Hardy-Sobolev Spaces Associated with Twisted Convolution

We first define the Hardy-Sobolev spaces associated with twisted convolution; then we give the atomic decomposition. As an application, we consider the endpoint version of the div-curl theorem for the twisted convolution.


Introduction
It is a well-established fact that, for the purposes of harmonic analysis or theory of partial differential equations, the right substitute for   (R  ) in case  ∈ (0, 1] is the (real) Hardy space   (R  ), or its local version ℎ  (R  ) (cf. [1]).The Hardy spaces, or their local versions if needed, behave nicely under the action of regular singular integrals or pseudodifferential operators.Moreover, in the case of Hardy spaces the Littlewood-Paley theory and interpolation results extend to the whole scale of Lebesgue exponents  ∈ (0,∞).It is hence natural to investigate Sobolev spaces where one (roughly speaking) demands that the th derivative belongs to a Hardy type space in the case  ≤ 1.After the fundamental work of Fefferman and Stein [2] this line of research was initiated by Peetre in early 70s, and it was generalized and carried further by Triebel and others.We refer to [3,4] for extensive accounts on general Besov and Triebel type scales of function spaces in the case  ∈ (0, 1]. Let   : S  /P → S  /P be the Riesz potential operators, where S  is the space of tempered distributions and P denotes the space of polynomials.We can define Sobolev space   (  ) ( > 1) to be the space of tempered distributions having derivatives of order  in   .The use of the Hardy-Sobolev spaces gives strong boundedness of some linear operators instead of the weak boundedness.For instance this is the case of the square root of the Laplace operator Δ 1/2 .The Hardy-Sobolev spaces were studied by many authors.In [5], the author investigated the spaces   (  ) (0 <  ≤ 1), where   denotes the Hardy spaces.The spaces   form a natural continuation of the   space to 0 <  ≤ 1, and so the spaces   (  ) which are called Hardy-Sobolev spaces are natural generalizations of the homogeneous Sobolev spaces   (  ) to the range 0 <  ≤ 1. Strichartz [5] proved that  / (  ) was an algebra and found equivalent norms for the Hardy-Sobolev spaces or, more generally, for corresponding spaces with fractional smoothness and Lebesgue exponents in the range  > /( + 1).Torchinsky [6] discussed the trace properties of the spaces   (  ).Miyachi [7] characterized the Hardy-Sobolev spaces in terms of maximal functions related to mean oscillation of the function in cubes, thus obtaining a counterpart of previous results of Calderon and of the general theory of DeVore and Sharpley [8].More recently there has been considerable interest in Hardy-Sobolev spaces and their variants on R  , or on subdomains.Chang et al. [9] consider Hardy-Sobolev spaces in connection with estimates for elliptic operators, whereas Auscher et al. [10] study these spaces with applications to square roots of elliptic operators.Koskela and Saksman [11] show that there is a simple strictly pointwise characterization of the Hardy-Sobolev spaces in terms of first differences.In [12], the authors gave the atomic decomposition of the Hardy-Sobolev space and proved the endpoint case of the div-curl theorem of [13].Also the papers of Cho and Kim [14], Janson [15], and Orobitg [16] are related to the theme of the present paper.Recently, functional spaces associated with operators are considered by more and more mathematicians.In [17], the authors studied the Sobolev spaces associated with the twisted Laplacian and the Global well posedness of nonlinear Schrödinger equation.In [18], the authors defined the Hardy spaces associated with twisted Laplacian by the heat maximal function.They also gave the atomic decomposition and Riesz transform characterizations for the Hardy spaces.In this paper, we first define Hardy-Sobolev spaces associated with twisted Laplacian based on [17,18] and then give the atomic decomposition of them.Finally, we give an application of the Hardy-Sobolev spaces associated with twisted Laplacian.
The paper is organized as follows.In Section 2, we give some results that we will use in the sequel; In Section 3, we prove some properties of the Hardy-Sobolev space, including atomic decomposition.In Section 4, some applications will be given.

Preliminaries
In this paper we consider the 2 linear differential operators 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 [3,19,20].
In [18], 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 Riesz transform.
We first give some basic notations about  1  (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  1  (C  ) can be defined by For any We define the atomic Hardy space where the infimum is taken over all decompositions  = ∑      and   are  1,  -atoms.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 Riesz transforms.The terminology is motivated by the fact that they are essentially the operators which are formally defined as The following result has been proved in [18].
Moreover, the following result has been proved in [18] or [21].
The dual space of Hardy space  1  (C  ) is defined in [18].
Definition 4. A locally integrable function  is said to be in the BMO type space BMO  if there exists a constant  > 0 such that, for every ball  = (, ), The norm ‖‖ BMO  of  is the least value of  for the above inequality.
The Sobolev spaces associated with  are defined as follows (cf.[17]).Definition 5. Given  ∈ (1, ∞) and  ∈ N, we define the Sobolev space of order  associated with twisted convolution, denoted by  , (C  ), as the set of functions  ∈   (C  ) such that Throughout the article, 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 ≤ .

Hardy-Sobolev Spaces
In this section, we define Hardy-Sobolev spaces associated with  and consider some properties of them.Definition 6.We define the Hardy-Sobolev space  1,1  (C  ) as the set of functions  ∈  1 (C  ) such that with the norm We can prove that  1,1  (C  ) is a Banach space.In order to do that, we need the following lemma (cf.P122 [22]).
with the norm ‖‖ H This gives the proof of Theorem 10.
By Proposition 3 and Theorem 10, we can get the following endpoint case of square root problem for  (for the elliptic second-order divergence operator see Theorem 40 in [10]).
Corollary 11.There exists  > 0 such that, for all  ∈ In the following, we consider the atomic decomposition of  1,1  (C  ).
The atomic quasi-norm in where the infimum is taken over all decompositions  = ∑     , where   are (1, )-atoms.
In order to prove the atomic decomposition of  1,1  (C  ), we need the following lemma.In the following, we consider the dual spaces of  1,1  (C  ).Our proof is motivated by [10].(36) ) to be the set of all tempered distributions of the form ∑      , and the sum converges in the topology of S  (C  ), where   are  1,  -atoms and ∑  |  | < +∞.The atomic quasi-norm in  1,  (C  ) is defined by