# Molecular Characterization of Hardy Spaces Associated with Twisted Convolution

Authors: Jizheng Huang; Yu Liu
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We give a molecular characterization of the Hardy space associated with twisted convolution. As an application, we prove the boundedness of the local Riesz transform on the Hardy space.


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## Body

## 1. Introduction

In this paper, we consider the2n linear differential operators
(1)Zj=∂∂zj+14z-j,Z-j=∂∂z-j-14zj,111111111111on⁡Cn,j=1,2,…,n.
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
(2)[Zj,Z-j]=-12I,j=1,2,…,n.
The operator L defined by
(3)L=-12∑j=1n‍(ZjZ-j+Z-jZj)
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
(4)(τwf)(z)=exp⁡(12∑j=1n‍(wjzj+w-jz-j))f(z)=f(z+w)exp⁡(i2Im⁡(z·sw-)).
The “twisted convolution” of two functions f and g on Cn can now be defined as
(5)(f×g)(z)=∫Cn‍f(w)τ-wg(z)dw=∫Cn‍f(z-w)g(w)ω-(z,w)dw,
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 HL1(Cn) associated with twisted convolution. They gave several characterizations of HL1(Cn) 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 HLp(Cn) associated with twisted convolution for 2n/(2n+1)<p<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 HLp(Cn). As an application, we prove the boundedness of the local Riesz transform on the Hardy space HLp(Cn).We first give some basic notations aboutHLp(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)Mσf(z)=sup⁡φ∈Bsup⁡0<t<σ|φt×f(z)|.
Then, the Hardy space HLp(Cn) can be defined by
(7)HLp(Cn)={f∈S′(Cn):M∞f∈Lp(Cn)}.
For any f∈HLp(Cn), define ∥f∥HLp(Cn)=∥M∞f∥Lp.Definition 1.
Let0<p≤1≤q≤∞ and p≠q. A function a(z) is a HLp,q-atom for the Hardy space HLp(Cn) associated to a ball B(z0,r) if (1)
supp⁡a⊂B(z0,r);(2)
∥a∥q≤|B(z0,r)|1/q-1/p;(3)
∫Cn‍a(w)ω-(z0,w)dw=0.We define the atomic Hardy spaceHLp,q(Cn) to be the set of all tempered distributions of the form ∑j‍λjaj (the sum converges in the topology of S′(Cn)), where aj are HLp,q-atoms and ∑j‍|λj|p<+∞.The atomic quasinorm inHLp,q(Cn) is defined by
(8)∥f∥L-atom=inf⁡{(∑j‍|λj|p)1/p},
where the infimum is taken over all decompositions f=∑j‍λjaj and aj are HLp,q-atoms.The following result has been proved in [4, 7].Proposition 2.
Let2n/(2n+1)<p≤1. Then, for a tempered distribution f on Cn, the following are equivalent:(i)
M∞f∈Lp(Cn);(ii)
for someσ, 0<σ<+∞, Mσf∈Lp(Cn);(iii)
for some radial functionφ∈S, such that ∫Cn‍φ(z)dz≠0, we have
(9)sup⁡0<t<1|φt×f(z)|∈Lp(Cn);(iv)
f can be decomposed as f=∑j‍λjaj, where aj are HLp,q-atoms and ∑j‍|λj|p<+∞.Corollary 3.
Let2n/(2n+1)<p≤1 and 1<q≤∞. Then, HLp,q(Cn)=HLp(Cn) with equivalent norms.In order to give the main result of this paper, we need the dual space of Hardy spaceHLp(Cn).Definition 4.
Let0≤α<1/2n; a locally integrable function f is said to be in the Campanato type space ΛαL if there exists a constant K>0 such that, for every ball B=B(z,r),
