# Hardy-Sobolev Spaces Associated with Twisted Convolution

Authors: Jizheng Huang; Weiwei Li; Yaqiong Wang
Journal: Journal of Function Spaces
 (2017)

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## 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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## Body

## 1. Introduction

It is a well-established fact that, for the purposes of harmonic analysis or theory of partial differential equations, the right substitute forLp(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].LetIα: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 IαLpp>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 Section2, 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 the2n linear differential operators(1)Zj=∂∂zj+14zj¯,Zj¯=∂∂zj¯-14zj,onCn,j=1,2,…,n.Together with the identity they generate a Lie algebrahn which is isomorphic to the 2n+1 dimensional Heisenberg algebra. The only nontrivial commutation relations are(2)Zj,Zj¯=-12I,j=1,2,…,n.The operator L defined by (3)L=-12∑j=1nZjZj¯+Zj¯Zjis 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 (4)τwfz=exp⁡12∑j=1nwjzj+wj¯zj¯fz=fz+wexp⁡i2Imz·w¯.The “twisted convolution” of two functions f and g on Cn can now be defined as (5)f×gz=∫Cnfwτ-wgzdw=∫Cnfz-wgwω¯z,wdw,where ωz,w=exp⁡i/2Imz·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 aboutHL1(Cn). Let B denote the class of C∞-functions φ on Cn, supported on the ball B(0,1) such that φ∞≤1 and ∇φ∞≤2. For t>0, let φt(z)=t-2nφ(z/t). Given σ>0,0<σ≤+∞, and a tempered distribution f, define the grand maximal function(6)Mσfz=supφ∈B⁡sup0<t<σ⁡φt×fz.Then the Hardy space HL1(Cn) can be defined by(7)HL1Cn=f∈S′Cn:M∞f∈L1Cn.For any f∈HL1(Cn), define fHL1Cn=M∞fL1≔M∞fL1Cn.Definition 1.
Let1≤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)
suppa⊂Bz0,r;(2)
aq≤B(z0,r)1/q-1;(3)
∫Cnawω¯z0,wdw=0.We define the atomic Hardy spaceHL1,q(Cn) to be the set of all tempered distributions of the form ∑jλjaj, and the sum converges in the topology of S′(Cn), where aj are HL1,q-atoms and ∑jλj<+∞.The atomic quasi-norm inHL1,q(Cn) is defined by(8)fL-atom=inf∑jλj,where the infimum is taken over all decompositions f=∑j‍λjaj and aj are HL1,q-atoms.Letψ be a C∞-function on Cn with compact support and such that ψ≡1 on a neighborhood of zero. Define(9)Rjz=zjz2n+1ψz,R¯jz=z¯jz2n+1ψzfor j=1,2,…,n.We refer to the singular integral operatorsRj, 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 ZjL-1/2, Z¯jL-1/2, j=1,2,…,n.The following result has been proved in [18].Proposition 2.
For a tempered distributionf on Cn, the following are equivalent:
(i)M∞f∈L1(Cn);
(ii) for someσ,0<σ<+∞,Mσf∈L1(Cn);
(iii) for some radial functionφ∈S, such that ∫Cn‍φ(z)dz≠0, we have (10)sup0<t<1⁡φt×fz∈L1Cn;
(iv)f can be decomposed as f=∑j‍λjaj, where aj are HL1,q-atoms and ∑j‍λj<+∞.
(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 transformsRj, R¯j,j=1,2,…,n, are bounded on HL1(Cn).The dual space of Hardy spaceHL1(Cn) is defined in [18].Definition 4.
A locally integrable functionf is said to be in the BMO type space BMOL if there exists a constant K>0 such that, for every ball B=B(z,r),(11)1B∫Bfv-1B∫Bfuω¯z,uduωz,vdv≤K.The norm fBMOL of f is the least value of K for the above inequality.The Sobolev spaces associated withL are defined as follows (cf. [17]).Definition 5.
