Hardy-Sobolev Spaces Associated with Twisted Convolution

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.

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Jizheng Huang 1 and Weiwei Li 1 and Yaqiong Wang 1

Academic Editor:Alberto Fiorenza

College of Sciences, North China University of Technology, Beijing 100144, China

Received 15 March 2017; Revised 21 May 2017; Accepted 28 May 2017; 21 June 2017

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.

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 ^Lp (^Rd ) in case p∈(0,1] is the (real) Hardy space ^Hp (^Rd ), or its local version ^hp (^Rd ) (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 p∈(0,∞). It is hence natural to investigate Sobolev spaces where one (roughly speaking) demands that the sth derivative belongs to a Hardy type space in the case p<=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 p∈(0,1].

Let _Iα :^{S[variant prime]} /P[arrow right]^{S[variant prime]} /P be the Riesz potential operators, where ^{S[variant prime]} is the space of tempered distributions and P denotes the space of polynomials. We can define Sobolev space _Iα (^Lp ) (p>1) to be the space of tempered distributions having derivatives of order α in ^Lp . 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 ^Δ1/2 . The Hardy-Sobolev spaces were studied by many authors. In [5], the author investigated the spaces _Iα (^Hp ) (0<p<=1), where ^Hp denotes the Hardy spaces. The spaces ^Hp form a natural continuation of the ^Lp space to 0<p<=1, and so the spaces _Iα (^Hp ) which are called Hardy-Sobolev spaces are natural generalizations of the homogeneous Sobolev spaces _Iα (^Lp ) to the range 0<p<=1. Strichartz [5] proved that _In/p (^Hp ) 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 p>n/(n+1). Torchinsky [6] discussed the trace properties of the spaces _Iα (^Hp ). 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 ^Rd , 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 2n linear differential operators [figure omitted; refer to PDF]

Together with the identity they generate a Lie algebra ^hn which is isomorphic to the 2n+1 dimensional Heisenberg algebra. The only nontrivial commutation relations are [figure omitted; refer to PDF] The operator L defined by [figure omitted; refer to PDF] is nonnegative, self-adjoint, and elliptic. Therefore it generates a diffusion semigroup {^TtL_}t>0 ={^e-tL_}t>0 . The operators in (1) generate a family of "twisted translations" _τw on ^Cn defined on measurable functions by [figure omitted; refer to PDF] The "twisted convolution" of two functions f and g on ^Cn can now be defined as [figure omitted; refer to PDF] where ω(z,w)=exp[...]((i/2)Im(z·w¯)). More about twisted convolution can be found in [3, 19, 20].

In [18], the authors defined the Hardy space ^HL1 (^Cn ) associated with twisted convolution. They gave several characterizations of ^HL1 (^Cn ) via maximal functions, the atomic decomposition, and the behavior of the Riesz transform.

We first give some basic notations about ^HL1 (^Cn ). Let B denote the class of ^C∞ -functions [straight phi] on ^Cn , supported on the ball B(0,1) such that _{([straight phi])∞} <=1 and _{(∇[straight phi])∞} <=2. For t>0, let _{[straight phi]t} (z)=^t-2n [straight phi](z/t). Given σ>0, 0<σ<=+∞, and a tempered distribution f, define the grand maximal function [figure omitted; refer to PDF] Then the Hardy space ^HL1 (^Cn ) can be defined by [figure omitted; refer to PDF] For any f∈^HL1 (^Cn ), define _{(f)^HL1 (^Cn )} =_{(M∞ f)^L1} [: =]_{(M∞ f)^L1 (^Cn )} .

Definition 1.

Let 1<=q<=∞. A function a(z) is a ^HL1,q -atom for the Hardy space ^HL1 (^Cn ) associated with a ball B(_z0 ,r) if

(1) supp a⊂B(_z0 ,r);

(2) _(a)q <=^{(B(_z0 ,r))1/q-1} ;

(3) ^∫Cn a(w)ω¯(_z0 ,w)dw=0.

We define the atomic Hardy space ^HL1,q (^Cn ) to be the set of all tempered distributions of the form _∑jλjaj , and the sum converges in the topology of ^{S[variant prime]} (^Cn ), where _aj are ^HL1,q -atoms and _∑j (_λj )<+∞.

The atomic quasi-norm in ^HL1,q (^Cn ) is defined by [figure omitted; refer to PDF] where the infimum is taken over all decompositions f=_∑j [...]_λjaj and _aj are ^HL1,q -atoms.

Let ψ be a ^C∞ -function on ^Cn with compact support and such that ψ≡1 on a neighborhood of zero. Define [figure omitted; refer to PDF] for j=1,2,...,n.