(10)|B|-α(∫B‍|f(υ)-(1|B|∫B‍f(u)ω-(z,u)du)ω(z,υ)|2ggggggg×dυ|B|∫B‍|f(v)-(1|B|∫B‍f(u)ω-(z,u)du)ω(z,v)|2)1/2≤K.
The norm ∥f∥ΛαL of f is the least value of K for which the above inequality holds.The dual space ofHL1(Cn) is the BMO type space BMOL(Cn) (cf. [4]). Note that Λ0L is identified with BMOL. Let HLp,2,a denote the space of finite linear combinations of HLp,2-atoms, which coincides with Lc2(Cn), the space of square integrable functions with compact support. By Proposition 2, HLp,2,a is a dense subspace of HLp(Cn). Set
(11)Lg(f)=∫Cn‍f(z)g-(z)dz,f∈HLp,2,a,g∈Lloc⁡2(Cn).Similar to the classical case in [10], we immediately obtain the following theorem which proves that Λ1/p-1L is the dual space of HLp(Cn) for 2n/(2n+1)<p<1.Theorem 5.
Let2n/(2n+1)<p<1. Then(a)
supposeg∈Λ(1/p)-1L; then Lg given by (11) extends to a bounded linear functional on HLp(Cn) and satisfies
(12)∥Lg∥≤C∥g∥Λ(1/p)-1L;(b)
conversely, every bounded linear functionalL on HLp(Cn) can be realized as L=Lg with g∈Λ(1/p)-1L and
(13)∥g∥Λ(1/p)-1L≤C∥L∥.Remark 6.
We may define the spaceΛ(1/p)-1,q′L,2n/(2n+1)<p<1, 1≤q′≤∞, by
(14)|B|1-(1/p)(∫B‍|f(υ)-(1|B|∫B‍f(u)ω-(z,u)du)ω(z,υ)|q′ggggggggggg×dυ|B|∫B‍|f(υ)-(1|B|∫B‍f(u)ω-(z,u)du)ω(z,υ)|q′)1/q′≤K,
where B=B(z,r). The norm ∥f∥Λ(1/p)-1,q′L of f is the least value of K for which the above inequality holds. Due to Theorem 5, Λ(1/p)-1,q′L is also identified with the dual space of HLp(Cn). The proof is almost the same as that of Theorem 5. Thus, the space Λ(1/p)-1,q′L coincides with Λ(1/p)-1L and ∥f∥Λ(1/p)-1,q′L~∥f∥Λ(1/p)-1L.Definition 7.
Let2n/(2n+1)<p≤1≤q≤∞, p≠q, and ϵ>(1/p)-1. Set a=1-(1/p)+ϵ, b=1-(1/p)+ϵ. A function M∈Lq is called a HLp,q,ϵ-molecule with the center z0 if (1)
|z|2nbM(z)∈Lq,(2)
N(M)=∥M∥Lqa/b∥|·-z0|2nbM∥Lq1-(a/b)<∞,(3)
∫Cn‍M(z)ω-(z0,z)dz=0.
Then, we can obtain a molecular characterization ofHLp(Cn) as follows.Theorem 8.
Givenp,q,ϵ as in Definition 7, then f∈HLp if and only if f can be written as f=∑jλjMj‍, where Mj are HLp,q,ϵ-molecules and ∑j|λj|p‍<∞. The sum converges in HLp norm and also in (Λ(1/p)-1L)* when 2n/(2n+1)<p<1. Moreover,
(15)∥f∥HLp~∥f∥HLp,q,ϵ,M=inf⁡{(∑j‍|λj|p)1/p},
where the infimum is taken over all decompositions of f into HLp,q,ϵ-molecules.Letψ be a C∞-function on Cn with compact support and such that ψ≡1 on a neighborhood of zero. Define
(16)Rj(z)=zj|z|2n+1ψ(z),R-j(z)=z-j|z|2n+1ψ(z),
for j=1,2,…,n.We refer to the singular integral operatorsRj,R-j 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 ZjL-1/2, Z-jL-1/2, j=1,2,…,n.As an application of Theorem8, we can prove the following.Theorem 9.
The local Riesz transformsRj,R-j, j=1,2,…,n are bounded on HLp(Cn), where 2n/(2n+1)<p≤1.Remark 10.
Whenp=1, Theorem 9 is proved by the connection between HL1(Cn) and Hardy space on the Heisenberg group H1(Hn) (cf. Lemma  4.9 in [4]).Throughout the paper, 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.