Givenp∈(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 (12)Zj1⋯Z¯jmf∈LpCn,1≤j1,…,jm≤n,1≤m≤k,with the norm(13)fWk,p=∑1≤j1,…,jm≤n,1≤m≤kZj1⋯Z¯jmfp+fp.Throughout the article, we will useC 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 withL and consider some properties of them.Definition 6.
We define the Hardy-Sobolev spaceHL1,1(Cn) as the set of functions f∈L1(Cn) such that (14)Zjf∈HL1Cn,Z¯jf∈HL1Cn,j=1,2,…,n,with the norm (15)fHL1,1=∑j=1nZjfHL1+Z¯jfHL1+fL1.We can prove thatHL1,1(Cn) is a Banach space. In order to do that, we need the following lemma (cf. P122 [22]).Lemma 7.
Let1≤p<∞, f∈Lkp(Cn), and {fm} be a sequence such that fm-fp→0. Then, for any α≤k, we have (16)∂αfm∂xα-∂αf∂xαp⟶0.By Lemma7, we can prove the following.Proposition 8.
H
L
1,1
(
C
n
) is a Banach space.Proof.
Letfm be a Cauchy sequence in HL1,1Cn. Then ∂fm/∂xj and {xjfm},j=1,2,…,n are Cauchy sequences in HL1(Cn). Let f be the limit of fm in L1Cn. Then, by Lemma 7,(17)∂fm∂xj-∂f∂xj1⟶0,xjfm-xjf1⟶0.Since HL1(Cn) is a Banach space, there exist g,h∈HL1(Cn) such that(18)∂fm∂xj-gHL1⟶0,xjfm-hHL1⟶0.By (17) and (18), we get g=∂f/∂xj and h=xjf. This proves Zjfm-ZjfHL1→0 and Z¯jfm-Z¯jfHL1→0 for j=1,2,…,n, that is, fm-fHL1,1→0, and then we get HL1,1(Cn) is a Banach space.Now, we give an equivalent characterization ofHL1,1(Cn).Definition 9.
LetHL1,1(Cn)=L-1/2(HL1(Cn)) or (19)HL1,1Cn=f∈L1Cn∣L1/2f∈HL1Cn,with the norm fHL1,1=L1/2fHL1+f1.Theorem 10.
The normsfHL1,1 and fHL1,1 are equivalent; that is, there exists a constant C>0 such that (20)C-1fHL1,1≤fHL1,1≤CfHL1,1.Proof.
Letf∈HL1,1(Cn). Then, Zjf∈HL1(Cn) and Z¯jf∈HL1(Cn) for j=1,2,…,n. Since HL1(Cn)⊂L1(Cn), we have Zjf1≤ZjfHL1 and Z¯jf1≤Z¯jfHL1 for j=1,2,…,n. Note that Zj=ZjL-1/2L1/2=RjLL1/2 and Z¯j=Z¯jL-1/2L1/2=R¯jLL1/2, by Proposition 2,(21)fHL1,1=L1/2fHL1+f1≤∑j=1nRjLL1/2f1+R¯jLL1/2f1+f1=∑j=1nZjf1+Z¯jf1+f1≤∑j=1nZjfHL1+Z¯jfHL1+f1≤CfHL1,1,that is, f∈HL1,1(Cn).
Iff∈HL1,1(Cn), then, by Proposition 3,(22)fHL1,1=∑j=1nZjfHL1+Z¯jfHL1+f1=∑j=1nRjLL1/2fHL1+R¯jLL1/2fHL1+f1≤CL1/2fHL1+f1≤CfHL1,1.This gives the proof of Theorem 10.By Proposition3 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 existsC>0 such that, for all f∈HL1,1(Cn), (23)C-1fHL1,1≤∇LfHL1≤CL1/2fHL1≤CfHL1,1≤CfHL1,1.In the following, we consider the atomic decomposition ofHL1,1(Cn).Definition 12.
We say a functionb(x) is an (1,q)-atom associates with a ball B(z0,r) for the space HL1,1(Cn), if (1)
suppb⊂Bz0,r,(2)
L1/2bq≤Bz0,r1/q-1,where 1<q≤∞.The atomic quasi-norm inHL1,1(Cn) is defined by (24)fL-atom=inf⁡∑cj,where the infimum is taken over all decompositions f=∑‍cjbj, where bj are (1,q)-atoms.In order to prove the atomic decomposition ofHL1,1(Cn), we need the following lemma.Lemma 13.
