Molecular Characterization of Hardy Spaces Associated with Twisted Convolution

where ω(z, w) = exp((i/2) Im(z ⋅ w)). More about twisted convolution can be found in [1–3]. In [4], the authors defined the Hardy space H L(C n ) associated with twisted convolution. They gave several characterizations of H L(C n ) 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 andWang [7] defined theHardy space


Introduction
In this paper, we consider the 2 linear differential operators on C  ,  = 1, 2, . . ., . ( Together with the identity they generate a Lie algebra ℎ  which is isomorphic to the 2 + 1 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 {   } >0 = { − } >0 .The operators in (1) generate a family of "twisted translations"   on C  defined on measurable functions by The "twisted convolution" of two functions  and  on C  can now be defined as where (, ) = exp((/2) Im( ⋅ )).More about twisted convolution can be found in [1][2][3].
In [4], the authors defined the Hardy space  1  (C  ) associated with twisted convolution.They gave several characterizations of  1  (C  ) 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    (C  ) associated with twisted convolution for 2/(2 + 1) <  < 1. 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    (C  ).As an application, we prove the boundedness of the local Riesz transform on the Hardy space    (C  ).

Journal of Function Spaces
We first give some basic notations about    (C  ).Let B denote the class of  ∞ -functions  on C  , supported on the ball (0, 1) such that ‖‖ ∞ ≤ 1 and ‖∇‖ ∞ ≤ 2. For  > 0, let   () =  −2 (/).Given  > 0, 0 <  ≤ +∞, and a tempered distribution , define the grand maximal function Then, the Hardy space    (C  ) can be defined by For any  ∈ We define the atomic Hardy space where the infimum is taken over all decompositions  = ∑      and   are  ,  -atoms.The following result has been proved in [4,7].Proposition 2. Let 2/(2 + 1) <  ≤ 1.Then, for a tempered distribution  on C  , the following are equivalent: (ii) for some , 0 <  < +∞,    ∈   (C  ); (iii) for some radial function  ∈ S, such that ∫ C  () ̸ = 0, we have The norm ‖‖ Λ   of  is the least value of  for which the above inequality holds.
The dual space of  1  (C  ) is the BMO type space BMO  (C  ) (cf. [4]).Note that Λ  0 is identified with BMO  .Let H ,2,  denote the space of finite linear combinations of  ,2  -atoms, which coincides with  2  (C  ), the space of square integrable functions with compact support.By Proposition 2, Similar to the classical case in [10], we immediately obtain the following theorem which proves that Λ  1/−1 is the dual space of    (C  ) for 2/(2 + 1) <  < 1.
Remark 6.We may define the space where  = (, ).The norm of  is the least value of  for which the above inequality holds.Due to Theorem 5, Λ  (1/)−1,  is also identified with the dual space of    (C  ).The proof is almost the same as that of Theorem 5. Thus, the space Λ  (1/)−1,  coincides with Λ  (1/)−1 and where the infimum is taken over all decompositions of  into  ,,  -molecules.
Let  be a  ∞ -function on C  with compact support and such that  ≡ 1 on a neighborhood of zero.Define for  = 1, 2, . . ., .
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    −1/2 ,    −1/2 ,  = 1, 2, . . ., .
As an application of Theorem 8, we can prove the following.Remark 10.When  = 1, Theorem 9 is proved by the connection between  1  (C  ) and Hardy space on the Heisenberg group  1 (H  ) (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  1 ∼  2 , we mean that there exists a constant  > 1 such that 1/ ≤  1 / 2 ≤ .

𝑝 𝐿 (C 𝑛 )
In this section, we prove the main result of this paper.Firstly, we have the following lemma.
where  > 0 and  is a positive constant that is independent of . Proof.Since we get Therefore, This proves that  is a molecule with center at  0 .
The following lemma is the key step for the proof of Theorem 8.
where  is independent of .
In the following, we will prove where  depends on , . Therefore, Let Then, Journal of Function Spaces 5 Thus, by Abel transform, Following from (34), we obtain Let   = 2 −2(+1) and Therefore, In fact, (44) implies that (42) holds in S  (C  ).
This proves (42) and the case of  = 2 for Lemma 12 is proved.Similarly, the case of  ̸ = 2 can be proved as the case of  = 2. Lemma 12 is proved.

The Boundedness of Local Riesz
Transform on  -molecule and the norm N(  ()) ≤ , where  is independent of .
) to be the set of all tempered distributions of the form ∑      (the sum converges in the topology of S  (C  )), where   are   |  |  < +∞.