## 2. Molecule Characterization ofHLp(Cn)

In this section, we prove the main result of this paper. Firstly, we have the following lemma.Lemma 11.
Ifa is a HLp,q-atom for 2n/(2n+1)<p≤1≤q≤+∞ supported in B(z0,r), then a is a HLp,q,ε-molecule centered at z0 and
(17)N(a)≤C,
where ϵ>0 and C is a positive constant that is independent of a.Proof.
Since(18)∥a∥q≤|B|1/q-1/p=|B|a-b,
we get
(19)∥|·-z0|2nba(·)∥q≤r2nb∥a∥q≤C|B|b|B|a-b=C|B|a.
Therefore,
(20)N(a)≤C|B|(a/b)(a-b)|B|a(1-(a/b))=C.
This proves that a is a molecule with center at z0.The following lemma is the key step for the proof of Theorem8.Lemma 12.
IfM is a HLp,q,ϵ-molecule with center at z0, then M∈HLp,q(Cn) and
(21)∥M∥HLp≤CN(M),
where C is independent of M.Proof.
Ifq=2, let σ=∥M∥21/(2n(a-b)), E0={z∈Cn:|z-z0|≤σ}, and Ek={z∈Cn:2k-1σ<|z-z0|≤2kσ}. Denote Mk=Mχk, where χk is the characteristic function of Ek.
Let(22)Pk(z)=1|Ek|∫Ek‍M(u)ω-(z0,u)duω(z0,z)χk(z).
Then
(23)∫Cn‍(Mk(z)-Pk(z))ω-(z0,z)dz=∫Ek‍M(z)ω-(z0,z)dz-∫Ek‍M(z)ω-(z0,z)dz=0.
Without loss of generality, we can assume that N(M)=1. Then
(24)∥|·-z0|2nbM(·)∥21-(a/b)=∥M∥2-a/b.
Therefore,
(25)∥|·-z0|2n((1/2)+ϵ)M(·)∥2=∥M∥2-a/(b-a)=σ2na.
Let Bk={z∈Cn:|z-z0|≤2kσ}. Then
(26)supp⁡(Mk-Pk)⊆Ek⊆Bk.
In the following, we will prove
(27)1|Bk|∫Bk‍|Mk-Pk|2dz≤C|Ek|∫Ek‍|Mk(z)|2dz.
In fact, by
(28)∫Ek‍(Mk(z)-Pk(z))P-k(z)dz=0,
we have
(29)1|Bk|∫Bk‍|Mk(z)-Pk(z)|2dz=1|Bk|∫Bk‍(Mk(z)-Pk(z))(M-k(z)-P-k(z))dz=1|Bk|∫Bk‍(Mk(z)-Pk(z))M-k(z)dz.
Since
(30)∫Ek‍M-k(z)Pk(z)dz=∫Ek‍M-k(z)1|Ek|∫Ek‍M(u)ω-(z0,u)duω(z0,z)dz=1|Ek|∫Ek‍M(u)ω-(z0,u)du∫Ek‍M-k(z)ω(z0,z)dz=|Ek||Pk(z)|2,
we get
(31)1|Bk|∫Bk‍|Mk(z)-Pk(z)|2dz=1|Bk|∫Ek‍|Mk(z)|2dz-∫Ek‍|Ek||Bk||Pk(z)|2dz≤1|Bk|∫Ek‍|Mk(z)|2dz≤C|Ek|∫Ek‍|Mk(z)|2dz.
Therefore, (27) holds true. In particular, we have
(32)1|B0|∫B0‍|M0(z)-P0(z)|2dz≤C|E0|∫E0‍|M0(z)|2dz≤Cσ-2nσ4n((1/2)-(1/p))=C|B0|-2/p.
For k≥1,
(33)1|Bk|∫Bk‍|Mk(z)-Pk(z)|2dz≤C|Ek|∫Ek‍|Mk(z)|2dz=C|Ek|∫Ek‍|Mk(z)|2|z-z0|2n(1+2ϵ)1111111111111×|z-z0|-2n(1+2ϵ)dz≤C(2kσ)-2n(2k-1σ)-2n(1+2ϵ)11111×∫Ek‍|Mk(z)|2|z-z0|2n(1+2ϵ)dz≤Cσ-4n(1+ϵ)2-4kn-4knϵσ4na≤C2-4kn-4knϵ-4kn/p(2kσ)-2n/p=C2-4kna|Bk|-2/p,