Leta(z) be a (1,q)-atom associated with ball B(z0,r) of HL1(Cn) and b(z)=L-1/2a(z). Then (25)Zjbz≤Crz-z02n+1,Z¯jbz≤Crz-z02n+1for z-z0≥2r and j=1,2,…,n.Proof.
Forj=1,2,…,n, since a(z) is a (1,q)-atom, we have (26)Zjbz=ZjL-1/2az=RjLaz=∫Cnauzj-uiz-u2n+1ψz-uω¯z,udu=∫Cnauω¯z0,uzj-uiz-u2n+1ψz-uωz0-z,u-zj-z0,jz-z02n+1ψz-z0ωz0-z,z0du,and in the last equality, we use the fact ω¯z,u=ω¯z0,uωz0-z,u, the definition of Riesz transform and atom.
whenu∈B(z0,r) and z-z0>2r, we have z-u~z-z0, so (27)Zjbz=∫Cnauω¯z0,uzj-uiz-z02n+1ψz-uωz0-z,u-zj-z0,jz-z02n+1ψz-z0ωz0-z,z0du.Let gu=zj-ujψz-uωz0-z,u. Then there exists a constant C>0 such that g′u≤C. Therefore,(28)gu-gz0=g′ξu-z0≤Cu-z0,Zjbz≤C∫Cnu-z0z-z02n+1audu≤Crz-z02n+1.The proof of Z¯jbz is similar to the proof of Zjbz.Now we can prove the following result.Theorem 14.
The normsfHL1,1 and fL-atom are equivalent; that is, there exists a constant C>0 such that (29)C-1fHL1,1≤fL-atom≤CfHL1,1.Proof.
To showf=∑‍λibi∈HL1,1Cn, it suffices to prove that for any (1,q)-atom b, with bHL1,1≤C independent of b. By Theorem 10 and Proposition 3, (30)bHL1,1=L1/2bHL1=∑j=1nRjL1/2b1+R¯jL1/2b1+L1/2b1=∑j=1nZjb1+Z¯jb1+L1/2b1=∑j=1n∫Bz0,rRjL1/2bzdz+∫Bz0,rR¯jL1/2bzdz+∫Bz0,rL1/2bzdz≤B1/q′∑j=1nRjL1/2bzq+R¯jL1/2bzqB1/q′L1/2bzq≤CB1/q′B1/q-1≤C.For the reverse, if f∈HL1,1Cn, there exists g∈HL1Cn such that f=L-1/2g. Since g=∑‍λiai, where ai are (1,q)-atoms in HL1, we get f=∑‍λiL-1/2ai with ∑‍λj<∞. Since L-1/2ai does not have compact support, it is not an atom for HL1,1Cn.
Leta be a (1,q)-atom of HL1 such that suppa⊂B(z0,r) and b(z)=L-1/2a. We choose a smooth partition of unity 1=ϕ0+∑m=1∞‍ϕm, where ϕ0≡1 and ϕ1≡0 on z-z0<2r.(31)suppϕ0⊂z∣z-z0≤4r,suppϕ1⊂z∣2r≤z-z0≤8rand ϕmz=ϕ121-mz for m≥2. Then b(z)=ϕ0b+∑m=1∞‍ϕmb. We will show ϕmb=λmbm for appropriate scalars λm, where bm are (1,q)-atoms in HL1,1(Cn) and ∑‍λm<C.
It is obvious thatsuppbm⊂Bz0,24+mr. Let (32)λm=24+mr2n1-1/qL1/2ϕmbq.For m=0, since L1/2bq=1, we get L1/2ϕ0bq≤C. For m≥1, by Lemma 13, we have (33)Zjϕmbq≤C2mr2n/qZjϕmb∞≤Cr2mr2n/q2mr-2n-1=C2-m2mr-2n/q′,Z¯jϕmbq≤C2-m2mr-2n/q′.So λm≤C2-m, which gives ∑‍λm≤C.In the following, we consider the dual spaces ofHL1,1(Cn). Our proof is motivated by [10].Definition 15.
