Abstract
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.
1. Introduction
It is a well-established fact that, for the purposes of harmonic analysis or theory of partial differential equations, the right substitute for in case is the (real) Hardy space , or its local version (cf. [1]). The Hardy spaces, or their local versions if needed, behave nicely under the action of regular singular integrals or pseudo-differential operators. Moreover, in the case of Hardy spaces the Littlewood-Paley theory and interpolation results extend to the whole scale of Lebesgue exponents . 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 . 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 .
Let be the Riesz potential operators, where is the space of tempered distributions and denotes the space of polynomials. We can define Sobolev space 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 . The Hardy-Sobolev spaces were studied by many authors. In [5], the author investigated the spaces , where denotes the Hardy spaces. The spaces form a natural continuation of the space to , and so the spaces which are called Hardy-Sobolev spaces are natural generalizations of the homogeneous Sobolev spaces to the range . 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 . 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 , 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.
2. Preliminaries
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 areThe 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 [3, 19, 20].
In [18], 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 Riesz transform.
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 functionThen the Hardy space can be defined byFor any , define .
Definition 1. Let . A function is a -atom for the Hardy space associated with a ball if(1);(2);(3)
We define the atomic Hardy space to be the set of all tempered distributions of the form , and the sum converges in the topology of , where are -atoms and .
The atomic quasi-norm in is defined bywhere the infimum is taken over all decompositions and are -atoms.
Let be a -function on with compact support and such that on a neighborhood of zero. Definefor .
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].
Proposition 2. 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 .
(v) and for .
Moreover, the following result has been proved in [18] or [21].
Proposition 3. The Riesz transforms , ,, are bounded on .
The dual space of Hardy space is defined in [18].
Definition 4. A locally integrable function is said to be in the BMO type space if there exists a constant such that, for every ball ,The norm of is the least value of for the above inequality.
The Sobolev spaces associated with are defined as follows (cf. [17]).
Definition 5. Given and , we define the Sobolev space of order associated with twisted convolution, denoted by , as the set of functions such that with the norm
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 , we mean that there exists a constant such that .
3. 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 as the set of functions such that with the norm
We can prove that is a Banach space. In order to do that, we need the following lemma (cf. P122 [22]).
Lemma 7. Let , , and be a sequence such that . Then, for any , we have
By Lemma 7, we can prove the following.
Proposition 8. is a Banach space.
Proof. Let be a Cauchy sequence in . Then and are Cauchy sequences in . Let be the limit of in . Then, by Lemma 7,Since is a Banach space, there exist such thatBy (17) and (18), we get and . This proves and for , that is, , and then we get is a Banach space.
Now, we give an equivalent characterization of .
Definition 9. Let or with the norm .
Theorem 10. The norms and are equivalent; that is, there exists a constant such that
Proof. Let . Then, and for . Since , we have and for . Note that and , by Proposition 2,that is, .
If , then, by Proposition 3,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 such that, for all ,
In the following, we consider the atomic decomposition of .
Definition 12. We say a function is an -atom associates with a ball for the space , if (1),(2),where .
The atomic quasi-norm in is defined by where the infimum is taken over all decompositions , where are -atoms.
In order to prove the atomic decomposition of , we need the following lemma.
Lemma 13. Let be a -atom associated with ball of and . Then for and .
Proof. For , since is a -atom, we have and in the last equality, we use the fact , the definition of Riesz transform and atom.
when and , we have , so Let . Then there exists a constant such that . Therefore,The proof of is similar to the proof of .
Now we can prove the following result.
Theorem 14. The norms and are equivalent; that is, there exists a constant such that
Proof. To show , it suffices to prove that for any -atom , with independent of . By Theorem 10 and Proposition 3, For the reverse, if , there exists such that . Since , where are -atoms in , we get with . Since does not have compact support, it is not an atom for .
Let be a -atom of such that and . We choose a smooth partition of unity , where and on .and for . Then . We will show for appropriate scalars , where are -atoms in and .
It is obvious that . Let For , since , we get . For , by Lemma 13, we have So , which gives .
In the following, we consider the dual spaces of . Our proof is motivated by [10].
Definition 15. We say a distribution belongs to the BMO-Sobolev spaces if there exist and such that . Define where the infimum is taken over all functions and such that , .
Theorem 16. The dual space of is isomorphic to . Moreover, given , the linear functional extends to a bounded linear functional on . Conversely, for any , there exists a unique such that, for all , one has .
Proof. Let and and such that and Then, for all , Since is dense in , this means that extends to a bounded linear form on , with Since this is true for all , one obtains .
Conversely, let be a bounded linear form on . Since is isometrically isomorphic to a subspace of , there exist and such that, for all ,Set and . Then and . This proves Theorem 16.
4. An Application: Div-Curl Lemma
In [13], the authors proved the following well-known div-curl Lemma: Let and . If with and with on , then . Now, we consider the case of ; as an application of Theorem 14, we give the endpoint version of the div-curl lemma.
Theorem 17. Let and with on . Then , where .
Proof. If , by Theorem 14, has the decompositionwhere the are -atoms and . Therefore, for , To prove , we need only to prove are atoms by the atomic decomposition of . Since is an -atom, there exists a ball such that and . Following from , Since satisfies the moment condition and , we get , and this yields the moment conditionWe complete the proof of Theorem 17.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 11471018).