where C depends on n,ϵ. This proves that Mk-Pk=λkak, where ak is a HLp,2-atom supported on Bk and |λk|≤C2-2kna.
Now, we prove that∑k=1∞‍Pk(z) has atomic decomposition. For k≥1,
(34)|Pk(z)|≤1|Ek|∫Ek‍|M(u)|du=1|Ek|∫Ek‍|u-z0|2nb|M(u)||u-z0|-2nbdu≤C(2kσ)-2nb1|Ek|∥|·-z0|2nbM(·)∥2|Ek|1/2≤C(2kσ)-2nb-nσ2na=C2-kn(1+2b)σ-2n/p.
Therefore,
(35)Pk(z)=∑l=k+1∞‍(Pl-1(z)-Pl(z)).
Let
(36)Nk=∑l=k∞‍∫El‍M(u)ω-(z0,u)du.
Then,
(37)N0=∑l=0∞‍∫El‍M(u)ω-(z0,u)du=∫Cn‍M(u)ω-(z0,u)du=0.
Thus, by Abel transform,
(38)∑k=0∞Pk(z)=∑k=0∞∑l=k+1∞(Pl-1(z)-Pl(z))=∑k=0∞Nk+1{|Ek|-1ω(z0,z)χk(z)11111111111-|Ek+1|-1ω(z0,z)χk+1(z)}.
Following from (34), we obtain
(39)|Nk+1{|Ek|-1ω(z0,z)χk(z)1-|Ek+1|-1ω(z0,z)χk+1(z)}|≤C2-2n(k+1)ϵσ2n-(2n/p)|Bk+1|-1=C2-2n(k+1)ϵσ2n-(2n/p)(2k+1σ)-1=C2-2na(k+1)|Bk+1|-1/p.
Let μk=C2-2na(k+1) and
(40)bk(z)=C-122na(k+1)Nk+1{|Ek|-1ω(z0,z)χk(z)11111111111111111-|Ek+1|-1ω(z0,z)χk+1(z)}.
Then, bk are HLp,∞-atoms, ∑k=0∞‍|μk|p<∞, and
(41)∑k=0∞‍Pk(z)=∑k=0∞‍μkbk(z).
Therefore,
(42)M(z)=∑k=0∞‍λkak(z)+∑k=0∞‍μkbk(z)
holds pointwise, where ak are HLp,2-atoms and bk are HLp,∞-atoms, and
(43)∑k=0∞‍{|λk|p+|μk|p}<∞.
When p=1, it is easy to see that the sum in (42) converges in L1.
To proveM∈HLp,q for 2n/(2n+1)<p<1, we need to show that, for every g∈Λ(1/p)-1L,
(44)∫Cn‍M(z)g(z)dz=lim⁡m→∞∑k=0m‍∫Cn‍{λkak(z)+μkbk(z)}g(z)dz.
In fact, (44) implies that (42) holds in S′(Cn).
For anyz∈Cn, there exists k≥0 such that z∈Ek. If k=0, then
(45)M(z)=λ0a0(z)+μ0b0(z).
If k≥1, then
(46)M(z)=λk(z)ak(z)+∑j=k-1k‍μjbj(z).
Therefore, when |z-z0|≤2mσ,
(47)∑k=0∞‍(λkak(z)+μkbk(z))=∑k=0m‍(λkak(z)+μkbk(z))=M(z).
Thus,
(48)∫{z:|z-z0|≤2mσ}‍∑k=0m‍(λkak(z)+μkbk(z))g(z)dz=∫{z:|z-z0|≤2mσ}‍M(z)g(z)dz.
Let m→∞; the right side is ∫Cn‍M(z)g(z)dz. The left side is
(49)lim⁡m→∞∑k=0m‍∫{z:|z-z0|≤2mσ}‍(λkak(z)+μkbk(z))g(z)dz=lim⁡m→∞∑k=0m‍∫Cn‍(λkak(z)+μkbk(z))g(z)dz.
This proves (42) and the case of q=2 for Lemma 12 is proved. Similarly, the case of q≠2 can be proved as the case of q=2. Lemma 12 is proved.Proof of Theorem8.
Theorem8 follows from Lemmas 11 and 12.