We say a distributionT∈D′(Cn) belongs to the BMO-Sobolev spaces BMOL-1(Cn) if there exist ϕ0∈L∞(Cn) and Φ∈BMOL(Cn,Cn) such that T=ϕ0-divΦ. Define (34)TBMOL-1Cn=inf⁡ϕ0∞+ΦBMOLCn,where the infimum is taken over all functions ϕ0∈L∞(Cn) and Φ∈BMOL(Cn,Cn) such that T=ϕ0-divΦ, divΦ=(Z1Φ,…,ZnΦ,Z¯1Φ,…,Z¯nΦ).Theorem 16.
The dual space ofHL1,1(Cn) is isomorphic to BMOL-1(Cn). Moreover, given T=ϕ0-divΦ∈BMOL-1(Cn), the linear functional (35)DCn∋f⟼∫Cnfzϕ0zdz+∑j=1n∫CnZjfzΦjz+Z¯jfzΦjzdz=T,fextends to a bounded linear functional LT on HL1,1(Cn). Conversely, for any L∈HL1,1Cn′, there exists a unique T∈BMOL-1(Cn) such that, for all f∈D(Cn),L(f)=T,f, one has LT~TBMOL-1(Cn).Proof.
LetT∈BMOL-1Cn,ϵ>0 and ϕ0∈L∞(Cn) and Φ∈BMOL(Cn,Cn) such that T=ϕ0-divΦ and (36)ϕ0∞+ΦBMOLCn≤1+ϵTBMOL-1Cn.
Then, for allf∈D(Cn), (37)T,f≤∫Cnfzϕ0zdz+∑j=1n∫CnZjfzΦjz+Z¯jfzΦjzdz≤inf⁡ϕ0∞+ΦBMOLCnfHL1,1.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 (38)LT≤1+ϵTBMOL-1Cn.Since this is true for all ϵ>0, one obtains LT≤TBMOL-1(Cn).
Conversely, letL be a bounded linear form on HL1,1(Cn). Since HL1,1(Cn) is isometrically isomorphic to a subspace of L1(Cn)⊕HL1(Cn)⊕⋯⊕HL1(Cn), there exist ϕ0∈L∞(Cn) and ϕ1,…,ϕd∈BMOL(Cn) such that, for all f∈D(Cn),(39)Lf=∫Cnfzϕ0zdz+∑j=1n∫CnZjfzΦjz+Z¯jfzΦjzdz,sup1≤j≤n⁡ϕ0∞,ΦjBMOL≤L.Set Φ=(ϕ1,…,ϕn) and T=ϕ0-divΦ∈D′(Cn). Then T∈BMOL-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 dive=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.
Letf∈HL1,1(Cn) and e∈L∞(Cn,Cn) with dive=0 on Cn. Then e·∇Lf∈HL1(Cn), where ∇L=(Z1,…,Zn,Z¯1,…,Z¯n).Proof.
Iff∈HL1,1(Cn), by Theorem 14, f has the decomposition(40)f=∑i=0∞λiai,where the ak are HL1,1-atoms and ∑k=0∞‍λi<∞. Therefore, for e∈L∞Cn,Cn, (41)e·∇Lf=∑i=0∞λie·∇Aai.To prove e·∇Lf∈HL1Cn, we need only to prove e·∇Lai are HL1-atoms by the atomic decomposition of HL1Cn. Since ai is an HL1,1-atom, there exists a ball Biz0,r such that suppe·∇Lai⊂Bi and ∇Lai2≤Bi-1/2. Following from e∈L∞(Cn,Cn), (42)e·∇Lai2≤CBi-1/2.Since ai satisfies the moment condition and dive=0, we get e·∇Lai=div(aie), and this yields the moment condition(43)∫Cne·∇Laiω¯z,z0dz=0.We complete the proof of Theorem 17.



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