We refer to the singular integral operators _Rj , _R¯j 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 _Zj^L-1/2 , _Z¯j^L-1/2 , j=1,2,...,n.

The following result has been proved in [18].

Proposition 2.

For a tempered distribution f on ^Cn , the following are equivalent:

(i) _M∞ f∈^L1 (^Cn );

(ii) for some σ, 0<σ<+∞, _Mσ f∈^L1 (^Cn );

(iii) for some radial function [straight phi]∈S, such that _∫^Cn [...][straight phi](z)dz≠0, we have [figure omitted; refer to PDF]

(iv) f can be decomposed as f=_∑j [...]_λjaj , where _aj are ^HL1,q -atoms and _∑j [...](_λj )<+∞.

(v) f∈^L1 (^Cn ) and _Rj ×f, _R¯j ×f∈^L1 (^Cn ) for j=1,2,...,n.

Moreover, the following result has been proved in [18] or [21].

Proposition 3.

The Riesz transforms _Rj , _R¯j , j=1,2,...,n, are bounded on ^HL1 (^Cn ).

The dual space of Hardy space ^HL1 (^Cn ) is defined in [18].

Definition 4.

A locally integrable function f is said to be in the BMO type space BM_OL if there exists a constant K>0 such that, for every ball B=B(z,r), [figure omitted; refer to PDF] The norm _(f)BMOL of f is the least value of K for the above inequality.

The Sobolev spaces associated with L are defined as follows (cf. [17]).

Definition 5.

Given p∈(1,∞) and k∈N, we define the Sobolev space of order k associated with twisted convolution, denoted by ^Wk,p (^Cn ), as the set of functions f∈^Lp (^Cn ) such that [figure omitted; refer to PDF] with the norm [figure omitted; refer to PDF]

Throughout the article, we will use C to denote a positive constant, which is independent of main parameters and may be different at each occurrence. By _B1 ~_B2 , we mean that there exists a constant C>1 such that 1/C<=_B1 /_B2 <=C.

3. Hardy-Sobolev Spaces

In this section, we define Hardy-Sobolev spaces associated with L and consider some properties of them.

Definition 6.

We define the Hardy-Sobolev space ^HL1,1 (^Cn ) as the set of functions f∈^L1 (^Cn ) such that [figure omitted; refer to PDF] with the norm [figure omitted; refer to PDF]

We can prove that ^HL1,1 (^Cn ) is a Banach space. In order to do that, we need the following lemma (cf. P122 [22]).

Lemma 7.

Let 1<=p<∞, f∈^Lkp (^Cn ), and {_fm } be a sequence such that _{(fm -f)p} [arrow right]0. Then, for any (α)<=k, we have [figure omitted; refer to PDF]

By Lemma 7, we can prove the following.

Proposition 8.

^{H L 1,1} ( ^{C n} ) is a Banach space.

Proof.

Let (_fm ) be a Cauchy sequence in ^HL1,1 (^Cn ). Then (∂_fm /∂_xj ) and {_xjfm }, j=1,2,...,n are Cauchy sequences in ^HL1 (^Cn ). Let f be the limit of (_fm ) in ^L1 (^Cn ). Then, by Lemma 7, [figure omitted; refer to PDF] Since ^HL1 (^Cn ) is a Banach space, there exist g,h∈^HL1 (^Cn ) such that [figure omitted; refer to PDF] By (17) and (18), we get g=∂f/∂_xj and h=_xj f. This proves _{(Zjfm -Zj f)^HL1} [arrow right]0 and _{(Z¯jfm -Z¯j f)^HL1} [arrow right]0 for j=1,2,...,n, that is, _{(fm -f)^HL1,1} [arrow right]0, and then we get ^HL1,1 (^Cn ) is a Banach space.

Now, we give an equivalent characterization of ^HL1,1 (^Cn ).

Definition 9.

Let ^HL1,1 (^Cn )=^L-1/2 (^HL1 (^Cn )) or [figure omitted; refer to PDF] with the norm _(f)^HL1,1 =_{(^L1/2 f)^HL1} +_(f)1 .

Theorem 10.

The norms _(f)^HL1,1 and _(f)^HL1,1 are equivalent; that is, there exists a constant C>0 such that [figure omitted; refer to PDF]

Proof.