## 3. The Boundedness of Local Riesz Transform onHLp(Cn)

In this section, we prove the boundedness of local Riesz transform onHLp(Cn) by using Theorem 8.Proof of Theorem9.
By Theorem8, it is sufficient to prove that, for any HLp,2-atom a, Rj(a) is a HLp,2,ε-molecule and the norm N(Rj(a))≤C, where C is independent of a.
Assume thatsupp⁡a⊂B(z0,r); then
(50)∫Cn‍Rj(a)(z)ω-(z0,z)dz=∫Cn‍(∫Cn‍a(z-u)ui|u|2n+1ψ(u)ω-(z,u)du)ggggggg×ω-(z0,z)dz=∫Cn‍ui|u|2n+1ψ(u)(∫Cn‍a(z-u)ω-(u+z0,z)dz)ggggggg×ω-(z0,z)du=0,
where the last equality is valid because a(·-u) is an atom supported on B(u+z0,r). This proves that Rj(a) satisfies moment condition.
DenoteM(z)=Rj(a)(z). Then, we have
(51)∥M∥2=∥Rj(a)∥2≤C∥a∥2≤|B|1/2-1/p=C|B|a-b.
Let B*={z∈Cn:|z-z0|≤2r}. Then,
(52)∫Cn‍|z-z0|2n(1+2ϵ)|M(z)|2dz=∫B*‍|z-z0|2n(1+2ε)M(z)2dzggggggg+∫(B*)c‍|z-z0|2n(1+2ϵ)|M(z)|2dz=I+II.
For I,
(53)I≤C|B|1+2ϵ∫Cn‍|M(z)|2dz≤C|B|2+2ϵ-(2/p)=C|B|2a.
For II, since
(54)|Rj(a)(z)|=|∫Cna(u)zj-ui|z-u|2n+1ψ(z-u)ω-(z,u)du|=|∫Cna(u)ω-(z0,u)(zj-ui|z-u|2n+1ψ(z-u)ggggggggggggggggggg×ω(z0-z,u)ggggggggggggggggggg-zj-z0,j|z-z0|2n+1ψ(z-z0)ggggggggggggggggggg×ω(z0-z,z0)zj-ui|z-u|2n+1)du∫Cn‍a|≤C∫Cn|u-z0||z-z0|2n+1|a(u)|du≤Cr|B|1-1/p1|z-z0|2n+1,
we get
(55)II≤Cr2|B|2-2/p∫(B*)c‍|z-z0|2n(1+2ϵ)|z-z0|4n+2dz.
Let 0<ϵ<1/2n. Then
(56)II≤C|B|2+2ϵ-2/p=C|B|2a.
Therefore,
(57)N(M)=∥M∥2a/b∥|·-z0|n(1+2ϵ)M(·)∥21-(a/b)≤C|B|(a/b)(a-b)|B|a(1-(a/b))=C.
This completes the proof of Theorem 9.



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