.