Let f∈^HL1,1 (^Cn ). Then, _Zj f∈^HL1 (^Cn ) and _Z¯j f∈^HL1 (^Cn ) for j=1,2,...,n. Since ^HL1 (^Cn )⊂^L1 (^Cn ), we have _{(Zj f)1} <=_{(Zj f)^HL1} and _{(Z¯j f)1} <=_{(Z¯j f)^HL1} for j=1,2,...,n. Note that _Zj =_Zj^L-1/2L1/2 =^RjLL1/2 and _Z¯j =_Z¯j^L-1/2L1/2 =^R¯jLL1/2 , by Proposition 2, [figure omitted; refer to PDF] that is, f∈^HL1,1 (^Cn ).

If f∈^HL1,1 (^Cn ), then, by Proposition 3, [figure omitted; refer to PDF] 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 L (for the elliptic second-order divergence operator see Theorem 40 in [10]).

Corollary 11.

There exists C>0 such that, for all f∈^HL1,1 (^Cn ), [figure omitted; refer to PDF]

In the following, we consider the atomic decomposition of ^HL1,1 (^Cn ).

Definition 12.

We say a function b(x) is an (1,q)-atom associates with a ball B(_z0 ,r) for the space ^HL1,1 (^Cn ), if

(1) supp b⊂B(_z0 ,r),

(2) _{(^L1/2 b)q} <=^{(B(_z0 ,r))1/q-1} ,

where 1<q<=∞.

The atomic quasi-norm in ^HL1,1 (^Cn ) is defined by [figure omitted; refer to PDF] where the infimum is taken over all decompositions f=∑[...]_cjbj , where _bj are (1,q)-atoms.

In order to prove the atomic decomposition of ^HL1,1 (^Cn ), we need the following lemma.

Lemma 13.

Let a(z) be a (1,q)-atom associated with ball B(_z0 ,r) of ^HL1 (^Cn ) and b(z)=^L-1/2 a(z). Then [figure omitted; refer to PDF] for (z-_z0 )≥2r and j=1,2,...,n.

Proof.

For j=1,2,...,n, since a(z) is a (1,q)-atom, we have [figure omitted; refer to PDF] and in the last equality, we use the fact ω¯(z,u)=ω¯(_z0 ,u)ω(_z0 -z,u), the definition of Riesz transform and atom.

when u∈B(_z0 ,r) and (z-_z0 )>2r, we have (z-u)~(z-_z0 ), so [figure omitted; refer to PDF] Let g(u)=(_zj -_uj )ψ(z-u)ω(_z0 -z,u). Then there exists a constant C>0 such that (^{g[variant prime]} (u))<=C. Therefore, [figure omitted; refer to PDF] The proof of (_Z¯j b(z)) is similar to the proof of (_Zj b(z)).

Now we can prove the following result.

Theorem 14.

The norms _(f)^HL1,1 and _(f)L-atom are equivalent; that is, there exists a constant C>0 such that [figure omitted; refer to PDF]

Proof.

To show f=∑[...]_λibi ∈^HL1,1 (^Cn ), it suffices to prove that for any (1,q)-atom b, with _(b)^HL1,1 <=C independent of b. By Theorem 10 and Proposition 3, [figure omitted; refer to PDF] For the reverse, if f∈^HL1,1 (^Cn ), there exists g∈^HL1 (^Cn ) such that f=^L-1/2 g. Since g=∑[...]_λiai , where _ai are (1,q)-atoms in ^HL1 , we get f=∑[...]_λi^L-1/2_ai with ∑[...](_λj )<∞. Since ^L-1/2_ai does not have compact support, it is not an atom for ^HL1,1 (^Cn ).

Let a be a (1,q)-atom of ^HL1 such that supp a⊂B(_z0 ,r) and b(z)=^L-1/2 a. We choose a smooth partition of unity 1=_[varphi]0 +^∑m=1∞ [...]_[varphi]m , where _[varphi]0 ≡1 and _[varphi]1 ≡0 on (z-_z0 )<2r. [figure omitted; refer to PDF] and _[varphi]m (z)=_[varphi]1 (^21-m z) for m≥2. Then b(z)=_[varphi]0 b+^∑m=1∞ [...]_[varphi]m b. We will show _[varphi]m b=_λmbm for appropriate scalars _λm , where _bm are (1,q)-atoms in ^HL1,1 (^Cn ) and ∑[...](_λm )<C.

It is obvious that supp _bm ⊂B(_z0 ,^24+m r). Let [figure omitted; refer to PDF] For m=0, since _{(^L1/2 b)q} =1, we get _{(^L1/2[varphi]0 b)q} <=C. For m≥1, by Lemma 13, we have [figure omitted; refer to PDF] So _λm <=C^2-m , which gives ∑[...](_λm )<=C.

In the following, we consider the dual spaces of ^HL1,1 (^Cn ). Our proof is motivated by [10].

Definition 15.

We say a distribution T∈^{D[variant prime]} (^Cn ) belongs to the BMO-Sobolev spaces BM^OL-1 (^Cn ) if there exist _[varphi]0 ∈^L∞ (^Cn ) and Φ∈BM_OL (^Cn ,^Cn ) such that T=_[varphi]0 -div Φ. Define [figure omitted; refer to PDF] where the infimum is taken over all functions _[varphi]0 ∈^L∞ (^Cn ) and Φ∈BM_OL (^Cn ,^Cn ) such that T=_[varphi]0 -div Φ, div Φ=(_Z1 Φ,...,_Zn Φ,_Z¯1 Φ,...,_Z¯n Φ).

Theorem 16.

The dual space of ^HL1,1 (^Cn ) is isomorphic to BM^OL-1 (^Cn ). Moreover, given T=_[varphi]0 -div Φ∈BM^OL-1 (^Cn ), the linear functional [figure omitted; refer to PDF] extends to a bounded linear functional _LT on ^HL1,1 (^Cn ). Conversely, for any L∈^{(HL1,1 (Cn ))[variant prime]} , there exists a unique T∈BM^OL-1 (^Cn ) such that, for all f∈D(^Cn ), L(f)=(T,f), one has (_LT )~_{(T)BM^OL-1 (^Cn )} .

Proof.

Let T∈BM^OL-1 (^Cn ), [...]>0 and _[varphi]0 ∈^L∞ (^Cn ) and Φ∈BM_OL (^Cn ,^Cn ) such that T=_[varphi]0 -div Φ and [figure omitted; refer to PDF]

Then, for all f∈D(^Cn ), [figure omitted; refer to PDF] Since D(^Cn ) is dense in ^HL1,1 (^Cn ), this means that f[...](T,f) extends to a bounded linear form _LT on ^HL1,1 (^Cn ), with [figure omitted; refer to PDF] Since this is true for all [...]>0, one obtains (_LT )<=_{(T)BM^OL-1 (^Cn )} .

Conversely, let L be a bounded linear form on ^HL1,1 (^Cn ). Since ^HL1,1 (^Cn ) is isometrically isomorphic to a subspace of ^L1 (^Cn )[ecedil]5;^HL1 (^Cn )[ecedil]5;[...][ecedil]5;^HL1 (^Cn ), there exist _[varphi]0 ∈^L∞ (^Cn ) and _[varphi]1 ,...,_[varphi]d ∈BM_OL (^Cn ) such that, for all f∈D(^Cn ), [figure omitted; refer to PDF] Set Φ=(_[varphi]1 ,...,_[varphi]n ) and T=_[varphi]0 -div Φ∈^{D[variant prime]} (^Cn ). Then T∈BM^OL-1 (^Cn ),L=_LT and (T)<=(2n+1)(L). This proves Theorem 16.

4. An Application: Div-Curl Lemma

In [13], the authors proved the following well-known div-curl Lemma: Let 1<p,q<∞ and 1/p+1/q=1. If f∈^Lp (^Rd ,^Rd ) with curlf=0 and e∈^Lq (^Rd ,^Rd ) with div e=0 on ^Rd , then e·f∈^H1 (^Rd ). Now, we consider the case of p=1; as an application of Theorem 14, we give the endpoint version of the div-curl lemma.

Theorem 17.

Let f∈^HL1,1 (^Cn ) and e∈^L∞ (^Cn ,^Cn ) with div e=0 on ^Cn . Then e·_∇L f∈^HL1 (^Cn ), where _∇L =(_Z1 ,...,_Zn ,_Z¯1 ,...,_Z¯n ).

Proof.

If f∈^HL1,1 (^Cn ), by Theorem 14, f has the decomposition [figure omitted; refer to PDF] where the _ak are ^HL1,1 -atoms and ^∑k=0∞ [...](_λi )<∞. Therefore, for e∈^L∞ (^Cn ,^Cn ), [figure omitted; refer to PDF] To prove e·_∇L f∈^HL1 (^Cn ), we need only to prove e·_∇Lai are ^HL1 -atoms by the atomic decomposition of ^HL1 (^Cn ). Since _ai is an ^HL1,1 -atom, there exists a ball _Bi (_z0 ,r) such that supp e·_∇Lai ⊂_Bi and _{(∇Lai )2} <=^{(_Bi )-1/2} . Following from e∈^L∞ (^Cn ,^Cn ), [figure omitted; refer to PDF] Since _ai satisfies the moment condition and div e=0, we get e·_∇Lai =div(_ai e), and this yields the moment condition [figure omitted; refer to PDF] We complete the proof of Theorem 17.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11471018).

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